University of Southern California Dissertations and Theses

Development and implementation of methods for sliced velocity map imaging. Studies of overtone-induced dissociation and isomerization dynamics of hydroxymethyl radical (CH₂OH and CD₂OH)

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Development and implementation of methods for sliced velocity map imaging. Studies of overtone-induced dissociation and isomerization dynamics of hydroxymethyl radical (CH₂OH and CD₂OH)

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Content DEVELOPMENT AND IMPLEMENTATION OF METHODS FOR SLICED VELOCITY MAP IMAGING. STUDIES OF OVERTONE-INDUCED DISSOCIATION AND ISOMERIZATION DYNAMICS OF HYDROXYMETHYL RADICAL (CH 2 OH AND CD 2 OH) by Mikhail Ryazanov A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA in Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CHEMISTRY: CHEMICAL PHYSICS) December 2012 Copyright 2012 Mikhail Ryazanov Table of Contents List of Tables v List of Figures vii Abstract xvi Chapter 1 Introduction 1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Chapter 2 Design and implementation of an apparatus for sliced velocity map imaging of H atoms 3 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Design goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 Exact relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.2 Approximate relationships . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Ion optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.4.1 Wiley–McLaren arrangement for TOF spatial focusing . . . . . . 30 2.4.2 Eppink–Parker arrangement for radial focusing . . . . . . . . . . 33 2.4.3 Minimalistic SVMI system . . . . . . . . . . . . . . . . . . . . . . . 39 2.4.4 Additional lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.5 Simulations and final design . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.5.1 Accelerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.5.2 Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 2.5.3 Focusing criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.5.4 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.5.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.6 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.6.1 Ion optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.6.2 Charged particle detector . . . . . . . . . . . . . . . . . . . . . . . . 102 2.6.3 Image acquisition and analysis . . . . . . . . . . . . . . . . . . . . 107 2.7 Experimental tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 2.7.1 Using O + and O + 2 from O 2 . . . . . . . . . . . . . . . . . . . . . . . 114 ii 2.7.2 Using H + from HBr and H 2 S . . . . . . . . . . . . . . . . . . . . . 120 2.7.3 Position mapping mode . . . . . . . . . . . . . . . . . . . . . . . . . 127 2.7.4 TOF-MS operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 2.9 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Chapter 3 Determination of velocity distributions from raw (S)VMI data 147 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3.2 Rectification of image distortions . . . . . . . . . . . . . . . . . . . . . . . 152 3.3 Reconstruction of velocity distributions from VMI data with finite slicing thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 3.3.1 Basis set definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 3.3.2 Sliced projections of basis functions . . . . . . . . . . . . . . . . . 166 3.3.3 Maximum-likelihood estimation . . . . . . . . . . . . . . . . . . . 172 3.3.4 Constrained solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 177 3.3.5 Experimental examples . . . . . . . . . . . . . . . . . . . . . . . . . 179 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 3.5 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Chapter 4 Overtone-induced dissociation and isomerization dynamics of the hydroxymethyl radical (CH 2 OH and CD 2 OH) 188 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 4.2 Experimental arrangement and procedures . . . . . . . . . . . . . . . . . 194 4.3 Experimental results and analysis . . . . . . . . . . . . . . . . . . . . . . 199 4.3.1 Action spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 4.3.2 Kinetic energy release distributions . . . . . . . . . . . . . . . . . 204 4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 4.4.1 Overtone spectroscopy and state interactions . . . . . . . . . . . . 210 4.4.2 Product state distributions . . . . . . . . . . . . . . . . . . . . . . . 212 4.4.3 Competition between O¡H bond fission and isomerization . . . . 217 4.5 O¡H bond fission and electronic states . . . . . . . . . . . . . . . . . . . . 218 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 4.7 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Bibliography 240 Appendices 250 A Listings of input files for SIMION simulations . . . . . . . . . . . . . . . 250 A.1 Geometry definition . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 iii A.2 Particles definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 A.3 Analysis and optimization program . . . . . . . . . . . . . . . . . 254 B Additional Pareto-optimal ion optics parameters and performance characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 B.1 With respect to¢K and¢t . . . . . . . . . . . . . . . . . . . . . . . 263 B.2 With respect to¢K and¢t 0 . . . . . . . . . . . . . . . . . . . . . . 265 C Mechanical drawings and photographs . . . . . . . . . . . . . . . . . . . . 267 D Electronic circuit diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 E High-voltage nanosecond pulser . . . . . . . . . . . . . . . . . . . . . . . . 281 F H fragment velocity distribution in photodissociation by the H- detection laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 G Details of spectral fittings . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 G.1 CH 2 OH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 G.2 CD 2 OH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 H Plots of kinetic energy distributions fits . . . . . . . . . . . . . . . . . . . 292 I Rotational excitation model . . . . . . . . . . . . . . . . . . . . . . . . . . 292 J Plots of vibrational densities . . . . . . . . . . . . . . . . . . . . . . . . . . 299 iv List of Tables 3.1 Antiderivatives of integrand terms in (3.30). Explicit expressions are given for several low n values, and general the form — for higher. . . . 171 4.1 Summary of experimental and theoretical (see [11]) vibrational param- eters of CH 2 OH and CD 2 OH. All energies are in cm ¡1 . Vibrational mode descriptions are approximate (for CD 2 OH, replace “CH” with “CD”); the mixed states in CH 2 OH are denoted by leading contribu- tions in their VCI wavefunctions. Experimental values without refer- ences were obtained in the present work; values in italics are from Ar matrix studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 4.2 Spectral fitting results. All spectroscopic values are in cm ¡1 ; tempera- tures are in K. Confidence intervals are given in x§¢x notation; stan- dard deviations of the fit in units of the least significant digit are giv- en in parentheses. Numbers in italics correspond to theoretical values from [11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 4.3 Dissociation energies (D 0 ) determined from kinetic energy distribu- tions of H and D fragments. The confidence intervals include energy calibration and KED fitting uncertainties. . . . . . . . . . . . . . . . . . 208 4.4 Formaldehyde fragment rovibrational distribution parameters ob- tained from fitting of H and D cofragment kinetic energy distributions (standard deviations in units of the least significant digit are given in parentheses). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 4.5 Formaldehyde vibrational populations predicted by the model. Devia- tions from the experimental values (see table 4.4) are given in paren- theses in units of the least significant digit. . . . . . . . . . . . . . . . . 215 v I.1 Inertial parameters of formaldehyde isotopologs. . . . . . . . . . . . . . 296 I.2 Parameters of the rotational excitation model. Tested hypotheses, with reference points shown in bold, are numbered in accordance with the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 vi List of Figures 2.1 General scheme of velocity map imaging setup. . . . . . . . . . . . . . . 4 2.2 Illustration of blurring due to a finite slice thickness. The thick arc shows the part of monochromatic particles that is detected by slicing. The recorded projection of the arc is shown by the thick straight seg- ment marked¢v x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Diagram illustrating the electric field approximation used in the derivation of the TOF velocity dispersion. . . . . . . . . . . . . . . . . . . 21 2.4 Ion optics configuration with a field-free drift region. . . . . . . . . . . . 26 2.5 Geometry of Wiley–McLaren TOF mass spectrometer as an example of a system with TOF spatial focusing. . . . . . . . . . . . . . . . . . . . . . 31 2.6 Focal length for TOF spatial focusing in the Wiley–McLaren system (see fig. 2.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.7 Cross-section of Eppink–Parker VMI setup for radial focusing. Electric field configuration is illustrated by gray equipotential contours. Sample ion trajectories are shown in black for v 0 parallel to the z axis and in blue for v 0 perpendicular to it. . . . . . . . . . . . . . . . . . . . . . . . . 34 2.8 Focal length for radial focusing in Eppink–Parker VMI system (see fig. 2.7; both apertures have40 mm, and L 0 Æ L 0 0 Å15 mm). . . . . . . 37 2.9 Interdependence of L 0 0 , L 1 and E 1 /E 0 for simultaneous TOF and ra- dial focusing in the minimal SVMI system (Eppink–Parker arrange- ment, see figure 2.7). Calculated points are connected by straight line segments for clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 vii 2.10 Performance characteristics for the minimal SVMI system under si- multaneous TOF and radial focusing conditions. . . . . . . . . . . . . . 44 2.11 Optical scheme for a positive lens added in the drift region, showing particle trajectories with (thick lines) are without (thin lines) the lens. 48 2.12 Relative magnification with additional lens and required focal lengths. 50 2.13 Cross-section of cylindrical electrostatic lens with equipotential con- tours shown by gray curves and particle trajectories by black ones. . . 55 2.14 Cross-section of accelerator. Electrodes and metallic holders are shown in black, nonsymmetric parts are schematically given in gray (insula- tors — by dotted lines). Red lines show an example of equipotential contours. The top part includes an electric circuit diagram of the resis- tive voltage divider. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.15 Example of smooth variation of KE and TOF resolutions for effective accelerator lengths changing in 1 mm steps. . . . . . . . . . . . . . . . . 62 2.16 Cross-section of additional lens. Blue lines show an example of equipo- tential contours for intermediate acceleration mode. . . . . . . . . . . . 64 2.17 Diagram illustrating initial position uncertainties and the initial vol- ume used in focusing calculations (coordinate system corresponds to figure 2.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.18 Velocity and position sampling used in focusing calculations. . . . . . . 68 2.19 Illustration of focusing with aberrations (vertical scale expanded and aberrations exaggerated for clarity). Positions of minimum blur spots for each initial velocity are indicated by vertical bars. . . . . . . . . . . 70 2.20 Relative overall kinetic energy and TOF resolutions as functions of magnification index. (Lengths L 0 and L 1 are in mm.) . . . . . . . . . . . 86 2.21 Interdependence of ion optics parameters and magnification index. (Lengths L 0 and L 1 are in mm.) . . . . . . . . . . . . . . . . . . . . . . . 87 2.22 Dependence of voltage scale on imaged kinetic energy range and mag- nification index. (Lengths L 0 and L 1 are in mm.) . . . . . . . . . . . . . 87 viii 2.23 Illustration of electrostatic potentials and particle trajectories for the extreme operating conditions. Top: lower KE limit (K max Æ 0.163 eV; L 0 Æ 40 mm, L 1 Æ 50 mm, V 0 Æ 320 V, V 1 Æ 231.1 V, V L Æ 0); equipotential contours drawn with 10 V steps. Bottom: upper KE limit (K max Æ 6.75 eV; L 0 Æ 75 mm, L 1 Æ 55 mm, V 0 Æ 3000 V, V 1 Æ 1891 V, V L Æ¡6600 V); equipotential contours drawn with 100 V steps (500 V steps below¡1000 V). In both cases¢¿ max ¼ 50 ns. Trajectories correspond to the initial sam- ple illustrated in figure 2.18. . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.24 Schematic cutaway view of the designed VMI ion optics inside the de- tection chamber. Connecting wires and other parts of the apparatus are not shown. (The actual implementation also has minor differences mentioned in the text.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.25 Schematic cutaway view (partial) of the accelerator assembly near the molecular beam entrance end. . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.26 Mass gating pulse shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2.27 Time profile of effective slicing pulse. Points represent experimental data. Fitted Gaussian profile is shown by dotted line. . . . . . . . . . . . 107 2.28 Diagram of using rotating plano-parallel plate for changing ionization region position in focusing tests. (Drawing not to scale.) . . . . . . . . . 115 2.29 Velocity map images of O + from O 2 photodissociation atºÆ 44444 cm ¡1 taken in two different ion optics regimes. (Raw event-counting data rendered at the native 1024£1024 px resolution is shown.) . . . . . . . 117 2.30 Kinetic energy distributions of O + extracted from sliced images shown in figure 2.29. Peaks due to 3-photon dissociation of neutral O 2 are marked according to atomic states of the two O fragments, other peaks correspond to O + 2 dissociation. . . . . . . . . . . . . . . . . . . . . . . . . . 118 2.31 Sliced velocity map images of H + from HBr photodissociation at ºÆ 44444 cm ¡1 taken at two different ion optics regimes. The plot shows kinetic energy distributions extracted from the images (peaks are marked according to the atomic states of the Br cofragment). . . . . . . 122 ix 2.32 Sliced velocity map image and extracted kinetic energy distribution of H + from H 2 S photodissociation atº¼ 42560 cm ¡1 . HS cofragment elec- tronic states are indicated in the plot. (Dotted line shows distribution from the HBr experiment for comparison.) . . . . . . . . . . . . . . . . . 124 2.33 Sliced velocity map image of H + from HBr photodissociation at ºÆ 44444 cm ¡1 and ºÆ 82260 cm ¡1 (near Ly-®). Plotted kinetic energy distributions were extracted separately from the horizontal and verti- cal parts of the image (within 45° sectors), and background was sub- tracted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 2.34 Images of the ionization region at various positions of the laser beam focus relative to the molecular beam recorded using O + 2 ions. . . . . . . 130 2.35 Images of the ionization region at various positions of the laser beam focus relative to the molecular beam recorded using O + ions. . . . . . . 131 2.36 “Raster” image of the molecular beam (using O + 2 ions) obtained by ver- tical scanning of the laser beam. (Compare to figure 4a in [7].) . . . . . 132 2.37 Example of mass spectrum (from CH 2 OH studies described in chap- ter 4). Some identified ions are marked under the peaks. . . . . . . . . 134 2.38 Example of mass spectrum in the region of higher masses. . . . . . . . 136 2.39 Dependence of Pareto-optimal performance characteristics on the total used length (in mm) of the accelerator. . . . . . . . . . . . . . . . . . . . 139 3.1 Transformation of polar grid in rectification of distortions. . . . . . . . 153 3.2 Examples of distorted raw velocity map images and its rectification. Lower part shows the intensity as a function of polar coordinates (plot- ted in rectangular coordinates). . . . . . . . . . . . . . . . . . . . . . . . . 157 3.3 Radial (speed) intensity and anisotropy parameter distributions ex- tracted from the raw and rectified images shown in figure 3.2. . . . . . 158 3.4 Schematic illustration of detected intensity contributions in full projec- tion and sliced projection. Useful signals are shown in solid colors, and interfering contributions — by hatched areas. . . . . . . . . . . . . . . . 160 x 3.5 Triangular basis function centered at grid point R i . . . . . . . . . . . . 163 3.6 Illustration for definition of variables ½ and µ and their relationships to r, z, andµ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 3.7 Illustration of various quantities involved in integration of one basis function within one slice segment. . . . . . . . . . . . . . . . . . . . . . . 169 3.8 Raw velocity map images of O + from O 2 photodissociation at º Æ 44444 cm ¡1 (see section 2.7 and figure 2.29 for details). . . . . . . . . . 180 3.9 Radial part of parallel (»cos 2 µ) and perpendicular (»sin 2 µ) compo- nents of intensity distribution for data shown in figure 3.8. . . . . . . . 181 3.10 Comparison of radial dependences of the parallel intensity component (see figure 3.9) extracted from data shown in figure 3.8 using different methods. (Left column shows the low-velocity part. Right column — the low-intensity part.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 4.1 Energy diagram for ground electronic state dissociation and isomer- ization of hydroxymethyl radical according to calculations in [11]. Ge- ometric configurations of all relevant stationary points are shown in figure 4.2. All energies are relative to the ground vibrational state of CH 2 OH (CD 2 OH). (VSCF/VCI zero-point energies are used for the rad- ical and dissociation products, and RCCST(T)/AVTZ harmonic ZPEs — for all other configurations; corresponding imaginary frequencies are indicated below the barriers). Experimental vibrational levels studied in the present work are also shown for comparison. . . . . . . . . . . . . 189 4.2 Geometric configurations for relevant stationary points of the potential energy surface (calculated at RCCSD(T)/AVTZ level of theory in [11]). All configurations, except the CH 2 OH minimum (a), have C S symme- try. Bond lengths are in angstroms, angles — in degrees (dihedral an- gles are shown in square brackets). (CH 2 O parameters for reference: r CO Æ 1.208 Å, r CH Æ 1.102 Å,\COHÆ 121.7°.) . . . . . . . . . . . . . . . 190 4.3 Example of full-projection velocity map image of H + from CH 2 OH pho- todissociation. The faint inner circle correspond to vibrational predis- sociation at ºÆ 13602 cm ¡1 , whereas most of the signal comes from photodissociation by the H-detection laser (º¼ 27420 cm ¡1 ) through the lowest excited electronic state (see appendix F). . . . . . . . . . . . 194 xi 4.4 Scheme of the experimental arrangement (top view, not to scale). . . . 195 4.5 Action spectra for CH 2 OH and CD 2 OH one-photon dissociation ob- tained by monitoring H and D products. Black curves show experi- mental data (without background subtraction for H and with — for D). Rovibrational band fits are represented by red curves (contours) and sticks (spectral line positions and intensities). Estimated band origin positions for unperturbed levels of CH 2 OH (see subsection 4.4.1) are indicated by dashed lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 4.6 Raw sliced velocity map images of H and D fragments used for deter- mination of kinetic energy release distributions (see figure 4.7). . . . . 205 4.7 Kinetic energy release distributions in CH 2 OH and CD 2 OH one-pho- ton dissociation extracted from SVMI data shown in figure 4.6. Ori- gins of vibrational levels of the formaldehyde cofragment (see table 4.4) are marked by vertical lines. Lower KER parts with expanded vertical scale are show on the right. . . . . . . . . . . . . . . . . . . . . . . . . . . 206 4.8 Part of potential energy surface showing the conical intersection at r O¡H ¼ 1.38 Å, ' OH Æ 0° and lowest barriers at r O¡H » 1.5 Å, ' OH Æ §90°. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 4.9 Relevant natural molecular orbitals of the lowest excited and the ground electronic states at the CH 2 OH PES minimum geometry (see figure 4.2(a)). Isosurfaces enclosing 1/2 (and 3/4 for ground-state¼ CO ) of the probability are plotted. (In natural-orbital representation the “¾ OH ” character is distributed among several orbitals, which are not shown here.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 4.10 Relative orbital energies and occupancies in planar geometries. . . . . 223 4.11 Orthogonal molecular orbitals in T-shaped geometries. . . . . . . . . . 224 4.12 Orbital ordering and occupancies in T-shaped geometries. . . . . . . . . 224 4.13 3-center natural orbitals from XMCQDPT2/SA-2-CASSCF calculations (plotted using 1/2 probability, as in figure 4.9). Compare to the schematic orbital shapes shown in figure 4.11. . . . . . . . . . . . . . . . 226 xii 4.14 SA-MCSCF energies as functions of O¡H distance and¡OH group tor- sion. Symbols correspond to values from the calculations. Fitted hyper- bolas are shown by solid curves, and their asymptotes — by dashed (ground state) and dotted (excited state) lines. . . . . . . . . . . . . . . . 227 4.15 2-state interaction parameters extracted from PES fittings. Symbols (solid for ground state and open for excited) correspond to fitted val- ues. Expected dependences, sin' OH for coupling, and constant (ground state) and sin 2 ' OH (excited state) for crossing energies, are shown by dashed curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 4.16 ROHF ground-state energies (compare to figure 4.14). . . . . . . . . . . 231 B.1 Relative overall kinetic energy and TOF resolutions as functions of magnification index. (Lengths L 0 and L 1 are in mm.) . . . . . . . . . . . 263 B.2 Interdependence of ion optics parameters and magnification index. (Lengths L 0 and L 1 are in mm.) . . . . . . . . . . . . . . . . . . . . . . . 264 B.3 Dependence of voltage scale on imaged kinetic energy range and mag- nification index. (Lengths L 0 and L 1 are in mm.) . . . . . . . . . . . . . 264 B.4 Relative overall kinetic energy and TOF resolutions as functions of magnification index. (Lengths L 0 and L 1 are in mm.) . . . . . . . . . . . 265 B.5 Interdependence of ion optics parameters and magnification index. (Lengths L 0 and L 1 are in mm.) . . . . . . . . . . . . . . . . . . . . . . . 266 B.6 Dependence of voltage scale on imaged kinetic energy range and mag- nification index. (Lengths L 0 and L 1 are in mm.) . . . . . . . . . . . . . 266 C.1 Accelerator assembly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 C.2 Top (installed) and bottom views of accelerator suspension and align- ment assembly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 C.3 Accelerator installed in the vacuum chamber (without extension). . . . 278 C.4 Additional lens. (Thin copper wires holding the parts together before installation are cut and removed after anchoring inside the vacuum chamber.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 xiii D.1 Voltage divider for accelerator. Connections for effective lengths L 0 Æ 45 mm (4.5 cm), L 1 Æ 60 mm (6 cmÆ 10.5 cm¡4.5 cm) are illustrated. 279 D.2 Bias voltage supply for detector front plate. . . . . . . . . . . . . . . . . 279 D.3 Voltage divider and signal decoupler for detector operation in electrical signal pickup mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 D.4 Phosphor screen voltage supply for detector operation in imaging mode. 280 D.5 Voltage supply for AV-HVX-1000A pulser. . . . . . . . . . . . . . . . . . . 281 D.6 Pulse and bias voltage combiner for AV-HVX-1000A pulser. . . . . . . . 281 E.1 Schematic circuit diagram and operation sequence. . . . . . . . . . . . . 282 E.2 Main board. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 E.3 Auxiliary board. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 E.4 Example of output electrical pulse (measured at the pulse monitor out- put). Note that fast oscillations after the main peak are mostly due to electrical interference in low-voltage signal lines induced by powerful electromagnetic radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 F.1 Raw full-projection and sliced velocity map images of H fragments. . . 288 F.2 Kinetic energy release distributions extracted from images shown in figure F.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 G.1 Experimental spectrum and best fitted simulation of 3º 1 Åº 2 Ã 0 band of CH 2 OH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 G.2 Experimental spectrum and best fitted simulation of 4º 1 Ã 0 band of CH 2 OH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 G.3 Experimental spectrum and best fitted simulation of 4º 1 Ã 0 band of CD 2 OH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 xiv H.1 Comparison of fits and modeled distributions with the experimental KEDs. Full KER range is shown in the left column, and expanded part corresponding to vibrationally excited formaldehyde — in the right. Rows correspond to figure 4.7. Horizontal scale shows the internal en- ergy of formaldehyde fragments in cm ¡1 . . . . . . . . . . . . . . . . . . . 294 J.1 2-dimensional vibrational density distributions. . . . . . . . . . . . . . . 300 xv Abstract An experimental setup for sliced velocity map imaging (SVMI) of H fragments produced in photodissociation of small molecules and radicals is designed, construc- ted and characterized experimentally. The setup uses an ion-optical arrangement consisting of an accelerator that allows creation of variable electrostatic field geome- tries near the photodissociation region (where the studied fragments are produced) for initial acceleration and focusing of the ionized fragments and an additional elec- trostatic lens inside the drift tube for independent control of the optical magnifica- tion. This approach permits largely independent control of the radial and temporal characteristics of the velocity mapping, making it possibile to achieve time-gated slicing with selectable relative thickness for a very broad range of fragment kinetic energies (from a fraction of an electronvolt to a few electronvolts) — an important ca- pability not available in other SVMI setups described in the literature. At the same time, the kinetic energy resolution (»1 %) obtainable in a large part of the operating range is comparable to that of the best known SVMI setups. Two methods for analysis of the raw data produced in VMI and SVMI experi- ments are developed and implemented. The first method is intended for correction of small distortions of raw velocity map images. It is shown that correction of these distortions, easily achievable by processing of the image using this method, might sig- nificantly increase the effective resolution for the speed or kinetic energy distribution xvi extracted from the image. In addition, artifacts in angular distributions, produced by the distortions, are eliminated in the corrected images. The method is universal, be- ing able to correct almost arbitrary distortions, and can be applied to full-projection and sliced velocity map imaging data represented as coordinate lists and as raster images. The second data-processing method allows reconstruction of SVMI data record- ed with slicing pulses of arbitrary shape and relative thickness (including the full- projection VMI as a trivial particular case). This ability permits to exploit the im- proved signal-to-noise ratio of the SVMI method and at the same time achieve the best resolution over the whole imaged velocity range, impossible with finite slicing without reconstruction. Testing of the method on synthetic and experimental images demonstrated that the results obtained by reconstruction of sliced images outper- form the results from sliced images without reconstruction and from Abel inversion of full-projection images in terms of the resulting signal-to-noise ratio and resolution simultaneously. The developed and implemented setup and methods were successfully used in studies of overtone-induced dissociation and isomerization dynamics of hydroxy- methyl radial CH 2 OH and its isotopolog CD 2 OH. While these studies, at first glance, largely repeat the previous work, the new experimental capabilities in the present work, significantly improving the resolution and detection efficiency, allowed to ob- tain quantitative and qualitative results at a completely new level. In particular, interaction of the third O¡H stretch overtone (4º 1 level) with a combination of O¡H and antisymmetric C¡H stretches (3º 1 Åº 2 level) was observed for the first time. Dissociation of the vibrationally excited radical to formaldehyde and hydrogen fragments (CH 2 OH¡ ! CH 2 OÅ H) was observed from both of these levels. In case of CD 2 OH excitation (for which only the 4º 1 level was observed), in addition to the H xvii products, a small amount of D fragments correlating with CHDO cofragments was observed for the first time, providing an explicit experimental demonstration of the CD 2 OH¡ ! CHD 2 O¡ ! DÅCHDO decomposition through isomerization to the methoxy radical. Analysis of the vibrational distributions of the formaldehyde products in all reactions suggests that while O¡H bond fission is responsible for the major part of the produced H fragments, a noticeable part of them in CH 2 OH and CD 2 OH de- composition is also due to the isomerization pathway. The relatively high kinetic energy resolution in the present experiments allowed an accurate direct determina- tion of the hydroxymethyl dissociation energies: D 0 (CH 2 OH¡ ! CH 2 OÅH)Æ 10160§ 70 cm ¡1 , D 0 (CD 2 OH¡ ! CD 2 OÅH)Æ 10135§ 70 cm ¡1 , D 0 (CD 2 OH¡ ! CHDOÅD)Æ 10760§60 cm ¡1 . xviii Chapter 1 Introduction This dissertation expounds the results of my work during the last few years of my Ph. D. studies. 1 It consists of three largely independent but at the same time closely connected parts: 1. Development of a new experimental setup for sliced velocity map imaging (SV- MI) of hydrogen atoms produced in photodissociation reactions (chapter 2). 2. Development of methods for extraction of velocity distributions from the raw data recorded with VMI instruments in both slicing and full-projection modes (chapter 3). 3. Finally, experimental studies of vibrational predissociation and isomerization of isotopomers of the hydroxymethyl radical (chapter 4). 2 Although only the last of these parts is directly related to chemistry, its completion would be impossible without the preceding parts, especially the first one. In addition, the results obtained in the first two parts have much broader applicability and, in my 1 The results of some other work in which I participated during the previous years were published in [1], [2] and [3]. 2 Most results of this work were published in [4]. 1 opinion, might present even larger methodological interest for the physical chemistry community. The parts are presented in the most logical order, even though the work described in chapters 2 and 3 significantly overlapped in time. Moreover, only a small part of the chapter 3 results was applied in the chapter 4 studies, while these methods were used in analysis of other experiments even before completion of the instrument described in chapter 2. As mentioned earlier, the parts can be considered as almost independent, and therefore each corresponding chapter has its own introduction and conclusion sec- tions. The general summary for the entire dissertation was already given in the ab- stract and does not need to be repeated here. References [1] H. Krovi, O. Oreshkov, M. Ryazanov, D. A. Lidar, Non-Markovian dynamics of a qubit coupled to an Ising spin bath, Phys. Rev. A 76(5), 052117 (2007). [2] B. Karpichev, L. W. Edwards, J. Wei, H. Reisler, Electronic spectroscopy and pho- todissociation dynamics of the 1-hydroxyethyl radical CH 3 CHOH, J. Phys. Chem. A 112(3), 412 (2008). [3] L. W. Edwards, M. Ryazanov, H. Reisler, S. J. Klippenstein, D-atom products in predissociation of CD 2 CD 2 OH from the 202–215 nm photodissociation of 2- bromoethanol, J. Phys. Chem. A 114(17), 5453 (2010). [4] M. Ryazanov, Ch. Rodrigo, H. Reisler, Overtone-induced dissociation and isomer- ization dynamics of the hydroxymethyl radical (CH 2 OH and CD 2 OH). II. Velocity map imaging studies, J. Chem. Phys. 136(8), 084305 (2012). 2 Chapter 2 Design and implementation of an apparatus for sliced velocity map imaging of H atoms 2.1 Introduction Velocity map imaging (VMI) is the most powerful method of momentum spec- troscopy — experimental investigation of mechanical momentum distributions of free particles, products of disintegration (dissociation, ionization, detachment) and scattering of atomic and molecular systems. While VMI is inherently applicable to electrically charged particles only, with the help of an appropriately selected ioniza- tion method (such that the ion retains the initial momentum of the neutral particle and does not decompose) it becomes suitable for neutral products as well. Moreover, highly selective ionization methods, such as resonance-enhanced multiphoton ion- ization (REMPI), allow state-resolved detection of the products. This makes VMI the ultimate tool in scattering experiments — assuming that the initial momenta are known and measuring the final momentum of one particle 1 , the differential scatter- ing cross-section can be directly obtained in the experiment. Employment of coin- 1 The other momentum is known from conservation of the total momentum. 3 cidence detection schemes allows to study more complicated processes, where more than two fragments are formed. The fundamental idea of VMI is that if the charged particles are formed in a specially shaped electric field (or the field is switched on at the right moment of time), they are accelerated towards the detector in such a way that their arrival positions and times depend on the initial velocities. In a coordinate system defined with the x and y axes parallel to the detector plane, and z perpendicular to it and coinciding with the symmetry axis of the VMI apparatus (see figure 2.1), this VMI mapping can detector x y δr=(δx,δy,δz) v=(v x ,v y ,v z ) t E laserbeam molecular beam x y z R θ Figure 2.1 General scheme of velocity map imaging setup. be written as vÆ (v x ,v y ,v z )7! (x, y, t). (2.1) (The mapping is approximately linear and is different for particles of different mas- ses, which will be discussed later.) Thus collecting the data from a position/time- resolving detector allows determination of the initial velocity vector of the particle. While such mapping can be achieved simply by a homogeneous electric field, a spe- cial shape of an inhomogeneous field is required to make the mapping insensitive to the initial positions ±r of the particles. This “focusing” is important, since in the 4 experiment the interaction/ionization region always has a finite size, sometimes not significantly smaller than the detector radius, and mapping of the initial positions to the arrival positions and times would lead to overlap between the images of different velocities, inevitably reducing the velocity resolution of the system. Historically, detectors resolving both position and time became available only re- cently, and they are still more expensive and complicated and have lower resolu- tion and/or saturation level 1 than detectors resolving only position (2-dimensional) or time (1-dimensional). 2 Ion current multipliers were known for a long time and, being connected to a fast oscilloscope, presented a tool for measurements of arrival time distributions. Among the instruments based on this kind of detection were time of flight (TOF) mass spec- trometers. Although intended to measure mass spectra instead of velocity spectra, they can be considered as precursors of modern VMI spectrometers. In 1955 Wiley and McLaren gave a detailed analysis [3] of TOF mass spectrometers, where they described the electric field configuration required to minimize the effect of the initial spatial spread of the particles on their arrival time (space focusing conditions) and discussed the effect of the initial kinetic energy. In connection with VMI, the TOF space focusing conditions are a subset of the overall VMI focusing conditions, and the kinetic energy effect on TOF is directly related to the v z 7! t part of (2.1), allow- ing determination of the 1-dimensional projection of the initial velocity distribution from the measured profile of the TOF signal. This is especially useful in combination with the so-called “core sampling” technique [4] where a mask placed in front of the detector selects a narrow range of the other two velocity components according to the (v x ,v y )7! (x, y) part of (2.1). Therefore, the determination of the full 3-dimensional 1 Maximum number of detectable particles per acquisition frame. 2 See for example [1] and section 3.2 in review [2]. 5 distribution is, in principle, possible by scanning the mask position. However, since only a small fraction of all particles is detected, the method either has poor resolution (in v x and v y ) or is very time-consuming. The advent of reliable microchannel plate (MCP) multipliers in 1970s [5] was an essential step in the development of position-sensitive detectors with high spatial resolution. The idea was to install a phosphor screen immediately after the MCP to convert the amplified space-resolved current to an optical image (as in MCP-based image intensifiers) and then record the resulting image with the already widely avail- able high-resolution optical imaging instruments. For example, modern systems use a digital videocamera with »1 Mpx resolution directly connected to a computer for real-time data analysis. However, this kind of detector does not have sufficient time resolution to obtain the v z component for each particle. In principle, the total MCP current can be measured simultaneously with the image acquisition and yield an equivalent of the TOF signal. Thus 2-dimensional distribution of (v x ,v y ) and 1-di- mensional distribution of v z can be determined, but the correlation between these distributions is lost, and determination of the initial 3-dimensional distribution of (v x ,v y ,v z ) is not possible in the general case. There are however a few special cas- es when the 3-dimensional distribution can be determined with acceptable accuracy from these incomplete data. First, in the limit when the average number of particles reaching the detector is very small, most of acquisition frames have at most one detection event. In that case the (v x ,v y ) components of the sole particle can be directly determined from the single spot location in the image frame, and the v z component can be directly de- termined from the single peak time in the MCP current, providing the full velocity vector (v x ,v y ,v z ) of the single detected particle. The obvious disadvantage of this case 6 is that the amount of information gained from each frame is very small, and thus an accurate determination of the distribution requires a very large number of frames. Since the frame rate is limited by the laser repetition rate (10 Hz in the present work) and/or the videocamera acquisition capabilities (usually.100 Hz), the overall required experiment time becomes extremely long. The situation can be somewhat improved by using a split anode (that is, instead of the total TOF signal, several TOF signals with coarse spatial resolution are record- ed, and the spatial resolution is subsequently improved using the imaging data; the number of events is then limited by one per each anode segment and thus can reach several events per acquisition frame) or a resistive or delay-line anode (which can give precise arrival time information and some position information, but only when no more than one particle arrives within the detection time interval; this “dead time” limits the total number of detectable events). A detailed description of such 3-dimen- sional techniques is available in a comprehensive review [2]. Unfortunately, they of- fer only a moderate increase in the allowed signal intensity and require much more specialized and expensive equipment, so they will not be considered here. Another very important case is when the 3-dimensional distribution is known to have some symmetry that reduces the amount of unknown information to 2 or 1 dimensions. For example, in photodissociation or photoionization by a single linear- ly polarized laser the distribution of the photofragments has cylindrical symmetry with the axis parallel to the laser polarization vector. 1 That is, assuming the sym- metry axis is the y axis, the 3-dimensional distribution I(v x ,v y ,v z ) can be described as 2-dimensional I(v ½ ,v y ), where v ½ Æ q v 2 x Å v 2 z . This reduction does not facilitate the measurements by itself, but suggests that there might exist a bijective (one-to-one) 1 The case of spherical symmetry is even simpler (and its angle-independent radial distribution can be measured directly by core-sampling TOF), but for the present discussion we can consider it as a subcase of cylindrical symmetry. 7 mapping between I(v ½ ,v y )Æ I ³ q v 2 x Å v 2 z ,v y ´ Æ I(v x ,v y ,v z ) and its measured projection P(v x ,v y )Æ Å1 Z ¡1 I(v x ,v y ,v z )dv z Æ Å1 Z jv x j I(v ½ ,v y ) v ½ dv ½ q v 2 ½ ¡ v 2 x . (2.2) The forward mapping I(v ½ ,v y )7! P(v x ,v y ) explicitly given by (2.2) is called “Abel transform”. 1 Its properties were studied in the 19th century, and the inverse trans- form was found in an explicit form: I(v ½ ,v y )Æ¡ 1 ¼ Å1 Z v ½ dP(v x ,v y ) dv x dv x q v 2 x ¡ v 2 ½ . (2.3) Therefore, in case of cylindrical symmetry, the initial 3-dimensional distribution in- deed can be reconstructed from the measured 2-dimensional projection. This approach was first applied to photochemical reactions by Chandler and Houston in 1987 [6]. Their setup did not use proper electric field configuration to perform VMI focusing (see p. 4), so the obtained velocity resolution was very limit- ed, but nevertheless the results were encouraging. The situation greatly improved in 1997 when Eppink and Parker demonstrated [7] that the focusing can be easi- ly achieved by using a simple electrostatic lens. Their arrangement is still widely used today in the original form or with minor modifications, and velocity resolution ¢v/v¼ 0.2 % has been reached 2 (see, for example, [8]). Despite all its success, the “projection–reconstruction” approach has a few serious drawbacks. The most obvious one — inapplicability to distributions lacking cylindri- 1 Strictly speaking, Abel transform connects 1-dimensional functions I v y (v ½ )´ I(v ½ ,v y ) and P v y (v x )´ P(v x ,v y ) for each particular v y . That is, mapping (2.2) is a direct product of Abel transform in the v ½ subspace and identity mapping in the v y subspace. 2 For monochromatic particles. This qualification is important since all ion-optical systems have chro- matic aberrations, and thus the more relevant parameter max¢v/v max calculated over the whole velocity range might be larger. A more detailed discussion is given on p. 42. 8 cal symmetry — is a major concern for general scattering experiments, but has no importance for the present work. However, two other important issues are related to the reconstruction part described by (2.3). First, the integral is taken from v ½ to infinity, meaning that in order to get I(v ½ ) at particular v ½ , knowledge of the pro- jection P(v x ) for all v x > v ½ is required. In other words, even if all the information interesting for a particular experiment is contained only in the low-velocity part of the distribution, its extraction without recording the complete projection of the whole distribution is impossible. As a consequence, the relative velocity resolution for the interesting part of the distribution is decreased. The second, partially related, issue is associated with the form of the integrand. The experimentally measured projection always has some noise, 1 and its differentiation leads to an extreme amplification of the noise. Although the resulting very noisy derivative is then integrated, the inte- gration weight 1/ q v 2 x ¡ v 2 ½ drops quite quickly near the lower integration limit, so that no efficient averaging takes place. At the same time, the weight drops slowly for large v x values, and the noise from the outer parts of the image makes a significant contribution to the inner parts of the reconstructed distribution. That is, the intensi- ty resolution (signal-to-noise ratio) of the low-velocity part of the distribution is also noticeably reduced. The third special case or, more precisely, approach to determination of the 3-di- mensional distribution from 2-dimensional measurements was developed under the name “sliced VMI” (SVMI) in 2001 by Gebhardt, Rakitzis, Samartzis, Ladopoulos and Kitsopoulos [9], and in 2003 in a better form by Lin, Zhou, Shiu and Liu [10] and Townsend, Minitti and Suits [11]. It is based on the fact that preselection of one v z value for all particles can be done much easier that resolution of v z for each of them. 1 The lower limit of the noise level is determined by the shot noise due to the discrete nature of the measured signal. In practice, the projection intensity (the number of events per resolved interval ¢v x ¢v y ) is of the order of 100, leading to the relative noise of the order of p 100/100Æ 10 %. 9 That is, 2-dimensional slices I v z (v x ,v y ) can be recorded for any chosen v z value. The whole 3-dimensional distribution then can be obtained by v z scanning — very sim- ilar to the core sampling approach, but much more efficient, since the 2-dimension spatial scanning with a mechanical mask is replaced with the 1-dimension temporal scanning with an electrical detector-gating pulse. Moreover, the application of SVMI to cylindrically symmetric distributions is very advantageous, because in that case the central slice (v z Æ 0) contains all the re- quired information. Namely, if the slicing function S(v z ) 1 is centered at v z Æ 0 (that is, R v z S(v z )dv z Æ 0), then the measured sliced projection is P(v x ,v y )Æ Å1 Z ¡1 I(v x ,v y ,v z )S(v z )dv z ¼ (2.4) ¼ I(v x ,v y ,0)¢¢v z , (2.5) where¢v z Æ R S(v z )dv z is the effective slice thickness. Therefore, if the slice is thin enough, the initial distribution can be obtained immediately as I(v ½ ,v y )¼ P(v x Æ v ½ ,v y )/¢v z for v ½ À¢v z . (2.6) A quantitative estimation of how thin the slice should be is readily obtained from the diagram shown in figure 2.2. The monochromatic particles with velocity v ½ produce the finite effective range¢v x in the sliced projection with the effective thickness¢v z . The relative blurring is then ¢v x v ½ Æ 1¡ s 1¡ µ ¢v z 2v ½ ¶ 2 ¼ 1 2 µ ¢v z 2v ½ ¶ 2 . (2.7) 1 Defined as the dependence of the relative detection efficiency on the v z velocity component. It is assumed to be normalized by amplitude: maxS(v z )Æ 1. 10 v x v z Δv z v ρ Δv x Figure 2.2 Illustration of blurring due to a finite slice thickness. The thick arc shows the part of monochromatic particles that is detected by slicing. The recorded projection of the arc is shown by the thick straight segment marked¢v x . That is, even if the effective slice thickness¢v z is»1/10 of the total velocity range 2v ½ (from ¡v ½ to Åv ½ ), the effective blurring is only »0.5 %. Of course, with a fixed slice thickness the relative resolution decreases for the low-velocity part of the distri- bution (when v ½ ¿¢v z , the “sliced” projection (2.4) degenerates to the regular projec- tion (2.2)). However, in contrast to the full projection approach, the expression (2.6) for “reconstruction” is local, meaning that the low-velocity part can be measured sep- arately with higher resolution. 1 Moreover, the transformation (2.4) is also invertible (see chapter 3), and the inverse transformation has better properties than the in- verse Abel transform (2.3). This means that SVMI always outperforms conventional 2-dimensional VMI. Several other special cases allowing determination of the 3-dimensional distribu- tion from its 2-dimensional projections can be found in the VMI review [12]. Some special techniques for selective detection are also mentioned there. However, these 1 The ratio ¢v z /v ½ for a fixed slicing pulse duration depends on the ion cloud TOF stretching. As discussed in subsections 2.4.3 and 2.5.5, the experimental conditions required to achieve large TOF stretching are more favorable for slower particles. Thus the achievable SVMI resolution is generally better for the low-velocity part of the distribution if it is measured separately. 11 techniques are less general than the described slicing approach 1 and hence are not considered in this work. The overview of the available VMI methods given above reveals that SVMI is the most effective choice for studies of product distributions in photodissociation reac- tions. Therefore, the following sections are focused on the desired performance of the SVMI setup, the general properties of ion-optical systems suitable for SVMI imple- mentation, and the selection of the optimal design and its parameters to meet the performance requirements. 2.2 Design goals The designed SVMI system was intended to replace the previously used TOF detection system in the free-radical molecular beam apparatus designed and built by Conroy and described in his Ph. D. dissertation [13]. Since the apparatus itself has shown good operability and satisfied the experimental needs, it was desirable to implement the retrofitting with minimal changes to the other parts of the machine. This imposed some geometric constraints on the ion optics parts, as discussed later. Another important geometrical parameter in the design was the detector diameter. Since the MCP detector assembly is the most expensive part of the SVMI system and the only part that has a finite lifetime and thus has to be replaced occasionally, it was decided, for unification purposes, to employ the same detector type as in the other VMI setup [14] used in the laboratory. The primary design goal of the new system was the ability to perform SVMI of atomic hydrogen products from photodissociation of small organic radicals. The ki- 1 For example, the “optical slicing” method requires a pump–probe arrangement in which the “pump” laser(s) should not cause ionization of the studied particles, and the “probe” laser should only ionize them and should not create such particles by itself 12 netic energy release (KER) in such processes ranges from a fraction of an electronvolt to a few electronvolts, and due to the large mass difference between the H fragment and the molecular cofragment, almost all the kinetic energy is carried by the H atom. Conversion of the produced H atoms to detectable H + ions is performed by the conve- nient and practically proven [13] near-threshold REMPI through the 2p state. The process is quite efficient and, since the recoil in near-threshold ionization is negli- gible, the produced H + ions retain the initial momentum of the H fragments. The achievable resolution is therefore determined by the uncertainties in the initial ex- perimental conditions and the accuracy of the SVMI setup itself. The experimental uncertainties involve position, velocity and time uncertainties. The initial position uncertainty comes from the finite size of the ionization region, determined by the intersection of the molecular beam with the ionizing laser beam, and the finite precision of the location of this intersection relative to the ion-optical system. The diameter of the skimmed molecular beam at the ionization region is 2¡3 mm, and the laser beam waist length is usually comparable to it. This defines the¢x uncertainty (see figure 2.1). The¢y and¢z uncertainties are mostly defined by the laser beam alignment relative to the ion optics, since the beam waist radius is usually negligibly small (of the order of several micrometers for the focused beam), and can be estimated as»1 mm each. The initial velocity uncertainty comes from the velocity distribution of the parent radical and is determined by the velocity spread in the molecular beam. The average speed in a “supersonic” helium beam does not exceed »1400 m/s. Given the nozzle diameter of 0.5 mm and the skimmed beam diameter of 2¡3 mm at»100 mm from the nozzle, the beam divergence multiplied by the average speed yields a limit for the transversal velocity components¢v x and¢v y of about 50 m/s which is less than 0.5 % of the characteristic speed of H fragments (»14000 m/s for 1 eV kinetic ener- 13 gy). The longitudinal velocity spread¢v z might be somewhat larger, but is still small compared to the H fragment velocities. Moreover, since v z is not measured, and¢v z affects only the effective slice thickness, its effect on the resolution should be negli- gible. The initial time uncertainty is due to the finite duration of the ionizing laser pulse and the timing jitter. The optical pulse is generated by frequency doubling of the radiation produced by a dye laser pumped with a Q-switched Nd:YAG laser and thus has a duration of a few nanoseconds. Its jitter directly replicates the jitter of the pump laser and generally does not exceed 1 ns. Thus the overall time resolution of the experiment is inherently limited at the few nanoseconds level, and this fact must be taken into account when the desired characteristics of the SVMI setup are consid- ered. In particular, slicing with detector gating pulses shorter than a few nanosec- onds will not lead to any improvement in the resolution but rather will decrease the recorded signal strength, according to (2.5). The effective relative slice thickness »1/10 required for good resolution (see p. 10) therefore demands TOF stretching for the imaged velocity range of the order of a few tens of nanoseconds. This criterion constitutes the main difficulty in the SVMI design for H ions, as discussed below. Another major requirement for the new system was not directly related to its SVMI capabilities but concerned its ability to operate in the TOF mass spectrometer (TOF-MS) mode with sufficient mass resolution. As mentioned above, VMI spectrom- eters grew out of TOF mass spectrometers and in principle can be used for the same purpose. However, the optimal design decisions and operation parameters for these two uses are quite different and even contradictory in some respects. Fortunately, the mass resolution required mostly for diagnostic purposes — observation of the radi- cals, their precursors and decomposition products — is very modest by the TOF-MS standards. The ions of interest contain a few C, O and H/D atoms and, possibly, one 14 halogen (Cl) atom. Therefore, a mass resolution of 1 Da (one H atom difference) for masses up to»100 Da should be sufficient. 2.3 General considerations As was briefly mentioned in section 2.1, there are two approaches to SVMI im- plementation. The first [9] is based on the so-called “delayed extraction”, where the ionization occurs in the absence of electric fields, the ion cloud is allowed to expand for a predetermined amount of time and is then accelerated towards the detector us- ing a pulsed electric field. The TOF dispersion (v z 7! t part) is achieved basically by mapping the positions at the moment of the field switching z(t delay )Æ±zÅ v z ¢ t delay (2.8) to the arrival times. This obviously requires that the setup does not meet Wiley– McLaren space-focusing conditions. However, (2.8) still contains the initial position ±zÆ z(0). Therefore, either the technique cannot be applied to VMI studies with ex- tended ionization regions, or a complicated analysis of the possibility to suppress the ±z term while retaining the v z ¢ t delay term in the z(t delay )7! t mapping is required. 1 Moreover, the transversal expansion of the ion cloud before the extraction makes the transversal focusing (for the (v x ,v y )7! (x, y) part) more demanding. Yet another potential complication is that for reliable operation of the system the time-depen- dent electric field must have well-defined and stable characteristics. This requires an electronic circuit that can switch a finely controllable kilovolt-range voltage in a few nanoseconds without producing noticeable ringing. The latter, in turn, requires very good impedance matching between the electric pulser and the ion-optical as- 1 No attempts of such analysis were known to me at the time of the present work. 15 sembly to which the voltage is applied. Since the dimensions of the system are not negligible compared to the “wavelength” 1 , the whole problem might require solution of time-dependent Maxwell equations in nontrivial geometry and with real nonper- fect conductors and dielectrics. As a result of these difficulties, the rational design, optimization and implementation of the system become an intricate task. Fortunately, the second approach [10, 11] to SVMI implementation involves only electrostatic fields for velocity mapping and focusing purposes and therefore avoids the complications described above. Therefore, all the following analysis deals only with electrostatic systems. 2.3.1 Exact relationships As a starting point, consider the behavior of a charged particle in an electric field with fixed geometry in a general case. The equation of motion with initial conditions is 8 > > > > > > < > > > > > > : m d 2 r(t) dt 2 Æ qE ¡ r(t) ¢ , dr(0) dt Æ v 0 , r(0)Æ r 0 , (2.9) where m is the mass of the particle, q is its electric charge, and E(r) is the electric field that depends on coordinates (but not on time explicitly). The general properties of the motion can be studied by finding the invariants of (2.9). For that purpose it is convenient to rewrite the electric field in the whole space as E(r)´ E¢f(r), (2.10) 1 The electric field propagation speed in vacuum is c¼ 30 cm/ns and is even smaller in waveguides, including the one formed by the ion optics, its surroundings and connections. 16 where E is a scalar constant describing the overall field strength (can be chosen as EÆjE(0)j), and f(r) is a vector function describing the coordinate dependence of the field. In addition, a scaling substitution 8 > > > > < > > > > : t 0 Æ®t, m 0 Æ¯m, q 0 E 0 Æ°qE (2.11a) (2.11b) (2.11c) with arbitrary constants®,¯ and° can be made into (2.9) to obtain 8 > > > > > > < > > > > > > : ¯ ® 2 m d 2 r(®t) dt 2 Æ° qE ¡ r(®t) ¢ , 1 ® dr(0) dt Æ v 0 0 , r(0)Æ r 0 0 . (2.12) It is obvious from comparison of (2.9) and (2.12) that the form of the solution does not change if 8 > > > > > > < > > > > > > : ¯ ® 2 Æ°, v 0 0 Æ 1 ® v 0 , r 0 0 Æ r 0 . (2.13a) (2.13b) (2.13c) The last equation, (2.13c), simply tells that in order to have the same trajectory the initial position of the particle should not change. The preceding equation (2.13b) defines the scaling of the initial velocity. It would also be useful to consider how the initial kinetic energy K 0 should scale: K 0 0 Æ m 0 (v 0 0 ) 2 2 Æ ¯ ® 2 mv 2 0 2 Æ ¯ ® 2 K 0 . (2.14) 17 Comparison of (2.14) with (2.13a) and (2.11c) immediately yields that K 0 0 q 0 E 0 Æ K 0 qE Æ const (2.15) is an invariant of (2.9). An important property of (2.15) is that it does not depend on the particle mass. That is, all singly charged ions with identical initial kinetic en- ergies (and identical initial velocity directions) have identical trajectories in a given electrostatic field. If the field direction is reversed (E 0 Æ¡E), electrons with the same K 0 will also have the same trajectories. This fact reveals that in the electrostatic im- plementation of VMI the initial kinetic energy is a more fundamental (and often more convenient) quantity than the initial velocity or the initial momentum. 1 Another im- portant observation is that the imaged kinetic energy range is directly proportional to the electric field strength. That is, the image magnification (the (v x ,v y )7! (x, y) part) can be easily controlled by proportional scaling of all voltages applied to the ion optics. Other useful invariants can be obtained if some parameters are fixed. The first invariant is related to the behavior of different particles in the same electric field (E 0 Æ E). 2 According to (2.11), it corresponds to°Æ 1 and thus, from (2.13a), to¯/® 2 Æ 1 The role of kinetic energy can be understood from Maupertuis’ principle (see §44 in [15]), which de- termines the trajectory of a mechanical system in a time-independent potential. The abbreviated action extremum conditions for a single particle in an external potential U(r) can be written ex- plicitly:± R r r 0 q 2m ¡ [K 0 ÅU(r 0 )]¡U(r) ¢ jdrjÆ 0. Obviously, the coordinate-independent mass m can be taken outside the integral and the variation without changing the variational equation and thus its solution — the trajectory. The same reasoning applies to a common scaling factor for K 0 and U. In other words, the trajectory is determined by the K 0 /U ratio, but not by K 0 or U separately, which is reflected in (2.15) where the denominator is related to U as dUÆ¡qEdr. These arguments do not hold for time-dependent electric fields (the interaction depends not only on the particle position, but also explicitly on time) and for magnetic fields (the interaction depends on velocity and cannot be described by a potential, although static magnetic fields do not affect the kinetic energy of electrically charged particles), thus the trajectories in such fields generally do depend on the particle mass. 2 Henceforth only singly charged ions or electrons are considered. It is also implicitly assumed that the particles are positively charged (cations); for negatively charged particles (anions, electrons) the 18 1, or K 0 0 Æ K 0 . That is, the only free parameter is the particle mass. Substitution of (2.11) into¯/® 2 Æ 1 yields t 0 p m 0 Æ t p m , or t 0 t Æ s m 0 m . (2.16) In other words, the TOF is directly proportional to the square root of the particle mass, if all other parameters are fixed. This forms the foundation of the TOF-MS operation. It also allows mass-selectivity in (S)VMI and will be important in the further analysis of the temporal characteristic of the SVMI system. The second invariant is related to the behavior of the same particle (m 0 Æ m) in different electric fields. This corresponds to ¯Æ 1 in (2.11). Although the kinetic energy scaling was already given by (2.15) for the general case, we will be mostly interested in the “central” TOF, t 0 , of the ion cloud, that is in the behavior of the particles with v 0 Æ 0, or K 0 Æ 0 (this case satisfies (2.13b) automatically). Substitution of (2.11) into 1/® 2 Æ° (from (2.13a)) yields t 0 0 p E 0 Æ t 0 p E, or t 0 0 t 0 Æ s E E 0 . (2.17) In other words, the central TOFs for particles of each mass are inversely proportional to the square root of the electric field strength (or, equivalently, the square root of the applied voltages), if the field geometry is fixed. Combination of (2.16) and (2.17) gives the general TOF dependence t 0 » r m E . (2.18) electric field should be reversed. 19 2.3.2 Approximate relationships Now the properties related to the velocity mapping (2.1) shall be studied. Unfortu- nately, this cannot be done in a general form, as in the previous subsection, and thus some restrictions and approximations appropriate for the considered SVMI systems must be made. One of such restrictions is that only electric fields with cylindrical symmetry around the z axis will be considered. This restriction does not reduce the design freedom and is even desirable for an SVMI system. Namely, it means that all directions in the (x, y) plane are equivalent, and the system does not introduce any artificial anisotropy in the (v x ,v y )7! (x, y) part of the mapping. It also entails that the mapping conserves the angle µ (see figure 2.1). With that in mind, the “radial” velocity v r ´ q v 2 x Å v 2 y (2.19) is introduced, 1 which is mapped to the radius RÆ p x 2 Å y 2 at the detector. The first task is a more detailed determination of the expression for TOF. Since (2.18) gives the TOF for particles with v 0 Æ 0, the general dependence can be written as tÆ t 0 Å¿(v 0 ), (2.20) where ¿(v 0 ) defines the TOF–velocity dispersion function (¿(0)Æ 0), and all terms implicitly depend on m and the electric field. Within the paraxial approximation 2 the exact equations of motion (2.9) can be approximately separated into the axial 1 The index “r” cannot be confused with the radius-vector rÆ (x, y, z), since the latter always appears as a vector. 2 Meaning that the particles do not depart from the z axis significantly, and the axial component E z of the electric field has very weak dependence on the radial distance r from the axis, at least in the reachable regions. 20 and radial parts. The axial part is then 8 > > > > > > > < > > > > > > > : m d 2 z(t) dt 2 Æ qE z ¡ z(t) ¢ , dz(0) dt Æ v z , z(0)Æ z 0 , (2.21a) ( 9 > > = > > ; 2.21b) and still contains the field E z (z) in an arbitrary form. In order to obtain an explicit (although approximate) dispersion function ¿(v z ), the field in the vicinity of the ionization region can be approximated with a homoge- neous field E z (z)j z¼z 0 ¼ E 0 as shown in figure 2.3. (The justification of this approx- E z (z) +v z −v z +v z E z (z)≈E 0 detector z=l z 0 Δz b z=L Figure 2.3 Diagram illustrating the electric field approximation used in the derivation of the TOF velocity dispersion. imation will be given after the formal derivations of its consequences.) In that case (2.21) can be analytically integrated up to zÆ l (lÀ z 0 ): z(t)Æ qE 0 2m t 2 Å v z tÅ z 0 , z6 l, (2.22) 21 and the TOF up to zÆ l can be found directly: t l ´ tj z(t)Æl Æ ¡v z Å r v 2 z Å2 qE 0 m (l¡ z 0 ) qE 0 m , (2.23) where the positive root of the quadratic equation (2.22) was chosen. This expression can be rearranged as t l Æ¡ mv z qE 0 | {z } t b Å p 2m qE 0 s mv 2 z 2 |{z} K z Å qE 0 (l¡ z 0 ) | {z } ¢U , (2.24) revealing the physical meaning of the participating terms. Namely, t b ´¡mv z /qE 0 is the time required to stop a particle initially flying with velocity¡v z from the de- tector (see figure 2.3), K z is the initial kinetic energy (K 0 ) component associated with motion along the z axis, and¢U is the energy gained by the particle from the electric field. Taking into account that¢UÀ K z (see below), the first term under the root can be treated as a small parameter, which yields t l Æ¡ mv z qE 0 Å s 2m(l¡ z 0 ) qE 0 ÅO µµ v z v ¢U ¶ 2 ¶ , (2.25) where v ¢U ´ s 2¢U m À v z (2.26) is the characteristic speed gained due to the acceleration, 1 and O(...) contains quad- ratic and higher-order terms in the small parameter v z /v ¢U . Notice that the velocity- independent term in (2.25) is proportional to p m/E 0 in full agreement with the gen- eral result (2.18). 1 In particular, v ¢U is the “central” (see p. 19) speed at zÆ l. 22 The subsequent behavior of the particle can be described by formal integration of (2.21a) starting from zÆ l, 8 > > > > > > < > > > > > > : m d 2 z(t) dt 2 Æ qE z ¡ z(t) ¢ , t> t l , dz(t l ) dt Æ v l , z(t l )Æ l, (2.27) where the velocity v l ´ v(t l )Æ vj zÆl can be obtained by plugging (2.25) into v(t)Æ qE 0 m tÅ v z or simply from energy conservation: v l Æ s 2(K z Å¢U) m Æ v ¢U ÅO ¡ (v z /v ¢U ) 2 ¢ . (2.28) For the TOF from zÆ l to the detector (zÆ L) the integration yields 1 t l!L Æ L Z l dz q 2q m R z l E z (z 0 )dz 0 Å v 2 l , (2.29) which depends on the initial velocity v z only through v l given by (2.28) and thus has the same weak dependence: t l!L (v z )Æ t l!L (0)ÅO ¡ (v z /v ¢U ) 2 ¢ , (2.30) where t l!L (0)Æ r m 2q L Z l dz q E 0 (l¡ z 0 )Å R z l E z (z 0 )dz 0 (2.31) 1 Since the transition from (2.27) to (2.29) is not so obvious, here is a hint: the first step involves multiplication of the first equation by dz and the rearrangement d 2 z dt 2 dzÆ d ³ dz dt ´ dt dzÆ ³ dz dt ´ d ³ dz dt ´ , after which the subsequent steps are trivial. See also §11 in [15] for derivations of an equivalent expression from energy conservation. 23 (notice again the p m/E factor). This, together with (2.25), allows to write the two terms in (2.20) for the total TOF as t 0 Æ s 2m(l¡ z 0 ) qE 0 Å t l!L (0) (2.32) and ¿(v 0 )¼¡ mv z qE 0 . (2.33) That is, the v z 7! t part of the mapping (2.1) is approximately linear: v z 7! t¼ t 0 ¡ m qE 0 ¢ v z . (2.34) The total TOF spread (ion cloud stretching) for particles with initial kinetic energy K 0 is therefore ¢¿ K 0 ´ max mv 2 0 2 ÆK 0 ¿(v 0 )¡min mv 2 0 2 ÆK 0 ¿(v 0 )Æ 2 p 2mK 0 qE 0 . (2.35) It should be noted that (2.35) is in fact a stronger result than the relation (2.33). Namely, a properly operating SVMI setup should not significantly distort the shape of the ion cloud, so that even if there are deviations from (2.33), the shortest TOF should still correspond to the particles initially flying directly towards the detector, and the longest — to those flying in the opposite direction, as illustrated in figure 2.3. The latter particles after the “turnaround time” t TA Æ 2t b (see (2.24)) return to their initial position but with the exactly reversed initial velocity, 1 and thus after that point their behavior is exactly identical to that of the former particles. It means that the TOF difference is exactly equal to t TA and is determined by the electric field only in the small interval¢z b . 1 The trajectory schematically shown in the figure is vertically unfolded for clarity. 24 Another important fact that should be mentioned here is that the TOF veloci- ty dispersion (2.33) (and the TOF stretching (2.35)) is inversely proportional to the electric field strength, in contrast to the central TOF (2.18), which is inversely propor- tional to its square root. This has important implications for both SVMI and TOF-MS, as will be shown in a moment. The justification of the assumptions involved in the derivation of the above re- sults should now be given. As discussed on p. 14, the required TOF stretching is few tens nanoseconds. Solving (2.35) to obtain, for example,¢¿ K 0 Æ 50 ns for hydro- gen ions with K 0 Æ 1 eV gives the required fieldE 0 ¼ 58 V/cm. The maximum value of¢z b Æ K 0 /(qE 0 ) is then only about 0.17 mm — even smaller than the initial po- sition uncertainty (see p. 13), which indicates that formula (2.35) can be actually considered exact for all practical purposes. The characteristic size of the VMI ion op- tics elements is of the order of a few centimeters, defining the length scale for the spatial electric field variation. Therefore, the choice of l» 1 cm gives a reasonable estimation for the region where the field can be treated as approximately homoge- neous (see figure 2.3). Thus¢U» 58 eV in (2.24) is indeed much larger than K 0 , and (v z /v ¢U ) 2 6 K 0 /¢U» 1/58 in (2.25) is also a small parameter. The numerical simu- lations described in section 2.5 in fact show that the main factors determining the inaccuracy in (2.33) are deviations from paraxial approximation, and they become noticeable only when the ion-optical system operates in unfavorable conditions. For investigation of the TOF-MS resolution and the v r 7! R part of the mapping a more explicit expression for the central TOF (2.32) is required. It can be obtained for the practically important field configuration shown in figure 2.4 where the particles are accelerated by the electric field only in a limited “acceleration” region from z 0 to L a and then fly to the detector through a field-free “drift” region. The subradical integral in (2.31) then changes only in the z 0 6 z6 L a region, and thus the main 25 E(r) detector z=L a z 0 z=L acceleration drift E=0 Figure 2.4 Ion optics configuration with a field-free drift region. integral can be split into two parts, R L l Æ R L a l Å R L L a , with the second part having no z- dependence. The total expression for the central TOF (from z 0 ) then becomes t 0 Æ r m 2q L a Z z 0 dz q R z z 0 E z (z 0 )dz 0 Å r m 2qV 0 (L¡ L a ), (2.36) where V 0 ´ L a Z z 0 E z (z)dz (2.37) is the total acceleration voltage (electric potential difference between the initial po- sition z 0 and the drift region). The expression (2.36) shows that the total TOF grows linearly with the total length of the VMI setup, and if L a is relatively short, t 0 ¼ r m 2qV 0 L» L¢ r m E . (2.38) The mass resolution is determined by the maximum mass for which the TOF distribution does not overlap with that of the next mass. Therefore, assuming ideal TOF spatial focusing and a characteristic kinetic energy K 0 for particles of mass m 26 and mÅ1 Da, the masses are resolved if 1 2 ¢¿ K 0 (m)Å 1 2 ¢¿ K 0 (mÅ1 Da)6 t 0 (mÅ1 Da)¡ t 0 (m). (2.39) Substitution of (2.38) and (2.35), with an assumption mÀ 1 Da, yields 1 2 ¢¿ K 0 (m)Å 1 2 ¢¿ K 0 (mÅ1 Da)» p mK 0 E , (2.40) t 0 (mÅ1 Da)¡ t 0 (m)» L s mÅ1 Da E ¡ L r m E » L p mE , (2.41) and hence the maximal resolved mass m max » L s E K 0 . (2.42) For each particular setup it can also be expressed more quantitatively through the parameters for hydrogen ion (mÆ 1 Da): m max ¼ t 0 (H Å ) 2¢¿ K 0 (H Å ) . (2.43) This dependence immediately shows one of the mentioned contradictory require- ments for SVMI and TOF-MS. Namely, the mass resolution is higher for stronger electric fields, where¢¿ K 0 is smaller, but the slicing conditions require a specific fi- nite¢¿ K 0 value and thus generally relatively weak fields (see p. 25). In principle, if the TOF-MS and SVMI modes are used independently, then a strong field can be set in one mode and a weaker field in the other. However, designs optimized for TOF-MS operation benefit from using a long drift region (according to (2.42)) and a very short acceleration region (figure 2.4) in order to maximize E 0 in (2.35) without inflating V 0 in (2.36). At the same time, such geometrical parameters are usually unfavorable 27 for SVMI because they lead to an expansion of the image size at the detector plane beyond the desirable limits. Namely, the radial velocity component v r (t) inside the acceleration region de- pends linearly (in paraxial approximation) on its initial value v r : v r (t)¼ v r ¢ g(t), (2.44) where g(t) is a system-dependent function, and is (exactly) conserved in the field-free drift region: v r (tÈ t a )Æ v r (t a ), or g(tÈ t a )Æ g(t a ), (2.45) where t a is the time spent in the acceleration region (z(t a )Æ L a ). Therefore, the image radius at the detector RÆ r 0 Å t(v 0 ) Z 0 v r (t)dtÆ r 0 Å v r 2 4 t a Z 0 g(t)dtÅ g(t a )¢(t 0 ¡ t a )ÅO ¡ ¿(v 0 )/t 0 ¢ 3 5 (2.46) is directly proportional to the radial component of the initial velocity as well. That is, the radial part of the velocity mapping v r 7! R¼ const t 0 ¢ v r (2.47) is also approximately linear (compare to (2.34)) with a system-dependent proportion- ality constant. For t a ¿ t 0 (that is, L a ¿ L) the image size scales as R» L s K r V 0 » L s K 0 E , (2.48) 28 where the “radial kinetic energy” K r ´ mv 2 r /2 and (2.38) were substituted. 1 Hence the use of a long drift region and very short acceleration region optimal for TOF-MS would lead to a very large image size due to the L proportionality in (2.48) and very small V 0 caused by the required weak field E 0 and short L a in (2.37). A few remarks should be made about the radial part (2.47) of the velocity map- ping (2.1). First, the original expression (2.46) contains a weak dependence on the initial axial velocity component through the t(v 0 ) dependence on it. If a more ac- curate TOF from (2.34) is substituted into (2.47) instead of t 0 , the overall velocity mapping becomes (v r ,v z )7! ³ const h t 0 ¡ m qE 0 ¢ v z i ¢ v r , t 0 ¡ m qE 0 ¢ v z ´ . (2.49) That is, a small cross-term v z v r appears in the radial part. This term is not related to any approximations and disappears only in the limit¢¿ K 0 /t 0 ! 0 (meaning t 0 !1, since ¢¿ K 0 must have a finite value). In practice it means that in the projection– reconstruction approach (see p. 8) the inverse Abel transform can be applied only when¢¿ K 0 ¿ t 0 , which is in fact usually satisfied in modern VMI setups. 2 The cross- term has no effect in SVMI of cylindrically symmetric distributions, since only the slice with v z Æ 0 is recorded. In SVMI of general distributions the correction according to the selected v z value can be easily done regardless of the¿(v 0 )/t 0 ratio. The second remark, related to the issue emphasized on p. 25, is crucial for SVMI and concerns the different dependences of the axial and radial parts of the mapping on the electric field strength. Namely, for a fixed field geometry the TOF spread (2.35) 1 Notice again that R depends on the K 0 /E ratio and does no depend on the particle mass, in agreement with the general result (see p. 18). 2 See [16] for a detailed discussion and a reconstruction method for the¢¿ K 0 6¿ t 0 case. Please note that the electrostatic optical system described there has no field-free drift region and thus has a poor¿(v 0 )/t 0 ratio, so that the values of their½ parameter (equal to qV 0 /K 0 in the present notation) required for nearly parallel projection would be somewhat overestimated for modern systems. 29 is proportional to p K 0 /E, whereas the image size (2.48) is proportional to p K 0 /E, which means that, in general, fulfillment of slicing conditions and selection of appro- priate image size at the detector cannot be achieved simultaneously. In other words, an SVMI setup intended for measurements in variable kinetic energy ranges must necessarily permit creation of electric fields with variable geometries. 2.4 Ion optics This section is devoted to an overview of ion optics concepts and historical exam- ples of their application to the VMI problem. This information serves as an illustra- tion of the general principles and is important for understanding the limitations of these simple systems and the means to overcome these limitations. 2.4.1 Wiley–McLaren arrangement for TOF spatial focusing As mentioned in the introduction, the conditions required for making the TOF independent on the initial positions of the ions were studied by Wiley and McLaren in 1955 [3] in the context of TOF mass spectrometer. Since they were not interested in the initial radial position and velocity distributions of the particles, a 1-dimensional electric field configuration was chosen for simplicity. In other words, the axial part (2.21) of the equations of motion provided an exact description of the problem. Another simplification of the problem was to limit the field configuration to a few regions of homogeneous electric field. From a practical perspective, such field con- figuration is easily created by applying different voltages to a few mutually parallel grids installed perpendicular to the TOF axis as illustrated in figure 2.5. The grids (meshes) must have sufficiently high open area ratio in order to transmit the par- ticles and, at the same time, sufficiently small cell size in order to keep the electric 30 E 0 detector L 1 z 0 =0 z 0 =δz z=L E 1 E=0 L 0 L 0 0 Figure 2.5 Geometry of Wiley–McLaren TOF mass spectrometer as an example of a system with TOF spatial focusing. field on both sides nearly homogeneous. 1 Such simple E z (z) dependence allows facile analytical evaluation of the integrals in (2.36) and (2.37) and therefore results in a relatively simple analytical expression for the total TOF. Without going into details, which can be found in [3], the idea of the focusing can be understood from very simple considerations. Namely, ions originated farther from the detector spend more time in the accelerator region than ions originated closer to the detector (the first term in (2.36) increases when z 0 decreases), but acquire higher energy from the electric field and thus fly through the drift region faster (the second term in (2.36) decreases with z 0 because V 0 in (2.37) grows). Therefore, at some point in time these ions should meet in the drift region. Existence of more than one region of homogeneous electric field allows to vary V 0 (which determines the final energy KÆ qV 0 ) independently of E 0 (which determines the energy difference ±KÆ¡qE 0 ±z) for matching the “focal length” of the system to the detector position. In fact, two regions (as depicted in figure 2.5) are already sufficient for that purpose. Quantitative analysis gives a relatively simple analytical expression (see [3]) for the dependence of the focal length F t on the two accelerator lengths L 0 0 and L 1 and 1 The depth (distance from the grid plane) of field inhomogeneities is comparable to the mesh open- ing diameter, so, for example, the field between two grids with densities of the order of »10 mm ¡1 separated by a few centimeters can be considered as homogeneous with very high accuracy. 31 the field strengths ratio E 1 /E 0 . The dependence is illustrated in figure 2.6, which 0 100 200 300 400 500 600 700 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 F t ,mm E 1 /E 0 L 0 0 =50mm 40mm 30mm 20mm L 1 =50mm 40mm 30mm 20mm Figure 2.6 Focal length for TOF spatial focusing in the Wiley–McLaren system (see fig. 2.5). makes clear that F t can be varied in a very broad range by adjusting any of the parameters, especially the E 1 /E 0 ratio. This last fact is quite useful from a practical perspective, since the applied voltages that determine E 0 and E 1 for given L 0 0 and L 1 are the most easily controllable parameters. The prominent increase of F t with L 1 is mostly due to the increase in the final kinetic energy — the particles have higher velocity in the drift region and thus meet at a longer distance. One interesting feature of the general dependence also visible in the plots is that in the case of a single acceleration field (that is, E 1 Æ 0) the focusing is still present, and F t Æ 3L 0 0 . 1 However, in light of the remark made on p. 29, such configuration is not very suitable for the SVMI needs because the field geometry depends on only one 1 In addition, when E 1 /E 0 Æ 1, the configuration is also effectively a single field region of length L 0 0 ÅL 1 . In that case F t Æ 3(L 0 0 Å L 1 ), which is also evident from the figure. 32 parameter L 0 0 Æ L/3, whereas the two-field configuration has 3 degrees of freedom {L 0 0 /L,L 1 /L,E 1 /E 0 }. In more complex geometries, including 2-dimensional field configurations 1 , the dependence of F t on the field geometry becomes more complicated, and its deter- mination usually requires numerical integration. However, the general qualitative relation to E 0 and V 0 is preserved. 2.4.2 Eppink–Parker arrangement for radial focusing The first ion-optical system able to perform spatial focusing in the radial direc- tion, that is to make the arrival positions of the particles independent on their initial positions, was studied by Eppink and Parker in 1997 [7] and is surprisingly similar to the Wiley–McLaren arrangement. The only difference is that the two grids are re- placed by apertures — flat electrodes with circular opening of finite radius, as shown in figure 2.7. The finite size of the opening breaks the homogeneity of the electric fields at finite scales comparable to the aperture radii. Since the electric potential '(r, z) in free space satisfies the Laplace’s equation 1 r @ @r µ r @' @r ¶ Å @ 2 ' @z 2 Æ 0, (2.50) any inhomogeneities must lead to appearance of a non-zero radial component of the electric field E r (r)Æ¡ @'(r) @r 6Æ 0. (2.51) 1 Meaning cylindrically symmetric geometry, see p. 20. 33 detector z=L V 0 E 0 E=0 V 1 V=0 L 1 L 0 L 0 0 z 0 =0 ±δy E 1 Figure 2.7 Cross-section of Eppink–Parker VMI setup for radial focusing. Electric field configuration is illustrated by gray equipotential contours. Sample ion trajecto- ries are shown in black for v 0 parallel to the z axis and in blue for v 0 perpendicular to it. Moreover, the fact that @ 2 ' @z 2 is finite, in conjunction with (2.50), leads to the paraxial dependence E r (r)» r, (2.52) which has the form exactly suitable for focusing of charged-particle beams. The focusing properties of these so-called “aperture electrostatic lenses” were ac- tually known since at least 1932 when Davisson and Calbick studied them theoreti- cally and experimentally [17] 1 , and were widely used in many charged-particle beam applications including imaging devices such as cathode ray tubes and electron micro- scopes (since the same early 1930s [19]). The exceptionally long delay in the adoption of these ideas in the VMI was probably caused by two factors. The first is that the charged-particle optics and reaction dynamics of free molecules were quite separat- ed fields. In principle, even though TOF spectrometers that were used in the latter 1 Their first communication appeared in 1931 as APS meeting abstract (p. 585 in [18]), but it was related to slit apertures which behave somewhat differently from circular apertures. 34 can have “optical” elements in electron guns and ion beam collimators and reflectors, they were not intended to produce a meaningful spatial distribution that could be analyzed. The second factor is that VMI systems are not imaging systems from the optical point of view. Namely, an optical system is classified as “imaging” if it pro- duces a real image of a real object. In other words, it should map points from the object space to points in the image space. On the contrary, VMI systems produce a real image of a velocity distribution, that is they map “points” from the velocity space (beam directions and speeds) to points in the image space. 1 In the context of optics such systems can be classified as telescopic objectives. At the same time, their input lens is of the immersion type. This is a quite extraordinary combination, meaning that even familiarity with the usual devices based on charged-particle optics would not immediately give an idea of the VMI possibility. The behavior of charged particles in electric field configurations formed by aper- tures can be studied using analytical approximations (see [22] for an example). In particular, they show that the focal length f of the aperture lens, in the first approxi- mation, does not depend on its diameter and is determined by the ratio of the particle kinetic energy at the lens plane to the difference between the electric field strengths in the regions separated by the aperture: f Æ 4K(z lens ) q ¡ E zÈz lens ¡ E zÇz lens ¢. (2.53) This dependence has two important properties. First, the focal length is directly pro- portional to the kinetic energy of the particle. This fact reveals that electrostatic lenses have very high “chromatic aberrations” (dependence of the refraction on the 1 Therefore, the terminology must be used carefully to distinguish the “velocity map imaging” from the truly imaging techniques such as “ion imaging” in “imaging mass spectrometry” (see, for example, [20]), as well as from various “velocity mapping” techniques, which provide spatial distribution of flow velocity in media (for example, of blood flow [21]). 35 energy). 1 In addition, since electric fields exist before and after the lens, the kinet- ic energy of the particles constantly changes as they fly through the system, so the properties of a lens substantially depend on its location. Second, the lens is asym- metric. Namely, it is converging (f È 0) if the field increases along the path, and diverging (f Ç 0) if the field decreases. 2 In application to the system shown in fig- ure 2.7 it means that the first aperture works as a positive lens, and the second one — as a negative, but since the kinetic energy at the second lens is higher than at the first, the overall result is positive (if the second field is sufficiently strong). All the mentioned effects can be noticed in the figure. Although the focusing does not depend on the lens diameter in the first approx- imation, some dependence emerges if the ions are created within a close distance (comparable to the aperture radius) from the lens plane, as shown in figure 2.7. In that case the electric field in the ionization region has a nonzero curvature, resulting in initial acceleration towards the axis, which leads to some additional focusing. 3 The position of the leftmost electrode in figure 2.7 might also affect the focusing, if it is placed close to the lens, since the field is flattened near the electrode surface. How- ever, if the distance from the lens plane exceeds the aperture diameter, the effect is minor. In any case, numerical simulations become the preferred method for quantita- tive analysis because they can be performed relatively quickly with modern comput- ers, work equally well in any geometry and have good accuracy. The lack of specif- 1 For example, electrostatic lenses have two times different focal lengths for particles with energies K and 2K, whereas “regular” lenses for visible light have only»4 % focal length variation for the whole visible spectrum (the same twofold difference in the photon energy hº). 2 More precisely, the field projection on the traveling direction is important. If particles are decelerated flying against the field, then “less negative” fields mean a field increase, so (2.53) and its description do not contradict the time reversal symmetry. 3 The curvature is nevertheless sufficiently small and can be safely ignored in the analysis given on p. 21. 36 ic restrictions additionally allows more detailed studies of various aberrations. All characteristics of ion-optical systems reported from this point onwards (as well as the electric potential contours and particle trajectories in figure 2.7) were obtained by numerical simulations in the ion optics simulation software package SIMION 8 [23, 24] complemented with customized user-programming scripts for analysis of the trajectories and (semi)automatic optimization of the electric field parameters. An illustration of the radial focusing focal length F R dependence on L 0 0 , L 1 and E 1 /E 0 for the two-aperture system is given in figure 2.8. As with the TOF focusing −400 −200 0 200 400 600 800 1000 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 F R ,mm E 1 /E 0 L 0 0 =20mm 30mm 40mm 50mm L 1 =20mm 30mm 40mm 50mm Figure 2.8 Focal length for radial focusing in Eppink–Parker VMI system (see fig. 2.7; both apertures have40 mm, and L 0 Æ L 0 0 Å15 mm). dependence F t (L 0 0 ,L 1 ,E 1 /E 0 ) (see p. 31), the radial focal length F R (L 0 0 ,L 1 ,E 1 /E 0 ) al- so can be varied in a very broad range by adjusting any of the parameters, among which the electric field strengths ratio again has the most prominent effect. Howev- 37 er, in contrast to F t , F R is negative for E 1 /E 0 . 2, meaning that the beams actually diverge. This fact is easily understood for E 1 /E 0 Ç 1, since in that case both lenses are diverging, as can be seen from (2.53). When E 1 /E 0 È 1, the first lens becomes converging, but its optical power (1/f ) is proportional to E 1 ¡ E 0 Æ ([E 1 /E 0 ]¡ 1)E 0 , which has smaller magnitude than the proportionality factor 0¡ E 1 Æ¡[E 1 /E 0 ]E 0 for the second, diverging, lens. Nevertheless, the kinetic energy factor in (2.53) de- creases the power of the second lens relative to the first one. The ratio of the kinetic energies strongly depends on both length parameters (L 0 0 and L 1 ), what explains why the threshold ratio min F R È0 [E 1 /E 0 ] at which the overall optical power becomes positive increases with L 0 and decreases with increasing L 1 . 1 For the sake of completeness, it should be pointed out that if the second aper- ture is replaced with a flat grid (as in the original Wiley–McLaren system), then only the first, positive, lens remains, making the focusing possible with lower E 1 /E 0 ra- tios. This arrangement, however, does not offer any practical advantages and is even detrimental, since maintaining the grid at sufficiently flat geometry is practically challenging, and the presence of the grid in the beam path decreases the transmis- sion of the system. 2 An additional problem with the grid is that its transmission has spatial dependence, which might introduce artificial patterns in the VMI result. Another possibility is to extend the second electric field region up to the detector, thus having only one positive lens and no grids. Or to use specially shaped electrodes instead of the flat leftmost plate in figure 2.7 and the aperture plate in order to create a suitable field curvature in the ionization region for overall focusing effect even with the single aperture working as a negative lens (see [25] for an attempt to use 1 The distance between the lenses, equal to L 1 , also affects the overall optical power through changing the relative position of the focal points of the two lenses, so the dependence on L 1 is not purely due to the kinetic energy dispersion effect. 2 The Eppink–Parker configuration actually has ideal transmission. 38 this approach). Such configurations, however, have the same drawback as the single- field TOF focusing (see p. 32), namely, very small freedom of electric field geometry variations. 1 2.4.3 Minimalistic SVMI system The fact that the Wiley–McLaren arrangement for TOF focusing and the Eppink– Parker arrangement for radial focusing are very similar suggests that both types of focusing can be achieved simultaneously in a system with two aperture lenses. In- deed, comparison of the F t dependence shown in figure 2.6 2 and the F R dependence shown in figure 2.8 reveals that for any given {L 0 0 ,L 1 } the former monotonically grows with E 1 /E 0 , and the latter monotonically descends. Therefore, for any lengths combi- nation {L 0 0 ,L 1 } there always exists exactly one electric fields ratio E 1 /E 0 at which the F t and F R curves cross, so placing the detector at LÆ F t Æ F R produces the sought VMI system that performs the mapping (2.1) independently of the initial ion posi- tions. Hence the parameters required for SVMI operation should satisfy the following conditions: 8 > > > > > > < > > > > > > : ¢t maxK 0 (E 0 )Æ¢¿ S , F t (L 0 0 ,L 1 ,E 1 /E 0 )Æ L, F R (L 0 0 ,L 1 ,E 1 /E 0 )Æ L, (2.54) where¢¿ S is the TOF stretching required for slicing (»10 times the effective slicing width, see p. 14). This system of 3 equations has 5 independent variables: {L,L 0 0 ,L 1 , E 0 ,E 1 /E 0 }. E 0 is determined by the first equation, and the remaining 2 equations 1 The variation of electrode shapes in the “single-field” case is too impractical. 2 As mentioned above, this dependence should be slightly different for the system with apertures in- stead of grids, but the difference is actually small and not qualitative. 39 with 4 variables leave 2 degrees of freedom in the solution. From a mathematical point of view, the total length L is a very good choice of a free parameter because, according to (2.48), it is directly related to the radial magnification constant in (2.47) and thus allows to change the overall image size independently of the TOF stretching (2.35). Unfortunately, this approach is rather inconvenient from a practical point of view because for good adjustability of the image size the total length of the appara- tus must be variable by several times, which is difficult to perform for a high-vacuum system. 1 Therefore, a more practical approach is to fix the total length at some rea- sonable value, mostly determined by the desired mass resolution, since L is also the most important factor in (2.42). This leaves only one free parameter in (2.54) that can be used for image size adjustments or minimization of aberrations. The choice of the parameter is not very important, as long as it uniquely determines the solution of (2.54). In the results reported below the initial acceleration length L 0 0 was chosen as the independent variable based on its role in affecting the electric field curvature in the ionization region and the kinetic energy at the first lens plane, and through that the aberrations of the whole ion-optical system. As discussed in sections 2.3.1 and 2.3.2, all performance characteristics have sim- ple scaling rules with respect to the overall electric field strength and particle mass. Therefore, numerical simulations for one particular type of ions and one electric field scale are sufficient to completely characterize the system. The natural choice of ions are H + , since they have “unit” mass (mÆ 1 Da) and are particularly important for the intended application of the designed setup. The natural measure of the overall elec- tric field strength is E 0 (see (2.54) and the discussion following it), which determines the TOF stretching. The choice E 0 Æ 20 V/cm, of the order of the lowest practical lim- 1 An example of a VMI system with variable total length is described in [26]. Notice, however, that the length change requires a partial disassembly of the vacuum chamber and hence is at least time- consuming. 40 it, yields¢¿ K 0 Æ 144 ns for H + ions with K 0 Æ 1 eV. This should be sufficient for SVMI of distributions with maximum kinetic energy down to a fraction of an electronvolt, thus satisfying the low-KER limit of the design goals (see p. 12). The remaining parameters required for numerical solution of (2.54) are the total length L, the aperture radii and the distance L 0 ¡ L 0 0 between the terminating elec- trode and the ionization region. As mentioned above, the total length has a strong effect on the mass resolution and thus must be sufficiently large. The maximum length that could be achieved with the existing apparatus (see the beginning of sec- tion 2.2) by attaching a standard-size extension nipple without additional vacuum pumping is about LÆ 600 mm, which was chosen for these preliminary simulations. Although the aperture radii have small effect on the focal length (see p. 35), they must be several times larger than the expected beam size in order to keep the aber- rations at a reasonably low level. A detailed account will be given later, and for now it will be assumed that an aperture diameter comparable to the detector diameter is a sensible choice. 1 Thus, for the sake of simplicity, both aperture diameters were chosen equal to the employed detector diameter, namely 40 mm. The L 0 ¡L 0 0 distance has similarly little importance and also does affect the aberrations. However, it is not obvious in advance whether placing the ionization region closer to the flat electrode (for flatter electric field) or farther from it (for more stable field curvature) is better. Therefore, the simulations were performed for several choices of L 0 ¡L 0 0 (fortunately, the utilized simulation procedure allows this without much additional effort). The family of solutions of (2.54) as dependences of L 1 and E 1 /E 0 on L 0 0 is shown in figure 2.9 for various L 0 ¡L 0 0 values. It is immediately evident that L 1 is an almost linear function of L 0 0 , and that any of these lengths indeed uniquely determines the 1 As explained on p. 28 and is evident from figure 2.7, the beam expands after the second lens, hence the beam diameter (for particles that can reach the detector) within the accelerator region should be smaller than the detector diameter. 41 0 20 40 60 80 100 0 10 20 30 40 50 60 70 1 2 3 4 5 6 7 L 1 ,mm E 1 /E 0 L 0 0 ,mm L 0 −L 0 0 =5mm 10mm 15mm 20mm 25mm 30mm Figure 2.9 Interdependence of L 0 0 , L 1 and E 1 /E 0 for simultaneous TOF and radial focusing in the minimal SVMI system (Eppink–Parker arrangement, see figure 2.7). Calculated points are connected by straight line segments for clarity. solution. E 1 /E 0 is also smooth but has a more nonlinear relation to the lengths. The parameters are essentially independent on the L 0 ¡ L 0 0 distance, except for the case when both L 0 ¡L 0 0 and L 0 0 are smaller than the aperture radius, which is the expected result (see p. 36). As a matter of fact, the solution allows a very broad selection of one of the parameters L 0 0 , L 1 or E 1 /E 0 (plus an almost free choice of L 0 È L 0 0 ). This freedom, as already mentioned, could be used for optimization of the impor- tant characteristics of the SVMI system. These characteristics include the total TOF, t 0 , (determining the mass resolution by (2.42), since¢¿ K 0 is fixed by fixing E 0 ), the overall image size at the detector, maxR, and the imperfections in TOF and radial focusing (determining the effective slicing width and the achievable perpendicular velocity resolution, respectively). The focusing imperfections are caused by unavoidable aberrations of ion optics, 42 which lead to mapping of a parallel beam to a finite-size “spot” instead of the ideal “point” (in both arrival time and position). Since the SVMI setup must work well for all kinds of initial velocity distributions, the most relevant description of the focus- ing imperfections is given by the maximum size of these spots produced for all initial velocities within the whole measured range. Additionally, for photodissociation pro- cesses KER is a more interesting quantity than the initial velocity, and hence the relative KER resolution (the ratio of the resolved kinetic energy interval¢K to the maximum measurable kinetic energy maxK) becomes the natural choice for charac- terization of the radial focusing quality. KER is directly related to the initial velocity of the detected particle. For example, for dissociation into two particles, KÆ ³ 1Å m M ´ K 0 Æ ³ 1Å m M ´ m 2 ¢ v 2 0 , (2.55) where m is the mass of the detected fragment, and M is the mass of the cofragment. Since the radial mapping v r 7! R is approximately linear (see (2.47), and v r Æjv 0 j for the central slice), the relative KER resolution can be expressed as ¢K maxK ¼ max mv 2 0 2 6maxK 0 2R(v 0 )¢R(v 0 ) (maxR) 2 , (2.56) where¢R is the radial spot size. The simulation results for the performance characteristics are summarized in figure 2.10. The first noticeable feature of the shown dependences is a somewhat larger influence of the L 0 ¡ L 0 0 distance, although for its values larger than 5 mm the effect is still rather small. The most important fact evident from the figure is that both the TOF resolution¢t and the relative KER resolution¢K have a distinct minimum located between L 0 0 Æ 30 and 40 mm. Perfect focusing is never reached, and 43 2.6 2.8 3.0 0 10 20 30 40 50 60 70 t 0 , μs L 0 0 , mm 0 0.2 0.4 0.6 0.8 0 10 20 30 40 50 60 70 Δt, ns L 0 0 , mm 40 42 44 46 48 50 0 10 20 30 40 50 60 70 maxR, mm L 0 0 , mm 0 0.5 1 1.5 0 10 20 30 40 50 60 70 ΔK, % L 0 0 , mm L 0 −L 0 0 =5 mm 10 mm 15 mm 20 mm 25 mm 30 mm Figure 2.10 Performance characteristics for the minimal SVMI system under si- multaneous TOF and radial focusing conditions. the resolution deteriorates noticeably when the free parameter is changed from its optimal value. The qualitative dependence of the resolution on the initial acceleration length L 0 0 is easily understood by considering the participating types of aberrations. The chro- matic aberrations — dependence of the focal length on the kinetic energy in (2.53) — increase when the relative range of kinetic energies at the first lens plane 1 increases. The absolute range is determined by the initial velocity distribution of the particles, but the average at the lens is governed by the energy gained from the electric field and is approximately equal to qE 0 L 0 0 . Therefore, for smaller L 0 0 values the relative ki- 1 The relative range at the second lens is always smaller and thus has a smaller effect. 44 netic energy spread is larger, and the chromatic aberrations are stronger. On the oth- er hand, the geometric aberrations — dependence of the focal length on the distance from the optical axis — are larger for larger beam radii at the lenses (breakdown of paraxial approximation). As can be seen from figure 2.7, the largest expansion of the beam occurs before the first lens, where the particles did not yet gain a substantial axial velocity but nearly conserved their radial velocities. Therefore, extension of the initial length L 0 0 increases the filling factor 1 of the lenses and thus leads to stronger geometrical aberrations. Although these two types of aberrations have opposite L 0 0 dependences, they are always present and have different origins, preventing their mutual cancellation. The aperture radius also plays an important role in the aberra- tions, mostly in the geometrical ones through the filling factor. This means that the location of the overall aberrations minimum should depend on the aperture radius. However, the simulations performed with aperture radii larger and smaller than the initially chosen 40 mm showed that the value of the minimum does not noticeably change with the radius. Therefore, the radius for which the locations of the minima for¢K and¢t coincide (as in figure 2.10) is a wise choice. As can be seen from the figure, the total TOF is always close to 3 μs and can- not be increased by variation of the parameters, meaning that the mass resolution remains constant and is determined by the total length L. Substitution of t 0 Æ 3 μs and¢¿ K 0 Æ 144 ns into (2.43) gives m max ¼ 10.4 Da, which is obviously sufficient for separation of the imaged H + ions from all other particles, but is also clearly low for the design goal formulated on p. 14. However, this m max value corresponds to the low-field limit used in the simulations and can be improved according to (2.42) by increasing the voltages applied to the ion optics. The maximum voltage used in the E 0 Æ 20 V/cm case is»300 V (also nearly independent on the parameters), thus rais- 1 Ratio of the beam radius to the aperture radius. 45 ing it to the upper practical limit of »5 kV should yield m max ¼ 43 Da for particles with 1 eV maximum initial kinetic energy. This is already acceptable for small organ- ic radicals, as well as for Cl atoms (and HCl), but not for organic molecules containing a Cl atom. Nevertheless, if these heavier molecules have lower initial kinetic energy, 1 the mass resolution for them will be larger. For instance, for maxK 0 Æ 0.1 eV masses up to m max ¼ 135 Da become resolved. The conclusion from these estimations is that the mass resolution with the chosen LÆ 600 mm can be considered acceptable, but any significant reduction of the total length would undermine the TOF-MS perfor- mance. A significant increase of L is also not particularly necessary regarding the mass resolution demands and would be undesirable due to practical considerations. The last and the most problematic dependence illustrated by figure 2.10 is that the overall image radius can be varied by only»20 %, and this variation would lead to deterioration of the kinetic energy resolution by »3 times, as can be seen from comparison of the corresponding plots. From a practical point of view this means that the adjustable parameter must be used to minimize the aberrations, and the ra- dial magnification can be controlled only by the overall electric field strength scaling according to (2.48). This unfavorable fact essentially shows that even the two-aper- ture system does not have sufficient flexibility to solve the problem with correlation between the radial magnification and the TOF stretching (see p. 29). The overall image radius of»46 mm for 1 eV ions means that, according to (2.48), only ions with K 0 . 0.2 eV can be captured by the employed40 mm detector. The TOF stretching¢¿ K 0 ¼ 64 ns for this K 0 is close to the desired value (see p. 14). That is, the design goals for the low-KER limit (see p. 12) are actually satisfied by the considered two-aperture system with a total length close to 600 mm. The situation 1 For example, all parent molecules in the molecular beam have essentially zero kinetic energy (rela- tive to the beam). Heavy fragments of dissociation processes should also have relatively small kinetic energy, as can be seen from inversion of (2.55). 46 for higher kinetic energies is worse. For example, even for K 0 Æ 1 eV the electric field strength must be increased from E 0 Æ 20 V/cm to»100 V/cm in order to fit the whole image into the detector, and at this E 0 value the TOF stretching drops, according to (2.35), to only¢¿ K 0 ¼ 27 ns, which is already insufficient for good slicing. For distri- butions with higher kinetic energies slicing becomes completely impossible, as the TOF stretching approaches the inherent time resolution of the experiment. This in- dicates that some additional means for the radial magnification control independent on the initial acceleration field strength E 0 are required. 2.4.4 Additional lenses Obviously, the radial image magnification can be controlled by adding more lenses to the ion-optical system. To be specific, only an image size reduction is required from the magnification control, since the original magnification provided by the system described in the previous subsection is large enough for the set experimental goals. The freedom in selection of the optical scheme for the “zoom” control is some- what limited by the general properties of electrostatic lenses. As can be seen from figure 2.8, the focal length is never shorter than a few aperture diameters. Moreover, the aberrations for lenses operating with such short focal lengths are significant. This means that construction of a compact “zoom lens” from many short-focus lens- es, as in regular optics, is impossible, at least if high resolution is required. In other words, magnification control by addition of as few as possible relatively weak lenses is desirable. A single negative lens added after the accelerator would not make any improvement. First, the output lens of the accelerator is already negative. Second, a negative lens increases the divergence of the beam and thus should lead to a higher magnification instead of a lower. 47 On the contrary, a single positive lens added after the accelerator can perform the desired magnification control, as illustrated in figure 2.11. The figure shows the L (z=L a ) F R L−l−F R l f l f l R R 0 detector source Figure 2.11 Optical scheme for a positive lens added in the drift region, showing particle trajectories with (thick lines) are without (thin lines) the lens. most pictorial case, where F R is positive and short, so that a real image is formed before the additional lens. This allows a trivial derivation of the zoomed image size R 0 directly from similar triangles in the drawing 1 : R 0 Æ¡ F R L R¢ l L¡ l¡ F R , (2.57) where the minus sign accounts for the reversed image orientation. Thus the relative magnification with the additional lens is M rel ´ R 0 R Æ¡ F R l L(L¡ l¡ F R ) . (2.58) Although (2.58) was obtained from consideration of one particular case, it remains valid even when no intermediate image is formed (that is, even when F R is negative 1 The treatment given here implicitly assumes that the magnification constant in (2.47) for the ac- celerator does not change when its focal length is varied, and that the virtual source formed by the accelerator is not displaced from the actual source (that is, the dotted continuations of the trajectories in the figure indeed come to the original positions of the particles). While these assumptions are not true in general (for example, if the straight parts of trajectories in figure 2.7 were continued to the left, the crossings would occur at zÇ 0 instead of z 0 Æ 0), they give sufficiently good approximations for the qualitative description, especially in the cases important for the present work. 48 or exceeds L), which can be checked by simple but somewhat lengthy manipulations with ray optics equations. One of these equations, 1 f l Æ 1 l Å 1 L¡ l¡ F R , (2.59) gives the general expression for the required focal length f l Æ l µ 1¡ l L¡ F R ¶ (2.60) of the additional lens. Now the parameters for the additional lens, namely at what distance l from the detector it should be placed and what focal length f l it should have, required to achieve a relative magnification M rel Ç 1 can be studied. Since it was decided to keep the total system length L fixed, both M rel and f l depend on two free parameters: the accelerator focal length F R and the lens position l. These relationships are illustrat- ed in figure 2.12 as dependences on l for a few F R values and on the optical power 1/F R for a few l values (both l and 1/F R , as well as f l , are given in relation to L, since this produces more convenient dimensionless plots). Reiterating the point that signif- icant geometrical changes in a high-vacuum system should be avoided, the principal question in the analysis of (2.58) is the choice of the optimal single position l that would allow the largest range of magnification variations achievable by adjusting F R in a reasonable range. According to figure 2.8 and the related discussion, F R can be varied from the original value L in (2.54) to infinity and further to negative values »(¡L) by relatively small variations of the electric field strengths ratio E 1 /E 0 . Values jF R jÇ L are also possible, but require larger changes in the accelerator parameters and lead to stronger aberrations (see p. 47). In other words, the optical power 1/F R 49 −1 0 1 0 0.25 0.5 0.75 1 M rel l/L 0 1 2 0 0.25 0.5 0.75 1 f l /L l/L −1 0 1 −2 −1 0 1 2 3 M rel L/F R 0 1 2 −2 −1 0 1 2 3 f l /L L/F R F R =0.5L 1.0L 2.0L ∞ −2.0L −1.0L −0.5L F R =0.5L 2.0L ∞ −2.0L −1.0L −0.5L l=0.75L 0.50L 0.25L l=0.75L 0.50L 0.25L Figure 2.12 Relative magnification with additional lens and required focal lengths. can be varied in a finite rangej1/F R j. 1/L without adverse effects, which explains why L/F R was chosen instead of F R for the plots in figure 2.12 (additionally, 1/F R is a continuous function of E 1 /E 0 and goes smoothly through zero where F R has an infinite discontinuity between the negative and positive branches). The first remarkable property of (2.58) is that for 0Ç F R Ç L (corresponding to the situation depicted in figure 2.11 and illustrated in figure 2.12 by the F R Æ 0.5L curves) the relative magnification is always negative and can be varied in the whole range from ¡0 (at l! 0) to ¡1 (at l! F R , or F R ! l if l is fixed). This would be 50 the general solution for the independent magnification control, but unfortunately it requires the shortest focal lengths F R and f l compared to all other cases, which means that this solution leads to the strongest aberrations and in practice has strong limitations, since the focal lengths cannot be made very short. Before discussing the other solutions, it should be mentioned that this case with intermediate focusing is the only one allowing an increase of the magnification and was successfully used for that purpose in experiments with low K 0 /(qV 0 ) ratios (either due to low produced K 0 or high required V 0 ), examples of which can be found in [22, 27, 28]. The need for increased magnification means that the original beam size was quite small, and therefore the mentioned problems with aberrations and relatively short focal lengths were not severe in these systems. 1 The two remaining cases, F R Ç 0 and F R È L, correspond to optical powers 1/F R Ç 1/L and can be considered together. They have the property that the relative magni- fication always remains positive and can be varied from unity, when the additional lens is inactive (f l Æ1), and all focusing is due to the accelerator (F R Æ L), towards zero, when the accelerator produces a strongly diverging beam (F R !¡0), and the image is produced by the additional lens with a finite focal length (f l ! l[1¡ l/L]). This lower limit is, of course, not reachable in practice, but even for a relatively mild variation of the accelerator optical power ¡ 1 L 6 1 F R 6 1 L (2.61) 1 The small beam diameter allows to make the aperture diameters large compared to it (thus main- taining low filling factors) but at the same time small compared to the required focal lengths (thus keeping the lenses not very strong). 51 the relative magnification range is l 2L¡ l 6 M rel 6 1 (2.62) with the required additional lens focal length range l µ 1¡ l 2L ¶ 6 f l 6Å1. (2.63) According to (2.62), the magnification range is broader when l is smaller, that is, the additional lens is placed closer to the detector. However, according to (2.63), in that case the required min f l ¼ l also becomes shorter, which places a practical limit on the range. Nevertheless, even for lÆ 0.25L (see figure 2.12) a substantial magnification range of 1 : 7 could be reached, with a seemingly reasonable minimal f l ¼ 0.22L. The word “seemingly” in the last phrase refers to the fact that although f l ¼ 0.22L is not too small compared to L and is large compared to the detector diameter (40 mm), which was used in subsection 2.4.3 for justification of the accelerator aper- ture diameters, the beam diameter (for the detectable particles) at the additional lens is actually 1 M rel L¡l L times larger than the detector diameter. For M rel Æ 1/7 and LÆ 600 mm this means 210 mm beam diameter for the lens with »132 mm focal length. That is, the additional lens must have an overall size comparable to the total length of the SVMI system and will have very large geometrical aberrations. In oth- er words, the practical magnification range expected for a system with such detector size is more modest. At the same time, the radial magnification by itself is not the most important performance characteristic of an SVMI system, since fitting a higher velocity (or ki- netic energy) range into the detector radius means that the total TOF spread of the 52 detectable particles also increases according to (2.35), and hence the slice made by a fixed-duration detection pulse becomes relatively thinner. In order to keep the rel- ative slicing thickness at the desired level 1 the overall electric field strength can be increased, which in turn leads to even higher kinetic energy range fitting into the detector radius (see (2.15)). This interdependence can be quantified using scaling re- lationships (2.35) and (2.48) and demanding that the TOF stretching and the overall image radius remain unchanged when M rel is applied: 8 > > > > > < > > > > > : q K 0 0 E 0 Æ p K 0 E , M rel s K 0 0 E 0 Æ s K 0 E , (2.64) where K 0 0 and E 0 correspond to SVMI with the additional lens, and K 0 and E — without it. Simple algebraic transformations lead to K 0 0 K 0 Æ 1 M 4 rel , (2.65) E 0 E Æ s K 0 0 K 0 . (2.66) The fourth power in (2.65) shows that the possible KER range variation is actually significantly larger than the range of radial magnifications created by the additional lens. For example, even a lens with M rel Æ 1/2 added to the SVMI system studied in subsection 2.4.3 should allow slicing for distributions with maxK 0 from 0.2 eV to 3.2 eV, thus satisfying the main goal for the designed system (see p. 12). Moreover, only a fourfold variation of the electric field strength from E 0 Æ 20 V/cm (and V 0 » 1 Too thin slices do not lead to an improvement of the overall resolution, since other limiting factors (aberrations) start to prevail over the finite-slice blurring (2.7), but do proportionally decrease the signal level according to (2.5). Therefore, the optimal relative slice thickness must have some finite value. 53 300 V) to E 0 Æ 80 V/cm (and V 0 » 1200 V) would be required. At M rel » 1/2 the maximum beam diameter at the lens should not exceed»80 mm, which is quite reasonable for the considered system dimensions. Nevertheless, since the aperture diameter must exceed the maximum beam diameter (and, if possible, be a few times larger, to avoid excessive aberrations), and the overall diameter (with the brim) of an aperture lens must be additionally at least 3¡5 times larger than the opening itself (to reduce the outer fringe effects), the required aperture lens would still remain rather bulky. Recalling that the designed ion-optical system was intend- ed for retrofitting of an existing setup (see the beginning of section 2.2), which has the vacuum chamber diameter of only»150 mm, the task appears challenging, if not impossible. Fortunately, apertures are not the only means of creating inhomogeneous elec- tric fields suitable for charged-particle beam lenses. The other simple type of lenses, called “cylindrical lenses”, is created by electrodes assuming a shape of cylindrical surfaces with generatrices parallel to the optical axis (instead of planes perpendic- ular to it). 1 In contrast to aperture lenses, which separate two regions of different electric fields, cylindrical lenses separate two field-free regions with different elec- trostatic potentials, as illustrated in figure 2.13. The electric field formed between the electrodes must satisfy the same equation (2.50), and thus the arguments about its paraxial properties remain the same. The most useful feature of cylindrical lens- es in the context of this subsection is that if the gap between the electrodes is much smaller that the cylinder radius, then the penetration of external fields inside the lens is negligible. Therefore, the overall diameter of the lens might be only slightly 1 Although the term “cylindrical lenses” is also used in regular (photon) optics and is similarly due to the use of cylindrical surfaces, the optical lenses have the cylinder generatrices perpendicular to the optical axis and thus perform only 1-dimensional focusing. However, the electrostatic cylindrical lenses are symmetric around the optical axis and focus equally in 2 dimensions. 54 V z<z lens E E=0 E=0 V z>z lens Figure 2.13 Cross-section of cylindrical electrostatic lens with equipotential con- tours shown by gray curves and particle trajectories by black ones. larger than its opening diameter. In other words, a cylindrical lens that fits into a 150 mm chamber might be expected to work with80 mm beams without enor- mous aberrations. Unlike aperture lenses, cylindrical lenses are optically thick, meaning that their front and rear principal planes and focal lengths do not coincide. No simple expres- sions for the focal lengths exist, but in the case of small gap and weak focusing both focal lengths behave as f » K ¡ K¡ q £ V zÈz lens ¡V zÇz lens ¤¢ ¡ q £ V zÈz lens ¡V zÇz lens ¤¢ 2 D, (2.67) where D is the cylinder diameter, and the proportionality factor is about 5 (see [29] for the expansions with more terms and actual numerical coefficients). As can be seen from (2.67) (and remains true in the general case), the lens is always converg- ing regardless of the applied potentials, as long as the particles have enough kinetic energy to pass through it. This allows a very useful combination of two sequential two-electrode lenses, which can be considered as a single three-electrode lens, with equal entrance and exit potentials and a different potential applied to the central electrode. Such lenses provide beam focusing without overall altering of the parti- 55 cle energies and are sometimes called “einzel” lenses. 1 They can be used in both modes — with the central potential either lower or higher that the outer potential. The particles are then temporarily accelerated or decelerated (depending on their electric charge) upon entering the lens. The intermediate acceleration mode is usual- ly preferred, since it leads to lower chromatic and geometrical aberrations than the intermediate deceleration mode. In practical applications the length of the central cylinder is usually made equal to its diameter D, and the gaps »D/10 between the cylinders are used. The outer electrodes of lengths D are also sufficient to contain the electric field inside the lens, so that the lens properties do not depend on its surroundings. Therefore, the overall length of the additional cylindrical lens can be slightly smaller than 3 vacuum cham- ber diameters (that is, »400 mm), occupying almost the whole “drift” region of the designed SVMI system but fitting into it without obstacles. The above discussion mentioned explicitly that the radial focusing focal length of the accelerator must be changed when the magnification control with the additional lens is utilized. However, since the particles are temporarily accelerated inside the additional lens, their total TOF is also affected, and the TOF focusing conditions for the accelerator must change accordingly as well. Therefore, the last two equations of (2.54) have to be replaced with expressions depending on the parameters of the additional lens. The new equations for the actual system are formulated and solved in the next section, but it is appropriate to make here a general remark about their properties. According to the discussion on p. 51, the required F R variation is relative- 1 The word “einzel” is a rather illiterate English adoption from the German term “die Einzellinse” meaning “the single lens” (the noun prefix “Einzel-” corresponds to the adjective “einzeln” — single, solitary, individual, discrete, separate and so on), which referred to the fact that this lens is self- contained, and its addition to a system does not change the electric fields and potentials outside the lens. A more meaningful English term used for these lenses is “unipotential”, alluding to only one adjustable potential (of the central electrode). 56 ly small. The central length of the additional lens, where the particles fly faster than they would without the lens, is also relatively small compared to the total length of the system, and thus the required F t correction should be relatively small as well. As can be seen from figures 2.6 and 2.8, relatively small changes in the accelerator pa- rameters should be sufficient to accommodate these variations. Therefore, although the solution for a system with the additional lens must be different from what is shown in figure 2.9, these differences are not outstanding. And a final remark before going to the description of the actual setup. Even though the idea of using a separate lens for independent magnification control is not completely new (see p. 51), all applications reported in the literature have been designed only for the 2-dimensional VMI without time resolution. Additionally, in on- ly one case [30] the problem was in reducing the magnification for distributions with high kinetic energies, as in the present work. The optical scheme chosen there was very similar to the one described in this section, 1 however, the authors did not provide its justification (literally: “we had no prior knowledge of the optimal configuration”) and obtained the final geometry solely by numerical optimizations. Some attempts of magnification control in SVMI were made in [11], but since the authors tried to achieve it by adding more lenses to the accelerator (contrary to what is said on p. 52), the resulting range was not much larger than the range shown in figure 2.10, despite the fact that no attention was paid to TOF focusing and aberrations (that is, the free parameter space was significantly larger). 1 This is not surprising, since similar problems often lead to similar solutions. (In reality, the devel- opment was largely independent: although [30] was published before the present work began, it was devoted to a rather different task, and I discovered it only after the general ideas discussed above were already formulated.) 57 2.5 Simulations and final design The actual design is heavily based on the ideas and results discussed in the pre- vious section. It was deliberately kept as simple as possible because of two factors. First, I had no previous experience in design or construction of ion optics. Second, the used numerical simulations do not predict the behavior of a real system exactly. Therefore, it was desirable to start with a minimal system, which could be built and experimentally characterized without additional complications. It should serve as a proof of concept and a test case for future developments, but at the same time must be usable as an instrument. Thus this section is focused on the issues that were abso- lutely necessary or at least very useful for the SVMI operation according to the goals stated in section 2.2. Some possible future improvements of the present system are discussed in section 2.9. Although SIMION can perform all calculations in a truly 3-dimensional geome- try, the required amount of computational resources is much larger than in the case of cylindrically-symmetric configuration of the electrodes. Taking into account that the real setup was designed to have deviations from cylindrical symmetry only in the outer (in the radial direction) parts, and these deviations are relatively small, the electric field in the important regions should be very well described by the solution obtained for the cylindrically symmetric configuration from which all nonsymmetric features were removed. Therefore, in all simulations reported below the relatively small dielectric holders and connecting wires were removed completely. 1 Nonsym- metric parts of metallic holders were also removed, but the large cylindrical parts were left intact. 1 In any case, the calculations with dielectrics became available in SIMION only since version 8.1, which appeared after the present system was already built. 58 2.5.1 Accelerator The minimal accelerator design requires two electric field regions, as discussed in section 2.4. That is, it should include one flat, terminating, electrode with a small opening for entrance of the molecular beam and two flat electrodes with larger aper- tures serving as electrostatic lenses. The preliminary results (see figure 2.9) indicate that the required separations L 0 and L 1 between the electrodes should be »50 mm each. It means that the overall diameter of the electrodes must be of the order of 240 mm (40 mm aperture, and at least 2£ 50 mm brim width required to avoid field distortions due to fringe ef- fects 1 and asymmetric environment), which creates the first design problem, since the internal vacuum chamber diameter is only»150 mm, as mentioned earlier. 2 The second problem arises from the fact that, according to the actual simulations (see subsection 2.5.5 below), the optimal values for both L 0 and L 1 depend on the magnification level. In other words, operation at different magnifications requires variation of the distances between the electrodes. Although the variation should be relatively small (from 30 to 80 mm), a precise mechanical movement of relatively thin plates inside a confined high-vacuum chamber and with high voltages applied is not a simple task. Both of these problems can be solved simultaneously by a simple modification of the electrode configuration. Namely, the fringe effects can be reduced not only by making the brims wider than the gaps, but also by making the electrode-free gaps smaller. Insertion of intermediate electrodes with applied potentials equal to the po- tentials that should have existed in the original field configuration does not change 1 No outer fringe effects are visible in figure 2.7 because the electrodes of the shown system actually extend vertically far beyond the depicted portion. 2 And the aperture lenses cannot be replaced by cylindrical lenses, as discussed in subsection 2.4.4, because the ionization region must have a nonzero and relatively homogeneous electric field. 59 the field inside the apertures, but allows to make the brims much narrower. For ex- ample, electrodes with 10 mm spacing do not require brims wider than 20 mm and thus can have the overall diameter of 80 mm, which should fit well with all addition- al holders into the given chamber. This configuration is shown in figure 2.14. 1 Since V 0 E=0 V 1 V=0 E 0 E 1 L 1 L 0 ΔL E 0,1 E 1,ff Figure 2.14 Cross-section of accelerator. Electrodes and metallic holders are shown in black, nonsymmetric parts are schematically given in gray (insulators — by dotted lines). Red lines show an example of equipotential contours. The top part includes an electric circuit diagram of the resistive voltage divider. the original fields E i (iÆ 0,1) between the corresponding adjacent electrodes were homogeneous, the original potential was a linear function of the z coordinate within 1 Please not that from this point onwards V 0 refers to the potential of the accelerator terminating electrode, instead of the potential at the initial position of the particles, as it was in the previous sections. 60 each gap. Therefore, the original potential differences must be split among the inter- mediate electrodes proportionally to the spacings. For equally spaced electrodes this splitting is trivially achieved by a resistive divider with identical resistors, as shown in the figure. In principle, if the intermediate apertures have the same diameter as the lens apertures, then the resulting electric field should be slightly different from the orig- inal one, 1 but this difference is rather small, at least in the most important region near the axis, and does not significantly affect the focusing properties of the lenses. At the same time, if all electrodes have the same shape, than the lengths L i can be chosen arbitrarily by purely electrical commutation of the applied potentials, with- out any mechanical movements inside the vacuum system. It is obvious that any distance divisible by the interelectrode step¢LÆ 10 mm can be created by this ap- proach. However, if¢L is small compared to L i , then arbitrary effective distances, not limited to multiples of¢L, can be created by applying potentials such that the electric field strengths in the “intermediate” regions (see figure 2.14) are E 0,1 Æ {L 0 /¢L}E 0 Å(1¡{L 0 /¢L})E 1 , E 1,ff Æ {(L 0 Å L 1 )/¢L}E 1 , (2.68) where { ² } denotes the fractional part of the number (06 { ² }Ç 1), E 0,1 corresponds to the transition between E 0 and E 1 , and E 1,ff — between E 1 and the field-free region. The values given by (2.68) are equal to the average field strengths in the respective regions calculated for the field configuration with real electrodes present at the po- sitions corresponding to the actual “aliquant” L 0 and L 1 . The required potentials at all electrodes can be generated by the same resistive network, but with the “inter- 1 As can be seen from figure 2.7, the equipotential contours between the electrodes do not become straight immediately at rÆ r aperture . 61 mediate” resistors replaced by potentiometers of equal total resistance, as shown in figure 2.14. Numerical simulations show that this approach indeed allows continu- ous variation of both accelerator lengths, and the resulting performance characteris- tic have smooth dependences on the effective lengths, as illustrated by an example shown in figure 2.15. 1 40 45 50 55 60 0.7 0.8 0.9 ΔK,% 40 45 50 55 60 0.0 1.0 2.0 3.0 Δt,% L 0 ,mm L 1 , mm ΔK,% L 0 ,mm L 1 , mm Δt,% Figure 2.15 Example of smooth variation of KE and TOF resolutions for effective accelerator lengths changing in 1 mm steps. For the final design the “round” step¢LÆ 10 mm was chosen, so that the lengths L i expressed in centimeters correspond to the numbers of interelectrode gaps, mak- ing the electric commutation more convenient for the user. All electrode plates were made 1 mm thick and with 80 mm external diameter (see above) for 40 mm internal diameter of the apertures (see p. 45). The small (3 mm) orifice in the terminating plate was used in the simulations, but its presence apparently does not affect the electric field at the ionization region that was placed at 15 mm from the terminating electrode 2 , in the middle between the first two apertures. According to figure 2.9, an 1 The absolute values of¢K and¢t from the plots shown in figure 2.10 should not be compared to the data presented here and below because the preliminary simulations described in section 2.4 were performed with a smaller initial volume and thus showed smaller blurring. 2 It was important to keep this distance small in order to make the accelerator more compact, but the 62 overall length L 0 Å L 1 Æ 110 mm would be sufficient for operation without the addi- tional lens, but for greater flexibility the length 130 mm was chosen, resulting in 14 electrode plates total. This number might appear excessive (especially for a configu- ration with only two field regions), but, for example, the system described in [10] used 29 plates for the same purpose. The only significant difference is that the design in [10] was not intended for lengths variations, and thus all voltage-dividing resistors could be placed inside the vacuum chamber, requiring only two high-voltage electri- cal feedthroughs for the input voltages, 1 whereas the present system requires a facile commutation of the plates and thus demands either a remote-controlled commutator inside the chamber or an external commutator with separate electrical connections to each plate. The latter approach was chosen as more practical for the given number of plates (see subsection 2.6.1 for details). 2.5.2 Lens The most “usual” configuration was chosen for the additional lens of unipotential type: all three cylindrical electrodes have equal diameters, all lengths equal to the diameter, and both gap widths equal to 1/10 of the diameter, as show in figure 2.16. The outer electrodes are kept at the vacuum chamber potential (that is, grounded), and the optical power of the lens is controlled by the potential V L applied to the cen- tral electrode. Since the electric field configuration inside the lens is defined mostly by the edges of the electrodes, the middle part of each electrode was made out of less rigid but much lighter sheet metal inserted into strong and well-shaped holders that form the electrode edges. To be on the safe side, this construction was preserved in field at 5 mm (between the terminating plate and the first aperture) might be noticeably distorted by the3 mm orifice. At the same time, according to figure 2.10, the distance L 0 ¡ L 0 0 Æ 15 mm is already sufficient to suppress the effect of the terminating plate on the system characteristics. 1 Maybe even one input voltage was divided — it is not clear from the published description. 63 V L D L D L D L D L D L /10 D L /10 Figure 2.16 Cross-section of additional lens. Blue lines show an example of equipo- tential contours for intermediate acceleration mode. the simulations (see the figure). Since the maximum voltage applied to the lens can be estimated at a few kilo- volts, the gap between the outer surface of the electrode and the grounded vacuum chamber wall must be at least a few millimeters for stable and safe operation. The electrode “wall thickness” must be also of the order of a few millimeters in order to have sufficient stiffness for maintaining the geometry of the lens. These factors de- termine the maximum internal diameter of the electrodes. For the present system with internal vacuum chamber diameter»150 mm the value of D L Æ 120 mm, close to the practical limit, was chosen. This led to the total length of the lens equal to 384 mm. The lens was placed at the detector end of the chamber, in accordance with the conclusion made on p. 52, leaving a small gap (20 mm) between the last electrode edge and the detector flange for better vacuum pumping of the space between the electrode and the chamber wall. 64 2.5.3 Focusing criteria Before proceeding to the actual simulations, it is important to define explicitly and formalize the characteristics that should be optimized. Some of these quantities, such as the central TOF and the maximum radius for a given kinetic energy, have relatively obvious definitions, and their values do not depend significantly on the particular method of estimation, as long as the ion-optical system accomplishes TOF and radial focusing. At the same time, the criteria determining whether the focusing is accomplished and how good it is are less obvious. The finite longitudinal size of the ionization region 1 and the existence of aberrations, which essentially are the causes of imperfect focusing, mean that at any given ion optics parameters the focusing for different parts of the initial momentum and position distribution is different. Thus the answer to the question, how good is the focusing, should depend substantially not only on the definition of “focusing”, but also on the distribution being mapped. As it was already mentioned on p. 42, the most reasonable requirement from the SVMI system is to work well for all possible distributions, since the system must have sufficient versatility in order to be usable in many experiments, even not yet planned. Therefore, the main goal chosen for the simulations in this section was optimization of the worst case, which means that the performance for all particular cases is guaranteed 2 to reach at least these “general” results, but might be somewhat better in favorable situations. The initial distribution has two independent factors: position distribution and ve- 1 Since the ionization region is located in a nonzero axial electric field, different initial positions of particles with equal initial velocities nevertheless lead to different kinetic energies of such particles in the system. 2 Of course, “guaranteed” only to the extent of the simulation accuracy. However, optimization for par- ticular cases has exactly the same problems with reliability of the simulations. Since the systematic errors are expected to be relatively small and similar for all interesting particular cases, the opti- mization of the worst case is still a sensible approach. 65 locity distribution. The uncertainty in the initial position, as mentioned on p. 13, is determined by the size of the ionization region and by the uncertainty in its posi- tion relative to the ion optics. In order to perform the worst-case optimization, the maximum possible uncertainty must be assumed in the calculations. As can be seen from figure 2.17, the¢x uncertainty comes from the molecular beam diameter. 1 An laserbeam molecular beam x y z 1mm 2mm 1mm molecular beam laserbeam Δz Δx Figure 2.17 Diagram illustrating initial position uncertainties and the initial vol- ume used in focusing calculations (coordinate system corresponds to figure 2.1). effective value¢xÆ 2 mm was used for it in the present calculations. The uncertainties ¢y and ¢z come from the laser beam diameter and the un- certainty in the beam alignment relative to the system. The laser beam diameter (for focused beams) is actually much smaller than the alignment uncertainty, and thus the values of¢y and¢z in each particular experiment practically do not exceed a small fraction of a millimeter. However, the alignment relative to the SVMI sys- tem and its stability and repeatability between experiments (for example, between the calibration of the mapping and the actual measurements) cannot be guaranteed with such precision. Therefore, the simulations are required to find the conditions that would be robust with respect to these possible variations in the alignment. This approach also guarantees that the possible deviations between the calibration and 1 Even for a focused laser beam its depth of focus is not short enough to be a limiting factor. Also, for one-photon processes the focusing has no effect on¢x, since the number of photoreaction events is determined by the integral flux, which remains constant along the beam. 66 the measurements performed with different alignments will not exceed the simulat- ed scatter. With this in mind, the values of¢y and¢z were set to the practical laser beam alignment precision estimated at 1 mm, leading to the initial volume depicted in figure 2.17. The origin of the coordinate system used in the simulations was set to the center of this volume (this obviously does not affect the results, but is a useful convention). In order to optimize the worst-case resolution, the sample of velocity distribution used for scatter calculations must be representative for all possible cases. Since the mapping is continuous and nearly linear (see (2.34) and (2.47)), the sampling does not need to be dense, but it must include the limiting cases, as well as at least a few intermediate points to account for nonlinear distortions. 1 Although there are many different methods of systematic sampling in spherical coordinates 2 , a regular grid was used in this work as the simplest one. It should be noticed that the shape of the trajectories can depend on the relative orientation of the initial velocity and position vectors, but, due to cylindrical symmetry of the electric field, it does not depend on their azimuthal angles (µ in figure 2.1) with respect to the system. Moreover, test simulations indicated that initial displacements from the z axis in the direction per- pendicular to it and to the initial velocity vector (that is, ±rÒ e z £ v 0 , where e z is the axial basis vector) have a very minor effect on the resolution in comparison to all other factors. This means that sampling of velocities in only one plane (xz in this 1 The following discussion is devoted to minimization of the sample size. A larger sample should, obviously, give more reliable and smooth estimations than a smaller sample, but the simulation time depends linearly on the number of particles in the sample. The time is not negligible even for a single run with a few tens of particles and becomes appreciable when multiple runs are performed in an optimization process. Therefore, selection of a small but representative sample is desired for speeding up the optimizations. In principle, more accurate performance estimations for each particular case can be obtained afterwards with a larger and more relevant sample. 2 The use of spherical coordinates here is dictated only by convenience: they provide separation of the kinetic energy (K 0 Æ mjv 0 j 2 /2) from the angular part (direction of v 0 ). The distribution is not required to be spherically symmetric, but its description in spherical coordinates facilitates the consideration in terms of kinetic energy. 67 case, since¢xÈ¢y) is sufficient for estimation of the maximum scatter. Additionally, due to the reflection symmetry of the system (ion optics and initial volume), veloci- ty samples with equal z-components and opposite x-components (that is, (v x ,v z ) and (¡v x ,v z )) should yield exactly the same results. Therefore, the variation of v x can be limited to v x > 0 without losing generality. Based on these considerations, the following velocity distribution sample was con- structed (see figure 2.18): 3 different values of kinetic energy K 0 Æ K max , 0.4K max , v x v z α 0 Δα 1mm x z 2mm 0.1K max 0.4K max K max Figure 2.18 Velocity and position sampling used in focusing calculations. 0.1K max define the speeds (magnitudes of velocities), spanning the range from the maximum speed to the zero 1 with approximately uniform intervals; for each speed value 5 velocity directions ®Æ® 0 Å n¢® (nÆ 0...4) relative to the z axis are used, starting from® 0 Æ 10° with¢®Æ 40° steps. 2 The final representative sample for position and velocity distribution was ob- 1 The zero velocity is not included by itself, since it would convey no information on the kinetic energy resolution (dK 0 Æ v 0 dv 0 Æ 0 when v 0 Æ 0). At the same time, the family of lowest speed particles (with K 0 Æ 0.1K max ) provides enough information for the lower part of the distribution in the important cases of not very large aberrations and distortions. 2 These values were selected in order to include ®Æ 90° (which should sample the central slice and reproduce maxR for each K 0 ) and perform symmetric sampling relative to it. Angles®Æ 0° and 180° were not included for the same reason as K 0 Æ 0 (they produce RÆ 0 for any K 0 ). 68 tained as direct product of this velocity distribution sample and a representative position sample. As mentioned above (p. 67), displacements along the y axis have minor effect on the resulting arrival positions and times and therefore do not need to be sampled, which means that taking initial positions from the rectangle¢x£¢z at y 0 Æ 0 (projection of the initial volume shown in figure 2.17) is sufficient. Moreover, the test simulations also demonstrated that the geometric shape of a group of parti- cles with a common initial velocity is distorted as they travel through the system 1 , but its convexity is approximately preserved. In other words, particles originated from internal points of the initial volume arrive at the detector in between (both in space and time) the “boundary” particles and thus make no contribution to the over- all spot size. Therefore, the space distribution sample consisting of only the 4 corner points of the rectangle is representative for the purpose of the worst-case optimiza- tion and is also the minimal representative sample. The final sample thus consists of 3£ 5£ 4Æ 60 particles. An example of the initial parts of their trajectories in the system is shown in figure 2.18. Now, after the initial conditions were defined, the focusing criteria can be dis- cussed. According to the idea of VMI (see p. 4), each group of “equivalent” particles, that is all particles with equal initial velocities, must arrive at the detector at a single point (position of which depends on the initial velocity) regardless of initial positions of the particles within the group. In practice this ideal situation is impossible, and each group forms a spot of a finite size instead of a single point. Therefore, it is nat- ural to define the quality of focusing through the spot size — the smaller is the spot, the better is the focusing. For example, for each group (parallel monochromatic ion beam, in terms of ion optics) the focal distance F R of the optical system is the distance 1 In the case of ideal focusing the whole initial volume should shrink to a single point when the group reaches the detector position. 69 at which the spot reaches its minimum size: F R (v 0 )´ argmin z ¡ ¢r v 0 ¯ ¯ z ¢ , (2.69) where ¢r v 0 ¯ ¯ z ´ max ±r ¡ r(±r,v 0 )j z ¢ ¡min ±r ¡ r(±r,v 0 )j z ¢ (2.70) is the scatter of radial distances r(±r,v 0 ) corresponding to particles with initial posi- tions±r and velocity v 0 , measured at distance z. This focal distance, however, depends on the initial velocity of the particles, as illustrated in figure 2.19, which constitutes the notion of chromatic aberrations in ion z=argmin z ¡ max v 0 r v 0 Δr v 0 ¢ r z Figure 2.19 Illustration of focusing with aberrations (vertical scale expanded and aberrations exaggerated for clarity). Positions of minimum blur spots for each initial velocity are indicated by vertical bars. optics. The worst-case optimization for the whole velocity distribution must therefore minimize the maximum spot size for all possible initial velocities to find the optimal 70 focal distance: F v ´ argmin z ¡ max v 0 ¢r v 0 ¯ ¯ z ¢ . (2.71) The subscript v of F v signifies that this optimum is for the overall velocity resolution. Namely, according to (2.47), the mapping from the velocity to the radius is nearly lin- ear, thus the scatter in the arrival positions is transferred directly to the scatter of the measured velocities by the inverse, also nearly linear, mapping. If the final goal of the measurements is to obtain the KER distribution, as is usually the case, the optimality criterion must minimize the kinetic energy scatter instead of the veloci- ty scatter. The difference between these two approaches comes from nonlinearity of the relationship K 0 Æ mv 2 0 /2, which means that dK 0 Æ mv 0 dv 0 , and the relative KE resolution ¢K 0 K max Æ 2v 0 ¢v 0 (v max ) 2 (2.72) is not directly related to the relative velocity resolution ¢v 0 v max . The focal distance opti- mal for kinetic energy measurements is therefore F K ´ argmin z ³ max v 0 ¡ r v 0 ¢r v 0 ¯ ¯ z ¢ ´ . (2.73) It is evident that for KE focusing the spot sizes corresponding to faster particles in the distribution are more important than those of slower particles (see also footnote on p. 68). Due to an almost monotonic character of chromatic aberrations (see fig- ure 2.19) the maxima in (2.71) and (2.73) are reached at v 0 Æ v max , 1 and thus the overall relative KE resolution at F v is max¢K 0 /K max ¼ 2max¢v 0 /v max — approx- imately two times worse than the overall relative velocity resolution. However, the test simulations showed that the value of max¢K 0 /K max at F K is actually compara- 1 Because of the minimization in these expressions the maxima must also be reached at some lower v 0 values, but this is not important here. 71 ble to the value of max¢v 0 /v max at F v . That is, even though the difference between the focal distances F K and F v in practically important cases is relatively small, the correct choice of the focusing criterion appropriate for a particular type of measure- ments makes a twofold difference in the resulting resolution. The focal distance defined by (2.73) or (2.71) is useful for general considerations (see, for example, figure 2.8), but not so convenient for optimization of ion optics parameters: the operation of an SVMI system requires fulfillment of the condition F K Æ L (see (2.54)) in any case, thus instead of adjusting ion optics parameters for simultaneous satisfaction of this equation and optimization of resolution, it is more rational to optimize the resolution by direct minimization of the maximum spot size at the detector. With this approach the overall relative kinetic energy resolution is calculated as ¢K K max ´ max v i [hR(v i )i¢R(v i )] (R max ) 2 , (2.74) where the average arrival position R for particles with initial velocity v i is hR(v i )i´ 1 N pos X ±r j R(±r j ,v i ), (2.75) and the corresponding scatter is ¢R(v i )´ max ±r j R(±r j ,v i )¡min ±r j R(±r j ,v i ). (2.76) The maxima, minima and sums are taken over the respective velocity or position samples described above (see p. 68 and figure 2.18; N pos Æ 4 is the number of initial positions in the sample). The maximum radius R max ´ max v i hR(v i )i corresponds to the maximum kinetic energy K max and usually should be equal to the detector radius. The overall TOF characteristics can be defined similarly, but with a small dif- 72 ference stipulated by the fact that the present SVMI system is not intended for re- covery of the initial kinetic energies from the TOF data and requires minimization of the TOF scatter only in the direct sense in order to minimize its contribution to the effective slice thickness. Therefore, the following expressions for the overall TOF resolution ¢t´ max v i ¢t(v i ), ¢t(v i )´ max ±r j t(±r j ,v i )¡min ±r j t(±r j ,v i ), (2.77) and the total TOF spread ¢¿ max ´ max v i ht(v i )i max v i cos® i Å min v i ht(v i )i min v i cos® i , ht(v i )i´ 1 N pos X ±r j t(±r j ,v i ) (2.78) were used (here t(±r j ,v i ) denote the arrival times of the corresponding particles). The seemingly complicated form of (2.78) is caused by the lack of the limiting angles ®Æ 0° and 180° in the used sample. Hence the total TOF spread is estimated from extrapolation of the extremal present angles according to (2.34) and the relationship v z Æ cos®¢v 0 . (For a sample with the limiting angles the expression for the total TOF spread becomes simply¢¿ max Æ max v i ht(v i )i¡min v i ht(v i )i.) Besides the TOF resolution (2.77), the operation of a system in SVMI mode is also sensitive to TOF distortions produced by the ion optics. These distortions man- ifest themselves as deviations from the simple mapping v z 7! t described by (2.34) with the arrival time becoming additionally dependent on the radial component v r of the initial velocity. While this dependence is relatively small and is not detrimental in case of 3-dimensional detection with time resolution 1 , its presence is very unde- sirable for slicing, since slices taken by selection of arrival times no longer result in images of (v x ,v y ) distributions corresponding to particular v z values. Therefore, this effect must be taken into account in assessment of the ion-optical system per- 1 In that case both radial and TOF distortions can be corrected by postprocessing of the recorded data. 73 formance. For that purpose one additional characteristic, the TOF distortion of the central slice (v z Æ 0), was introduced in the simulations. It is defined as ¢t 0 ´ max v i ?e z ht(v i )i¡ min v i ?e z ht(v i )i, (2.79) where the maximum and minimum are taken over the velocities with zero z-compo- nent. In the actual simulations, due to the lack of v 0 Æ 0 in the sample, the following estimation ¢t 0 Æht(v max ,®Æ 90°)i¡ 1 2 £ ht(v min ,® min )iÅht(v min ,® max )i ¤ (2.80) was used. It is based on the observations made from the test simulations that: 1) the TOF distortion is monotonic with respect to v r , 2) particles with higher v r have longer flight times (due to longer flight paths and shorter intermediate acceleration in the additional lens, which is the main source of the distortion), 3) the average of TOFs of the particles with v min and extremal angles provides a better TOF estimation for v 0 Æ 0 than the TOF of the particle with v min , ®Æ 90° (the particles with extremal angles have smaller v r and thus are less affected by TOF distortions). 2.5.4 Simulation details As mentioned earlier, all numerical simulations were performed using SIMION 8 package. In general terms, the simulation process begins from defining the geometry of all electrodes (that is, all conductive bodies, including the vacuum chamber walls, the detector and so on). This geometry is projected onto a suitable spatial grid, and SIMION numerically solves Laplace’s equations on this grid in order to find the de- pendence of the electrostatic potential in the whole space on the electrode potentials. 74 Due to the linearity of Laplace’s equation this procedure requires solution of N el (the number of electrodes) Laplace’s equations each of which has boundary conditions with all, except one, electrodes grounded (zero potential). The potential distribution in free space is then obtained for arbitrary combination of electrode potentials as the corresponding linear combination of these particular solutions. The particular solutions are found only once, when the geometry is defined, and stored for further use. This makes any adjustments of electrode potentials during the simulations an easy and fast process. (On the other hand, any modifications of the geometry require complete recomputation of all Laplace’s equations and therefore is time-consuming.) Once the electrostatic potential in free space is available, the particles with de- fined initial positions and velocities can be propagated according to the equations of motion (2.9). SIMION uses an adaptive numerical integration method to solve these equations 1 and reports the arrival times and positions for each particle. These da- ta can be saved for further analysis by external means or passed to a user-defined subroutine within the simulation process. The latter method was chosen in the present work because it allows smoother integration of the preparation of the initial conditions, the simulation itself and the analysis of its results. In particular, all these steps can control each other, providing the possibility of automatic optimization of some characteristics with minimal user’s intervention. These capabilities were not only useful during the design process, but are also quite important in the practical use of the system, since, as shown in the next subsection, the system offers a broad range of operation conditions, and the complete summary of optimal parameters and characteristics for each particular case cannot 1 The electric field at trajectory integration points is found by interpolation of the numerical derivative of the electrostatic potential. The trajectory integration points generally do not coincide with the potential grid points and are usually denser and irregular. Moreover, the equations of motion are always solved in the full 3-dimensional space (in rectangular coordinates), even for potentials defined in restricted geometries (for example, cylindrically symmetric, as in the present case). 75 be provided here. Therefore, the most practical approach is to use the most important plots (given in the next subsection) for selection of the desired regime of operation and then to find the detailed data for it by simulations and/or optimizations stating from the chosen point. Listings of all files required for SIMION simulations are given in appendix A. They consist of the geometry definition file, the particles definition file and the user program file. The geometry definition file (appendix A.1) formalizes the descriptions of the ac- celerator and the lens given in subsections 2.5.1 and 2.5.2. Additionally, it describes the internal shape of the vacuum chamber and the detector, with some simplifications immaterial to the problem. Since the design optimization involved some changes of the geometry, the file was also changing during the development, but the present listing corresponds to the final version, which should be very close to the geometry of the physical implementation. As mentioned above, cylindrical symmetry was used for the electric potential, and thus the geometry is defined in cylindrical coordinates 1 . A grid step of 0.5 mm was chosen to adequately represent all relevant details (the finest of which is the 1-mm thickness of the accelerator plates). The file defines 17 logical electrodes: 1 grounded and 16 adjustable, out of which 14 are the accelerator plates, 1 is the central element of the lens, and 1 is the front plate of the detector. The reason why the latter is not grounded is explained in subsection 2.6.2 below. The particles definition file (appendix A.2) defines the initial states of the par- 1 Attention! Cylindrical geometry in SIMION is always defined with the x axis as the cylindrical axis. This does not coincide with the coordinate system used in the present work. Therefore, (x, z) coor- dinates used throughout this chapter correspond to (Y , X) coordinates in SIMION. This should not lead to a confusion in reading, nor in using the provided program, but must be kept in mind if modi- fications to any of the files are to be made. 76 ticles as described on p. 68 and illustrated in figure 2.18. In principle, as shown in subsections 2.3.1 and 2.3.2, all characteristics of the system in any mode of operation can be extracted, using the corresponding scaling relationships, from simulations based on a sample with fixed kinetic energies and particle mass, if the electric poten- tials are scaled appropriately. On the other hand, the simulations can be performed with the actual potentials and parameters of the particles. The choice between these two approaches is merely a matter of convenience. The convenience has two aspects — for the end user and for the implementation of the simulation algorithm. It should be noted that SIMION does not allow dynamic creation of particles for the simulations. That is, all needed particles must be defined beforehand, using only a static description. At the same time, the potentials can be varied freely, as mentioned above, which suggests that the first approach is simpler to implement, even though it is less convenient for the user. 1 However, parameters of each defined particle can be changed during the simula- tions in the same way as the electrode potentials. It means that if all particles in the desired sample are defined in some “generic” manner, their parameters can be adjusted at the beginning of the simulation according to the actual properties (par- ticle mass and maximum kinetic energy) of the sample, and then the simulations can proceed with the actual values of all parameters, presenting a totally consistent picture to the user. This approach is obviously preferred from the user’s perspective, and since it is also not more difficult than the first approach from the programming point of view, it was chosen in this work. Specifically, K max Æ 1 eV and mÆ 1 Da were used for the sample described by the 1 The drawbacks come from the fact that while the user-defined subroutine can present the analyzed characteristics in the appropriately scaled form, it cannot affect other useful information available through the SIMION interface, such as isopotential contours, electric field readings and time markers on the trajectories. 77 particles definition file. The particle initialization subroutine described below scales the particle mass and the resulting velocity 1 for each particle according to the user’s input at the beginning of each simulation cycle. The initial spatial distribution of the particles was defined symmetrically around the origin (as shown in figure 2.18). It is not modified in the simulations — instead, the potential array that describes the ion-optical system is placed in the workbench in such a way that the origin (and thus the center of the ionization region) falls into the desired position relative to the accelerator. On the one hand, this placement is fully controlled by the standard means from the SIMION interface, on the other hand, the coordinate system in this approach is always associated with the ionization region center, which is a natural reference point. The user-program file (appendix A.3) contains definitions of all necessary vari- ables and subroutines used for control of the simulations, calculation of performance characteristic from the simulation outcome and optimization of some of these char- acteristics. Details of working with the program and some SIMION-specific aspects are given in the appendix. The optimizations are performed with the help of the SIMION built-in implemen- tation of the simplex method (Nelder–Mead algorithm [31]). Although the simplex op- timization is extremely universal (it works for all continuous objective functions and even for some discontinuous ones) and robust, this generality comes from the fact that only numerical values of the objective function at several points of the parame- ter space are used. Hence, even if the objective function has some “good” properties (for example, is smooth or doubly differentiable), that information is not used, and the required number of iterations is generality large compared to more specialized 1 SIMION internally works with velocities only, although the initial conditions can be defined by spec- ifying kinetic energies, which is more convenient for the present problem. 78 optimization methods. Nevertheless, the practical use of the program showed that this number is not overwhelming (normally .200 optimization steps, resulting in the overall time of a couple minutes), and the simplicity of use and modifications outweighs the imperfect time efficiency. The present program can optimize the overall kinetic energy resolution¢K and solve for the maximum measurable kinetic energy maxK. The latter is also achieved through minimization, with the objective function jR max ¡ R det j, where R det is the detector radius. Since the condition R max Æ R det can always be satisfied by an appro- priate value of K max , the objective function ¢KÅ¸jR max ¡ R det j (2.81) is used for optimization of the overall resolution under the constraint that the image exactly fills the detector. 1 The constant factor¸ compensates the difference in dimen- sions of the two terms and plays an important role in ensuring that the second term indeed imposes a constraint. Namely, both ¢K and R max depend on all adjustable parameters (see p. 255) and therefore, if¸ is sufficiently small, then the decrease in ¢K upon variation of the parameters might overpower the increase in¸jR max ¡R det j, leading to shifting of the minimum of (2.81) to a point where R max 6Æ R det (obvious- ly, in the limit ¸! 0 the optimization becomes unconstrained). On the other hand, the dependences are nonlinear, and thus very large values of¸ would cause (2.81) to form a narrow curved valley, which is known to present difficulties for all numerical optimization methods. The empirical value ¸Æ 10 (for¢K in % and R in mm) was found to be a good choice for all relevant simulations in the present work. The selection of¢K as the optimized characteristic is explained by its importance 1 The optimization result in that case obviously includes the maximum detectable kinetic energy maxKÆ K max as well. 79 for the SVMI performance. Indeed, with the present ion-optical system the imaged kinetic energy range and the TOF stretching for it can be set to the desired values, and the optimization goal is to maximize the resolution for the selected regime. It should be clear from the previous discussion that among the characteristics deter- mining the total resolution of an SVMI system the radial focusing, characterized by ¢K, has the greatest effect. Since simultaneous optimization of all contributions is not possible (see p. 88 for a detailed discussion), this most important one was chosen for the “elementary” optimization by the program. Only one parameter, V 1 , is varied in the present program for the minimization of ¢K. The reasoning behind this choice is simple. The problem has five adjustable ion optics parameters: L 0 , L 1 , V 0 , V 1 and V L . The lens voltage V L is associated mostly with the magnification factor and has a relatively small effect on the focusing. The highest voltage applied to the accelerator is equal to V 0 and is a less flexible parame- ter than V 1 from an experimental point of view. Although the lengths L 0 and L 1 are also adjustable, their variation for optimization of focusing at fixed V 1 is not a good idea — first, the variation of the lengths is much more limited than that of the volt- age, 1 second, the precision of adjusting the voltage at fixed chosen lengths is much higher than of adjusting the lengths to some general values. Hence V 1 seems to be the most convenient adjustable parameter for minimization of¢K, both theoretically and practically. In principle, variation of V 1 at fixed values of all other ion optics parameters achieves variation of the focusing by changing both E 1 and E 0 electric field strengths and in that way also leads to changes in R max and¢¿ max for a given K max . Therefore, if a¢K minimization under strictly fixed maxK and¢¿ max is required, the varied 1 For example, configurations with L i Ç 0 or L 0 ÅL 1 È 130 mm are not physically feasible, whereas the voltage can be set to any value. 80 parameters must also include V 0 (for E 0 compensation, to keep¢¿ max constant) and V L (to compensate the magnification changes). However, in practice ¢K is a very strong function of V 1 , and thus no significant differences in maxK and¢¿ max upon the¢K minimization by V 1 adjustment should be expected, unless the starting point for the optimization was chosen very far from the working conditions. That is, for practical applications, which require only coarse settings of maxK and ¢¿ max , the difference between the simple approach with only V 1 and the more complicated one with {V 0 ,V 1 ,V L } is not important. Nevertheless, if a more constrained optimization is needed, the modification of the objective function (2.81) and inclusion of additional variables in the parameter space are straightforward. 2.5.5 Simulation results The goals of the simulations were the study of the operational limits, determi- nation of the resolution and optimal conditions for each achievable imaged kinetic energy range, and verification of the chosen geometrical parameters. The exact scaling relationships discussed in subsection 2.3.1 allow to limit vari- ation of the ion optics parameters to 4 variables: {L 0 ,L 1 ,V 1 /V 0 ,V L /V 0 }. That is, the maximum voltage on the accelerator, V 0 , can be used as the overall electric field strength scale, with the shape of the electric field described by the geometrical pa- rameters L 0 and L 1 and the relative potentials V 1 /V 0 and V L /V 0 . Then, according to (2.15), the imaged kinetic energy range should scale proportionally to V 0 . The TOF, according to (2.18), should be proportional to p m/V 0 , and the maximum TOF stretching, according to (2.35) and the maxK » V 0 scaling, should have the same 81 p m/V 0 dependence. It means that the “magnification index” 1 introduced here as M´¢¿ max p K max /m (for K max Æ maxK) (2.82) does not depend on V 0 and thus can be determined as a function of the 4 free variables mentioned above at any fixed V 0 value. The magnification index plays the major role in selection of the ion optics param- eters for SVMI operation. Namely, it is completely and unambiguously determined by the experimental conditions: the particle mass m, the desired imaged KE range K max and the TOF stretching¢¿ max required for slicing. Therefore, expressing oth- er performance characteristics and the required ion optics parameters as functions of the magnification index should provide the most convenient representation of the simulation results. The performance characteristics related to resolution also can be expressed in a more universal form than they were defined above. The relative overall kinetic ener- gy resolution¢K defined by (2.74) is already a dimensionless quantity independent on the V 0 and m scalings, but the overall TOF resolution¢t defined by (2.77) and the central slice distortion¢t 0 defined by (2.79) are absolute quantities (with the dimen- sion of time) and scale with V 0 and m as the TOF itself. However, being divided by ¢¿ max they become dimensionless, independent on V 0 and m, and directly represent- ing the quantities important for slicing — the relative time resolution of the axial velocity mapping (2.34) and the shape distortion in realization of the entire mapping (2.1). Therefore, the relationships between the ion optics parameters and the resulting 1 This quantity is not directly related to the relative magnification M rel used in subsection 2.4.4. The index rather describes the combination of the temporal and radial characteristics and is numerically equal to the total TOF stretching for a distribution of mÆ 1 Da particles with K max Æ 1 eV under the voltages scaled such that the image fills the whole detector. 82 performance characteristics are represented here as follows. The magnification index M is taken as the main independent variable, which determines a subset of the elec- tric field configuration parameters {L 0 ,L 1 ,V 1 /V 0 ,V L /V 0 } corresponding to this value of M and with V 1 /V 0 satisfying the KE focusing condition (see (2.81) and the dis- cussion about it). This subset has two free parameters, 1 which were chosen simply as L 0 and L 1 , and corresponds to a set of relative resolutions and distortions {¢K, ¢t/¢¿ max ,¢t 0 /¢¿ max } obtainable for the given value of M. Selection of the optimal combination from the latter set thus determines the required L 0 and L 1 lengths and, through them, the required voltage ratios. The absolute voltage scale is determined by the desired K max or¢¿ max in (2.82). The relationship between maxK and V 0 was chosen for this purpose in the present work, since it allows a more evident repre- sentation, and selection of maxK is more relevant from the experimental point of view. 2 Presentation of characteristics dependent on three parameters (M, L 0 and L 1 ) is a difficult task, especially if all these parameters can vary continuously and in broad ranges. The decision made in this work was to use graphical representation and to plot the characteristics as functions of the main variable M for a discrete set of L 0 and L 1 values. Namely, a regular grid with 5 mm steps in each variable was used for {L 0 ,L 1 } combinations. This produces a reasonable balance between the density of the parameter space sampling and the number of plotted curves. Additionally, the current implementation of the system uses a voltage divider that employs fixed resis- tors instead of continuously variable potentiometers (see p. 101) and allows lengths 1 Out of 4 degrees of freedom one (V 1 /V 0 ) is determined by the KE focusing condition, and one more is constrained by M, on which V L /V 0 has the largest effect. 2 Namely, maxK must be set according to K max in the studied distribution, which should be appropri- ately known from preliminary experiments or theoretical estimations, while¢¿ max must simply have some value determined by the desired slice thickness and not directly dependent on the particular distribution. 83 selection with exactly these 5 mm steps, thus the plots are directly related to the operation of the actual system. Since the program used for generation of the simulated characteristics and de- scribed in the previous subsection operates in terms of the {L 0 ,L 1 ,V 0 ,V L }7! {maxK,¢¿ max ,¢K,¢t,¢t 0 ;V 1 } (2.83) transformation instead of the {M,L 0 ,L 1 }7! ½ ¢K, ¢t ¢¿ max , ¢t 0 ¢¿ max ; V 1 V 0 , V L V 0 , qV 0 maxK ¾ (2.84) one required for the chosen data representation, the algorithm for constructing the (2.84) dependences from (2.83) data was organized in the following way. A fixed value V 0 Æ 1000 V was used in all simulations. 1 For every combination of lengths {L 0 ,L 1 } (with L i divisible by 5 mm, as mentioned above) the lens voltage V L was sampled from 0 to 2700 V (with variable steps, from 50 to 200 V, adaptively selected for suffi- cient smoothness of the results), and for each V L value the optimal values of V 1 and K max were found, as described on p. 79, together with the corresponding performance characteristics. This resulted in a large set of data in the form (2.83). Then the val- ues of M were computed from maxK and¢¿ max for each sampled combination of the ion optics parameters, and the resulting sampling of the mapping V L 7! M for each {L 0 ,L 1 } was inverted to obtain the sampling in the form M7! V L , as needed for (2.84). The relative characteristics in the right side of (2.84) were subsequently calculated from the corresponding absolute values in (2.83). This resulted in transformation of 1 This value is arbitrary, but lies roughly in the middle of the actual working V 0 range and thus should be suitable for all simulations (meaning that the simulated condition would not differ drastically from the actually used ones and therefore should not lead to unexpected numerical errors in the simulations). 84 the initial simulations data set from the form (2.83) to the desirable form (2.84), al- though with nonuniform (and irregular with respect to {L 0 ,L 1 }) sampling over M. Nevertheless, since all interdependences are continuous and relatively smooth, and the original sampling was sufficiently dense, the results can be resampled to any desired grid using interpolation. In the present work a resampling to a regular grid with 0.1 ns p eV/Da step in M with linear interpolation was used for plotting and analysis of the data. The resulting performance characteristics are presented in figure 2.20, and the corresponding operational parameters — in figures 2.21 and 2.22. The quantity “total ¢t” plotted in figure 2.20 is defined as the sum of¢t and¢t 0 and is a more relevant temporal characteristic for SVMI operation than¢t and¢t 0 by themselves. Namely, the quality of slicing performed by time gating of the detector is sensitive to the total deviation of the actual arrival times of particles with v z Æ 0 from the arrival time ex- pected for perfect velocity mapping. This deviation has contributions from both the imperfect TOF focusing described by¢t and the distortion of the v z 7! t mapping de- scribed by¢t 0 . The maximum TOF scatter important for the worst-case performance should therefore include both these components and for the whole imaged distribu- tion is equal to the sum of the worst-case estimations of these deviations. 1 1 In principle, the overall TOF scatter described by ¢t is calculated for the whole velocity distribu- tion, and thus the scatter for particles with v z ¼ 0 — the only important part of the distribution for recording of the central slice — might be smaller. However, for the vast majority of the practically important conditions the velocity-dependent part of the TOF scatter is negligible (as a reminder, the scatter is due to the initial position distribution, assumed to be velocity-independent), and the scatter is nearly symmetric around the average TOF. Thus the overall scatter¢t gives a good estimation of the TOF scatter for any initial velocity value, and the “total¢t” for the central slice can be expressed as max ±r,v?e z t(±r,v)¡min ±r,v?e z t(±r,v)Æ max v?e z ¡ max ±r t(±r,v) ¢ ¡ min v?e z ¡ min ±r t(±r,v) ¢ ¼ ¼ max v?e z µ ht(v)iÅ ¢t 2 ¶ ¡ min v?e z µ ht(v)i¡ ¢t 2 ¶ Æ ¡ max v?e z ht(v)i¡ min v?e z ht(v)i ¢ Å¢tÆ Æ¢t 0 Å¢t, (2.85) 85 1.0 1.5 2.0 20 40 60 80 100 120 ΔK,% M,ns p eV/Da 0 2 4 6 20 40 60 80 100 120 TotalΔt,% M,ns p eV/Da 0 1 2 3 4 20 40 60 80 100 120 Δt,% M,ns p eV/Da 0 1 2 3 4 20 40 60 80 100 120 Δt 0 ,% M,ns p eV/Da 1.0 1.5 2.0 20 40 60 80 100 120 ΔK,% M,ns p eV/Da 0 2 4 6 20 40 60 80 100 120 TotalΔt,% M,ns p eV/Da 0 1 2 3 4 20 40 60 80 100 120 Δt,% M,ns p eV/Da 0 1 2 3 4 20 40 60 80 100 120 Δt 0 ,% M,ns p eV/Da L 0 =30 35 40 45 50 55 60 65 70 75 80 85 90 95 L 1 =30 35 40 45 50 55 60 65 70 75 80 85 90 95 Figure 2.20 Relative overall kinetic energy and TOF resolutions as functions of magnification index. (Lengths L 0 and L 1 are in mm.) 86 0.0 0.5 1.0 1.5 2.0 20 40 60 80 100 120 −V L /V 0 M,ns p eV/Da 0.6 0.7 0.8 20 40 60 80 100 120 V 1 /V 0 M,ns p eV/Da 0.0 0.5 1.0 1.5 2.0 20 40 60 80 100 120 −V L /V 0 M,ns p eV/Da 0.6 0.7 0.8 20 40 60 80 100 120 V 1 /V 0 M,ns p eV/Da L 0 =30 35 40 45 50 55 60 65 70 75 80 85 90 95 L 1 =30 35 40 45 50 55 60 65 70 75 80 85 90 95 Figure 2.21 Interdependence of ion optics parameters and magnification index. (Lengths L 0 and L 1 are in mm.) 500 1000 1500 2000 20 40 60 80 100 120 qV 0 /maxK M,ns p eV/Da 500 1000 1500 2000 20 40 60 80 100 120 qV 0 /maxK M,ns p eV/Da L 0 =30 35 40 45 50 55 60 65 70 75 80 85 90 95 L 1 =30 35 40 45 50 55 60 65 70 75 80 85 90 95 Figure 2.22 Dependence of voltage scale on imaged kinetic energy range and mag- nification index. (Lengths L 0 and L 1 are in mm.) 87 Nevertheless, the dependences for the constituents¢t and¢t 0 are also plotted, since they are determined by different factors and might have different contribu- tions and degrees of importance in particular cases. For example,¢t is due to finite dimensions of the initial volume, and the plotted scatter corresponds to their specif- ic values (see figure 2.18). If the ionization region or the uncertainty in its location in a particular experiment is smaller, then the TOF scatter¢t would be accordingly smaller and hence having a smaller effect on the total¢t for SVMI. At the same time, the plotted¢t 0 values correspond to the whole velocity distribution from 0 to v max (or KE from 0 to K max ). If the actual imaged distribution has a narrower range of KEs (or only a narrow range is interesting for a particular measurement), then the distortion only for that range is relevant from the practical point of view. Such “par- tial”¢t 0 would be of course smaller than the overall¢t 0 and even becomes negligible for nearly monochromatic particles (not a very common but sometimes encountered case). Moreover, from a broader perspective, the designed ion-optical system can be used with a complete 3-dimensional detection with time resolution instead of slicing, and in that case the TOF distortion has no effect on the performance, since it can be corrected by data processing, but the TOF scatter becomes crucial for the final resolution. These considerations show that the relative importance of different character- istics depends on the specific application. Comparison of the dependences plotted in figure 2.20 reveals that they often have mutually opposite behaviors, so that no conditions can reach the best performance in all respects at once. Therefore, it is fundamentally impossible to provide “the best” ion optics parameters for all cases. 1 where definitions (2.77) and (2.79) were used. 1 Simultaneous optimization of more than one characteristic is generally infeasible because simultane- ous minimization of N objective functions {f n } N nÆ1 dependent on M common parameters {p m } M mÆ1 ´ p requires simultaneous satisfaction of N£M equations @f n @p m Æ 0, which clearly cannot be done if NÈ 1. However, since such multi-objective optimization problems emerge quite often, some approaches to 88 Instead, the data shown in figures 2.20–2.22 provide the Pareto-optimal character- istics and parameters, out of which a specific combination should be selected based on the specific requirements in a particular experiment. The Pareto optimality with respect to¢K and total¢t was used here, since these are the most important criteria for SVMI operation, but the plots optimal with respect to {¢K,¢t} and {¢K,¢t 0 } are also provided in appendix B. Although the figures might appear too small for practical use, this is not a prob- lem in reality. First of all, the operation of the system is impossible without a comput- er (at least for data collection and analysis), and the figures in the electronic version of this dissertation can be magnified as much as necessary. If a greater precision in the numbers in needed, the tabular data used for the plots is also available on the working system. In addition, these data can be easily generated using SIMION and the files provided in appendix A, and the presence of this simulation framework is practically indispensable for efficient operation of the system. Second, as mentioned on p. 76, the main purpose of these plots for practical use is to provide sufficient in- formation for selection of the starting point in optimization that should be performed for each particular situation. From this point of view, the precision of the plots is satisfactory even in the printed form. their solution were devised. In maximalistic approach, where the finite goal is well-understood and can be formalized, a single overall “aggregate objective function” is constructed from the individual objective functions (for example, as a linear combination f Æ P N nÆ1 w n f n , where the weights w n might depend on p as well), reducing the mathematical problem to a regular single-objective optimization. In the opposite, minimalistic, approach the trade-off decisions are postponed, and only the subset of parameter space is found from which the final “optimal” selection must be made based on some ad- ditional considerations. This subset, called the Pareto set, contains all parameter combinations that are optimal in the sense that for them no single characteristic can be improved without deteriorating other characteristics. All definitely worse parameter combinations p, for which an unambiguously better p 0 exists (such that f i (p 0 )6 f i (p) for every i, and the inequality is strong for at least one i), are eliminated. However, giving a preference to one Pareto-optimal p over another is a trade-off decision and can be done only after an assessment of the relative importance of each characteristic, which might be different in different situations. So, although this approach does not provide a univocal answer to the optimization problem, it is still very useful because it maximally reduces the amount of data for further analysis. 89 For example, if a distribution of H fragments with maxK¼ 0.5 eV is to be recorded using SVMI with 5 ns effective slice thickness, a total TOF stretching of »50 ns is required. These conditions lead to MÆ 50 ns¢ p 0.5 eV/1 Da¼ 35 ns p eV/Da. Using figure 2.20, many possible combinations of ion optics parameters can be chosen for this value of M. One of such combinations, corresponding to ¢K ¼ 0.8 % and total ¢t. 4.5 % (¢t¼ 2.5 %,¢t 0 . 2 %) — intermediate values for all these performance characteristics, has L 0 Æ 40 mm, L 1 Æ 70 mm and, according to figure 2.21, V L /V 0 ¼ ¡0.95 and V 1 /V 0 ¼ 0.76. From figure 2.22, the ratio qV 0 /maxK ¼ 1350, and thus V 0 Æ 1350£ 0.5 eV/1eÆ 675 V should be used. This corresponds to V 1 Æ 513 V and V L ¼¡640 V (rounded for convenience). Optimization starting from these parameters yields a more accurate value V 1 Æ 516 V and the characteristics: maxKÆ 0.506 eV, ¢KÆ 0.786 %, ¿ max Æ 49 ns,¢tÆ 1.24 ns (»2.5 %),¢t 0 Æ 0.92 ns (»1.9 %) — all very close to the initial values taken from the plots. 1 The temporal characteristics obtained for this combination of parameters indicate that if the worst case is indeed realized in the experiment, the total effective slice thickness would be&7 ns (5 nsÅ total¢t), which comprises»1/7 of the total TOF stretching and thus leads to an inner “wing” in the recorded distribution due to finite slicing thickness (see figure 2.2) with »1 % ¢v/v extent for the maximum velocity, or »2 % ¢K/maxK for the whole kinetic energy range 2 . Although this additional 1 Molecular beam velocity v beam Æ 0 was used in these illustrative simulations, as well as for generation of the plots. This parameter has a relatively minor effect on the results for H fragments but becomes increasingly important for heavier particles (for example, for v beam Æ 1600 m/s the initial kinetic energy due to the motion with the beam reaches 1 eV at m¼ 80 Da; this clearly causes a shift in the total TOF comparable to the TOF spread, but also has a noticeable effect on other characteristics due to chromatic aberrations of the ion optics). Optimization with the actual molecular beam velocity is recommended for all practical simulations. 2 The extent of the wing becomes larger for lower velocities in the velocity scale, as can be un- derstood from figure 2.2 and expression (2.7) multiplied by v ½ , but remains constant everywhere in the KE scale. Namely, using figure 2.2 and relationship (2.48), for any given KE the maxi- mal radius is max v z R(K)» p K, and the minimal radius satisfies min v z [R(K)] 2 Æ max v z [R(K)] 2 ¡ (R max ¢t s /¢¿ max ) 2 , where ¢t s denotes the absolute time slice thickness. Substitution of the first relationship into the second and rearrangement of the terms shows that the range of projections 90 broadening seems large compared to the resolution quoted above (¢K Æ 0.786 %), these 2 % correspond to the total spread, and the actual effective broadening is much smaller. Nevertheless, the wing length can be reduced to, say,»1 % in KE by choosing longer TOF stretching. For instance, according to figure 2.20, at M¼ 71 ns p eV/Da required for¢¿ max ¼ 100 ns the ion optics parameters can be chosen to yield¢K¼ 1 % and total¢t¼ 5.5 % (that is, the total effective relative slice thickness will be about (5 ns/100 ns)Å 5.5%¼ 1/10 leading to ¢ s K ¼ 1/100). These parameters are: L 0 Æ 45 mm, L 1 Æ 85 mm, V L /V 0 ¼¡1.72, V 1 /V 0 ¼ 0.77 and V 0 ¼ 760£ 0.5 eV/1eÆ 380 V, from where V 1 ¼ 293 V and V L ¼ 655 V. Optimization starting from these values results in V 1 Æ 292 V and maxKÆ 0.497 eV (¢KÆ 0.99 %) with¢¿ max Æ 100 ns (total ¢tÆ 5.8 %). 1 On the other hand, if the initial position distribution is known to be narrow along the ion-optical axis (that is, the ions are created by a focused laser) and stable during the experiment, then the experimental TOF scatter should be much smaller than the ¢t values used for the present plots and thus can be mostly ignored. In that case the plots in figures B.4–B.6 (appendix B) would be more helpful. For example, parameters leading to minimal¢t 0 and maximal (but still reasonable)¢KÇ 0.9 %, namely, L 0 Æ 55 mm and L 1 Æ 70 mm, can be selected for given MÆ 35 ns p eV/Da. This presents an example of operation without the additional lens (V L Æ 0 according to figure B.5). The optimized parameters V 0 Æ 865 V and V 1 Æ 645 V (starting from V 1 ¼ 650 V) yield maxKÆ 0.499 eV (¢KÆ 0.875 %) and¢¿ max Æ 49.7 ns with¢t 0 Æ 0.15 ns (only in the KE scale does not depend on K and is simply ¢ s K Æ (¢t s /¢¿ max ) 2 K max . Obviously, for K6 (¢t s /¢¿ max ) 2 K max the distribution is not sliced at all (for the considered example this limit is»0.01 eV). 1 The same parameters, but with all voltages increased by 2 times (V 0 Æ 760 V, V 1 Æ 584 V, V L Æ¡1310 V) can be used for D fragments with K max ¼ 1 eV (since M Æ 100 ns¢ p 1 eV/2 Da¼ 71 ns p eV/Da has the same value), or with all voltages increased by 4 times (V 0 Æ 1520 V, V 1 Æ 1168 V, V L Æ¡2620 V) — for SVMI of H distribution with K max ¼ 2 eV with 50 ns stretching. However, the relative TOF characteristics in the latter case will be not as good, leading to the worst-case ¢ s K/maxK¼ [(5 ns/50 ns)Å5.8%] 2 ¼ 2.5 %. 91 »0.3 %). If, in addition to the initial position distribution properties specified above, the position of the ionization region is also known with precision better than the assumed§0.5 mm, then the actual kinetic energy resolution should be better than this worst-case value. However, it would be advisable to redo the optimizations with a sample more relevant to the actual distribution, since the optimal parameters for such tightened conditions might be somewhat different and thus result in even better performance. Of course, the ion-optical system can also be used for “regular” full-projection VMI without slicing. All temporal characteristics are unimportant for such application, 1 hence the best selection of ion optics parameters should correspond simply to the lowest¢K value. As can be seen from the plots, these parameters are L 0 Æ 45 mm, L 1 Æ 75 mm, V 1 /V 0 ¼ 0.77 for operation without the additional lens (V L Æ 0) or L 0 Æ 35 mm, L 1 Æ 95 mm, V 1 /V 0 ¼ 0.815, V L /V 0 ¼¡1.5 for operation with the lens. The latter option results in slightly better resolution, somewhat higher imaged KE range for the same maximum voltage (or lower required voltages for the same KE range), but»1.5 times worse mass resolution due to greater TOF stretching. Unless the KE range is too large, requiring very high voltages applied to the accelerator, operation without the lens is preferred, since, besides providing better mass separation, it uses higher acceleration potentials, which leads to higher detection efficiency by the MCP detector. Overall, the simulation results presented in figures 2.20–2.22 demonstrate that the magnification index can be changed from »20 to »130 ns p eV/Da while keep- ing the worst-case overall relative KE resolution better than 2 % and the total rela- tive TOF deviations below»6 %. For a broad range of magnifications (namely, below 1 However, mass resolution, determined by some of the temporal characteristics, might be important in certain cases. 92 Figure 2.23 Illustration of electrostatic potentials and particle trajectories for the extreme operating conditions. Top: lower KE limit (K max Æ 0.163 eV; L 0 Æ 40 mm, L 1 Æ 50 mm, V 0 Æ 320 V, V 1 Æ 231.1 V, V L Æ 0); equipotential contours drawn with 10 V steps. Bottom: upper KE limit (K max Æ 6.75 eV; L 0 Æ 75 mm, L 1 Æ 55 mm, V 0 Æ 3000 V, V 1 Æ 1891 V, V L Æ¡6600 V); equipotential contours drawn with 100 V steps (500 V steps below¡1000 V). In both cases¢¿ max ¼ 50 ns. Trajectories correspond to the initial sample illustrated in figure 2.18. 93 »80 ns p eV/Da) relative KE resolution of 1 % or better is achievable. For SVMI of H fragments using 50 ns TOF stretching, this¢K6 2 % range corresponds to kinet- ic energies from»0.16 to»6.8 eV (or to»2.5 eV for¢K6 1 %), although the upper limit might pose some practical difficulties due to relatively high required voltages (V L » 7000 V), and the low acceleration voltages (V 0 » 300 V) for the lower KE lim- it might lead to decreased detection efficiency. Nevertheless, the design goals set in section 2.2 can be considered satisfied. Example of simulations for operation near the lower and upper KE limits are shown in figure 2.23. The optimizations described above dealt only with electrical parameters of the system at a fixed geometry of the electrodes, 1 and therefore are not truly complete. As mentioned at the beginning of subsection 2.5.4, any changes to the geometry of the electrodes require complete recomputation of the electrostatic potential arrays. While this process is not more computationally demanding than a single voltage- optimization run, its integration with user programs in SIMION 8 is less smooth. Moreover, in contrast to the studied “electrical” optimizations, the geometry opti- mization must be performed for the whole operational range of the system, since by design the geometrical parameters cannot be adjusted in a working machine. That is, every geometry optimization cycle must involve a complete recomputation of all the datasets presented in this subsection. And even though their dependences on the varied geometrical parameters must be relatively smooth, which might be used for speeding up the recomputations to some extent, the effort required for such com- plete optimization and analysis significantly exceeds the amount of work needed to produce the partially optimized results described above. Therefore, instead of the full optimization, it was decided to perform only several 1 Although the accelerator lengths were extensively varied, this was achieved by purely electrical means and thus is not an electrode geometry optimizations in the strict sense. 94 test simulations for a few variations of the geometry starting from the one described in subsections 2.5.1 and 2.5.2 in order to ensure that the chosen layout is not too far from the possible optimum. The accelerator was not modified from the arrange- ment optimal for the low-KE range operation without the additional lens (see sub- section 2.4.3), but the influence of the additional lens position and the length of its central element on the resolution for operation at large M values was studied. It was found that a shorter lens (0.5 : 1 length-to-diameter ratio) leads to higher aberrations than the “standard” one (1 : 1 ratio), in accordance with the general results (see [29]), regardless of its position. At the same time, neither an increase of the length within the available space, nor small possible shifts of the lens within the drift region led to evident improvements of the performance. Since the lens already had the maximum practical diameter, it could be varied only towards smaller values, but simulations with 100 mm diameter expectedly showed larger aberrations. Based on these tests, it was concluded that even if the present geometry is not per- fectly optimal, it is still quite close to the optimum, and no significant improvement of the performance characteristics can be achieved within the used optical scheme. As mentioned at the beginning of this section, this scheme was chosen as the simplest approach to the posed problem. Thus after a satisfactory solution was obtained in the numerical simulations, it was appropriate to perform experimental tests of the designed system and refine the practical demands from it before considering more complicated arrangements. 2.6 Implementation As mentioned in section 2.2, the designed SVMI system was intended for retrofit- ting of the free-radical molecular beam apparatus used in the laboratory since 1998. 95 A description of the apparatus construction is given in [13] and will not be repeated here. The only modifications to the vacuum chamber done in the present work were related to the “detection region” part and included removal of the TOF spectrometer assembly mounted on the top flange and replacement of the end flange 1 by a nip- ple 2 of the same diameter for axial extension of the chamber. The SVMI components added to the machine are described in the following subsections. 2.6.1 Ion optics Since the arrangement of the electrodes in the designed ion-optical system is very simple, its implementation was relatively straightforward. However, some practical aspects had to be taken into account in the design of the supporting structures. The fact that the system must operate in high vacuum poses some restrictions on the selection of materials and requires to minimize the number of mating surfaces and avoid tight or enclosed spaces in order to reduce outgasing. The high voltages (up to a few kilovolts) applied to the electrodes require careful electrical insulation and sufficient spacings among them and from the grounded parts. For operation at lower voltages the possible electric charge accumulation 3 can lead to significant electric field distortions and thus must be eliminated at least in the vicinity of the important regions. From a mechanical point of view, the supporting structures must provide reliable and precise positioning of the ion optics elements with respect to each other, as well as relative to the rest of the system — most importantly, the ionization region and 1 Which had no function except hosting the vacuum gauge port. 2 A standard6 00 (»154 mm), 13.12 00 (336 mm) long nipple with 8 00 ConFlat-type flanges modified by addition of one port for vacuum gauge and one small flange for electrical feedthrough was ordered from MDC Vacuum Products. The whole piece was welded from nonmagnetic type 304 stainless steel standard components, and the inner surface electropolished by the company. 3 From spurious charged particles hitting electrically insulating surfaces and from photoelectron emis- sion caused by scattered light. 96 the detector. For example, 1 mm displacement of the accelerator relative to the vac- uum chamber will lead to the same 1 mm displacement of the ionization region from the designed position, which exceeds the initial volume dimensions assumed in the design and simulations. The 20 mm radius of the detector located »600 mm away from the ionization region means that even»2° misalignment of the accelerator axis from the vacuum chamber axis will place the center of the formed image outside the detector, and for efficient use of the whole detector area very careful centering of the image on the detector is required, leading to much tighter alignment tolerances. Nevertheless, all of these problems are not unusual and can be solved with com- monly available materials and manufacturing facilities available in a regular ma- chine shop. For example, all custom-made parts required for assembly of the present system were made in the USC machine shops using materials ordered from McMas- ter-Carr Supply Company. The mechanical drawings of these parts are given in ap- pendix C for reference. A schematic cutaway view of the whole ion-optical system in- stalled in the “detection chamber” part of the apparatus is also shown in figure 2.24. Accelerator assembly The accelerator plates were made from stock oxygen-free copper sheets (1 mm thickness) by machining. All plates, except the first one, have the same shape as in the simulations (see p. 62): 80 mm outer diameter and 40 mm aperture diameter. The first plate has the same outer diameter, but instead of the aperture uses an orifice with3 mm opening and 90° chamfer on the inner side to admit the molecular beam inside the accelerator. All plates were electrochemically polished in the laboratory us- ing the standard method [32] of anodic dissolution in 85 % (14 M) H 3 PO 4 . Although copper has a relatively poor machinability, this was not a problem for parts with such 97 Figure 2.24 Schematic cutaway view of the designed VMI ion optics inside the detection chamber. Connecting wires and other parts of the apparatus are not shown. (The actual implementation also has minor differences mentioned in the text.) 98 simple shapes. At the same time, it has excellent vacuum properties and allows very simple attachment of wire leads by soldering. Moreover, in contrast to many other metals and alloys, copper oxides are electrically conductive, which makes accumula- tion of stray surface charges impossible. Mechanical fixation and electrical insulation of the plates was achieved by insert- ing them between four MACOR 1 bars with shallow grooves (see figure C.1). The bars are held between two nonmagnetic stainless steel rings, fixed and centered in radial direction by set screws on each side. The rings are tied together with four partially threaded bronze screws and nuts. The screws have an additional purpose of pushing out the bronze pins that anchor the entrance side of the whole accelerator assem- bly inside the vacuum chamber, as illustrated in figure 2.25. 2 Such design solution vacuum chamber wall tightening screw insulator bar accelerator plates pin ring set screw Figure 2.25 Schematic cutaway view (partial) of the accelerator assembly near the molecular beam entrance end. allows very simple alignment of the accelerator entrance with the molecular beam 1 A variety of glass-ceramic material with good machinability and vacuum properties. 2 The original design, reflected in figure 2.24 and the drawings in appendix C, involved leaf springs between the pins and the chamber wall, but upon the implementation it became evident that the anchoring works fine even without these additional details, and they were not installed. The only practical disadvantage of the lack of these springs is that installation into (and removal from) the vacuum chamber must be done carefully in order to prevent the lower pins from falling through the vacuum pump port. 99 skimmer (and hence the apparatus axis) by adjusting the four screws from the exit end (see figure C.3). This ability is very important in the present setup because the existing vacuum chamber has only a small (»2 cm) opening for the skimmer in the wall separating the source and detection regions, practically preventing any access from that end. Also, hanging the whole accelerator assembly on the top flange (see figure 2.24) in a sufficiently rigid and precisely adjustable manner would required a much more complicated construction. On the other hand, the exit side of the accelerator could not be supported in the same fashion due to the large-diameter vacuum pump port located below it. More- over, the tight angular alignment requirements (see p. 97) meant that the ability to perform adjustments from outside the vacuum chamber during system operation (for prompt feedback) would be very desirable. Therefore, the exit side was attached to a translational stage installed on the smaller top flange, as shown in figure 2.24. The stage allows small vertical and horizontal movements (perpendicular to the main apparatus axis) 1 of the exit end of the accelerator, and through that the accelerator axis can be aligned with the desired direction. While this alignment system does not provide a good precision or accuracy by itself, practical operation showed that its use with real-time feedback from the VMI acquisition system allows very simple and fast centering of the image with radial deviation.1 %, and centering within»1 pixel (at 1024£ 1024 px resolution) is possible with meticulous adjustments. The alignment stability was also found sufficient for all practical purposes. 2 1 Figure C.2 shows close-up pictures of this assembly. Rotation of the big brass nut moves the central screw vertically. The screw passes through the rectangular plate that can slide on the flange surface and thus displace the central screw horizontally. The bellows providing a vacuum seal between the central screw and the flange can be seen in the bottom view. The accelerator assembly is attached to the bottom of the central screw by means of a hook, the lower part of which can be seen in figure C.3. 2 It appears that for the ion-optical system without magnetic shielding, image shifts due to variable stray magnetic fields might be more important than the alignment instability. 100 Figure 2.24 also shows mu-metal 1 magnetic shielding (in dark gray) of the accel- erator consisting of an additional plate at the entrance and a cylindrical sheet sur- rounding the plates and insulator bars. It was included in the design as an option, but not installed in the present machine, since no operations with photoelectrons were planned in the initial stages of the present work, and the necessity of magnetic shielding for ions is relatively small. However, the shielding can be installed in the future if needed. All electrical connections to the electrodes were made by means of high-vacu- um compatible high-voltage insulated wires soldered to the outer edges of the copper plates. The wires, collected in one bunch (see figure C.1), were routed through the top flange of the vacuum chamber to a pair of multipin electrical feedthroughs. 2 Selection of the effective lengths L 0 , L 1 and appropriate division of the voltages V 0 , V 1 (provid- ed by two commercial adjustable high-voltages power supplies) is performed with a voltage divider (see figure D.1 in appendix D) installed outside the vacuum chamber. Lens assembly Implementation of the lens was more straightforward than that of the accelerator. It basically followed the construction described in subsection 2.5.2 and is clearly seen in figure C.4. All rings were made from the same nonmagnetic stainless steel as the accelerator rings. The central parts of each electrode were made from cylindrically bent metal sheets. 3 The grounded outer elements use four brass set screws in each 1 A nickel-iron alloy with high magnetic permeability. 2 Since all accelerator electrodes (except the last one, which is always grounded) require applications of different potentials, a high-voltage feedthrough with at least 13 pins was needed. However, no commercially available feedthroughs with such parameters were found at the time of construction, and thus a pair of 8-pin feedthroughs (from Insulator Seal, a part of MDC Vacuum Products) installed on a tee adapter (see figure 2.24) was used instead. 3 Perforated nonmagnetic stainless steel sheet was used in the present machine, but it can be replaced with mu-metal sheet, if magnetic shielding of the drift region is desired. 101 ring for anchoring inside the vacuum chamber. The rings of the central element are attached to the inner rings of the outer elements by MACOR insulator strips fastened with stainless steel (to the grounded parts) and nylon (to the high-voltage parts) screws. The whole assembly was radially centered in the vacuum chamber by symmetric adjustment of all 16 set screws such that the assembly can be inserted into the cham- ber freely but without radial play, then positioned correctly in the axial direction relative to the open detector end of the chamber extension, and fixed by symmetric tightening of the set screws. 2.6.2 Charged particle detector A standard40 mm double-stack imaging MCP detector assembly (Galileo Elec- tro-Optics 1 3040FM series;10 μm pore) already kept in the laboratory as a spare for the other VMI setup (see [14]) was used in the initial stage of the work. The CF6 00 flange of the detector was mounted at the end CF8 00 flange of the extended vacuum chamber through a “zero-length” adapter 2 , as shown in figure 2.24. An electric shield made from oxygen-free copper sheet was attached at the vacuum side of the adapter in order to protect the near-detector part of the drift region from electric fields created by the unshielded high-voltage wires of the detector assembly. The detector can be used in two different modes. The main, imaging, mode is em- ployed in all VMI measurements. The other, electrical signal pickup, mode, where the total amplified ion (or electron) signal is collected from the phosphor screen elec- trode, is used for TOF-MS measurements, TOF spectrum examination in preparation for VMI (and especially SVMI) measurements, and auxiliary diagnostics. Despite sig- 1 Now PHOTONIS USA. 2 Ordered from MDC Vacuum Products and slightly modified in the USC machine shops to accommo- date the electrical feedthroughs of the detector. 102 nificant differences between these two modes, they share some common properties. For example, the detector input (front plate of the MCP stack) is always kept at the same voltage. This voltage was assumed to be zero in the simulations (see subsec- tions 2.5.4 and 2.5.5) in order to have a field-free region near the detector. However, experimental tests of the implemented system (see next section) revealed that dur- ing operation with negative voltages applied to the additional lens, spurious “tails” can appear in the TOF signal after otherwise normal peaks. The origin of this effect is very simple. The particles (ions) being detected produce secondary electrons upon impact on the detector input surface. Those electrons that enter the MCP channels create the useful signal by avalanche amplification. At the same time, some elec- trons are ejected from the detector and do not contribute to the signal immediately. In absence of electric fields they fly away and are eventually absorbed by the ground- ed vacuum chamber walls or other surfaces. But when a negative voltage is applied to the lens, some of the electrons are diverted by it back to the detector and upon arrival are “detected” as genuine particles. The initial velocity distribution of these secondary electrons and the electric field strength in the lens determine the tempo- ral extent and the intensity of the produced signal tails. However, the experiments indicated that the electrons contributing to the unwanted effect have relatively small kinetic energies, and application of a small negative voltage (¡5 V in the presented case) to the detector input creates a sufficient electric field to divert the majority of them 1 , so that the spurious tails disappear. Yet the employed potential is much small- er than characteristic energies of the actual particles accelerated from the ionization region and therefore does not introduce noticeable changes in the system properties. 2 1 Obviously, no voltage can prevent the particles from returning, since the total energy is conserved in electrostatic fields, and the returning particles will regain their initial kinetic energies at the detector, regardless of its absolute potential. 2 Minor decreases in the total TOF were observed only starting from¡20 V. No effect on VMI operation even at low ion optics voltages was observed for the¡5 V detector bias. Nevertheless, the simulations 103 The circuit diagram of the bias voltage supply is given in figure D.2 (appendix D). Be- sides the DC bias voltage, it provides AC grounding for the front of the MCP stack, so that the transient performance (for both signal pickup and gating) is not affected. Detector operation in electrical signal pickup mode requires application of up to »2 kV potential difference across the MCP stack and »200 V bias of the phosphor screen, working as the anode, relative to the MCP output for effective collection of the produced electrons. The signal is usually picked up as anode current. Operating the whole assembly with input at nearly ground potential means that the output in- evitably has an»2 kV potential above ground, and the small useful signal must be separated from this large DC voltage offset. All these demands are handled by a sim- ple divider–decoupler (see figure D.3). It rejects high-frequency noise from the single supplied high voltage, divides the potential between the anode and the MCP stack, and decouples the anode current from the high-voltage supply. The input impedance (50 Ohm) of a fast voltage amplifier (KOTA Microcircuits KE104, 1100 MHz band- width, 5£ voltage gain) connected to the circuit output converts this current to volt- age, and the amplified signal is transmitted to the measurement system. The operation in imaging mode requires application of a much higher voltage (»5 kV) to the phosphor screen for improved conversion efficiency. This voltage is pro- vided by a simple custom-built power supply (see figure D.4). Mass gating in “regular” VMI experiments and slicing time selection in SVMI experiments are performed by application of short (»100...200 ns for mass gating and»5 ns for slicing) high-volt- age pulses to the back of the MCP stack at appropriate times. Two different HV pulse generators are used for this purpose. The mass gating pulse for full-projection VMI is produced by an AV-HVX-1000A framework (see subsection 2.5.4) allows variation of the detector potential, and the actual value¡5 V was used in all simulations mentioned below in this chapter. 104 high-voltage pulser commercially produced by Avtec 1 . The pulser essentially consists of a fast high-voltage MOSFET switch with triggering electronics and connects an external voltage supply to the load for an amount of time specified by the incoming TTL trigger pulse. While the voltage rise time provided by the switch is»10 ns, the pulser has no discharge circuit, and thus the voltage fall time for loads with nonzero capacitance (such as the MCP stack) is much longer. Moreover, the maximum output voltage is strictly limited to 850 V, which is insufficient for operation of the detector. Nevertheless, with a simple additional circuit (see figure D.6) that provides quick discharging of the load and impedance matching between the pulser and the detec- tor (to avoid electrical ringing), and allows addition of a DC bias voltage V bias to the output from another voltage supply, electrical pulses of trapezoidal form shown in figure 2.26 with rise and fall times t rise ¼ t fall ¼ 20 ns and sustained voltages V max up 0 V bias V max t rise t on t fall Outputvoltage Time Figure 2.26 Mass gating pulse shape. to 2 kV for adjustable duration t on from»70 ns to»1 μs can be obtained. A simple custom-built power supply (see figure D.5) installed with the pulser and the com- biner circuit near the detector provides voltage for the fixed 700 V pulse amplitude 2 , 1 Now Directed Energy (DEI). 2 An amplitude somewhat below the maximum allowed 850 V is used in order to protect the MOSFET switch from breakdown by possible voltage overshots during transient processes after the fast switch opening. 105 whereas the bias voltage is provided and controlled by a commercial power supply in- stalled in the common electronics rack. Despite the»1200 V bias constantly supplied to the MCP stack, no detectable background signal is produced outside the active pulse phase thanks to the exponential voltage–gain characteristic of the detector. 1 The slicing pulse for SVMI measurements is provided by a fast high-voltage elec- trical pulser built in the laboratory, under guidance of Dr. András Kuthi from the Pulsed Power Laboratory at USC. A detailed description of the pulser is given in ap- pendix E. The features relevant to SVMI applications include the ability to gate the employed detector with a relatively clean electrical pulse 2 having fixed duration with »5 ns halfwidth and amplitude adjustable in the whole working range of the detector (that is, up to »2 kV). Although the delay between the trigger input and the high- voltage pulse output is»900 ns, the jitter of the output pulse timing is.1 ns, which does not exceed the timing jitter of the employed lasers used for fragment ionization. The effective slice thickness is determined by convolution of the ionization inten- sity time profile, the detector gain time profile, and the relative timing jitter distribu- tion (see p. 14). Although the optical intensity of the ionizing laser pulse also has tem- poral halfwidth of a few nanoseconds, 3 the nonlinearities in the ionization process, as well as in the voltage–gain dependence of the detector, lead to some narrowing of the effective temporal distribution. The characteristics of the resulting effective slicing pulse can be easily determined experimentally by “slicing” the photoelectron distribution produced by the ionizing laser. Due to the very small mass of the elec- trons, their TOF spread, especially if high acceleration fields are used, is also small, 1 The gain increase is approximately two times per each 100 V, thus 700 V difference leads toÈ100-fold gain suppression, lowering the signal intensity well below the event counting threshold. 2 The electrical pulse consists of one narrow high-amplitude peak and some surrounding stray oscilla- tions that have a much lower amplitude and thus do not affect the detector gating. 3 Which means that the»5 ns halfwidth of the detector gating pulse is a good match in regard to the intensity–resolution tradeoff, as mentioned on p. 14. 106 as evident from (2.35). For example, electrons with initial energies K 0 Æ 1 eV created in electric field E 0 Æ 100 V/cm have TOF spread¢¿ K 0 ¼ 0.7 ns, and it can be reduced even more by using stronger electric fields. Therefore, the total photoelectron signal intensity measured as a function of the delay between the ionizing laser trigger and the slicing pulse trigger reproduces the effective slicing pulse profile withÇ1 ns reso- lution. An example of such profile measured during experimental tests of the present system (see next section) using H atom ionization laser is shown in figure 2.27. As 0 −10 −5 0 5 10 Relativeintensity Relativetime,ns Figure 2.27 Time profile of effective slicing pulse. Points represent experimental data. Fitted Gaussian profile is shown by dotted line. can be seen from comparison with the fitted curve, the actual profile has a nearly Gaussian shape. The halfwidth of the fitted Gaussian is 4.24§0.13 ns in this partic- ular case, and this value is typical for all experiments conducted using the present experimental setup. 2.6.3 Image acquisition and analysis The optical image produced on the detector phosphor screen is captured by a regular digital videocamera (PixeLINK PL-B741F) connected to a personal computer. The camera has a 8.58 mm£6.86 mm CMOS sensor with 1280£1024 px resolution, out of which the central 1024£ 1024 px region is used for image acquisition in the 107 present system. Although the quantum efficiency of CMOS sensors is generally lower than that of CCD sensors, they are several times cheaper (for the same physical size) and allow several times faster readout (up to»30 frames per second in the present case), which facilitates synchronous real-time analysis of the data for each laser pulse (10 Hz repetition rate in the current experiments). The objective lens was carefully selected for maximum performance of the image transfer from the detector screen to the camera sensor. The Tamron 23FM25SP lens with 25 mm focal length and f/1.4 aperture was chosen for this purpose. It has extremely low geometrical distortions and guarantees megapixel resolution over the whole field of view, perfectly matching the employed camera sensor characteristics. The minimal object distance of 15 cm results in exact matching of the produced image size with the detector size, maximizing both the resolution and the light collection efficiency. The images captured by the camera for each laser pulse are transferred to the ex- periment-controlling computer for real-time processing. Out of 10-bit intensity reso- lution of the camera only 8 bits are used, as this was found sufficient for analysis and reduces the computational costs 1 . An event-counting algorithm used for determina- tion of particles arrival positions from the raw image data is similar to the approach described in [33]. Each frame, after subtraction of the background (see below), is scanned for con- nected groups of pixels 2 with intensities above a predefined threshold. Each found group is analyzed in the following way. If the number of pixels in the group is below one predefined value, it is considered as a noise fluctuation and ignored. On the other 1 However, with the current implementation the processor load of the employed regular desktop com- puter (AMD Athlon II X2 250, 3 GHz processor) usually does not exceed»30 %. 2 4- and 8-connectivity were tested, and 4-connectivity was found to perform better, in particular with respect to separation of close-lying spots and rejection of near-threshold noise. 108 hand, if the number exceeds another predefined value, the spot is treated as a clus- ter of multiple close-lying particle hits and is written to the output file as raw data for further analysis. 1 If the group passes these preliminary tests, it is regarded as a candidate for a single-hit spot, and additional statistical parameters are calculated from the pixel values. The most important parameters are the hit coordinates (X,Y ) determined as weighted averages of pixel coordinates (X i ,Y i ): XÆ P I i X i P I i , YÆ P I i Y i P I i , (2.86) where I i are the pixel intensities, and the sums are taken over all pixels in the group. Besides that, the geometrical form of the spot is estimated from the standard devia- tions ¾ X Æ s N N¡1 P I i (X i ¡ X) 2 P I i , ¾ Y Æ s N N¡1 P I i (Y i ¡Y ) 2 P I i (2.87) (where N is the number of pixels in the group) of the intensity distribution, and if they do not fall into a predefined interval, the group is again treated in the same way as in the initial coarse size filtering. For all groups that passed these tests the determined parameters (plus the max- imal and total intensities within each group) are recorded to the output file grouped by frames. The frame sequence number and time of acquisition are also recorded. While in the simplest case further processing of the data in order to obtained a ras- terized velocity map image uses only the hit coordinates of all spots, the additional 1 For example, more complicated multipeak fitting algorithms can be used to extract the positions of each hit within the cluster. In the present work, however, the number of such clusters was much smaller than the number of single hits, and therefore it was decided to simply ignore them. This might be not the best solution, since rejection of multihit spots, in principle, leads to nonlinear in- tensity response of the system, decreasing the measured intensity of the most intense image parts (where the multihit probability is higher). On the other hand, another simple treatment, where a single hit is assigned to the cluster center, is not much better from the intensity perspective, but ad- ditionally introduces spurious hit positions (which were not present in the actual distribution) into the measurement results. 109 information turns out to be very useful for more advanced filtering and analysis of the data, including diagnostic and optimization purposes. Each acquired frame is also displayed on the computer screen in real time, with the analyzed spots high- lighted by different colors according to their types, facilitating initial experimental preparations and providing convenient visual monitoring during data collection. The background subtraction mentioned above is implemented with a fully au- tomatic adaptive method. It is based on background estimation using a pixelwise average of a certain number 1 of previous frames. The average, however, is not un- conditional: from each frame only the pixels that were not identified as belonging to particle hit spots are taken, but all detected spots are excluded from the background estimation. This approach prevents estimated background intensity buildup leading to desensitization in the high-intensity areas of the image, and at the same time completely eliminates any fixed-pattern background, as well as all background con- tributions slowly changing in time. From a practical perspective, this allows reliable event counting operation even in the presence of ambient light. 2 2.7 Experimental tests The newly built system required experimental characterization before it could be used in actual scientific experiments. The test program carried out for that pur- pose included: experimental validation of the numerical model (see subsection 2.5.4), studies of the reliability of the optimal parameters obtained in the simulations (see subsection 2.5.4), development of experimental procedures, and calibration of the ve- 1 8 was found to be a good choice, at least for the present system. 2 For example, regular room lighting in the laboratory has no obvious detrimental effect on the present setup. However, a few frames immediately after abrupt changes in the lighting conditions might be spoiled, and therefore optical shielding of the detector–camera pair from external light sources is still preferable. 110 locity mapping for further measurements. The experimental tests were performed with ions created by well-studied pro- cesses and thus having well-known properties. For most tests the multiphoton dis- sociation of O 2 molecule at ºÆ 44444 cm ¡1 (¸Æ 225 nm) eventually producing O + ions was used. This reaction has been widely employed for characterization of VMI systems since their invention [7] and has several appealing properties. First, the initial step involves a resonant 2-photon absorption [34], which allows selection of a single rotational level as the initial state of the molecule by proper tuning of the excitation laser. Second, the following steps involve either dissociation of the excited neutral molecule into atomic fragments created in a few different electronic (spin– orbit) states and subsequent photoionization of the produced excited atoms, or pho- toionization of the molecule with formation of O + 2 ion in several vibrational states 1 and its subsequent photodissociation. In both cases the internal energy of the parent molecule (or molecular ion) immediately before the photodissociation step is well- defined, and the possible internal energies of the fragments (O ¤ or O + ) are known precisely. Therefore, the kinetic energy distribution of the dissociation products con- sists of several sharp peaks, 2 positions of which are determined by the excitation photon energy and the energy conservation law and thus can be accurately predict- ed. 3 These peaks have different intensities and anisotropies and span a KE range from»0.1 to»1.65 eV, presenting a very convenient distribution for characterization of VMI setup capabilities. Although the creation of O + from O 2 requires a total of 4 1 The rotational state associated with nuclear motion is not strictly preserved upon ionization, but the probability of changing it by more than a few quanta is very low. 2 The velocity change due to electron recoil upon ionization is quite small because of the large ion/elec- tron mass ratio and does not exceed the initial velocity spread in the molecular beam. 3 The dissociation energy of O 2 involved in the energy balance equation has been experimentally determined with accuracy exceeding the VMI resolving capabilities (namely, D 0 (O 2 ¡ ! 2O( 3 P 2 ))Æ 41268.6§1.1 cm ¡1 , according to the analysis reported in [35]). Other energies are known with even better “spectroscopic” accuracy. 111 photons (in either case), the overall efficiency is high enough, so that laser pulse en- ergies well below 1 mJ (focused) and only a few percent of O 2 in the molecular beam 1 are sufficient for the experiments. Moreover, only one laser is required, eliminating mutual alignment and synchronization issues. In addition to all these favorable prop- erties the process has a useful side effect: some of the created “intermediate” O + 2 ions do not dissociate. They retain essentially zero velocity and also present a convenient tool for other types of studies (for example, TOF properties and spatial effects). Testing of performance for H fragments — the main object of the intended ap- plications — was more difficult. Unfortunately, no convenient sources of them were found. Among all stable volatile diatomics containing H, only HBr was suitable from the experimental point of view. On the one hand, it has favorable properties: pho- todissociation in a broad range of photon energies, including ºÆ 44444 cm ¡1 (facili- tating switching from experiments with O 2 ), and formation of fragments in only two possible states, H( 1 S 1/2 )Å Br( 2 P ± 3/2 ) and H( 1 S 1/2 )Å Br ¤ ( 2 P ± 1/2 ), with well known en- ergies 2 . On the other hand, the resulting KE distribution with only two peaks is too sparse for complete characterization of VMI. Photodissociation of HBr is a 1-photon process with excitation directly to a repulsive electronic state. Therefore, no selectivi- ty of initial states is possible. Nevertheless, the characteristic rotational temperature in the employed molecular beam is of the order of 10...20 K, meaning that the initial internal energy distribution is already sufficiently narrow. 3 Dissociation of polyatomic molecules usually produces H fragments with broad kinetic energy distributions due to the high density of states available in the molec- ular cofragment and the absence of strong selectivity among them. The resulting 1 Even ordinary atmospheric air at somewhat higher concentration can be used instead of pure oxygen. 2 HBr dissociation energy is 30210§40 cm ¡1 [36] for the HÅBr channel and 3685 cm ¡1 higher [37] for the HÅBr ¤ channel. 3 The first vibrational level of HBr is»2600 cm ¡1 above the ground state and should not be populated even at room temperature. 112 kinetic energy spectrum is therefore too dense to be resolved by VMI instruments. A relatively sharp distribution is possible only when the cofragment is a diatomic with broad vibrational and rotational structures. That is, triatomic volatile hydrides might be suitable sources of well-structured H fragments distributions. The most common of them, H 2 O, could be a good choice. As shown in [38], it is photodissoci- ated by Ly-® radiation (ºÆ 82259 cm, ¸Æ 121.57 nm) — the same radiation that is used in detection of H fragments (by REMPI through the 2p state), allowing experi- ments without additional light sources — and produces a distribution with rich and sharp structure in the 0...»3.1 eV KE range. 1 However, the overall process of creat- ing H + ions from H 2 O requires two Ly-® photons. Unfortunately, the intensity of the VUV radiation produced by the employed tripling cell (see [13] and also [39] for more details), being acceptable for H detection purposes, was found too low for obtaining a sufficient signal level in such experiments. Attempts to perform multiphoton dis- sociation of H 2 O with a separate laser also did not lead to a ready success and were abandoned. On the other hand, photodissociation of H 2 S in the same UV range as in O 2 and HBr experiments is quite efficient and was successfully used in this work. It produces H(1s) atom and HS radical in two spin-orbit states X 2 ¦ 3/2,1/2 separated by only»380 cm ¡1 (»0.05 eV) 2 and with some rotational excitation. 3 This situation is somewhat closer to the expected applications of the system (photodissociation of small polyatomic organic radicals) and allows to examine the SVMI performance for close-lying peaks broadened by rotational envelopes. 1 The distribution actually extend to the limiting KE value hº¡ D 0 ¼ 5.1 eV, but its intensity above »3.1 eV drops significantly. 2 The more accurate value 376.8 cm ¡1 and the H 2 S dissociation energy 31451§4 cm ¡1 are summarized in [40]. 3 Experiments at higher excitation energy »52000 cm ¡1 (193 nm from ArF excimer laser) indicate that excitation of a few vibrational levels in HS (! e » 2700 cm ¡1 ) is also possible (see, for example, [41]). However, it was not observed in the present work, perhaps due to a difference in dissociation dynamics at different excitation energies. 113 2.7.1 Using O + and O + 2 from O 2 The tests started from obtaining O + 2 and O + signals in mass-spectrometer mode. After the experimental condition “external” to the VMI system were established, the detection was switched to imaging mode, and the accelerator configuration and volt- ages were set according to one of the suitable simulations results. 1 A preliminary velocity-map image was observed similar in appearance to the expected one, indicat- ing that the system is functioning correctly at the basic level. The first VMI test was the validation of the simulated parameters required to satisfy the focusing conditions. The radial and TOF focusing parts were tested sep- arately, but using very similar approaches. As discussed in subsections 2.4.1 and 2.4.2, the radial focusing deals mostly with the initial position uncertainty in the ra- dial direction (±x and±y), whereas the TOF focusing — with displacements along the ion-optical axis (±z). The intersection of the focused laser beam with the molecular beam creates an ionization region narrow in the y and z directions but elongated in the x direction, as depicted in the left part of figure 2.17. While the¢x elongation (or position along the x axis) cannot be easily varied in the experiment, the position of the ionization region along the y and z axes can be changed simply by parallel shifting of the laser beam before in enters into the vacuum chamber. In practice this shift was achieved by placing a 3.2-mm thick fused silica plano-parallel plate 2 after the laser beam focusing lens, as shown in figure 2.28. Rotation of the plate around its horizontal axis displaces the transmitted beam, and therefore the ionization region, vertically without significantly affecting other parameters. The displacement±y can 1 Since the tests were organized in a stepwise manner, to make sure that simple components work as expected before studying the behavior of more complicated arrangements, the initial operation did not involve the additional lens (that is, its central element was grounded). 2 A regular 1/8-inch thick spare vacuum chamber window was used. 114 laserbeam focusinglens vacuumwindow molecularbeam δy plano-parallelplate α d Figure 2.28 Diagram of using rotating plano-parallel plate for changing ionization region position in focusing tests. (Drawing not to scale.) be controlled quickly and to a good precision by adjusting the rotation angle®: ±yÆ sin(®¡¯) cos¯ d, where ¯Æ arcsin sin® n , (2.88) d is the thickness of the plate and n is the refractive index of its material (n¼ 1.52 at ¸Æ 225 nm). For example, rotation by 5° shifts the beam by»0.1 mm, and the shift at 60° is»1.7 mm. If the focusing conditions are met, the image produced at the detector should not depend on the initial position of the particles. Conversely, if the conditions deviate from the proper ones, then the images taken at the laser beam displaced upward and downward are shifted with respect to each other in the vertical direction. Hence studying the dependence of this relative shift on the ion optics parameters allows to find the experimental values at which the shift turns to zero, and therefore the focusing is achieved. This procedure was performed for several combination of the accelerator lengths using fixed V 0 and varying V 1 , first without the additional lens (V L Æ 0) and then at a few different V L values. Likewise, the TOF focusing was tested by displacing the laser beam in the hori- zontal direction (accordingly, the plano-parallel plate was rotated around its vertical 115 axis for that purpose) and observing the temporal shifts in the TOF detection mode. The O + 2 signal was very convenient in these studies, since it produces a single narrow peak, whereas the time resolution of the system 1 did not allow to distinguish suffi- cient structure in the O + signal. Otherwise, the experiments were performed similar to the radial focusing tests and for the same {L 0 ,L 1 } and V L combinations. In all cases the agreement between the experimentally found V 1 values and the values obtained by numerical optimizations (see section 2.5) was excellent, with the deviations being well below 1 %. The central TOF values were also in.1 % agree- ment with the simulations. Accurate TOF stretching measurements were not possi- ble due to the limited time resolution, but no deviations from the predictions were observed either. Therefore, it was concluded that the simulations yield reliable opti- mal conditions for both radial and TOF focusing in all modes of operation and also satisfactorily predict the TOF properties. A few velocity map images of O + were taken at selected ion optics parameters to test the actual operation in VMI and SVMI modes. One of the settings tested the operation without the additional lens, and the other — with it. Due to the “large” mass of the O + ion (in comparison to that of H + ) the TOF stretching was sufficient in both cases, so that the »5 ns slice was thin (»6 %) in one case and even thinner (.4 %) in the other. “Regular” full-projection velocity map images were also taken at the same conditions for comparison. The resulting raw images are presented in figure 2.29. The data were collected for »100000 laser shots (»3 hours) for each full-projection image and for»70000 laser shots (»2 hours) for their sliced counterparts. It should be noted that the experimen- tal conditions (molecular beam parameters and the laser power) were not exactly the 1 Several nanoseconds, determined mostly by the laser pulse duration and to a smaller extent by the electrical detection bandwidth. 116 L 0 Æ 50 mm, L 1 Æ 50 mm, V 0 Æ 3000 V, V 1 Æ 2117 V, V L Æ 0 (maxK¼ 1.7 eV, max¢¿¼ 82 ns): full projection (»1000000 events) central slice (»96000 events) L 0 Æ 35 mm, L 1 Æ 60 mm, V 0 Æ 1500 V, V 1 Æ 1130 V, V L Æ¡2400 V (maxK¼ 1.7 eV, max¢¿¼ 131 ns): full projection (»630000 events) central slice (»22000 events) Figure 2.29 Velocity map images of O + from O 2 photodissociation at º Æ 44444 cm ¡1 taken in two different ion optics regimes. (Raw event-counting data ren- dered at the native 1024£1024 px resolution is shown.) 117 same for each of the experiments and even varied slightly during the measurements. Therefore, the integrated intensities of the images should not be compared quanti- tatively. Moreover, due to the multiphoton nature of the involved processes and com- petition among different channels leading to O + products the intensity distributions within each image are very sensitive to the used laser power and thus differ between the experiments. 1 Nevertheless, the overall structure of the kinetic energy spectrum does not depend on these variations and can be compared. Figure 2.30 shows the kinetic energy distributions obtained from the sliced im- 0 1 0.0 0.5 1.0 1.5 Relativeintensity K,eV Withoutlens Withlens 3 P 2,1,0 +3s 5 S ◦ 2 3 P 2,1,0 +3s 3 S ◦ 1 3 P 2,1,0 +3p 5 P J 3 P 2,1,0 +3p 3 P J 1 D 2 +3p 5 S ◦ 2 O+O + 4hνlimit Figure 2.30 Kinetic energy distributions of O + extracted from sliced images shown in figure 2.29. Peaks due to 3-photon dissociation of neutral O 2 are marked according to atomic states of the two O fragments, other peaks correspond to O + 2 dissociation. ages. The raw images were rectified as described in chapter 3, and the intensity 1 In addition, the images in figure 2.29 are shown with enhanced contrast for better demonstration of the structured features. That is, their visual intensities should not be relied upon. 118 distributions were integrated over the polar angles (with the appropriate weighting factor) to obtain radial distributions. The results were converted to KE distributions using videocamera calibration (pixels7! physical dimensions) and SIMION simula- tions (radii 7! kinetic energies). As can be seen from the figure, in both cases this procedure yielded KEs in good agreement with the predicted peak positions. Since the fine structure 1 cannot be fully resolved, it is difficult to make a precise conclu- sion about the discrepancies, but they definitely do not exceed the observed peak halfwidths 2 and can be estimated to lie within»1 %. However, the raw images had radial distortions»1.3 % for operation without the lens and»1.9 % — with the lens, and the choice of the reference direction that must be made in the rectification pro- cess could lead to that much uncertainty in the speed transformation (and accord- ingly, up to »3.5 % in the KE scale). In both studied cases the horizontal direction was found to give results consistent with the expectations and between each other, whereas the mentioned distortions corresponded mainly to various degrees of verti- cal shrinking of the images. The causes of this shrinking, which apparently depends on the ion optics parameters, remained unknown. The two distributions plotted in figure 2.30 can serve as an illustration to the discussion on p. 90 about the relative slicing thickness and its effect of the resulting KE resolution. Comparison of the low-KE parts of the plots clearly shows that even for the peaks near K» 0.1maxK, having relative slice thickness 5 ns/ ¡p 0.1¢82 ns ¢ ¼ 1 The ground electronic term 2s 2 2p 4 3 P J of O atom is split into three sublevels with energies 0, 20, and 28 meV (0, 158 and 227 cm ¡1 ) for JÆ 2, 1 and 0 respectively [37]. The splitting for each of the excited terms 2s 2 2p 3 ( 4 S ± )3p 5 P J and ...3p 3 P J does not exceed§3 cm ¡1 and can be neglected. All other participating terms are not split. 2 One of the broadening mechanisms in addition to VMI focusing imperfections is the initial velocity distribution in the molecular beam. The assessment of »50 m/s given on p. 13 and found not very important for H fragments becomes more critical for the heavier O fragments, constituting »1.4 % of the speed for O with K¼ 1.13 eV and »2.5 % for K¼ 0.33 eV (»2000 m/s). That is, this effect is likely to limit the final KE resolution regardless of the ion optics performance. In addition, the peaks corresponding to the 3s states are apparently overlapped by peaks from O + 2 dissociation. 119 19 % in the one case and 5 ns/ ¡p 0.1¢ 131 ns ¢ ¼ 12 % in the other, the “wing” hard- ly stands out against the overall resolution and is not significantly affected by the appreciable (»1.6 times) variation of the relative slicing thickness. However, this situation holds as long as the slice thickness for the whole distribution remains rea- sonable. For example, at»30 % slicing of the whole KE range the central part of the “sliced” image would effectively look like that of the full-projection image (left side of figure 2.29) with the sharp structure completely washed out. On the other hand, the signal-to-noise ratio drops for thinner slicing, and from that point of view the»6 % slice is definitely better than the.4 % one, which is also evident from figures 2.29 and 2.30. 1 2.7.2 Using H + from HBr and H 2 S The slicing performance for H + ions was tested mostly with HBr photodissocia- tion. Although the same laser as in the O 2 experiments (ºÆ 44444 cm ¡1 ) was used for HBr excitation, the H fragments cannot be ionized by this radiation, and an addi- tional separate “H-detection” arrangement was employed. 2 It consists of a laser pro- ducing tunable radiation nearºÆ 27420 cm ¡1 (¸ vac Æ 364.7 mn) and a Kr/Ar tripling cell attached to the vacuum chamber for production of radiation near the Ly-® line (2sÃ 1s transition of H atom, ºÆ 82259 cm ¡1 ). This frequency-tripled VUV radia- tion passes to the vacuum chamber through a MgF 2 lens 3 and excites the H atoms to the 2p state. The unconverted part of the fundamental UV radiation then ionizes the 1 Strictly speaking, the intensity of the image taken with the additional lens is lower than what should be expected from the TOF stretching considerations. This is partly due to the different experimental conditions, as mentioned above, and partly because of lower detection efficiency for ions accelerated by the lower electric potential. Nevertheless, the point is that reduction of the signal level without noticeable improvement of the resolution makes identification of the signal structure more difficult, and this is well illustrated by the figures. 2 This arrangements is in fact used for H fragment ionization in all actual experiments and thus was incorporated in the original apparatus. Its description can be found in [13]. 3 Also serving as a window separating the gas-filled cell from the vacuum chamber. 120 excited atoms: H(2p) hº ¡ ¡ ! H + Å e – . Since the energy difference between H + and H(2p) is exactly 1/3 of the difference between the 2p and the ground 1s states of H, the ion- ization produces no recoil, and the formed H + ions inherit the velocity distribution of the H fragments without additional broadening. 1 Unlike the O 2 experiments, where O + was produced by O + 2 dissociation or nonresonant ionization, the 1Å1 0 REMPI of H is possible only in a relatively narrow frequency range. Moreover, the small mass of H atoms means that their velocity distribution has a large v x spread (along the laser beam direction), and the resulting Doppler shifts of the resonant frequency exceed the bandwidth of the employed laser even for K¿ 1 eV. That is, frequency scanning of the “detection” laser over the whole Doppler profile is necessary for recording of the complete velocity distribution. Therefore, all H + images in this subsection were taken with the detection laser continuously sweeping the required range multiple times during the imaging data acquisition (to reduce effects from possible drifts of the experimental conditions). Figure 2.31 shows the results of two SVMI experiments, in which the same veloc- ity distribution was recording using different ion optics modes. The first one involved operation without the additional lens, thus resembling the usual Eppink–Parker ar- rangement. The accelerator voltages were chosen to provide sufficient kinetic energy range, but the resulting TOF stretching of only 19 ns was too small for satisfactory 1 In contrast, the broadening due to electron recoil is an important issue in another widely used H ionization scheme by 2Å1 REMPI atºÆ 41130 cm ¡1 , where the excess energy of 13710 cm ¡1 (1.7 eV) must go into translational energy. Although the electron carries most of this energy, the H + velocity is changed by»420 m/s, which is well above the resolution of VMI systems and thus either becomes a limiting factor or requires deconvolution of the measured distributions. It should be noted that 3Å1 REMPI using the sameºÆ 27420 cm ¡1 radiation without tripling is also possible (see, for example, [42]) but is a much less efficient process. For instance, without tripling no H + signal could be observed in the present setup at the usual UV laser pulse energies (»2...3 mJ). Production of pulses with significantly higher energy is associated with more experimental difficul- ties on the one hand and might have undesired consequences, since many studied organic radicals have 1- or 2-photon absorption in this UV region, on the other. 121 L 0 Æ 50 mm, L 1 Æ 50 mm, L 0 Æ 30 mm, L 1 Æ 70 mm, V 0 Æ 3500 V, V 1 Æ 2468 V, V L Æ 0 V 0 Æ 1000 V, V 1 Æ 781 V, V L Æ¡2200 V (maxK¼ 2 eV, max¢¿¼ 19 ns): (maxK¼ 1.9 eV, max¢¿¼ 50 ns): (»130000 events) (»39000 events) 0 1 1.0 1.2 1.4 1.6 1.8 Relativeintensity K,eV Withoutlens Withlens (Fullprojection) 2 P ◦ 3/2 2 P ◦ 1/2 Figure 2.31 Sliced velocity map images of H + from HBr photodissociation at ºÆ 44444 cm ¡1 taken at two different ion optics regimes. The plot shows kinetic energy distributions extracted from the images (peaks are marked according to the atomic states of the Br cofragment). 122 slicing with the»5 ns pulse. As can be seen from the recorded image and the extract- ed kinetic energy distribution, the resulting peaks have rather long wings on the low- er-KE side, although the extent of the wings is still limited compared to the full-pro- jection case. 1 The second mode involved operation with the additional lens, and the parameters were chosen to attain the TOF stretching of 50 ns for approximately the same KE range. The resulting image and the distribution extracted from it evidently have much more localized peaks, with very small lower-KE wings. The halfwidths of the peaks are»20 meV, that is»1.2 % of the KE range, in good agreement with the worst-case¢KÆ 1.45 % predicted by simulations. Unlike the O 2 images, rectification using the vertical reference direction (as plotted in figure 2.31) gave a better match of the absolute energies, although the discrepancies did not exceed »1.5 % in KE with either direction for all H + images. Moreover, even the images (namely, the slice “without lens” and the “full projection”) taken at supposedly the same experimental conditions show a small difference in the peak positions. This observation suggests that the discrepancies between the simulations and the experimental results come from small unaccounted variations of the experimental parameters, such as the volt- ages applied to the ion optics and the exact position of the ionization region. Another SVMI test was carried out with H fragments from H 2 S photodissociation atº¼ 42560 cm ¡1 (¸¼ 235 mn) 2 and using the same ion optics parameters as in the HBr experiment with the additional lens. As mentioned above (p. 113), the velocity 1 The “kinetic energy distribution” plotted for the full projection is not a physically meaningful prop- erty. Instead, it is simply the recorded full-projection intensity distribution converted to the KE scale using the procedure that is valid only for distributions recorded by sufficiently thin slicing. The pur- pose of showing it with the sliced data is to illustrate the extreme case in the dependence of peak shapes on the slice thickness. 2 This experiment was performed in process of preparations for the experiments described in chap- ter 4. The laser wavelength was changed from the 225 nm used in the O 2 and HBr experiments, but not yet calibrated. Hence the “approximately equal” notation is used as a reminder. The lack of calibration prevents the quantitative comparison with the simulations, but the assessment based on the estimated uncertainty agrees well with results for all other cases. 123 distribution produced in this process has close-lying peaks with extended rotational envelopes, resembling the distributions in the anticipated actual experiments. The experimental results are presented in figure 2.32 and show that the two peaks with 1.0 1.1 1.2 1.3 1.4 1.5 Relativeintensity K,eV 2 Π 1/2 2 Π 3/2 Figure 2.32 Sliced velocity map image and extracted kinetic energy distribution of H + from H 2 S photodissociation atº¼ 42560 cm ¡1 . HS cofragment electronic states are indicated in the plot. (Dotted line shows distribution from the HBr experiment for comparison.) »47 meV (»2.5 % of maxK) separation are indeed well resolved by the system. At the same time, the noticeable wings are caused by the rotational excitation of the SH cofragment. They are much longer than the wings due to the finite slice thick- ness (compare to the distribution obtained for monochromatic particles in the HBr experiment) and completely mask the effect of the latter. Finally, figure 2.33 shows an example of operation in an extreme regime with maxK Æ 6.55 eV and ¢¿ maxK Æ 58 ns (L 0 Æ 85 mm, L 1 Æ 30 mm, V 0 Æ 1900 V, V 1 Æ 886 V, V L Æ¡5000 V) 1 . In this image the H fragments with K» 6 eV produced 1 These characteristics correspond to MÆ 58 ns¢ p 6.55 eV» 150 ns p eV/Da, which lies beyond the range plotted in figures 2.20, 2.22 and B.1–B.6. The parameters were selected to satisfy both radial 124 0 1 2 3 4 5 6 Normalizedintensities K,eV Horizontal Vertical 2 P ◦ 3/2 2 P ◦ 1/2 2 P ◦ 3/2 2 P ◦ 1/2 Figure 2.33 Sliced velocity map image of H + from HBr photodissociation at ºÆ 44444 cm ¡1 and ºÆ 82260 cm ¡1 (near Ly-®). Plotted kinetic energy distributions were extracted separately from the horizontal and vertical parts of the image (within 45° sectors), and background was subtracted. by HBr photodissociation by Ly-® radiation from the H-detection laser can be seen in addition to the KÇ 2 eV fragments. In contrast to the full inner rings corresponding to dissociation by the ºÆ 44444 cm ¡1 laser, only a small part of the outer ring is visible. The cause of this deficiency is that the efficient production of the VUV radia- tion requires phase-matching conditions in the tripling cell, which are satisfied only in a limited frequency range. While the H REMPI needs only one VUV photon, the production and detection of the»6 eV H fragments requires two VUV photons and is therefore much more sensitive to the VUV intensity. Hence, even though the tripling efficiency is sufficient for H detection over the Doppler profile, 1 the intensity outside and TOF focusing conditions (that is, the solution with minimal ¢t was chosen from the Pareto set) and should lead to ¢K Æ 3.3 %, ¢tÆ 0.65 ns (»1.1 %) and ¢t 0 Æ 4 ns (»7 %) according to the simulations. While this combination is not optimal from the SVMI perspective, it is in line with the conditions used for figure 2.31, which were also chosen to satisfy the TOF focusing, and provides an example of another kind of L 0 /L 1 combination (namely, “long/short” in addition to “medium/medium” and “short/long”). 1 The resonant 2pÃ 1s transition of H atom is very strong and was probably almost saturated in this 125 the phase-matching resonance cannot produce a detectable photodissociation signal. At the same time, the fragments produced at the peak efficiency can be ionized only if their v x velocity component does not lead to Doppler detuning. As a consequence, only a small fraction of the high-KE fragments was observed in the experiment. The high-KE fragments correlated with Br in the ground state were not detected at all, either due to the Br ¤ /Br branching ratio significantly shifted towards Br ¤ or strong perpendicular anisotropy of the Br( 2 P 3/2 ) channel at this high photon energy. 1 Nevertheless, the kinetic energy distribution extracted from the available part of the image shows that the simulations are again in good agreement with the experi- ment, reproducing KEs with»1.8 % accuracy. The inner rings also vividly illustrate the anisotropic (horizontal) broadening due to imperfect radial focusing and the ver- tically thin but horizontally elongated ionization region. For a more qualitative ex- amination of this effect the kinetic energy distribution was calculated based only on the vertical (using the (0§ 22.5)° and (180§ 22.5)° sectors) and only on the horizon- tal (using the (90§ 22.5)° and (270§ 22.5)° sectors) 2 parts of the image, and the re- sults are shown for compared in figure 2.33. The “horizontal” distribution has peaks broadened by »0.24 eV, which is in good agreement with the predicted worst-case resolution (0.22 eV), whereas the “vertical” distribution has much narrower peaks. Although the horizontal part of the outer rings is not present in the recorded image, their horizontal broadening is expected to be comparable to the vertical one, since the ion optics parameters were optimized mostly for focusing of the high-KE part of experiment. 1 Unfortunately, no experimental data on HBr dissociation at these high photon energies was found in the literature. The published theoretical investigations are also not conclusive and usually do not go beyond »75000 cm ¡1 , since the absorption cross-section drops quickly at high energies. Never- theless, the available information allows to expect that the Br( 2 P 3/2 ) channel indeed has a strong perpendicular anisotropy over the whole energy range, and the Br ¤ /Br branching ratio, according to [43], might be quite large. 2 Polar angles are measured from the positive direction of the y axis, see figure 2.1. 126 the distribution, as explained in subsection 2.5.3. A striking difference of the image shown in figure 2.33 from all other images pre- sented in this section is the presence of a nearly uniform background over the whole detector area. This background was observed even when no ions were generated by the lasers, and its source was traced to the high negative voltage applied to the ad- ditional lens. The study of this effect revealed that the background starts appearing at V L »¡2500 V, and its intensity quickly increases with the lens voltage, 1 however, no background is observed at any positive voltages. The strong dependence of the intensity on the voltage and independence on other experimental conditions suggest that the background is due to electrons created at the central element of the addi- tional lens by field emission and/or field ionization of the residual gas in the vacuum chamber and accelerated by the large potential difference between the electrode and the detector. This unfortunate circumstance means that in actual experiments, un- less the signal is strong and structured enough to be confidently separable from the background, the working lens voltage is limited by ¡2500...¡3000 V, and thus the achievable maximum imaged KE range also has an upper limit unrelated to the ion optics aberrations. 2.7.3 Position mapping mode The dependence of the arrival positions of the particles on their initial positions is very undesirable in VMI operation, since it decreases the achievable velocity reso- lution for spatially extended ionization regions. However, this dependence can be put to good use, namely, for imaging (now in the optical sense) of the ionization region 1 The intensity at V L Æ¡5000 V was so high that it overwhelmed all useful signals in the “regular” VMI mode with»100 ns mass-gate detection pulse and saturated the MCP detector in the TOF mass- spectrometer mode. 127 shape and position. The experimental diagnostics of these characteristics might be useful for optimization of the overlap between the laser and the molecular beam, and among different lasers in multi-laser experiments, as well as for alignment of the molecular beam relative to the ion optics. This kind of imaging is similar to electron microscopy and requires operation in “position mapping mode” with initial positions mapped to arrival positions, desirably, regardless of the initial velocities. These requirements are in some sense opposite to the VMI requirements and therefore lead to significantly different ion optics designs and operating conditions. Electron microscopes, for example, use much more compact optics (at a few-millimeter scale) near the object and much higher voltages (a few tens of kilovolts). Nevertheless, the first electrostatic electron microscope (see figure 6 in [19]) had a simple lens system consisting of two apertures, very similar to the accelerator of the present VMI setup, and used moderate voltages, but still offered resolution at the micrometer scale. Therefore, there is a hope that even the existing VMI accelerator, designed without the microscopy applications in mind, might be configured for a reasonable performance in position mapping. SIMION simulations and experimental tests showed that although for particles with K» 1 eV satisfactory velocity-independent focusing cannot be achieved at prac- tically allowable accelerator voltages, the spread due to initial velocities can be re- duced significantly and made a few times smaller than the size of the ionization region image. That is, the spatial parameters can be measured at least qualitatively. At the same time, the blurring for ions formed without appreciable kinetic energy 1 is negligible, which allows qualitative measurements even with the achievable per- formance. Among the studied configurations suitable for position mapping, one is 1 Such as O + 2 in the O 2 experiment, or ionization of H atoms present in the molecular beam or as a background in the vacuum chamber. 128 particularly convenient from a practical point of view: L 0 Æ 30 mm, L 1 Æ 60 mm, V 0 Æ V 1 , V L Æ 0. The V 0 Æ V 1 condition can be easily achieved by disconnecting the V 0 voltage supply from the voltage divider. The position mapping does not depend on the absolute value of the single applied voltage, but the blurring decreases and the detection efficiency increases at higher acceleration voltages, so it is advisable to apply the highest allowable voltage to the V 1 input. Figure 2.34 shows an example of operation in the position mapping mode with V 0 Æ V 1 Æ 2000 V for O + 2 ions produced by photoionization of O 2 (see subsection 2.7.1). The images were taken with the laser beam displaced vertically by »1.7 mm from the central position (using the arrangement shown in figure 2.28) and its focal point moved by»2 mm along the beam direction (by moving the focusing lens). Figure 2.35 shows results obtained in the same way, but for O + ions. It is evident that the O + 2 images provide very sharp pictures of the ionization region, 1 although even for the O + ions with K» 0.2...0.4 eV (the most intense part) the KE blurring does not mask the initial position information. The molecular beam shape at z¼ 0 can be measured by vertical scanning of the ionizing laser beam (or using a sufficiently wide beam in case of 1-photon ionization), as demonstrated in figure 2.36. This image was obtained by monitoring O + 2 ions and shifting the laser beam vertically by rotating the plano-parallel plate (see figure 2.28) in»5° steps. The nonlinearity in (2.88) led to nonuniform sampling in terms of ver- 1 The blurring seen near the focus of the laser beam is probably caused by space charge effects. Since the electrons formed in the ionization process have enough kinetic energy to leave the ionization vol- ume and do so quickly due to their small mass, the remaining system of N ions acquires electrostatic energy N P iÆ1, jÈi q 2 r i j , where q is the electric charge, and r i j are the initial distances between the ions. This energy is eventually converted to additional kinetic energies of the ions, resulting in»Nq 2 /hri KE change per ion, wherehri is the average inter-ion distance (comparable to the laser beam diameter). The effect must be stronger near the laser beam focus, where the ionization region radius is smaller, and the produced ion density is larger (due to higher energy density and multiphoton nature of the ionization), which agrees with the observed behavior. 129 Focusing farther centered closer higher Vertical position centered lower Figure 2.34 Images of the ionization region at various positions of the laser beam focus relative to the molecular beam recorded using O + 2 ions. tical positions, but the displacements can be easily calculated, and the fitting of the measured positions to the calculated beam displacements yields 182 px/mm (that is, 5.5 μm/px) scaling factor for the native 1024£1024 px imaging resolution. Therefore, the maximum molecular beam diameter determined from this image is »3.8 mm, which indicates that the3 mm orifice in the terminating plate of the accelerator 130 Focusing farther centered closer higher Vertical position centered lower Figure 2.35 Images of the ionization region at various positions of the laser beam focus relative to the molecular beam recorded using O + ions. (see figure 2.14 and p. 97) was acting as the limiting factor in this experiment. 1 The 5.5 μm/px imaging sensitivity and the ability to determine the positions with »1 px resolution (even in presence of blurring, by finding the center of the produced 1 This result could be expected from simple geometrical considerations: the1 mm skimmer located at »1 cm from the nozzle outlet provides only very rough collimation. A later visual inspection of the system also showed a spot a few millimeters in diameter left by the molecular beam on the plate around the orifice. 131 Figure 2.36 “Raster” image of the molecular beam (using O + 2 ions) obtained by vertical scanning of the laser beam. (Compare to figure 4a in [7].) symmetric intensity distribution) mean that the laser beams can be aligned with pre- cision comparable to their focused diameters. The limitation of the method is that the laser beam being aligned must produce at least some ions. It is desirable that ions of the same mass, or similar masses, 1 are used for alignment of all beams, since in presence of stray magnetic fields trajectories of ions of different masses are deflected differently. Nevertheless, this deflection is relatively small, 2 and the obtained accu- racy is usually better than the accuracy achievable by regular measurements outside the vacuum system (especially with the attached tripling cell). Although the 2-dimensional images provide information only about the distribu- tion along the x and y axes, the information related to the z axis displacements can be obtained from the TOF measurements, since the TOF focusing is also absent in 1 They are not required to be the ions on which the VMI measurements will be done. 2 Trajectories of H + ions accelerated by 2 kV have curvature radius r g Æ mv/qB» 130 m due to the Earth’s magnetic field in the worst case. The deflection at the detector after 0.6 m flight path there- fore should not exceed»1.5 mm, or»35 px. Heavier particles are affected even less. 132 this mode of operation, and thus the overall TOF depends on the initial positions of the ions. These measurements, however, are much less precise due to the inherently limited time resolution of the system. 1 It should be noted that the possible applications of the “position mapping” dis- cussed in this subsection were already mentioned in the first article [7] about the VMI method by Eppink and Parker. However, they described it merely as an “out of focus” operation of the VMI setup and gave only examples for O + 2 ions without initial kinetic energies, making no attempt to optimize the operating conditions to better suit the position mapping requirements. As far as I know, since then the influence of initial positions of the particles on their arrival positions in VMI experiments was considered rather as a curious effect of imperfect focusing and a resolution-decreas- ing annoyance, than as an opportunity for diagnostics and optimization of experi- mental conditions. 2.7.4 TOF-MS operation Testing of the mass-spectrometer mode was not performed during the experimen- tal characterization described in this section. However, for the sake of completeness, an example of a relatively rich mass spectrum from a later experiment with organic radicals (see chapter 4) is provided here. Figure 2.37 shows the TOF signal record- ed using ion optics parameters L 0 Æ 50 mm, L 1 Æ 50 mm, V 0 Æ 3000 V, V 0 Æ 2115 V, V L Æ 0 and converted to the ion mass scale according to (2.16). The experiment was devoted to studies of the CH 2 OH radical, hence the molecular beam contained these 1 Moreover, the V 0 Æ V 1 regime leads to a relatively low electric field strength at the ionization region, and hence, according to (2.35), a significant TOF spread for particles with nonzero initial kinetic energies. For example, in the O + experiment (figure 2.35) the spread for K 0 » 0.4 eV ions was¢¿ K 0 ¼ 150 ns, whereas the TOF sensitivity was only about 55 ns/mm. It means that if accurate position determinations from TOF measurements are required, then more favorable ion optics parameters must be considered. As the usability of these measurements with the given time resolution (see p. 14) was questionable, no optimizations for this task were performed in the present work. 133 0 0 5 10 15 20 25 30 35 40 45 Relativeintensity 75 m/z,a.u. H + He + C + CH + CH + 2 CH + 3 CO + CHO + CH 2 O + CH 2 OH + ? 35 Cl + H 35 Cl + 37 Cl + H 37 Cl + ? ? Figure 2.37 Example of mass spectrum (from CH 2 OH studies described in chap- ter 4). Some identified ions are marked under the peaks. radicals as well as unreacted precursor molecules CH 3 OH and Cl 2 , and intermediate Cl atoms together with HCl and H byproducts, all without appreciable kinetic ener- gy. The º» 42700 cm ¡1 radiation used for REMPI of CH 2 OH was also resonant for HCl and near-resonant for Cl. In addition to “direct” ionization, it also dissociated all present polyatomic species and ionized the fragments, resulting in the observed COH + nÆ 0...2 and CH + nÆ 0...3 progressions of ions with various kinetic energies. As can be seen from the plot, all observed masses were completely resolved. The ions without kinetic energy produced narrow peaks, whereas for other ions the peaks were broadened to various degrees. The case of Cl + is particularly interesting, since its peak 1 had a complex shape and the greatest width. The shape is determined 1 Both 35 Cl + and 37 Cl + isotopes behave similarly, but the 37 Cl + signal has»3 times lower intensity (due to the lower natural abundance of 37 Cl) and is contaminated with the electrical ringing produced by 134 by contributions from different types of Cl atoms prior to ionization. First, some Cl atoms with practically no kinetic energy are present in the molecular beam initially. They give the central sharp part of the peak. Second, Cl atoms might be produced by Cl 2 or HCl photodissociation. While the 1-photon energy is in principle sufficient to dissociate both Cl 2 and HCl, 1 the additional broader part of the peak was produced by resonant photodissociation of HCl, as revealed by scanning the laser frequency. The maximum observed width of the peak is consistent with 2-photon dissociation, which leads to Cl atoms with K¼ 3.1 eV: according to the SIMION simulations, the TOF spread for such fragments should be about §80 ns, slightly less than the »100 ns difference between the central arrival times of H 35 Cl + and 35 Cl + . This example al- lows to make a conclusion that the mass resolution of 1 Da even for particles with relatively high kinetic energies is possible up to at least m» 40 Da. Figure 2.38 shows another example of a mass-spectrum from the same experi- ment, but taken atº¼ 42720 cm ¡1 (one of the Cl atom REMPI frequencies), 2 clearly illustrating that for “cold” particles the mass resolution is significantly better even at higher masses. Although no particles heavier than 75 Da were observed in the con- ducted experiments, it is reasonable to expect that the resolving power of the system should be sufficient for all conceivable experiments on the existing apparatus. the preceding strong and sharp H 35 Cl + signal. Therefore, only the 35 Cl + signal will be discussed. 1 D 0 (Cl 2 )¼ 19990 cm ¡1 [44], D 0 (HCl)¼ 35750 cm ¡1 [45]. Both molecules have repulsive states cor- relating with ground state fragments, but the absorption cross-sections are actually very low near 42700 cm ¡1 . 2 The mass-spectrum part between 70 and 75 Da was very sensitive to the laser frequency and showed different combinations of 70, 72 and 74 Da peaks with various intensities. Although these peaks certainly correspond to the isotopomers of Cl 2 , their relative intensities did not correlate with the expected abundances. The mechanism leading to this high ionization selectivity was unclear. More- over, the 75 Da peak was always present in the spectra, sometimes alone, and could not be identified entirely. 135 0 66 68 70 72 74 76 78 Relativeintensity m/z,a.u. 35 Cl + 2 35 Cl 37 Cl + 37 Cl + 2 ? Figure 2.38 Example of mass spectrum in the region of higher masses. 2.8 Conclusion The simulations and experimental tests described in the previous sections demon- strated that the designed and built SVMI system completely satisfies the demanded requirements (see section 2.2). Specifically, it is suitable for sliced velocity map imag- ing of H fragments with kinetic energies from a fraction of an electronvolt to a few electronvolts, providing kinetic energy resolution.2 % (and»1 % for the range below »2 eV). The comparison of the experimental results with the results of numerical simula- tions indicated that the employed simulation framework provides a reliable descrip- tion of the actual system performance. In particular, all parameters of the optimal operating conditions can be set based solely on the results obtained by optimization of the simulated performance. The predicted velocity mapping in that case does not differ from the actual one by more than 1 %, and therefore the kinetic energy cali- bration can be obtained with 1¡2 % accuracy from the simulations alone. This fact 136 greatly facilitates experimental studies in which such accuracy is sufficient, since it allows to choose the best conditions for each particular measurement without worry- ing about the issues with additional experimental calibration. On the other hand, if necessary, experimental recalibration can be also performed, although care must be taken to use as similar experimental conditions as possible if absolute KE accuracy .1 % is desired. While the main intention of the developed ion-optical system was the application for VMI with sliced detection, its advanced temporal characteristics are potentially useful for 3-dimensional detection with time resolution as well. According to the sim- ulations, a temporal resolution of the velocity mapping.1 % — in good concordance with the radial resolution — can be achieved for a very broad range of operating conditions. That is, the present ion-optical system provides extensive possibilities for independent adjustments of the radial and temporal extents of the particles cloud at the detector, which can be utilized for better matching of these characteristic with the resolving capabilities of the detection system and therefore can help 1 to improve the resolution or enhance the range of 3-dimensional VMI setups. 2.9 Future work Even though the present SVMI system showed good performance and usabili- ty during the tests and in subsequent experiments, some of which are described in chapter 4, there is always a room for improvement. In addition, it would be desirable to correct the design and implementation flaws that led to the problems mentioned above. Several possible modifications of the system are summarized in this sections. 1 Although the present system was designed with relatively specific goals in mind and might be not optimal for other uses, I hope that the ideas described in this chapter will help in optimal design of similar systems for other specific cases. 137 The changes of the existing hardware required to implement them would range from none to a complete redesign. The most obvious, and probably most important, revision is related to the par- asite electron signal generated by the additional lens at high voltages (see p. 127). Since there are good indications that the electrons are generated near the surface of the central electrode due to a high local electric field strength, and such processes are known to be more efficient near surface roughnesses, careful polishing of the rel- evant surfaces might eliminate or at least considerably reduce the electron emission intensity. The most important surface is that of the lens ring (closest to the detec- tor 1 ) of the central element, but polishing of all other high-voltage parts and their immediate neighbors might be also necessary. 2 The next set of possible modifications is related to the accelerator assembly. A close examination of the Pareto-optimal plots for the performance characteris- tics of the system (see figure 2.20) reveals that at high magnification indices M& 80 ns p eV/Da (that is, for high imaged KE ranges or large TOF stretchings) all opti- mal parameters require the use of the whole available length of the accelerator. (For better visualization the data from figure 2.20 are replotted in figure 2.39 in terms of the total length L a ´ L 0 Å L 1 .) This situations contrasts with the diversity seen at lower M values and suggests that a longer accelerator could lead to somewhat better performance at high KE ranges by allowing a greater variety of conditions for the optimization. While only a minor improvement is expected (based of the overall behavior of the envelopes of the presented data), the possibility is worth checking by performing additional numerical simulations with more electrodes in the accelerator subsystem. 1 Electrons emitted from the farthest ring are accelerated in the opposite direction and thus should not reach the detector. 2 At least, it will enhance the high-vacuum performance. 138 1.0 1.5 2.0 20 40 60 80 100 120 ΔK,% M,ns p eV/Da 0 2 4 6 20 40 60 80 100 120 TotalΔt,% M,ns p eV/Da L a =85 90 95 100 105 110 115 120 125 130 Figure 2.39 Dependence of Pareto-optimal performance characteristics on the to- tal used length (in mm) of the accelerator. More complicated electric field configurations in the accelerator also deserve an extended study. The simplest extension of the two-region concept used throughout this chapter is to use three regions with different electric field strengths. 1 This will add two parameters (length L 2 and electric field strength E 2 ) to the optimization space, but since the available two-region results can be used as starting points, the complete optimization 2 seems doable. If it leads to a noticeable improvements in at least some of the performance characteristics, then the general configuration with independent voltages applied to all apertures should be studied as well. The large number of parameters would require a very different optimization approach. Per- haps, formulating a single optimization criterion in each particular case and starting one of the usual local minimization methods from the most suitable 2- or 3-region conditions would be the most practical approach. An appealing feature of these stud- ies in regard to the existing apparatus is that no modifications of the parts inside the 1 Although many VMI setups used at present go beyond the basic Eppink–Parker arrangement and in- corporate more than two apertures, the benefits of such designs were not quantified in the literature known to me. 2 “Complete” in terms of the parameter space. Simultaneous optimization of many characteristics is impossible with any number of parameters (see p. 88), however, all characteristics optimized with more parameters can be simultaneously better than those optimized with less. 139 vacuum chamber are required. The 3-region mode will need only one more voltage supply connected to the existing voltage divider. A practically usable implementa- tion of the “general” case, however, must involve a computer-controlled version of a variable voltage divider (which is still a relatively simple electronic device). The studies of more complicated fields could be combined with a more detailed study of the aperture radii. As described on p. 45, in the present work the radius was chosen mostly based on the performance without the additional lens. All apertures had the same radius, first, to limit the number of parameters in the geometry opti- mization, second, to facilitate manufacturing of the parts and assembly of the appa- ratus. Nevertheless, the “upper KE limit” example in figure 2.23 illustrates that the filling factor of the last apertures under operation in high-M regimes is very large, which might be the main source of the increased aberrations. Therefore, utilization of apertures of different radii, at least larger towards the exit end of the accelerator (especially if the accelerator is to be extended, as mentioned before), might improve the performance characteristics for operation with high KE ranges and high TOF stretchings. 1 Another set of modifications concerns the additional lenses in the drift region or closer to the detector. As explained at the beginning of section 2.5, the present sys- tem was intentionally kept as simple as possible, and therefore only one additional unipotential lens was installed after the accelerator (see subsection 2.4.4). The simu- lations, however, showed that a very short unipotential lens performs worse than an extended one (see p. 94). Since the thick lens in principle has two different “refractive surfaces”, a stack of shorter but weaker lenses (that is, the use of multiple refractive surfaces) might show a better performance than the single utilized lens. For exam- 1 Since regimes with moderate M values generally use shorter accelerator lengths (see figure 2.39), they should not be affected by increased radii of the “inactive” apertures. 140 ple, the optimization of a double unipotential lens described in [30] did not converge to a single lens, which suggests that multielectrode compound lenses indeed have some advantages. The optimization applied to the present system may be organized as geometry optimization of a few electrodes, similar to the approach used in [30], 1 or employ the method that was used for the accelerator design and optimizations in the present work, with multiple fixed electrodes and variable voltages. In conjunction with a more complicated configuration of the additional lens, the requirement that the electrostatic potential at the exit of the accelerator is equal to the detector potential can be lifted. This can be done either by keeping the detector grounded and extending the first electrode of the additional lens to the accelera- tor exit (to prevent the grounded vacuum chamber walls from distorting the electric field) or by using an electrically biased detector. Besides providing more freedom for ion optics optimizations, the larger possible potential difference between the ioniza- tion region and the detector should lead to higher kinetic energies of the particles at the detector and hence to higher detection efficiency, which is especially important for operation near the lower KE limit, where the acceleration voltage is relatively small due to the requirement of low electric field strength in the accelerator region. Preliminary simulations showed that using a negative (for positive ions) bias of the detector in the present system is also very effective in reducing the image size, in accordance with the analysis given on p. 52, and requires lower voltages applied to the additional lens for similar resulting characteristics. 2 The main practical problem with the bias is that the employed detector encloses the MCP stack in a holder that 1 However, using independent voltages for all electrodes should be preferred over the configuration with multiple unipotential lenses with grounded intermediate electrodes. 2 This fact is notable in connection with the problem of parasite electron emission (see p. 127 and p. 138). On the one hand, lower lens voltages should reduce the emission intensity, on the other — a significant negative potential of the detector should repel at least some of the produced electrons. 141 has a complicated shape and exposes all electric potentials 1 near the detector input surface. In the present design a grounded electric shield covering these holder parts was sufficient to maintain a nearly field-free region around the detector, but cre- ation of a well-behaved electric field configuration would require more careful design considerations and thus was not included in studies of the minimalistic system. Nev- ertheless, biased detectors are often used in TOF measurements, 2 for which the field distortions are less crucial, and there are no compelling reasons against improving the field geometry for VMI use. The last part of this section is devoted to more fundamental considerations. Al- ready in the early days of charged particle optics Scherzer has published his famous article [46] with the proofs that chromatic and spherical aberrations are unavoid- able in systems with cylindrically symmetric time-independent electrostatic fields. While these theorems did not pose any particular limit on the aberrations, electron microscopy — the main field of applications — has eventually reached a resolving power limit dictated by the aberrations in practically reasonable arrangements, and the study of methods to circumvent the troublesome theorems started to develop. The ideas included addition of optical elements without cylindrical symmetry (“mul- tipoles”), alternating electric fields and combinations with magnetic lenses. A rel- atively recent theory for some of these methods for aberrations corrections can be found in [47]. Although not all of the achievements in electron microscopy are ap- plicable to the VMI case, the possibility of using such ideas for improvement of VMI performance characteristics is definitely interesting. 3 1 Potentials applied to the front, middle and back of the MCP stack, as well as potentials of the phos- phor screen and the (grounded) mounting vacuum flange. 2 Mainly due to the convenience of operating the detector with grounded output. 3 Some attempts to use magnetic fields for VMI applications were already made. For example, in [48] the measured kinetic energy range was significantly increased by application of an axial magnetic field. (The electrostatic part in that case used Wiley–McLaren arrangement and actually did not 142 References [1] I. M. Ismail, M. Barat, J.-C. Brenot, J. A. Fayeton, V. Lepere, Y. J. Picard, A zero dead-time, multihit, time and position sensitive detector based on micro-channel plates, Rev. Sci. Instrum. 76(4), 043304 (2005). [2] A. I. 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Instrum. 76(1), 013105 (2005). 146 Chapter 3 Determination of velocity distributions from raw (S)VMI data 3.1 Introduction The raw data produced by sliced velocity map imaging setups, such as the one de- scribed in chapter 2, consist of a list of Cartesian coordinates of the particles arrival positions or of a raster image describing the integral recorded intensity distribution on a rectangular (usually square) grid. In both cases the information is represented in terms of Cartesian components of the initial velocity vector. The analysis of pho- todissociation and photoionization processes, however, is most rationally performed in form of speed or kinetic energy release (KER) distributions, as well as angular distributions, which in case of excitation of non-oriented molecules by a linearly polarized laser can be shown to consist of a combination of a few lowest spherical harmonics [1]. Therefore, a transformation of the raw data to the more convenient distributions is required before analysis of the SVMI results. Using the notation introduced in section 2.1, the speed distribution (that is, ve- 147 locity distribution integrated over all angles) I(v) can be calculated as 1 I(v)´ Z jv 0 jÆv I(v 0 x ,v 0 y ,v 0 z )dv 0 x dv 0 y dv 0 z Æ Æ 2¼ Z 0 ¼ Z 0 I(v ½ Æ vsinµ,v y Æ vcosµ)v 2 sinµdµ@'» » 2¼ Z 0 I ¡ xÆ R(v)sinµ, yÆ R(v)cosµ ¢¡ R(v) ¢ 2 jsinµjdµÆ Æ 2¼ Z 0 I ¡ R(v),µ ¢¡ R(v) ¢ 2 jsinµjdµ, (3.1) where I(x, y) is the spatial intensity distribution at the detector (also expressed as I(R,µ) in polar coordinates 2 ), and R(v) is the radial part of the velocity mapping (see p. 28), which can be determined by experimental calibration or from charged parti- cle optics simulations. Usually this mapping is very close to linear, and the radial intensity distribution I(R)Æ 2¼ Z 0 I(R,µ)R 2 jsinµjdµ (3.2) can be obtained first and then converted to the speed distribution as I(v)» I ¡ R(v) ¢ . (3.3) The distribution of kinetic energies of the detected fragment can be obtained from 1 Since only relative intensity distributions are usually required instead of absolute intensity measure- ments, the common proportionality factors consisting of constants and unit conversions are omitted in these and subsequent equations. In case of absolute measurements they all can be absorbed in the experimental intensity calibration factor. 2 The polar angleµ varies between 0 and 2¼ for the 2-dimensional slice. Therefore, the upper integra- tion limit was changed in order to include both halves of the image, and the weight was written as jsinµj to remain positive for theµÈ¼ half. 148 its speed distribution and the relationships I(K)dKÆ I(v)dv and dKÆ mv dv as I(K)Æ 1 v(K) I ¡ v(K) ¢ , v(K)Æ p 2K/m, (3.4) or directly from the experimental intensity distribution (in case of nearly linear v r 7! R mapping) as I(K)» 1 R(K) I ¡ R(K) ¢ Æ 2¼ Z 0 I ¡ R(K),µ ¢ R(K)jsinµjdµ, (3.5) where the R(K) dependence can also be calibrated experimentally or, in case of the present setup, is conveniently provided by the simulations framework described in chapter 2. The primary quantity of interest in photodissociation experiments is not the ener- gy of the detected fragment but the kinetic energy release (KER). KER is defined as the difference between the total kinetic energy of the fragments and the initial kinet- ic energy of the parent particle (molecule or radical). 1 In the center of mass (COM) reference frame it is simply equal to the sum of the fragment KEs. Conservation of the total mechanical momentum in photodissociation 2 means that in the COM frame (where the parent particle is stationary) the momentum p 0 of the cofragment has its direction opposite and the magnitude equal to that of the detected fragment momen- tum p. In case of SVMI experiments with a molecular beam source (directed along the ion-optical axis) the radial component of the parent particle momentum is negli- gible, the radial component of the detected fragment p r Æ mv r is measured directly, and the axial component p z ¼ 0 (in the COM frame) is selected by slicing. Therefore, 1 As a reminder, “kinetic energy” here means the translational energy. Rotational and vibrational excitations are assumed to be a part of the internal, quantized, energies of the involved particles. 2 Momentum of the excitation photon is very small and can be safely neglected. 149 the total KER is KERÆ KÅ K 0 Æ p 2 r 2m Å p 02 r 2m 0 Æ p 2 r 2m ³ 1Å m m 0 ´ Æ ³ 1Å m m 0 ´ K, (3.6) where m 0 and K 0 are the mass and the KE of the cofragment. That is, the KER distribution can be obtained from the measured KE distribution of one fragment by linear scaling of the independent variable: I(KER)» I ³ KÆ m 0 mÅm 0 KER ´ (3.7) (notice also that mÅ m 0 here is the mass of the parent particle). As mentioned above, the raw SVMI data is not in the form I(R,µ) required for (3.2) and (3.5), but nevertheless the integrals in these equations can be easily evalu- ated using the raw data. In the case of Cartesian coordinates list the coordinates of each hit can be transformed to polar representation, binned by small radial intervals (determined by the desired sampling resolution), and then within each interval the radial intensity can be calculated as I(R i Ç RÇ R iÅ1 )Æ X R i ÇRÇR iÅ1 jsinµj, (3.8) where the sum is taken over all events with R in the given range. In the case of raster image defined on a rectangular grid the integrals can be evaluated by resam- pling the data in polar coordinates. If the initial rasterization has pixels smaller than the SVMI resolution of the system, and the resampling is also performed with suf- ficiently small steps, the results obtained by (3.8) directly from hits coordinates and by resampling of the rasterized image are practically indistinguishable, and any of the methods can be used. 150 The anisotropy parameters for angular distributions can be obtained by fitting the corresponding trigonometric polynomials to the intensities sampled at several values of the polar angle. The exact procedure depends on the studied process (for example, how many photons and with which polarization are absorbed before the dissociation) and whether the radial anisotropy distribution or only integral param- eters for individual KE ranges (that is, corresponding to products in particular states) are required. However, before the actual extraction of the physically meaningful distributions from the raw data the latter might require some additional preprocessing. The first reason is related to deviations from the velocity mapping that was expected to have angle-independent properties. In reality, small asymmetries in the instrument and asymmetric stray fields lead to geometric distortions of the produced images. These distortions might be small and not immediately obvious in the raw image, but nev- ertheless decrease the resulting resolution of speed or KE distributions obtained by integration over angles, since in the presence of distortions each particular speed is projected to somewhat different radii at different angles, which causes blurring of the integrated radial distributions. The method for detection and rectification of such distortions is described in section 3.2. The second kind of additional processing is related to the finite slicing thickness in SVMI experiments. As mentioned in chapter 2, the velocity distribution can be extracted from the sliced image directly only if the slice is sufficiently thin. However, besides being unrealizable in practice, infinitely thin slices would provide infinitely small signals. Therefore, the actual images always have some radial blurring (see (2.7) and figure 2.2), which is more pronounced in the low-velocity part. 1 In case of 1 As explained on p. 90, this blurring remains constant over the whole distribution in the KE scale. Nevertheless, a better KE resolution could be obtained in its absence. 151 full-projection VMI this “blurring” becomes extreme, but application of inverse Abel transform allows to reconstruct the initial distribution by removing these overlap- ping projection parts. Therefore, it might be expected that an analogous transform can be performed for the sliced projection in order to remove the radial blurring and reconstruct the original distribution with maximal resolution. This transform is dis- cussed in section 3.3, where its implementation is also described with examples of application to SVMI data. Since the present SVMI setup described in chapter 2 produces relatively small distortions, and the distributions obtained in the present experiments did not have structured low-velocity parts suffering from finite-slice blurring, the illustrative ex- amples for both methods of imaging data processing are based on other experiments performed on other setups, where the addressed problems were more evident. 3.2 Rectification of image distortions Geometric distortions in velocity map images manifest themselves as deviations from perfectly circular shapes of the rings corresponding to monochromatic parti- cles. The angular dependence of velocity distributions in photodissociation of non- oriented molecules by one or few photons is quite smooth. This means that angular distortions, even if present, are difficult to detect. But at the same time, the effect of small distortions 1 on determination of anisotropy parameters is also small. The radi- al structure, however, is often quite sharp, and this circumstance means that radial distortions are both important and easily detectable. Therefore, in the present image rectification method the correction is performed only in the radial direction, leaving the polar angles unaffected. 1 Which are indeed small in all practical cases. 152 The essence of the description of distortions and their correction is illustrated in figure 3.1. If an orthogonal polar grid 1 is defined in the original velocity space y x θ i R j y x θ i R i,j distorted rectified Figure 3.1 Transformation of polar grid in rectification of distortions. (v x ,v y ), its mapping to the detector plane (x, y) by a nonperfect system is distorted and resembles an asymmetric spiderweb, as shown on the left in the figure. This distorted grid, however, can be transformed back to an orthogonal grid shown on the right by angle-dependent radial stretching/shrinking, restoring the perfect mapping for the initial grid. A continuous transformation that corrects the grid distortions can be applied to the entire image and, if the grid is sufficiently dense, should correct the mapping for all velocities even between the grid lines. As a concomitant effect, the overall radial nonlinearity of the v r 7! R mapping can be corrected in the same procedure. From the practical perspective, the distorted grid is defined based on the observed radial structure of the image. Several reference polar angles µ i are picked in a way that allows adequate description of the observed radial distortions, 2 and then for each angle the radii R i, j are measured for several rings well-recognized in the raw 1 That is, the grid lines correspond toµÆ const (polar rays) and rÆ const (concentric circles). The grid is not required to be uniform, which is demonstrated in the figure. 2 For example, the polar angles can be sampled densely in more distorted regions and sparsely in regions with smaller distortions. 153 image. 1 The radii R j that should be produced in the rectified image are set to values corresponding to the expected velocities, if these are known, 2 or according to simu- lated mapping properties and radii measured in the least distorted parts of the raw image otherwise. Of course, an actual experimental calibration with a known distri- bution is the preferred method, but as mentioned in chapter 2, it must be done at the same experimental conditions, which is not always possible. In case of relative- ly small distortions, determination of R j values from the same raw image without calibration might lead to small discrepancies in the resulting KEs after the rectifica- tion. However, since KEs are not calibrated anyway, and these discrepancies cannot exceed the blurring of the distribution extracted from the nonrectified image, the rectification is still profitable due to an improvement in the final resolution, which is often more important than the absolute energies. The continuous transformation that rectifies the grid, (µ i ,R i, j )7! (µ i ,7! R j ), was chosen to be piecewise bilinear in µ and R. Specifically, within each grid cell the radius R in the original raw images is transformed to radius R 0 in the resulting rectified image as R7! R 0 Æ (R¡ R µ, j )R jÅ1 Å(R µ, jÅ1 ¡ R)R j R µ, jÅ1 ¡ R µ, j , (3.9) 1 It might be difficult to measure these radii reliably at all angles for rings with strongly anisotropic intensity distributions (such that the intensity approaches zero at some angles). In case of perpen- dicular anisotropy (zero intensity at poles and maximal near the equator) an additional image can be taken for calibration with the laser polarization switched to the ion-optical axis direction, producing isotropic distribution in the (x, y) plane with the same radial distribution. For rings with parallel anisotropy this, obviously, cannot be done. Nevertheless, it should be clear that the quality of rectifi- cation for the image parts without detectable signals has no effect on the final results, and therefore any reasonable radius (for example, an average of radii at the adjacent angles) can be picked at such low-intensity angles. 2 That is, the rings are already identified, and the corresponding fragment KEs can be accurately determined form previously measured quantities. 154 where R µ, j Æ (µ¡µ i )R iÅ1, j ¡(µ iÅ1 ¡µ)R i, j µ iÅ1 ¡µ i , (3.10) and µ i 6µÇµ iÅ1 , R µ, j 6 RÇ R µ, jÅ1 (and hence R j 6 R 0 Ç R jÅ1 ). 1 This transforma- tion can be applied directly to the hits positions (after conversion of their Cartesian coordinates to polar coordinates) in the case of raw data represented by a hits list. On the other hand, if a rasterized image is to be rectified, the inverse transfor- mation RÃ R 0 must be applied for each pixel in the resulting image to determine its intensity from the corresponding pixel in the original image. This inverse transfor- mation is also linear and can be easily obtained by rearrangement of (3.9): RÆ (R 0 ¡ R j )R µ, jÅ1 Å(R jÅ1 ¡ R 0 )R µ, j R jÅ1 ¡ R j . (3.11) In principle, in rectification of rasterized images the intensities must be multiplied by the Jacobian determinant of the transformation. In practice, however, the stretch- ing or shrinking applied for distortion corrections is relatively minor, and thus this determinant is close to unity, introducing only a relatively small intensity correction. Moreover, when considering fine details of the intensity distribution, the nonunifor- mity in detection efficiency 2 might play a larger role, but if it is corrected by exper- imental calibration, the effect of the Jacobian determinant can be incorporated in 1 It is implied that R i,1 Æ R 1 ´ 0 for all i, and the R i,max j values are large enough to satisfy RÇ R i,max j for all experimental data. In addition, for the last angular segment the values µ (max i)Å1 ´µ 1 and R (max i)Å1, j ´ R 1, j are used in accordance with continuity requirements. The computer program that was created in the present work for implementation of this transformation ensures that all these conditions are satisfied. 2 Although event counting methods used in all modern VMI systems can significantly reduce spatial nonuniformity in the MCP detector gain, they do not eliminate it completely. In addition, the de- tection efficiency of MCPs is always below unity [2], meaning that a fraction of the particles does not produce any signal, and this fraction might be different for different parts of the detector (for example, due to contamination or erosion of the channels). Another factor affecting the “detection” nonuniformity is that the production efficiency of detectable ions from neutral fragments might be also nonuniform with respect to fragment velocity (two different examples of such effect are given in chapter 4 and appendix F). 155 the same experimental intensity correction factor. Therefore, the Jacobian intensity correction was not implemented in the present work. Although the piecewise linear transformations (3.9) and (3.11), being continu- ous, are not smooth, this fact does not affect their practical usability. Namely, any smooth transformation can be approximated by piecewise linear transformations to any given accuracy. Experience with various experimental images showed that even relatively sparse grids (for example, with»45° angular steps and a few radial posi- tions) allow to reduce the distortions to a level below the actual velocity resolution of the system and are therefore sufficient for the task. At the same time, the simplicity of definition and evaluation of piecewise linear functions makes the implementation and use of the method more convenient than if some smooth transformations (for example, bicubic splines) were used. The performance of the method will be illustrated by one example using full- projection VMI data shared with us by the Fleming Crim group and shown in fig- ure 3.2. 1 The left part of the figure, corresponding to the raw velocity map image, demonstrates that the radial distortions that can be not very obvious in the original image become quite evident when the recorded intensity distribution is plotted as a function of the polar angle and radius in rectangular coordinates. The explanation of this visual effect is that it is much easier to distinguish a curved line from a straight line than an approximately “round” line from a perfect circle. The right part shows the same image and its polar intensity dependence after the rectification. As can be seen in the lower pictures, the bent curves corresponding to individual rings became much more straight. Careful comparison of the images in their “normal” representa- 1 This full-projection velocity map image was recorded by detecting H fragments in one-photon disso- ciation of ammonia NH 3 hº ¡ ¡ ! NH 3 ( ˜ A 1 A 00 2 )¡ ! NH 2 Å H at 46158 cm ¡1 for studies reported in [3], but was not included in the final publication. The authors were surprised by the unusual anisotropy oscillations obtained from reconstruction of their images and turned to our group for advice. 156 Raw image: Rectified image: Radius! Radius! Polar angle! Polar angle! Figure 3.2 Examples of distorted raw velocity map images and its rectification. Lower part shows the intensity as a function of polar coordinates (plotted in rectan- gular coordinates). tion also reveals that the raw one is less circular, especially in the top left part. Construction of the grid, however, was performed based not only on the visual inspection. The angular part of the grid was chosen consisting of 10 polar rays: the most distorted, top, part was sampled with 30° steps (angles 0°, 30°, 60°, 90° and 270°, 300°, 330°), and in the remaining, bottom, part sampling with 45° steps was used (angles 135°, 180°, 225°). Radial intensity profiles at each of these angles were 157 calculated from the raw image and compared to each other. Since this particular im- age has a clear radial cutoff, its value in the obtained radial distributions for each of the angles was used for the R 2, j values in the transformation, 1 and the value for R 2 was selected as an average of these R 2, j values. Examination of the correspond- ing profiles obtained from the image rectified using this grid showed that they have matching structures in the whole radial range. The intensity plot for the rectified image (shown in figure 3.2) also demonstrates that the structural features look prac- tically straight at all radii and angles. Therefore, this grid was already sufficient for this image, and no addition of radial or polar lines to it was necessary. 2 Figure 3.3 shows a comparison of the radial and angular distributions obtained 0 0 50 100 150 200 250 300 Intensity R, px −1 0 1 β Raw Rectified Figure 3.3 Radial (speed) intensity and anisotropy parameter distributions ex- tracted from the raw and rectified images shown in figure 3.2. from the raw and the rectified images after inverse Abel transform. 3 The most strik- 1 As a reminder, R 1, j Æ 0 implicitly. 2 In very few cases encountered in the present work addition of 1 or 2 more radii in the grid was justified. At the same time, regular angular sampling with 45° steps (that is, 8 rays) was sufficient in most cases, since the distortions were usually more smooth than in this example. 3 The BASEX [4] program was used for the reconstruction and extraction of the plotted distributions. 158 ing difference between the two radial distributions is the resolution in the higher-R part. The structure for the last few peaks is completely washed out in the distribu- tion obtained from the raw image, whereas the rectified image yields good resolution for the whole distribution. The difference for lower R values is smaller, since the central part of the image is less distorted, and the peaks there are broader. Anoth- er important effect is that the anisotropy parameter 1 obtained from the raw image shows large oscillations and attains positive values for some of the outer peaks, even though the image itself does not show any indications of parallel character for them. 2 This artifact is caused by the fact that for each ring in the distorted image the in- tensity peaks at different radii for different angles, so that when angular profiles are considered at constant radii, they appear more “parallel” at one R values and more “perpendicular” at the other, even if the actual intensity is isotropic within the ring. This example clearly illustrates that image distortions not only decrease the final speed (or KE) resolution of VMI systems, but might also lead to misleading results regarding the angular velocity distributions. Therefore, detection and rectification of such distortions is always desirable and even indispensable in some cases. 3 3.3 Reconstruction of velocity distributions from VMI data with finite slicing thickness The advantages of sliced velocity map imaging (SVMI) over the “regular” full- projection VMI were already described in the introduction to chapter 2. The most 1 See [1] (or [4]) for its definition. 2 Moreover, the only anisotropy seen in the image has perpendicular character for the background. No- tice that the anisotropy parameter obtained from the rectified image is consistent with this picture — it shows¯¼ 0 for the peaks and¯Ç 0 for the background. 3 Strictly speaking, distortions in full-projection velocity map images violate the assumptions of the direct Abel transform, and hence application of the inverse Abel transform for reconstruction of the initial velocity distribution is inherently unreliable in such cases. In particular, it can produce artifacts in the radial distribution more complicated than a simple blurring. 159 important of them is the experimental separation of contribution from high and low kinetic energies (KE), which enables measurements of only the interesting part of the distribution and significantly improves the signal-to-noise ratio (SNR) of the result by rejecting the unnecessary signal that would overlap with the lower-KE parts in the full-projection case (see figure 3.4). Fullprojection: x y z Slicedprojection: x y z detector detector Figure 3.4 Schematic illustration of detected intensity contributions in full pro- jection and sliced projection. Useful signals are shown in solid colors, and interfering contributions — by hatched areas. Extraction of the radial distribution from the raw data, however, becomes prob- lematic closer to the center of the image (the low-velocity part of the distribution), where the relative slice thickness becomes large. Thick slicing means that the contri- butions from different KEs are no longer well-separated, and therefore the distribu- tion “naively” extracted from such part of the image shows a significant blurring. For the lowest velocities (or KEs) the slicing essentially degenerates to a “regular” full projection, since the signal from all velocitiesjvj.¢v z (see p. 10 for the notation) is detected. If the full projection takes place for the whole distribution, then the resulting im- age can be used in inverse Abel transform for extraction of the initial velocity distri- bution. Since this was the case for all VMI data before the invention of SVMI, many methods and their implementations were created for realization of this reconstruc- 160 tion, among which BASEX [4] (developed previously in our group) and its variant pBasex [5] (developed elsewhere based on the BASEX ideas) are probably the most widely used, even up to the present days. 1 It is obvious, however, that none of them can be applied directly to the thick slicing case, since the assumptions required by these methods are not satisfied everywhere in the sliced images. 2 The only known attempt to develop a method for reconstruction of velocity distributions from sliced images was described in [7]. That method, however, was extremely limited, assuming only rectangular slicing pulses and isotropic velocity distributions, and apparently did not find any noticeably use afterwards. This situation is of course worrisome, especially for SVMI of H fragments, thin slicing of which presents experimental difficulties. As can be seen from results de- scribed in subsection 2.5.5, larger TOF stretching required for thiner slicing leads to worse KE resolution due to increased aberrations in the ion-optical system. On the other hand, full-projection operation can give better KE resolution, but leads to much worse SNR (and hence much lower intensity resolution) in the inner part, which might make it equally unusable. Therefore, if a method for inversion of the sliced projection transformation (see (2.4)) P(v x ,v y )Æ Å1 Z ¡1 I(v x ,v y ,v z )S(v z )dv z (3.12) with experimentally suitable slicing functions S(v z ) existed, it would allow to com- bine the best properties of the SVMI (namely, good SNR) and reconstructed full- 1 A more recent method described in [6] should also be mentioned here. It is based on several progres- sive ideas, such as taking into account the experimental noise explicitly and producing physically meaningful results without negative intensities, many of which were adopted in the present work. Unfortunately, analysis of references to the publication [6] reveals that the method did not attract any appreciable attention. 2 Even for the central part, which itself if fully projected but has only incomplete projections of the other parts. 161 projection VMI (namely, good velocity resolution) and avoid their weaknesses. The analysis given below indicates that such method can be created and also provides an example of one of the possible approaches. 3.3.1 Basis set definition The first thing that must be done in discussions about integral transforms of functions defined on discrete grids is the definition of the implied basis. Essential- ly, all numerical integration methods are based on approximation of the integrand by a sum of certain basis functions for which analytical integrals are known. 1 This fact, being obvious for mathematicians, 2 was apparently so surprising for the VMI community that the first VMI reconstruction method where the basis functions were defined explicitly was named simply “BASEX” for “BAsis Set EXpansion”. Although it became customary to use Gaussian functions in the basis for the radi- al part, argumenting this choice by their smooth properties, in practice this smooth- ness is not used, and the reconstructed distributions have a significant random noise, requiring additional smoothing before further analysis. Therefore the present method uses simpler functions, which being not smooth, are continuous, localized and much more suitable for analytical evaluation of the required integrals. Specif- ically, the radial basis consists of “triangular” functions of equal width centered at grid points, as shown in figure 3.5. A uniform radial grid is used in the present work for convenience, and hence the radius is measured simply in grid units (which usu- ally correspond to one pixel size in the original image, but can be smaller or larger). 1 The basis functions can range from rectangular functions (constant within some range and zero outside it) used in the “rectangle rule” to nonlocal polynomials or polynomials multiplied by some function in various Gaussian quadrature methods. 2 See, for example, a relatively old article [8] discussing various approaches to numerical evaluation of direct and inverse Abel transforms. 162 I r R i R i+1 R i−1 1 0 Figure 3.5 Triangular basis function centered at grid point R i . The basis function centered at radius R is therefore b R (r)´ 8 > > > > > > < > > > > > > : r¡(R¡1), r2 [R¡1,R] (RÅ1)¡ r, r2 [R,RÅ1] 0, otherwise. (3.13) Expansion over this basis produces a piecewise linear approximation for the radial part of the distribution. The angular part of the initial distribution is traditionally described by Legen- dre polynomials in cosµ. These polynomials are closely connected to the spherical harmonics, which in turn are eigenfunctions of the angular momentum operator. Therefore, dissociation of particles with finite rotational quantum numbers by a fi- nite number of photons leads to product states with angular parts described exactly by a finite number of spherical harmonics. It was also shown (see [1]) that in case of thermal initial rotational distribution and dissociation by N photons actually on- ly the squares of the lowest harmonics up to lÆ N are present in the final angular intensity distribution. Moreover, for linearly polarized lasers the distribution must have cylindrical symmetry around the laser polarization axis and reflection symme- try with respects to the perpendicular plane (that is, belong to the totally symmetric representation A 1g of the D 1h symmetry group), so that the angular intensity dis- tribution is described by the harmonics with only lÆ 2n (n6 N) and mÆ 0, that 163 is, I(µ)Æ 1 4¼ N X nÆ0 ¯ 2n P 2n (cosµ), (3.14) where P l are Legendre polynomials, and ¯ l are coefficients (¯ 0 ´ 1) that are called “anisotropy parameters” and depend on details of the particular process. Nevertheless, the use of this basis in the reconstruction of experimental data is not very convenient. Namely, these polynomials are orthogonal and hence all of them (except P 0 Æ 1) possess both positive and negative values. The total distribution, how- ever, must be nonnegative everywhere. Therefore, the coefficients ¯ l in (3.14) must be constrained in order to produce physically meaningful results. In case of 1-photon dissociation the sum has only two terms, one of which is simply 1, and the well- known range ¡16¯ 2 6 2 can be easily established. But in N-photon processes all N coefficients are interdependent, and constraining them to physically meaningful combinations becomes more elaborate. At the same time, the angular dependence of the intensity in dissociation has its roots in the angular dependence of the transition dipole moment matrix elements: I(µ)» ¯ ¯ ¯ X { j} a { j} hÃ f j ˆ μ¢EjÃ j N¡1 ihÃ j N¡1 j ˆ μ¢EjÃ j N¡2 i...hÃ j 1 j ˆ μ¢EjÃ i i ¯ ¯ ¯ 2 , (3.15) where the sum goes over all combinations { j} of the intermediate states, i denotes the initial state (parent particle) and f — the final state (dissociation fragments), ˆ μ and E are the dipole moment operator and the laser electric field vector, and a { j} are coefficients depending on the energy spectrum. In most practically important cases only one term in the sum has a large contribution to the final amplitude, 1 and then 1 This is not true in specially designed experiments for so-called “phase control”, where, for instance, simultaneous coherent 1- and 2-photon dissociations take place. In such cases the two contributing amplitudes interfere, and the resulting angular distribution might exhibit a more or less strong asymmetry (see, for example, a theoretical study [9] and experiments [10–13]). Therefore, the basis described below would not be suitable for reconstruction of this kind of distributions. Notice, however, 164 the distribution becomes I(µ)» Y n ¯ ¯ hÃ n j ˆ μ¢EjÃ n¡1 i ¯ ¯ 2 Æ Y n ¡ c n cos 2 µÅ s n sin 2 µ ¢ (3.16) with coefficients c n and s n depending on the orientation of the n-th transition dipole moment relative to the asymptotic recoil direction and the degree of rotational av- eraging. That is, opening the brackets and collecting cos 2 µ and sin 2 µ products, the distribution can be represented as a sum of nonnegative functions: I(µ)» N X nÆ0 a n cos 2(N¡n) µsin 2n µ (3.17) with some coefficients a n . These are the functions that were chosen for the angular basis set in the present work. An observation that sin 2 µÆ 1¡cos 2 µÆ cos 0 µ¡cos 2 µ, (3.18) and thus cos 2(N¡n) µsin 2n µÆ cos 2(N¡n) µ ¡ 1¡cos 2 µ ¢ n Æ n X mÆ0 (¡1) m C m n cos 2[(N¡n)Åm] µ, (3.19) where C m n ´ n! m!(n¡ m)! (3.20) are the binomial coefficients, allows to rewrite (3.17) as a polynomial in cos 2 µ only: I(µ)» N X nÆ0 a 0 n cos 2n µ (3.21) that the customary Legendre-polynomial expansion (3.14) would not suit either. 165 with coefficients a 0 n simply related to the coefficients a n in (3.17) through (3.19). This means that the “usual” expansion over the even Legendre polynomials in cosµ (which obviously contain only the cos 2n µ terms) can be easily obtained from the expansion (3.17). Another important fact related to (3.21) is that projections of the cos 2n µ func- tions are readily expressed through the same cos 2n µ functions, as explained in the next subsection. 3.3.2 Sliced projections of basis functions Sliced projection of each basis function (product of the radial and angular func- tions defined above) must be calculated in order to perform the sought reconstruction of the initial distribution. This problem is complicated by the fact that the slicing function might be different in different experiments, but development of a relative- ly general method that would be suitable for all experiments is highly desirable. In principle, the projections for arbitrary slicing functions can be easily performed by numerical integration. This easiness, however, is only conceptual, since the actually required computational resources are quite large. Namely, reconstruction of an im- age with N£N pixels requires»N basis functions. For each of them the contributions of the projection to all N£N pixels of the image must be calculated. The contribution to each pixel, in turn, requires integration in the z direction with also»N integra- tion points. That is, the total computation requires O(N 4 ) time. Even though the calculations can be optimized, 1 the actual computation time observed for N¼ 1000 in the present work 2 was of the order of one hour. This might be acceptable for anal- ysis of many experiments performed in the same conditions (so that the calculated 1 For example, by using the left–right and top–bottom reflection symmetries and not calculating the projection intensity at the pixels where it is certainly zero. 2 These calculations were performed during the early stages of the work in order to study the viability of the SVMI reconstruction idea and explore various particular approaches to its implementation. 166 basis projections can be reused), but would be very inconvenient in all other cases. Therefore, a more efficient approach with analytical calculation of the projections is desirable. Since all further discussion will be given in coordinates related to the projection (that is (x, y, z) instead of (v x ,v y ,v z )), it is appropriate to repeat here the main projec- tion equation (3.12) expressed in these spatial variables: P(x, y)Æ Å1 Z ¡1 I(x, y, z)S(z)dz. (3.22) It is also very useful to consider it in polar coordinates: 1 P(r,µ)Æ Å1 Z ¡1 I(½,µ 0 )S(z)dz, (3.23) where 8 > > < > > : ½Æ p r 2 Å z 2 , cosµ 0 Æ µ r ½ ¶ cosµ, (3.24a) (3.24b) as can be easily derived from the diagram shown in figure 3.6. The last equation y x θ r z 0 θ 0 ρ P(r,θ) I(ρ,θ 0 ) Figure 3.6 Illustration for definition of variables ½ and µ and their relationships to r, z, andµ. 1 In contrast to the notation used in the previous sections, the lower-case r is used for the radius because the upper-case R now denotes the radial grid points, see (3.13). 167 (3.24b) makes obvious that projections of each angular intensity component I(r,µ)Æ ±(r¡ R)¢cos n µ with any slicing functions are simply P(r,µ)Æ Å1 Z ¡1 ±(½¡ R)cos n µ 0 S(z)dzÆ Æ Å1 Z ¡1 ± ³p r 2 Å z 2 ¡ R ´ µ r p r 2 Å z 2 ¶ n cos n µ S(z)dzÆ Æ 2 4 Å1 Z ¡1 ± ³p r 2 Å z 2 ¡ R ´ µ r p r 2 Å z 2 ¶ n S(z)dz 3 5 ¢cos n µ (3.25) and have the same cos n µ angular dependences. Therefore, the total projection can be expressed in the same form P(r,µ)Æ N X nÆ0 f 0 n (r)cos 2n µ (3.26) as the initial intensity distribution (3.21) and even in a form P(r,µ)Æ N X nÆ0 f n (r)cos 2(N¡n) µsin 2n µ, (3.27) using the same angular basis functions as in (3.17). 1 The computation of the b R (r)¢cos n µ basis functions projections, however, requires some suitable representation of the slicing function S(z). This representation must allow analytical integration in (3.23) on the one hand and be sufficiently flexible to encompass all practically important cases of S(z) on the other. A piecewise linear ap- proximation on an arbitrary grid was chosen in the present implementation based on 1 The radius-dependent coefficients f 0 n in (3.26) are obviously nonnegative, since the integral in (3.25) is positive. The nonnegativity of the coefficients f n in (3.27) can be also shown by somewhat lengthy manipulations involving (3.19) and the fact that r/½6 1 in (3.24). Notice also that in the latter case the terms become intermixed due to different contributions of different (r/½) powers within each of them. 168 the already mentioned approximation properties and the simplicity of integration. 1 Collecting all the basis functions together, the projection of slice segment [Z 1 , Z 2 ] of one radial–angular function is p R;n (r,µ; Z 1 , Z 2 )Æ Z 2 Z Z 1 b R (½)cos n µ 0 S lin (z)dz, (3.28) where S lin (z) is the linear approximation of S(z) for z2 [Z 1 , Z 2 ]. Since the radial function b R (r) has two linear segments: “internal” (rÇ R) and “external” (rÈ R), the integral in (3.28) can be represented as a sum of integrals of the internal and external parts (see figure 3.7): r ρ R R+1 R−1 0 z Z 1 Z 2 Z R ext int Figure 3.7 Illustration of various quantities involved in integration of one basis function within one slice segment. p R;n (r,µ; Z 1 , Z 2 )Æ £ I int;n (r;R¡1,R¡1,R, Z 1 , Z 2 )Å ÅI ext;n (r;RÅ1,R,RÅ1, Z 1 , Z 2 ) ¤ cos n µ. (3.29) 1 The choice of the arbitrary grid in this case is dictated by the fact that the analytical expressions for the involved integrals have the same form for any grid, but the freedom of choice of the grid points allows to decrease the total number of grid points required to approximate the original function with a given accuracy and therefore speed up the calculations. For example,§1 % approximation of Gaussian pulses S(z)» exp ¡ ¡(z/w) 2 ¢ with any width parameter w requires only 8 linear segments, and§0.1 % approximation — only 24. That is, approximation of any practical S(z) with any practical accuracy should be possible with only»10 intervals. 169 These partial integrals are (with “Å” for the internal and “¡” for the external parts): I int,ext;n (r;R 0 ,R 1 ,R 2 , Z 1 , Z 2 )Æ§ Z £ Z 0 R 1 ,Z 0 R 2 ¤ (½¡ R 0 ) µ r ½ ¶ n (s 1 zÅ s 2 )dzÆ Æ§ £ s 1 F z½;n ¡ s 1 R 0 F z;n Å s 2 F ½;n ¡ s 2 R 0 F 1;n ¤ ¯ ¯ ¯ Z 0 R 2 Z 0 R 1 , (3.30) where the integration interval is the intersection of the integrand support © z :½Æ p r 2 Å z 2 2 [R 1 ,R 2 ] ª with the slicing interval: £ Z 0 R 1 , Z 0 R 2 ¤ Æ £ Z R 1 , Z R 2 ¤ \[Z 1 , Z 2 ], Z R Æ 8 > < > : p R 2 ¡ r 2 , rÇ R 0, otherwise (3.31) (if r> R 2 , Z R 1 > Z 2 or Z R 2 6 Z 1 , then the integral is obviously zero), the linear approximation coefficients of the slicing function are s 1 Æ S lin (Z 2 )¡ S lin (Z 1 ) Z 2 ¡ Z 1 , s 2 Æ S lin (Z 1 )¡ a 1 Z 1 , (3.32) and the antiderivatives F ...;n are listed in table 3.1. The full expression for the pro- jected basis functions is therefore p R;n (r,µ; Z 1 , Z 2 )Æ ³ s 1 n 2 £ F z½;n (Z 0 R )¡ RF z;n (Z 0 R ) ¤ ¡ ¡ F z½;n (Z 0 R¡1 )Å(R¡1)F z;n (Z 0 R¡1 )¡ ¡ F z½;n (Z 0 RÅ1 )Å(RÅ1)F z;n (Z 0 RÅ1 ) o Å Å s 2 n 2 £ F ½;n (Z 0 R )¡ RF 1;n (Z 0 R ) ¤ ¡ ¡ F ½;n (Z 0 R¡1 )Å(R¡1)F 1;n (z 0 R¡1 )¡ ¡ F ½;n (Z 0 RÅ1 )Å(RÅ1)F 1;n (z 0 RÅ1 ) o´ cos n µ. (3.33) 170 Table 3.1 Antiderivatives of integrand terms in (3.30). Explicit expressions are given for several low n values, and general the form — for higher. n F 1;n ´ R ³ r ½ ´ n dz F z;n ´ R z ³ r ½ ´ n dz 0 z 1 2 z 2 2 rarctan z r r 2 ln½ 4 1 2 z ³ r ½ ´ 2 Å 1 2 rarctan z r ¡ 1 2 r 2 ³ r ½ ´ 2 6 1 4 z ³ r ½ ´ 4 Å 3 8 z ³ r ½ ´ 2 Å 3 8 rarctan z r ¡ 1 4 r 2 ³ r ½ ´ 4 8 1 6 z ³ r ½ ´ 6 Å 5 24 z ³ r ½ ´ 4 Å 5 16 z ³ r ½ ´ 2 Å 5 16 rarctan z r ¡ 1 6 r 2 ³ r ½ ´ 6 2m> 4 m¡1 P kÆ1 a k z ³ r ½ ´ 2k Å a 1 rarctan z ½ , a k Æ Q m¡2 lÆk (2lÅ1) Q m¡1 lÆk (2l) ¡ 1 2m¡2 r 2 ³ r ½ ´ 2m¡2 n F ½;n ´ R ½ ³ r ½ ´ n dz F z½;n ´ R z½ ³ r ½ ´ n dz 0 1 2 £ z½Å r 2 ln(zÅ½) ¤ 1 3 ½ 3 Æ 1 3 r 3 ³ r ½ ´ ¡3 2 r 2 ln(zÅ½) ½r 2 Æ 1 1 r 3 ³ r ½ ´ ¡1 4 zr 2 ½ ¡ r 4 ½ Æ¡ 1 1 r 3 ³ r ½ ´ 1 6 zr 2 ½ · 1 3 ³ r ½ ´ 2 Å 2 3 ¸ ¡ 1 3 r 3 ³ r ½ ´ 3 8 zr 2 ½ · 1 5 ³ r ½ ´ 4 Å 4 15 ³ r ½ ´ 2 Å 8 15 ¸ ¡ 1 5 r 3 ³ r ½ ´ 5 2m> 4 m¡2 P kÆ0 a k zr 2 ½ ³ r ½ ´ 2k , a k Æ Q m¡2 lÆkÅ1 (2l) Q m¡2 lÆk (2lÅ1) ¡ 1 2m¡3 r 3 ³ r ½ ´ 2m¡3 Although this already looks complicated and after substitution of all F n and Z 0 expressions from the previous formulas becomes quite cumbersome, the important point to notice here is that this is a closed-form analytical expression 1 for exact (within the numerical precision) computation of the projections with arbitrary piece- wise-linear slicing functions. The computer implementation of the above equations was straightforward and demonstrated that they can be evaluated relatively quickly 1 Compare to the huge sums and products used in BASEX [4] for the Gaussian-form basis functions. 171 whenever needed (that is, no precomputation is required, allowing to analyze SVMI data with arbitrary slicing within»1 minute). 3.3.3 Maximum-likelihood estimation The reconstruction of the initial distribution from the experimental data requires solution of the problem in some sense opposite to the projecting considered in the previous subsection. In principle, linear equations connecting coefficients of the ini- tial distribution expansion over the basis with the measured pixel intensities can be easily written, and their solution should recover the initial distribution. The prob- lem encountered with this “direct” approach is that the number of pixels usually exceeds the number of basis functions, and the experimental data inevitably has noise, 1 which means that the obtained system of linear equations is inconsistent in all practical cases. Although this problem with the noise is well-known in general, and all previous developments in VMI data reconstruction tried to circumvent it by finding a solu- tion (estimation of the initial distribution) that describes the experimental data in some “average” sense and discussed the influence of the experimental noise on the obtained results (see also section 2.1), only one example of a systematic approach from the statistical perspective is known in the available literature. Namely, in [6] the experimental data were assumed to have Poissonian noise, and the maximum likelihood principle was applied in order to obtain the most probable initial velocity distribution for the given measured projection. The assumption of Poissonian noise is quite reasonable for the rasterized event- counting data. Specifically, the size of the initial (ionization) volume in VMI experi- 1 At least, the statistical fluctuations of the intensity (so-called “shot noise”). 172 ments is usually much larger than the distances at which the particles can interact. That is, all dissociation events can be considered as completely independent. The probability of detecting a particle by a certain pixel of the detector therefore does not depend on detection of other particles by other pixels. 1 Although Poissonian statis- tics implies a continuous-time process, which is not true for VMI experiments, where the events are counted only in a finite number of discrete acquisition frames, the total number (n) of frames is usually so large and the event probability (p) at each pixel in each frame is so low 2 that the resulting binomial distribution is practical- ly indistinguishable from the corresponding Poissonian distribution (with parameter ¸Æ np). The maximum-likelihood estimation (MLE) method is closely related to Bayesian inference and is based on Bayes’ theorem, which connects the probability distribu- tion of measurements under certain values of model parameters with the probability distribution expected for the values of the model parameters given the specific mea- sured data. Since the “forward” probability distribution (of expected measurements at given model parameters) usually can be easily calculated, 3 and a reasonable prior distribution for the model parameters can be assumed, 4 application of Bayes’s rule immediately yields the probability distribution for values of the unknown model pa- rameter estimated from the available experimental data. Location of the maximum in this distribution gives the most probable values of the model parameters (hence 1 This is usually true only for non-overlapping events. However, operation in conditions leading to a large number of occurrences of such inseparable events has other adverse consequences and thus should be avoided in any case. 2 Again, large probabilities would lead to missing of multiple hits, resulting in nonlinear intensity response, and thus must be also avoided. 3 It is probably needless to say that meaningful analysis of experimental data is impossible without an adequate mathematical model describing the relationships between the measured numbers and the investigated quantities. 4 In most cases the prior knowledge is very poor, and hence the prior distribution is also very broad and smooth, meaning that its effect on the final result is almost negligible. Therefore, in many cases simply a uniform prior distribution is used, leading to the classical MLE. 173 the “maximum likelihood” name), and the overall shape of the distribution allows to estimate the uncertainties (for example, standard deviations and correlation co- efficients) of the obtained values. As can be seen, the MLE ideas are very general and quite reasonable, and many well-known statistical methods are actually based on them. For example, the widely used weighted averaging method and the linear least-squares method in general are particular cases of MLE for normally distribut- ed measurements. The actual application of the maximum-likelihood principle to a particular prob- lem requires definition of the “likelihood function”L , which is proportional to the posterior probability distribution, and location of its maximum, which in continuous cases can be done by solving the system of equations gradL Æ 0. For independent pixels the likelihood function is simply a product of intensity probability distribu- tions for each pixel. As a reminder, the probability of recording exactly k events in case of Poissonian distribution with expectation value¸ is Pr(kj¸)Æ ¸ k e ¡¸ k! . (3.34) In our problem the expectation value is the expected total projection intensity P(x, y) (a sum of all projection (3.28) of all basis functions with corresponding coefficients over the whole slice) integrated over the corresponding pixel area {X,Y }: ¸Æ P(X,Y )´ Ï {X,Y } P(x, y)dx dy, (3.35) and the measured number of events is given by the corresponding pixel intensity in the experimental image: kÆ P E (X,Y ), (3.36) 174 where the index “E” means “experimental”. Therefore, for given P E distribution the likelihood function (ignoring the constant factors k!) is L Æ Y X,Y h ¡ P(X,Y ) ¢ P E (X,Y ) e ¡P(X,Y ) i (3.37) with the product taken over all pixels of the image. The customary technique is to consider the logarithm ofL , which has the same maximum location but converts the product to a more convenient sum: lnL Æ X X,Y £ P E (X,Y )lnP(X,Y )¡ P(X,Y ) ¤ . (3.38) Substituting the P(X,Y ) by its decomposition over the basis functions projections P(X,Y )Æ X R,n ® R;n p R;n (X,Y ) (3.39) (also integrated over the corresponding pixels) into (3.38) and its differentiation with respect to each expansion coefficient® R;n yields @ @® R;n lnL Æ X X,Y · P E (X,Y ) p R;n (X,Y ) P(X,Y ) ¡ p R;n (X,Y ) ¸ . (3.40) At the maximum likelihood solution (combination of the expansion coefficients) all these derivatives must be equal to zero. Direct substitution of (3.39) into (3.40) shows that the resulting equations are nonlinear in® R;n , which makes their solution problematic. Nevertheless, they can be converted to a more suitable form 0Æ X X,Y · P E (X,Y ) p R;n (X,Y ) P(X,Y ) ¡ p R;n (X,Y ) ¸ Æ 175 Æ X X,Y · P E (X,Y )p R;n (X,Y )¡ P(X,Y )p R;n (X,Y ) P(X,Y ) ¸ Æ Æ " X X,Y P E (X,Y )p R;n (X,Y ) P(X,Y ) # ¡ X R 0 ,n 0 " X X,Y p R 0 ;n 0(X,Y )p R;n (X,Y ) P(X,Y ) # ® R 0 ;n 0 (3.41) resembling a system of linear equations in ® R 0 ;n 0. These equations still contain the unknown P(X,Y ) in the denominator, but are convenient for iterative solution pro- cedure. Namely, the projection P(X,Y ) of the sought solution must be close to the actual experimental projection P E (X,Y ) if the solution describes the experimental data well. It might seem therefore that substitution of P E (X,Y ) in place of P(X,Y ) in (3.41) is a reasonable first approximation. This choice, however, is not the best possible, since each particular pixel in the experimental projection P E (X,Y ) contains a relatively large noise amplitude. From this point of view, fitting the raw data with an expression (3.27) or (3.26) and using that fit should significantly reduce the ex- perimental noise in the substituted P(X,Y ) approximation. Numerical experiments performed with synthetic and real-world images demonstrated that this approach indeed works perfectly, producing practically exact solutions at the first iteration, so that subsequent iterations (with substitution of the P(X,Y ) calculated from the preceding solution) did not lead to any noticeable changes. 1 Practical implementation of the considered MLE method is therefore straightfor- ward and, besides the previously described computations of the basis functions pro- jections, 2 requires only a solution of the system of linear equations (3.41) with a well- 1 In fact, even using P(X,Y )´ 1 in (3.41) yields reasonable results already at the first iteration. This approach, however, is equivalent to ignoring the noise statistics in the data (notice that the standard deviation for Poissonian distribution is ¾Æ p ¸, so that P(X,Y ) in the denominator serves as the familiar 1/¾ 2 weighting factor) and therefore leads to larger noise level in the result. 2 The integration over pixels areas, however, was not yet described. Taking into account that the in- volved functions have relatively simple analytical forms, and the integration domains (square pixels) are also simple, it is likely that this integration can be also performed analytically. Nevertheless, nu- merical experiments showed that simple numerical integration using very few points (for example, on a uniform 3£ 3 grid within each pixel) is sufficient. Since it was quite fast as well, no analytical derivations were performed in the present work. 176 behaved symmetric matrix with dimensions»1000£1000, which is easily performed using modern personal computers. The results, however, showed that while the new method has a substantial advantage allowing reconstruction of sliced VMI data, its performance for the full-projection VMI data is not significantly different from oth- er linear methods, such as BASEX or pBasex. Therefore, additional “regularization” ideas had to be added. 3.3.4 Constrained solutions The most obvious constraint that must be imposed on the solution was already mentioned and is that of the nonnegativity. Since the negative parts in the initial ve- locity distribution are physically meaningless, the expansion coefficients in the em- ployed positive basis must be also nonnegative, as explained earlier. From the MLE perspective this constraint means that the prior distribution is not just uniform, but is uniform only in the positive subspace (where all® R;n > 0) and is zero everywhere else (where at least one ® R;n Ç 0). Therefore, location of theL maximum requires constrained optimization instead of the unconstrained solution of gradL Æ 0. In can be shown that this problem effectively reduces to constrained optimization of a quadratic form with linear constraints, and that problem has relatively simple methods of solution. One of them (see, for example, [14]) is based on so-called “active set” approach and leads to a system of linear equations with look very similar to (3.41). The methods requires several iterations to find all “active” constraints, that is, the ® R;n coefficients that must be tied to zero (this is performed by replacing the corresponding equation in (3.41) with ® R;n Æ 0 and appropriately distributing the removed terms among other equations), but is proven to converge in a finite number of steps, which is usually small. Since updating of the inverse matrix required at each 177 iteration can be done in only O(N 2 ) time instead of O(N 3 ) required for the complete inversion, 1 the whole procedure takes only a little longer than obtaining the linear solution described in the previous section. Implementation of this nonnegativity constraint showed that it actually can sig- nificantly reduce the reconstruction noise, but, obviously, only in cases where the initial distribution approaches zero intensity. Specifically, the unphysical negative in- tensities are then effectively redistributed among the adjacent points, reducing their positive overshoots and thus making the overall distribution more smooth. Howev- er, in presence of a broad background, which shifts all intensities to positive values, even the noisy solution is positive, and the addition of nonnegativity constraint can- not improve it because that constraint is already satisfied. This observation suggests that it might be profitable to narrow the class of al- lowed solutions even more, such that the solutions with strong high-frequency os- cillations become excluded. The MLE method then should find the most probable solution among the permitted ones. The best method of limiting the high-frequency oscillations, however, is not so obvious. On the one hand, it is desirable to suppress them as much as possible, but on the other, the resolution should not be affected adversely, that is, separate narrow peaks must not be smoothed. In an attempt to achieve such selectivity the constraints ® R;n > min(® R¡1;n ,® RÅ1;n ) (3.42) were added to the positivity constraints (® R;n > 0). The reasoning behind (3.42) is simple: it does not affect separate narrow peak and has little effect on relatively smooth valleys (including at least 2 grid points) but prohibits single-point dips in 1 Here N is the matrix rank equal to the total number of coefficients ® R;n . The fast inverse matrix update method is known as “Sherman–Morrison formula”. 178 the distribution. 1 Since the main part of the noise consists of alternating high/low intensities, which are not allowed by (3.42), it might be expected that such simple additional constraints should achieve the sought goal, at least to some extent. Implementation of these additional constraints requires only moderate extension of the procedures designed for nonnegative solutions. However, in contrast to the nonnegativity constraints, the domain formed by inequalities (3.42) is not convex. This means that the procedures are not guaranteed to converge to the global maxi- mum ofL . Nevertheless, they can be modified to find all local maxima and choose the global one among them. Also, experience with various practically reasonable data showed that actually the difference even between the solution corresponding to the first local maximum found by the implemented greedy algorithm and the solution at the global maximum is well within the experimental uncertainties, and hence simply the first found maximum can be used. The computational complexity in that case is again usually O(N 3 ), and the whole procedure takes about twice more time than the linear solution. Examples of application of this approach to some experimental data are given in the next section. 3.3.5 Experimental examples The full-projection and sliced velocity map images of the same O 2 4hº ¡¡! OÅO + re- action as in subsection 2.7.1 but taken on the old VMI apparatus described in [15] are presented here for illustration of the new reconstruction method performance. The raw images shown in figure 3.8 were rectified as described in section 3.2 before further analysis. The initial velocity distribution was extracted from the full-projection image us- 1 The fact that any physically meaningful distribution can be considered as a sum of positive peaks of various widths means that presence of sharp dips is very unlikely. 179 Full projection: Central slice: Figure 3.8 Raw velocity map images of O + from O 2 photodissociation at º Æ 44444 cm ¡1 (see section 2.7 and figure 2.29 for details). ing BASEX without regularization (for best resolution) and with regularization pa- rameter equal to 1000 (for noise reduction). The same image was also reconstructed using the present method in the linear variant and with the noise-reducing con- straints (3.42) (and, of course, nonnegativity). The linear solution was almost indis- tinguishable from the BASEX solution without regularization and is therefore not shown in the comparisons below. In addition, the initial velocity distribution was extracted directly from the sliced image (by weighted integration over polar angles) and using the present method with the slicing function described by a Gaussian function fitted to the experimentally measured effective slicing pulse 1 . Since the strongest peaks (see figure 3.9) were reproduced well in all cases (both images and all methods), the comparison of performance is given in figure 3.10 only 1 See figure 2.27 and discussion in section 2.6.2 about the method of measurements. 180 0 1 0 50 100 150 200 250 300 Relativeintensity R,px Parallelcomponent Perpendicularcomponent Figure 3.9 Radial part of parallel (»cos 2 µ) and perpendicular (»sin 2 µ) compo- nents of intensity distribution for data shown in figure 3.8. for the most problematic parts of the distribution. These parts are the lower-velocity part, where reconstructions of the full-projection image show large noise, and the unreconstructed sliced image shows strong blurring, and the higher-velocity part with two peaks having very low intensities 1 . As can be seen from figure 3.10, the BASEX reconstruction of the full-projection image without regularization has the highest noise level, completely masking the low-intensity peaks and making the structure of the small peaks in the lower-velocity region indiscernible. Addition of regularization sufficient for reducing the noise leads to broadening of all peaks, especially significant in the lower-velocity part. The reconstruction of the same image by the present method shows much lower noise compared to the BASEX reconstruction without regularization and, at the same time, better resolution compared to the regularized BASEX reconstruction. Most im- 1 Notice, however, that these peaks can be clearly seen in the raw image, and therefore their presence is not an artifact of the methods that show them in the obtained distributions, but rather their absence from the distribution obtained by BASEX is an indication of its worse performance. 181 0 0.1 0.2 60 80 100 Relativeintensity R,px 0 0.02 200 220 240 Relativeintensity R,px 0 0.1 0.2 60 80 100 Relativeintensity R,px Presentmethod (fullprojection) 0 0.02 200 220 240 Relativeintensity R,px Presentmethod (fullprojection) 0 0.1 0.2 60 80 100 Relativeintensity R,px 0 0.02 200 220 240 Relativeintensity R,px BASEX BASEXregularized BASEX BASEXregularized Fromslicedimage Presentmethod Fromslicedimage Presentmethod Figure 3.10 Comparison of radial dependences of the parallel intensity component (see figure 3.9) extracted from data shown in figure 3.8 using different methods. (Left column shows the low-velocity part. Right column — the low-intensity part.) portantly, the two low-intensity peaks become clearly evident in the reconstructed distribution. The structure in the lower-velocity part, however, still remains not very clear. The distribution extracted directly from the sliced image is the smoothest among 182 all considered ones, but has relatively intense and long “wings” on the lower-velocity side of each peak, which significantly deteriorate the extracted distribution quality in the low-velocity region. Notice however, that the structure of the RÆ 60...80 px part is very similar to what was observed in the regularized BASEX reconstruction. Finally, the distribution reconstructed from the same sliced image by the present method shows only slightly more noise than the “unreconstructed” one and at the same time has excellent resolution in all parts. The low-intensity peaks near RÆ 220 px are much cleaner than in the reconstruction of the full-projection image, and the structure in the lower-velocity part is now also very clear, showing well-resolved peaks with quite low noise level. Notice that this part of the distribution demon- strates the two-point interpeak valleys mentioned in the previous subsection (see (3.42) and the discussion about it). 3.4 Conclusion Examples of application of the two developed and implemented methods for cor- rection of geometric distortions in the raw data (section 3.2) and reconstruction of the initial velocity distributions from the experimental data (section 3.3) to real-world problems provided in the corresponding sections clearly illustrate that both these methods can significantly enhance the resulting performance of the measurement– analysis process in VMI experiments using both full-projection and slicing detection techniques and eliminate at least some of the artifacts that can be observed in results of analysis performed without these methods. The comparison of the distributions reconstructed from the full-projection and sliced velocity map images of the same photodissociation process taken on the same experimental setup demonstrates explicitly that the reconstruction of the sliced im- 183 age has unambiguously better quality in all respects compared to the reconstruction of the full-projection image and the direct integration of the sliced image without re- construction. Since the developed reconstruction method is applicable to slicing with pulses of arbitrary shape, and its implementation works practically as fast as other currently used methods for full-projection reconstruction, the approach based on re- construction of SVMI data with the present method seems to be the best choice for all experimental studies. 3.5 Future work Although the methods described in this chapter demonstrated very good perfor- mance and were proven to be very useful, their present implementation was not convenient for actual use. Basically, the implementations did not go far beyond the “proof of concept” level. They used only file-based input/output and command-line interface, providing no interactivity, meaning that all preparations of the input data and analysis of the output results had to be done by external means. This situation resulted in tedious multiple iterations required to find the appropriate parameters in each particular case (mostly the location of the image center and definition of the grid points for the rectification). Therefore, implementation of a graphical user inter- face with interactive capabilities for data preparation and real-time analysis of the results should make the methods much more user-friendly and convenient for their actual application. Another important aspect of the reconstruction method, mentioned casually in subsection 3.3.3, is that more detailed analysis of the constructed likelihood function permits determination of uncertainties of the obtained most-probable solution. Some developments in that direction were already done in the present work, indicating 184 that reliable determination of the uncertainties is possible, but the actual implemen- tation was not completed. It should be clear that determination of uncertainties of the obtained values is as important as determination of the values themselves, since no meaningful comparison of the data with unknown amount of errors can be done. The implementation of the uncertainty estimation is therefore one of the main objectives of further developments. In addition, improvement of the final results quality might be achieved by re- moval of artifacts, such as “dark” or “bright” spots, from the raw VMI data. If the initial velocity distribution is reconstructed from the whole image, these damaged areas contribute to the obtained distribution, introducing spurious features. More- over their presence in the input data, in principle, breaks the assumptions made in derivation of the employed equations and therefore might make the reconstruction unreliable. 1 At the same time, examination of the likelihood function (3.37) and the subsequent equations reveals that there is no direct connection between the set of expansion coefficients and the set of pixels used for their determination. This sug- gests that if the damaged pixels are removed from the consideration, the resulting equations should be still solvable, 2 and the quality of the resulting solution must be better. In particular, it will not include the artifacts due to these damaged parts. This direction of further developments also seems very promising. Moreover, it is naturally combined with the uncertainty determinations. Namely, comparison of the deviations between the reconstructed projection and the experimental data should indicate the problematic areas with intensity deviations greatly exceeding the de- viations expected for the Poissonian distribution, helping in their identification and 1 This is somewhat similar to the case of geometrical distortions, see footnote on p. 159. 2 If the amount of the removed pixels is not too large, and the remaining pixels contain enough infor- mation. For example, complete removal of all pixels from the central part of the image, obviously, will make the determination of the lowest-velocity components impossible, since they do not contribute to the remaining outer parts of the image. 185 exclusion from the analysis. References [1] R. N. Zare, Photoejection dynamics, Mol. Photochem. 4(1), 1 (1972). [2] J. L. Wiza, Microchannel plate detectors, Nucl. Instr. and Meth. 162(1–3), 587 (1979). [3] M. L. Hause, H. Y. Yoon, F. F. Crim, Vibrationally mediated photodissociation of ammonia: product angular distributions from adiabatic and nonadiabatic dissociation, Mol. Phys. 106(9–10), 1127 (2008). [4] V. Dribinski, A. Ossadtchi, V. A. Mandelshtam, H. Reisler, Reconstruction of Abel-transformable images: the Gaussian basis-set expansion Abel transform method, Rev. Sci. Instrum. 73(7), 2634 (2002). [5] G. A. Garcia, L. Nahon, I. Powis, Two-dimensional charged particle image in- version using a polar basis function expansion, Rev. Sci. Instrum. 75(11), 4989 (2004). [6] F. Renth, J. Riedel, F. Temps, Inversion of velocity map ion images using iterative regularization and cross validation, Rev. Sci. Instrum. 77(3), 033103 (2006). [7] A. V. Komissarov, M. P. Minitti, A. G. Suits, G. E. Hall, Correlated product dis- tributions from ketene dissociation measured by dc sliced ion imaging, J. Chem. Phys. 124(1), 014303 (2006). [8] E. W. Hansen, P.-L. Law, Recursive methods for computing the Abel transform and its inverse, J. Opt. Soc. Am. A 2(4), 510 (1985). [9] H. L. Kim, R. Bersohn, Control of photofragment angular distribution by laser phase variation, J. Chem. Phys. 107(12), 4546 (1997). [10] Y.-Y. Yin, C. Chen, D. S. Elliott, A. V. Smith, Asymmetric photoelectron angular distributions from interfering photoionization processes, Phys. Rev. Lett. 69(16), 2353 (1992). [11] Y.-Y. Yin, D. S. Elliott, R. Shehadeh, E. R. Grant, Two-pathway coherent con- trol of photoelectron angular distributions in molecular NO, Chem. Phys. Lett. 241(5–6), 591 (1995). [12] B. Sheehy, B. Walker, L. F. DiMauro, Phase control in the two-color photodisso- ciation of HD + , Phys. Rev. Lett. 74(24), 4799 (1995). 186 [13] H. Ohmura, T. Nakanaga, Quantum control of molecular orientation by two- color laser fields, J. Chem. Phys. 120(11), 5176 (2004). [14] D. Goldfarb, A. Idnani, A numerically stable dual method for solving strictly convex quadratic programs, Math. Prog. 27(1), 1 (1983). [15] V. Dribinski, Photoelectron and ion imaging studies of the mixed valence/Ryd- berg excited states of the chloromethyl radical, CH 2 Cl, and the nitric oxide dimer, (NO) 2 , Ph. D. dissertation, University of Southern California, 2004. 187 Chapter 4 Overtone-induced dissociation and isomerization dynamics of the hydroxymethyl radical (CH 2 OH and CD 2 OH) 4.1 Introduction The hydroxymethyl radical (CH 2 OH) and its less stable isomer, the methoxy rad- ical (CH 3 O), are relevant to atmospheric and combustion chemistry, being implicated as intermediates in reactions such as OÅCH 3 [1–7] and CHÅH 2 O [8]. Recent studies also suggest that CH 2 OH and CD 2 OH play an important role in the unusual deuteri- um enrichment of interstellar methanol [9], and deuterium-substituted isotopologs of the CH 3 O isomer are involved in processes that strongly affect the HD/H 2 ratio in the Earth’s atmosphere [10]. The high-level electronic structure calculations reported in [11] 1 show that the barrier heights for CH 2 OH¡ ! CH 2 OÅ H dissociation and CH 2 OH¡ ! CH 3 O isomer- ization are very close in energy (»14000 cm ¡1 above the CH 2 OH ground state, 1 These calculations were performed by our collaborators as a computational part of the present study. 188 within »400 cm ¡1 of each other, see figure 4.1), but the rates for each of these 0 5000 10000 15000 Energy,cm −1 D···CHDO12614 H···CH 2 O 12161 H···CD 2 O 12059 CH 2 (H)O13839 CD 2 (H)O13723 CH 2 O···H14205 CD 2 O···H14135 D+CHDO10787 H+CH 2 O 10188 H+CD 2 O 10167 725i 967i 959i CH 3 O 3535 CHD 2 O3433 1934i 1934i CH 2 OH0 CD 2 OH0 13598,13657 13617 1756i 1755i CH 2 O+H10188 CD 2 O+H10167 Figure 4.1 Energy diagram for ground electronic state dissociation and isomer- ization of hydroxymethyl radical according to calculations in [11]. Geometric config- urations of all relevant stationary points are shown in figure 4.2. All energies are relative to the ground vibrational state of CH 2 OH (CD 2 OH). (VSCF/VCI zero-point energies are used for the radical and dissociation products, and RCCST(T)/AVTZ har- monic ZPEs — for all other configurations; corresponding imaginary frequencies are indicated below the barriers). Experimental vibrational levels studied in the present work are also shown for comparison. two processes should strongly depend on the character of the excited vibrational states, and thus might be highly state-specific. The isomerization of vibrationally excited CH 2 OH to CH 3 O must be quickly followed decomposition, since the bar- rier for CH 3 O¡ ! HÅ CH 2 O dissociation is lower than the isomerization barrier by »1700 cm ¡1 . Therefore, both decomposition pathways of CH 2 OH eventually lead to the same CH 2 OÅ H products, although very different distributions of final states 1 might be expected for these channels. The large difference between the dissociation and isomerization barriers in CH 3 O 1 Rovibrational excitation of CH 2 O and translational energies, as well as isotopic combinations in case of deuterium-substituted CH 2 OH. 189 1.083 1.083 0.962 0.962 1.371 1.371 1.079 1.079 10 8.8° 11 3.5° 11 8.7° 12 1.2° [22.9°] [174 . 7 °] (a) CH 2 OH 1.080 1.080 0.962 0.962 1.368 1.368 1.077 1.077 10 9.3° 11 5.0° 12 0.6° 12 4.4° ( b) CH 2 OH planar 1.244 1.244 1.491 1.491 1.096 1.096 12 0.0° 12 0.9° 11 8.1° [88.5°] (c) CH 2 OH ↔ CH 2 O + H 1.388 1.388 1.217 1.217 1.088 1.088 1.262 1.262 57.5° 54.4° 11 7.6° 11 9.1° 11 5.9° [103 . 4 °] ( d) CH 2 OH ↔ CH 3 O 1.379 1.379 1.096 1.096 1.104 1.104 10 5.0° 11 2.6° 11 1.6° 10 7.3° [116 . 4 °] (e) CH 3 O 1.228 1.228 1.102 1.102 1.780 1.780 10 3.1° 12 1.4° 11 6.4° 87.7° [95.4°] ( f ) CH 3 O ↔ H + CH 2 O Figure 4.2 Geometric configurations for relevant stationary points of the poten- tial energy surface (calculated at RCCSD(T)/AVTZ level of theory in [11]). All con- figurations, except the CH 2 OH minimum (a), have C S symmetry. Bond lengths are in angstroms, angles — in degrees (dihedral angles are shown in square brackets). (CH 2 O parameters for reference: r CO Æ 1.208 Å, r CH Æ 1.102 Å,\COHÆ 121.7°.) means that isomerization CH 3 O¡ ! CH 2 OH by vibrational excitation of CH 3 O is un- likely. For example, in [12, 13] the state-specific dissociation of the methoxy radi- cal was studied using stimulated emission pumping from an excited electronic state to reach high vibrational levels of the ground electronic state, mainly in the C¡O stretch. The state-specific decay rates in the tunneling regime and up to»1500 cm ¡1 above the dissociation barrier, almost reaching the isomerization barrier top, were determined. The rates showed large state-to-state fluctuations and were consistent with statistical theories that considered surmounting and tunneling through the CH 3 O¡ ! HÅCH 2 O barrier. However, no indications of isomerization to CH 2 OH were observed, in agreement with the relative barrier heights. Due to experimental difficulties, there are relatively few studies of the ground 190 Table 4.1 Summary of experimental and theoretical (see [11]) vibrational parame- ters of CH 2 OH and CD 2 OH. All energies are in cm ¡1 . Vibrational mode descriptions are approximate (for CD 2 OH, replace “CH” with “CD”); the mixed states in CH 2 OH are denoted by leading contributions in their VCI wavefunctions. Experimental val- ues without references were obtained in the present work; values in italics are from Ar matrix studies. Vibrational level CH 2 OH CD 2 OH Expt. Theor. Expt. Theor. zero-point energy 7956 6747 º 1 O¡H stretch 3675 [16] 3662 3650 [14] 3677 º 2 C¡H antisymmetric stretch 3162 [16] 3116 — 2362 º 3 C¡H symmetric stretch 3043 [16] 3020 — 2198 º 4 CH 2 scissors 1459 [14] 1448 — 1283 º 5 HCOH in-phase bend 1334 [14] 1336 1019 [15] 1013 º 6 C ¡ ¡ O stretch 1176 [15] 1171 1208 [15] 1201 º 7 HCOH out-of-phase bend 1048 [14] 1032 842 [14] 830 º 8 HCOH in-phase torsion 420 [14] 468 347 [15] 461 º 9 HCOH out-of-phase torsion 234 [15] 244 221 [15] 160 2º 1 7158 [16] 7171 — 7191 3º 1 10484 [17] 10514 — 10535 4º 1 j1 4 iÅj1 3 2 1 i 13598 13666 13617 13747 j1 3 2 1 i¡j1 4 i 13657 13767 electronic state of CH 2 OH in spite of its importance. Most fundamental vibrations of CH 2 OH and its isotopologs were observed in an Ar matrix [14] using infrared (IR) ab- sorption spectroscopy. A few fundamentals and several overtones and combinations of out-of-plane modes were observed in effusive molecular beam experiments [15] by means of resonance enhanced multiphoton ionization (REMPI) detection. More re- cent studies [16] in our group employed a supersonic molecular beam and IR–UV double-resonance techniques to observe partially resolved rotational structure of the antisymmetric and symmetric C¡H stretch fundamentals (º 2 andº 3 respectively, see table 4.1), as well as the O¡H stretch fundamental and its first overtone (º 1 , 2º 1 ). In the subsequent work [17] these investigations were extended to 3º 1 and 4º 1 levels of CH 2 OH and 4º 1 of CD 2 OH. The 4º 1 levels of the two isotopologs (»13600 cm ¡1 ) lie 191 above the dissociation thresholds (»10200 cm ¡1 ) and could be detected only by moni- toring H photofragments, indicating a relatively fast predissociation (lifetime»25 ps estimated from the spectral linewidths) of the hydroxymethyl radical at this energy. That work was the only known experimental study of the predissociation dynamics of the hydroxymethyl radical. The observation of predissociation induced by O¡H stretch overtone excitation was intriguing because the O¡H stretch is the reaction coordinate leading to CH 2 OÅ H. This situation is remarkably different from other cases of overtone-induced disso- ciation (see, for example, [18–23]), where the initially excited bond was not broken in the dissociation and retained its bound character even above the dissociation thresh- old. The existence of a barrier for CH 2 OH dissociation was assumed to be the reason why the O¡H stretch preserved its character of a bound vibration. Despite the fact that the calculations [1, 24] available at that time placed the barrier for isomerization to methoxy lower than the barrier for O¡H bond fission, no D products from CD 2 OH were detected. 1 Therefore, it was concluded that isomerization to CHD 2 O was at best a minor channel, and that the dissociation proceeded by tunneling through the bar- rier to CH 2 OÅH. In these previous studies the investigation of dynamics of the hydroxymethyl radical predissociation was not possible because the time-of-flight (TOF) detection method that was employed did not have sufficient resolution to reveal structure in the H fragment kinetic energy distribution (KED). The upgrade of the apparatus with sliced velocity map imaging (SVMI) detection system described in chapter 2 al- lowed to repeat and extend the previous studies at a new level in the present work. In particular, a small amount of D products was observed, providing direct evidence 1 The expected process leading to D products is CD 2 OH¡ ! CHD 2 O¡ ! DÅCHDO. Breaking of the C¡D bond in CD 2 OH with formation of DÅ CDOH was impossible from energy considerations (see sec- tion 4.4). 192 of isomerization in the CD 2 OHÃ ! CHD 2 O system. In addition, internal energy dis- tributions of CH 2 O, CHDO and CD 2 O cofragments created in dissociation of CH 2 OH and CD 2 OH were obtained 1 for the first time. Their analysis showed a qualitative difference for the two decomposition channels (O¡H bond fission and isomerization followed by dissociation), which allowed to estimate the state-specific branchings be- tween these two pathways. The major experimental complication in studies of vibrational predissociation on the ground electronic state involved contributions from very efficient dissociation up- on electronic excitation to the excited electronic states (see [25–29]) by 2- and 3-pho- ton absorption at the high laser intensities needed to effect excitation of the 4º 1 Ã 0 and higher transitions, as well as 1-photon absorption of the H-detection laser ra- diation. While these excitations gives rise predominantly to H fragments with high velocities, in the usual VMI detection the projections of these high velocities on the 2- dimensional detector contribute to signal at both large and small radii of the raw im- age (see chapter 2). These large interfering signals lead to unacceptable background noise in the region of the image that contains contributions from vibrational predis- sociation, as illustrated in figure 4.3, 2 making image reconstruction with sufficient signal-to-noise ratio (SNR) practically impossible. In this situation the introduction of the SVMI method was crucial because, in case of sufficiently thin slicing, contri- butions of different velocities are already separated in the image, eliminating the need for image reconstruction. It was therefore possible to record only the low-KE re- gion of the distribution, effectively discriminating against background signals at high 1 From sliced images of H + and D + 2 Note that in this experiment the H-detection laser polarization was perpendicular to the detector, and thus the contribution from dissociation by electronic excitation, which has strong perpendicu- lar anisotropy (see appendix F), to the central parts of the image was even somewhat smaller than it would be in an experiment with vertical polarization required for the velocity distribution recon- struction by inverse Abel transform. 193 Figure 4.3 Example of full-projection velocity map image of H + from CH 2 OH photodissociation. The faint inner circle correspond to vibrational predissociation at ºÆ 13602 cm ¡1 , whereas most of the signal comes from photodissociation by the H-detection laser (º¼ 27420 cm ¡1 ) through the lowest excited electronic state (see appendix F). kinetic energies. The energy resolution in the raw SVMI data additionally allowed to obtain photofragment yield spectra with resolved translational energies (that is, correlated with specific vibrational levels of the formaldehyde cofragment) by contin- uous scanning of the excitation laser. 4.2 Experimental arrangement and procedures The experimental procedures for the preparation of CH 2 OH samples and detec- tion of H and D atoms were similar to those used in the previous studies (see articles [16, 17] and dissertations [30–32]) and are described only briefly here. The overall scheme of the experiments is illustrated in figure 4.4 and can be described as follows. A molecular beam of CH 2 OH (CD 2 OH) radicals was created in the source chamber, skimmed, and introduced into the differentially pumped detection chamber, where 194 Ly-αand3Ly-α “H-detection”: λ=3Ly-α(∼2mJ) ionoptics detector pulsed nozzle 3%CH 3 OH 1%Cl 2 inHe(∼1.5bar) Cl 2 dissociationlaser (Nd:YAG3rdharmonic) ∼20mJ quartz tube cylindrical lens triplingcell video- camera slicing pulse ∼13500...14100cm −1 (740...710nm) “pump”: ∼35mJ (Kr/Ar) sourceregion detectionregion skimmer MgF 2 lens Figure 4.4 Scheme of the experimental arrangement (top view, not to scale). the radicals were dissociated by focused visible laser radiation. H (D) products were ionized by a counterpropagating laser radiation using REMPI, and the H + (D + ) ions were detected by SVMI. Two kinds of measurements were performed: 1) the depen- dence of fragment yields on excitation frequency (action spectra); 2) velocity distri- butions of the fragments at several fixed excitation frequencies. CH 2 OH (CD 2 OH) radicals were produced as before [30] by the reaction of CH 3 OH (CD 3 OH) with Cl atoms. Typically a mixture of 3 % methanol 1 and 1 % Cl 2 (by vol- ume) in »1.5 bar of helium was introduced into a piezoelectrically driven pulsed nozzle that has a quartz tube attached to it. Cl 2 molecules were dissociated inside the quartz tube (1 mm inner diameter, »7 mm length) by 355 nm laser light (3rd harmonic of Quanta-Ray GCR11 Nd:YAG laser, »20 mJ, focused with 15 cm focal 1 CH 3 OH and CD 3 OH (Aldrich, 99.5 atom % D) were used without further purification. 195 length (f. l.) cylindrical lens): Cl 2 hº ¡ ¡ ! Cl ¤ 2 ¡ ! ClÅCl. (4.1) The produced Cl atoms reacted with CH 3 OH, mostly by H abstraction from the CH 3 group (see [33] and also [34]): ClÅCH 3 OH¡ ! HClÅCH 2 OH, (4.2) forming the CH 2 OH radicals. While possible subsequent reactions of the radicals with Cl and among themselves: CH 2 OHÅCl¡ ! CH 2 OÅHCl, (4.3) CH 2 OHÅCH 2 OH¡ ! OHCH 2 CH 2 OH, (4.4) reducing the net yield are known to have reaction rates comparable and even exceed- ing that of (4.2), the maximum radical yield was obtained at the maximum achiev- able Cl 2 -photolysis laser pulse energy (»20 mJ) and with the laser radiation focused in the last few millimeters of the quartz tube. Since the present experiments used resonance excitation of radical vibrational levels and energy-resolved detection of H fragments, the byproducts (including CH 2 O) of these and other side reactions did not interfere with the measurements. After expansion at the exit of the quartz tube the molecular beam was introduced into the detection chamber through a skimmer (Beam Dynamics,1 mm). The rotational temperature of the radicals in the detec- tion region was 10¡15 K in (see table 4.2), and no apparent vibrational excitation was observed. The tunable radiation for radical excitation was generated by a dye laser (Contin- 196 uum ND6000, using LDS 722 dye in methanol, linewidth »0.1 cm ¡1 ) pumped with the second harmonic of a Q-switched Nd:YAG laser (Continuum PL8000). The ra- diation was focused in the molecular beam by a 30 cm f. l. lens. Pulse energies of »30¡40 mJ (measured before the vacuum chamber window) were used. H (D) fragments were ionized using a 1Å 1 0 REMPI scheme through the 2pÃ 1s transition. The radiation near Lyman-® (82290 cm ¡1 , 121.6 nm) was generated by frequency tripling 2¡3 mJ of the doubled output of a Nd:YAG-pumped dye laser (Continuum NY-81C/ND6000, LDS 751 dye, Inrad Autotracker-II doubler with KDP- D crystal) in a Kr/Ar gas cell (see [30] and [35]). The unconverted part of the radiation passing through the tripling cell was used for near-threshold ionization of the excited H (D) atoms. Experiments with the tripling cell gas mixture showed that the tripling efficiency rises with the total gas pressure, however, at sufficiently high pressures the intense radical-excitation radiation passing through the vacuum chamber and focusing inside the cell caused an optical breakdown in the gas, leading to production of very powerful broad UV/VUV spectrum and disturbing normal operation of the system. Therefore, the total gas pressure had to be limited by »1 bar (»800 Torr). The Kr:Ar ratio near 1 : 3 was adjusted for phase matching in the desired frequency range, different for H and D ionization. Because the bandwidth of the detection laser was narrower than the Doppler pro- file of the H (D) fragments, velocity map images were recorded by frequency scanning of the laser across the Doppler profile in order to achieve unbiased detection efficien- cy. However, for recording of action spectra such scans would take too long, 1 and since qualitative KEDs were sufficient, the detection frequency was fixed at a value 1 That is, the H-detection laser has to be scanned at least once for each value of the excitation laser frequency, which, in turn, should be changed in steps no larger than the laser linewidth (»0.1 cm ¡1 ) in order to prevent omission of sharp features in the spectrum. With the low repetition rate (10 Hz) of the employed lasers, such 2-dimensional scans would be impractical even for frequency ranges covering only one vibrational band. 197 optimized for best one-photon signal. The H + (D + ) ions produced from the photodissociation H (D) fragments were de- tected by SVMI arrangement described in chapter 2. The operation conditions of the ion optics were chosen in each case to provide an imaged KE range somewhat larger than the maximum KE of the studies fragments and»50 ns TOF stretching of the relevant part of the ion cloud, resulting in»1/10 relative slice thickness, which led to reasonable collection efficiency with sufficient KE resolution (»1 %). Kinetic energy release (KER) distributions were obtained by binning all recorded ion positions into small radial intervals, converting the obtained radial distributions to KE distributions using KE–radius dependences obtained from numerical simu- lations (see chapter 2), and finally changing the scale from H (D) kinetic energy to kinetic energy release in the reaction (see chapter 3). Energy-resolved action spectra were obtained by changing the excitation laser frequency in small steps while continuously running SVMI detection. Time stamps for each frequency were recorded, allowing subsequent binning of the detected ions by kinetic energies, as described above, and by laser frequency intervals. The result- ing 2-dimensional spectra (intensity dependence on the excitation photon energy and the kinetic energy release) were analyzed directly by visual inspection of plotted 2- dimensional intensity maps and as state-resolved photofragment yield spectra, ob- tained by integration of the 2-dimensional spectrum over KER intervals correspond- ing to specific vibrational excitations in the molecular cofragment. Besides the direct separation of parasite signals by energy difference, mentioned on p. 193, the application of SVMI in the present experiments significantly enhanced the detection sensitivity because of the following: 1. The use of event counting essentially eliminates detection background noise; 198 only statistical fluctuations of the signal remain. This allowed to observe small sig- nals and significantly improve the SNR by increasing the accumulation time, limited only by stability of the experimental conditions. 2. Although some interfering signals, for example, from photodissociation of un- reacted methanol in the molecular beam, overlapped in kinetic energy with the sig- nal from the studied processes, they had very smooth KE distributions in the studied range, and the sufficiently high KE resolution of the system allowed easy separation of the relatively sharp useful signal from such background contributions. 3. The measured KEDs allowed unambiguous identification of the dissociation cofragment through its vibrational structure, and the parent radical by the observed kinetic energy maximum (which should be consistent with the corresponding dis- sociation energy). These capabilities mean that it was possible to increase the rad- ical production yield without the risk of possible signal contamination by reaction byproducts. Even if such contaminations were observed, they could be detected and separated from the main signal. 4.3 Experimental results and analysis 4.3.1 Action spectra Figure 4.5 shows action spectra for photodissociation of CH 2 OH and CD 2 OH ob- tained by monitoring H and D photofragments in the region where a band identified as 4º 1 Ã 0 transition was observed in the previous work [17]. Each spectrum is de- rived from data obtained as described on p. 198. In monitoring H fragments, only the signal correlated with one-photon excitation and the formaldehyde cofragment in the ground vibrational state (which is the most populated, see table 4.4 below) is plotted. Signals correlated with vibrationally excit- 199 0 13580 13600 13620 13640 13660 13680 H + signal (a)CH 2 OH− →CH 2 O+H 0 13580 13600 13620 13640 13660 13680 H + signal (b)CD 2 OH− →CD 2 O+H 0 13580 13600 13620 13640 13660 13680 D + signal Excitationwavenumber,cm −1 (c)CD 2 OH− →D+CHDO (4ν 1 ) (3ν 1 +ν 2 ) Figure 4.5 Action spectra for CH 2 OH and CD 2 OH one-photon dissociation ob- tained by monitoring H and D products. Black curves show experimental data (with- out background subtraction for H and with — for D). Rovibrational band fits are represented by red curves (contours) and sticks (spectral line positions and intensi- ties). Estimated band origin positions for unperturbed levels of CH 2 OH (see subsec- tion 4.4.1) are indicated by dashed lines. 200 ed cofragments were significantly smaller but had identical excitation wavenumber dependences. Background in the images was relatively small at kinetic energies cor- responding to one-photon signals and thus is not subtracted in the plots. The bands near 13600 cm ¡1 for CH 2 OH and 13620 cm ¡1 for CD 2 OH are in good agreement with the previous results reported in [17] but have much better SNR. In experiments with CH 2 OH an additional band near 13660 cm ¡1 was observed in the present work. It has narrower linewidths and 4¡5 times lower integrated intensity than the main band. 1 In contrast, CD 2 OH did not show other detectable bands within §100 cm ¡1 of the observed band. As discussed in subsection 4.4.1 below, the two bands in CH 2 OH arise from an accidental resonance with another state involving the antisymmetric CH stretch. The higher SNR achieved in the present work allowed to detect also a small amount of D products from CD 2 OH dissociation. Figure 4.5(c) shows the D fragment yield spectrum from CD 2 OH, which is correlated with CHDO cofragments. As with the other bands, all state-resolved action spectra had identical shapes. However, in contrast to the H fragment channel, the fraction of vibrationally excited cofragments was significant, and the total intensity was very low. Therefore, the signal correlat- ed with all vibrational states of CHDO was collected in order to improve the SNR. When monitoring the small D photofragment signal, it was necessary to subtract background contributions. They were estimated by interpolation between the back- ground signal obtained at lower and higher kinetic energies than those produced by one-photon predissociation of CD 2 OH and smoothing over excitation wavenumbers. The background in case of D detection was coming mostly from multiphoton dissoci- ation by the excitation laser. 2 1 This probably explains why it was not distinguished from the more intense noise in the lower-reso- lution spectra recorded in [17]. 2 The electronic state excited by the D-detection laser has repulsive character for the O¡H bond 201 It is evident from comparison of the H and D action spectra of CD 2 OH shown in figure 4.5 that the D and H fragments come from the same source, namely, CD 2 OH overtone-induced dissociation. The small deviations in the shapes of the bands are probably due to the high noise level in the D spectrum and not to rotational state specificity in the processes producing H and D. For accurate determination of the band origins and assessment of the rotational structure, 1 the action spectra were analyzed using PGOPHER, a program for sim- ulating rotational structure [38]. Each band was fitted separately to a single vibra- tional transition under the rigid asymmetric rotor approximation and assuming ther- mal rotational distribution of the radicals. The rotational constants for the ground vibrational state were fixed at the most recent theoretical values [11] (they are in reasonable agreement with previously reported calculations [39]). The band origin, the excited vibrational state rotational constants, the Lorentzian linewidth and the rotational temperature were found by nonlinear least-square fitting (NLSF) of the partially resolved rotational structure. Excitation laser bandwidth was accounted for by Gaussian broadening with FWHM of 0.1 cm ¡1 . Since the CD 2 OH spectrum lacks a pronounced structure, the fit was relatively insensitive to variations of rotational con- (see discussion for figure 4.9 below) and thus can be expected to lead to direct dissociation CD 2 OH¡ ! CD 2 OÅ H, producing H atoms (for example, in experiments conducted in [31] with the “opposite” isotopolog CH 2 OD only D but not H atoms were detected; see also [36] for a theoretical discussion). However, on the way from the Frank–Condon region to the products the excited state has a conical intersection with the ground electronic state, and there is a probability that, instead of continuing O¡H bond stretching, the motion will be directed towards O¡H bond shortening on the ground electronic state (see the later theoretical work [37]). Then the excessive vibrational energy can lead to isotopic scrambling through isomerization to CHD 2 O (see figure 4.1) and eventual dis- sociation producing D and CHDO. The D signal from photodissociation by the D-detection laser was observed in the present experiment, although it was not clear whether it came from such dynamics or simply from¡OD contamination of the radical (either inherited from¡OD contamination of the precursor methanol or produced in the radical-forming reaction). 1 Analysis of rotational structure is helpful in corroboration that the observed spectral profiles corre- spond to rovibrational bands of the studied radicals. The spectral resolution (determined mostly by lifetime broadening) is not sufficient for distinction of individual rotational lines and hence does not allow accurate determination of the rotational constants, but the approximately estimated constants must be in reasonable agreement with theoretical estimations. 202 stants, and the upper-state constants for it were also fixed at the theoretical values. In addition, the NLSF of the CD 2 OH spectrum failed to converge for the linewidth, thus its confidence interval was determined by visual inspection. No attempt to fit the noisy D spectrum was made. A more detailed description of the procedure and comparison of the fitted contours with the experimental data is given in appendix G. The results of the fittings are summarized in table 4.2 1 and plotted in figure 4.5 Table 4.2 Spectral fitting results. All spectroscopic values are in cm ¡1 ; tempera- tures are in K. Confidence intervals are given in x§¢x notation; standard deviations of the fit in units of the least significant digit are given in parentheses. Numbers in italics correspond to theoretical values from [11]. Parameter CH 2 OH CD 2 OH 4º 1 3º 1 Åº 2 4º 1 º 0 13596.9§1 13656.2§1 13616.7§1 13 666 13 767 13 747 A v 6.10(3) 6.09(2) — 6.052 6.074 3.606 B v 0.964(4) 0.995(3) — 0.981 0.977 0.841 C v 0.841(5) 0.822(4) — 0.844 0.843 0.684 w L 1.80(3) 0.80(2) 1.2§0.1 T rot 12.9(1) 13.3(2) (fixed at 13) (see also figures G.1–G.3 in appendix G). Both bands of CH 2 OH were fitted using only a-type transitions, as addition of b-type transitions did not improve the goodness of the fits. At the same time, the best least-square fit for the CD 2 OH spectrum was obtained with about 25 % (by intensity) of b-type transition included. The determined spectroscopic constants are in reasonable agreement with the previous experimental 1 The confidence intervals for band origins include the laser calibration uncertainty. Please note that the quoted standard deviations for other parameters were obtained from a statistical analysis that assumes no systematic discrepancies between the data and the model, and therefore they must be considered with caution. Basically, they indicate the lower limits of the uncertainties. The actual uncertainties are expected to be a few times larger. 203 results [17] and with the recent theoretical values [11]. However, all the fits showed noticeable systematic deviations from the observed bands shapes. A possible reason of this discrepancy is discussed in subsection 4.4.1. 4.3.2 Kinetic energy release distributions KEDs correlated with one-photon dissociation were obtained for H and D frag- ments as described in section 4.2. No noticeable variations in the KEDs within the frequency span of each spectral band were observed. Therefore, quantitative mea- surements were performed only at excitation frequencies of maximum signal in each band: 13602 cm ¡1 and 13662 cm ¡1 for CH 2 OH and 13621 cm ¡1 for CD 2 OH (both H and D channels). The raw sliced velocity map images obtained in these experiments are presented in figure 4.6. The intense outer ring in each image correlates with vi- brationless formaldehyde cofragment. The weak inner rings correspond to various degrees of methylene group rock, wag and scissors excitations in the formaldehyde. The innermost ring in all images, except the last one, correlates with formaldehyde cofragment with excited C ¡ ¡ O stretch. The KEDs extracted from these images are shown in figure 4.7. In all experiments the background was unstructured in the relevant kinetic energy ranges; its intensity was relatively small for the H signals and comparable in size to the D signal. The background estimated by recording data in the same conditions, but with the exci- tation laser detuned from the resonance vibrational transition, 1 is subtracted in all shown plots. 1 A mentioned earlier, multiphoton absorption of the excitation laser contributed to the background in the studied kinetic energy range. Therefore, adequate background estimation only from images recorded without the excitation laser was not possible. At the same time, the KED from multiphoton dissociation did not show structural changes (and was relatively smooth in the overlapping KER range) in the relevant excitation frequency range, and hence could be determined from measurement immediately outside the one-photon band. 204 CH 2 OH¡ ! CH 2 OÅH,ºÆ 13602 cm ¡1 : CH 2 OH¡ ! CH 2 OÅH,ºÆ 13662 cm ¡1 : (maxK¼ 0.62 eV, max¢¿¼ 59 ns) (maxK¼ 0.455 eV, max¢¿¼ 51 ns) »730000 events »200000 events CD 2 OH¡ ! CD 2 OÅH,ºÆ 13621 cm ¡1 : CD 2 OH¡ ! DÅCHDO,ºÆ 13621 cm ¡1 : (maxK¼ 0.62 eV, max¢¿¼ 59 ns) (maxK¼ 0.45 eV, max¢¿¼ 53 ns) »1300000 events »820000 events Figure 4.6 Raw sliced velocity map images of H and D fragments used for deter- mination of kinetic energy release distributions (see figure 4.7). 205 0 1 Relativeintensity CH 2 OH− →CH 2 O+H,ν=13602cm −1 0 1 Relativeintensity CH 2 OH− →CH 2 O+H,ν=13662cm −1 0 1 Relativeintensity CD 2 OH− →CD 2 O+H,ν=13621cm −1 0 1 0 1000 2000 3000 Relativeintensity Kineticenergyrelease,cm −1 CD 2 OH− →D+CD 2 O, ν=13621cm −1 0 0.05 0 0.05 0.1 0 0.02 0.04 1500 2500 gnd C−Hsym C− −O CH 2 scis CH 2 wag C−Hasym CH 2 rock gnd C−Hsym C− −O CH 2 scis CH 2 wag C−Hasym CH 2 rock gnd C−Dsym C− −O CD 2 scis C−Dasym CD 2 rock CD 2 wag gnd C−H C−D C− −O CHDscis CHDrock CHDwag C− −O CH 2 scis CH 2 wag CH 2 rock C− −O CH 2 scis CH 2 wag CH 2 rock C−Dsym C− −O CD 2 scis C−Dasym CD 2 rock CD 2 wag Figure 4.7 Kinetic energy release distributions in CH 2 OH and CD 2 OH one-photon dissociation extracted from SVMI data shown in figure 4.6. Origins of vibrational levels of the formaldehyde cofragment (see table 4.4) are marked by vertical lines. Lower KER parts with expanded vertical scale are show on the right. 206 The velocity distributions for overtone-induced dissociation were isotropic within experimental accuracy. 1 Rotational structure in the formaldehyde cofragments was not resolved. Vibrational structure was resolved, but due to significant rotational excitation in the product the rotational envelopes of most of the vibrational states were partially overlapped. Each of the KEDs was analyzed separately by fitting the rovibrational contour with PGOPHER [38]. The rotational constants and energies of vibrational levels of formaldehyde isotopologs used in the fitting were compiled from experimental da- ta available in the literature (see table 4.4). The dissociation energy (D 0 ), the ex- perimental broadening (modeled with a Gaussian function) and the ground state rotational temperature were determined by least squares fitting of the KED part corresponding to vibrationless formaldehyde. Then the remaining part of the KED was fitted to determine the rotational temperature of vibrationally excited levels and their relative populations. The results are summarized in tables 4.3 and 4.4. The fits and their comparison to the experimental KEDs are shown in appendix H. The dissociation energies for all three observed reactions were determined direct- ly in the presented experiments for the first time. They are consistent with estimat- ed isotopic zero-point energy differences. The latest theoretical predictions [11, 39] of the dissociation energies also lie within the relatively tight confidence intervals of these experimental values. 1 The small apparent anisotropy seen in figure 4.6 is caused by nonuniform detection of H fragments created in the dissociation region with different velocity directions. Namely, the light H atoms are fast enough (»9 μm/ns) to travel a distance comparable to the laser beam waist radius in the few nanoseconds between the radical dissociation and their ionization. Therefore, the detected fraction of vertically ejected fragments is smaller than that of fragments ejected horizontally (and thus re- maining within the ionization laser beam). The employed lasers cannot produce pulses shorter than a few nanoseconds, so the average delay can be reduced only by reducing the temporal overlap, when the end of the ionization pulse overlaps with the beginning of the dissociation pulse. This indeed reduced the apparent anisotropy, but at the expense of significant reduction in the detected signal. SVMI and TOF studies did not show any noticeable dependence on the laser polarization, and hence it was concluded that the velocity distributions were isotropic. 207 Table 4.3 Dissociation energies (D 0 ) determined from kinetic energy distributions of H and D fragments. The confidence intervals include energy calibration and KED fitting uncertainties. Reaction D 0 , cm ¡1 CH 2 OH¡ ! HÅCH 2 O 10160§70 CD 2 OH¡ ! HÅCD 2 O 10135§70 CD 2 OH¡ ! DÅCHDO 10760§60 The observed rotational profiles were fit fairly well by thermal distributions. How- ever, since there is no reason to expect local thermodynamic equilibrium among the rotational levels within each vibrational state, these effective temperatures should be considered only as a measure of the extent of rotational excitation. Significant over- laps of the contours for vibrationally excited levels did not allow meaningful determi- nation of their individual temperatures, and thus a common temperature, different from that of the ground state, was used. The overlaps also caused large correlations among the derived populations, especially for rock and wag excitations, which are close in energy. Because of these problems the total fraction of excited state popula- tions obtained from the fits was also compared with estimation by direct integration of the excited states part of the KED, and these values showed good agreement, as can be seen from table 4.4. The uncertainties quoted in the table do not include sys- tematic errors due to nonuniform detection efficiency, which can be assessed at a few percent. Nevertheless, these errors are consistent for all KEDs and thus should not affect the analysis discussed in subsection 4.4.2 below. 4.4 Discussion The most intriguing result of the present study is the observation of D products from overtone-induced dissociation of CD 2 OH, which explicitly indicates that in ad- 208 Table 4.4 Formaldehyde fragment rovibrational distribution parameters obtained from fitting of H and D cofragment kinetic energy distributions (standard deviations in units of the least significant digit are given in parentheses). Fragment excitation Energy a , cm ¡1 Relative population (fit) b CH 2 OH¡ ! HÅCH 2 O CD 2 OH (13 621 cm ¡1 ) CH 2 O CD 2 O CHDO 13 602 cm ¡1 13 662 cm ¡1 HÅCD 2 O DÅCHDO ground 0 0 0 1 1 1 1 C=O 1746 1702 1724 0.0219(5) 0.0294(10) 0.0333(2) 0 scissors 1500 1100 1396 0.0179(5) 0.0300(11) 0.0103(3) 0.062(3) rock 1249 989 1028 0.0212(7) 0.0466(17) 0.0083(6) 0.151(9) wag 1167 938 1059 0.0211(7) 0.0377(17) 0.0091(5) 0.185(9) Excited fraction, % fit 7.6 12.6 5.7 28.8 integration 7.8 12.8 5.9 28.4 T eff rot , K ground 145(10) 155(10) 120(5) 320(20) excited 110(10) 140(20) 130(15) 180(15) a Experimental vibrational energies taken from Refs. [40] (CH 2 O), [41, 42] (CD 2 O) and [43, 44] (CHDO). b Population estimations for rock and wag are strongly correlated (½»¡0.8) in all cases due to a significant overlap of their rotational envelopes. 209 dition to dissociation by O¡H bond fission, isomerization to methoxy also takes place. As shown in previous work [45], the C¡H bond strength in CH 2 OD is»27400 cm ¡1 , 1 much greater than both the O¡H bond strength in CD 2 OH (»10100 cm ¡1 ) and the one-photon excitation energy (Ç14000 cm ¡1 ). That is, direct C¡D bond fission in CD 2 OH was impossible in the present experiment. 2 Yet the barrier height for iso- merization to CHD 2 O is only»13700 cm ¡1 , and the barrier for CHD 2 O¡ ! DÅCHDO dissociation is even lower (»12600 cm ¡1 ), as shown in figure 4.1. It can be concluded therefore that D atoms are produced by isomerization followed by dissociation. Very different KEDs (figure 4.7) of the H and D channel support the hypothesis that these products are created by two different processes after radical photoexcitation. 4.4.1 Overtone spectroscopy and state interactions The action spectra and their analyses are in agreement with previous results reported in [17], except that with better SNR in the present experiment an addition- al band near 13660 cm ¡1 was observed. Based on the previous analysis, it appears that the zeroth-order 4º 1 state carries most of the oscillator strength for the ob- served transitions. However, the present work together with the theoretical study [11] demonstrates that it is mixed in CH 2 OH with another close-lying vibrational state. The mixing does not occur in CD 2 OH, reinforcing the interpretation that the interaction derives from an accidental near-resonance specific to CH 2 OH. A very sim- ilar situation has been reported in [46] for the overtone spectroscopy of CH 3 OH, where 5º 1 is strongly mixed with 4º 1 Åº 2 (º 1 is the O¡H stretch andº 2 is one of the asymmetric C¡H stretches), and this interaction is also affected by isotopic substitu- 1 The isotopic difference between this value and the C¡D bond strength in CD 2 OH is negligible for this discussion. 2 The resonant character of the action spectrum shown in figure 4.5 also excludes the possible ¡OD contamination as a source of the D fragments. 210 tions. The available experimental data (see table 4.1) can be uses to estimate the ener- gies of the unperturbed zeroth-order states of CH 2 OH. For the O¡H stretch overtone 4º 1 this can be done by fitting the anharmonic expression G(v 1 )Æ! 1 (v 1 Å1/2)Å x 11 (v 1 Å1/2) 2 (4.5) to the lower O¡H stretch overtones (v 1 Ç 4), which are not perturbed, and substitu- tion of v 1 Æ 4 into the resulting expression. The value (4º 1 ) est Æ G(v 1 Æ 4)¡ G(v 1 Æ 0)¼ 13625 cm ¡1 was obtained in this process. The energy of the zeroth-order 3º 1 Åº 2 level, which is the best candidate for strong coupling, can be estimated from the ex- pression 1 G(v 1 Æ 3,v 2 Æ 1)¼ G(v 1 Æ 3)ÅG(v 2 Æ 1). (4.6) The predicted value (3º 1 Åº 2 ) est Æ G(v 1 Æ 3,v 2 Æ 1)¡G(v 1 Æ 0,v 2 Æ 1)¼ 13646 cm ¡1 is indeed very close to (4º 1 ) est . The positions of these estimated band origins are indi- cated in figure 4.5. As can be seen, they both lie in between the two observed bands, as expected for two states that repel each other. Since the lower observed transition has 4¡5 times higher intensity than the upper one, the corresponding excited state must carry a larger 4º 1 coefficient. 2 The vibrational configuration-interaction (VCI) calculations reported in [11] support this analysis, showing that both bright states found in this region have large contributions from 4º 1 and 3º 1 Åº 2 basis functions (see table 4.1). Based on the similarity in the linewidths of the transitions to 4º 1 in CH 2 OH and 1 Although this expression should have an anharmonic cross-term, no experimental data for its esti- mation is available. 2 According to [16] and [39], the º 2 mode has several times weaker IR intensity than the º 1 mode. Although the zeroth-order 3º 1 Åº 2 state could be excited directly, its dissociation without coupling to 4º 1 is unlikely, as discussed in subsection 4.4.3 below. 211 CD 2 OH, it was suggested previously [17] that the linewidths are dominated by the dissociation rate. Otherwise, one might expect different broadenings for the two iso- topologs, which have different densities of states and couplings. The present work confirms that the bands that are mostly 4º 1 have comparable linewidths in both isotopologs. At the same time, among the two bands of CH 2 OH the band with a large 3º 1 Åº 2 contribution has noticeable narrower lines than the “4º 1 ” band (see table 4.2). This suggests state-specificity in the dissociation, which is discussed in subsection 4.4.3. The inability to fit well the action spectra with isolated transitions is quite likely caused by the presence of other weakly absorbing states arising from weaker cou- plings of the bright states discussed above with combinations of lower frequency vi- brations. The inclusion of a b-type transition to improve the CD 2 OH spectrum fit might therefore be misleading and simply serve as apparent compensation of small contributions from other bands. 4.4.2 Product state distributions The analysis of the H and D KEDs summarized in table 4.4 sheds light on the dissociation dynamics of the hydroxymethyl radical in the ground electronic state. The dissociation of the CD 2 OH isotopolog enables a distinction between the H atom initially in the ¡OH group (which carries most of the optical excitation) and the D atoms initially attached to the carbon atom, and thus observation of D products explicitly demonstrates the presence of the isomerization channel. Such distinction, however, is impossible for CH 2 OH. Nevertheless, analysis of the experimental results indicates that isomerization is responsible for a fraction of H products in both radical isotopologs. 212 Note that KED for the CD 2 OH¡ ! CHDOÅ D reaction shows no C ¡ ¡ O stretch ex- citation in the formaldehyde cofragment (see figure 4.7 and table 4.4), whereas all H-producing reactions lead to a small but quite consistent population (»3 %) in this mode. In contrast, methylene group deformation modes (scissors, rock and wag) are significantly excited (»28 % total population) in the D-producing reaction but show smaller and variable excitations when monitoring H fragments. These observations suggest that direct O¡H bond fission produces formaldehyde that is predominantly vibrationless or has small C ¡ ¡ O stretch excitation, but excitation of the deformation modes is associated mainly with the isomerization channel contribution, which is dif- ferent in each case. The positive correlation between the effective rotational temper- atures and the extent of the deformation modes excitations indicates that rotational excitation is lower for direct O¡H bond fission and higher for dissociation after iso- merization. 1 In order to estimate the branching between isomerization and direct dissociation, the experimental vibrational population distributions were analyzed in terms of a simple model with the following assumptions: 1. The products observed in each experiment come from two independent disso- ciation channels (direct and through isomerization), and thus the observed distribu- tions are linear combinations of the distributions for each channel. 2. The distribution for each of the two channels is the same in all experiments. That is, it does not depend on isotopic substitution effects and variations in the rad- ical excitation energy in the narrow range (»100 cm ¡1 ) used in the present experi- 1 To be precise, only the two cases of CH 2 OH dissociation can be compared directly, because different masses of the H and D fragments and different moments of inertia of the formaldehyde cofragments should affect the energy partitioning between translations and rotations in the isotopically substi- tuted species. However, these isotopic effects alone cannot explain the almost threefold difference between the rotational temperatures of vibrationless CD 2 O and CHDO. See appendix I for further discussion regarding this issue. 213 ments. These assumptions allow to write 16 equations (4 vibrational populations in 4 experiments) for the observed relative populations p e,v of state v in experiment e: p e,v Æ (1¡ k e )p d,v Å k e p i,v , (4.7) where p d,v and p i,v are the populations of state v for the direct and isomerization channels respectively, and k e is the fractional contribution of isomerization channel in experiment e. Normalization to p e,0 Æ 1 is automatically satisfied by setting p d,0 Æ p i,0 Æ 1. For the CD 2 OH¡ ! DÅCHDO experiment the value k e Æ 1 is known, and thus the 16 equations have 11 unknowns: 4 excited vibrational populations for each of the 2 channels plus the isomerization contributions k e in 3 experiments. The system of nonlinear equations was solved numerically by minimization of weighted squared error (using covariance matrices for the experimental populations obtained in the KED fittings) under the constraints p d,v > 0, p i,v > 0, and 06 k e 6 1. Despite the fact that the number of unknowns is smaller than the number of equa- tions, the system has one degree of freedom. Namely, (4.7) can be recast as p e,v Æ (1¡ k e )p d,v Å k e p i,v Æ (1¡ k 0 e )p 0 d,v Å k 0 e p i,v , (4.8) 8 > < > : p 0 d,v Æ (1Å®)p d,v ¡®p i,v , k 0 e Æ (k e Å®)/(1Å®) (4.9) for any ®. This variation, however, is limited by p d,v > 0, k e > 0 restrictions, which are reached at p d,v Æ 0 for rock population on one side and k e Æ 0 for CD 2 OH¡ ! HÅ CD 2 O experiment on the other. The model results are summarized in table 4.5, where the percentage of products from isomerization refers to the total yield and was calcu- 214 Table 4.5 Formaldehyde vibrational populations predicted by the model. Deviations from the experimental values (see table 4.4) are given in parentheses in units of the least significant digit. Fragment excitation Relative population CH 2 OH¡ ! HÅCH 2 O CD 2 OH (13 621 cm ¡1 ) Direct dissociation a 13 602 cm ¡1 13 662 cm ¡1 HÅCD 2 O DÅCHDO ground 1 1 1 1 1 C=O 0.0293(Å74) 0.0251(¡43) 0.0320(¡13) 0.000(Å00) 0.0320...0.0333 scissors 0.0160(¡20) 0.0234(¡65) 0.0113(Å10) 0.068(Å12) 0.0113...0.0089 rock 0.0218(Å06) 0.0450(¡16) 0.0072(¡11) 0.183(Å07) 0.0072...0.0000 wag 0.0218(Å07) 0.0414(Å37) 0.0095(Å04) 0.158(¡12) 0.0095...0.0034 % isomerization 11 . . . 15 26 . . . 30 0 . . . 5 100 0 a Isomerization channel distribution can be seen in the DÅCHDO column. 215 lated as k e P v p i,v P v p e,v . (4.10) Detailed plots comparing the experimental KEDs and those obtained by using the model are shown in appendix H. As can be seen from the obtained parameter ranges in table 4.5, the uncertainty caused by underdeterminacy of the system is relatively small and does not affect the usability of the results. 1 Comparison of populations described by the model with those obtained by direct fitting of the experimental data, given in table 4.5 (see also figure H.1 for visual comparison of KEDs reproduction), shows that this simple model reproduces the experimental results fairly well, except the excited C ¡ ¡ O stretch population for the main band of CH 2 OH, for which the discrepancy is »16 standard deviations of the fit and »1/3 of the observed value. This disagreement is most probably a result of deviations from the second assumption of the model and does not undermine the qualitative conclusions. The model outcome is that in CH 2 OH dissociation 11¡15 % of the events proceed by isomerization when the excited level is dominated by 4º 1 , whereas the corresponding branching for the level dominated by 3º 1 Åº 2 is larger, 26¡30 %. In dissociation of CD 2 OH from a purer 4º 1 level, isomerization accounts for Ç13 % events (estimated fromÇ5 % fraction for the H channel). It is noteworthy that no excitation of C¡H (C¡D) stretch vibrations in the formal- dehyde products was observed in either reaction, even though the VCI calculations in [11] showed that the involved CH 2 OH (CD 2 OH) excited states contain considerable antisymmetric C¡H (C¡D) stretch contributions. This is yet more surprising in the 1 A quantitative measurement of D/H product ratio in CD 2 OH dissociation would allow unambiguous estimation of the direct dissociation distribution. However, an accurate measurement is difficult, and the model assumptions do not warrant a more quantitative evaluation. Qualitatively, the D yield was at least an order of magnitude smaller than the H yield, which is consistent with the branching estimated from the model. 216 light that the C¡D stretch frequencies in CD 2 O and CHDO are rather similar to the C ¡ ¡ O frequencies. 1 4.4.3 Competition between O¡H bond fission and isomeriza- tion The radical decomposition mechanism in the experiments reported above is not immediately obvious. On the one hand, one might expect that excitation of vibrations with principally O¡H stretch character and energies very close to the barrier height for O¡H bond fission should lead to relatively quick dissociation. However, while this fission is actually observed, its rate is too low (the barrier height estimated from the dissociation rate in [17] is»1000 cm ¡1 higher than the present theoretical value). On the other hand, the isomerization barrier is lower by»400 cm ¡1 than the barrier for O¡H bond fission, and nevertheless, the isomerization rate is even smaller than the O¡H bond fission rate, although is not negligible. The calculations [11] in fact show that the lowest barrier for O¡H bond fission corresponds to the geometry with the¡OH group twisted from the quasiplanar struc- ture of the radical [11, 39], such that the HOC plane becomes perpendicular to the HCH plane (see figure 4.2). The isomerization pathway involves the same¡OH group torsion, but additionally requires a large change of the HOC angle. Therefore, the O¡H stretch overtone excitation is not an effective way to promote dissociation and is even less effective for isomerization, leading to relatively long lifetimes of the vi- brationally excited states. Assuming that the observed spectral linewidths are determined by upper state 1 The small deviation that can bee seen in figure H.1 between the fits and the data for the CD 2 OH¡ ! CD 2 OÅ H case in the range »2000...2400 cm ¡1 might actually correspond to a minor excitation of the C¡D stretch modes (see figure 4.7), but it has very small amplitude and can be caused by a small excess of high rotational excitations in the C¡ ¡O stretch mode. 217 lifetimes dominated by dissociation rates, it appears that the overall dissociation rate of CH 2 OH from the level that is of purer 4º 1 composition is nearly twice high- er than the rate from the level with smaller 4º 1 contribution. At the same time the observed fraction of isomerization products shows the opposite trend (approximately twice the total hydrogen yield from isomerization for the longer-lived state). These observations suggest that the isomerization rate is largely insensitive to the initial state composition, whereas the O¡H bond fission rate decreases when the state has a smaller 4º 1 component. Indeed, the state dominated by 3º 1 Åº 2 must have smaller wavefunction amplitudes at large O¡H distances than the state dominated by 4º 1 (see appendix J), which explains its lower dissociation rate (regardless of whether tunneling or passing just above the barrier is involved), even though the total ex- citation energy is slightly higher. Since the isomerization transition state has less extended O¡H distance (see figure 4.2) the isomerization rate should be less sensi- tive to the degree of O¡H stretch excitation. A quantitative description of these decomposition processes would require quan- tum-mechanical calculations in at least 3 dimensions (O¡H stretch, ¡OH torsion, and HOC bending), and perhaps full dimensionality to adequately describe the prod- uct state distributions (even though C¡H (C¡D) stretches are not excited in the prod- ucts, they must participate in the CD 2 OH¡ ! CHD 2 O¡ ! DÅCHDO process). 4.5 O¡H bond fission and electronic states An accurate theoretical description of the CH 2 OH¡ ! CH 2 OÅ H dissociation pro- cess might be difficult due to specific properties of the potential energy surface (PES) connecting the initial radical geometry, through geometries with an extended O¡H bond, to geometry of the products, where the O¡H bond is completely broken. 218 The main reason why the CH 2 OH¡ ! CH 2 OÅ H reaction cannot occur in nearly planar geometries is that the ground electronic state ˜ X 2 A 00 of the radical is antisym- metric with respect to reflection in the plane (see figure 4.2(b)), whereas the total electronic state of the products, CH 2 O( ˜ X 1 A 1 ) and H(1s 2 S 1/2 ), is symmetric. There- fore, if considered exactly in the plane, the ground state of the radical correlates with products in excited states, and ground states of the products correlate with the first excited state (3s 2 A 0 ) of the radical. Different symmetries imply that these states can intersect, and they indeed form a seam of symmetry-allowed conical intersections with planar geometries and extended O¡H distances [36]. However, upon out-of-plane distortions of the nuclear configuration, such as¡OH group torsion, the symmetry is broken, and the two electronic states become of the same trivial A symmetry and are allowed to interact. 1 The interaction pushes the states apart in energy, lowering the energy of the lower state. In addition, the elec- tronic configuration can now change smoothly, so that the radical ground state be- comes continuously connected to the products ground state by a path going around the conical intersection. A part of the PES near the lowest conical intersection point of the two electronic states is shown in figure 4.8. 2 The plot corresponds to a 2-di- 1 The transition state configuration shown in figure 4.2(c) actually again has C S symmetry, but both electronic states are symmetric (A 0 ) with respect to reflections in that symmetry plane and thus also can interact. As explained below, the interaction strength in fact reaches its maximum in this symmetric configuration. 2 All calculations reported in this section were performed with computational chemistry programs GAMESS [47] by me and with Firefly [48] by Dr. Ksenia Bravaya, to whom I am very grateful for the provided computational results and extremely helpful discussions. The calculations were performed using SA-MCSCF and EA-EOM-CCSDt methods with aug-cc-pVDZ and aug-cc-pVTZ basis sets in GAMESS (see references in [47]), and using XMCQDPT2/SA-2-CASSCF(13/10) method and NBO analysis with basis sets 6-311+G* and a combination of DZP+Rydberg(C,O) and DZP+Diffuse(H) in Firefly (see references in [48]). It should be noted that all these methods with all basis sets showed qualitatively very similar results, and results of the most sophisticated methods with the largest basis sets were in good quantitative agreement with the results published in [11] regarding station- ary points of the ground-state PES and [37] regarding the conical intersection. Therefore, only the most suitable results will be given, with details mentioned only when needed. For example, both leaves of the surface shown in figure 4.8 were calculated using the EA-EOM-CCSDt method with the aug-cc-pVDZ basis set. 219 −90 0 90 ϕ OH , ◦ 1.2 1.3 1.4 1.5 1.6 1.7 r O−H ,Å 15000 20000 E,cm −1 Figure 4.8 Part of potential energy surface showing the conical intersection at r O¡H ¼ 1.38 Å,' OH Æ 0° and lowest barriers at r O¡H » 1.5 Å,' OH Æ§90°. mensional cut of the 9-dimensional PES with all degrees of freedom, except the O¡H bond length (r O¡H ) and the¡OH group torsional angle (' OH ), fixed at the values re- ported in [37]. 1 In principle, the energy of the lowest barrier is somewhat lower than that of the lowest saddle point in this cut, since all other degrees of freedom were not allowed to relax in these calculations, but the actual difference in geometries, and hence the energies, is quite small compared to the features demonstrated by the plot. The overall behavior of both electronic states near the conical intersection is rea- sonably well explained by very simple considerations. The electronic structure of the 1 r C¡H Æ 1.086 Å (cis), 1.083 Å (trans), r C¡O Æ 1.236 Å, µ HCH Æ 120.3°, µ HCO Æ 119° (trans), µ HOC Æ 107.7°. 220 CH 2 OHÃ ! CH 2 OÅH system can be described in terms of molecular orbitals as 1 ground: (1s C ) 2 (1s O ) 2 (¾ CH ) 2 (¾ CH ) 2 (¾ CO ) 2 (lp O ) 2 (¼ CO ) 2 (¾ OH ) 2 (¼ ¤ CO ) 1 , (4.11) excited: (1s C ) 2 (1s O ) 2 (¾ CH ) 2 (¾ CH ) 2 (¾ CO ) 2 (lp O ) 2 (¼ CO ) 2 (¾ OH ) 2 (¾ ¤ OH ) 1 (4.12) at shorter O¡H distances, closer to the CH 2 OH geometry, 2 and as ground: (1s C ) 2 (1s O ) 2 (¾ CH ) 2 (¾ CH ) 2 (¾ CO ) 2 (lp O ) 2 (¼ CO ) 2 (¾ OH ) 2 (¾ ¤ OH ) 1 , (4.13) excited: (1s C ) 2 (1s O ) 2 (¾ CH ) 2 (¾ CH ) 2 (¾ CO ) 2 (lp O ) 2 (¼ CO ) 2 (¾ OH ) 2 (¼ ¤ CO ) 1 (4.14) at longer distances, towards the CH 2 OÅ H limit, in which the delocalized orbitals transform to localized ones as¾ OH 7! lp O and¾ ¤ OH 7! 1s H . As can be seen, the differ- ences are only in the populations of the antibonding ¼ ¤ CO and ¾ ¤ OH orbitals and the ordering of the states by energy. A consistent picture, however, requires consideration of 4 orbitals, including the “paired” bonding orbitals ¼ CO and ¾ OH , and 3 electrons distributed among them. In case of planar configuration all 4 involved molecular orbitals are orthogonal, since ¼ (¤) CO are antisymmetric with respect to reflection in the molecular symme- 1 “lp O ” denotes a lone-pair orbital on the O atom (combination of 2s O and 2p O ). Different methods of molecular orbital localization might converge to representations (2p C ) 2 (2p O ) 1 or (2p O ) 2 (2p C ) 1 in- stead of the combination (¼ CO ) 2 (¼ ¤ CO ) 1 , but this delocalized representation is more useful for our purposes. Please note also that all these “bonding” and “antibonding” orbitals are not necessarily symmetric. Moreover, the relative contributions of the atomic orbitals in them changes as the molec- ular geometry changes (see below). Their “strengths” might be also different. For example, the state described as ...(¾ OH ) 2 (¾ ¤ OH ) 1 is not weakly bound in O¡H, as might be expected, but is actually weakly repulsive. 2 The lowest excited state at the CH 2 OH geometry is historically classified as “3s Rydberg” [49] be- cause its singly-occupied molecular orbital has a relatively large contributions from the 3s orbitals of the O and C atoms, which determine its long-range behavior. Nevertheless, a significant part of that molecular orbital is much more compact and has a¾ ¤ OH character (see figure 4.9) responsible for the rapid O¡H bond breaking upon excitation to this state. Moreover, its ¾ ¤ OH character increases, and the 3s character decreases with elongation of the O¡H bond, so that in the considered region near the conical intersection the orbital becomes almost purely valence antibonding¾ ¤ OH . 221 Lowest excited state: ¼ CO (shifted towards O) ¾ ¤ OH Å3s O,C ¼ ¤ CO (shifted towards C) occupancy: 1.93 occupancy: 1.00 occupancy: 0.08 Ground state: ¼ CO (shifted towards O) ¼ ¤ CO (shifted towards C) ¾ ¤ OH Å3s O,C occupancy: 1.99 occupancy: 1.01 occupancy: 0.02 Figure 4.9 Relevant natural molecular orbitals of the lowest excited and the ground electronic states at the CH 2 OH PES minimum geometry (see figure 4.2(a)). Isosurfaces enclosing 1/2 (and 3/4 for ground-state ¼ CO ) of the probability are plot- ted. (In natural-orbital representation the “¾ OH ” character is distributed among sev- eral orbitals, which are not shown here.) try plane, and ¾ (¤) OH are symmetric. The C¡O distance undergoes a relatively small change in the considered dissociation process, 1 but the O¡H distance changes signif- icantly, from»1 Å to infinity. The energy splitting between¼ CO and¼ ¤ CO therefore re- mains almost constant, whereas the splitting between¾ OH and¾ ¤ OH is large at small O¡H distances and decreases as the distance increases. This situation is schemati- cally illustrated in figure 4.10, from which the orbital populations and the ordering 1 From»1.37 Å to»1.21 Å, see figure 4.2. In the present 2-dimensional calculations near the lowest conical intersection geometry it was fixed at 1.236 Å, as mentioned earlier. 222 σ ∗ OH σ OH π ∗ CO π CO σ ∗ OH →1s H σ OH →lp O π ∗ CO π CO σ ∗ OH σ OH π ∗ CO π CO σ ∗ OH σ OH π ∗ CO π CO shortO−H longO−H groundstate excitedstate Figure 4.10 Relative orbital energies and occupancies in planar geometries. of the states is immediately obvious. 1 At the same time, the situation in T-shaped geometries 2 is more complicated. The ¼ CO and ¾ OH orbitals (as well as their antibonding pairs) are no longer orthogonal but have a significant overlap near the O atom. However, the 4 involved atomic or- bitals (2p C , 2p O in the symmetry plane, 2p O perpendicular to the symmetry plane, and 1s H ) can be separated by symmetry into 3 symmetric orbitals and 1 antisymmet- ric orbital (the perpendicular 2p O ). The antisymmetric orbital is already orthogonal to the 3 remaining atomic orbitals, out of which 3 orthogonal molecular orbital can be formed by appropriate linear combinations, as illustrated in figure 4.11. One of these orbitals is bonding in both C¡O and O¡H and can be considered as a¼ CO Å¾ OH combi- nation, another is totally antibonding, being¼ ¤ CO Å¾ ¤ OH , and the third is nonbonding, consisting of atomic orbitals combination 2p C Å 1s H . The obvious energy ordering of these orbitals is shown in figure 4.12. The ground state electronic configuration is also obtained trivially, but selection between the two possible configurations shown 1 The limiting orbitals for infinite O¡H distance are not shown for the excited state on purpose. As- sumption that the indicated electronic configuration conserves as the orbitals transform to lp O and 1s H leads to unphysical dissociation products CH 2 O – Å H + . In reality, at longer O¡H distances the 223 symmetricbonding 2p C 2p O 1s H 2p C 2p O 1s H 2p C 1s H + + + − + + + + + 0 + − + + − + − + + − symmetricantibonding nonbonding − − − − Figure 4.11 Orthogonal molecular orbitals in T-shaped geometries. a ground state excited state b n Figure 4.12 Orbital ordering and occupancies in T-shaped geometries. in the figure for the lowest excited state requires additional reasoning. Taking into account that double occupation of the nonbonding orbital would lead to more local- ized electron density, especially near the H atom, 1 the configuration shown by dotted lines is expected to lie higher in energy than the configuration obtained by the n! a excitation from the ground state. In contrast to the planar case, variations in the O¡H distance do not change the order of the 3 orbitals, and therefore the electronic states configurations are pre- configuration starts to interact with other states, and the character of this excited state changes. 2 That is, when HOC and HCH planes are perpendicular, as in figure 4.2(c). 1 If the antibonding orbital has equal contributions from 2p O and 1s H , allocating 2 electrons to it would mean that 1 of these electron resides on the H atom. Since one more electron must be placed in the bonding orbital, which also has nonzero density near the H atom, the total H atom charge would become negative. 224 served. The orbital shapes, however, undergo changes caused by distance-dependent variation of the¼ (¤) CO and¾ (¤) OH contributions and the fact that the¾ (¤) OH pair decompos- es into atomic orbitals at long O¡H distances. These transformations can be summa- rized as short O¡H long O¡H ¾ OH Ã [ sym. bonding 7! ¼ CO ¾ ¤ OH Ã [ sym. antibonding 7! ¼ ¤ CO 2p C Ã [ nonbonding 7! 1s H (4.15) Notice that the resulting ground state, as expected, correlates with CH 2 O(...(¼ CO ) 2 ) and H(1s) on one side and the radical ground state on the other, although the latter is now described as ...(l p O ) 2 (¾ OH ) 2 (2p C ) 1 instead of ...(¼ CO ) 2 (¾ OH ) 2 (¼ ¤ CO ) 1 given by (4.11). This change in orbital classification is caused by the fact that the ¾ (¤) OH and ¼ (¤) CO orbitals cannot coexist in T-shaped geometries (see above). The orbital picture at intermediate ¡OH group torsion angles is more complex, since all 4 atomic orbitals become mixed in all 4 orthogonal molecular orbitals, and there are no symmetry considerations that can be used for qualitative reason- ing. Nevertheless, the main idea that the state interaction comes from the overlap of nonorthogonal “primitive” ¾ (¤) OH and ¼ (¤) CO orbitals holds and allows to conjecture that the interaction strength should depend on' OH through the overlap integral as sin' OH , leading to a PES shape near the conical intersection point similar to what is shown in figure 4.8. The qualitative discussion presented above is completely confirmed by electronic structure calculations. For example, the 3-center natural orbitals obtained at r O¡H Æ 1.37 Å (corresponding to maximal state interaction and therefore orbital mixing) and ' OH Æ 90° are shown in figure 4.13. Notice that the natural orbitals reproduce the MO–LCAO compositions predicted in figure 4.11 very well. The natural bonding or- 225 Lowest excited state: sym. bonding sym. antibonding nonbonding occupancy: 1.84 occupancy: 0.92 occupancy: 0.26 Ground state: sym. bonding nonbonding sym. antibonding occupancy: 1.92 occupancy: 1.00 occupancy: 0.09 Figure 4.13 3-center natural orbitals from XMCQDPT2/SA-2-CASSCF calcula- tions (plotted using 1/2 probability, as in figure 4.9). Compare to the schematic or- bital shapes shown in figure 4.11. bital (NBO) analysis performed for both states left these orbitals almost unaffected. 1 Both states (ground and excited) expressed in both these sets of orbitals are also rel- atively pure, having occupancies of corresponding orbitals close to 2 and 1, although the excited state shows a noticeable contribution from the nonbonding orbital. 2 1 The main change was a reduction of the O atom orbital component in the nonbonding orbital, making it even closer to the qualitative picture shown in figure 4.11. The occupancies also did not change significantly, being (1.92, 1.02, 0.08) for the ground state and (1.84, 0.88, 0.33) for the lowest excited. It should be mentioned that an attempt to perform NBO analysis without 3-center bonds produced absolutely unsatisfactory results with poor decomposition, additionally changing discontinuously up- on O¡H distance variations. This circumstance reinforces the qualitative conclusion made above that adequate description of the adiabatic electronic states in the transition region (near the barrier) re- quires 3-center orbitals. 2 This contribution, as well as the decreased occupancy of the bonding orbital, actually might be ex- 226 The PES shape was also reproduced by all methods that take interaction of the two electronic states into account. Among them the 2-state state-averaged MCSCF with active space consisting of only two orbitals and one electron is the simplest and therefore the most illustrative. 1 The r O¡H profiles at several ' OH values obtained in these calculations are shown in figure 4.14. As can be seen, the main features of −114.35 −114.30 −114.25 −114.20 1.1 1.2 1.3 1.4 1.5 1.6 1.7 E MCSCF ,Ha r O−H ,Å ϕ OH = 0 ◦ 40 ◦ 90 ◦ Figure 4.14 SA-MCSCF energies as functions of O¡H distance and ¡OH group torsion. Symbols correspond to values from the calculations. Fitted hyperbolas are shown by solid curves, and their asymptotes — by dashed (ground state) and dotted (excited state) lines. the PES plotted in figure 4.8 are reproduced correctly even with such minimalistic electronic structure calculations, which means that the major factor determining the PES shape near the conical intersection and the lowest dissociation barriers is indeed the interaction between the two electronic states. plained by a small admixture of the “dotted” excited configuration shown in figure 4.12. 1 Note that 1 : 1 state averaging is crucial in this case. First, it ensures that both states are described with equal quality. Second, if only the lower-state energy is optimized, the MCSCF solution degen- erates simply to the ROHF case (see below) because the two configurations (in orthogonal orbitals) different by 1-electron excitation would not interact. 227 An assumption can be made that the energies of the zero-order diabatic states approximately described by configurations (4.11)–(4.14) have small angular depen- dences (see below) and change almost linearly in the considered range of O¡H dis- tances. In addition, as mentioned earlier, the coupling matrix element between them is determined by the overlap of the nonorthogonal¾ (¤) OH and¼ (¤) CO orbitals and can be expected to have a sin' OH dependence (assuming that the overlapping zero-order orbitals preserve their characteristic shapes near the O atom). Under these assump- tions the adiabatic energies of the coupled states should behave as E 0 § s µ ¢k 2 (r O¡H ¡ r 0 ) ¶ 2 Å(V 0 sin' OH ) 2 , (4.16) where r 0 and E 0 are the distance and energy at which the diabatic states cross, ¢k is the difference in their “slopes” (energy–distance proportionality coefficients), and V 0 is the maximal interaction strength (achieved at T-shaped geometries). An attempt to fit the calculations results with this expression led to remarkably good results. For example, fitting of the SA-MCSCF energy r O¡H profiles, a part of which is shown in figure 4.14, at fixed' OH values reproduces the data almost exactly. The analysis of angular dependences of the fitted parameters given in figure 4.15 shows somewhat worse agreement, but at the same time demonstrates that the expected »sin' OH trend for the coupling strength is indeed observed. The results of the EA- EOM-CCSDt PES (see figure 4.8) analysis given in the same figure for comparison also show very similar behavior, 1 even from a quantitative point of view, reinforcing the conclusion that the considered two-state model provides adequate description of the electronic structure for the two relevant states in the examined region of the PES. 1 The larger deviation between the ground and excited state fits at large angles is probably due to the fact that EA-EOM includes interactions among all electronic states, whereas the used SA-MCSCF calculations and (4.16) consider only two adjacent states. 228 0 10 20 0 30 60 90 V,mHa ϕ OH , ◦ SA-MCSCF 0 10 20 30 0 30 60 90 V,mHa ϕ OH , ◦ EA-EOM −114.32 −114.30 −114.28 0 30 60 90 E crossing ,Ha ϕ OH , ◦ SA-MCSCF −114.70 −114.68 −114.66 0 30 60 90 E crossing ,Ha ϕ OH , ◦ EA-EOM Figure 4.15 2-state interaction parameters extracted from PES fittings. Symbols (solid for ground state and open for excited) correspond to fitted values. Expected dependences, sin' OH for coupling, and constant (ground state) and sin 2 ' OH (excited state) for crossing energies, are shown by dashed curves. However, the difference in crossing energies obtained from the upper and lower states (see the right column in figure 4.15) is rather suspicious. Unfortunately, the exact source of this angle-dependent additional gap was not determined. Neverthe- less, some reasoning regarding this effect can be given from general considerations. Namely, if the ground-state electronic configurations of CH 2 OH and CH 2 OÅ H are considered in terms of only bonding orbitals and lone pairs, that is, CH 2 ¡O¡H : (1s C ) 2 (1s O ) 2 (¾ CH ) 2 (¾ CH ) 2 (¾ CO ) 2 (lp O ) 2 (lp O ) 2 (¾ OH ) 2 (2p C ) 1 , (4.17) CH 2 ¡ ¡ OÅH : (1s C ) 2 (1s O ) 2 (¾ CH ) 2 (¾ CH ) 2 (¾ CO ) 2 (lp O ) 2 (lp O ) 2 (¼ CO ) 2 (1s H ) 1 , (4.18) 229 they become angle-independent, and hence their energies also should not significant- ly depend on the angle, 1 which is indeed observed for asymptotes (representing the diabatic states in (4.16)) of the ground state fits in figure 4.14. At the same time, the excited states are less symmetric in this respect and might have noticeable an- gular variations of the energy. Recall that in the planar geometries the two states actually cross rigorously, since they have different symmetries. However, description of the upper-state electronic configurations in terms of the same zero-order bonding and antibonding orbitals as in the planar case leads to nonorthogonal wavefunctions with overlap integral behaving as sin' OH (see above). Therefore, the upper-state di- agonal Hamiltonian matrix element will have a sin 2 ' OH term, which coincides with the energy dependence observed from figure 4.15. An interesting peculiarity of the electronic structure in the CH 2 OHÃ ! CH 2 OÅH transition region of the PES, mentioned already in [50] 2 , is that the Hartree–Fock method in both restricted open-shell (ROHF) and unrestricted (UHF) forms fails to provide a reasonable description of the O¡H bond breaking. The UHF method produces wavefunctions with large spin contamination (hS 2 i¼ 1 instead of 3/4 at the barrier) and a barrier energy much lower than all other methods. The ROHF method tries to converge to either (4.17) or (4.18) configuration at any angle. More- over, at each geometry in the vicinity of the conical intersection point both solutions can be obtained, depending on the initial orbital guess used in the calculations (see figure 4.16), producing an intersection of the two states instead of a smooth barri- er. A possible reason of such behavior is that while HF methods account for some 1 In chemical terms, the ¡OH group can rotate around the single C¡O bond in CH 2 ¡O¡H almost freely, and the H atom brought to the O atom of CH 2 ¡ ¡ ¢¢ O ¢¢ from any direction experiences nearly the same repulsion. 2 Notice however, that figure 2 in [50] shows potential energy curves that apparently were made up from just 3 points and are quite misleading in terms of their shapes. 230 −114.38 −114.36 −114.34 −114.32 1.2 1.3 1.4 1.5 1.6 E ROHF ,Ha r O−H ,Å ϕ OH = 0 ◦ 40 ◦ 90 ◦ Figure 4.16 ROHF ground-state energies (compare to figure 4.14). correlations among electrons 1 , they do not include more subtle correlations that are important in this situation of interacting states. This leads to an unbalanced treat- ment of various contribution to the total energy, and the energy minimization for the ground state leads to more localized orbitals in form (4.17) or (4.18) instead of the orbitals shown in figure 4.11. An important consideration, however, is that Hartree– Fock energies calculated for Slater determinants constructed from correct orbitals (specifically, natural orbitals obtained with high-level methods) show correct depen- dences on r O¡H and ' OH , confirming again that a 1-determinant description with these orbitals is quite reasonable. The PES constructed in such way is roughly par- allel to the original high-level PES, indicating that all other electron correlations are practically geometry-independent. These shortcomings of Hartree–Fock results might be not very important by themselves, since other methods (even the minimal MCSCF described above) can be 1 Meaning that the produced wavefunction is totally antisymmetric, and that electron repulsions are present in form of a mean-field description. 231 used for a correct description. However, the discontinuous change of ROHF orbitals in this important PES region has dire consequences for all high-level methods that use ROHF orbitals and ground-state ROHF determinant as the reference. For example, trial RCCSD calculations performed near the intersection point produced ground- state PES that was not smooth or even discontinuous at small angles. 1 This means that methods especially designed for description of near-degenerate states or use better orbitals (obtained from SA-CASSCF or simply RHF orbitals of the CH 2 OH + cation, which behave normally at all relevant geometries and actually resemble the “correct” orbitals of the neutral CH 2 OH) must be used even if calculations of only the ground-state PES are required. 4.6 Conclusion The overtone-induced dissociation of CH 2 OH and CD 2 OH was studied experi- mentally at energies near the barrier tops for O¡H bond fission and isomerization to methoxy. Hydrogen products were monitored by SVMI, and action spectra and KEDs were measured in the region of excitation to 4º 1 . The action spectrum of CD 2 OH showed only a transition to the 4º 1 overtone, whereas two bands were observed in CH 2 OH in the corresponding excitation region. These bands were assigned as tran- sitions to two coupled states, the more intense band dominated by 4º 1 and the less intense one by 3º 1 Åº 2 . KEDs obtained from SVMI data show distinct dynamical signatures in the form- aldehyde cofragment associated with O¡H bond fission and isomerization to methoxy. Dissociation proceeding by O¡H bond fission leads mainly to formaldehyde in the vi- 1 Inclusion of non-iterative treatment of triple excitations was improving the PES shape, but did not eliminate the problems completely. The situation with ground-state calculations near the intersec- tion is also complicated by near-degeneracy of the two states, which is known to adversely affect performance of many methods. 232 brationless state, with a small (»3 %) excitation in the C ¡ ¡ O stretch and yet smaller excitations in the methylene deformation modes (scissors, rock and wag). On the other hand, dissociation following isomerization leads to formaldehyde that is ro- tationally hotter and has a much larger fraction (»28 %) of the energy deposited in methylene deformations, but not in the C ¡ ¡ O stretch. No excitation in the C¡H (C¡D) stretches was observed in formaldehyde from either reaction. Analysis of the data al- so suggests that the dissociation rate for O¡H fission depends more strongly on the initially excited level than the isomerization rate, at least for the levels examined in this study. The CH 2 OHÃ ! CH 3 O dissociative system has proven to be amenable to detailed experiments on its dynamics in the ground electronic state. Most of the previous experimental and theoretical work [12, 13, 51] has dealt with initial excitation of CH 3 O, for which isomerization is unlikely by energetic considerations. The present work is the first detailed study of excitation and dissociation of CH 2 OH and CD 2 OH that demonstrates the role of isomerization. 4.7 Future work A promising direction of future experimental studies is an extension of the vibra- tional excitation to higher energies. As mentioned in [37], depositing more energy in O¡H stretch vibrations on the ground electronic state should access the conical intersection with very low relative energies (“from below” on the ground electronic state or “from above” after electronic excitation of the vibrationally excited radical) and thus might lead to very interesting dynamics. Since one-photon excitation becomes less and less efficient for higher overtones, and at the same time multiphoton absorption becomes more efficient at higher pho- 233 ton energies and intensities, the most practical way for excitation of high vibrational level might be IR–IR double-resonance excitation through fundamental vibrations and/or lower overtones. Such techniques have been already successfully used in stud- ies of CH 3 OH [46] and provide an additional advantage that initial rotational level for the second, dissociating, excitation can be preselected by the first resonant ex- citation to a metastable state 1 having sharp rovibrational spectrum. In addition to pure overtones, studies of various combination bands, especially involving the vibra- tions that should promote or impede the two radical decomposition pathways (see subsection 4.4.3), are appealing. The experimental studies, however, must be backed by high-quality theoretical calculations. As can be seen from discussions in subsection 4.4.3 and section 4.5, this part appears to be quite challenging. First of all, a good potential energy surface is re- quired for proper description of the vibrational dynamics. The presence of singularity on the way from the “reactant” to the products (see figure 4.8) means, for example, that the smooth single-valued PES fit that was used in [11] cannot be adequate. 2 Even if the immediate vicinity of the conical intersection is not reached from energy considerations, the PES shape at small angles, corresponding to the initial state of the radical (see figures 4.2 and J.1), is significantly influenced by the cone even at much shorter O¡H distances and therefore must affect at least the initial dynamics after excitation. 3 Another complication might come from the fact that, as explained in [52], the very presence of a conical intersection introduces a geometric phase difference be- 1 With lifetimes determined by radiative decay, which is very slow for lower states. 2 Take into account that the conical intersection, which appears as a single point in the 2-dimensional PES cut shown in figure 4.8, is actually a 7-dimensional seam of intersections in the full 9-dimen- sional picture, since all 7 in-plane motions preserve the A 00 and A 0 symmetries of the two electronic states. 3 This part of the cone protruding into the CH 2 OH potential well is probably the main driving force leading to the coupling between the O¡H stretch and the¡OH torsion. 234 tween the paths bypassing it from two sides (at any distance!), affecting the overall dynamics. 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R., Two-pathway coherent control of photoelectron angular distributions in molecular NO, Chem. Phys. Lett. 241(5–6), 591 (1995). Zare, R. N., Photoejection dynamics, Mol. Photochem. 4(1), 1 (1972). Zhang, Y., Yang, C.-H., Wu, S.-M., Roij, A., van, Zande, W. J., van der, Parker, D. H., Yang, X., A large aperture magnification lens for velocity map imaging, Rev. Sci. Instrum. 82(1), 013301 (2011). Zhou, W., Yuan, Y., Chen, Sh., Zhang, J., Ultraviolet photodissociation dynamics of the SH radical, J. Chem. Phys. 123(5), 054330 (2005). Minutes of the Pasadena meeting, June 15 to 20, 1931, Phys. Rev. 38(3), 579 (1931). SIMION 8.0,http://www.simion.com, Scientific Instrument Services, Inc., Ringoes, NJ. 249 Appendices A Listings of input files for SIMION simulations The following listings provide short descriptions and source codes of input files required for numerical simulations of the ion optics performance in SIMION 8. Im- portant points specific to the problem are commented in the code, but the SIMION documentation should be consulted regarding the general information about the for- mats, semantics and how to use these files. A.1 Geometry definition Geometry of the electrodes (including grounded walls and holders) is defined as close as possible (within cylindrical symmetry) to the actual dimensions of the real setup. The resulting potential array must be refined in the “fast adjust” mode and inserted into the workbench using 0.5 mm/gu scale and¡51 mm X-shift. real.gem 1 ; This geometry file uses 0.5 mm grid. 2 ; All coordinates and sizes below are specified in millimeters, 3 ; and "locate(0,0,0,2)" sets the appropriate scaling. 4 ; The potential array size is defined accordingly. 5 6 pa_define(1370,152,1,cylindrical,y_mirror) 7 locate(0,0,0,2) { 250 8 ; vacuum chamber walls 9 electrode(0) { 10 fill {within{box(0,0,1,76)} ; skimmer side wall 11 within{box(0,75,676,76)} ; tube 12 within{box(684,0,685,76)}} ; MCP side wall 13 } 14 ; accelerator 15 locate(25.5) { 16 ; 1st ring 17 locate( 0) {electrode( 0) {fill{within{box(-5,36.5,0,55.5)} 18 within{box(0,51,14,55.5)}}}} 19 ; 1st plate 20 locate( 10) {electrode( 1) {fill{within{box(0,1.5,1,40)}}}} 21 ; other plates 22 locate( 20) {electrode( 2) {fill{within{box(0,20,1,40)}}}} 23 locate( 30) {electrode( 3) {fill{within{box(0,20,1,40)}}}} 24 locate( 40) {electrode( 4) {fill{within{box(0,20,1,40)}}}} 25 locate( 50) {electrode( 5) {fill{within{box(0,20,1,40)}}}} 26 locate( 60) {electrode( 6) {fill{within{box(0,20,1,40)}}}} 27 locate( 70) {electrode( 7) {fill{within{box(0,20,1,40)}}}} 28 locate( 80) {electrode( 8) {fill{within{box(0,20,1,40)}}}} 29 locate( 90) {electrode( 9) {fill{within{box(0,20,1,40)}}}} 30 locate(100) {electrode(10) {fill{within{box(0,20,1,40)}}}} 31 locate(110) {electrode(11) {fill{within{box(0,20,1,40)}}}} 32 locate(120) {electrode(12) {fill{within{box(0,20,1,40)}}}} 33 locate(130) {electrode(13) {fill{within{box(0,20,1,40)}}}} 34 locate(140) {electrode(14) {fill{within{box(0,20,1,40)}}}} 35 ; 2nd ring 36 locate(145) {electrode( 0) {fill{within{box(5,36.5,0,55.5)} 37 within{box(0,51,-14,55.5)}}}} 38 } 39 ; lens 40 locate(251.5) { 41 ; 1st element 42 locate(0) {electrode(0) {fill{within{box(0,60,19,69)} 43 within{box(0,61,120.5,62)} 44 within{box(101.5,60,120.5,69)}}}} 45 ; middle element 46 locate(131.5) {electrode(15) {fill{within{box(0,60,19,69)} 47 within{box(0,61,120.5,62)} 48 within{box(101.5,60,120.5,69)}}}} 49 ; last element 50 locate(263) {electrode(0) {fill{within{box(0,60,19,69)} 51 within{box(0,61,120.5,62)} 52 within{box(101.5,60,120.5,69)}}}} 251 53 } 54 ; MCP end 55 locate(677.5) { 56 electrode(0) {fill{within{box(-22.5,51,22.5,76)}} ; reducer flange 57 fill{within{box(-6,22,-7,51)}}} ; MCP shield 58 ; MCP (front only) 59 electrode(16) {fill{within{box(0,0,1,32)} 60 within{box(0,24,-1,32)}}} 61 } 62 } A.2 Particles definition Initial positions and velocities of the particles are defined for relatively uniform representation of the distribution. The mass and maximum kinetic energy are set to “unit” values (1 Da and 1 eV respectively) and are scaled according to the actual user settings by the program (see listing A.3) real.fly2 1 --[[ 2 3 Particle definitions for "real.lua" program. 4 3 * 4 * 5 = 60 protons are defined for all combinations of: 5 3 kinetic energies: 1, 0.4, 0.1 eV, 6 4 positions: (+-0.5 mm, +-1 mm), 7 5 angles: 10, 50, 90, 130, 170 degrees. 8 9 Colors: KE x = -0.5 x = +0.5 10 ----- --------------- ---------- 11 1.0 5 light red 1 red 12 0.4 6 light green 2 green 13 0.1 7 light blue 3 blue 14 15 --]] 16 17 particles { 18 coordinates = 0, 19 -- KE = 1 20 standard_beam { 252 21 ke = 1, color = 5, 22 position = vector(-0.5, -1, 0), tob = 0, 23 n = 5, az = 0, el = arithmetic_sequence {first = 10, step = 40}, 24 mass = 1.00727647, charge = 1, 25 }, 26 standard_beam { 27 ke = 1, color = 5, 28 position = vector(-0.5, 1, 0), tob = 0, 29 n = 5, az = 0, el = arithmetic_sequence {first = 10, step = 40}, 30 mass = 1.00727647, charge = 1, 31 }, 32 standard_beam { 33 ke = 1, color = 1, 34 position = vector(0.5, -1, 0), tob = 0, 35 n = 5, az = 0, el = arithmetic_sequence {first = 10, step = 40}, 36 mass = 1.00727647, charge = 1, 37 }, 38 standard_beam { 39 ke = 1, color = 1, 40 position = vector(0.5, 1, 0), tob = 0, 41 n = 5, az = 0, el = arithmetic_sequence {first = 10, step = 40}, 42 mass = 1.00727647, charge = 1, 43 }, 44 -- KE = 0.4 45 standard_beam { 46 ke = 0.4, color = 6, 47 position = vector(-0.5, -1, 0), tob = 0, 48 n = 5, az = 0, el = arithmetic_sequence {first = 10, step = 40}, 49 mass = 1.00727647, charge = 1, 50 }, 51 standard_beam { 52 ke = 0.4, color = 6, 53 position = vector(-0.5, 1, 0), tob = 0, 54 n = 5, az = 0, el = arithmetic_sequence {first = 10, step = 40}, 55 mass = 1.00727647, charge = 1, 56 }, 57 standard_beam { 58 ke = 0.4, color = 2, 59 position = vector(0.5, -1, 0), tob = 0, 60 n = 5, az = 0, el = arithmetic_sequence {first = 10, step = 40}, 61 mass = 1.00727647, charge = 1, 62 }, 63 standard_beam { 64 ke = 0.4, color = 2, 65 position = vector(0.5, 1, 0), tob = 0, 253 66 n = 5, az = 0, el = arithmetic_sequence {first = 10, step = 40}, 67 mass = 1.00727647, charge = 1, 68 }, 69 -- KE = 0.1 70 standard_beam { 71 ke = 0.1, color = 7, 72 position = vector(-0.5, -1, 0), tob = 0, 73 n = 5, az = 0, el = arithmetic_sequence {first = 10, step = 40}, 74 mass = 1.00727647, charge = 1, 75 }, 76 standard_beam { 77 ke = 0.1, color = 7, 78 position = vector(-0.5, 1, 0), tob = 0, 79 n = 5, az = 0, el = arithmetic_sequence {first = 10, step = 40}, 80 mass = 1.00727647, charge = 1, 81 }, 82 standard_beam { 83 ke = 0.1, color = 3, 84 position = vector(0.5, -1, 0), tob = 0, 85 n = 5, az = 0, el = arithmetic_sequence {first = 10, step = 40}, 86 mass = 1.00727647, charge = 1, 87 }, 88 standard_beam { 89 ke = 0.1, color = 3, 90 position = vector(0.5, 1, 0), tob = 0, 91 n = 5, az = 0, el = arithmetic_sequence {first = 10, step = 40}, 92 mass = 1.00727647, charge = 1, 93 }, 94 } A.3 Analysis and optimization program This user-programming script performs low-level initialization of simulation pa- rameters from user-defined high-level parameters, analysis of the simulation results and, possibly, optimization of some user parameters. It can be run in either interac- tive or batch mode. The user-controlled parameters are defined as SIMION “adjustable variables” and can be set through the corresponding interface (“Variables” window). These pa- 254 rameters are: • effective accelerator lengths L 0 and L 1 (in cm), • effective accelerator voltages V 0 and V 1 (in V), • lens voltage V L (in V), • maximum kinetic energy K max in the sample (in eV), • particle mass m (in Da), • molecular beam velocity v beam (in m/s). A few other variables control the automatic optimization parameters: • V 1 step — initial step size (in V) of V 1 variation for resolution optimization (zero value disables the optimization); • K step — initial step size (in eV) for maximum measurable K max determination (zero value disables the determination); • batch — nonzero value of this variable activates the batch mode, in which sets of the simulation parameters (L 0 , L 1 , V 0 , V 1 , V L and K max ) are read from the external file (in.dat placed in the working directory), and the optimization results for each set are recorded to another external file (out.dat in the same directory). Other parameters that do not need to be changed during the ordinary use of the pro- gram (the detector radius, the number of plates in the accelerator and the structure of the particle distribution sample) are defined simply as local variables and are hid- den from the interface but can be changed in the source code if adjustments to these properties are required. There are 3 subroutines, which are called by SIMION at various moments during the simulation process and perform the actions necessary for the control and analy- sis: 1. segment.initialize() is called upon the “initialization” of each particle and 255 scales its mass and initial velocity from the generic values according to the specified m and K max (namely, the default mass 1 Da is multiplied by m, and the initial velocity vector (corresponding to the sample with K max Æ 1 eV at mÆ 1 Da) is multiplied by p K max /m). Additionally, if requested, it prepares the optimization process or recalls the values of the varied parameters from the previous optimization step. 2. segment.init_p_values() is called once after all particles were initialized and sets the potentials of all accelerator electrodes based on L 0 , L 1 , V 0 and V 1 ac- cording to the procedure described in subsection 2.5.1. The potential of the central element of the lens is also set to V L . 3. segment.terminate() is called for each particle upon hitting an electrode (in normal operation — the detector) and collects the arrival times and radial positions. Upon “termination” of the last particle, when the data gathering is completed, these “raw” data are analyzed according to equations (2.74)–(2.80). A summary of the simu- lation parameters and the obtained results is printed to the “Log” window. If any opti- mization was enabled, the optimization state is updated, and, depending on whether the convergence is reached or not, either the optimization results are written to the output file, or a new simulation cycle is started. Also, when convergence is reached in the batch mode, a new set of the simulation parameters is read from the input file, and a new optimization run is requested (see item 1). In addition to these necessary actions the user program disables the display of particle trajectories during the optimization and enables it after the optimization is finished. The reason behind these operations is that while the trajectories are very helpful for visualization of the system operation, their display takes a considerable amount of time. Therefore, the display is disabled in order to speed up the optimiza- tion process and enabled only to show the final result. 1 1 If drawing of equipotential contours is enabled in the interface, they are updated for each simulation 256 Warning! The program does not validate the settings and user’s input. The user is responsible for providing sensible values for constants and adjustable variables. real.lua 1 --[[ 2 3 Analysis and parameter-optimization program. 4 5 (Requires electode definitions from "real.gem", patricle definitions from 6 "real.fly2" and the ionization region being located at the origin.) 7 8 Parameters are described below (see program code). Optimized parameters are: 9 10 V1 if V1step > 0, 11 Kmax if Kstep > 0. 12 13 If "batch" is not 0, then file "in.dat" is opened, from each line there 14 parameters 15 16 L0 L1 V0 V1 VL Kmax 17 18 are read (must be separated by spaces or tabs), and the optimization results 19 are appended to file "out.dat" in format (separated by tabs) 20 21 L0 L1 V0 V1 VL Kmax DK tave Dt tau Dt0 Rmax 22 23 where DK [%] is kinetic energy resultution, 24 tave [us] average TOF, 25 Dt [ns] TOF resolution, 26 tau [ns] full TOF spread, 27 Dt0 [ns] v_z = 0 slice distortion, 28 Rmax [mm] radius for KE = Kmax 29 (for convergency checks and calibration). 30 31 --]] 32 33 simion.workbench_program() cycle, which also takes noticeable time during optimizations. This behavior is not controlled by the program, however, the drawing can be easily disabled and enabled from the interface, even when the optimization process is active. Moreover, the optimized potentials are retained when the optimization finishes, so the corresponding contours can be plotted even afterwards (which is impossible for the trajectories). 257 34 35 adjustable L0 = 5 -- initial length [cm] 36 adjustable L1 = 5 -- secondary length [cm] 37 adjustable V0 = 3000 -- initial potential [V] 38 adjustable V1 = 2000 -- secondary potential [V] 39 adjustable VL = 0 -- lens potential [V] 40 adjustable V1step = 0 -- step size for V1 optimization [V] 41 adjustable Kmax = 1 -- maximum kinetic energy (detector edge) [eV] 42 adjustable Kstep = 0 -- step size for Kmax optimization [eV] 43 adjustable m = 1 -- ion mass [Da] 44 adjustable batch = 0 -- batch mode switch (0 = off, any other = on) 45 adjustable vbeam = 0 -- molecular beam velocity [m/s] 46 47 local Rdet = 20 -- detector radius [mm] 48 49 -- This constant must agree with electode definitions in "real.gem": 50 local Nacc = 14 -- number of accelerator electodes; lens must be Nacc + 1 51 52 -- These constants must agree with patricle definitions in "real.fly2": 53 local N = 5 -- number of particles per group (# of angles, must be odd) 54 local a0 = 10 -- minimum angle (for "tau"; max. must be = 180 - min.) 55 local Np = 4 -- number of positions 56 local NK = 3 -- number of energies (highest must be 1st) 57 58 local SimplexOptimizer = require "simionx.SimplexOptimizer" 59 local opt -- optimizer object 60 61 local indat -- input file descriptor 62 63 -- Read parameters from the input file (batch mode) 64 function read_input() 65 L0, L1, V0, V1, VL, Kmax = indat:read("*n", "*n", "*n", "*n", "*n", "*n") 66 end 67 68 -- Initialization of particles (called for each particle) 69 function segment.initialize() 70 if (V1step ~= 0) or (Kstep ~= 0) then -- optimization enabled 71 if (batch ~= 0) and not indat then -- first run in batch mode 72 indat = io.open("in.dat", "r") 73 read_input() 74 end 75 if not opt then -- first run (any mode) 76 opt = SimplexOptimizer {start = {V1, Kmax}, step = {V1step, Kstep}, 77 minradius = 1e-4} 78 sim_trajectory_image_control = 3 -- don’t show or record 258 79 end 80 V1, Kmax = opt:values() 81 end 82 -- scale initial conditions: 83 ion_mass = m * ion_mass 84 local vscale = sqrt(Kmax / m) 85 ion_vx_mm = vscale * ion_vx_mm + 1e-3 * vbeam 86 ion_vy_mm = vscale * ion_vy_mm 87 end 88 89 -- Initialization of accelerator voltages 90 function segment.init_p_values() 91 for n = 1, Nacc do 92 adj_elect[n] = 0 93 end 94 local dV0 = (V0 - V1) / L0 -- field in the initial region 95 local dV1 = V1 / L1 -- field in the secondary region 96 local fL = 1 - (L0 - floor(L0)) -- fraction for the intermediate region 97 -- initial region: 98 local V = V0 99 for n = 0, floor(L0) do 100 adj_elect[n + 1] = V 101 V = V - dV0 102 end 103 -- treatment of intermediate region: 104 V = V + (dV0 - dV1) * fL 105 -- secondary region: 106 for n = floor(L0) + 1, floor(L0 + L1) do 107 adj_elect[n + 1] = V 108 V = V - dV1 109 end 110 -- lens: 111 adj_elect[Nacc + 1] = VL 112 end 113 114 local y = {} -- hit radii (positions at the detector) 115 local t = {} -- hit times 116 local lastN = N * Np * NK -- last ion number 117 118 local iter = 0 119 -- Positions at the detector and arrival times are recorded for each ion. 120 -- All analysis is performed after the last ion hit. 121 function segment.terminate() 122 local n = ion_number 123 y[n] = ion_py_mm 259 124 t[n] = ion_time_of_flight 125 if n < lastN then return end 126 127 -- These 2D arrays accumulate statistics for each group of equivalent ions 128 local Ey = {} -- average radius 129 local Dy = {} -- radius spread 130 local Et = {} -- average time 131 local Dt = {} -- time spread 132 -- statistics for groups: 133 for K = 0, NK - 1 do -- KEs 134 Ey[K] = {} 135 Dy[K] = {} 136 Et[K] = {} 137 Dt[K] = {} 138 for a = 0, N - 1 do -- angles 139 local nKa = K * (N * Np) + a + 1 140 local yKa, miny, maxy = 0, math.huge, -math.huge 141 local tKa, mint, maxt = 0, math.huge, -math.huge 142 for p = 0, Np - 1 do -- initial positions 143 local yKap = y[nKa + p * N] 144 yKa = yKa + yKap 145 miny, maxy = min(miny, yKap), max(maxy, yKap) 146 local tKap = t[nKa + p * N] 147 tKa = tKa + tKap 148 mint, maxt = min(mint, tKap), max(maxt, tKap) 149 end 150 Ey[K][a] = yKa / Np 151 Dy[K][a] = maxy - miny 152 Et[K][a] = tKa / Np 153 Dt[K][a] = maxt - mint 154 end 155 end 156 157 -- Calculations of overall characteristics 158 local Rmax = 0 -- maximum radius 159 local maxDK = 0 -- KE resolution 160 local tave = 0 -- average TOF 161 local maxDt = 0 -- TOF resolution 162 local tau -- full TOF spread (estimated) 163 local Dt0 -- distortion of v_z = 0 slice (estimated) 164 for K = 0, NK - 1 do -- KEs 165 for a = 0, N - 1 do -- angles 166 Rmax = max(Rmax, Ey[K][a]) 167 maxDK = max(maxDK, Ey[K][a] * Dy[K][a]) 168 tave = tave + Et[K][a] 260 169 maxDt = max(maxDt, Dt[K][a]) 170 end 171 end 172 -- final calculations and scaling: 173 tave = tave / (NK * N) 174 maxDK = 200 * maxDK / Rdet ^ 2 175 maxDt = maxDt * 1e3 176 tau = (Et[0][N-1] - Et[0][0]) / math.cos(a0 / 180 * math.pi) * 1e3 177 Dt0 = (Et[0][(N-1)/2] - (Et[NK-1][N-1] + Et[NK-1][0]) / 2) * 1e3 178 -- report to log window: 179 print() 180 iter = iter + 1 181 print(iter) 182 print("L0 =", L0, " L1 =", L1) 183 print("V0 =", V0, " VL =", VL) 184 print("V1 =", V1, " Kmax =", Kmax) 185 print(string.format("Rmax = %.4g mm", Rmax)) 186 print(string.format("maxDK = %.3g %%", maxDK)) 187 print(string.format("tave = %.5g us", tave)) 188 print(string.format("maxDt = %.3g ns", maxDt)) 189 print(string.format("tau = %.3g ns", tau)) 190 print(string.format("Dt0 = %.3g ns", Dt0)) 191 192 -- Optimization... 193 if (V1step ~= 0) or (Kstep ~= 0) then 194 local res = 0 195 if V1step ~= 0 then 196 res = maxDK 197 end 198 if Kstep ~= 0 then 199 res = res + 10 * abs(Rmax - Rdet) -- (factor "10" for sharper cusp) 200 end 201 opt:result(res) 202 if not opt:running() then -- end of optimization 203 -- write result to the file: 204 local fout = io.open("out.dat", "a") 205 fout:write(string.format( 206 "%g\t%g\t%g\t%.5g\t%g\t%.5g\t%.4g\t%.5g\t%.3g\t%.4g\t%.4g\t%.5g\n", 207 L0, L1, V0, V1, VL, Kmax, maxDK, tave, maxDt, tau, Dt0, Rmax)) 208 fout:close() 209 -- next task or stop: 210 if batch ~= 0 then 211 opt = nil 212 if indat:read(0) ~= nil then 213 read_input() 261 214 else 215 indat:close() 216 V1step, Kstep = 0, 0 -- stop optimization 217 sim_trajectory_image_control = 0 -- show and retain 218 end 219 else 220 V1step, Kstep = 0, 0 -- stop optimization 221 sim_trajectory_image_control = 0 -- show and retain 222 end 223 end 224 sim_rerun_flym = 1 225 else 226 sim_rerun_flym = 0 227 end 228 229 sim_retain_changed_potentials = 1 230 231 end -- program 262 B Additional Pareto-optimal ion optics parameters and performance characteristics B.1 With respect to¢K and¢t 1.0 1.5 2.0 2.5 20 40 60 80 100 120 ΔK,% M,ns p eV/Da 0 1 2 3 4 20 40 60 80 100 120 Δt,% M,ns p eV/Da 0 2 4 6 8 20 40 60 80 100 120 Δt 0 ,% M,ns p eV/Da 1.0 1.5 2.0 2.5 20 40 60 80 100 120 ΔK,% M,ns p eV/Da 0 1 2 3 4 20 40 60 80 100 120 Δt,% M,ns p eV/Da 0 2 4 6 8 20 40 60 80 100 120 Δt 0 ,% M,ns p eV/Da L 0 =30 35 40 45 50 55 60 65 70 75 80 85 L 1 =30 35 40 45 50 55 60 65 70 75 80 85 90 95 Figure B.1 Relative overall kinetic energy and TOF resolutions as functions of magnification index. (Lengths L 0 and L 1 are in mm.) 263 0.0 0.5 1.0 1.5 2.0 2.5 20 40 60 80 100 120 −V L /V 0 M,ns p eV/Da 0.5 0.6 0.7 0.8 20 40 60 80 100 120 V 1 /V 0 M,ns p eV/Da 0.0 0.5 1.0 1.5 2.0 2.5 20 40 60 80 100 120 −V L /V 0 M,ns p eV/Da 0.5 0.6 0.7 0.8 20 40 60 80 100 120 V 1 /V 0 M,ns p eV/Da L 0 =30 35 40 45 50 55 60 65 70 75 80 85 L 1 =30 35 40 45 50 55 60 65 70 75 80 85 90 95 Figure B.2 Interdependence of ion optics parameters and magnification index. (Lengths L 0 and L 1 are in mm.) 500 1000 1500 2000 20 40 60 80 100 120 qV 0 /maxK M,ns p eV/Da 500 1000 1500 2000 20 40 60 80 100 120 qV 0 /maxK M,ns p eV/Da L 0 =30 35 40 45 50 55 60 65 70 75 80 85 L 1 =30 35 40 45 50 55 60 65 70 75 80 85 90 95 Figure B.3 Dependence of voltage scale on imaged kinetic energy range and mag- nification index. (Lengths L 0 and L 1 are in mm.) 264 B.2 With respect to¢K and¢t 0 1.0 1.5 2.0 20 40 60 80 100 120 ΔK,% M,ns p eV/Da 0 1 2 3 4 20 40 60 80 100 120 Δt 0 ,% M,ns p eV/Da 0 1 2 3 4 20 40 60 80 100 120 Δt,% M,ns p eV/Da 1.0 1.5 2.0 20 40 60 80 100 120 ΔK,% M,ns p eV/Da 0 1 2 3 4 20 40 60 80 100 120 Δt 0 ,% M,ns p eV/Da 0 1 2 3 4 20 40 60 80 100 120 Δt,% M,ns p eV/Da L 0 =30 35 40 45 50 55 60 65 70 75 80 85 90 95 L 1 =30 35 40 45 50 55 60 65 70 75 80 85 90 95 Figure B.4 Relative overall kinetic energy and TOF resolutions as functions of magnification index. (Lengths L 0 and L 1 are in mm.) 265 0.0 0.5 1.0 1.5 2.0 20 40 60 80 100 120 −V L /V 0 M,ns p eV/Da 0.6 0.7 0.8 20 40 60 80 100 120 V 1 /V 0 M,ns p eV/Da 0.0 0.5 1.0 1.5 2.0 20 40 60 80 100 120 −V L /V 0 M,ns p eV/Da 0.6 0.7 0.8 20 40 60 80 100 120 V 1 /V 0 M,ns p eV/Da L 0 =30 35 40 45 50 55 60 65 70 75 80 85 90 95 L 1 =30 35 40 45 50 55 60 65 70 75 80 85 90 95 Figure B.5 Interdependence of ion optics parameters and magnification index. (Lengths L 0 and L 1 are in mm.) 500 1000 1500 2000 20 40 60 80 100 120 qV 0 /maxK M,ns p eV/Da 500 1000 1500 2000 20 40 60 80 100 120 qV 0 /maxK M,ns p eV/Da L 0 =30 35 40 45 50 55 60 65 70 75 80 85 90 95 L 1 =30 35 40 45 50 55 60 65 70 75 80 85 90 95 Figure B.6 Dependence of voltage scale on imaged kinetic energy range and mag- nification index. (Lengths L 0 and L 1 are in mm.) 266 C Mechanical drawings and photographs This appendix includes scaled down drawings submitted to the USC machine shops for manufacturing of the custom-made parts. No descriptions for standard parts (such as nuts, set screws, washers and so on) that were purchased from Mc- Master-Carr are included. If necessary, their parameters can be derived from the parameters of the matching parts. The drawings have captions and comments included within and therefore are given below “as is” without figure numbers and captions. Photographs of the most important subassemblies are provided in figures C.1– C.4 after the drawings. They were taken in laboratory directly during the assembly process, without any special preparations, and therefore have relatively poor quality but should be sufficient for illustration purposes. 3/8"(stock) 3/8"(stock) 0.164" 10mm* 10mm* 1mm** 1/16"* 145mm round bottom smooth bottom View A View A 1/16" 45° INSULATOR BAR Alumina or Macor 6 pcs. * Dimensions of the grooves and their possitions are critical. If possible, machine all 6 pcs. at once (clamped together). ** Matches thickness of the plates. Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu 267 FRAME RING 1 Stainless steel (non-magnetic) 1 pc. Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu 1/4"* 1/4"* 3/4" 3/16" 45° 45° 4"(critical) 2 7/8" R2 3/16" 8-32UNC A A Section A-A * Positions af all holes are critical 1/4-28UNF 4-40UNC 1/4" drilled 3/16" (stock) 3/4" 2 3/4" 1/4" FRAME RING 2 Stainless steel (non-magnetic) 1 pc. Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu 1/4"* 1/4"* 1/2"(stock) 1/4" 45° 45° 4"(critical) 2 7/8" R2 3/16" R3/8" 1/4" 8-32UNC 3/16" 10-32UNF 3/16" A A Section A-A * Positions af all holes are critical 268 SCREW Bronze (non-magnetic, no zinc) 4 pcs. Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu 6 3/8" 5/8" 3/4" 90° screwdriver slot 1/16" width, 1/16" depth 1/4-28UNF thread 1/4-28UNF thread 1/4"(stock) PIN Bronze (non-magnetic, no zinc) 4 pcs. Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu 3/4" 1/4"(stock) 45° 269 1ST PLATE OFE Copper 1 pc. * External diameters of all plates are critical. If possible, machine all plates at once (clamped together). ** Use sheet stock with closest thickness (+ see notes for "INSULATOR BAR"). Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu 1mm(stock)** 80mm(critical)* 3mm 90° PLATE OFE Copper 15 pc. * External & internal diameters of all plates are critical. If possible, machine all plates at once (clamped together). ** Use sheet stock with closest thickness (+ see notes for "INSULATOR BAR"). Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu 1mm(stock)** 80mm(critical)* 40mm(critical) 270 ALIGNMENT HOOK Stainless steel 1 pc. Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu 1/2" 1/4" 1.32" [33.5mm] 0.2" [5mm] 0.6000 1.2" [30 mm] 1/4" 1/8" Was made in the lab to fit the chamber dimensions. ALIGNMENT SCREW Stainless steel 1 pc. * The screw should fit into the bellows. Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu 3/8-24UNF 1 3/4" 1/2" 4" .75* 1/4-28UNF Assembly: solder solder Blank flange and bellows are provided. vacuum side 271 ALIGNMENT SLIDER Stainless steel 1 pc. each * The smaller part should slide in the bigger one with no significant play. ** "ALIGNMENT SCREW" should fit into this hole with no significant play. Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu 2 3/4" 1 3/4"* 1 3/4* 1 3/4" 1/2"(stock) 1/2"(stock) 1/4" 1/4" 1 5/32" 1/4" 1/4-28UNF 3/8"** ALIGNMENT NUT Brass 1 pc. Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu 1/2"(stock) 3/8"-24UNF 1 1/2" Knurled surface 8-32UNC 272 LENS RING 1 Stainless steel (non-magnetic) 2 pcs. Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu Note: "RING 1", "RING 2" and "RING 3" all have similar shapes. R60mm R62mm B B 3/8" Section B-B 10-32UNF (tight) 3/4"(stock) R69mm 1/4" LENS RING 2 Stainless steel (non-magnetic) 2 pcs. Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu * These dimensions and positions af all holes are critical 68mm* R60mm* R62mm 3/16" 3/8" 8-32UNC (tight) Section A-A B B 45° 45° 3/8" A A Section B-B 10-32UNF (tight) 3/4"(stock) R69mm 1/4" 273 LENS RING 3 Stainless steel (non-magnetic) 2 pcs. Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu * These dimensions and positions af all holes are critical 68mm* R60mm* R62mm 3/16" 3/8" 8-32UNC (tight) Section A-A B B A A Section B-B 3/4"(stock) R69mm 1/4" LENS INSULATOR Macor 10 pc. Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu Positions af all holes are critical 2"(stock) 1/2"(stock) 1/8"(stock) 3/8" 13/16" 3/8" 100° 0.312" 0.164" (for #8 screw) 274 LENS SHIELD Stainless steel (non-magnetic) perforated sheet. Thickness: about 1mm; Hole diameter: < 4mm. 3 pcs. Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu * The "SHIELD"s are intented to be inserted into the "LENS RING"s. R62mm* 95mm [3 3/4"] small gap MCP SHIELD OFE Copper 1 pc. Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu 4" 1 3/4" 1" 0.138 (for #6 screw) 1/8" 1" 1" 3/8" Thickness .04" (sheet stock) 275 REDUCER FLANGE (Flange is provided) Front view: Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu 5/8", 1/4"deep (flat bottom) 2 bores 6-32UNC 1/4" deep 4 holes 2 1/4" Hanna Reisler group Misha Ryazanov x04105, ryazanov@usc.edu REDUCER FLANGE (Flange is provided) Rear view: 276 Figure C.1 Accelerator assembly. Figure C.2 Top (installed) and bottom views of accelerator suspension and align- ment assembly. 277 Figure C.3 Accelerator installed in the vacuum chamber (without extension). Figure C.4 Additional lens. (Thin copper wires holding the parts together before installation are cut and removed after anchoring inside the vacuum chamber.) 278 D Electronic circuit diagrams Figure D.1 Voltage divider for accelerator. Connections for effective lengths L 0 Æ 45 mm (4.5 cm), L 1 Æ 60 mm (6 cmÆ 10.5 cm¡4.5 cm) are illustrated. Figure D.2 Bias voltage supply for detector front plate. 279 Figure D.3 Voltage divider and signal decoupler for detector operation in electrical signal pickup mode. Figure D.4 Phosphor screen voltage supply for detector operation in imaging mode. 280 Figure D.5 Voltage supply for AV-HVX-1000A pulser. Figure D.6 Pulse and bias voltage combiner for AV-HVX-1000A pulser. E High-voltage nanosecond pulser The pulser is used for fast gating of the MCP-based particle detector in SVMI experiments. It produces electrical pulses of fixed duration (halfwidth »5 ns) with amplitude adjustable in the whole operating range of the detector (up to»2 kV) and is self-contained, requiring only a low-voltage (11¡14 V, 3 A) power supply and an input TTL triggering signal (&100 ns duration). The generated pulse is delivered to the detector through a short (»20 cm) cable with an SHV connector that must be con- nected directly to the detector electrical feedthrough. A 100-fold attenuated (.20 V amplitude) pulse is also provided through a BNC connector for monitoring purposes. 281 Figure E.1 Schematic circuit diagram and operation sequence. 282 It can be fed to any device with 50 Ohm input impedance using an appropriate cable of arbitrary length. The design was suggested by Dr. András Kuthi from the Pulsed Power Laboratory at USC and essentially represents a downscaled version of a megavolt-range pulser described in [1] (see also [2]). The pulser consists of an initial energy storage ca- pacitor, a relatively slow semiconductor switch followed by a 3-stage magnetic pulse compressor and ends with a step-recovery diode opening switch. The basic circuit diagram is shown in figure E.1 (compare to figures 5 and 6 in [1]). The main principles of the pulse forming sequence can be described as follows. A slow charging (phase A in figure E.1) of the energy storage capacitor C1 through the current-limiting resistor R1 and the primary winding of the transformer T pre- pares the system before the pulse generation can be initiated by an incoming trigger signal. Upon triggering (phase B), C1 starts discharging through the opened SCR switch and the saturable transformer T, charging capacitors C2 and C3 connected in parallel through the diode D1. When the maximum current through T is reached, the transformer core becomes saturated. At that point the coupling between the primary and secondary windings is lost, and the inductance of the secondary winding drops to a low value, leading to a quick recharging of C2 in the reverse polarity (phase C). Then the transformer core gets out of saturation, significantly increasing the sec- ondary winding impedance. At the same time, the current produced by C2 and C3 (now connected in series) discharging through the diode D2 and the inductor L1 sat- urates the core of L1, 1 which leads to a quick charging of the capacitor C4 (phase D). During this process the p–n junction of D2 is saturated with charge carriers, which makes possible backward discharging of C4 through the inductor L2 and D2 using 1 A preceding slow discharge of C3 through the secondary winding of T, L2 in parallel with D2–C4, and L1 helps this saturation. 283 its reverse recovery effect. The discharge current saturates the L2 core (phase E), 1 which in turn increase the C4 discharge rate. The charge stored in D2 is eventually depleted, and the current abruptly stops flowing through the diode. Since the high current in L2 cannot disappear immediately, its path becomes closed through C4 and the load connected to the pulser output (phase F), which causes a high voltage spike due to the load resistance. The LC circuit formed by L2, C4 and the load leads to subsequent recharging of C4 in the opposite direction, which causes an equally quick decrease of the load voltage (phase G). In principle, these oscillations could continue for several periods, creating unwanted ringing instead of a single pulse. However, with correctly chosen parameters of the circuit, desaturation of the L2 core after the first cycle and damping in the load prevent further spikes. The actual circuit diagram of the present pulser implementation is shown in fig- ure E.2. 2 The lack of specifications for electrical parameters of the detector, as well Figure E.2 Main board. 1 This is facilitated by the currents already flowing through L2 in the same direction during phases C and D. 2 Main part. The additional circuits are shown in figure E.3. 284 Figure E.3 Auxiliary board. as the usage of regular rectifier diodes, which are made not for step recovery oper- ation and hence also have no specifications for parameters relevant to such opera- tion, 1 required matching of some characteristics by trial and error. This especially pertains to capacitor values, the number and type of D2 diodes, and saturable in- 1 The choice of diodes was based simply on the fact that the availability and prices of “real” step- recovery diodes rated for such high voltages are beyond reasonable. At the same time, high-voltage rectifier diodes are relatively cheap and were already successfully used for pulser constructions in the Pulsed Power Laboratory at USC. Among them the models with sufficient forward current ratings (for phase D in figure E.1) must be chosen. An addition selection criterion is that devices with larger physical size of the p–n junction generally have larger stored charge. 285 ductor parameters. The bias current circuit that was added to the last core (of L2) allows to create some premagnetization and therefore to adjust the saturation mo- ment in time. This helps in achieving the maximum energy efficiency, which not only maximizes the output pulse amplitude, but also prevents afterpulses due to the un- used energy circulating in the magnetic compressor stages. The pulse amplitude is adjusted mostly by changing the primary charge voltage, but at each voltage the bias current must be also adjusted for best performance. Typical values in the present set- up are 560 V charge and 750 mA bias. The produced pulses have»1800 V amplitude 1 and»5 ns halfwidth of the main peak. The electrical pulse has some lower-frequen- cy oscillations before and some high-frequency oscillations after the main peak (see figure E.4), but their total amplitude does not exceed»500 V and has no effect on the 0 1000 2000 750 800 850 900 950 Outputvoltage,V Timeaftertrigger,ns Figure E.4 Example of output electrical pulse (measured at the pulse monitor output). Note that fast oscillations after the main peak are mostly due to electrical interference in low-voltage signal lines induced by powerful electromagnetic radia- tion. detector operation. 1 Due to the lack of correct impedance matching in the system (the main reason for which is that the detector itself was not properly designed for fast switching) the actual pulse amplitude at the detector cannot be measured accurately. The pulse monitor installed at the pulser output (before the cable) apparently can give up to »20 % error depending on the propagation characteristics. The quoted amplitude was estimated by comparison of the produced signal intensities with a slower pulser, for which the voltage can be measured without complications. 286 Figure E.3 shows the auxiliary circuits required for pulser operation. They in- clude the adjustable high voltage supply for the pulser charging (U6 and R4), the SCR gate driver (U1 and T) and the adjustable current source for the bias current (U3, Q, U5 and R5). The latter uses a voltage source U3 and linear current regu- lation by the transistor Q, which dissipates the excess voltage. Since the dissipated power is equal to the output bias current multiplied by the dissipated voltage, it might become substantial if the U3 output voltage is too high. At the same time, low voltage will limit the maximal possible output current. Therefore, a coarse voltage adjustment by the resistor R3 is provided. To minimize heat dissipation at any given bias current setting the voltage should be set to the minimal value ensuring reliable current stabilization (which can be easily checked by output pulse stability). References [1] Yu. A. Kotov, G. B. Mesyats, S. N. Rukin, A. L. Filatov, S. K. Lyubutin, A novel na- nosecond semiconductor opening switch for megavolt repetitive pulsed power tech- nology: experiment and applications, Digest of technical papers: Ninth IEEE in- ternational pulsed power conference, vol. 1, p. 134–139, Albuquerque, NM, USA, 1993. [2] A. Kuthi, P. Gabrielsson, M. R. Behrend, P. Th. Vernier, M. A. Gundersen, Na- nosecond pulse generator using fast recovery diodes for cell electromanipulation, IEEE Trans. Plasma Sci. 33(4), 1192 (2005). F H fragment velocity distribution in photodissoci- ation by the H-detection laser The raw velocity map images of H fragments produced in CH 2 OH photodisso- ciation by the H-detection laser (º¼ 27420 cm ¡1 , vertical polarization) are shown in figure F.1. The full-projection image was accumulated for »170000 laser shots 287 Full projection: Sliced: (maxK¼ 2.11 eV) (maxK¼ 2.18 eV, max¢¿¼ 48 ns) »6130000 events »720000 events Figure F.1 Raw full-projection and sliced velocity map images of H fragments. (»4.7 hours), and the sliced one — for »120000 laser shots (»3.3 hours). 1 The ex- tracted KEDs are plotted in figure F.2. Notice that the lower-KE part of the KED obtained from reconstruction of the full-projection image has very poor SNR, much worse that that of the SVMI data, even though the full-projection image was taken for longer time and has significantly larger total intensity. Small KE deviation in the two KEDs are caused by imperfect KE calibration (part of which is due to rectifica- tion of small raw image distortions). The CH 2 OH¡ ! CH 2 OÅ H dissociation energy obtained from these KEDs is D 0 Æ 10120§300 cm ¡1 , in excellent agreement with the 1 The left–right intensity asymmetry noticeable in the images is due to imperfect matching of focal positions for the fundamental (¸Æ 3Ly-®) and the tripled (Ly-®) radiation components (see footnote on p. 207), and imperfect centering of the phase-matching curve of the tripling mixture, which were slightly different in these two experiments. The image analysis, however, uses an average of the left and right parts (as well as of the top and bottom) and thus should not be affected by this detection artifact. The small low-intensity spots seen near the center in the full-projection image are due to detector burn-out caused by intense signals of low-KE C + ions in previous experiments. Their presence does not affect the results obtained for the KE ranges relevant in the present work. 288 0 5000 10000 15000 Intensity,a.u. Kineticenergyrelease,cm −1 Fromprojectionreconstruction Fromslice Figure F.2 Kinetic energy release distributions extracted from images shown in figure F.1. results reported in table 4.3. G Details of spectral fittings The contour fitting option in PGOPHER was employed for fittings of each band in the observed spectra. The program solves the rotational Hamiltonian to calcu- late transition frequencies (spectral line positions) and probabilities (line strengths), and generates a continuous simulated spectrum by convoluting the resulting “stick spectrum” with Gaussian and Lorentzian contours. The simulated spectrum can be compared with the experimental one visually or numerically by the magnitude of the deviation (total squared intensity difference between the simulation and the experi- mental data points). Any of the simulation parameters can be changed manually or optimized automatically to minimize the deviation. These parameters were: the band origin, the excited level rotational constants, Lorentzian spectral linewidth, rotation- al temperature, plus background level and overall intensity. The model was limited 289 to the rigid rotor approximation, since only the lowest rotational levels (J6 5) are appreciably populated at the low temperatures in the molecular beam (»13 K), and no individual lines were resolved in the spectrum for meaningful fitting of additional parameters. Throughout the spectral fitting the ground state rotational constants were fixed at the theoretically predicted values (A 00 Æ 6.457, B 00 Æ 0.982, C 00 Æ 0.855 for CH 2 OH and A 00 Æ 3.785, B 00 Æ 0.842, C 00 Æ 0.692 for CD 2 OH, all values in cm ¡1 ). The excit- ed state rotational constants were allowed to adjust, starting from the theoretically predicted values. Initial guesses for temperature and linewidths were made by visual comparison of the simulations and the experimental spectra. 1 Optimized parameters were obtained after several rounds of iterative nonlinear least-square fitting proce- dure that was repeated until the error reached the minimum and all parameters were converged. G.1 CH 2 OH The fitting of the most resolved transition, 3º 1 Åº 2 Ã 0, was achieved without difficulties. The experimental spectrum and the best fitted simulation are plotted in figure G.1 for comparison. Fitting of the 4º 1 Ã 0 transition was carried out using the same procedure. However, it was impossible to fit satisfactory the whole band, mostly due to the middle region marked gray in figure G.1. This region was therefore excluded from fitting to achieve better convergence for the rest of the spectrum. Both observed bands were fitted as pure a-type transitions. Inclusion of b-type transitions did not improve the simulation. 1 The approximate temperature range was known from previous experiments performed on the same machine, and the approximate linewidths for the previously observed bands were also known. 290 0 13640 13645 13650 13655 13660 13665 13670 13675 H + signal Excitationwavenumber,cm −1 Data Fit Figure G.1 Experimental spectrum and best fitted simulation of 3º 1 Åº 2 Ã 0 band of CH 2 OH. G.2 CD 2 OH The H fragment yield spectrum was fitted exclusively with theoretically predict- ed ground and excited state rotational constants without further optimization be- cause the attempt to optimize all parameters failed to converge to a reasonable fit. In addition, the rotational temperature was adopted to be 13 K based on the re- sults of CH 2 OH fittings. In contrast to the CH 2 OH spectra, the inclusion of b-type transition in the least-squares fit showed a noticeable improvement. The fraction of b-type transitions contribution was optimized by least-square fitting at different Lorentzian linewidths (which were varied manually). The confidence interval for the linewidth, 1.2§ 0.1 cm ¡1 , was established based on the reproduction of the visible spectral features. The b-type contribution was determined to be 25.6(5) % by intensi- 291 0 13575 13580 13585 13590 13595 13600 13605 13610 13615 13620 H + signal Excitationwavenumber,cm −1 Data Fit Figure G.2 Experimental spectrum and best fitted simulation of 4º 1 Ã 0 band of CH 2 OH. ty. 1 Figure G.3 shows the comparison between the simulation and the experimental data. H Plots of kinetic energy distributions fits Plots of KED fits (see section 4.3.2) in terms of internal energy of the formalde- hyde fragment and modeled rovibrational distributions (see section 4.4.2) are shown in figure H.1 in comparison with the experimental KEDs (see figure 4.7). I Rotational excitation model The dependence of the energy partitioning on the isotopic effects, under an as- sumption of invariable geometric parameters, can be found from a model based on 1 Please see a remark on p. 212. 292 0 13595 13600 13605 13610 13615 13620 13625 13630 13635 13640 H + signal Excitationwavenumber,cm −1 Data Fit Figure G.3 Experimental spectrum and best fitted simulation of 4º 1 Ã 0 band of CD 2 OH. conservation of energy, momentum and angular momentum. Let m be the mass of the atomic fragment, M — the mass of the molecular cofrag- ment, and I — its moment of inertia. The impact parameter b measured from the center of mass (COM) of the molecular cofragment is assumed to be a constant for each type of dissociation, independent on isotopic substitutions. The conservation laws (in the frame of reference associated with the nonrotating parent radical and centered at the COM of the molecular cofragment part) after the dissociation can be written as follows. For energy: kÅ KÅ RÆ E, (5.19) where k is the atomic translational energy, K and R are the molecular translational 293 0 500 1000 1500 2000 CH 2 OH− →CH 2 O+H,ν=13602cm −1 0 500 1000 1500 2000 CH 2 OH− →CH 2 O+H,ν=13662cm −1 0 500 1000 1500 2000 CD 2 OH− →CD 2 O+H,ν=13621cm −1 0 500 1000 1500 2000 CD 2 OH− →CHDO+D,ν=13621cm −1 1000 1500 2000 2500 1000 1500 2000 2500 1000 1500 2000 2500 1000 1500 2000 2500 Experiment Fit(ground) Fit(excited) Expt. Fit Model Figure H.1 Comparison of fits and modeled distributions with the experimental KEDs. Full KER range is shown in the left column, and expanded part corresponding to vibrationally excited formaldehyde — in the right. Rows correspond to figure 4.7. Horizontal scale shows the internal energy of formaldehyde fragments in cm ¡1 . 294 and rotational energies, and E is the total energy release. For momentum: pÅ PÆ 0, (5.20) where p and P are the momenta of the atomic and molecular fragments respectively. And for angular momentum: lÅ LÆ 0, (5.21) where lÆ bp is the atomic angular momentum (with respect to the origin), and L is the molecular rotational momentum. Substitution of kÆ p 2 2m , KÆ P 2 2M and RÆ L 2 2I in (5.19) and using (5.20) and (5.21) leads to p 2 2m Å p 2 2M Å (bp) 2 2I Æ E, (5.22) from where p 2 2 Æ E 1 m Å 1 M Å b 2 I , (5.23) and thus RÆ p 2 2 b 2 I Æ E I b 2 m Å I b 2 M Å1 . (5.24) The fraction of energy deposited into the molecular cofragment rotation is therefore R E Æ 1 1Å I b 2 ¡ 1 M Å 1 m ¢. (5.25) This fraction is mostly affected by isotopic differences in the atomic mass m (twofold difference for H and D) and the molecular moment of inertia I (also reaching two times for I A of CH 2 OH and CD 2 OH). The quantitative estimations can be done using m H Æ 1 Da, m D Æ 2 Da and formaldehyde parameters given in table I.1. Total 295 Table I.1 Inertial parameters of formaldehyde isotopologs. Molecule Rotational constants, Moments of inertia, (mass in Da) cm ¡1 Da¢Å 2 CH 2 O AÆ 9.40 I A Æ 1.79 MÆ 30 BÆ 1.30 I B Æ 13.0 CÆ 1.13 I C Æ 14.9 CD 2 O AÆ 4.72 I A Æ 3.57 MÆ 32 BÆ 1.08 I B Æ 15.6 CÆ 0.87 I C Æ 19.4 CHDO AÆ 6.61 I A Æ 2.55 MÆ 31 BÆ 1.16 I B Æ 14.5 CÆ 0.99 I C Æ 17.0 energies for each process can be taken from the KED fits (see subsection 4.3.2). Their values for the ground state formaldehyde cofragments were taken directly for each of the CD 2 OH experiments, but for CH 2 OH an average from the two experiments was used. The values for the vibrationally excited formaldehydes were estimated also as an average of the values for the rock and wag modes. 1 The resulting numbers are provided in table I.2 together with the reference effective temperatures copied from table 4.4. The extent of rotational excitation observed in the experiments was analyzed in the following way. First, a reference point was chosen, for example, ground state ef- fective temperature in the CD 2 OH¡ ! HÅ CD 2 O process. Then an impact parameter b eff matching the characteristic rotational energy RÆ kT eff rot was found from (5.25). After that this b eff value was used to calculate the effective rotational temperatures that can be expected for the other isotopologs under the same dissociation geome- try. This procedure was performed for several reference points for testing several hypotheses: 1 The intention of both averagings was to reduce the amount of analyzed data. They should not ad- versely affect the analysis, since the values being averaged are quite similar. 296 Table I.2 Parameters of the rotational excitation model. Tested hypotheses, with reference points shown in bold, are numbered in accordance with the text. Effective rotational temperature, K CH 2 OH¡ ! CD 2 OH¡ ! CH 2 OÅH CD 2 OÅH DÅCHDO Ground Exc. Ground Exc. Ground Exc. E, cm ¡1 3476 2268 3487 2524 2861 1818 Expt. fit (4º 1 ) 145(10) 110(10) 120(10) 130(10) 320(20) 180(15) (3º 1 Åº 2 ) 155(10) 140(20) 1. I A , bÆ 0.300 Å 233 120 257 2. I B , bÆ 0.628 Å 143 120 200 3. I B , bÆ 0.755 Å 133 124 180 1. Can purely isotopic effects explain the large difference between the ground state temperatures in the CD 2 OH¡ ! HÅ CD 2 O and CD 2 OH¡ ! DÅ CHDO process- es? According to (5.25), the largest difference can be expected if the most different moments of inertia, that is I A , are used. 1 The CD 2 OH¡ ! HÅCD 2 O ground state tem- perature was taken as the reference, and temperatures for CD 2 OH¡ ! DÅCHDO and CH 2 OH¡ ! HÅ CH 2 O were calculated. It can be seen from table I.2 that even in this extreme case the CD 2 OH¡ ! DÅ CHDO temperature is appreciably underestimated. At the same time, the CH 2 OH¡ ! HÅCH 2 O temperature is appreciably overestimated. This means that the observed large temperature difference cannot be explained by isotopic differences in the inertial parameters alone. 2. If the model for decomposition into “direct” and “isomerization” channels (see subsection 4.4.2) is reasonable, are the ground state temperatures for the CH 2 OH¡ ! HÅ CH 2 O and CD 2 OH¡ ! HÅ CD 2 O processes (which are mostly “direct”) consistent with the isotopic effect? The temperature for CD 2 OH¡ ! HÅ CD 2 O (the purest one, according to the model) was taken as the reference, and the other temperatures were 1 It that case the rotation must be excited mostly around the a principal axis of inertia, which coincides with the C¡ ¡O bond — not a very realistic assumption when the transition state geometries (see figure 4.2.) are considered 297 calculated with the assumption that the rotation is excited mostly around the b ax- is 1 . Table I.2 shows that the calculated CH 2 OH¡ ! HÅ CH 2 O temperature is now in excellent agreement with the experiment. The calculated CD 2 OH¡ ! DÅCHDO tem- perature is now, expectedly, even farther from the observed value, supporting the conclusion that the “isomerization” channel (the only source of D fragments) is qual- itatively different from the “direct” channel (the main source of H fragments). 3. Are the excited states temperatures, determined mostly by the “isomerization” channel, consistent with the isotopic effect? Now the CD 2 OH¡ ! DÅCHDO (pure “iso- merization”) excited states temperature was taken as the reference, and the other temperatures were calculated assuming rotation around the same b axis (see fig- ure 4.2(f) for the transition state geometry). It is evident from table I.2 that all calcu- lated excited states temperatures are consistent with the experimental values. No- tice, however, that according to table 4.5, the excited states are partially populated by the “direct” channel as well. Therefore, the agreement between the isotopically adjusted temperatures and the observations shows only that they do not contradict each other, and that the excited states temperatures for the “direct” channel are probably close to those for the “isomerization” channel. The conclusion that can be drawn from this analysis is that the temperature vari- ations within each channel are consistent with the simple isotopic effect, but “direct” and “isomerization” channels indeed lead to different rotational excitation, at least in the vibrationless formaldehyde. It is interesting that the impact parameters cal- culated from the transition state structures shown in figs. 4.2(c)/(f) for impulsive dis- sociation along the O¡H/C¡H bond are 0.535 Å and 0.601 Å respectively, being not unreasonably far (for such rough models) from the estimated 0.628 Å and 0.755 Å and 1 Lying in the CH 2 O plane, perpendicular to the C¡ ¡O bond — the most reasonable assumption for the transition state geometry shown in figure 4.2(c). 298 having the correct trend, which suggests that both processes do not require unusual dissociation or isomerization mechanisms. J Plots of vibrational densities Figure J.1 shows 2-dimensional projections of vibrational probability density dis- tributions obtained in VCI calculations summarized in table 4.1. 1 Quantities q i are mass-weighted normal coordinates in atomic units ( p m e a 0 ). Their numbering corre- sponds to normal modes given in table 4.1: q 1 and q 2 are O¡H and asymmetric C¡H stretches, q 7 , q 5 — COH bending, and q 9 , q 8 — out-of-plane motions. Note the pure O¡H stretch (vibrations along q 1 ) character of thej4 1 i wavefunc- tion of CD 2 OH and the complicated shapes (in q 1 , q 2 coordinates) of the excited CH 2 OH wavefunction, indicating strong interaction between the O¡H and antisym- metric C¡H stretch modes. Plots in the out-of-plane coordinates (q 9 , q 8 ) show very broad density distribu- tions even for the ground vibrational states of both isotopologs. This delocalization is due to a very flat bottom of the potential well in these coordinates. Even though the potential energy surface has two mirror-like minima corresponding to nonplanar ge- ometry (see figure 4.2), the barrier connecting them through a planar configuration is below the zero-point energy in these modes, which explains why the densities have their maxima at the planar geometry (q 8 Æ 0, q 9 Æ 0). Nevertheless, these maxima are also very flat, and therefore the radical does not have a well-defined “equilibrium geometry”, being most accurately described as quasiplanar with large out-of-plane motions. 1 The data for these distributions were kindly shared with me by Dr. Eugene Kamarchik during the collaborative studies of the hydroxymethyl radical vibrational spectroscopy and predissociation, al- though they were not included in the published articles due to size considerations. 299 CH 2 OH |0〉 CH 2 OH |1 4 〉+|1 3 2 1 〉 CH 2 OH |1 3 2 1 〉−|1 4 〉 CD 2 OH |0〉 CD 2 OH |4 1 〉 −20 0 20 40 q 1 −20 −10 0 10 20 q 2 −20 0 20 40 q 1 −20 −10 0 10 20 q 2 −20 0 20 40 q 1 −20 −10 0 10 20 q 2 −20 0 20 40 q 1 −20 −10 0 10 20 q 2 −20 0 20 40 q 1 −20 −10 0 10 20 q 2 −40 −20 0 20 40 q 7 −30 −20 −10 0 10 20 30 q 5 −40 −20 0 20 40 q 7 −30 −20 −10 0 10 20 30 q 5 −40 −20 0 20 40 q 7 −30 −20 −10 0 10 20 30 q 5 −40 −20 0 20 40 q 7 −30 −20 −10 0 10 20 30 q 5 −40 −20 0 20 40 q 7 −30 −20 −10 0 10 20 30 q 5 −40 −20 0 20 40 q 9 −40 −20 0 20 40 q 8 −40 −20 0 20 40 q 9 −40 −20 0 20 40 q 8 −40 −20 0 20 40 q 9 −40 −20 0 20 40 q 8 −60 −40 −20 0 20 40 60 q 9 −40 −20 0 20 40 q 8 −60 −40 −20 0 20 40 60 q 9 −40 −20 0 20 40 q 8 Figure J.1 2-dimensional vibrational density distributions. 300

Abstract (if available)

Abstract An experimental setup for sliced velocity map imaging (SVMI) of H fragments produced in photodissociation of small molecules and radicals is designed, constructed and characterized experimentally. The setup uses an ion-optical arrangement consisting of an accelerator that allows creation of variable electrostatic field geometries near the photodissociation region (where the studied fragments are produced) for initial acceleration and focusing of the ionized fragments and an additional electrostatic lens inside the drift tube for independent control of the optical magnification. This approach permits largely independent control of the radial and temporal characteristics of the velocity mapping, making it possibile to achieve time-gated slicing with selectable relative thickness for a very broad range of fragment kinetic energies (from a fraction of an electronvolt to a few electronvolts)—an important capability not available in other SVMI setups described in the literature. At the same time, the kinetic energy resolution (∼1%) obtainable in a large part of the operating range is comparable to that of the best known SVMI setups. ❧ Two methods for analysis of the raw data produced in VMI and SVMI experiments are developed and implemented. The first method is intended for correction of small distortions of raw velocity map images. It is shown that correction of these distortions, easily achievable by processing of the image using this method, might significantly increase the effective resolution for the speed or kinetic energy distribution extracted from the image. In addition, artifacts in angular distributions, produced by the distortions, are eliminated in the corrected images. The method is universal, being able to correct almost arbitrary distortions, and can be applied to full-projection and sliced velocity map imaging data represented as coordinate lists and as raster images. ❧ The second data-processing method allows reconstruction of SVMI data recorded with slicing pulses of arbitrary shape and relative thickness (including the fullprojection VMI as a trivial particular case). This ability permits to exploit the improved signal-to-noise ratio of the SVMI method and at the same time achieve the best resolution over the whole imaged velocity range, impossible with finite slicing without reconstruction. Testing of the method on synthetic and experimental images demonstrated that the results obtained by reconstruction of sliced images outperform the results from sliced images without reconstruction and from Abel inversion of full-projection images in terms of the resulting signal-to-noise ratio and resolution simultaneously. ❧ The developed and implemented setup and methods were successfully used in studies of overtone-induced dissociation and isomerization dynamics of hydroxymethyl radial CH₂OH and its isotopolog CD₂OH. While these studies, at first glance, largely repeat the previous work, the new experimental capabilities in the present work, significantly improving the resolution and detection efficiency, allowed to obtain quantitative and qualitative results at a completely new level. In particular, interaction of the third O–H stretch overtone (4v₁ level) with a combination of O–H and antisymmetric C–H stretches (3v₁ + v₂ level) was observed for the first time. Dissociation of the vibrationally excited radical to formaldehyde and hydrogen fragments (CH₂OH→CH₂O + H) was observed from both of these levels. In case of CD₂OH excitation (for which only the 4v₁ level was observed), in addition to the H products, a small amount of D fragments correlating with CHDO cofragments was observed for the first time, providing an explicit experimental demonstration of the CD₂OH→CHD₂O→D + CHDO decomposition through isomerization to the methoxy radical. Analysis of the vibrational distributions of the formaldehyde products in all reactions suggests that while O–H bond fission is responsible for the major part of the produced H fragments, a noticeable part of them in CH₂OH and CD₂OH decomposition is also due to the isomerization pathway. The relatively high kinetic energy resolution in the present experiments allowed an accurate direct determination of the hydroxymethyl dissociation energies: D₀(CH₂OH→CH₂O + H) = 10160±70 cm⁻¹, D₀(CD₂OH→CD₂O + H) = 10135±70 cm⁻¹, D₀(CD₂OH→CHDO + D) = 10760±60 cm⁻¹.

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Creator Ryazanov, Mikhail (author)

Core Title Development and implementation of methods for sliced velocity map imaging. Studies of overtone-induced dissociation and isomerization dynamics of hydroxymethyl radical (CH₂OH and CD₂OH)

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Chemistry (Chemical Physics)

Publication Date 11/14/2012

Defense Date 07/23/2012

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag hydroxymethyl radical,inverse Abel transform,isomerization,OAI-PMH Harvest,sliced velocity map imaging,vibrational predissociation

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Reisler, Hanna (committee chair), Haas, Stephan W. (committee member), Wittig, Curt F. (committee member)

Creator Email mikhail.ryazanov@gmail.com,ryazanov@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-112619

Unique identifier UC11290246

Identifier usctheses-c3-112619 (legacy record id)

Legacy Identifier etd-RyazanovMi-1291.pdf

Dmrecord 112619

Document Type Dissertation

Rights Ryazanov, Mikhail

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA