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Evaporative attachment of slow electrons to free sodium clusters

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Evaporative attachment of slow electrons to free sodium clusters

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Content EVAPORATIVE ATTACHMENT OF SLOW ELECTRONS TO FREE SODIUM CLUSTERS by Roman Mikhailovich Rabinovich A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) May 2008 Copyright 2008 Roman Mihailovich Rabinovich i Acknowledgements It is my great pleasure to acknowledge all the people who helped and supported me on this project. First of all I would like to thank Professor Vitaly V. Kresin, my research advisor. Not only did Vitaly guide me through this research project, but he was helping me on every step of my way through the graduate school, from application to USC in Fall 2001 to final stages of my dissertation and defining my future career. No matter what difficulties I encountered in the last six years when turned to Vitaly he always offered me his invaluable assistance, wise advice, and persuasive encouragement. I am very glad that I had a rare chance to be a student of such a truly good person as Vitaly V. Kresin. I want to thank my good fiend Oleg Kornilov for all the discussions, advice, and encouragement he gave me on this project. I cannot imagine anybody else devoting so much time to listening to all my ideas and being so enthusiastic about it. Oleg’s genuine interest in science has been always the admiration of mine. I also want to thank Dr. Klavs Hansen for his weighty advice on the statistical treatment of the cluster fragmentation. I am truly grateful to my friends Boris Karpichev, Vadim Mozhayskiy, Maksim Medvetsky, and Piotr Pieniazek, who tolerated me all these years and made living in Los Angeles so enjoyable. I want to thank all members of our research group, especially Chunlei Xia, Sascha Vongehr, and Ramiro Moro. Chunlei’s help was very valuable. It arrived ii exactly when most needed. Everything Sascha taught me, I find exceptionally useful and important. Ramiro’s optimism and enthusiastic attitude were very contagious and helped me to get through the toughest parts of this project. Above all I am grateful to my parents, Mikhail Rabinovich and Marina Prezent, for their infinite, unquenchable support and my daughter Faina for her love and unbelievable patience and understanding. iii Table of Contents Acknowledgements .........................................................................................................ii List of Tables..................................................................................................................vi List of Figures................................................................................................................vii Abstract..........................................................................................................................xii Chapter 1. Introduction.................................................................................................... 1 1.1 Alkali Clusters....................................................................................................... 1 1.2 Cluster Mass Spectra and Evaporative Ensemble ................................................. 3 1.3 Electron Attachment.............................................................................................. 7 1.4 Outline ................................................................................................................. 10 Chapter 2. Experiment................................................................................................... 12 2.1 General Outline ................................................................................................... 12 2.2 Cluster Source and Beam Collimation ................................................................ 13 2.3 Detector of Neutral Clusters................................................................................ 14 2.4 Electron Gun and Scattering Region ................................................................... 16 2.5 Detector of Negative Ions.................................................................................... 17 2.6 Optimization and Calibration .............................................................................. 20 2.7 Data Acquisition.................................................................................................. 23 2.8 Experimental Procedures..................................................................................... 24 2.9 Experimental Results........................................................................................... 26 2.10 Related Experimental Data of Other Groups .................................................... 31 Chapter 3. Evaporative Attachment .............................................................................. 33 3.1 Physics of Attachment and Outline of the Calculation ....................................... 33 3.2 Evaporation Statistics Overview ......................................................................... 36 3.2.1 Evaporation Rates......................................................................................... 36 3.2.2 Evaporation Probability................................................................................ 46 3.3 Temperatures of the Neutral Clusters.................................................................. 49 3.3.1 Sequence of Evaporations ............................................................................ 49 3.3.2 Single Channel Sequence (Monomer Evaporations Only)........................... 51 3.3.3 Two Channel Sequence (Monomer and Dimer Evaporations)..................... 54 3.3 Temperatures of the Cluster Anions.................................................................... 63 3.4 Fragmentation Patterns of Cluster Anion............................................................ 69 3.5 Anion Mass Spectra............................................................................................. 75 3.6 Dissociation Energies .......................................................................................... 76 3.7 Calculation Results.............................................................................................. 84 iv Chapter 4. Future Work................................................................................................. 87 4.1 Large Cluster Limit ............................................................................................. 87 4.2 Electron Attachment to Cluster Cations.............................................................. 90 References ..................................................................................................................... 93 Appendices .................................................................................................................... 97 Appendix A. Energy of Electrons ............................................................................. 97 Appendix B. Vertical Electron Detachment Energies............................................. 107 Appendix C. Correction to the Precursor Beam Mass Spectra ............................... 110 Appendix D. Sodium Cluster Cation Dissociation Energies................................... 115 Appendix E. Calculated Quantities ......................................................................... 116 v List of Tables Table 3.1 Corrected values of the sodium cluster cation dissociation energies for dimer evaporation.......................................................................................................... 86 Table B.1 Vertical electron detachment energy values for sodium cluster anions. After Ref. [KOS’07].................................................................................................... 107 Table D.1 Dissociation energies of sodium cluster cations......................................... 115 Table E.1 Calculated dissociation energies of neutral sodium clusters....................... 116 Table E.2 Calculated dissociation energies of sodium clusters anions. ...................... 117 Table E.3 Calculated boundary values for the temperature distributions of neutral sodium clusters prior to electron attachment and the weights of the distributions of clusters produced by the monomer and dimer evaporation channels. .................... 117 Table E.4 Calculated boundary values for the temperature distributions of sodium cluster anions after the electron attachment and the weights of the distributions of clusters produced by the monomer and dimer evaporation channels.......................... 118 vi List of Figures Figure 1.1 Sodium cluster abundance spectrum demonstrating magic numbers: 8, 20, 40, 58 After [KNI’84]............................................................................................... 4 Figure 1.2 Electron shells in self consistent effective potential corresponding to Na 40 [KNI’84].................................................................................................................. 5 Figure 1.3 Evaporative cooling diagram ......................................................................... 7 Figure 2.1 The outline of the experimental setup.......................................................... 12 Figure 2.2 Two connections schemes for the channeltron electron multiplier used for detection of negative ions. ....................................................................................... 20 Figure 2.3 Electron bombardment ionizer..................................................................... 22 Figure 2.4 Data acquisition system ............................................................................... 24 Figure 2.5 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: TC 665 reservoir = ° 761 =° 400 PkPa = 675 , TC , . ............................................................................................................... 27 nozzle Ar Figure 2.6 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: TC reservoir = ° 767 =° 315 PkPa = 665 , TC , ................................................................................................................. 28 nozzle Ar Figure 2.7 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: TC reservoir = ° 761 =° 300 PkPa = 662 , TC , . ............................................................................................................... 28 nozzle Ar Figure 2.8 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: TC reservoir = ° 781 =° 400 PkPa = , TC , . ............................................................................................................... 28 nozzle Ar Figure 2.9 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: 664 reservoir TC = ° =° Pa = , , . ............................................................................................................... 29 788 nozzle TC 400 Ar Pk vii Figure 2.10 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: 663 reservoir TC = ° =° Pa = 663 , , . ............................................................................................................... 29 786 nozzle TC 400 Ar Pk Figure 2.11 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: TC reservoir = ° 786 =° 400 PkPa = 661 , TC , . ............................................................................................................... 29 nozzle Ar Figure 2.12 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: TC reservoir = ° 765 =° 570 PkPa = 661 , TC , . ............................................................................................................... 30 nozzle Ar Figure 2.13 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: TC reservoir = ° 765 =° 570 PkPa = 15 1 ω − =⋅ = 23 0.74 De , TC , . ............................................................................................................... 30 nozzle Ar Figure 2.14 Combined mass spectra of sodium cluster anions (at the top) and neutral cluster (at the bottom)........................................................................................ 31 Figure 2.15. Size distribution of K n - , obtained from mass spectrum of the potassium cluster anions. After [NAG’94].................................................................... 32 Figure 3.1 Electron attachment energy diagram............................................................ 34 Figure 3.2 The monomer evaporation rates of sodium clusters calculated using , , 1 '410 s 15 0.69 DeV V 36 0.90 DeV = , = , , and 150K Θ= 8 2.1 10 WS R cm − − ≈⋅ ........................................................................................................ 43 Figure 3.3 Calculated probabilities of clusters to evaporate a fragment within the lifetime of 2 ms.............................................................................................................. 47 Figure 3.4 Calculated monomer and dimer evaporation probabilities for Na 16 during the experimental time of flight of 2 ms. Rates were calculated for the dissociation energies of 0.82 eV and 0.92 eV for the monomer and dimer respectively.................................................................................................................... 49 Figure 3.5 Step-like shape of the decay probability function. The displayed function was calculated for Na 17 using 17 0.86 D eV = ..................................................... 52 Figure 3.6 Evaporation probabilities and uniform temperature distribution of the clusters. Calculated for Na 17 using 17 0.86 D eV = and 18 0.90 D eV = ............................... 54 viii Figure 3.7 Calculated probabilities of cluster decay. Monomer and dimer evaporation channels are considered. Calculated for Na 16 using () m 16 0.82 D eV = and () 0.92 d 16 D eV = CC = () m = 2171 21 10 4 10 s ωω .......................................................................................................... 55 Figure 3.8 Calculated probabilities of monomer and dimer evaporation channels with different ratios of dissociation energies. Calculation parameters: , , 16 0.8 DeV − =⋅ = ⋅ () m = 2171 21 10 4 10 s ωω ....................................................................... 56 Figure 3.9 Experimentally measured branching coefficients for sodium cluster cations [BRE'89] (a) and evaporation chains for neutral sodium clusters (b)............... 58 Figure 3.10 Two consecutive evaporations in assumption of broad initial temperature distribution................................................................................................. 59 Figure 3.11 Calculated branching coefficients of monomer and dimer evaporation channels with different ratios of dissociation energies. Used parameters: , , 16 CC = 0.8 DeV − ≈≈⋅ ........................................ 60 Figure 3.12 Total temperature distributions of the clusters directly produced by both: monomer and dimer evaporation channels........................................................... 63 Figure 3.13 Comparison of photoelectron spectra of Na 20 - acquired at cluster temperatures of 100 K and 300 K After [KOS’05]. ...................................................... 66 Figure 3.14 Frank-Condon effect: “vertical” photoelectron detachment and adiabatic attachment of the electron.............................................................................. 66 Figure 3.15 Transformations of the temperature distribution during fragment evaporation. ................................................................................................................... 72 Figure 3.16 Anion fragmentation “tree” constructed of all possible combinations of the monomer and dimer evaporation......................................................................... 74 Figure 3.17 Reconstruction of the dissociation energies of Na 10 + and Na 22 + from the theoretical values calculated using semiempirical shell-correction method. Solid circles represent the theoretically calculated values [YAN’95] and open square represent the dissociation energies obtained from experimental data [BRE'89]. After [YAN’95]............................................................................................ 80 Figure 3.18 Anion mass spectra calculated using the original cation dimer dissociation energies from [BRE'89] and corresponding pre-factors in evaporation rate expressions, compared to the experimental data. ............................... 85 ix Figure 3.19 Anion mass spectra calculated using accurate pre-factor values and corrected dimer dissociation energies compared to the experimental data. .................. 85 Figure 4.1 Outline of the suggested experimental setup. .............................................. 91 Figure A.1 The scheme of the measurement of the electron distribution using the retarding potential techniques........................................................................................ 97 Figure A.2 A typical current vs. voltage curve obtained by the retarding potential measurement.................................................................................................................. 98 Figure A.3 The diagrams for the kinetic energy distribution of electrons in potential ϕ . ................................................................................................................. 100 Figure A.0.4 The current of electrons passing the scattering region as function of the potential inside the region...................................................................................... 101 Figure A.5 The electron energy diagrams of the retarding potential measurement in at different values of the potential barrier height. ................................................... 103 Figure A.6 The electron energy diagrams of inside the electron gun during the anion mass spectra acquisition. ................................................................................... 105 Figure B.1 Photoelectron spectra. The experimental spectra are shown in black, and theoretically calculated spectra are shown in red. After [MOS’03]. .................... 108 Figure B.0.2. Photoelectron spectrum of Na 20 - . After [KOS’05]. ............................... 108 Figure B.3 Photoelectron spectra. After [KOS’05]..................................................... 109 Figure C.1 Experimentally measured mass spectra of sodium cluster cations. (a) Spectrum obtained by our group on the experimental setup described in Chapter 2 by ionizing a beam of neutral sodium clusters with a coaxial UV laser beam of 350 nm wavelength. (b) A cation mass spectrum published in Ref. [BRE’94] (solid bars represent experimental data)...................................................................... 111 Figure C.2 Experimentally measured abundances of sodium cluster anions (corresponding to the spectra on the left side of Figures 2.5, 2.7, and 2.8, on the left). ............................................................................................................................. 112 Figure C.3 Mass spectra of neutral sodium clusters (corresponding to the spectra on the right side of Figures 2.6 and 2.8). Open bars represent the original measured abundances and solid bars represent the abundances after the correction..................................................................................................................... 114 x Figure E.1 Fragmentation patterns of sodium cluster anions. N is the size of the mother anion prior to................................................................................................... 119 Figure E.2 Fragmentation patterns of sodium cluster anions. N is the size of the mother anion prior to the evaporations........................................................................ 120 xi Abstract We have carried out a measurement of the mass spectra of sodium cluster anions formed in the collisions of free neutral sodium clusters with beam of low energy (0.1 eV) electrons. Anions covering the size ranges from to and from to were observed. The anion mass spectra were recorded simultaneously with those of the precursor cluster beam, which allowed us to monitor the effect of electron capture on the relative abundances of various cluster sizes. The anion mass spectra demonstrated significant restructuring with respect to the precursor beam: a downshift of the shell-closing magic numbers, a change in the shape of the overall intensity envelope, and, significantly, an alteration in the relative intensities of the open-shell peaks located between the magic numbers. This alteration did not represent a simple pattern shift by one electron number, and required an accurate analysis. - 7 Na - 92 Na - 132 Na - 144 Na The restructuring of the mass spectra was treated theoretically by means of an evaporative attachment model, consisting of three steps: (a) electron capture by the strong polarization potential of the cluster, (b) rapid dissipation of the electron energy into the internal vibrational energy of the cluster (cluster heating), and a statistical evaporative cooling process (monomer and dimer evaporations). The analysis yielded results in good agreement with the experimental data and explained the fine structure of the observed abundance restructuring patterns. Based on an accurate statistical description of dimer evaporation we derived an adjustment to the previous literature values of sodium cluster dimer evaporation xii energies, which had been obtained from cluster photodissociation experiments. The exponential sensitivity of the evaporation process to the cluster evaporation energies allowed us to verify the validity of this adjustment. The corrected evaporation energies for dimers were found to be approximately 20% higher than the original values. The results demonstrate that slow-electron capture offers a useful window into the statistical and binding properties of metal clusters. Conversely, they show that in interpreting electron capture and transfer reactions it is essential to account for the accompanying fragmentation effects. xiii Chapter 1. Introduction 1.1 Alkali Clusters In the last several decades, atomic and molecular clusters have attracted significant attention of scientists working in different fields of physics, chemistry and materials science. The roots of this interest lie both in the fundamental theory of finite body systems as well as in possible technological applications of clusters. Being small particles comprised of several to many thousands of atoms or molecules, clusters fall into a gray area between well studied fields of the gas phase and solid state. Traditionally, scientists approach studying matter in two different ways, considering ether interactions of a few elements and describing atoms and molecules in a gas phase or applying statistics to analyze very large numbers of interacting particles, describing solids and liquids. Studying the systems of intermediate sizes which cannot be well described by either of two approaches has always been a challenge with a promise of discovery of new physics. Atomic and molecular clusters belong to this category. Advances in theoretical and experimental techniques allowed researchers to step into this new area. Cluster research creates a link between single atoms or molecules and solid or liquid aggregate states. Tracing the structure and properties as the size of clusters changes reveals the transition of the atomic and molecular properties into properties of solid or liquid states and adds another dimension to the description of matter: size. Not only does it contribute to understanding of the fundamental science, it also gives a 1 powerful technological tool with wide variety of applications. By varying the cluster size desired properties of particles can be achieved. Additionally, some clusters have revealed unique properties which other aggregate states of the same material do not. Vivid examples include dielectric properties of mercury clusters [BUS'98], catalysis on clusters of noble metals [BND'06 ], and ferroelectricity in niobium clusters [MOR’03]. To utilize these properties, clusters can be used as building blocks for creation of new materials and devices, for example crystal structures with clusters instead of atoms, or cluster based nanoelectronic devices on surfaces. While there is significant progress towards developing applications, two fundamental issues result in serious obstacles. First one is the stability of the clusters. Many clusters are not stable and quickly decay under certain conditions. Second issue is the change of cluster’s structure and therefore properties as the cluster is deposited on a surface or interacts with other particles. Overcoming these problems calls for more experimental and theoretical research which will contribute to better understanding of the cluster structure and dynamical processes inside the clusters. Clusters of alkali metals make a convenient benchmark for studying properties of metal clusters. Similarly to the bulk solid state clusters of alkali atoms are primarily bound by delocalized valence electrons [HEE'93]. It has been shown that these electrons are nearly free and are organized into shells. The electronic shell structure was beautifully illustrated by observations of magic numbers in mass abundance spectra [KNI’84]. 2 Alkali clusters are among the most studied species and their basic structure has been well explained, however dynamical processes within the cluster as well as interaction of the cluster with other particles are the next level problems. Even such basic problems as fragmentation, light absorption, and electron scattering are not completely described by existing theories. We hope this work will contribute to solving the problems of interaction of clusters with slow electrons, relaxation of electronic excited states, and fragmentation in alkali clusters as well as better general understanding of structure and dynamic processes in clusters and nanoparticles. 1.2 Cluster Mass Spectra and Evaporative Ensemble As it was discovered in 1984 by W. D. Knight et al [KNI’84] the mass spectrum of sodium clusters produced by supersonic seeded cluster source exhibits a series of peaks of outstanding intensities or steps in intensities corresponding to the clusters of sizes 8, 20, 40, 58, 92, 138 (see Figure 1.1). These “magic numbers” were explained by the enhanced stability of these clusters due to their electronic structure using one-electron shell model [HEE'93]. Namely, in a spherically symmetric potential these numbers correspond to the closing of the electron shells (for illustration see Figure 1.2) and therefore the clusters of these sizes have higher binding energies than others. While the fact that more stable clusters have higher abundances is easily intuitively understood and one-electron approximation gives a spectacular qualitative description of this stability, it neither gives insight into the processes which lead to cluster formation and decay, nor does it provides a quantitative description of the mass spectra. It was shown later that evaporative cooling plays a key role in the formation of 3 cluster mass spectra and that detailed analyses can be performed within “evaporative ensemble” formalism [KLO’85]. Figure 1.1 Sodium cluster abundance spectrum demonstrating magic numbers: 8, 20, 40, 58 After [KNI’84] According to this formalism metal clusters detected in the supersonic beam are the products of decay – series of evaporation of dimers and monomers. Indeed, being formed in the source nozzle region at the temperatures about 1000K, clusters are metastable. They possess enough internal vibrational energy to fall apart: emit electrons, evaporate fragments, or single atoms, and as soon as sufficient amount of energy is concentrated in a particular vibrational mode or transferred to an electron the decay occurs. It was shown that for alkali clusters the primary decay channels are evaporations of dimers and single atoms [BRE'90, BRE'89]. From basic thermodynamics it follows that the decay probability should be very sensitive to the vibrational temperature of the cluster and to the amount of energy needed to break the 4 bond and evaporate the fragment (below this energy is referred to as evaporation or dissociation energy). In case of an infinitely big particle in which the canonical ensemble concept is applicable, the evaporation rate is proportional to the exponent of the ratio of the temperature to the evaporation energy: exp( / ) Bv rDkT ∝ , where is the Boltzmann constant and is evaporation (dissociation) energy. Unfortunately, for small particles the canonical ensemble is not applicable and evaporation statistics requires careful consideration, however the general character of the dependence holds and this formula is good enough for qualitative description of the evaporation ensemble. B k D Figure 1.2 Electron shells in self consistent effective potential corresponding to Na 40 [KNI’84] 5 The evaporation causes cooling of the cluster. Indeed the total energy of the parent particle is reduced by the dissociation energy and split between the internal and kinetic energies of the daughter particle and the fragment: parent daugter fragment fragment EE DKE E = ++ + (1.1) Each fragment evaporation occurrence reduces the internal energy of the cluster and results in decrease of the evaporation rate for the next fragment. Continued indefinitely, this process would cool the clusters until their internal energy is lower than the energy of evaporation, however in practice the mass spectra are usually acquired at a certain distance from the source which means that the clusters have certain lifetimes before they reach a detector. In other words, experimentalists usually observe beams of clusters of certain “age” which defines their properties such as their size, vibrational temperature etc. This concept of evaporative ensemble is presented graphically in Figure 1.3. This diagram schematically depicts the change of the cluster size, temperature and the vibrational energy as a function of time. In this view it is easy to see what happens if the cluster gets reheated, i.e. if additional energy is reported to the cluster vibrations after the cluster is sufficiently cold to be considered stable on the experimental timescale. The evaporation rate increases again causing the cluster to dump the additional energy through the fragmentation process. In alkali clusters this effect has been observed experimentally in sodium clusters [BRO'99 , MAR’91]. The clusters were reheated by the absorption of photons (the cluster beam was illuminated by a laser beam) and the products of 6 fragmentation were observed. Another process in which a similar effect can be expected is the electron attachment. Figure 1.3 Evaporative cooling diagram 1.3 Electron Attachment Due to mobility of delocalized valence electrons alkali clusters are highly polarizable. Both experimental and theoretical studies [BRR'02, HEE'93, KRE’92, TIK’01 1 ] have proven that polarizability of alkali clusters exceeds that of an ideal conducting sphere of same radius 3 R α = , and can be approximated as ( ) 3 cluster cluster R δ ≈+ . α δ , a measure of electron density spill-out, can be classically viewed as the difference between the radius of the cloud of delocalized electrons and the radius of ionic cluster core 1/3 cluster W S R RN − =⋅ , where WS R − is Wigner-Seitz radius 7 of the atom. Notably, the electron spill-out stays relatively constant through the entire cluster size range and for sodium clusters is approximately equal to 1.3A, which is slightly above a half of is Wigner-Seitz radius , 2.1 WS Na R = Å. High polarizability of alkali clusters causes appearance of strong electrostatic forces between the clusters and charged particles [KRE’99]. Applying electrostatics to the problem of interaction between an electron and alkali cluster results in the interaction potential: 2 , 4 () 2 cluster e e Vr r α =−, (1.2) which is generally valid for cluster-electron distances much larger than the cluster radius: . It was shown by Langevin that the problem of classical scattering for a particle in attractive potential has a solution in which the particle falls into the center of force [LNG’05, MCD’89]. Applied to electron-cluster interaction, that means that electron capture is a possible channel for collisions of alkali clusters with free electrons. Classical solution for the capture cross section is given by the following expression: cluster rR 4 r − 22 2 capture e E π α σ =, (1.3) where E is the kinetic energy of electron far from the cluster. Quantum mechanical solution of this problem recovers essentially similar result if the following condition is satisfied [VOG’54]: 4 2 8 e E em α ≥ , (1.4) 8 where is the mass of electron. For alkali clusters this condition holds for energies down to several micro electron volts. e m Experimental studies confirmed theoretical results both qualitatively and quantitatively [KAS’99, KAS’00 2 ]. First experiments on measurement of the cross section of the electron attachment to sodium clusters were performed using beam depletion technique. In these experiments the intensities of clusters passed through the beam of electrons but not affected by electron beam were measured. Depletion in the intensities correlated with electron beam current was attributed to inelastic collision resulted in electron capture. The experiments proved that sodium clusters have very large electron capture cross sections for the particles of their size which rises steeply as the electron energy approaches 0. Experimental values of the cross section were in good agreement with Langevin’s formula. Another fundamental question is what happens after the electron spirals down to the cluster. Various experiments with alkali clusters showed that the electron capture leads to the formation of cluster anions. In experiments with potassium clusters the anions were observed in the capture of low energy electrons [SEN’00] as well as in collisions with krypton atoms in high Rydberg state [NAG’94]. In our previous experiments negative particles formed in the collisions of sodium clusters with low energy electrons were detected, and though the measurement lacked mass resolution it demonstrated that the anions are products of the electron capture [KAS’00 1 ]. Not only did these experiments prove that the attachment takes place, but also they showed that resulting anions live long enough to be detected. This raises new questions: how is the 9 kinetic and binding energy of the captured electron dissipated; what are the relaxation channels; is there an effect on the cluster mass spectrum? For example, given that the beam of neutral clusters displays “magic numbers” at the electron-shell-closing sizes of , will the daughter anions have abundance maxima at these same sizes – which have an enhanced population of neutral precursors – or will they be able to reorganize in accordance with the shell closing sequence of ? 20,40,58,... Na . 19,39,57,.. Na − 1.4 Outline In this work we report the results of the experimental study and theoretical treatment of the attachment of slow electrons to free neutral sodium clusters through analysis of the mass spectra of the formed cluster anions. In chapter 2 the experimental measurement of the mass spectra of the cluster anions in the cluster-electron collisions is reported. Sections 2.1 through 2.8 are devoted to detailed description of the experimental method, setup, and procedures we used to obtain the mass spectra. In section 2.9 we show the results of the experiment: the observed spectra of the cluster anions compared side by side with the mass spectra of the parent cluster beam. In Section 2.10, the results of a previous experimental study are discussed from the point of view of the framework developed here. In chapter 3 the theoretical analysis of the obtained mass spectra based on the evaporative attachment is presented. In section 3.1 we discuss the physical processes of the attachment and outline the calculation approach we used. In sections 3.2 through 10 3.6 we describe details of the calculations. In section 3.7 we present the calculation results and compare them to the experimentally measured anion mass spectra. In chapter 4 we discuss possible future work in this area. In section 4.1 we consider the evaporative attachment in the limit of large particles and in section 4.2 we discuss experiential study of the electron attachment to cluster cations. 11 Chapter 2. Experiment 2.1 General Outline We have performed a direct measurement of the abundance mass spectra of sodium cluster anions formed in the collision of a collimated beam of neutral sodium clusters with a beam of slow electrons, simultaneously with the measurement of the abundance mass spectra of the precursor neutral cluster beam. The outline of the experimental setup is shown in Figure 2.1. Figure 2.1 The outline of the experimental setup Neutral sodium clusters were produced by a supersonic source, passed a set of the skimmer and collimators and were directed into the scattering region of the electron gun where they collided with the low energy electrons. Ions formed in the region passed to the quadrupole mass filter and were detected by the channeltron mass detector. However most of the clusters (above 99%) remained not ionized in the scattering region and passed the detector of negative ions undisturbed into the end part of the apparatus, where they were ionized by focused UV light and mass analyzed by a 12 second quadrupole mass spectrometer, providing the abundance mass spectra of the neutral cluster beam. Further in this section we present a detailed the description of the apparatus parts and experimental procedures. 2.2 Cluster Source and Beam Collimation The clusters were produced by a supersonic seeded cluster source similar to one described in the previous publications [HEE'85, HEE'93, KAS’01 2 ]. The source was made of stainless steel and equipped with a Laval shaped nozzle with the hole of 75 μm diameter. During the experimental runs the sodium reservoir was kept at the temperature of 660˚C and the nozzle was heated approximately 50˚C to 100˚C higher to avoid clogging. Argon was used as a carrier gas. The gas pressure varied depending on the size of the clusters studied in order to boost the intensity of clusters in the desired region. It is empirically known that pressure increase causes a shift of cluster intensities towards higher masses, so the carrier gas pressure in our experiments ranged from 300 kPa for Na 8 and Na 20 through 600 kPa for Na 138 . This high value of the source temperature (660˚C) was unusual for the source of this kind and was used to achieve maximum intensity of the cluster beam. Operation in this regime was often unstable and resulted in the partial or complete clogging of the nozzle. It was found that a temporary drop of the reservoir temperature by approximately 100˚C combined with increase of the nozzle temperature by 50˚C for one hour usually helped to clean the nozzle and restore the beam of clusters. 13 Within 8 mm from the nozzle, the cluster beam passed a molecular beam skimmer (Beam Dynamics, Inc.) of 0.4 mm diameter heated to approximately 330˚C. We employed a heated skimmer design to avoid deposits of sodium on skimmer’s surface. The beam was further collimated to a square 2 x 2 mm profile by a pair of collimator plates at the distance of 0.65 m from the source, and passed through a square 1.4x1.4 mm aperture directly before entering the scattering region of the electron gun (1.3 m from the source). 2.3 Detector of Neutral Clusters This part of the apparatus had been deigned and built some time ago and complete description of it can be found in [HEE'85, HEE'93]. The beam of neutral clusters that reached the end part of the setup was ionized by focused UV light. The UV light was produced by an Hg-Xe arc lamp (Oriel Instruments) and passed through a UV filter transparent for the light of the wavelength of 250 to 400 nm. The resulting radiation had the photon energy of 3.2 to 5 eV which slightly exceeds the ionization potential of the sodium clusters (the IP values of sodium clusters lie within the range of 3.2 to 3.8 eV [DUG'97, YAN’95]). Alkali clusters irradiated by photons of energy slightly exceeding their ionization potential were ionized almost without fragmentation [HEE'85]. Combining with the fact that the ionization cross section is a smooth function of the cluster size [HEE'85] it allowed us to assume that the abundances of the produced cluster cautions mirror the abundances of the neutral cluster beam. 14 The formed cations passed through a quadrupole mass filter and accelerated towards a high voltage dynode at 24 kV. The dynode was machined from stainless steel and coated by 5000 Ǻ layer of aluminum. The signal was detected by a scintillator plate attached to a photoelectron multiplier. The detector was run in a pulse regime registering single counts from cluster ions. Typical intensities of the cluster signals were in the range of 10 2 through 10 4 counts per second per cluster size. The dark count of the detector (dynode-scintillator-photomultiplier system) was approximately 20 counts per second. To reduce the noise a glow discharge in argon was performed between the dynode and the other electrodes every time after the chamber was exposed to atmosphere (see detailed description in [HEE'85]). Other sources of noise were signals of ions of the residual gases in vacuum chamber and electrical noises. The residual gas signal appeared at certain masses in the spectra (the most noticeable peak overlapped with the signal of Na 9 at ~205 AMU) and decreased as the vacuum in the detector chamber improved. The electrical noises appeared as rare sharp peaks in mass spectra of the amplitude of up to several thousand counts. During the experiments the detector was run in a scanning regime acquiring the mass spectra of the neutral cluster beam or was constantly set to detection of cluster of certain mass (usually Na 8 , Na 20 , Na 40 , Na 58 ). The later regime served to monitor the intensity of the cluster beam produced by the source and was used primarily during preliminary measurements or in-between of mass scan measurements. 15 2.4 Electron Gun and Scattering Region The construction of the electron gun was based on original design of a research group from New York University [COL'70] and described in details in dissertations of V. V. Kresin and V. Kasperovich [KAS’01 2, KRE’84]. The electrons were emitted by an indirectly-heated rectangular dispenser cathode (Spectra-Mat Inc.). Extracted and collimated by a series of grids and masks, the electrons were directed into the scattering region where they crossed the beam of clusters at a right angle. All walls surrounding the scattering region were made of molybdenum and were kept in electrical contact to insure equipotentiality inside the scattering region. The dimensions of the electron beam were 25 x 1.4 mm and precise alignment provided complete overlap of the cluster beam with the beam of electron. A collinear magnetic field of 400 Gauss was used to prevent the dispersion of electrons due to the space charge (the magnetic field used in the current modification of the electron gun was lower than in previous experiments in order to decrease the deflection of the formed cluster anions in the magnetic field). The body of the cathode was grounded so the emitting surface was kept at a constant potential. To vary the energy of electrons in the scattering region the potential applied to the walls of the region was controlled. The potential was chosen to maximize the cluster anion signal. The energy distribution of the electrons was analyzed using the retarding potential technique. The total electron current through the scattering region was ~10 μA and the average cross section weighted energy of the attached electrons was calculated to be 0.1 eV. The details of the application of the 16 retarding potential method and calculation of the average electron energy are presented in Appendix A. 2.5 Detector of Negative Ions Negative cluster ions born in the scattering region kept moving along with neutral clusters towards the detector. The electron gun magnetic field, being parallel to the electron beam and perpendicular to the beam of clusters, resulted in the appearance of the Lorentz force and deflected the formed anions. Two plates parallel to both the cluster beam and the magnetic field spaced by ~ 4 mm were placed outside the scattering region of the electron gun to form an electrostatic deflector. The voltage applied to the plates (~ 1 V) was selected to compensate the effect of the magnetic field and direct ions towards the cylindrical Einzel lens which was placed within 4 mm from the scattering region down the beam and focused the ions at the mass filter entrance. For the ion mass selection a quadrupole mass filter (ABB Extrel, QPS9000) was used. This model was designed for selection of ions of masses of 10 to 10,000 AMU. The quadrupole power supply had been customized to permit separate control over the RF and DC components of the voltage applied to the rods (usually the dependence between these two components is linear and set by the power supply). This customization allowed controlling the resolution of the quadrupole filter independently in different parts of the mass range and avoiding the excessive discrimination against lower ion masses [PED’04]. Unfortunately, the enhanced flexibility of this regime came at the expense of additional calibration procedures required along the entire mass rage. Since these calibrations were troublesome at higher ion masses, we used the 17 independent control over RF and DC components only for preliminary calibrations with positive ions (see further in the chapter). Past the mass filter, the ions of a designated mass reached a custom-designed ion detection unit (Detector Technology, Inc) which consisted of a high voltage conversion dynode and a channeltron electron multiplier. The unit was specifically engineered for the effective detection of heavy negative ions, which requires significantly higher voltage applied to the conversion dynode than in the case of positive ions [KRI’91]The unit was designed to operate at the conversion dynode voltage up to 20 kV, with the noise of less than 1 cps. In practice achieving such a low noise level at this voltage proved to be impossible in the existing experimental setup. The sources of the noise included field ionization of the molecules of residual gases in the vacuum, contamination of the dynode’s surface, and electrical noises. To reduce the noise the following actions were taken: (I) In the pulse detection regime the channeltron output signal is fed directly into a preamplifier of high sensitivity (usually detection threshold was set in between of 20 and 100 mV). In the standard connection scheme the channeltron output is connected through a capacitor to the output of a high voltage power supply (Figure 2.2 (a)). Consequently, even small instability in the applied high voltage can result in significant noise at the preamplifier input. To avoid this problem an alternative connection scheme was used (Figure 2.2 (b)). In this scheme high voltage has no direct connection to the channeltron’s output, so the output signal is more stable with respect to the high voltage noise. The experimental comparison of the two connections 18 revealed that the use of scheme (b) reduced the dark count of the channeltron from ~ 50 cps to less than 0.5 cps using low-noise HV power supplies (see (II)). (II) Despite the fact that using the above connection scheme isolates the channeltron output from the high voltage circuits, using low-noise high voltage power supplies proved to be important. In the experiment we used commercially available models of power supplies: Spellman 50P50 and Bertan 9201. (III) To reduce the noise associated with the dynode, a preconditioning procedure was used: The dynode voltage was raised slowly till the noise reached ~ 20 cps and left at the same level until the noise dropped to l cps (usually took from half an hour to several hours), then the voltage was raised again. The steps were repeated till the desired dynode voltage reached. A combination of all three steps allowed operation of the detector unit at the dynode voltage at 16 kV and the noise level as low as 1 cps. Another kind of noise, observed during the experiments, was the noise associated with the cluster beam. This noise amplitude varied from less than 1 cps to 10 2 cps depending on the source temperature and the pressure of the carrier gas (the higher the pressure and the temperature the bigger the noise level) and gave the main contribution to the background of the anion mass spectra. Experiments with a chopped cluster beam showed that the noise signal was modulated with the same frequency and phase as mass selected cluster signal. We assumed that signal originated from cluster anions of higher masses (M »10,000 AMU) which were too heavy to be filtered out by 19 the quadrupole mass filter. It is also conceivable that the noise could be the result of the field ionization of heavy clusters in the field of the conversion dynode. Figure 2.2 Two connections schemes for the channeltron electron multiplier used for detection of negative ions. 2.6 Optimization and Calibration The biggest experimental challenge was the low intensity of the cluster ions which arose from several reasons: First, the intensity of the parent cluster beam was restricted by the capability of the cluster source. The described source conditions allowed producing neutral cluster beam intensities up to several thousands of counts at a particular cluster size, which is high for continuous alkali cluster beams. However these cluster counts were still orders of magnitudes smaller than the intensities which could be reached for atomic or molecular beams. Second, the flow of the low-energy 20 electrons in the scattering region was limited due to the formation of space charge. Despite the optimization of the electron beam to achieve the maximum yield of the cluster anions, and high values of the attachment cross sections, a measurement of depletion showed [KAS’99] that less than 1% of the neutral clusters were ionized by the attachment. Finally, not all ions formed in the scattering region reached the detector. Magnetic field necessary to focus the electron beam also had a side effect of deflecting ions away from the exit of the region. Probability of cluster ions to escape the scattering region was approximately proportional to the square root of cluster’s mass [KAS’01 2 ] and the estimated values for Na 7 - , Na 19 - , Na 39 - , Na 57 - were 30%, 45%, 66%, and 80% respectively. Combination of these factors resulted in expected anion intensities of less than 10 2 cps at a selected cluster mass (experiment showed the estimate was correct within an order of magnitude). Optimization of signals of such a low intensity is a complicated problem, especially in the first stage when the setting required for signal detection are unknown and even slight parameter shift from the position of maximum may result in inability to resolve the signal from the background noise. To overcome this difficulty the optimization procedures were done in three steps: First, we used positive ions to perform rough optimizations of ion optics and calibration of the quadrupole mass filter. The electron gun was removed from the setup and replaced with a homebuilt cylindrical electron bombardment ionizer (see Figure 2.3). The channeltron detector was switched to regime of detection of positive ions and the voltages of reversed polarities were applied to all ion optics and quadrupole 21 elements. In the initial stage the signal from residual gases was used, and later the optimization was performed with the atomic beam of argon and the molecular beam of SF 6 , expanded through the same nozzle which was used in the sodium cluster source. Figure 2.3 Electron bombardment ionizer In the second stage we detected cluster anions operating the quadrupole mass filter in an integral (ion guide) regime. In this regime only the oscillating part of the voltage (RF) is applied to the quadrupole rods, which makes the filter transparent for ions of a wide mass range. In our experiment it allowed cluster anions of most masses to be detected simultaneously. Once a signal was observed, we employed a chopper and a multi channel analyzer (ORTEC, MCS-pci) to verify that the signal was originated from the cluster beam. This step allowed us to define the needed corrections to the ion optics voltages originally determined for positive ions. The corrections were due to the presence of the magnetic field of electron gun and to differences in the kinetic energy of the cluster anions and the positive ions formed in the ionizer. 22 The final steps of optimizations were performed directly before the acquisition of every anion mass spectrum. The quadrupole filter was set to the mass of a specific cluster within the studied mass range and the voltage applied to the ion optics was adjusted to maximize the anion yield. Short mass scans which covered two adjacent cluster peaks were used to fine-tune the resolution of the mass filter. A preliminary mass calibration of the quadrupole analyzer was also performed using positive ions formed in the electron bombardment ionization of argon and SF 6 beams. The final calibrations were done with the use of the sodium cluster anion signal itself, tracing the pattern of regularly displaced cluster peaks through the entire scanning range from Na 7 to Na 140 and referencing it to the points calibrated with the positive ions. 2.7 Data Acquisition To automate the collection of mass spectra, a data acquisition system was designed. The outline of the system is presented in Figure 2.4. The central part of the system was a PC computer equipped with a data acquisition board (National Instruments, PCI-6034E) and an analog output board 16-bit (National Instruments, PCI-6703), which controlled the quadrupole power supply and collected the ion count signal. The mass command voltage was generated by an analog output and set the RF and DC amplitude, defining the transparency mass range of the quadrupole filter (1 mV of change in the command signal roughly corresponded to 1 AMU shift of the permitted ion mass range). The output signal of the channeltron was converted by a 23 preamplifier (Advanced Research Instruments, F-100TD) to TTL pulses of 5ns width and collected by a 12-bit counter incorporated in the data acquisition board. Figure 2.4 Data acquisition system LabView-platform-based software was developed to control the mass spectrum acquisition process. The software allowed obtaining the mass spectra through the scanning of a preselected mass range, saving, exporting, and analyzing the acquired data as well as constant monitoring the signal at a particular ion mass. An important feature of the software was a separate storage of every mass scan data in a multiple scan regime, which provided extended flexibility for analysis of low intensity signals. 2.8 Experimental Procedures A typical experimental run lasted from 2 to 15 hours. After the source was heated to the operating temperature and the beam of neutral clusters was stabilized, the 24 optimization procedures were performed to achieve maximum cluster intensity in the desired mass region. The optimization included: cluster source alignment; cluster source carrier pressure adjustment; the adjustment of the ion optics and quadrupole settings in the integral regime; the adjustment of the ion optics and quadrupole settings for ions of selected size; and the quadrupole filter resolution adjustment. The parameters of the electron gun such as the cathode temperature and the voltage applied to the grids and masks usually stayed unchanged during the entire series of the experiments; however, all parameters as well as the characteristics of the electron beam were monitored, and recorded prior to every experimental run. Typical values of the detected anion yield at a certain cluster mass ranged from 1 to 10 2 cps. Usual mass spectrum acquisition consisted of 10 to 30 scans and lasted from 30 min to 3 hours. Depending on the intensity of the ion yield the mass spectrum covered from 4 to 35 cluster peaks. The acquisition was often interrupted earlier because of an unexpected drop in the cluster signal or significant increase in the noise level. In many cases the cause the problem was eliminated by cleaning the source nozzle (cluster signal drop) or performing a preconditioning procedure for the dynode (noise increase), however sometimes the problem could not be fixed without complete shutdown of the experiment. The analysis of the precursor beam was constantly performed during the experimental runs using the detector of neutral clusters. The mass spectra and intensity information of the parent beam were used for monitoring the performance of the cluster source. 25 2.9 Experimental Results The obtained mass spectra of the collision products are presented in Figures 2.5-2.13, accompanied by the mass spectra of the precursor cluster beams. Due to their measured mass and regular displacement in the spectra we identified the peaks as sodium cluster anions. Combined, the measured spectra covered the cluster size range from Na 7 to Na 92 , and from Na 132 to Na 144 . For better visual representation and easier comparison to the mass spectrum of the neutral clusters, we combined the anion spectra in to one graph in figure 2.6. Note that the resulting graph is not strictly accurate because different parts of the mass spectra, shown in different colors, were acquired under slightly different experimental conditions (see the captions of Figures 2.5, 2.7-2.13). The mass spectra of sodium cluster anions demonstrate significant restructuring with respect to the parent beam of neutral clusters: (I) The spectra exhibit enhanced intensities of the cluster anions of sizes from approximately from 20 to 60 atoms and the rapid fall-off of the peak intensities of smaller clusters. (II) Abundance magic numbers related to electron shell closings (8, 20, 40, 58, 92, and 138) are clearly shifted by one towards smaller sizes (7, 19, 57, 91, and 137), which corresponds to electron shell closings in anions. (III) The signal of Na 8 - is too low to be observed and the peak of Na 20 - very small. 26 (IV) Between magic numbers the relative intensities of the anion peaks are in an inverse correlation with the intensities of the peaks of neutrals, which is not a simple pattern shift by one electron number. While the increased intensities of the anion peaks of the middle range and the fall-off of the intensities at lower masses could be explained by enhanced deflection of the lighter clusters in the magnetic field of the electron gun and discrimination of the detector against heavier ions, understanding of the magic number shift as well as restructuring of the spectra between them required careful treatment. In the next chapter we present theoretical calculations where the observed anion spectra were analyzed based on evaporative attachment model. Figure 2.5 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: , 665 reservoir TC =° 761 nozzle TC = ° , 400 Ar PkPa = . 27 Figure 2.6 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: , 675 reservoir TC =° 767 nozzle TC = ° , 315 Ar PkPa = . Figure 2.7 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: , 665 reservoir TC =° 761 nozzle TC = ° , 300 Ar PkPa = . Figure 2.8 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: , 662 reservoir TC =° 781 nozzle TC = ° , 400 Ar PkPa = . 28 Figure 2.9 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: , 664 reservoir TC =° 788 nozzle TC = ° , 400 Ar PkPa = . Figure 2.10 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: , 663 reservoir TC =° 786 nozzle TC = ° , 400 Ar PkPa = . Figure 2.11 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: , 663 reservoir TC =° 786 nozzle TC = ° , 400 Ar PkPa = . 29 Figure 2.12 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: , 661 reservoir TC =° 765 nozzle TC = ° , 570 Ar PkPa = . Figure 2.13 Mass spectra of cluster anions (on the left) and neutral clusters (on the right). The cluster source conditions: , 661 reservoir TC =° 765 nozzle TC = ° , 570 Ar PkPa = . 30 Figure 2.14 Combined mass spectra of sodium cluster anions (at the top) and neutral cluster (at the bottom). 2.10 Related Experimental Data of Other Groups To our knowledge, the only other experimental study of the formation of alkali cluster anions by adding electrons to neutral clusters, in which the mass spectra of the products were reported, was performed by Nagamine et al [NAG’94]. They observed the spectrum of anions produced by the electron transfer from high-Rydberg krypton atoms to potassium clusters. The size distribution of the resulting cluster anions 31 reported in Ref. [NAG’94] is presented in Figure 2.15. This distribution exhibits noticeable similarities to those of sodium cluster anions obtained in the present work (Figures 2.5 -2.8). Figure 2.15. Size distribution of K n - , obtained from mass spectrum of the potassium cluster anions. After [NAG’94]. The authors assumed that the pattern was due to strong cluster-shape related variations in the attachment cross section, and neglected post-transfer evaporation effects. We find, however, that the long-range cross sections are relatively smooth functions of size, and mass spectrum restructuring is in reality due to evaporation cascades (see Chapter 3). 32 Chapter 3. Evaporative Attachment 3.1 Physics of Attachment and Outline of the Calculation The observed mass spectra of the cluster anions indicate that the capture of electrons by cluster polarization potential is followed by electron attachment. That means that a bound state between the cluster and the electron is formed (see Figure 3.1). In principle the electron can spend some time in a metastable state formed by a superposition of the polarization and centrifugal potentials, but the probability that the electron would stay in such a state long enough for the ions to be detected in our experiment (~10 -4 sec) is negligible 1 . The only other possibility is that the electron falls into an available shell state inside the cluster. The consequent question is what the dissipation channels for the energy of the electron are, i.e. where this energy goes. The radiative electron attachment to metal clusters was studied theoretically and it was shown that the radiation channel is weak for the collision energies below the collective electron resonances (the resonance energies of sodium clusters are on the order of several eV [CNE'96]. Other possibilities include the energy transfer to the cluster ions directly or the excitation of single or collective electron states, which will quickly thermalize through the interaction with the cluster core. Since the electron state density in sodium clusters of considered size is significantly lower than the inverse typical temperature ( ( ) 1/ B gE k T < ) [HEE'93], the electron heat capacity is much 1 Another argument against this scenario is good agreement of the Langevin cross section with the experimental data. Indeed, the Langevin cross sections can be obtained using classical mechanics, which prohibits tunneling. That means that only electrons with energies exceeding the height of the centrifugal potential barrier are captured. For these electrons the metastable state does not exist. 33 smaller than that of the ion vibrations and all energy is transferred to the vibrations of cluster ions. All these channels cause the energy of the captured electron to transform into vibrational energy of the cluster, i.e. to heat the cluster. Figure 3.1 Electron attachment energy diagram Heating the cluster affects the decay rate and boosts the evaporative cooling, causing rapid fragmentation of the cluster. Rough estimates show that the energy of 1.2 eV (the low limit for electron affinities to sodium clusters) results in the change in clusters temperature for Na 8 , Na 20, and Na 40 of 580 K, 230 K, and 120K respectively, which corresponds to the increase in the evaporation rate by factors of ~10 5 , ~10 3 , and ~10 2 [ 2 ]. Such a large increase in the evaporation rates suggests that the evaporative cooling indeed follows the electron attachment. Combined with the fact that an B k 2 In this estimate we used approximate RRK expressions (explained further in the chapter) for the cluster heat capacity and evaporation rate 3 N CN = ( ) ( ) 2/3 0 exp B rT rN D k T ≈ . The initial temperatures of the clusters were calculated from the cluster lifetime solving the equation i T () 1 il rT t = for . D ≈ 1 eV, r i T 0 ≈ 10 12 sec -1 , t l ≈ 2x10 -3 sec 34 additional electron affects the dissociation energy of the cluster (clusters’ binding energies vary significantly as the number of electrons changes) it leads to the restructuring of the cluster mass spectra. Since the attachment of electrons finally results in fragment evaporations, the described above mechanism is often referred to as “Evaporative Attachment” [WEB’99] or “Electron Capture Dissociation” (ECD) [ZUB’98]. Further in this chapter we present the detailed analysis of the restructuring of the anion mass spectra caused by statistical evaporations of monomers and dimers following the electron attachment. The steps of our calculations can be summarized as follows: (i) The beam of the neutral clusters was analyzed based on the evaporative ensemble approach. The vibrational temperature distributions of the clusters were calculated using clusters’ dissociation energies and lifetimes (time of flight before the attachment). (ii) The temperature distributions of the cluster anions immediately after the attachment were calculated using the energy distributions of the parent clusters, kinetic energies of electrons, and the electron affinities which were deduced from published experimental photoelectron spectra. (iii) The evaporative ensemble approach was applied to cluster anions. Using the initial energy distributions of the anions the fragmentation patterns for clusters were calculated. 35 (iv) The mass spectra of the anions were obtained using the experimentally measured mass spectra of the neutral clusters, Langevin electron attachment cross sections, and the calculated fragmentation patterns of the cluster anions. 3.2 Evaporation Statistics Overview 3.2.1 Evaporation Rates The evaporation rate of monomers and dimers in clusters (statistical frequency of the fragment evaporation) can be described by the formalism originally developed for nuclear reactions by Weisskopf [WEI’37] and later adapted for cluster evaporations [GSP'82, HAN’99 2, KLO’85, KLO’87, LEP’04]. Using the principle of detailed balance the rate of evaporation and the rate of the reverse process (fragment attachment) can be connected using general expression: decay k attachment k producrs decay formation parent kk ρ ρ =⋅ , (3.5) where parent ρ is the density of the parent cluster state and producrs ρ is the density of state of the products of decay. The latter is the product of the state densities of the daughter cluster daughter ρ and evaporated fragment fragment ρ (monomer or dimer) and the density of translational kinetic energy states t ρ after the evaporation: producrs daughter fragment t ρ ρρ ρ = ⋅ ⋅. (3.6) Considering the equilibrium system of a cluster and its fragments in a finite volume and taking into account the energy conservation law (total energy of the system must 36 conserve) an expression for the differential evaporation rate can be obtained (for details see [HAN'99 1 ]): ()() 23 () (,) () tt fragment t fragment t daughter fragment t fragment fragment fragment parent gd kd d ED d E μσ ε ε ε εεε ε π ρε ερε ε ρ =× −− − ×⋅ , (3.7) where fragment ε is the total internal energy of the fragment, t d ε is the translational kinetic energy of both products after evaporation, is fragment’s spin degeneracy factor, g μ is the reduced mass of the system after evaporation: (1/ 1/ ) daughter fragment fragment mm m μ=+ ≈ , is the mass of the daughter cluster and daughter m fragment m is the mass of the fragment, () σ ε is the cross section of the reverse (association) process, E is the energy of the cluster before the evaporation, and D is the activation energy of the channel (fragment evaporation energy). In considered temperature range the electronic excited states of both clusters and fragments are either unavailable or very unlikely due to the high energy required to excite such states. Therefore the densities of electronic states can be accounted as delta functions and after integration result in factor of 1. The angular momentum conservation law also creates limitations for the variety of possible states available to the system (the daughter and the fragment). However, due to the fact that the moment of inertia of the cluster is much larger than fragment’s moment of inertia, the density of cluster’s rotational energy states is much higher than that of the fragment and, consequently, for every state of the fragment there is a state of the daughter which compensates for the change in fragment’s angular momentum. 37 Therefore, the angular momentum conservation results in no limitation for the rotational state of the fragment but for each of these states allows only one rotational state of the daughter cluster. First we consider the statistics of the monomer evaporation process. Monomers do not have rotational and vibrational degrees of freedom so the evaporation rate expression for monomers can be reduced: () 11 1 1 1 23 () ( ) () () m tt N t t N gED kd d E μσ ε ε ρ ε εεε πρ − −− = , (3.8) where is the monomer spin degeneracy, 1 2 g = 1 Na m μ ≈ is the mass of the monomer (sodium atom), 1 () σ ε is the cross section of the monomer attachment, ( ) N E ρ is the state density of cluster of size N, and D (m) is the monomer evaporation energy. It can be rewritten as: () 1 11 1 1 23 exp ln ( ) () () () m N tt t t N ED g kd d E ρε μσ ε ε t εεε πρ − ⎡⎤ −− ⎣⎦ =⋅ , (3.9) Because translational energy ε is much smaller than the energy of the daughter (see explanation in section 3.3) the logarithm can be expanded to the first order of () m t ED ε − t ε : () 1 () () 1 1 () () 1 1 exp ln ( ) ln ( ) exp ln ( ) ln ( ) ()exp m Nt m m N Nt m m N Nt ED dED ED dE dED ED dE ρε ρ ρε ρ ρε − − − − − ⎡⎤ −− ⎣⎦ ⎡⎤ − =−+ ⎢⎥ ⎣⎦ ⎡ ⎤ − =− ⋅ ⎢ ⎥ ⎣ ⎦ . (3.10) Introducing the temperature of the daughter cluster: 38 1 () 1 1 ln ( ) () m N dB B dED dS E Tk k dE dE ρ − − − ⎛⎞ ⎡ ⎤ − ⎛⎞ ⎣ ⎦ ⎜⎟ =⋅ =⋅ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ , (3.11) the rate can be written as () 11 1 1 1 23 () ( ) () exp () m tt N t NB gED kd d EkT μσ ε ε ρ ε d εεε πρ − ⎛⎞ − =⋅ ⋅ ⎜⎟ ⎝⎠ . (3.12) Substituting the geometrical cross section for the formation process (fragment attachment) and assuming a sticking coefficient of unity: 2 22 2 3 111 () NNWS R RR R N σπ π π − =+ ≈ ≈ − , where R N-1 , R 1 , R N , and R W-S are the radii of the daughter, atom, mother, and the Wigner-Seitz radius of one atom respectively, the differential monomer evaporation rate can be integrated: () () 1 2 2 () 11 () 23 1 1 3 0 ( ,() () ED m WS B d m N N gR kT ED kED k d N E μ ρ εε πρ − − − − == ×⋅ ∫ ) . (3.13) The density of vibrational states of the daughter can be approximately expressed through that of the mother in the approximation of a system of harmonic oscillators. The state density of the system of 1 n + oscillators of energy E can be directly found from the state density of the system of oscillators by effectively dividing the system into two parts (n oscillators and one) and doing summation over all possible distributions of the total energy between these parts: n () ( ) () / / 1 1 0 E E nn n j E Ei Ed ω ω ρρω ρ ω + = ⎛⎞ =≈ ⎜ ⎝⎠ ∑ ∫ ⎟ . (3.14) 39 Substituting the state density ( ) E ρ in form () 1 n n E Ec ρ ω − ⎛⎞ = ⎜⎟ ⎝⎠ , where c is a constant, the integral can be evaluated: () () () 1 1 0 1 n E n nn n cEE EEdEc nn ρρ ωω ω − + ⎛⎞ ≈= = ⎜⎟ ⎝⎠ ∫ E. (3.15) Putting the frequency of vibrations ω equal to the phonon frequency in bulk sodium, the Debye temperature can be used ( B k ω = Θ ) resulting in expression: () () 1n T n E E ρρ + ≈ Θ . (3.16) From here the formula for the daughter state density is: () 3 () () 1 () ( m NdN ED T ED ρρ − −≈Θ − ) m , (3.17) which results in expression for evaporation rate: () () 2 2 () 11 () 23 1 3 () 23 1 () , () () ' () m WS B m N dN m N dN gR k ED kED N TE ED N TE μ ρ πρ ρ ω ρ − Θ ⎛⎞ − Θ =× ⎜⎟ ⎝⎠ ⎛⎞ − Θ = ⎜⎟ ⎝⎠ , (3.18) where () 2 2 11 1 3 ' WS B gR k μ ω π − Θ = . We can rewrite the state density ratio as: () ( () () () exp ln ( ) ln ( ) () m m N N N ED ED E E ρ ρ ρ − ) N ρ ⎡ ⎤ =−− ⎣ ⎦ , (3.19) and expand the logarithms to the first order of D/2 at the midpoint : * /2 EE D =− 40 () () () () () () () 2 () () () () 2 () ( ) () () 2 () () () () ln( ( / 2)) exp ln ( / 2) / 2 / 2 ln( ( / 2)) exp ln ( / 2) / 2 / 2 ln( ( / 2)) exp / 2 m N N m mm m N N m mm m N N m mm N ED E dED ED D OD dE dED ED D OD dE dED DOD dE ρ ρ ρ ρ ρ ρ ρ − ⎡⎤ − =− − ⋅ + ⎢⎥ ⎣⎦ ⎡⎤ − ×− + ⋅ + ⎢⎥ ⎣⎦ ⎡⎤ − =⋅ + ⎢⎥ ⎣⎦ × (3.20). Further introducing the effective temperature of evaporation: 1 () * ln ( / 2) m N B dED Tk dE ρ − ⎛⎞ ⎡ ⎤ − ⎣ ⎦ ⎜ = ⎜ ⎝⎠ ⎟ ⎟ , (3.21) we write: () () * 1 () exp () m m N B N ED DkT E ρ ρ − ⎡ ⎤ =− ⎣ ⎦ . (3.22) Finally the atom evaporation rate can be rewritten: () *() 23 () * 11 ,' exp m B d kT D N D kT T ω ⎛⎞ Θ 1 m ⎡ ⎤ =⋅ − ⎜⎟ ⎣ ⎦ ⎝⎠ . (3.23) The exponential factor is much more sensitive to the change of temperature than the ratio ( d T Θ ) so the first factor of the expression is often considered constant ( 11 ' d T ωω =Θ ) and the expression is further reduced: ( ) *() 23 () * 11 ,exp m B kT D N D kT ω 1 m ⎡ ⎤ =⋅ − ⎣ ⎦ . (3.24) This result essentially reproduces the expression for the atom evaporation rates recovered by the RRK method [JAR’94, KSL’28, RIC’27, ROB’72], based on approximation of the cluster ions with a system of harmonic oscillators. The important 41 difference is that the pre-factor 1 ω in RRK approximation is taken equal to the vibrational frequency which differs significantly in value from the pre-factor described. If the heat capacity CdEdT = is considered constant in the energy interval from to E, the effective temperature can be expressed: () m ED − ( ) ( ) * 1 /2 m mother TE T E D C =− () C B k . (3.25) The fact that effective temperature used in the rate expression is lower by than the actual temperature of the mother cluster is sometimes referred as “finite size temperature correction”. Approximating ion vibrations in clusters as a system of harmonic oscillators results in formula for the heat capacity of the cluster . In principle this approximation may be not accurate, especially at higher temperatures where the clusters are rather in the liquid than in the solid state. However, as it was theoretically shown in [FRO'97] that for alkali metals the inaccuracies appear to compensate each other and the resulting heat capacity is close to that of the system of oscillators. * T () /2 m D (3 6) N CN =− In the Figure 3.2 we present the comparison for the atom evaporation rates of sodium clusters of sizes 15, 23, and 36 calculated using the exact expression for evaporation rate in which the pre-factor is a function of temperature and expressions with a constant pre-factors: ω defined above and a pre-factor obtained with the RRK theory . It is clear from the pictures that using a constant pre-factor itself does not alter the evaporation rate significantly and the rates calculated with 13 1 310 RRK s ω − ⋅ ∼ RRK ω differ by the pre-factor ratio: 2 10 RRK ωω ≈ . 42 Figure 3.2 The monomer evaporation rates of sodium clusters calculated using 15 1 1 '410 s ω − =⋅ , , , 15 0.69 DeV = 23 0.74 De = 36 0.90 De V V 150K Θ = , and 8 2.1 10 WS R cm − − ≈⋅ = , To obtain the dimer evaporation rate the densities of the fragment internal states are to be taken into account. This density is the product of the rotational and vibrational energy states densities ( r ρ and v ρ respectively): ( ) ( ) ( ) fragment fragment r r v v ρ ερερ = ε, (3.26) where r ε and v ε are the rotational and vibrational energies of the fragment respectively: fragment r v ε εε = +. (3.27) By substituting the fragment state density into the evaporation rate expression we obtain: 43 () () () 22 2 2 23 () 2 () (, , ) () tt rv t v t t d Nrvtrrvv rv N gd kddd ED dd E μσε ε ε εεε ε ε ε π ρ ε ε ε ρε ρε ε ε ρ − =× −− − − ×⋅ , (3.28) where is the dimer evaporation energy. The daughter state density factor () d D ( () 2 d Nr ED ) vt ρ εεε − −− − − can be expanded to the first orders of r ε and v ε at and by substituting the temperature of the daughter similarly to how it was done in the evaluation of the monomer evaporation rate we get: () d ED − d T () ( ) ( ) ( ) ()()()() () () 22 exp exp dd NrvtrrvvN v v Bd v r Bd ED ED kT kT t ρ εεε ρ ε ρ ε ρ ε ρε ε ρ ε ε −− −− − − = − − ×− − × . (3.29) The expression can be integrated over the vibrational and rotational energies of the fragment: () () ( ) () () ()() ()() () 2 00 () 2 00 exp exp d Nrvtrrvv d Ntt rrBdr v vBd ED d d ED kT d kT d vr v ρ εε ε ρ ε ρ ε ε ε ρερε ρεε ερεε ∞∞ − − ∞∞ − −−− =− − × ×− ⋅ − ∫∫ ∫∫ ε (3.30) The obtained integrals are, in fact, the partition functions of vibrational and rotational states of the fragment: ()() ( () ) () ( 0 0 exp exp rrBdrr vvBdvv kT d Z T kT d Z T ρε ε ε ρε ε ε ∞ ∞ −= −= ∫ ∫ ) d d , (3.31) so using known formulae for the partition functions of a diatomic molecule [HAN'99 1 ]: 44 () () ,2 ,2 1 2 2 v v rd T ZT h T ZT B ν = + = . (3.32) where v ν is the vibrational frequency and B is the rotational constant of the dimer, the integral can be expressed: () () ( ) () () 2 00 () 2 1 22 d Nrvtrrvv d Bd d Nt v ED d d kT T ED Bh vr ρ εεε ρ ε ρ ε ε ε ρε ν ∞∞ − − − −−− ⎛⎞ =+ −− ⎜⎟ ⎝⎠ ∫∫ . (3.33) Since the vibrational frequency of the sodium dimer is close to the phonon frequency in the bulk sodium the Debye temperature can be used: 11 22 dd v TT h ν +≈ +≈ d T Θ Θ , (3.34) and the expression can be further reduced: () () ( ) () () 2 00 22 () 2 2 d Nrvtrrvv d Bd Nt ED d d kT ED B vr ρ εεε ρ ε ρ ε ε ε ρε ∞∞ − − − −−− ≈−− Θ ∫∫ . (3.35) From here the dimer evaporation rate can be evaluated in analogy to that for the monomer case and the final expression for the dimer evaporation rate is: () () 2 3 2 22 () 23 () * 2 3 2 23 ( ) * 2 ,e 2 'exp WS Bdd B d d B d gR k kED N D kT BT NDkT T μ π ω − Θ ⎛⎞ Θ xp ⎡ ⎤ =×− ⎜⎟ ⎣ ⎦ ⎝⎠ ⎛⎞ Θ ⎡⎤ =× − ⎜⎟ ⎣⎦ ⎝⎠ , (3.36) 45 where () 3 2 22 2 3 ' 2 WS B gR k B μ ω π − Θ = and ( ) ( ) * 2 /2 d TE TE D C =− () . Similarly to the monomer case the pre-factor varies slowly with the change of temperature in comparison to the exponential factor and can be considered constant. So the expression takes a simple form: () 23 ( ) * 22 2 ,exp d B kED N D kT ω ⎡ ⎤ =× − ⎣ ⎦ . (3.37) Note that due to the high density of dimer states the pre-factor value for the dimer evaporation 2 ω is significantly larger than that for the monomer evaporations 1 ω : 222 111 2 B d gk gBT ω μ ωμ Θ Θ = ⋅⋅. (3.38) Substituting the values for the sodium dimer: 2 1 g = , 21 2 μ μ ≈ , 1 0.15 B cm − = in the considered temperature range ( ) we obtain: 500 T ∼ K 2 21 10 ωω ≈ . 3.2.2 Evaporation Probability The probabilities to evaporate a fragment in a defined time interval can be easily obtained from k(T). If only one evaporation channel is possible, considering an ensemble of n 0 clusters of size N at temperature T, a differential equation for the number of clusters of the original can be written: N n () NN dn n k T dt =−⋅ , (3.39) It has a solution: ( ) 0 () exp ( ) N nt n kTt =−. (3.40) 46 From this probability of the cluster to stay in the original state (size) after time t can be written as: ( ) () exp ( ) N Pt kTt =−, (3.41) and the probability to evaporate a fragment: ( ) () 1 exp ( ) Nn Pt kTt − =− −. (3.42) Taking into account that the velocity of sodium cluster beams produced by a supersonic source is ~10 3 m/s and the flight path is a few meters long, the characteristic life time of the cluster is on the order of milliseconds. By substituting the time into the expression the probability of decay as a function of cluster’s temperature can be obtained. In Figure 3.3 the calculated evaporation probabilities as functions of temperature for clusters of sizes 15, 23, and 36 are presented. Figure 3.3 Calculated probabilities of clusters to evaporate a fragment within the lifetime of 2 ms. 47 In the case of two decay channels of comparable probabilities (many clusters can evaporate dimers as well as monomers) the evaporation equation takes the form: ( ) () () ( ) () ( ) () () () () md md dn n k T dt n k T dt nk T k T dt =− ⋅ + ⋅ =− ⋅ + ⋅ , (3.43) where indices (m) and (d) denote the rates of monomer and dimer channels, respectively. One can see that the total evaporation rate is simply the sum of the monomer and the dimer rates and the probability of a particular channel is proportional to the channel’s rate: ( ) ( ) () ( ) 0 ( ,) exp () () md PT t k T k T t = −+ ⋅ , (3.44) and ( () () 0 () ( ) (,) (,) 1 (,) (,) ( ,) m m md kTt ) P Tt P Tt kTt k Tt = + − , (3.45) () () () () ( ) (,) (,) 1 (,) (,) ( ,) d d md kTt P Tt PTt kTt k Tt = + − . (3.46) To illustrate the character of the functions, the probabilities of transitions for Na 16 are shown on the Figure 3.4. Note, that due to the difference in pre-factor values, the evaporation rates are of the same order of magnitude when the dimer dissociation energy is higher than that of monomer. 48 Figure 3.4 Calculated monomer and dimer evaporation probabilities for Na 16 during the experimental time of flight of 2 ms. Rates were calculated for the dissociation energies of 0.82 eV and 0.92 eV for the monomer and dimer respectively. 3.3 Temperatures of the Neutral Clusters 3.3.1 Sequence of Evaporations Alkali clusters produced by the supersonic source are formed in the vicinity of the nozzle [HAB'94], as a result of fast cooling of the metal vapor in expanding beam of the carrier gas [MIL’88]. They grow by attaching atoms and dissipating the released binding energy through collisions with the carrier gas atoms. It was proved experimentally [BRE'90, BRE'89, BRO'99 , MAR’91] that the closed-shell magic 49 numbers in the alkali mass spectra are the result of the fragment evaporation process which indicates that the clusters formed in the supersonic source are sufficiently hot to evaporate fragments: dimers and monomers. Exact evaporation patterns depend on the dissociation energies, initial temperatures, and heat capacities of the clusters, but generally because heat capacity grows with the cluster size, while D does not vary much, the bigger the cluster the more evaporations it undergoes when starting from the same initial temperature. Every act of evaporation cools the cluster reducing its vibrational energy by the value of dissociation energy, internal energy of the evaporated fragment, and the kinetic energy of the channel: ( ) daughter mother mother fragment kinetic EEEE DE E =−Δ= −− + . (3.47) For monomer evaporations, the fragment internal energy is zero and the kinetic energy ; for the dimer channel 2 Bd kT ≈ 6 fragment kinetic B daughter EE kT + ≈ . In the considered cluster size interval (calculations are performed for the clusters sized from 9 to 39) the vibrational temperature reduction caused by a single evaporation (3 6) mother daughter B TT EkN −≈Δ − ranges from several tens to several hundreds of Kelvins. As a result, every subsequent evaporation in the sequence has significantly lower rate: () ( ) ( ) (3 6) dm B kT kT E k N k T =−Δ − m , (3.48) so it requires significantly longer time. For our calculations this produces a convenient result: In a sequence of several evaporations the total time of the sequence is essentially equal to the time of the last evaporation act: 50 12 ... nn n nk n tt t t t k − −− − + ++ + ≈. (3.49) That means that the entire lifetime of the cluster in the setup can be applied towards the last act of evaporation. From this we can estimate the limits for the temperature of clusters in the beam. 3.3.2 Single Channel Sequence (Monomer Evaporations Only) Temperature limits Initially we consider only monomer evaporations. That implies that for every cluster state only one decay channel is possible. In this case the evaporation probability function (see Figure 3.5) exhibits a step like shape with a pronounced threshold temperature. Below this temperature the evaporation probability is essentially zero, and unity above it. If probability is calculated for the lifetime of the beam, the threshold temperature defines the maximum temperature of the clusters in the beam. Indeed, if the cluster has higher temperature it will evaporate a fragment and cool down. Below this temperature it can survive long enough to be detected. To find the threshold temperature the probability equation needs to be solved with respect to temperature: l t () ( ) 23 () exp exp /2 1/2 NB Pt N Dk T D C t ω ⎡⎤ =−⋅ − − ⋅ = ⎣⎦ l , (3.50) which results in the expression for cluster’s maximum temperature: () () max 2/3 2/3 2 ln ln 2 2 ln Bl Bl DD T C ktN DD C ktN ω ω =+ − ≈+ . (3.51) 51 Figure 3.5 Step-like shape of the decay probability function. The displayed function was calculated for Na 17 using 17 0.86 D eV = The low limit of temperatures, on the other hand, arises from the fact that every cluster is a product of evaporation process itself so its existence requires its mother to have evaporated a fragment, i.e. requires mother’s temperature to exceed its threshold temperature. Expressing the temperature of the mother through the temperature of the daughter we write: ( ) () min max 36 mother B TT Ek N ≈ −Δ −. (3.52) Form of the temperature distribution (single channel sequence) Now, when the maximum and minimum temperatures are defined, the next question is the temperature distribution within these limits. First we consider the temperature distribution of the initial cluster beam. It is a complicated issue closely linked to the dynamics of the cluster nucleation in the supersonic beams, which has 52 been discussed for a long time [KAP’88] but an unequivocal answer was not found. Taking into account that the amount of energy released by atoms binding into a cluster (~1 eV per atom) is dissipated though a large number of collisions with the carrier gas ( /collision, at 1500K) we assume that initial distribution is sufficiently smooth and can be considered uniform within the range which translates into the temperature span displayed by the clusters flying in the beam after evaporations. 2 0.26 B kT eV ≈ The next problem is how this distribution transforms through the evaporation process. If the ensemble of same-size clusters of a certain initial temperature distribution (uniform in our case) follows a defined evaporation chain, i.e. all the clusters undergo the same evaporations, the energy loss by each cluster at one evaporation can be written as sum: ( ) fragment kinetic ED E E Δ= + + . The first term – the dissociation energy – is approximately the same for all clusters and the other two are of the order of . For sodium clusters of sizes above 10 atoms the sum B kT fragment kinetic EE + is small 3 and the temperature distribution of the clusters can be considered simply shifted at every evaporation of the chain. From this, we conclude that the shape of the initial temperature distribution of an ensemble transforms essentially almost unchanged through any particular evaporation path with no branching points, provided the clusters are given enough time to evaporate, which is 3 In order for the temperature dependent terns not to introduce noticeable change to final cluster’s temperature the total energy transferred to fragment’s internal and kinetic energies must be much lower than the final internal vibrational energy of the cluster: () B initial final final nk T T CT + , where n is the number of evaporations, T initial and T final are the initial an final temperatures of the cluster. Taking into account that , , and we obtain condition: , which holds well if cluster’s size N exceed 10 . ~ 400 final TK ~ 1000 initial TK 3 B CNk nN 53 true for all clusters of temperature above T max . So the final temperature distribution of the clusters can be considered uniform within T min and T max limits (see Figure 3.6 for illustration). Figure 3.6 Evaporation probabilities and uniform temperature distribution of the clusters. Calculated for Na 17 using 17 0.86 D eV = and 18 0.90 D eV = 3.3.3 Two Channel Sequence (Monomer and Dimer Evaporations) Temperature limits The picture is more complicated if the existence of a second channel – dimer evaporations is taken into account. First let us see how the possibility of dimer evaporations affects the temperature limits. Since the probability to find the cluster in the original state has essentially the same form as in the one channel case (see Figure 3.7) we can similarly calculate: 54 () max 2/3 2 ln eff eff Bleff DD T C kt N ω = +, (3.53) where D eff and eff ω are effective dissociation energy and the pre-factor, which are approximately equal to those of the more probable channel. Figure 3.7 Calculated probabilities of cluster decay. Monomer and dimer evaporation channels are considered. Calculated for Na 16 using () 16 0.82 m D eV = and () 16 0.92 d D eV = The lower temperature limit is affected by the presence of dimer channels in two ways: First, the evaporation probability through a particular channel is expressed by a function which incorporates evaporation rates of both channels and depends on cluster’s temperature: 55 Figure 3.8 Calculated probabilities of monomer and dimer evaporation channels with different ratios of dissociation energies. Calculation parameters: 16 CC = , , () 0.8 m DeV = 2171 21 10 4 10 s ωω − =⋅ = ⋅ 56 ( ) ( ) () () () ( ) () ( ) () (,) 1 exp ( ) ( ) () () m mm md kT PTt t k T k T kT k T d ⎡ ⎤ =−−⋅+ ⎣ ⎦ + (3.54) and ( ) ( ) () () ( ) ( ) () ( ) () (,) 1 exp ( ) ( ) () () d dm md kT PTt t k T k T kT k T d ⎡ ⎤ =−−⋅+ ⎣ ⎦ + . (3.55) These functions have non-trivial shapes which depend on the ratio of dissociation energies of the channels (see Figure 3.8). However, even though function forms are different from those in the single channel case (not clearly defined step functions) we still can the lower temperature limit similarly to how it was done in the single case, e.g. via defining the threshold temperature: () () min max () () ( ()2/3 (3 6) 2(3 ln mother B channel channel channel channel B Bl TT Ek N DD E Ck N kt N ω ≈−Δ − Δ =+− ) 6) − , (3.56) where index “channel” means the considered channel of evaporation (monomer or dimer, see further). The second effect of dimer evaporation on the lower temperature limit is that the clusters could be produced from more than one of potential mothers. The lower limit in this case depends on which exact mother it was originated from. That means that the temperature distribution is to be seen as the superposition of two distinct distributions with two different lower limits. To determine the weights of these distributions we consider possible evaporation chains of the sodium clusters (See Figure 3.9). Fortunately, these chains are very simple because, as was observed 57 experimentally [ 4 ], only clusters of even sizes in considered mass range can exhibit substantial probability to evaporate a dimer. That means that only even-sized clusters can have two possible mothers. Figure 3.9 Experimentally measured branching coefficients for sodium cluster cations [BRE'89] (a) and evaporation chains for neutral sodium clusters (b). Further we use the assumption that the initial cluster temperature distributions (hot clusters) are broad and the most of the clusters evaporate many times before they reach the temperature of a long-lived state (for illustration see Figure 3.10). In that case at every evaporation instance only a small portion of the clusters have temperatures below the threshold and do not decay further: . ( ) ( ) N threshold N threshold nT T nT T ≤> 4 We constructed the evaporation chains based on experimental values of the branching coefficients. The coefficients as well as dissociation energies were measured by Bréchignac et al. for sodium cluster cations [BRE'89] C. Bréchignac, P. Cahuzac, J. Leygnier, and J. Weiner, J. Chem. Phys. 90, 1492 (1989).. Later in this chapter we will discuss the correlation between the dissociation energies of neutral and charged clusters. So far it is important to point out that the dissociation energies and the branching coefficients of the cations are approximately equal to those of neutral clusters with the same number of valence electrons. 58 Figure 3.10 Two consecutive evaporations in assumption of broad initial temperature distribution. Consequently, in a sequence of two evaporations: 1 N Na Na N − → , and the numbers of evaporating clusters are approximately the same: 1N Na Na − → 2N− 2 ) 11 1 11 () ( () N N N threshold N threshold N threshold N N nnTT nTT nT T n →− − − −−→− = >+ ≤ ≈> = . (3.57) From here for a cluster which can be created by two possible mothers we obtain: 59 Figure 3.11 Calculated branching coefficients of monomer and dimer evaporation channels with different ratios of dissociation energies. Used parameters: 16 CC = , , . () 0.8 m De = V 71 21 21 10 4 10 s ωω − ≈≈⋅ 60 12 1 22 NN N N NN NN nn nn −→− →− →− →− ≈, (3.58) which is exactly the branching ratio for clusters of size N: () 1 () 2 m NN N d NN N nk nk →− →− =. (3.59) Therefore, the weights of the two superimposed distributions approximated by simply the branching coefficients of its dimer evaporating mother (for example for Na 14 it is branching coefficients of Na 16 ). In our calculations we used the aforementioned experimentally measured branching coefficients of sodium cluster cations [BRE'89] (see Figure 3.9. (a)). Form of the temperature distribution (two channel sequence) Dimer evaporations affect the form of the cluster temperature distributions as well. Because the branching coefficients in actuality depend on the temperature, the temperature distribution is changing as the clusters undergo evaporations. To understand the character of the distribution transformations we used the evaporation rate formulas to calculate branching coefficients as functions of temperature with different ratios between the dissociation energies of two channels () ( ) md DD (see Figure 3.11). From the analysis we conclude that (i) the branching coefficients are smooth monotonic functions of temperature; (ii) if the dissociation energies are equal the dimer evaporation is the more probable channel. (iii) the clusters of higher temperatures are more likely to follow the channel with higher evaporation energy than the colder 61 cluster. So starting with a uniform distribution the channels of higher evaporation energy will produce daughter temperature distributions ( ) FT with positive slope: () 0 dF T dT > and vice versa. In this view we conclude that the resulting temperature distributions are not uniform, but smooth functions which reflect the cumulative effect of the particular sequence of evaporations that has formed the ensemble. In our understanding, obtaining the described temperature distributions is a complicated computational problem which goes beyond the limits of this work. In our further calculations we assumed that similarly to the single channel evaporations the temperature distributions of the clusters stay unchanged as clusters evaporate fragments. Therefore the final temperature distributions of the clusters of a certain size are either square box like (uniform with the boundaries of T min and T max ) if the clusters of this size can be directly produced only by one channel, or can be seen as a superposition of two square box like distributions with the same maximum temperature T max and distinct minimum temperatures. T min1 and T min2 corresponding to the formation channels (see Figure 3.12). The described method was applied to find the temperature distributions of the neutral clusters prior to the collisions with the electrons. The advantage of the method is that it allows one to calculate the temperature distributions of the alkali clusters based only on the dissociation energies of the clusters and the experimental lifetime of 62 the cluster beam. The time of flight was directly calculated based on the length of beam’s path (2 m) and the cluster velocity 5 : /2 l tV L ms = ≈. (3.60) The other input, the dissociation energies, is discussed in the section 3.6 of this chapter. Figure 3.12 Total temperature distributions of the clusters directly produced by both: monomer and dimer evaporation channels. 3.3 Temperatures of the Cluster Anions Provided that the temperature and consequently the internal vibrational energy of the neutral cluster are known it is easy to calculate the temperature of the anion, formed by the electron attachment. As it was mentioned earlier we assume that the 5 The velocity of the cluster beam was measured in previous experiments of our group, using a mechanical chopper wheel and a multi channel analyzer [TIK’01 2 ] G. Y. Tikhonov, Ph. D. Thesis, University of Southern California, 2001. It was found to be approximately 1000 m/s. 63 attached electron ends up in the lowest available state in the cluster and all the excessive energy of the electron is transformed into ion vibrations: () ( neutral anion electron EE E += ) . (3.61) Since the electrons do not contribute to the heat capacity of the clusters in the considered range of temperatures and the heat capacity of the ion core is approximated by that of the system of harmonic oscillators, the heat capacities of the anions and neutral clusters are considered the same: ( ) 36 anion neutral B CC N == −k, (3.62) and the temperature change due to the electron attachment is expressed as: () () ( ) 36 anion neutral electron B E TT Nk =+ − . (3.63) The energy contributed by the electron consists of electron’s kinetic energy and the electron affinity to the neutral cluster. The attachment cross section of the clusters is described well by the Langevin formula, so it does not depend on cluster’s temperature and has the same form for clusters of different sizes: 22 2 () N kinetic N kinetic e E E π σα =⋅. (3.64) This allows us to apply the same cross-section-weighted average kinetic energy of the electrons EK electron , which was found to be 0.1 eV (see Appendix A), to all clusters. The other part of electron’s energy, electron affinity, is the property of the cluster which can either be calculated theoretically or measured experimentally. Published theoretical data is primarily limited to the results obtained with the 64 semiempirical version of shell-correction method [YAN’95]. While this data exhibit good general agreement with the experiments [YAN’95] accuracy is an issue, and taking into account that the fragment evaporation process is very sensitive to cluster’s temperature, the theoretical values may serve not very well for our calculations. Therefore, we have chosen to use published values of the electron binding energies recovered from experimentally measured photoelectron spectra of sodium cluster anions [KOS’07, KOS’05, MOS’03]. Indeed, the anion photo detachment is the reverse process with respect to the electron attachment studied here: NN N N electron Na Na e Na e Na E ω −− −− +→ + +→ + . (3.65) So if the neutral cluster and anion are in the same electronic and vibrational states in these two processes, the energy released by the electron attachment is equal to the difference between the energy of the photon ( ) E ω and kinetic energy of the emitted electron in photoionization. Excited electron states in alkali clusters decay quickly due to interaction with the ions so it is assumed that that the neutral clusters produced by supersonic source are in the ground electron state. In the photoelectron spectra the transition to the ground state is associated with the detachment of the least bound electron and appears in the spectra as the peak of the highest electron energy (see Figure 3.13). So the position of this peak defines the least energy required to detach an electron: the electron detachment potential. () Ee − 65 Figure 3.13 Comparison of photoelectron spectra of Na 20 - acquired at cluster temperatures of 100 K and 300 K After [KOS’05]. Figure 3.14 Frank-Condon effect: “vertical” photoelectron detachment and adiabatic attachment of the electron. 66 Concerning the vibrational states, two aspects require attention. First one is the Frank-Condon principle [CON'26, FRA'26]. Even though the considered clusters are large particles, adding an electron can result in substantial changes of the geometry of the cluster. This effect is clearly pronounced when the initial or the final state is closed shell, because closed shell clusters are of spherical shape while the clusters with non- zero orbital momentum lose spherical symmetry due to Jann-Teller effect [CLE'85 1 ]. The electron photo detachment is a “vertical” process, i.e. after the electron leaves the cluster stays in an excited vibrational state (see Figure 3.14). Contrarily, the electron attachment is an adiabatic process and the amount of the energy transferred to heat is exactly equal to the electron binding energy. Therefore the electron affinities are smaller than the binding energy values obtained in Ref. [KOS’07]. The difference between them should be of the same order of magnitude as the value of the energy level splitting in non-spherical clusters, the upper limit of which can be roughly estimated as follows [CLE85 2 ]: () non spherical F N E E N − Δ ∼. (3.66) Another estimate for this value can be obtained from the peak broadening in the photoelectron spectra, which is on the order of ~0.1 eV (for an example see Figure 3.13). For smaller clusters this difference can be significant but the exact values are cluster size specific and obtaining them is a complex unsolved problem. Consequently, the described effect was not included in our calculations but we admit that this can result in inaccuracies for clusters in the lower side of the considered mass range. 67 Second aspect is that the photoelectron spectra were obtained for clusters produced in aggregation source and cooled with liquid nitrogen down to ~ 100 K, so the temperatures of the clusters in the electron attachment experiment are significantly higher. Although the comparison of the mass spectra of clusters of these two temperatures revealed some differences (see Figure 3.13) the position of the least bound peak was found to differ insignificantly, and we assume that the detachment energies obtained from the “cold” spectra are suitable for our calculations. Therefore we put the electron affinity of the neutral sodium clusters equal to the electron detachment potentials of the cluster anions. The experimental values of the vertical electron detachment energies values are listed in Appendix B. Finally we express the anion temperature as: () () ( ) 36 anion neutral Nelectr B EDP EK TT Nk on + =+ − , (3.67) where EDP is the electron detachment potential. So we conclude that the temperature distribution of the formed anions is the same as that of its precursor (neutral cluster) up to a constant terms defined by cluster’s size. Taking into account that the distribution of the neutral clusters is uniform and limited by T min and T max (or the superposition of two of such distributions) the anion distribution is similarly uniform with the lower and higher temperature limits defined as: () () () ( ) max max () ( ) min min 36 36 anion neutral Nelectr B anion neutral Nelectr B EDP EK TT Nk EDP EK TT Nk on on + =+ − + =+ − . (3.68) 68 3.4 Fragmentation Patterns of Cluster Anion To calculate the fragmentation patterns of the anions we used the same formalism as for the neutral cluster evaporations. The probabilities to evaporate the monomer and the dimer and the probability not to evaporate a fragment for an anion of temperature T within time t are: ( ) ( ) () () () ( ) () ( ) () (,) 1 exp ( ) ( ) () () m mm anion anion anion md anion anion kT PTt t k T k T kT kT d ⎡ ⎤ =−−⋅+ ⎣ ⎦ + , (3.69) ( ) ( ) () () ( ) () () ( ) () (,) 1 exp ( ) ( ) () () d dm anion anion anion md anion anion kT PTt t k T k T kT kT d ⎡ ⎤ =−−⋅+ ⎣ ⎦ + , (3.70) and ( ) (0) ( ) ( ) ( ,) exp () () mm anion anion P Tt tkT kT ⎡ ⎤ =−⋅ + ⎣ ⎦ . (3.71) Note that the evaporation rates of anions are different from those of neutrals because the dissociation energies of the anion and neutral clusters are not the same (see explanation in section 3.6 of this chapter). The time of flight for the anion clusters is defined by the time spent in the quadrupole filter. Since the average potential in the quadrupole region was approximately by 0.8 UeV Δ ≈ higher than in the scattering region the anions we decelerated in the filter and the time of flight can be defined as: ( ) 2/ QMA beam N tL V U m =−⋅Δ, (3.72) where is the length of the filter, is the velocity of the neutral cluster beam, and is the mass of the cluster of size N. Calculated values of t ranged from 10 30 QMA L =cm beam V N m -4 to 3.5x10 -4 s. 69 For an ensemble of clusters of the same size, and a temperature distribution F(T) the total number of clusters which followed a certain scenario (evaporated monomer, evaporated dimer, or stayed unchanged) can be defined by integration over the temperature: () ( ) () 1 0 (,) , m N nTt P TtFTdT ∞ − = ∫ , (3.73) () ( ) () 2 0 (,) , d N nTt P TtFTdT ∞ − = ∫ , (3.74) and () ( ) (0) 0 (,) , N nTt P Tt FT dT ∞ = ∫ . (3.75) If distribution F(T) between and is uniform: ( min anion T ) ) ( max anion T min max min max , () 0, , fT TT FT TT T T ≤≤ ⎧ = ⎨ < < ⎩ , (3.76) where ( ) () () 0max min anion anion fn T T =− is a normalization factor and is the total number of clusters in the ensemble, the integrals can be reduced: 0 n () () max () min () 1 (,) , anion anion T m N T nTt f P TtdT − =⋅ ∫ , (3.77) () () max () min () 2 (,) , anion anion T d N T nTt f P TtdT − =⋅ ∫ , (3.78) and 70 () () max () min (0) (,) , anion anion T N T nTt f P Tt dT =⋅ ∫ . (3.79) Another part of the problem is defining the resulting temperature distributions of the products of the decay. Of course the exact values can be computed through similar integration, however that would complicate the following steps of the calculations and the accuracy of this approach exceeds the accuracy of the entire formalism. So instead, we again employ the threshold temperature of the evaporation: () 2/3 2 ln eff eff threshold Bleff DD T C kt N ω = +, (3.80) where D eff and eff ω are dissociation energy and the pre-factor, respectively, of the more probable channel, and assume that the anions with the temperature above it decay, and the others do not. This concept is illustrated in Figure 3.15. Three distinct scenarios are considered: First, the entire range of the anion’s temperature is above the threshold ((a) in Figure 3.15). In that case all the anions decay and the resulting temperature distribution is just shifted. () () () ( ) min min 36 channel daughter mother B E TT kN Δ =− − , (3.81) and () () () ( ) max max 36 channel daughter mother B E TT kN Δ =− − . (3.82) 71 Figure 3.15 Transformations of the temperature distribution during fragment evaporation. 72 Second, the entire temperature range is below the threshold and evaporations do not occur ((b) in Figure 3.15). In the third scenario the threshold temperature lies within the temperature range of the anions and only the clusters with the temperatures above the threshold decay ((c) in Figure 3.15). Their temperatures are shifted but T min is defined by the threshold temperature: () () () min 36 channel daughter threshold B E TT kN Δ =− − , (3.83) and ( ) () () ( ) max max 36 channel daughter mother B E TT kN Δ =− − . (3.84) As before, () () 2 mm NN B E Dk Δ= + T T for the monomer evaporation channel and for the dimer channel. () () 6 dd NN B ED k Δ= + Once the channel probabilities and resulting temperature distributions are defined, one particular instance of anion decay is fully described. The formed daughter anions however can undergo the evaporation process again. Therefore in order to construct the full fragmentation pattern of the anion we traced all the channels until every particular evaporation product had a negligible probability of decay (the entire temperature range is below the threshold temperature). Figure 3.16 illustrates the idea of the evaporation “tree” constructed of all possible combinations of the monomer and dimer evaporation. The abundances of the resulted “stable” anions (circled in the figure) form the fragmentation pattern. The calculation showed that no anion underwent more than three consecutive evaporations. 73 Figure 3.16 Anion fragmentation “tree” constructed of all possible combinations of the monomer and dimer evaporation. The last aspect is the treatment of the anion ensemble with non-uniform distributions (anions formed from neutral clusters with two mothers). Since its distribution is the superposition of two uniform distributions, the ensemble can be treated as two independent ensembles with uniform distributions resulting in two separate fragmentation pattern. The fragmentation patterns are to be combined with the weights, corresponding to the weights of the temperature distributions in the original ensemble. 74 3.5 Anion Mass Spectra To construct the resulting anion mass spectra the fragmentation patterns were convoluted with the abundances of the anion clusters prior to the onset of fragmentation. The latter can be obtained up to a constant factor from the mass spectra of the neutral cluster beam, multiplying each cluster mass’ intensity by the attachment cross section value for the cluster. As we mentioned before, experimental studies confirm [KAS’99] that the attachment cross section values are in good agreement with Langevin capture cross sections, which are proportional to the square root of the cluster’s polarizability: 1/ 2 22 2 NN e E π σα ⎛⎞ = ⎜⎟ ⎝⎠ . (3.85) Expressing the polarizability through cluster’s radius ()() 3 3 1/3 NN WS RNR α δδ − ≈+ ≈ + and neglecting the electron spill-out δ , we see that the polarizability is proportional to the cluster size , so the electron attachment cross section is proportional to the square root of the size of the cluster: 3 NW NR α − ≈⋅ S N N σ ∝ . Therefore we write: () ( anion neutral NN IcNI =⋅ ⋅ ) ) ) , (3.86) where and are intensities of the anion and neutral clusters, and c is a constant, same for all clusters. Since we are interested the relative abundances of anions, factor c can be simply omitted: (anion N I (neutral N I () ( anion neutral NN INI =⋅ ) . (3.87) 75 The intensities of the neutral clusters were measured experimentally, simultaneously with anion mass spectrum acquisition. Taking into account that neutral clusters’ intensities do not change between the scattering region of the electron (~1.5 m from the cluster source) gun and the detector ionization region (~2 m from the cluster source) 6 we used the intensities obtained from the experimental mass spectra. The possibility of fragmentation of the clusters as a result of the ionization with the UV light was considered (see Appendix C), however resulting corrections did not introduce a noticeable change to the calculation results. Another issue to be taken into account is a non-uniform probability of an anion to escape the scattering region, which as was mentioned in section 2.1 is roughly proportional to the square root of anion’s mass. We included this effect in our calculations by multiplying the final anion intensities by the square root of the anions size. 3.6 Dissociation Energies An important input to our calculations is the dissociation energies of the neutral sodium clusters and cluster anions. The evaporation rates are exponentially sensitive to the dissociation energies, so obtaining accurate values is a significant part of this work. Similarly to the electron detachment energies, the published theoretical results of dissociation energy values arise from the application of the semiempirical version of 6 The exponential character of dependence of the evaporation probability on time allows us to state with certainty that the evaporations happening between the scattering region and the detector have negligible effect on the cluster abundances. 76 the shell-correction method [YAN’95] and lack desired accuracy. On the other hand the experimentally measured values for the dissociation energies of sodium clusters are available only for cations. In view of this we employed a hybrid approach: We used the shell correction method and the liquid drop model to calculate the dissociation energies of the neutral clusters and anions from published experimentally measured energies of cations. The dissociation energies of the cations were found by Bréchignac et al [BRE'89]. In their experiment sodium clusters were ionized by a laser beam of 3.8 eV photon energy and mass selected using a time of flight mass spectrometer. The mass pre-selected beam of cations was sent to a drift region for a time period of several milliseconds, where the clusters underwent fragment evaporations. The intensities of the formed products of evaporations (cations of reduced size) were measured with another mass spectrometer. Appling techniques similar to those described earlier in this chapter, the monomer and dimer evaporation rates were calculated obtaining the probabilities of cation fragmentation. From those the dissociation energies were extracted using a modification of RRK method, which incorporates the principle of detailed balance, namely applying the following expression for both monomer and dimer evaporation rates [ENG'87]: () 38 31 (, ) 3 78 N ED kE D N g E E νπμσ − − − ⎛⎞ =− ⎜⎟ ⎝⎠ , (3.88) where ν is the vibrational frequency of ions 12 1 3.1 10 s ν − ≈⋅ , is the cluster size, N μ is the reduced mass, σ is the geometrical cross section, and E is the vibrational 77 internal energy of the mother. It is easy to see that this expression is approximately equal to the expression in Eq. (3.14) (obtained in Section 3.2.1) in the approximation of a system of harmonic oscillators: () () () () 38 31 33 2 38 23 2 23 2/3 1 (, ) 3 78 37 1 8 2 () () () () N N B B B N B B N N N N ED kE D N g E E NkT kE g kT E E gk ED TE gk ED TE ED N E νπμσ πμ σ π μσ π μσ ρ πρ ρ ω ρ − − 8 D − − − ⎛⎞ =− ⎜⎟ ⎝⎠ − Θ− ⎛⎞ ⎛⎞ =⋅ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ Θ Θ− ⎛⎞⎛ ⎞ = ⎜⎟⎜ ⎟ ⎝⎠⎝ ⎠ Θ − Θ ⎛⎞ ≈⋅ ⎜⎟ ⎝⎠ − =⋅ ⋅ . (3.89) As was discussed previously, this formula gives accurate results for the monomer evaporation rate but a careful treatment of the dimer evaporation process requires taking into account the densities of rotational and vibrational states of the dimer. As a result, the accurate value of the pre-factor for dimer evaporations differs from that used in Ref. [BRE'89] by a factor of ~10 2 . Consequently the obtained dimer evaporation energies might exhibit a systematic error which would affect the results of out calculations. Recalculating the dissociation energies from raw experimental data published in [BRE'89] based on correct values of the pre-factor cannot be a part of this work, so in order account for this inaccuracy we performed our calculations in two different ways: First, we used the original dimer dissociation energies from [BRE'89] and in order for the calculation to be self-consistent we employed the same pre-factor for both monomer and dimer evaporations: 78 16 1 12 410 s ωω − == ⋅. (3.90) Second, we used the accurate pre-factor value for the dimers: 21 21 10 4 10 s ωω 81 − ≈=⋅, (3.91) and corrected the dimer dissociation energies in such a way that evaporation rates of dimers in the experiment described in Ref [BRE'89] stay unchanged (so the correction will not affect the evaporation probabilities and the branching ratios calculated in [BRE'89]): () ( ) ()( ) ( ) 22 2 1 , dimer corrected dimer N kD k D ωω =, N . (3.92) From this condition we obtain the correction: () ()( ) () 21** ()( ) ( ) * 2 1 exp exp ln dimer corrected dimer NN B dimer corrected dimer NN B DD kT kT DD kT ωω ω ω ⎛⎞ ⎛ −=− ⎜⎟ ⎜ ⎝⎠ ⎝ ⎛⎞ =+ ⎜⎟ ⎝⎠ B ⎞ ⎟ ⎠ . (3.93) The approximate values of the temperature T * to be used here can be obtained from the flight time of the mother cluster prior to entering the drift region : 6 210 l ts − ≈⋅ () () * 2/3 1 ln dimer Bl D T kt N ω ≈ . (3.94) The experimental dissociation energies of cations are available for sodium cluster anions sized from 3 to 37 (see Appendix D for the values), but there is no information regarding Na 10 + and Na 22 + . The reason is that the cations of these sizes are 79 missing from the mass spectrum [BRE'89] [ 7 ]. So in order to complete the sequence we had to use aforementioned theoretical values. As shown in the Figure 3.17 the dissociation energies calculated using the semiempirical shell-correction method soundly reproduce the data up to a factor 1.2. Therefore the values of cluster cations of sizes 10 and 22 were constructed by dividing the computed energies by 1.2: and . () 10 0.45 cation DeV = () 22 0.5 cation DeV = Figure 3.17 Reconstruction of the dissociation energies of Na 10 + and Na 22 + from the theoretical values calculated using semiempirical shell-correction method. Solid circles represent the theoretically calculated values [YAN’95] and open square represent the dissociation energies obtained from experimental data [BRE'89]. After [YAN’95]. Next we define the relation between the dissociation energies of the cations and those of the neutral clusters. According to the shell-correction method [YAN’93, YAN’95], the total binding energy of an alkali cluster or cluster ion of size N (number of ions) with N’ shell electrons can be expressed as a sum of two functions: ( ) () ( ') binding LDM N SC E NE R E N =+Δ. (3.95) 7 This fact is consistent with the evaporation chains of the clusters, because the evaporations which would result in formation of such clusters are either very weak or inexistent: Na 23 + and Na 11 + evaporates dimers only and Na 24 + and Na 12 + – only monomers. 80 The first term, , is the energy of the cluster obtained using a classical liquid droplet model [BRE'94]. Being defined through the volume and the surface energies of the cluster, is a perfectly smooth function of cluster’s radius R LDM E LDM E N . The second term, , is the quantum correction arising from the shell structure of the electron states in alkali clusters. The shell-correction oscillates significantly as the number of shell electrons in the cluster changes. (') SC EN Δ Within the liquid drop model approximation, the binding energy of a cluster ion can be obtained from the binding energy of the neutral cluster of the same size by solving a simple electrostatic problem of taking an electron from a position near a conducting sphere of radius R to infinity [STA’87]. For the initial position 0 rR δ = + , where R δ using the image charge method it can be written: () 32 2 2 () ( ) 22 2 00 5 48 2 cation neutral LDM LDM Ree EE e R rr R δ −= ≈− − . (3.96) For large particles ( ) this expression should be equal to the work function so divergent term R →∞ 2 4 e δ should be replaced with the value of the work function ϕ . That results in formula: 2 () ( ) 5 8 neutral cation LDM LDM e EE R ϕ = +−. (3.97) According to the shell correction method we express the total binding energy of the neutral cluster of size N as follows: 81 () ( ) ( ) () () () () () 2 () ( ) ' 5 ' 8 neutral neutral neutral binding LDM N SC cation neutral LDM N SC E NE R E N e ER E N R ϕ =+ =+−+ . (3.98) Then assuming that the shell correction does not depend on the total charge of the cluster (the energies of the shell electron states are smooth functions of the number of ions) () ( ) () ( ) ' neutral cation SC SC ' E NE N ≈ , and to leading order: () ( ) () ( ) 1/ 1 2 3 cation cation LDM N LDM N v s ERER aaN − + ≈−+ 3 eV eV , (3.99) where and 1.12 v a = 1.02 s a = are the volume and surface energy term factors, respectively, from the liquid drop model approximation [BRE'94], we can write further: ( ) ( ) () ( ) () () ( ) () ( ) 1/ 1 () 1/3 ' 2 ' 3 2 1 3 cation neutral LDM N SC cation cation LDM N SC v s cation binding v s ERE N ER E NaaN EN aaN − + − + ≈+ −+ ≈+−+ 3 , (3.100) and, finally: () ( ) 2 () ( ) 1 52 1 83 neutral cation binding binding v s N e ENE N aa R ϕ − =++−−+ /3 N . (3.101) Similarly we find the expression for anions: () ( ) 2 () ( ) 1/ 32 1 83 anion neutral binding binding v s N e ENE N aa R ϕ − =+−+−+ 3 N . (3.102) Constructing a series of monomer evaporations: 12 ... ... Nn n N Na Na Na Na Na Na Na Na −− →+ → + + →→ ++ , (3.103) 82 the total binding energy of a cluster can be presented as a sum of dissociation energies: , so the dissociation energy can be expressed as the difference between adjacent total binding energies: 2 () N binding k k EN = = ∑ D () ( 1) N binding binding DE N E N = − −. (3.104) From this the dissociation energy of a neutral cluster can be found from that of a cation: () () () () () () 2 () 1/3 2 1/3 () 1 () ( 1) 52 1 83 52 1 83 neutral neutral neutral N binding binding cation binding v s N cation binding v s N DE NE N e EN aaN R e EN aaN R ϕ ϕ − − − =− ⎛⎞ =++−−+ ⎜⎟ ⎝⎠ ⎛⎞ −+−−+ ⎜⎟ ⎝⎠ − − − . () 2 () 1 4/3 1 2 () 1 1/3 1/3 4/3 2 () 1 4/3 51 1 2 89 51 1 2 89 1 52 24 9 cation s N NN cation s N WS WS cation s N N a e D RR N a e D NR N NR a e D NR N + − + − − + ⎛⎞ =+ − − ⎜⎟ ⎝⎠ ⎛⎞ ⎜ =+ − − ⎜ ⋅ −⋅ ⎝⎠ ≈− − ⎟ ⎟ . (3.105) Similarly obtained formula for anions is: 2 () ( ) 1 4/3 2 89 anion neutral s NN N a e DD NR N + ≈+ −. (3.106) Thereby we obtain a predictable result: the dissociation energies of the clusters with the same number of electrons are equal up to the liquid drop surface energy term plus the electrostatic correction associated with the relocation of the electron charge 83 caused by fragment evaporation. For dimer evaporations this correction differ by factor of 2: 2 ()( ) ( )( ) 1 4/3 54 12 9 neutral dimer cation dimer s NN N a e DD NR N + =− − , (3.107) 2 ( )() ( )() 1 4/3 2 4 49 anion dimer neutral dimer s NN N a e DD NR N + ≈+− . (3.108) 3.7 Calculation Results Because the experimentally measured dissociation energies of cations are available only in a limited interval of cluster sizes, our calculations were constrained to the cluster anion range from Na 3 - to Na 33 - . There was no single experimental mass spectrum of anions which would cover the entire range, so we performed analysis in two intervals: from size 9 to 19 and from size 19 to 33 (corresponding mass spectra are shown in Figure 2.5 and 2.8). A comparison of the calculated anion mass spectra obtained using original dissociation energies from Ref. [BRE'89] and the corresponding pre-factors in evaporation rate expressions (same for monomer and dimer evaporations rates) with the experimental data is presented in Figure 3.18. As it is seen from the comparison the calculation results exhibit good agreement with the experimental data. The mass spectra computed using the accurate value of the pre-factor in the dimer evaporation rate expression and accordingly corrected dimer dissociation energies (see Section 3.6) are shown in Figure 3.19. Even though the spectra are generally very similar to those presented in Figure 3.18 they exhibit slightly better 84 agreement with the experimental data, especially for larger clusters (from 20 to 32 atoms), where the dimer evaporation channel has stronger influence on the mass spectra shape. Note that both calculations did not employ adjustable parameters except the normalization factors common for all cluster masses in the spectra. Figure 3.18 Anion mass spectra calculated using the original cation dimer dissociation energies from [BRE'89] and corresponding pre-factors in evaporation rate expressions, compared to the experimental data. Figure 3.19 Anion mass spectra calculated using accurate pre-factor values and corrected dimer dissociation energies compared to the experimental data. The corrected values of the dimer evaporation energies for sodium cluster cations are presented in Table 3.1. All other data obtained in the course of the calculations such as the dissociation energies of the neutral clusters and anions, the 85 initial cluster temperatures, the temperatures of cluster anions prior to evaporation, and fragmentation patterns of the anions are presented in Appendix E. Based on the demonstrated agreement between the calculation results and the experimentally measured data we conclude that the presented evaporative attachment model accurately describes the generation of negative cluster ions by low-energy electron attachment. We also conclude that the obtained corrections for the dimer evaporation energies are valid. Table 3.1 Corrected values of the sodium cluster cation dissociation energies for dimer evaporation. Cluster (dimer N D ) ) , corrected (dimer N D , original from [BRE'89] Na 7 + 1.19 eV 0.95 eV Na 9 + 1.72 eV 1.31 eV Na 11 + 0.83 eV 0.60 eV Na 13 + 1.09 eV 0.84 eV Na 15 + 1.03 eV 0.81 eV Na 17 + 1.02 eV 0.80 eV Na 19 + 1.34 eV 1.10 eV Na 21 + 1.29 eV 1.05 eV Na 23 + 0.82 eV 0.65 eV Na 25 + 1.08 eV 0.87 eV Na 27 + 1.12 eV 0.91 eV Na 29 + 1.14 eV 0.92 eV Na 30 + 1.20 eV 0.98 eV Na 31 + 1.18 eV 0.96 eV Na 33 + 1.19 eV 0.96 eV Na 35 + 1.29 eV 1.05 eV 86 Chapter 4. Future Work 4.1 Large Cluster Limit It was shown in chapter 3 that statistical fragment evaporation is an essential part of the electron attachment process for clusters in the size range from 9 to 33 atoms. We expect that this is true as well for clusters of other sizes measured in our experiments. However for significantly larger particles the picture can be different. The evaporation process in anions can be triggered by two circumstances: the cluster heating by electron’s energy or the decrease in anion’s dissociation energies with respect to the neutral cluster. Both of them vanish as the cluster size tends to infinity. Indeed, as the heat capacity of the cluster grows the electron binding energy can be absorbed without substantial temperature increase; and oscillations in dissociation energies arising from the shell electron structure should disappear at larger cluster sizes. Below we estimate the size at which the evaporations following the electron attachment become insignificant. Characteristic time of fragment evaporation can be found as: () () 2/3 11 ,e , B D tDT kDT N kT ω ⎛⎞ == ⎜⎟ ⎝⎠ xp . (4.1) In order for the evaporations to be unsubstantial the evaporation time change due to the electron attachment should be less than the evaporation time itself: tt Δ< . We estimate time change due to heating by means of differentiation of this expression with respect to the temperature: 87 2/3 2/3 2 2 1 exp exp B BB dt tT dT d T dT N k T TD D Nk T kT ω ω Δ≈Δ ⋅ ⎛⎞ ⎛⎞ =Δ ⋅ ⎜ ⎜⎟ ⎜ ⎝⎠ ⎝⎠ ⎛⎞ Δ =− ⎜⎟ ⎝⎠ D ⎟ ⎟ . (4.2) And can be expressed in the limit of large clusters as: T Δ 3 electron B E EA T CN Δ= ≈ k . (4.3) So we can rewrite the above condition as: 5/3 2 2 2/3 1 exp exp BB EA D D D Nk T kT N kT ωω ⎛⎞ ⎛⎞ ⋅ < ⎜⎟ ⎜ ⎝⎠ ⎝⎠B ⎟ . (4.4) and obtain 22 B EA D N kT ⋅ >. (4.5) Further, expressing the temperature from the time of flight of the neutral precursor beam we find: l t ( ) 2/3 ln l tN D NE ω < A eV . (4.6) Substituting the bulk limit for the dissociation energy ( 0.94 bulk D ≈ ) and expressing electron affinity as work function minus the finite size correction [WON’02]: 2 1/3 5 8 WS e EA N R ϕ − − =−, (4.7) we find that this condition is satisfied if cluster’s size exceeds approximately 3000 atoms. 88 As concerns time change due to the shift of dissociation energies, from setting dt tD dD t Δ ≈Δ <, (4.8) we similarly find a condition: anion neutral B DD k −<T. (4.9) The electrostatic part arising from the Liquid Drop Model: 4/3 2 WS NeR − − ⋅ ∼ is clearly negligible for . The shell correction portion 3 10 N > ( ') ( ' 1) DDN DN Δ=− − is estimated based on the distance between electron shells in clusters. One-electron self consistent potentials of large clusters of adjacent sizes are considered essentially equal. In that case the is approximately equal to the distance between the top two occupied electron energy levels, which is bound by the distance between two shell levels. Further assuming that they are roughly equally distributed in the depth of the potential well we write: D Δ (') ( '1) F E DN DN n −−≈, (4.10) where is the Fermi energy, and is the number of the electron shells. Similar estimate can be found in Ref [CLE85 ~5 F EeV 1/3 ~ nN 2 ]. So we write: 1/3 F B E kT N < (4.11) Substituting the approximate value of cluster’s , we obtain . ~ 0.05 B kT eV 6 10 N > From the experimental point of view the formation of the magic numbers is a reliable sign that the condition (') ( '1) B DN DN k T −−< is not fulfilled. Indeed the 89 variations in intensities of cluster peaks in the mass spectra arise from the fact that the evaporation probabilities of the clusters of adjacent sizes are not equal. Therefore we conclude that the evaporative attachment effect on abundance spectrum structure should not be substantial at cluster sizes above 10 6 . For clusters comprised of 10 3 through 10 6 atoms, while the temperature does not increase substantially as a result of the attachment, the effect could be essential due to the change in the binding energy of the clusters upon electron capture. These conclusions can be verified experimentally, using a setup similar to that employed in the present work and a source of large cluster (for example a condensation source used in Ref [WON’02]). This experiment can be proposed for a possible future work. 4.2 Electron Attachment to Cluster Cations Another possible direction of a future work in the area of evaporative attachment is studying the electron attachment to positively charged cluster ions. The outline of the suggested setup is presented in Figure 4.1. Clusters produced by a supersonic source can be ionized by a UV lamp, mass selected by a quadrupole filer, accelerated, and focused in the scattering region of the electron gun. The attachment products (neutral cluster) can be reionized by UV light, mass selected, and collected. 90 Figure 4.1 Outline of the suggested experimental setup. Working with cations gives two significant advantages over the experiments with neutral clusters: First, cations can be easily mass selected prior to the collision, so the fragmentation patterns of clusters of every size can be directly measured in the experiment. This will allow one to look separately into fragment evaporation processes from selected clusters and observe the resulting fragmentation patterns (similar to those presented for anions in Appendix E) directly. Second, the electron capture cross section of a cluster cation is expected to be larger than that of a neutral cluster. Indeed, the Coulomb potential falls off with the distance as 1/ while the potential between an electron and an induced dipole (neutral cluster) falls off as . Consequently, the attractive Coulomb potential is much more efficient at “grabbing” electrons at larger distance and bringing them closer to the cluster, from where the electrons will spiral down to the cluster due to dominating induced dipole term in the potential. As a result, the yield of attachment products is expected to be significantly higher than that in the current experiment. This can allow: (i) to improve accuracy of the measurement r r 4 1/ r 4 1/ r 91 and (ii) to vary the kinetic energy of the captured electrons (this was impossible with attachment to neutral clusters due to very weak yield of produced anions). Combined together these two advantages will allow capturing the details of the evaporative attachment process which has been unreachable so far. 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Am. Chem. Soc. 120, 3265 (1998). 96 Appendices Appendix A. Energy of Electrons Not only does the formation of space charge create a substantial obstacle in generation of low energy and high intensity electron currents it also significantly complicates measurements of the energy distribution of produced electron beams. The space charge of electrons inside the electron gun scattering region makes it impossible to directly measure the distribution using the retarding potential method. In this work we used an approach similar to that described in [KAS’01 2, KRE’84]: we performed the retarding potential measurement and analyzed the obtained curve using general knowledge of the energy distributions of the electrons produced by thermoionic emission. Figure A.1 The scheme of the measurement of the electron distribution using the retarding potential techniques. 97 The scheme of the retarding potential measurement is presented on Figure A.1. The current of electrons collected by the anode and slit anode (I A ) was recorded as a function of the voltage applied to the plates surrounding the scattering region and Grid 3 (V SR ). Figure A.2 A typical current vs. voltage curve obtained by the retarding potential measurement. Grid 2 was pulsed with a square shaped voltage ranging from -3 to + 3 volts at the frequency of 157 Hz. The electron beam was gated in order to separate the electron current from the leakage currents between the electrodes. The latter proved to appear after long usage of the electron gun due to the formation of a conducting layer on the surfaces of the insulators separating the electrodes. While leakage currents did not change the potential landscape inside the electron gun and consequently did not interfere with the scattering process in any way, they did offset the measured values of 98 electron currents significantly. The value of the electron current was recovered up to a constant factor by a lock-in amplifier (Stanford Research Systems, SR510). A typical curve obtained by the retarding potential measurement is presented in Figure A.2. The electron currents produces by dispersion cathodes can be described well using by Richardson’s law of thermoionic emission: 2 exp B WF jT kT ⎛ ∝ ⎜ ⎝⎠ ⎞ ⎟ , (A.1) where j is the current density and WF and T are the work function and temperature of the cathode. Taking the vacuum potential of cathode’s surface for zero, the energy distribution of electrons for Richardson’s currents in potential a ϕ can be expressed as (for illustration see Figure A.3): () () () max max max () exp , 0, B e ce kT D e εϕ ϕ ε ϕϕ ε ε ϕϕ ⎧ ⎛⎞ +− ⋅−> ⎪⎜⎟ = ⎨⎝⎠ ⎪ − ≤ − ⎩ , (A.2) where c is a constant, ε is the electron kinetic energy, T is the temperature of the cathode, and max ϕ is the maximum potential the current passes through. The total electron current passing though the retarding potential R ϕ is then obtained by integrating the distribution over all electron energies: () max 0 0m exp R R B R e I I kT I ϕϕ max ax ϕ ϕ ϕ ϕ ⎧ ⎛⎞ − −> ⎪⎜⎟ = ⎨ ⎝⎠ ⎪ ≤ ⎩ . (A.3) 99 Here max ϕ is the maximum the potential the current passes through other than R ϕ (usually the potential of cathode’s surface: zero), and 0 I is the total current emitted by the cathode. Figure A.3 The diagrams for the kinetic energy distribution of electrons in potential ϕ . 100 In presence of the space charge, the total retarding potential can be expressed as a sum of the potential as if it would be without the electron current, and Coulomb’s potential of the space charge C ϕ Δ . In the present measurement, the retarding voltage was applied to electrodes surrounding the scattering region (scattering region plates and grid 3). Without the formation of space charge the potential within the region would be equal to the potential of the electrodes’ surfaces so the total retarding potential can be expressed as: ( ) RP SR U C ϕϕ ϕ = +Δ, (A.4) where P ϕ is the potential of the surface (vacuum level) of the scattering region plates. Figure A.0.4 The current of electrons passing the scattering region as function of the potential inside the region 101 Taking into account that we put the potential of cathode’s surface equal to zero we can write: P SR W U e ϕ Δ =−, (A.5) where is the difference between the work functions of the material of the scattering region plates and the cathode (further in the text we describe how it’s value was obtained). So the retarding potential can be expressed: W Δ RSR W U e C ϕ ϕ Δ = −+Δ. (A.6) Using the Equation (A.5) the measured current-voltage curve was transformed to the function of the electron current depending on the value of potential P ϕ (see Figure A.4). The obtained function can be effectively divided in three parts corresponding to three different regimes (see Figure A.4 and A5): (i) Saturated regime. In this regime the electron current reaches its maximum value. This indicates that the potential inside the scattering region is below that of the surface of the cathode, consequently it does not create a barrier and all electrons approaching the scattering region pass to the anode. (ii) Limiting regime. Only the electrons of energies exceeding the height of the potential barrier pass to the anode. The distribution of the electrons is “cut” at the lower energy side. Since the “cutting point” correspond to zero kinetic energy the kinetic energy distribution of the electrons within the scattering region has the form an exponential tail: 102 Figure A.5 The electron energy diagrams of the retarding potential measurement in at different values of the potential barrier height. 103 () exp B D kT ε ε ⎛⎞ ∝− ⎜⎟ ⎝⎠ , 0 ε > (A.7) (iii) Low current regime. This regime is essentially similar to the regime II with the distinctive property of very low electron current passing through the retarding barrier. The space charge within the scattering region is a functional of the current and it vanishes as the current value approaches zero. If current is low the potential of the space charge is small and the total potential with the scattering region is close to the potential of the electrodes’ surfaces: R P ϕ ϕ ≈ . Therefore the current in this regime can be well described by expression: ( ) ( ) 0 0 exp exp SR C B SR B WU I II kT WU I kT ϕε ⎛⎞ Δ− +Δ =−⎜⎟ ⎜⎟ ⎝ ⎛⎞ Δ− ≈− ⎜⎟ ⎝⎠ ⎠ V . (A.8) By fitting the experimental current function with this formula in the low current regime and extrapolating the fit up to the point of the intersection with the current saturated value, the difference between the work functions of the material of the scattering region plates and the cathode can be found. W Δ During the anion mass spectra measurements the scattering region voltage value was , which corresponds to the second of the three considered regimes. The difference between the settings during the electron current measurements and during the mass spectra acquisitions was in the voltage applied to Slit Anode (see Figure A.6). While measuring the electron current the anode potential was set to collect the electrons (+10V) and during the spectra acquisitions the anode was kept at 0 V to 0.96 SR U = 104 reflect the electrons back to the scattering region. This resulted in higher value of the space charge potential and further increased the height of the retarding barrier. Another effect of it is that within the scattering region the potential was monotonically increasing along the trajectory of an electron moving away from the cathode. Therefore the exponential energy distribution of the electrons held at every point of the trajectory for all electrons (the distribution of electrons reflected from the anode is exactly the same). Figure A.6 The electron energy diagrams of inside the electron gun during the anion mass spectra acquisition. The average kinetic energy of the electrons captured by clusters can is obtained through by integration: 105 () () () () 0 0 Dd Dd ε σε ε ε ε ε σε ε ∞ ∞ = ∫ ∫ , (A.9) where () 1/ 2 σεε − ∝ is the Langevin capture cross section. Substituting the exponential energy distribution of the electrons we obtain integrals, which can be evaluated analytically: 1/ 2 0 1/ 2 0 exp 2 exp B B B d kT kT d kT ε εε ε ε εε ∞ ∞ − ⎛⎞ − ⎜⎟ ⎝⎠ = ⎛⎞ − ⎜⎟ ⎝⎠ ∫ ∫ =. (A.10) The temperature of the cathode, during the experiments was approximately 1500 K, which results in the average electron kinetic energy of: 0.1eV ε ≈. (A.11) 106 Appendix B. Vertical Electron Detachment Energies The values of the vertical electron detachment energies for sodium cluster anions obtained from the experimental photoelectron detachment spectra are listed in Table B.1. Corresponding photoelectron detachment spectra (taken from Ref [KOS’05, MOS’03]) are presented in Figures B.1, B.2, and B.3. Table B.1 Vertical electron detachment energy values for sodium cluster anions. After Ref. [KOS’07]. Cluster size (atoms) Detachment energy (eV) Cluster size (atoms) Detachment energy (eV) 51.14 22 1.47 61.27 23 1.48 71.28 24 1.47 80.92 25 1.62 91.39 26 1.41 10 1.19 27 1.52 11 1.37 28 1.49 12 1.24 29 1.58 13 1.56 30 1.51 14 1.20 31 1.54 15 1.44 32 1.63 16 1.56 33 1.63 17 1.70 34 1.45 18 1.50 35 1.66 19 1.64 36 1.58 20 1.34 37 1.74 21 1.36 38 1.65 107 Figure B.7 Photoelectron spectra. The experimental spectra are shown in black, and theoretically calculated spectra are shown in red. After [MOS’03]. Figure B.0.8. Photoelectron spectrum of Na 20 - . After [KOS’05]. 108 Figure B.9 Photoelectron spectra. After [KOS’05]. 109 Appendix C. Correction to the Precursor Beam Mass Spectra As we mentioned in Section 2.3, many measurements of the size-dependent properties of alkali clusters have shown that mass spectra produced by filtered near- threshold UV ionization closely reflect the population of the original neutral beam. However in the view of the discussion of evaporation chains presented in Section 3.3 we gave some more careful consideration to this matter. As it follows from the branching ratios and evaporation chains presented in Figure 3.9 in Subsection 3.31 there is no fragment evaporation process that would lead to formation of neutral clusters of sizes 9 and 21. Indeed, clusters with 10 and 22 valence electrons (neutrals of sizes 10 and 22, cations of sizes 11 and 23, and anions of sizes 9 and 21) decay only by evaporating dimers, while clusters with 11 and 23 valence electrons (neutrals of sizes 11 and 23, cations of sizes 12 and 25 and anions of sizes 10 and 22) evaporate monomers only. Therefore the clusters Na 9 and Na 22 could be present in the beam only if they had formed cold and had not evaporated any fragments at all. In view of the discussion presented in Subsection 3.3.1 this scenario seems unlikely, so we assume that the appearance of the peaks corresponding to Na 9 and Na 21 in the mass spectra of clusters ionized by UV light is an indication of the partial fragmentation of clusters during or after ionization. This assumption is further confirmed by the comparison of the previously discussed mass spectra with the spectra of sodium cluster anions and cations (which do not require ionization prior to detection). As it is seen from Figures C.1 and C.2 the peaks of cations of sizes 10 and 22 and anions of sizes 8 and 20 are either very small 110 comparing to those of other clusters or non-existent. On the contrary the abundances the cations Na 9 + and Na 21 + , produced by filtered near-threshold UV ionization (which are expected to reflect the abundances of the neutral clusters of the same size) are quite significant (see Figure C.3, open bars). For example the intensity of Na 21 + exceeds that of Na 22 + . Figure C.10 Experimentally measured mass spectra of sodium cluster cations. (a) Spectrum obtained by our group on the experimental setup described in Chapter 2 by ionizing a beam of neutral sodium clusters with a coaxial UV laser beam of 350 nm wavelength. (b) A cation mass spectrum published in Ref. [BRE’94] (solid bars represent experimental data). 111 Figure C.11 Experimentally measured abundances of sodium cluster anions (corresponding to the spectra on the left side of Figures 2.5, 2.7, and 2.8, on the left). 112 To take into account the effect of partial fragmentation of clusters in our calculations, we introduced a correction to the mass spectra of the neutral clusters. Instead of assuming that the measured mass spectrum of the cations M (cations) (N) produced by near-threshold UV ionization is completely identical to the spectrum of neutral clusters in the beam, we viewed it as a superposition of two spectra: the spectrum of unfragmented cations M (neutral) (N) (identical to the spectrum of neutral clusters in the beam), and the mass spectrum of cations which has restructured completely M (restructured) (N). From this the neutral cluster mass spectrum can be found as follows: () ( ) () ( ) () ( ) neutrals cations restructured M NM N M N =− (C.1) We assumed that the form of the restructured mass spectrum () () restructured M N is close to that of the mass spectrum of cations () ( ) experimental restructured M N measured experimentally by ionizing a beam of neutral sodium clusters with UV laser of 350 nm wavelength (Figure C.1 (a)): () ( ) ( ) () ( ) experimental restructured restructured M NFN M N ≈⋅ , (C.2) where () F N is a smooth function of the cluster size N. We put this function to be linear in the considered cluster size range: ( ) FN N α β = +, (C.3) and found coefficients α and β by demanding the abundances of neutral clusters Na 9 and Na 21 in the resulting spectrum to be equal to zero: 113 () () () ( ) ( ) () () () () () () ( ) () () 999 9 20 20 20 20 0 experimental neutrals cations restructured experimental neutrals cations restructured MM M MM M αβ αβ =−+⋅ 0= = −+ ⋅ = . (C.4) The corrected mass spectra of the neutral clusters (corresponding to the original spectra presented on Figures 2.6 and 2.8) are presented in Figure C.3. While the method we used to define the correction might seem to be rough we found it applicable because the correction is relatively small in comparison to the original abundances for most cluster sizes. Figure C.12 Mass spectra of neutral sodium clusters (corresponding to the spectra on the right side of Figures 2.6 and 2.8). Open bars represent the original measured abundances and solid bars represent the abundances after the correction. 114 Appendix D. Sodium Cluster Cation Dissociation Energies In table D.1 the dissociation energies of sodium cluster cations published by Bréchignac et al in [BRE'89] are presented. Table D.2 Dissociation energies of sodium cluster cations. 115 Appendix E. Calculated Quantities In this appendix the data obtained in the course of the calculations is presented. In Tables E.1 and E.2 the calculated dissociation energies of the neutral and negatively charged sodium clusters obtained as described in Section 3.6 are listed. In Table E.3 we present the calculated maximum and minimum temperatures of the neutral sodium clusters prior to the electron attachment. Separate minimum temperature values are given for clusters produced by different evaporation channels (monomer and dimer evaporation) along with the proportional weights of the abundances of clusters produced by these channels, , (for details see Section 3.3.3). Analogous values for the cluster anions after the electron attachment and before evaporation cooling are given in Table E.4. Anion fragmentation patterns, i.e. abundances of the daughter clusters produced by the sequences of evaporations, are presented in Figure E.1. () monomer N W (dimer N W ) Table E.3 Calculated dissociation energies of neutral sodium clusters. NN 7 0.58 23 0.70 8 1.16 1.41 24 0.78 1.01 9 0.32 25 0.76 10 0.67 0.60 26 0.81 1.06 11 0.58 27 0.76 12 0.81 0.92 28 0.84 1.08 13 0.64 29 0.83 1.14 14 0.72 0.88 30 0.86 1.13 15 0.62 31 0.81 16 0.76 0.90 32 0.89 1.14 17 0.80 33 0.90 18 0.85 1.23 34 0.93 1.24 19 0.80 35 0.83 20 0.89 1.20 36 0.88 21 0.46 37 0.82 22 0.64 0.75 ()( ) , neutral monomer N D eV ()( ) , neutral dimer N DeV ()( ) , neutral monomer N D eV ()( ) , neutral dimer N D eV 116 Table E.4 Calculated dissociation energies of sodium clusters anions. NN 7 1.15 1.39 22 0.70 8 0.31 23 0.78 1.00 9 0.66 0.59 24 0.75 10 0.57 25 0.81 1.05 11 0.81 0.90 26 0.76 12 0.63 27 0.84 1.08 13 0.71 0.87 28 0.83 1.14 14 0.62 29 0.86 1.13 15 0.76 0.89 30 0.81 16 0.80 31 0.88 1.14 17 0.84 1.23 32 0.90 18 0.80 33 0.93 1.24 19 0.88 1.20 34 0.83 20 0.46 35 0.88 21 0.64 0.74 36 0.82 ()( ) , anion monomer N D eV ()( ) , anion dimer N D eV ()( ) , anion monomer N DeV ()( ) , anion dimer N D eV Table E.5 Calculated boundary values for the temperature distributions of neutral sodium clusters prior to electron attachment and the weights of the distributions of clusters produced by the monomer and dimer evaporation channels. N 7 439 71 100% 0% 8 804 36 0% 59 100% 9 207 0 100% 0% 10 332 102 20% 138 80% 11 339 143 100% 0% 12 457 136 60% 132 40% 13 349 162 100% 0% 14 380 148 25% 171 75% 15 322 188 100% 0% 16 384 205 90% 142 10% 17 399 222 100% 0% 18 413 215 100% 189 0% 19 385 242 100% 0% 20 419 127 0% 173 100% 21 214 0% 0% 22 296 201 80% 200 20% 23 321 227 100% 0% 24 355 221 85% 210 15% 25 339 239 100% 0% 26 359 226 83% 228 17% 27 335 253 98% 218 2% 28 368 252 87% 237 13% 29 362 263 100% 0% 30 373 250 70% 251 30% 31 350 273 100% 0% 32 378 278 90% 261 10% 33 380 289 100% 0% 34 392 259 100% 0% 35 348 276 100% 0% 36 368 95% 5% () dimer N W () monomer N W () max, , neutral N TK ()( ) min, , neutral dimer N TK ()( ) min, , neutral monomer N TK 117 Table E.6 Calculated boundary values for the temperature distributions of sodium cluster anions after the electron attachment and the weights of the distributions of clusters produced by the monomer and dimer evaporation channels. N 7 1475 1106 100% 0% 8 1436 667 0% 690 100% 9 1008 783 100% 0% 10 910 705 20% 742 80% 11 954 757 100% 0% 12 945 639 60% 634 40% 13 918 732 100% 0% 14 786 554 25% 577 75% 15 768 634 100% 0% 16 831 652 90% 590 10% 17 852 676 100% 0% 18 790 592 100% 566 0% 19 771 629 100% 0% 20 719 428 0% 474 100% 21 503 0% 0% 22 591 497 80% 496 20% 23 604 511 100% 0% 24 624 490 85% 479 15% 25 621 521 100% 0% 26 596 463 83% 465 17% 27 579 497 98% 462 2% 28 598 482 87% 467 13% 29 596 498 100% 0% 30 590 466 70% 467 30% 31 563 486 100% 0% 32 596 495 90% 478 10% 33 591 499 100% 0% 34 574 441 100% 0% 35 549 477 100% 0% 36 554 95% 5% () dimer N W () monomer N W () max, , anion N TK ()( ) min, , anion monomer N TK ()( ) min, , anion dimer N TK 118 Figure E.13 Fragmentation patterns of sodium cluster anions. N is the size of the mother anion prior to 119 Figure E.14 Fragmentation patterns of sodium cluster anions. N is the size of the mother anion prior to the evaporations. 120

Abstract (if available)

Abstract We have carried out a measurement of the mass spectra of sodium cluster anions formed in the collisions of free neutral sodium clusters with beam of low energy (0.1 eV) electrons. Anions covering the size ranges from Na-7 to Na-92 and from Na-132 to Na-144 were observed. The anion mass spectra were recorded simultaneously with those of the precursor cluster beam, which allowed us to monitor the effect of electron capture on the relative abundances of various cluster sizes. The anion mass spectra demonstrated significant restructuring with respect to the precursor beam: a downshift of the shell-closing magic numbers, a change in the shape of the overall intensity envelope, and, significantly, an alteration in the relative intensities of the open-shell peaks located between the magic numbers. This alteration did not represent a simple pattern shift by one electron number, and required an accurate analysis.

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Creator Rabinovich, Roman Mikhailovich (author)

Core Title Evaporative attachment of slow electrons to free sodium clusters

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 04/15/2008

Defense Date 03/24/2008

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

Tag alkali clusters,electron attachment,evaporative attachment,evaporative cooling,evaporative ensemble,OAI-PMH Harvest

Language English

Advisor Kresin, Vitaly (committee chair), Chang, Tu-nan (committee member), Haas, Stephan (committee member), Thompson, Richard (committee member), Vilesov, Andrey (committee member)

Creator Email rrabinov@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1112

Unique identifier UC189341

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Rights Rabinovich, Roman Mikhailovich

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