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Temperature-dependent photoionization and electron pairing in metal nanoclusters

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Temperature-dependent photoionization and electron pairing in metal nanoclusters

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Content 1 Temperature-dependent photoionization and electron pairing in metal nanoclusters By Avik Halder 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, 2015 2 to my parents and wife 3 Acknowledgements It has been a thrilling experience over the past several years during my graduate study at University of Southern California. I would like to take the opportunity to thank my advisor, colleagues and family members whose encouragement, support and motivation have rooted towards the successful completion of the thesis. I would like to bestow my deepest gratitude to my advisor, Prof. Vitaly Kresin. He is a wonderful mentor and a fascinating scientist. It has been a great experience to work under his guidance as he introduced me to the intriguing field of nanocluster research. He has always been patient at difficult times and taught me how to troubleshoot while venturing through the challenging project. He encouraged me to extend my knowledge to a wide horizon of topics and would always appreciate new ideas and helped to take the right path. My research experience during the last six years would be treasured all my life, and would hope to have him as my friend, philosopher and guide for many more years to come. I am indebted to Dr. Chunrong Yin for his help in building and mastering the nanocluster source and Dr. Anthony Liang for his support towards development of the experiment tools and large scale data analysis. I am thankful to my fellow colleagues Dr. Nicholas Guggemos, Chuanfu Huang, Malak Khojasteh and Daniel Merthe who havehelped in keeping a lively working atmosphere in the lab.I would like to acknowledge my friends Dr. Amit Choubey, Dr. Siddhartha Santra, Dr. Ansuman Adhikary, Dr. Nick Chancellor and Jose Alonso who had always been motivating. 4 I am grateful to Prof. Vladimir Z. Kresin, Yu. N. Ovchinnikov, Andrey Vilesov, Bernd v. Issendorff, Martin Schimdt and Klavs Hansen for having the opportunityto share scientific ideas that have helped in resolving various complicated issues. I would also like to thank Prof. Aiichiro Nakano, Moh El-Naggar and Stephen Cronin for numerous valuable comments and suggestions. I would like to extend my regards to USC Machine shop staffs Donald Wiggins, Mike Cowan and Ramon Delgadillo for their constant help and support in building our apparatus. I am obliged to my parents Gora Chand Halder and Mili Halder who provided me the first taste of science and this achievement would never be possible without them. I am grateful to my grandparents Madan Mohan Banerjee and Late Namita Banerjee whose constant encouragement propelled me towards my goal. Finally, I would like to express my warm appreciation to my wife Sayanti Ghosh who has always been patient to help me get through the tough times and overcome every challenges. 5 Contents Acknowledgement..........................................................................................................................3 Abstract.........................................................................................................................................14 1 History of Clusters ............................................................................................................... 17 1.1 Rise of clusters ........................................................................................................... 17 1.2 Shell structure in metal clusters ................................................................................ 19 1.2.1 Spherically symmetric clusters ........................................................................... 21 1.2.2 Deformed clusters .............................................................................................. 25 2 Nanocluster Superconductivity ............................................................................................ 29 2.1 Bulk superconductor ................................................................................................ 29 2.2 Pairing Mechanism .................................................................................................. 30 2.2.1 Intuitive description ......................................................................................... 30 2.2.2 Mathematical viewpoint ................................................................................... 34 2.3 Low dimensional superconductors ......................................................................... 37 2.3.1 Two dimension ................................................................................................ 37 2.3.2 One dimension ................................................................................................. 38 2.3.3 Zero dimensionalsuperconductor ...................................................................... 41 2.4 Nanocluster superconductor .................................................................................... 44 2.4.1 Electrons in cluster pair akin nucleons .............................................................. 45 2.4.2 Electron pairing in grains to clusters ................................................................. 46 2.4.3 Experimental techniques .................................................................................. 48 3 Experiments .......................................................................................................................... 52 3.1 Overview of cluster nucleation ............................................................................... 52 3.2 Details of clustering process ................................................................................... 52 3.3 Thermalizing tube .................................................................................................... 57 3.3.1 T>90K ............................................................................................................. 58 3.3.2 T<90K ............................................................................................................. 60 6 3.4 Cluster thermalization ............................................................................................. 63 3.4.1 Method1........................................................................................................... 64 3.4.2 Method2 ................................................................................................................... 67 3.5 Optimization of inner design of thermalizing tube ................................................ 69 3.6 Details of the experiment ........................................................................................ 72 3.6.1 Photoionization experiment .............................................................................. 72 3.6.2 Electron impact ionization ................................................................................ 83 4 High T c pairing in Al n nanoclusters ..................................................................................... 88 4.1 Closed shell clusters ................................................................................................ 89 4.2 Incomplete shell clusters ......................................................................................... 90 4.3 Discussion ................................................................................................................ 94 4.4 Conclusion ............................................................................................................. 101 4.5 Possible application of nanocluster based networks ........................................... 102 5 Temperature dependent appearance energy for Al n clusters ........................................... 104 5.1 Appearance energy measurement ......................................................................... 106 5.2 Results and discussions ......................................................................................... 108 5.2.1 Thermal shift of clusters’ appearance energy .................................................. 108 5.2.2 Temperature dependence of bulk work function ............................................. 111 5.2.3 Anomalous shell structure .............................................................................. 112 5.3 Conclusion ............................................................................................................. 115 6 Temperature dependent appearance energy of Cu n clusters ............................................. 117 6.1 Results .................................................................................................................... 118 6.2 Discussions............................................................................................................. 121 6.2.1 Shell structure, density of states and metallicity.............................................. 121 6.2.2 Temperature dependence of appearance energy and density of states .............. 127 6.3 Conclusion ............................................................................................................. 129 7 7 Ionization of cold aluminium clusters: Electron impact vs multiphotoionization ........... 132 7.1 Ionization modes .................................................................................................... 132 7.1.1 Photoionization at low and high fluence .......................................................... 133 7.1.2 Photoabsorption and Photofragmentation ....................................................... 136 7.1.3 Electron impact ionization ............................................................................. 139 7.2 Determination of photoabsorption cross section................................................. 141 7.3 Conclusion ............................................................................................................. 143 8 Energetics of Quantum dots............................................................................................... 144 8.1 Shape of 2D asymmetric quantum dot ................................................................ 147 8.1.1 Internal energy of the quantum dot ................................................................ 151 8.1.2 Physical properties of the quantum dot .......................................................... 155 8.1.3 Conclusion .................................................................................................... 158 8.2 Shape of asymmetric 3D ellipsoidal quantum dot .............................................. 159 8.2.1 Internal energy of the quantum dot ................................................................ 162 8.2.2 Physical properties of the quantum dot .......................................................... 167 8.3 Conclusion ............................................................................................................. 170 REFERENCE ........................................................................................................................ 172 Appendix ............................................................................................................................... 187 Appendix A............................................................................................................................. 187 Appendix B ............................................................................................................................. 189 Appendix C ............................................................................................................................. 240 Appendix D............................................................................................................................. 242 Appendix E ............................................................................................................................. 244 E.1: Calculation of proportionality constant F for n(x,y) ........................................ 244 E.2: Solution for {µ,γ 1 ,γ 2 } ..................................................................................... 244 E.3: Integrals used for deriving E p ......................................................................... 247 E.4: Solution for {µ,γ 1 ,γ 2 } ...................................................................................... 247 E.5: Integrals used for deriving E p ......................................................................... 249 8 List of Figures 1.1 Metal clusters in metal stained glass. .................................................................................. 17 1.2 Semiconducting carbon clusters comprising the soot exhaust from factory. ........................ 17 1.3 Shell structure in Na N abundance spectra ............................................................................ 19 1.4 Different shapes of the metal clusters: closed and open electronic shelltronic shell. ............ 20 1.5 Perfect icosahedral and cuboctaheral cluster structures ....................................................... 20 1.6 Shape of Woods-Saxon potential 21,23 .................................................................................. 22 1.7 Energy level occupation for electrons/nucleons in clusters/nuclei 22 . ..................................... 23 1.8 Schematic of nuclear shell model with spin-orbit coupling 22 ............................................... 24 1.9 Relative binding energy, Δ 2 (N) vs. cluster number N for (a) Li (b) Na (c) K ...................... 25 1.10 Na N mass spectra: experiment and theory ......................................................................... 27 1.11 Nilsson diagram with dimensionless cluster energy levels ................................................. 28 2.1 Intuitive picture for electron pairing - I ............................................................................... 31 2.2 Intuitive picture for electron pairing - II .............................................................................. 32 2.3 Universal temperature dependence of the energy gap parameter ......................................... 36 2.4 Pairing transitions in one dimension ................................................................................... 40 9 2.5 Observation of shell effects in nanoparticles of Sn .............................................................. 44 2.6 Energy levels close to E F with the highly degenerate HOS and LUS ................................... 48 3.1 Aluminum target after sputtering ........................................................................................ 54 3.2 The rise in pressure inside the aggregation chamber and the thermalizing tube. .................. 56 3.3 The rate of pressure rise addition of Ar and He gas in source .............................................. 57 3.4 Schematic of thermalizing tube made out of oxygen-free high purity copper ...................... 59 3.5 Thermalizing tube with temperature gradient ..................................................................... 60 3.6 Schematics of thermalizing tube used for T > 90 K ............................................................. 61 3.7 Brass thermalizing tube ...................................................................................................... 62 3.8 Temperature variation of the thermalizing tube ................................................................... 63 3.9 The change in the mass spectra with temperature for two different tubes ........................... 70 3.10 Thermalizing tube for pressure measurement inside tube and aggregation chamber ........... 71 3.11 Photoionization experimental setup .................................................................................. 74 3.12 Simion simulation for beam spread for TOF plates ........................................................... 75 3.13 Schematic of the laser alignment design ........................................................................... 77 3.14 520nm output from OPO and frequency doubled UV wavelength 260 nm ........................ 79 10 3.15 Typical TOF mass spectra at 90 K or Al n clusters ............................................................. 81 3.16 Multi-gaussian peak fitting to the TOF mass spectra. ........................................................ 83 3.17 Electron impact ionization setup ....................................................................................... 84 3.18 Mass spectra of silver clusters by electron-impact ionization mass spectrometry. ............ 86 4.1 Photoionization of aluminium nanoclusters, for Al n (n=64,…,67) ....................................... 90 4.2 Temperature dependence of the Al 66 spectrum and the density of states. ............................. 91 4.3 Photoionization yield curvesfor copper nanoclusters ........................................................... 92 4.4 Raw data for the photoyield spectrum of incomplete shell clusters Al n (n=37, 44, 68)......... 93 4.5 Evolution of the Al 37 spectral feature with decreasing temperature ..................................... 94 4.6 Derivatives of the near-threshold portion of the photoionization yield for Al 37 .................... 95 4.7 Evolution of the Al 44 spectral feature with decreasing temperature. .................................... 95 4.8 Derivatives of the near-threshold portion of the photoionization yield for Al 44 . ................... 96 4.9 Evolution of the Al 68 spectral feature with decreasing temperature ..................................... 96 4.10 Derivatives of the near-threshold portion of the photoionization yield for Al 68 . ................. 97 5.1 Linearity plot for Al 67 ...................................................................................................... 106 5.2 Time of flight dependence on temperature for Al n ............................................................ 107 11 5.3 Similar plot at a few wavelengths along with multi Gaussian fit for Al n ............................ 107 5.4 Temperature dependence of appearance energy for Al n (n=48-63) .................................... 109 5.5 IP with temperature shift represented by the bar ............................................................... 110 5.6 Thermal shift in appearance energies ................................................................................ 110 5.7 WF extrapolation for AE vs n -1/3 plot at 65K and 230K .................................................... 113 6.1 Cu n cluster ion yield as a function of laser fluence at =216 nm. . .................................... 118 6.2 Cu n Spectra at 216nm and 245nm. .................................................................................... 119 6.3 Yield spectra of Cu clusters marking the sharp drop in IP at 49 and 61 ............................. 121 6.4 DOS from derivative of Cu n Yield plot for size 49-71. ...................................................... 122 6.5 Photoionization yield curvesfor copper nanoclusters ........................................................ 123 6.6 Appearance energies at low and high temperature for Cu n clusters .................................... 124 6.7 The appearance energies of Cu n clusters referred to the Kubo gap 214 ................................. 125 6.8 Kubo gap with HOMO-LUMO gap for Cu n ...................................................................... 126 6.9 IP with n -1/3 based on conducting droplet model for Cu n . α≈0.2, ........................................ 129 7.1 Fragmentation in photoionized Al clusters ......................................................................... 133 7.2 Intensity of ionized Al clusters with photon energy flux density ....................................... 134 12 7.3 Arrows mark the point of inflexion in the Al cluster intensity with energy flux plot .......... 135 7.4 Theoretical Poisson distribution resembling pick up event of photons by clusters ............. 137 7.5 Monomer decay rate for Al 67 (left) and Al 210 (right).......................................................... 138 7.6 Cluster evaporation with laser pulse ................................................................................. 139 7.7 Normalized Al cluster intensity with impact electron energy ............................................ 140 7.8 Mass spectrum of Al cluster with varying energy of impact electrons ............................... 140 7.9 Linearity of Al cluster intensity with energy flux of electrons ........................................... 141 7.10 Theoretical plot of absorption cross-section from Drude theory for Al nanoparticles 270 .... 142 8.1 Electron distribution inside an asymmetric 2D quantum dots. ........................................... 151 8.2 The physical properties of deformedInGaAs 2D quantum dots ......................................... 154 8.3 Internal energy and chemical potential of 2D quantum dots ............................................. 155 8.4 Capacitive energy of 2D quantum dots .............................................................................. 157 8.5 Capacitive energy for 2D quantum dots in InGaAs with N=20. ......................................... 158 8.6 Variation of ratio of semi-minor to semi-major axes, k = c/R with δ for 3D quantum dot .. 162 8.7 The physical properties of GaAs 3D quantum dots with N = 100. ..................................... 164 8.8 Internal energy calculated classically for 3D GaAs quantum dot. ....................................... 165 13 8.9 Numerical vs Classical energy calculation for 3D QDs. .................................................... 166 8.10 Internal energy of the deformed three dimensional quantum dot. .................................... 167 8.11 Classical and spin DFT results 3d QDs ........................................................................... 168 8.12 Capacitive energy plotted for symmetric 3D QDs ............................................................ 169 8.13 Capacitive energy for the deformed 3D quantum dots. ..................................................... 170 14 Abstract A unique property of size-resolved metal nanocluster particles is their “superatom”-like electronic shell structure. The shell levels are highly degenerate, and it has been predicted that this can enable exceptionally strong superconducting-type electron pair correlations in certain clusters composed of just tens to hundreds of atoms. In our experiment we observed a spectroscopic signature of such an effect. A bulge-like feature appears in the photoionization yield curve of a few free cold closed-shell (magic clusters) or near closed shell (clusters with slight Jahn-Teller distortion) aluminum clusters and shows a rapid rise as the temperature approaches ≈100 K. The novel behavior, previously not reported for clusters, implies an increase in the effective density of statesclose to Fermi level and the closing of energy spacing between highest occupied (HOS) and lowest unoccupied shells (LUS). It is consistent with a pairing transition and suggests a high-temperature superconducting state with T c ~  100 K.Unlike bulk superconductors the phase transition has a noticeable broadening due to quantum fluctuations. Our results highlight the promise of metal nanoclusters as high-T c building blocks for materials and networks. Photoionization yield (Y(E)) curves obtained for small aluminum and copper clusters show a quadratic rise around the threshold which is consistent with the theoretical Fowler prediction for the bulk surfaces at low temperature. The data can be fitted to this model to derive the appearance energies (AEs) of the clusters. Sharp drop in AEs have been observed in certain specific sizes that can be related to electronic shell closings. Self-consistentshell model and ellipsoidal Clemenger-Nilsson model have been quite successful in predicting the shell structure for alkali clusters. The clusters under survey are much more complicated as they undergo strong 15 perturbation to the free electron behavior. Careful analysis of these features helps in understanding the electronic and geometric structures of these clusters. A controlled variation of the cluster temperature within the range 60K to 230K has been performed to derive accurate value for the thermal coefficient of appearance energies.The size dependent evolution of AEs is consistent with the electrostatic spherical droplet model and extrapolates to bulk work function (WF). The thermal coefficient is a good match with the theoretically derived value for WF. The AEs measured using this method is free from any contamination which has been the main source of error for the temperature dependent measurements from bulk surfaces. The photoionization yield curves have been used for a long time in determining the AEs for clusters and WF of bulk metals. We have shown for the first time that the derivative of Y(E) curves gives a very good measure for the density of states (DOS) of the clusters. The energy gap, δ, appearing between the highest occupied shell (HOMO) and the lowest unoccupied shell (LUMO) have been referred to the Kubo gap to observe the metallic transition in copper clusters. We also performed a comparative study of the two ionization mechanisms by bombarding the cold aluminum clusters at ≈90K with photons from a nanosecond tunable laser or by electrons from an electron-impact ionizer. By monitoring the ionization yield of the same cluster beam under the action of these two different probes, we identified some interesting differences. Whereas multiphoton ionization produces singly-charged cations accompanied by copious fragmentation, the impact of electrons with the same total energy as n photons is substantially “softer”. Specifically, electrons produce multiply charged cations when their total impact energy exceeds the corresponding ionization threshold, but do not give rise to massive 16 shifts in the abundance spectra that are characteristic of extensive fragmentation. Therefore, it appears that the ionizing electrons do not deposit significant excess internal energy in the cluster. Furthermore, saturation curves corresponding to the transition from single-photon to multiple successive absorption events allowed us to estimate absolute cluster photoabsorption cross sections at these frequencies. They were found to be in good agreement with off-resonance surface plasmon values derived from the aluminum dielectric function. Theoretical analysis has been carried out of another prototype finite Fermi system where the electrons are confined within an anisotropic harmonic potential background. The two- dimensional (2D) electron “puddle” represents quantum dot geometry whereas the three- dimensional (3D) counterpart represents a metal cluster. The systems have the relaxation to attain any general shape with azimuthal symmetry. A classical electrostatic approach lets us derive the electronic distributions, which are in good agreement with the numerical density functional theory (DFT) calculations and the classical limit solution for the Thomas-Fermi artificial atoms.We also calculated the physical observables like the chemical potential, the electron affinity, the ionization potential and the capacitance (more familiar as “addition energy” to the experimentalists) for the electronic distributions. The experimentally measured value of addition energyfor the 2D electronic systems can be reproduced with excellent accuracy by our simple results. 17 Chapter 1 History of Clusters 1.1 Rise of clusters Figure 1.1: Metal clusters in metal stained glass.http://www.barcelonaphotoblog.com Can we seeclusters? Yes!Figure 1.1 is a picture of the stained glass ceiling from Palau de la Música at Barlcelona built in 1908 and guess what provides the beautiful color; fine metal clusters. Let us see another example in Figure 1.2, the exhausts from factory to car or even in the kitchen corner are comprised of semiconductor carbon clusters. Figure 1.2: Semiconducting carbon clusters comprising the soot exhaust from factory. 18 From the most beautiful architecture to the environment polluting gases, clusters are present everywhere and we encounter them in our daily world even without realizing their presence. So what are clusters? Nanoclusters are agglomerates of atoms or molecules ranging from a few to several hundred in size. It plays a unique role in the world of physics, as it helps to understand the evolution from quantal atomistic regimeto classical bulk like features.The history of clusters goes back to Middle Ages with the discovery of techniques to stain glasses with fine metal clusters producing beautiful colors as is shown in Figure 1.1. These have been studied in details for electromagnetic properties of clusters deposited on dielectric matrix [1, 2]. The central challenge with this technique was to control the size of the clusters. With the development of molecular beam techniques [3], gas phase clusters could be produced with precise size control up to a few atoms. The appearance energies measured for the small alkali clusters demonstrated sharp drop with increase in size tending towards the bulk work function [4]. The results were curious but were mostly restricted towards a few atomic systems, and no particular pattern was observed with size evolution. After about a decade, in late 1983 Walter Knight’s group at University of California, Berkeley produced and detected individualmetal clusters up to hundreds of atoms [5, 6]. The photoionization abundance spectra maintained striking order till large sizes with peaks appearing for N=8, 20, 40, 58, and 92 as is shown in Figure 1.3. The electronic structure of the cluster was immediately recognized to display periodicity as of electrons in a spherical potential well [5].Simultaneous, theoretical work by W. Ekardt on metal clusters based on self-consistentab initio jellium treatment with the assumption that the delocalized electrons in the clusters confined by a uniform positive background supported the shell structure discovered in the alkali clusters [7, 8]. After these early discoveries, the field of cluster physics got heavily explored over the next decades and produced wealth of data in pursuit of fundamental 19 understanding and development of new nano-materials like fullerenes [9], carbon nanotubes [10]and graphene [11, 12]. Figure 1.3: Shell structure in Na N abundance spectra discovered by W.D. Knight et. al. [5]. Picture taken from Rabinovitch et. al. [13] 1.2 Shell structure in metal clusters In our present study the shell structure of the clusters play an indispensible role, so we will do an exhaustive review regarding its origin and fundamentals. From the free-electron model the electrons inside a metallic lattice are like free particles with their wavefunctions defined by plane waves with any wavelengths. For a finite system nanocluster due to size quantization the wavefunctions get modified to standing waves with specific wavelength which results in splitting the valence band to discrete density of states [14, 15]. Thus, even though the electronic structure for most of the clusters e.g. alkali and noble metals follow a free electron model, but for complete understanding of the features the atomic structure should also be taken into account.One obvious demonstration of the coexistence has been in the electronic and atomic shell closings. In Figure 1.4(a) we present the structure of the clusters with electronic shell closings where the confining potentials automatically give rise to spherical symmetry and are 20 very stable structures. As electrons get added or removed from the cluster the symmetry is lost and the clusters attain a prolate or oblate shape as presented in Figure 1.4 (b) and (c) respectively. a. b. c. Figure 1.4: Different shapes of the metal clusters: (a) spherical: closed electronic shell. (b) prolate and (c) oblate: open electronic shell. Figure 1.5: Perfect icosahedral and cuboctaheral structures demonstrated in the top and bottom panel respectively for Lennard-Jones clusters in their minimum energy configurations [16] In Figure 1.5 we see the atomic shell closings structures with the top and bottom panel representing the icosahedral and cubocthedral structures. The existence of these highly symmetric structures has been observed in alkali and noble metal clusters [16]. Later we will 21 show from our photoionization measurement data that Cu 55 demonstrates itself as perfect icosahedra. From the threshold ionization data it has been shown for the simplest single electron Na N clusters that the oscillations observed in a mass spectra is a result of electronic shell closings for small sizes and for larger clusters a periodicity in the geometrical shell closings have been observed (see the review by T. P. Martin [17]). We will focus on the electronic shell closings as they play the key role for the observed effects that will be discussed in the following Chapters. 1.2.1 Spherically symmetric clusters The shell structure was previously observed for the electrons in an atom and inside the atomic nuclei for nucleons. Though originating from different physical interactions, both atoms and nuclei demonstrated unusual stability while containing “magic number” electrons and nucleons respectively [18, 19].The nucleons were assumed to move in an effective Woods-Saxon potential(which is an intermediate between the harmonic and square well potentials) [20, 21] and being fermion follows Pauli’s exclusion principle.The form of the nuclear potential is given in eq.(1.1),                   a R r V r V S W 0 0 . . exp 1 (1.1) where R 0 =r 0 A 1/3 defines the outer boundary of the nuclei and {V 0 , a, r 0 } are constants with a typical set of value {50Mev, 0.5fm, 1.2fm}. The potential is represented in Figure 1.6 where we can see that the potential is a rounded edge square well and has a finite but negligible slope at the boundary. Due to the non-trivial nature of the potential, there is no direct closed form for the eigenfunctions that can be derived; the two closest ideal potentials are the harmonic oscillator potential and the square well potential. A comparative measure of the energy levels for the three 22 different potentials is presented in Figure 1.7 [for detailed discussion see books by M. G. Mayer & J. H. D. Jensen [22], and by P. Ring & P. Schuck [23]]. Figure 1.6: Shape of Woods-Saxon potential [21, 23] It is not surprising that the Woods- Saxon potential has the orbital energy ordering intermediate between that of the two idealized potentials. Experimentally, shell closings have been observed for nucleon size (N) 2, 8, 20, 28, 50 and 82 [24, 25]. From Figure 1.7 we can find that the closings for N=2,8 and 20 are correctly predicted, but some other major shell closings are missing. This is a result of the fact that in all these derivations, the spin of the nucleon have been neglected. On introducing the spin-orbit coupling the degeneracy in the energy levels shown in Figure 1.7 is lifted and we can see from Figure 1.8 the missing major shell closings have been recovered. Cluster physics is comprised of systems which are composed of multi-atoms but at the same time small enough so as not to reach the bulk regime. Many of its models are thus developed starting from the atomic, nuclear or solid state sciences. The nuclear shell model developed to understand the structure and stabilities of atomic nuclei are successfully applied to explain the shell structure in metal clusters. Shell structure have been initially discovered in Na cluster [5], were later observed in other monovalent clusters; e.g. alkali K and Li clusters [26–30] 23 and noble metal clusters like Cu, Ag and Au [31]. It is also observed in divalent Cd, Zn, Mg [32, 33, 34] and trivalent Al clusters [35,36]. Figure 1.7: Energy level occupation for electrons/nucleons in clusters/nuclei [22]. Unlike nuclei, the fundamental interactions in clusters can be conceived explicitly: (a) the electron-electron mutual interactions have been extensively studied and (b) the electrons are delocalized and are subject to an effective smooth potential due to ionic background [31].The positive charge density of the background is assumed to be the same as that of bulk and is given by (4/3πr s 3 ) -1 where r s is the Wigner-Seitz radius.The radius of the jellium sphere is given by n 1/3 r s , where n is the number of electrons in the cluster [37]. The total energy of the interacting electron gas can be defined via the local density functional scheme [38, 39], E el =E kin +E xc +E coul +E ext (1.2) 24 where the first four terms are the kinetic, exchange-correlation, Coulomb and external-field energy terms respectively. Under the approximation of a Woods- Saxon type electrostatic potential (see Figure 1.6) due to the ionic background which is flat within the cluster and goes as r -1 for large r. Cohen and co-workers derived the energy of the monovalent alkali clusters [40] and multivalent clusters [41]. Figure 1.8: Schematic of nuclear shell model with spin-orbit coupling [22] The relative stability of the clusters in terms of the total energy of the clusters are given be Δ 2 (N), Δ 2 (N)=E(N+1)+E(N-1)-2E(N) (1.3) 25 where as we can see from Figure 1.9 the peaks in the stability plot well reproduce the experimental result for shell closings [42].The spherical assumption reduces computational costs, and at the same time efficiently determines the major features of the mass spectra. However, there it is incapable to deduce the fine structures for which the knowledge of the shape of the clusters is important. In the following Section we will discuss the spheroidal model developed for clusters which can recover the additional peaks in the mass spectra. Figure 1.9: Relative binding energy, Δ 2 (N) vs. cluster number N for (a) Lithium, (b) Sodium and (c) Potassium [42] 1.2.2 Deformed clusters Inside nuclei the spherically symmetric self consistent confining potential can only be ideally realized close to the magic sizes. For general sizes, there is shape distortion and whose experimental manifestation has been seen in the existence of rotational bands and very large 26 quadrupole moments (See book by Ring& Schuck [23]). A general form of the deformed Woods-Saxon potential is discussed in Ref. [43,44]. An approximation to this model is provided by S. G. Nilsson [45] as a shift in the energy levels for an anisotropic oscillator background. Keith Clemenger [46] furthered this model to the metal clusters in order to have a complete representation of the subshell features. Under spherical symmetry the energy levels are 2l+1 degenerate, which washes minor features of the abundance spectrum. The states of the valence electrons in the clusters was demonstrated by a single-particle Hamiltonian, eq.(1.4) which assumes axial symmetry.     3 2 3 1 2 2 0 2 2 2 2 2 0 2 2 2 2 2 1 2 2                                        z n z l l U z m m p H  (1.4) assuming the constant volume constraint 1 2     z . The l 2 term acts as the perturbation to the oscillator potential, thus splitting the energy levels corresponding to different lassociated with each n. <l 2 > n term is a constant n(n+3)/2 which keeps the average spacing between the different major shells constant. With this we can see from Figure 1.9 that the peaks in the stability diagram, for scaled Δ 2 (N) with cluster size, N accurately reproduces the peaks in the abundance spectra of Na clusters arising due to several subshells. The adjustable parameter, U from the Hamiltonian in eq.(1.4) is taken as 0.04. The energy corresponding to each N used for Figure 1.10 is derived from the energy diagram in Figure 1.11 where the value of δ assigned to each cluster size is such that it achieves minimum ground state energy. The existence of deformed equilibrium states was later confirmed within the self-consistent jellium model calculations by Ekardt and Penzar [31,47]. 27 Figure 1.10: (a) Experimental Na N mass spectra and (b) second difference of the sum of single particle energies as a function of N with U=0.04 reproducing the minor peaks from the experimental data [48] Based on the Clemenger-Nilsson model, the clusters with incomplete shells undergoes Jahn-Teller distortion and deforms away from sphere. With increase in perturbation the clusters first becomes prolate and once the topmost level is more than half filled, it changes to oblate [31,48, 49]. Even though such configurations are convincing as they follow minimum energy configurations, it is difficult to envision such an abrupt geometry change. It has recently been conceived that the shape evolution in the clusters can be a manifestation of quantum oscillations between the quasiresonant prolate and oblate configurations and a function of the degree of shell filling [49]. As an experimental verification, a measurement for the surface plasmon resonance focused on small Li clusters is current in pursuit in Prof. Vitaly V. Kresin’s group. The quasiresonant structures should in principle result in a peak split of the absorption 28 cross-section. The newly discovered phenomenon should also have interesting effect on the expected dipole moment for the clusters and a theoretical calculation for this finding is undertaken as a current ongoing project [50]. Figure 1.11: Nilsson diagram with dimensionless cluster energy levels as a function of distortion parameter δ, for U=0.04 [46] 29 Chapter 2 Nanocluster Superconductivity The physical properties of nanoclusters show noticeable differences relative to their bulk counterparts due to the finite size effect. The common characteristic features like polarizability, appearance energies, density of states measured by photoelectron spectroscopy, optical properties etc [31,51–53] has already been studied in details. Superconductivity is the newest entry in the field of nanoclusters. In the following sections we will provide a theoretical overview of superconductivity in bulk and nanoclusters, followed by the first experimental evidence for the existence of pairing. 2.1 Bulk superconductor Superconductor, one of the major discoveries in the era of modern physics dates back to 1911, when Kamerlingh Onnes and his student Gilles Kolst [54] discovered that on cooling mercury down to ~4K its resistivity vanishes. The metal enters a new phase where the existence of persistent currents could be demonstrated and thus an example of a perfect conductor. The next hallmark was the discovery of Meissner effect in 1933 [55] where it was observed that the magnetic field is expelled from a metal as it enters the superconducting phase. The demonstration of perfect diamagnetism was manifested in different ways inside different superconductors which introduced the classification of Type I and Type II superconductors. The difference in the observation was supported by the phenomenological description of Ginzburg and Landau [56]. It introduces the idea of magnetic vortices inside the superconductors that for Type I superconductors completely exclude the magnetic field below a certain critical strength, exhibiting pure Meissner state. For Type II superconductors [57] the magnetic flux is partially repelled thus giving rise to a mixed state with quantized flux entities called vortices. 30 The phenomenon of superconductivity was developed by various imminent physicists [58]. The widely accepted microscopic theory describing the superconducting state was developed in 1957 by Bardeen, Cooper and Schrieffer [59] based on phonon-mediated pairing of electrons with opposite momentum and spin. In the next Section, I will provide a physical explanation for the origin of this attractive force between electrons causing it to bind together as Cooper pairs. In 1964 W. Little [60] introduced a new idea for high T c superconductivity in organic polymers. Here the pairing is not initiated by phonon exchange but rather by electronic mechanism. It describes the presence of coexistence of a pair of electronic states where the excitation in one group gives rise to pairing in the other group [58]. This lead to the search for high T c superconductors and since then several high T c superconductors have been discovered; most noticeable in high T c oxides (cuprates) [61]. The highest T c has been observed in HgBaCaCuO under high pressure at ~150K [58,62,63]. Superconductivity has also been observed in other systems like MgB 2 compounds, pnictides, intermetallic compounds, fullerenes and other organic compounds. 2.2 Pairing Mechanism 2.2.1 Intuitive description The phenomenon of superconductivity is manifested by the formation of weakly bound electron pairs. This is a very interesting effect as the electrons should in principle repel each other due to Coulomb forces; so where does the attractive force originate? I will provide an intuitive illustration for explaining this effect based on the description provided by Weisskopf [64]. 31 Let us define a system with positive ions forming a lattice with interionic distance d and filled with electron gas at zero temperature. The momentum of the fastest electron at the top of Fermi distribution could be written as p F =ah/d and the Debye frequency of ions assuming they are independent oscillator of mass M is 2 d mM h b D   , with m as the mass of the electrons, h is the Plank’s constant and a,b are constants of the order unity. Thus the electron moving inside the lattice spends a time τ=d/v e within a distance d from an ion and transfers a momentum p=τe 2 /d 2 to the neighboring ion which gets displaced by δ=p/Mω D ~ d/β where β is the Mach number,√(M/m)of electron motion; since the sound velocity obviously is v s ~dω D . v e is the speed of the electrons inside the lattice and may be given as md h a v e  . We must remember that at T=0 (by assumption) the electrons cannot transfer energy to the ion vibrations which would return to original state within a time ~ 1  D  . So in this situation we have a ‘quasiparticle’ where an electron is accompanied be a tail of negative attractive potential, U of diameter d extending over a distance l=v e /ω D =βd as shown in Figure 2.1. Figure 2.1: There is a negative potential behind the electron e moving towards the right, within a cylinder of length l and diameter ~d The potential U in the tube due to a displacement δ changes the ordinary potential ~e 2 /d of an ion at a distance ~d by U ~ -(e 2 /d 2 )δ that may be written as ~ -ћω D . With this development of the ion behavior in response to the electron, let us see how this affects the interaction between two electrons. The effect of the potential tube U can only be felt by the electrons lying close to 32 the Fermi surface and moving in a relatively opposite direction with their closest distance of approach less than d which is of the order of wavelength  of the electrons close to the Fermi sea. Thus the two pairing electrons must be in a relative S state (L=0, where Lis the relative orbital angular momentum quantum number) and have opposite spins in order to satisfy the Pauli principle since the S- state is symmetric. The potential within the tube U as a function of r, the distance from the electron is shown in Figure2.2 by the bold line. Figure2.2: The approximate potential felt by another electron as a function of the distance r within the tube of length l. The dashed line provides more practical picture as the ions has a time delay to respond to the passing electrons and to displace by the distance δ The dashed line curve shows a more realistic situation for the potential since the ions has a finite response time to the electron interaction. The electrons in this potential well of depth U produce a lowest bound state of energy, 2 2 2 2ml U       as 2 2 2ml U    (2.1) Thus based on this model the two electrons on the top of the Fermi distribution would be bound by a binding energy Δ~U, for which a rigorous derivation is shown by Weisskopf [64]. But here my plan is to provide a physical argument towards the formation of Cooper pairs: First, two binding electrons would have a relative kinetic energy of 2ε F and it might appear that since ε F >>U, these electrons cannot form a bound state since they will lie high up in the unbound continuum. However, because of Pauli principle this is the lowest possible achievable state, and 33 thus the above argument may not be applied in this situation. Secondly, the bound electron pair can only be formed due to a linear superposition of free electron wavefunctions in the unoccupied states above the Fermi energy and not those sitting below the Fermi energy. This condition needs to be satisfied while solving the Schrödinger equation for the pair of electrons on the top of the Fermi distribution. Now I have provided an intuitive picture for the situation where the electrons form a bound pair and there energy drops to 2ε F -2Δ from 2ε F where, A F D d e      4 2 10 ~ 2 exp 2               (2.2) where ε A is the atomic energy given by e 2 /d of the order of a few eV. The microscopic theory for the weak coupling approximation was developed by Bardeen, Cooper and Schrieffer as published in their landmark paper [58,59,65] and is widely accepted as the BCS model. In three-dimension it can be shown from quantum mechanical treatment that a bound state between particles can only exist if there exists an energy level which describes attraction. This is only possible if the depth of the potential well exceeds a certain minimum value. The beauty of Cooper’s theorem is that it has demonstrated how the electrons at the Fermi sea can form pairs no matter how weak the attraction. Let us digress a bit to recall what happens for a two dimensional case. It has been shown [66] that for a two dimensional problem a bound state can be formed for arbitrarily weak attraction. Now for electrons in a metal, the electrons on the Fermi surface are essentially performing a two dimensional motion in momentum space, as due to Pauli exclusion principle none of the electrons can make a transition to the states below the Fermi level as they are completely occupied. It can be shown that the form of the energy gap, Δ 34 from eq. (2.2) that can be represented as   1 2   e D  , where λ is the coupling constant identical to the two-dimensional case [58,66]. The characteristic length can also be derived quite easily as the Fermi energy m p E F F 2 2  with the superconducting phase characterized be an energy gap at the Fermi level which can be connected to the momentum spread as   m p p F  . Using this relation in conjugation to the uncertainty relation   p r   we obtain   F v r   . Introducing the correct numerical factor we obtain the coherence length as nm v F 1000 0       which is ≈10 4 lattice spacing. However, as found in many bulk and low dimensional superconducting materials the coupling strength increasesand 1   , for which the BCS formalism fails. But we can qualitatively say that Δ(0)/E F which gives the number of paired electrons is much larger than that for a conventional superconductor. In the following Section we will discuss a mathematical formalism which can be generally applied for phonon mediated electron coupling in both weak and strongly coupled superconductors. 2.2.2 Mathematical viewpoint In the general formulation we will find that all the important properties can be described in terms of a single parameter, the critical temperature T c . The fundamental analytical equation describing pairing was derived by Eliashberg [67,68],       ' ' 2 ~ ; ~ ' , ' n s n n n F D dk T Z T n              (2.3) where, Δ(ω n ,T) is the order parameter, Z is the renormalization constant, T is the temperature, k’ defines the phonon momentum and ' , ~ ~ k k    defines the matrix element. D is the phonon propagator defined as,   2 ' 2 2 n n D        , where Ω is the phonon frequency and the 35 Matsubara frequencies ω n =(2n+1)T [58,69].       ' 2 2 ' 2 ' , , , n n n k n n s T T T F               is the Gor’kov pairing function [70] with ξ k’ is the electron energy referred to the Fermi level. With considerable manipulation eq.(2.3) reduces to the following form,         ' ' 0 ' ' ' 0 ' ' ' ~ ; 1 ~ ; 1 , n n n n n n n n n n n D T Z D T Z T                              (2.4) where, λ is the coupling constant given by,            1 2 F d   , α 2 (Ω) is the electron- phonon interaction and F(Ω) is the phonon density of states. For weak coupling λ<<1, this reduces to the usual BCS form,    1 exp ~    c T (2.5) This dependence changes with increase in λ. For intermediate coupling λ~1.5 the T c is still exponentially dependent on λ though with a different functional form. But for strong coupling with λ>>1the dependence becomes linear. Special cases for λ<<1: The excitation energies of the quasi-particles in the BCS ground state can be written as,   2 1 2 2 k k k E     , where Δ k is the energy gap or the minimum excitation energy. This can be understood from the fact that even at the Fermi surface where ξ k =0, 0    k k E . For ξ k ≠ 0, E k represents the energy of an excited quasi-particle of momentum ħk. At T=0, the energy gap is related to the transition temperature as,   764 . 1 0   c kT (2.6) The gap parameter follows an universal temperature dependence for the weak-coupling superconductors. 36 Figure 2.3: Universal temperature dependence of the energy gap parameter and comparison with experimental data for Nb Near T=0, the gap is almost constant given by Δ(0) and near T c it monotonously drops to zero as shown in Figure 2.3 with the following dependence,     2 1 1 74 . 1 0             c T T T (2.7) The prefactor varies within the range 1.5 to 2.25 for the known weakly coupled superconductors [37]. For, T>T c , Δ(T) → 0 and the excitation spectrum becomes same as that for a normal state. We should keep in mind these relations are not valid for the intermediate- and strong-coupled superconductors. In the following Section, I will provide an overview of the low dimensional superconductors for which one or more of its dimension becomes comparable to the coherence length. These systems let one venture through several novel features that are not present in bulk superconductors. 37 2.3 Low dimensional superconductors The dimensionality for a superconductor is determined by comparing the size of a superconductor with the coherence length ξ 0 . For a bulk superconductor ξ 0 is in the order of several 100nm. The transition temperature is a property of the metal in three dimensions, however as the physical dimensions of the superconductor are reduced so that it becomes comparable with the characteristic length scale its properties changes significantly [71]. I will discuss sequentially the change in property as we go down to two and one dimension and finally the granular superconductor also known as zero dimensional superconductors. 2.3.1 Two dimension Two dimensional superconductors have been developed by several techniques like molecular beam epitaxy [72, 73], evaporation from bulk metal ingots [74,75, 76,77] sputtering [78,79] and quench condensation [80,81]. Let us consider the example of conventional superconductors, bulk Al and Sn for which the T c is 1.2K and 3.7K respectively. For, Al T c enhancement is observed on reducing the thickness below ~40nm and is observed in two independent measurements: in Ref.[82,83] the transition was observed at 2.1K and later in Ref. [84] the T c was reported to peak for a three – monolayer film at ~6K. The T c rise was attributed to the disorder within the metal itself.With further decrease of thickness the transition temperature shows a monotonic drop [74]. A similar non-monotonic behavior was also observed for Sn thin films. However, other superconductors like Nb [77,85], In [86] and Pb [87]shows a monotonic drop in T c from their bulk value of 9.5K, 3.4K and 7.2K respectively [37].The T c suppression was related to proximity effect and weak localization for thickness down till ~40nm and for thinner films the effect is dominated by bulk resistivity effect [85]. The boundary effects 38 at the normal – superconductor junctions and that for “dirty superconductors” have been discussed in the review by P. G. De Gennes [88]. Recent advancement in the field of thin film superconductor lead to the discovery of oscillations of T c with film thickness by Guo et. al. [89,90]. Thin Pb films of the order of a few nm confines the motion of the electrons leading to the formation of quantum well states. From the functional dependence of T c in eq. (2.4) and (2.5), its direct relation to the density of states and the electron-phonon coupling can be noticed [90–93]. These results provide convincing proof of quantum size effects and an evidence for electronic structure. The oscillations stem from the filling of quantum well states as more electrons gets added with increase in film thickness, which is akin the shell filling in atoms. The periodicity in this behavior with the film thickness is over half a Fermi wavelength as it is related to half the electron density and the crystal structure [93]. The absence of this effect from previous studies is possibly due to structural defects [94]. Recent work by Gennady Logvenov at BNL has discovered the thinnest high-temperature superconductor in few atomic thickness cuprate superconductorwhich is a proof of the fact that in the cuprate superconductors the superconducting behavior can be localized into a single superconducting plane. This can help understanding the origin ofhigh transition temperatures and create the possibility to develop tunable high-temperature superconductors for variety of electronic devices [95]. 2.3.2 One dimension In bulk three dimensional superconductor we have observed complete disappearance of resistance below the transition temperature. In one dimensional nanowires thermal and quantum fluctuations of the order parameter plays a dominating role and as a result below ~50nm, finite 39 resistance is observed down till very low temperatures. For narrower wires of diameter down to ~10nm, high resistance is expected even at T=0 as a manifestation of a new phenomenon observed at low dimension due to quantum phase slippage [96–98]. Due to phase slip which occurs over a length scale of coherence length in the superconducting nanowires, a resistive tail is observed below critical temperature [99]. A phase slip event in a superconducting nanowire is a dual process to Cooper-pair tunneling in a Josephson junction where a transfer of 2e charge is associated with a phase shift of 2π between the two superconducting regions [100]. A broad resistive transition is observed increasing with decreasing cross-sectional area of the nanowire. Based on the LAMH theory developed by Langer, Ambegaokar, McCumber and Halperin [101,102] phase slips occur via thermal activation as the system passes over a free- energy barrier proportional to nanowire cross-sectional area. During phase slip the superconducting parameter vanishes due to fluctuation at a certain point of the wire resulting in a relative phase slip by 2π across this point, resulting in a voltage pulse the sum of which results in a resistive voltage. M. Tinkham and co-workers [98] performed a careful measurement with more than 20 nanowires of nominal diameter ranging from 10 to 22nm and lengths 100 to 1000nm. The transitions are observed to widen with decreasing nanowire diameter. The data has been observed to be in good agreement with the theoretical model that includes thermally activated phase slips close to T c and quantum phase slips at lower temperatures (<T c /2) [103]. The homogeneity of the quasi 1D-superconductor nanowires is of primary concern in interpreting the experimental data related to the quantum fluctuations. Theoretical models assume: (i) spatial constancy of the transition temperature T c , (ii) the cross-section of the wire to be constant along the length of the tube and (iii) the probing instruments are non-invasive. However, simultaneously attaining all these conditions is quite challenging. The imperfections in 40 the design could lead to varying local critical temperature (T c ) along the length of the tube, though if the imperfections are not too strong the critical temperature is averaged over on the coherence length scale (ξ). With this a broad smooth-looking temperature dependence of resistance is observed [104]. It is a well known fact that the low dimensional superconductors depend of fabrication process details and size dependent effects. With lowering of characteristic dimension varying temperature dependence have been observed: (i) T c rise has been observed for indium, aluminum and zinc, (ii) T c drop has been observed for lead, niobium and MoGe [97,105–108] and (iii) no observable T c variation has been observed in tin nanowires [71,109–111]. The origin of the counter-dependence of T c is not clear while theoretical models have been developed explaining both enhancement [112] and depression of [113] of T c . For example I am providing the variation of T c with nanowire cross- section in Figure 2.4 [96] where we observe a T c rise with decrease of nanowire cross-section accompanied by a broadening of R(T) below the critical temperature. Similar effects have also been observed in Zn and In nanowires. a. b. Figure 2.4: (a) Critical temperature dependence with the cross-section of the nanowire (Inset) Distribution of nanowire cross-sections for an effective nanowire diameter 75nm with 10µm length (b) Variation of resistance around transition with dots    and diamonds (◊) representing the simulated data and    representing the experimental result [96,104] 41 2.3.3 Zero dimensionalsuperconductor Finally, we will discuss about zero dimensional superconductor where we confine the particle dimension to the order of the characteristic length scale known as the coherence length ξ 0 . Let us recall the free electron model of metals developed by Drude and Lorentz and later described by the more sophisticated quantum mechanical description. Based on these the characteristic feature of the metals is described by the partially filled topmost conduction band and on satisfying the Pauli principle, the excitation spectrum forms a continuum starting from infinitesimal energies. With miniaturization of the material the continuum spectrum becomes discretized with the average spacing between the quantum level given by the Kubo gap δ, N E F 3 4   (2.8) where E F is the Fermi energy level and N, the number of free electrons in the particle. The Fermi energies are typically in the order of a few eV. The energy gap Δ(0) for elemental superconductors is in the order of ≈1-2meV. This implies from the above equation the energy gap between the quantum levels is in the same order as the superconducting energy gap, δ~Δ(0) for a particle with a few thousand electrons. Based on the “Anderson criterion” [114] the superconductivity in a finite system will manifest itself if the pairing gap, Δ, is not significantly less than the spacing of the quantized electronic levels, δ. Within the assumption of either an approximately equidistant or a random level spacing, this on occasion has led to assertions that a superconducting transition would not occur in very small particles. This feature can be intuitively understood by the fact that in bulk superconductors, plenty of electrons are present within the energy gap Δ close to the Fermi level which condenses during superconducting phase transition to form the BCS ground state. However for particles with δ>>Δ(0) there is no enough 42 electrons close to E F within the energy interval Δ in order to stabilize the superconducting ground state. The granular superconductors thus forms the lowest dimension superconductors, as on further size reduction the pairing phenomenon completely disappears. So the grain sizes of these novel superconductors will be comparable with the coherence length with the lowest being ~10nm. The grains are often found embedded in a normal metal or insulator matrix a common example for which would be e.g. Pb islands in Ag matrix [115] or Al grains in Al oxide matrix respectively [58,116]. The islands couple via S-I-S or S-N-S coupling. Based on the coupling strength and temperature, the superconducting islands can behave as zero dimensional superconducting particles above the bulk T c with strong fluctuations [116, 117] or as a granular array of junctions whose superconducting properties mimic a single very high quality junction [118]. Due to the inherent inhomogeneity of the granular system containing grains of varying size and coupling strengths, the phase transition in not sharp but rather broad. At the onset of the transition the resistance is dominated by the intergranular resistance, but with lowering of temperature more of the grains become superconducting and the islands couple to form small networks. This manifests in observation of small Meissner effect with shielding current over finite regions of the sample. With further lowering temperature superconductivity percolates the resistance finally vanishes well below the transition in highest T c grains. This feature has been observed in NbN grains embedded in BN matrix [119] and Pb island in Ag matrix [115]. These could serve as a model for the superconductors that become superconducting on doping e.g. cuprates and Fe-based superconductors.An important example would be that of Al grains in Al oxide matrix with a T c of 3.1 K. The origin of this effect was predicted due to size quantization [113,120] and was later confirmed by spectroscopic and T c measurements [90]. 43 T c has been found to increase on small particles of most elemental superconductors like Al, Sn, In, Ga etc, [121,122] though Pb a strong coupling superconductors showed no variation in T c down till ~10nm, below which it sharply dropped to zero [123]. The two main reason behind this have been the phonon softening for small particles as they have a high surface to volume ratio and secondly the quantum size effect which has already been discussed above. Molecular dynamics simulations by Dickey and Paskin showed the existence of low frequency phonon modes for the fine particles [124]. This effect results in an enhanced electron-phonon coupling, λ for which a theory has been proposed by [125] and later modified by Dynes [126]. There is a competition between λ increase and the effective electron-electron repulsion term μ. Enhancement of T c occurs as a result of λ rising faster than μ whereas depression of T c happens if μ rises faster than λ. The first experimental observation of discrete energy level spectrum due to size quantization was observed as a result of the tunneling measurement on Al nanoparticles of ~10nm was performed by Ralph et. al. [127]. In a recent measurement from July 2010 paper [128] entitled “Observation of shell effects in superconducting nanoparticles of Sn,” the Nanoscale Science group at the Max Planck Institute in Stuttgart reported on STM differential conductance measurements on metal nanoparticles of 5-30 nm height grown on a surface (see Figure 2.5(a)). In Sn particles they observed “giant oscillations in the superconducting energy gap with particle size leading to enhancements as large as 60%,” see Figure 2.5(b). The fact that size quantization can produce a T c and the energy gap in ~5-50 nm superconducting grains has been known for over forty years [120], but “the large gap oscillations are the first experimental proof of coherent shell effects in nanoscale superconductors” [128]. The explanation, as shown in Figure 2.5(c), lies in the fact that for particles with a discrete energy spectrum, even relatively 44 small changes in size and shape can lead to large variations in the number of electronic energy levels within the pairing window, and thereby to strong variations in the superconducting energy gap. a. b. c. Figure 2.5: Observation of shell effects in nanoparticles of Sn, adapted from [128].(a) Conductance spectra of particles grown on a surface are measured with the help of an STM tip. (b) Many Sn particles exhibit superconducting energy gaps strongly exceeding the bulk value (dots: data, lines: guide to the eye). Furthermore, many variations are found with only small changes in the particle size. This is schematically explained within the shell effect picture in (c). The diagrams show nanoparticles with a finite number of discrete energy levels within the pairing region (~vibrational frequency around the Fermi level). Even relatively small variations in particle morphology and Fermi level position can strongly alter the number of levels within the window, dramatically changing the magnitude of the superconducting gap 2.4 Nanocluster superconductor As we reviewed in the previous Sections, since the discovery of superconductivity in the early twentieth century, the field of research has undergone great advancement and is heavily 45 explored. One noticeable fact is the interest in high T c and small scale superconductors. The smallest system where electron pairing has been observed contains 10 4 - 10 5 electrons [127, 129,130], also organic superconductors seems prospective in the field of small superconductors [131]. As we discussed in Section 2.3.3 for zero dimensional superconductors, the coupling vanishes with the decrease in size below ~10nm when the pairing gap Δ becomes significantly smaller than the average electronic level spacing δE (e.g. for Al, E F,bulk ~11eV and δE=1-10K for a system with 10 4 -10 5 electrons) as is consistent with the “Anderson criterion” [114]. In the following Sections we will discuss the special case of metal nanoclusters comprised of ~10 2 -10 3 electrons which can host strongly coupled paired electrons at high temperatures. However, this can be expected in special sizes with (a) closed shells known as magic clusters and (b) incomplete shell clusters with Jahn-Teller distortion. 2.4.1 Electrons in cluster pair akin nucleons The discovery of shell structure in nanoclusters, in the mid 1980s [6] has revealed that the energy levels in these "superatoms" [132] are discrete and not equidistant. This is analogous to what has been observed in atomic nuclei for long [23]. The nucleons bear similarity to the atomic clusters in several aspects: (i) electrons in clusters comprise a well defined Fermi system with shell structure similar to nucleons, (ii) the shell closings can be well explained in both systems by spherical jellium models [5,46,133] and (iii) the splitting of the HOS into sublevels for an incomplete shell cluster given by the projection of the angular momentum operator has also been observed in the atomic nuclei [134]. Nucleon pairing is a very intriguing effect which got its' first theoretical explanation from BCS theory, which was discovered to explain the pairing of electrons inside bulk superconductors [23]. Keeping all these in mind it is perfectly realistic to expect pairing of electrons in the same way as for nucleons inside a nuclei [135]. This has been 46 realized in the early 1990s by some pronounced scientists like J. Friedel [136], W.D. Knight [137] and B. Mottelson [138], but at that time material community’s attention was wholly absorbed in the newly discovered high T c -cuprates. Only recently this has got into attention and is studied in gory details. 2.4.2 Electron pairing in grains to clusters The underlying physics for the superconducting grains, also known as zero dimensional superconductors [74,121,122] undergoes sharp modification beyond the cotemporary grand canonical treatment of BCS theory. In these semi-finite systems the average spacing between the energy levels δE becomes greater than the pairing order parameter Δ [139]. Weakly coupled superconductors in bulk enter the strong coupling regime along with a noticeable rise in transition temperature [120]. As is discussed in Section 2.3.3 Al grains in Al oxide matrix has a transition temperature of ~3.1K [116], noticeably enhanced relative to the bulk T c ~1.2K. The classic work by M. Tinkham and his group for the first time experimentally probed the discrete energy level spectrum in a nanoparticle (of size ~5-10nm) and its effect on the superconducting properties via tunneling spectroscopy [129]. In a more recent work on STM differential conductance measurements of Sn nanoparticles (~5-30nm) at T c ~1.4K, giant oscillations in the pairing order parameter has been observed with particle size. The gap parameter Δ could show an enhancement of up to 60% and is the first proof of coherent shell effects in nanoscale superconductors [128]. The study on Sn nanoparticles revealed the powerful influence of shell bunching (and antibunching) of size-quantized states on the pairing gap as we show in Figure 2.5. 47 On a separate study Ga 84 -based cluster metal has been reported to have a T c ~7.2K which is noticeably higher than that for bulk Ga (T c ~1K) [140]. Our present study comprises particles in the sub-nm size range with spherical or near-spherical shapes, has the symmetry to possess high angular momentum degeneracy of approximately 2(2l+1) where l is the shell angular momentum quantum number. With this certain clusters are predicted to have an order of magnitude gap enhancement. A couple of experiments have been reported with encouraging results with regard to this effect. Measurement of electric susceptibility for cold Nb clusters at ~20K for size range ~30- 100 shows prominent odd-even effects. This peculiar effect has been predicted to be a result of pairing [141,142]. However, the Nb clusters shows no prominent shell structure which has been one of the prime force behind the theory that has been developed for these high T c nanoclusters. Another measurement on the specific heat capacity, reported small jumps at ~200K for Al 45 - and Al 47 - . These observations were suggested to be a result of pairing. But the statistics available on this measurement was extremely limited [143]. For certain size selected closed shell magic clusters the HOS close to the Fermi level are highly degenerate along with closely spaced LUS. The scenario is akin to Van Hove singularity which is represented by a sharp peak in the density of states (DOS) at the Fermi level. This in turn increases the effective pairing coupling constant which results in a significant enhancement of Δ and transition temperature. T c enhancement is also predicted for clusters which has a slight Jahn-Teller deformation. For these sizes, the HOS and LUS splits into bunches of energy levels close to Fermi level along with a significant restructuring in the energy spectrum. The bunching results in a spread of the density of states over several levels which is detrimental to T c , however 48 simultaneous shift in the position of the chemical potential, results in a T c increase. A cartoon for the highly degenerate states and its energy levels on splitting is plotted in Figure 2.6. Figure 2.6: Energy levels close to E F with the highly degenerate HOS and LUS (left) and bunching of HOS and LUS in distorted Jahn-Teller clusters (right) Theoretical prediction has only been subjected for elements which are superconductors in bulk and displays a prominent shell structure in their clusters [135,144]. Now, that this feature is theoretically well established, it would be extremely interesting to attempt a direct experimental verification for this new class of high T c superconductors. 2.4.3 Experimental techniques It is apparent from the foregoing that to pursue the promise of high T c in nanocluster systems, a key first step is to corroborate and pinpoint the effect in individual metallic clusters over a well-defined range of temperatures, materials, and sizes. In particular, it is essential to detect the direct influence of pairing on the electronic spectrum of the nanocluster. This is by no means straightforward when one is working with free nanoparticles in a beam (which at the moment is the only established way to ensure precise size selectivity and the persistence of electronic shell structure). Not only is it, of course, impossible to apply electrodes to the flying 49 clusters to measure the electrical conductivity, but infrared absorption spectroscopy (such as that used to detect gap opening in bulk superconductors) or detection of rotational bands (used to monitor nucleon pairing in nuclei) also so far remain impractical for metal nanocluster beams. There does exist, however, a signature of nanocluster pair correlation which is realistically accessible to experimental detection. There are a couple of ways that may be thought of in order to probe the pairing correlation. Firstly, the unusually strong effect of the pairing interaction on the electrons in the highest occupied discrete levels (shells), the shell spacing δE near the Fermi level will be significantly modified: calculation [135,145] predicts an increase by ~20-40 meV below T c . For example, for Zn 84 ,δE would change from ≈70 meV above 95 K to ≈95 meV below, and for Cd 83 from ≈6 meV above 90 K to ≈34 meV below. (Notice that in these examples the pairing-induced shift is of the same order of magnitude as δE, hence consistent with observability under the Anderson criterion.) The spacing change will translate into a shift in the position of the highest occupied electronic shell with respect to the vacuum level, and this shift can be detected by a careful measurement of the cluster ionization potential  0 . The analysis should be made carefully as this effect may be contaminated with the thermal effect which would also result in a similar order of magnitude increase of appearance energy. Though the change in appearance energy due to thermal effect is more gradual, but we should remember that due to size quantization the phase transition in finite size superconductors is also relatively broad. A complementary analysis would be to look for the change in density of states close the ionization threshold (lying right below the Fermi level) that can be deduced from photoionization measurements. Similar spectroscopic manifestation has been observed in bulk high T c superconductors [146,147]. 50 Indeed, photoionization measurements on metal clusters in a beam are capable of sufficiently high precision for this purpose. For example, as measured for alkali clusters in Ref. [148], we have demonstrated that resolution on the order of 5 meV can be obtained by fitting the cluster ion yield curves Y(hν) to a “Fowler plot.” The high resolution also helps is capturing the features in the photoyield curves occurring in the order ofmeV. So any changes in the spectroscopic feature resulting due to a temperature dependent electronic phase transition can be easily observed. Odd-even alternations of ≈4 meV magnitude have been detected in the ionization potentials of cold ferroelectric Nb n clusters (and proposed to be related to electron pairing) [142]. It is appropriate once again to point out a parallel with nuclear physics, where nuclear level densities exhibit noticeable odd-even alternations attributed to pairing correlations in the energy spectra [23]. Electron pairing in the bulk superconductors has spectroscopic manifestation in that the density of states close to the Fermi level peaks across phase transition. Similar peak should also be observed for nanoclusters representing compression of shell around the Fermi level. Thus a temperature dependent study of the nanoclusters needs to be performed. Any fluctuation in the density of states resulting due to restructuring of the electronic levels needs to be carefully monitored. Another idea would be measuring the near-threshold photoemission profiles for a particular cluster for a series of its internal temperatures and recording the corresponding  0 vs. T values we’ll have the sensitivity required to detect a restructuring in the electron level spectrum occurring below T c . Such a pairing-induced shift in the position of the highest occupied electronic level should be evident for certain specific cluster sizes, below a critical temperature characteristic of the number and type of atoms in the nanocluster. Furthermore, a pronounced 51 odd-even effect should be present (e.g., a cluster M n would display a distinctive jump in  0 with decreasing temperature but M n±1 would not). Thus, the central aim of the project was to map out the ionization threshold behavior for a prospective metal cluster over a range of cluster temperatures and sizes for which we will be able to unambiguously ascertain the presence of a high-temperature pairing transition and to investigate its fundamental dependence on nanocluster parameters. For our present experiment we focused on Al which is a monoisotopic element, hence is free of isotope mixing and Al n clusters show well defined shell structure for n>30 [53,149,150]. Furthermore, aluminum clusters has been explicitly studied theoretically and has proven to be a promising candidate for high T c cluster pairing [135,144]. The details of the experiment will be described in the following Chapters. Here we should mention that other possible candidates for this study were Zn, Cd, Ga and In. All these elements are superconductors in bulk and in addition there clusters are known to have well defined shell structure which makes these clusters perfect candidates in order to have electron pairing. 52 Chapter 3 Experiments 3.1 Overview of cluster nucleation The nanoclusters are formed in a magnetron sputtering gas aggregation cluster source [151,152]. This incorporates an alternative method for producing metal vapor in contrast to thermal evaporation which requires melting the metal by heating over 1000C [153–155]. The magnetron head is housed inside liquid nitrogen cooled chamber and the metal vapor is produced by argon ion sputtering of a 1 inch diameter target tablet, with a thickness of 1/8 inch. The hot sputtered atoms collide with the cold buffer gas molecules at liquid nitrogen temperature. The pressure inside the chamber is maintained at ≈1 mbar in order to ensure enough collisions between the atoms and the gas molecules to initiate nucleation. The metal atomssubsequently dimerizes by three body collision, where the gas molecule acts as the seeding agent, M+M+Ar/He=M 2 + Ar/He (3.1) The cluster initially grows by subsequent addition of atoms and further away from the target as the atom density decreases, the growth process is dominated by single atom absorption or cluster-cluster coagulation [153–156], hence preceding via the following two mechanisms, M n +M=M n+1 (3.2) M n +M p =M n+p (3.3) 3.2 Details of clustering process The magnetron sputtering source has the capability to produce neutral as well as charged clusters. For our experiment we have focused on the neutrals whereas for the charged clusters detailed study has been pursued in Refs. [156–159].There are several source parameters that can 53 be tuned to optimize the cluster intensity and size distribution. It must be realized that the effect of all the parameters are convoluted. We will provide a qualitative picture for the effect of these parameters on the cluster abundance spectra. The gas flow introduced inside the condensation source plays multiple roles: (1) it controls the pressure inside the aggregation chamber and hence the probability of collisions between the gas molecules and metal vapor, (2) the dwell time of the clusters inside the tube is inversely proportional to the gas flow rate and (3) Ar acts as the sputtering gas generating atoms from the metal surface. Thus on one hand increasing the gas flow rate has a positive effect on cluster growth and it tends to enhance coalescence among clusters, while on the other hand the dwell time of the clusters inside the aggregation tube decreases and the clusters are swept out faster. Overall the increase of gas flow rate has a positive impact on cluster growth and beam formation. The flow of Ar and He gasare individually controlled by Alicat Scientific flow controllerswhichare subsequently mixed before getting into the chamber and opens right behind the magnetron head through a ~2mm diameter orifice. The inlet gas pressure at the controller is maintained at ~70 psi. The magnetron head is comprised of a set of ring magnet placed under the target material and separated by a thin stainless steel cup. The assembly is kept at a temperature of ~15-20 C by a constantly flowing cold water from a chiller. The magnetron head acts as the cathoderelative to the grounded cupwhich acts as a shield around the head andis kept at a separation of ~1 mm. A strong electric field of ~(2–3)∙10 5 V/mis generated between the electrodes by the application of a negative DC voltage of ~200-250 V at the cathode and a self sustained discharge is initiated. The strong ring magnet generates a magnetic field of ~75 Gauss above the target surface and lies perpendicular to the electric field. This creates a region of 54 enhanced electron density which drifts along a circle parallel to the cathode. The Ar gas molecules gets easily ionized and the strong electric field acting on it makes the Ar + ions blast on the metal target along the ring [157]. This is clearly marked by the corrosion on the target surface as you can see in Figure 3.1. Figure 3.1: Aluminum target after sputtering At the beginning of sputtering it takes ~30secs for the accelerated ions to sputter off the oxidized surface from the target, following which a constant power could be maintained. Since the metal target gets heated at the surface due to sputtering, even a small amount of oxygen if leaking into the system would oxidize the metal surface (this can be seen as a dark layer on the surface of the target, after we take the target out following the experiment) and interfere in maintaining a constant discharge. This was a major source of problem for producing the Al, Zn and Cd clusters whereas for the noble metal clusters like Ag and Cu this was not too much of a problem. Thus special care had been taken to get rid of any source of leak or contamination in the system. A constant power of ~40 W was maintained for generating Al clusters which has low 55 sputter yield, whereas for Ag and Cu it was maintained within ~10W-20W. The sputter yield of Al:Cu:Ag is ~1:2:3. The experiment is performed in the controlled power mode, thus the ion current between the electrodes are constant. The cluster intensity increases with increase of power to an optimal values after which it saturates and starts to drop. We have done a time of flight measurement where the complete mass spectra originating at the source can be recorded which spans over the size range of a few to tens of thousands of atoms. It must be realized that this is not the case for several other commonly used detection techniques e.g. quadrupole mass spectrometer where a relatively small mass range is scanned. With increase in power it is usual for the mass spectra to shift with the average distribution shifting towards larger mass [156]. As the magnetron power is increased to very high value, an overall intensity drop is noticed which could be a possible result due to softening of the metal surface which has a negative impact on the sputtering yield. This is also consistent with the fact that the sputtering yield efficiency is qualitatively higher for harder metals with higher melting point. An alternative argument might be that with increase in power the probability of ionized clusters formation increases at the expense of neutral clusters. However, in a separate experimental setup it has been observed by us and also from other groups that the intensity of the ionized clusters also shows a similar drop beyond a certain magnetron power. Thus the later possibility is less likely. The sputtering yield also goes down as the groove shown in the Figure 3.1deepens, thus after prolonged use the beam intensity would gradually go down [157]. Helium, being the lighter gas and having a stronger flowplays the main role as a career gas in the beam formation and the clustering process. Monitoring the rate of increase of pressure inside the aggregation chamber with the Ar/ He gas flow rate we can see from 56 Figure 3.2 and Figure 3.3 that Ar being the heavier gas, has a much stronger effect on the pressure rise. As we will show later, the pressure inside the chamber is directly proportional to the collision among the molecules. Increasing the Ar flow rate has a direct effect on the intensity of the sputtered atoms, so we see a rapid rise in the cluster intensity with Ar flow rate. Figure 3.2: (Left) The rise in pressure inside the aggregation chamber and the thermalizing tube. The circled peaks refer to the positions corresponding to Ar addition (Right) The increase in the pressure is plotted along with its derivative. We notice spikes at the points where Ar is added resulting in a sharp rise in pressure The other parameter that has an active role in the clustering process is the aggregation length. As we tune the length we find that there is a critical distance beyond which the cluster intensity rapidly rises. Qualitatively, the cluster intensity for all size increases with aggregation length while the peak of the abundance spectra tends to shift towards higher mass. This is a result of increase in the residence time and the competition between the atoms developing into smaller clusters and the smaller clusters coalescing into larger ones. The gas flow and the aggregation length can be controlled together to optimize the intensity of any specific size range. Even though these parameters are element specific, an aggregation length of ~10 cm and an Ar:He flow rate of 1:3 can be considered as a good starting point for cluster optimization. Keeping the 57 gas flow ratio same, the absolute value needs to increase for a metal which has a lower sputtering yield. Figure 3.3: The rate of pressure rise due to individual gases where red and blue lines correspond to the addition of Ar and He gas respectively in the aggregation chamber (left) and thermalizing tube (right) The aggregation chamber has 3 inch inner diameter and is made out of stainless steel [The system has been developed at USC by Dr. Chunrong Yin adapting the original design from Haberland/ Issendorff’s group at U Freiburg]. The chamber was mounted on a flange and could be controlled from outside for alignment purpose via a cross-diagonal motion. There is also the option of controlling the exit aperturefor the clusters from the aggregation chamber and is designed in the form of an iris. The chamber is wrapped with about 20 layers of superinsulation for reducing the heat loss via conduction to outside of the source chamber. 3.3 Thermalizing tube The clusters formed inside the aggregation chamber are at liquid nitrogen temperature. For our purpose we want to vary the temperature of the clusters over a wide range. For that we attached a thermalizing tube at the exit of the source. The gas pressure inside the tube is 58 maintained such that enough collision can be guaranteed with the metal clusters so as to ensure thermalization. Two separate setups have been used for attaining a temperature higher or lower than 90 K. The design of the tubes and the collision formula for thermalization determination are described below: 3.3.1 T>90 K A customed OFHC Copper tube is made for enhanced heat conduction. The tube has an inner and outer diameter of 16 mm and 38 mm respectively as shown in Figure 3.4. The bottom of the tube stays in contact with the front ring of the aggregation chamber (see Figure 3.5), through which a continuous flow of liquid nitrogen is maintained during the experiment. An extra inner and outer lip of ~0.25” and ~2” depth is made to increase the contact surface area between the thermalizing tube and the aggregation chamber as shown in Figure 3.6. The contact surfaces are smeared with a layer of silver filled electrically and thermally conductive paste from Lesker Co. (P/N VPU075736) to ensure better heat transfer. This is only necessary when trying to achieve the lowest temperature of ~90 K. Closer to this equilibrium point the rate of heat exchange tends towards a quasistatic limit. It takes ~1 hour for the temperature of the tube to drop from 300 K to 90 K. The thermalizing tube is covered with ~20 layers of superinsulation in order to reduce the heat loss to the chamber and to attain better equilibrium along its length. For higher temperatures a rope heater from McMaster Co. or band heater from Watlow Co.is attached towards the base of the thermalizing tube extending over a length of ~1”. This design has been adapted based on the isotherm produced in 59 Figure 3.5 where we can see than the source of heat sink is only towards the base of the thermalizing tube directly connected to the aggregation chamber which stays at a temperature of ~80-90K. a. b. Figure 3.4: (a) Side view and (b) Rear view of thermalizing tube made out of oxygen-free high purity copper for better temperature conduction. The legs sticking out at the bottom is for enhanced thermal contact with the liquid nitrogen cooled aggregation chamber for better thermal conduction. The Teflon spacers at the base are for tuning the rate of cooling based on the target temperature The heater is wrapped with a high temperature cloth while the rest of the tube has the superinsulation cover. For higher temperature we tried to reduce the heat flow to the aggregation chamber by placing variable thickness Teflon spacer ranging between 1/8” to ¼” between the tube and the aggregation chamber. This minimizeslocal heating around the base of the tube as the heater would need to supply much less heat to maintain the high temperature as the heat drain through the base is minimized.Furthermore, with this the liquid nitrogen consumption is minimized which is important for long experiments continuing for 30-40 hours. With this we could attain temperatures within the range 90 K - 230 K by keeping the heater power to ~30 W. Three RTD sensors from Omega Engineering(Model.RTD-3-1PT100KN1515CLA-36-T) placed along the length of the tube are used to monitor the temperature. Thus to ensure 60 maximum contact with the tube and stay closest to the inner diameter of the tube (this provides the closest approximation to the cluster temperature), a hole has been drilled parallel to the tube diameter grazing the inner surface as has been shown in Figure 3.6. The hole diameter is marginally wider than the RTD sensor diameter. The conductive silver paste is filled inside the hole in order to have enhanced contact between the sensor and the tube. A PID controller (Model.CNi3244) works in conjugation with the sensor attached closest to the base to maintain a stable fixed tube temperature. We could achieve a very precise temperature along the length of the tube with an accuracy of ±1 K. For our experiment we went only up till 230 K, but as we testedcould reliablyattain much higher temperature. Figure 3.5: Thermalizing tube with temperature gradient where the base stays at a temperature of ~90 K being directly attached to the aggregation chamber 3.3.2 T<90 K In order to reach temperatures less than 90 K, we use a cold head refrigerator (Serial. S/C Air from CTI Cryogenics. P/N D8032-225). The situation here is tricky as the cold head refrigerator acts as a heat sink and the aggregation tube is the heat source. The first stage of the 61 refrigerator has been used which has higher cooling power of ~8 W and could reach only down till ~50 K. The cold stage is attached to the thermalizing tube from the top using four Ag coated Cu braid of ¾” width and 1/8” thickness. The braids are hard soldered to a couple of OFHC rings custom made to fit the cold stage and attached using an In foil and Ag paste for better heat conduction. The thermalizing tube is only about 1/8” in thickness and is made out of Brass to reduce the heat loss towards the aggregation chamber (see picture in Figure 3.7). Figure 3.6: Schematics of thermalizing tube used for T>90 K The braids are uniformly wrapped around the tube with Ag paste in between the braid and the tube for better connection. The lengths of the braids are kept as short as possible to minimize heat loss.The tube and braids are covered with about 20 layers of superinsulation to prevent heat loss. It is important to keep in mind here that the second stage which gets cooled down till about ~10 K lies close to the thermalizing tube andcould condense the Ar gas in the 62 chamber which might result in a pressure rise. Hence this stage itself has beenisolated from the system by covering it withmultiple superinsulation layers. Figure 3.7: Brass thermalizing tube During the cooling phase the liquid nitrogen cooled aggregation tube acts as the heat sink till a temperature of 90 K. You may notice in Figure 3.8 when the aggregation tube is at room temperature and the brass tube is cooled only by the cold head the temperature tends to saturate at ~100 K. On starting the liquid nitrogen it acts as an assist and a lower minimum is reached. The Brass tube is kept separated from the aggregation tube by a Teflon spacer. This reduces the heat loss to the aggregation chamber below 90 K, while at the same time keeping the Teflon which is in contact with the Brass tube at a low temperature, thus reducing the temperature gradient at the base region. We have done the experiment with Teflon layer thickness of 1/8”, ¼”, ½”. For the two thinner spacers the tube could be cooled down to 65 K but within about 14-17 hours the LN2 inside the aggregation chamber started to freeze which blocked the LN2 flow. Using the ½” spacer the lowest T attained was 73 K, but no freezing of the LN2 was observed over an interval 63 of ~24 hours. The temp has been measured using a Si diode sensor from LakeShore (Model.DT- 470-DI-13) which is very precise down to a few K. Figure 3.8: Temperature variation of the thermalizing tube 3.4 Cluster thermalization The clusters entering the thermalizing tube are at liquid nitrogen temperature. The gas molecules passing through the tube under a pressure of ~0.8 mTorr collides with the tube wall and thermalizes. These molecules subsequently acts as the heat bath for the metal clusters which are present in relatively much lower density and being heavier should be much less mobile. The clusters are basically dragged along with the gas molecules, and close to the exit attains similar velocity as the gas molecules [156]. Measurement of cluster temperature has always been challenging, hence there have been devoted alternative experiments where an indirect determination of the cluster temperature has been achieved [141]. The clusters depending on they being charged or neutrals could be thermalized in two ways. For the charged clusters a common techniques has been to hold the cluster inside a 64 temperature controlled ion trap where the clusters collide with the gas molecules to thermalize [160]. The clusters that we have been focusing on are neutrals and thus we used an alternative similar method where the gas molecules under high pressure passes through the thermalizing tube attached at the exit of the aggregation chamber. Similar method have previously been employed by Ref. [161] where caloric behavior has been measured by observing the photofragmentation spectra of Na clusters and the results are in good agreement with that observed by Ref [160]. These results cross verify the thermalization of the clusters. We will provide a theoretical calculation as a proof for the cluster thermalization below. 3.4.1 Method 1 For our calculation we will assume the gas flow through the thermalizing tube to be homogenous. Realistically though the flow along the axis will be somewhat faster and close to the wall will be slower, but to first order of approximation this should not create any observable difference. At thermodynamic equilibrium the gas should follow the condition, r r r n n n T p T p    (3.1) where   n n n T p  , , are the pressure, temperature and influx of He and Ar gas at standard temperature and pressure conditions (STP) into the aggregation chamber. As these gas fluxes pass through the thermalizing tube,   r r r T p  , , are the pressure, temperature and flux of gas molecules inside the tube. The flux of gas in the thermalizing tube can be easily written as r r r r t l d        4 2   where t r is the time the gas spends inside the thermalization tube, l r and d r are the length and the diameter of the thermalizing tube. Inserting this in the above eq. (3.1) we get the following expression: 65 n r n r n r r r p T p T d l t   4 2  (3.2) We used a specially customized thermalizing tube (see Figure 3.10 below) in order to monitor the pressure variation as a function of gas load. The collected data are presented below which are subsequently used in order to calculate the residual time of the clusters inside the tube and the number of collisions with the gas molecues, (1) l r = 10 cm ; length of thermalization tube (2) d r = 2.5 cm ; Diameter of thermalization tube (3) p r ,T r :: 1: 0.77 mbar, 294 K, 2: 0.56 mbar, 130 K; Pressure measured inside thermalization tube at low and high Temperature (5) p n =1 bar ; STP conditions at which the gas flow is calibrated (6) T n = 273 K ; STP conditions at which the gas flow is calibrated (7) n  = 0.36 slm ; Incoming flux of He (0.3 slm) +Ar (0.06 slm) gas in the aggregation chamber For the two sets of conditions for {p r ,T r } measured inside the thermalization tube t r comes out to be, 5.8496 msec and 9.6211 msec for conditions 1 and 2 respectively. The number of collisions that the clusters undergo with the gas molecules is given by, r t v N    (3.3) Since the flow rate of He is three times higher than Ar, for estimating the number of collisions we will use the parameters for He. This will provide us a lower estimate for the number of 66 collisions since Ar being the heavier gas would have higher collision cross sections and correspondingly more probability to collide with the clusters. Thus, σ is the He atom - cluster collision cross section and is proportional to the cluster hard sphere cross section. ρis the density of the gas molecules inside the thermalizing tube, v is the most probable velocity of He gas. ρand v can be estimated as given below, r b r T k p   (3.4) He r b m T k v 2  (3.5) where k b = 1.38 · 10 -23 J/K is the Boltzmann constant and m He is the mass of a He atom. We will derive the number of collisions for Al 32 and Al 88 , which is the size range of Al that we will be focusing on in the following Sections. Let us first estimate the collision cross section for the clusters which is given by, 2 r    , r being the cluster radius. For clusters the radius can be written as, r=r s n 1/3 [162]. Here r s is the Wigner - Seitz radius and n is the number of electrons in the cluster. For Al, r s =2.07a B [163] and using this we get the cross section for Al 32 and Al 88 as 68 Å 2 and 133 Å 2 respectively. {ρ, v } at high and low temperatures are found to be {0.1898 · 10 23 m -3 ,1105.5 m/s} and {0.3122 · 10 23 m -3 , 735.09 m/s} respectively. Using these numbers in eq.3.3 we get the number of collisions for Al 32 and Al 88 to be {8.3· 10 4 | 294K , 1.5 · 10 5 | 130K } 32 and {1.6· 10 5 | 294K , 3· 10 5 | 130K } 88 . As has been shown by molecular dynamics simulation for palladium clusters with helium atoms, these collisions are enough for equilibrating the metal clusters to the wall temperature well within a few K [160,164,165]. In the following 67 Section we will show an alternative derivation where by direct calculation we can we will discuss the cluster thermaltization. 3.4.2 Method 2 From the molecular dynamics simulations performed by P.Ballone and M.Mareschal [166] on Fe 43 , we obtain the relaxation time of the vibrational temperature to be τ = 0.1 µs. Here the clusters were present in an atmosphere of Helium with p ref = 10bar, T ref = 300 K, D = 4 · 10 -6 m, where {p ref , T ref , D} are the stagnation conditions with D as the nozzle diameter. Starting from this relaxation time we can scale it to our experimental parameter as [167,168], 3 3 ref ref ref ref ref r T p Tr p      (3.6) where we used the fact that the two body collisions between He atoms with the clusters within the thermalizing tube scales as T p . The relaxation time is inversely proportional to the number of collisions that a cluster encounters. Also in the first approximation the energy required for a certain increase in the cluster temperature scales as the number of atoms in the clusterwith 3 r N  , withras the cluster radius. The cross section of the He atom - cluster collision process is proportional to the cluster hard sphere section: 3 / 2 2 N r     . Thus the relaxation time for the cluster vibration goes as 3 / 1 N . Now we will scale the relaxation time found for Fe 43 using simulations to find the relaxation time for Al 32 and Al 88 of Al clusters, which is the size range of interest. The scaling ratio 135 . 1   Fe Al r r  , where we used the atomic radius for Al and Fe to be 143 pm and 126 pm respectively. Thus, the above eq.3.6 can be written as ref at ref ref at ref ref r N pT r TN p , 3 / 1 3 / 1    , where r at is the 68 atomic radius for r Al and r at,ref is for r Fe . Using these scaling the relaxation times for the clusters at high and low temperature comes out to be {1.3 msec | 294K , 0.8 msec | 130K } 32 and {1.8 msec| 294K , 1.1 msec| 130K } 88 . After knowing the correct relaxation time and the dwell time that the clusters spend within the thermalizing tube we can calculate how much the cluster temperature equilibrates with the ambient temperature of He gas. The thermalization process can be described as, ) ( 1 ) ( t T dt t T d      (3.7) where gas vib T T T    . The above eq. can be integrated to provide,  / 0 t e T T     (3.8) Thus, using the proper values of τ from above and t=t r , the residence time of the clusters inside the thermalizing tube from Section. 3.4.1, we find τ/t as {0.2222 | 294K , 0.0883 | 130K } 32 and {0.3077| 294K , 0.1143| 130K } 88 . So we gather the parameters as follows: ΔT 0 = 217 K and 53 K for the high and low tube temperatures given by 294 K and 130 K respectively, starting from the original cluster temperature of 77 K. Using these we get ΔT as, {2 K | 294 K , 0 K | 130 K } 32 and {8 K| 294 K , 0 K| 130 K } 88 . We mostly focused in the temperature range 60 K to 230 K and thus we can conclude that over this temperature range the clusters will equilibrate to the wall temperature with a few K. 69 3.5 Optimization of inner design of thermalizing tube The response of the cluster abundance spectra to temperature of the thermalizing tube has been quite interesting. We find the abundance spectra behave quite differently at high and low temperature. With increase in the tube temperature a rapid increase of the intensity of large clusters has been noticed while that forsmall clusters tends to drop. On the contrary at low temperatures the intensity of the small clusters strengthens while the large clusters vanish. The overall total count rate increase up to an order of magnitude on comparing the total intensity at 60 K and 230 K. From equilibrium thermodynamics, we know that the pressure inside a chamber is directly proportional to temperature and inversely to the volume. Thus, at high temperature there is an increased collision rate among clusters and also between clusters and gas molecules. So the small cluster intensity drops due to two possible reasons: (1) increased collision with the gas molecules kills the small clusters and (2) collision among the clusters leads to formation of bigger clusters. This feature is probabilistically low since the pressure inside the tube is lower than that inside the aggregation chamber. For larger clusters entering the tube, the scenario is different. At lower temperatures these clusters tend to go towards the tube wall and stick to it in order to reduce its entropy. It has been observed to vanish completely with further lowering of temperature [169]. For smaller cluster, scenario is congenial at low temperature as they encounter lesser collisions with the gas molecules and fellow clusters. The pressure inside the tube drops by ~ 0.15 mbar for a temperature drop of ~ 200 K. Thus most of the clusters entering the tube survive the passage to the exit. The shift in the mass spectra due to temperature can be observed from Figure 3.10. For the small clusters which would be size separated, we observe a flight time reduction with cluster temperature, which is likely to be due to increase of pressure 70 inside the thermalization tube as a result of temperature rise as is discussed in more detials in Chapter 5. However the shift is representative and the exact magnitude should not be taken seriously. This is because the internal energy change of the cluster is proportional to the vibrational degrees of freedom ≈ (3N-7)k b T [13] where N is the number of electrons in the cluster. The thermal shift was found to be larger for bigger clusters as can be seen from the mass spectra in Figure 3.9. Figure 3.9: (a) The change in the mass spectra with temperature for two different tubes. In both cases we observe the mass spectra shift to higher mass with temperature rise: (i) Wide thermalizing tube at 150 K and 280 K (blue and red respectively) (ii) Narrow tube with 90 K and 170 K (green and magenta respectively) (b) Representative picture with the blue and red lines are at cold and hot T respectively. Look at the decrease in TOF with T rise. This has been observed for all days For our experiment we are mostly concerned about the small clusters which are produced in the low end of the mass spectra and hence are relatively weak under any temperature conditions. In order to reduce the pressure inside the tube we have increased its inner diameter which increases the volume. However, care has been taken so that there is enough pressure to ensure the number collision required for cluster thermalization. The minimum pressure inside the tube 71 has been measured using a dummy tube with the pressure gauges attached to it as shown in Figure 3.10(a). a. b. Figure 3.10: (a) Dummy thermalizing tube for pressure measurement inside tube and aggregation chamber with temperature variation (b) Thermalizing tube with bypass valve for reducing pressure inside the tube The entry into the thermalization tube is smooth without any lip, so that there is no stagnation pressure inside the tube causing partial expansion which could change the cluster internal temperature. With an inner diameter of 16 mm that we used for our experiment, we could go up till 230 K while maintaining a decent number of small clusters in the beam. An alternative method would be to introduce a leak created symmetrically around the tube as shown in the schematic in Figure 3.10(b) [170]. The leak zone opens at a bypass valve connected to an external mechanical pump or roots blower. This can independently control the pressure inside the tube keeping the gas influx constant. For the low temperature regime below 90 K, we have used a thermalizing tube which is 8 mm in diameter. This is done in order to compensate for the reduction of the number of collisions between the gas molecules and the clusters due to low temperature. 72 3.6 Details of the experiment Clusters exit the thermalizing tube through a 6 mm diameter aperture towards a 2 mm diameter conical skimmer located ~ 1.5 cm away. The source chamber is pumped by a 3000 l/s diffusion pump (Varian VHS-10 with an extended cold cap), and is maintained at a pressure of approximately 7×10 -3 mbar. The following chambers are maintained at lower pressure of ~10 -7 Torr. The clusters are subsequently ionized by electron or photon impact as necessary for the specific experiment. We will discuss these ionization techniques explicitly below: 3.6.1 Photoionization experiment The modified Wiley-McLaren time of flightmass spectrometer (TOFMS) is located about 50cm downstream from the skimmer [171, 172,173]. The spectrometer separates clusters flying in vacuum based on the mass-to-charge ratio (m/z) which controls the flight time for the clusters from the ionization region to the detector. All the different size clusters in the beam are simultaneously ionized by ns laser pulse and are accelerated towards the detector by a constant electric field. The velocities of the clusters are thus inversely proportional to their mass. These results in a mass dispersion in the time domain where the lighter clusters has a shorter flight time relative to the heavier clusters. There major advantages of TOFMS are: (1) the complete mass spectrum gets collected with every single Laser pulse. This increases the speed of data collection as an entire spectrum is generated within several msec (2) this is ideal for the measurements in which the knowledge of the relative intensities is important at every time instant as the source conditions can have fluctuations over time (3) the design is simple and does not require extremely precise mechanical alignment. The spectrometer is composed of a set of three plates producing a pair of constant electric field zones as shown in Figure 3.11. The space between the first two plates is the ionization 73 region where the neutral clusters flying in are ionized by the laser pulses. The intensity of the pulse is maintained spatially uniform by collimating a wider beam. The beam is initially multimodal and the collimator picks the central zone of the beam which has a uniform intensity distribution. The ionized clusters are accelerated towards the detector by double field acceleration as shown in Figure 3.11. This introduces an extra controlling parameter and the resolution can be much improved by controlling the relative field intensities between the ionization and the acceleration zones. In the first zone the field strength is kept low. The laser pulse ionizing the cluster beam has a finite width of ~ 2 mm, so the clusters ionized closer to the middle plate gains lower energy relative to the ones born closer to the source backing plate. Thus over the flight path the clusters with higher energy and hence larger velocity overtakes the slower clusters by the time detector is reached. Thus the space resolution leads to a time spread, ΔT Δs . The laser pulse cannot be focused too much since high laser fluence leads to multi- photoionization and fragmentation of the clusters. Thus the time spread is minimized by reducing the applied field in this region. As we will see below the space resolution can be expressed as a function of the ratio of the field strength in the acceleration to the ionization region E d /E s and the width of the acceleration region, d. Another possibility of time spread occurs due to the initial velocities of the clusters. It is possible that a pair of clusters ionized at an identical position by the laser pulse has the same speed but directed in opposite direction. This can be minimized if the initial energy is a small fraction of the total energy the clusters attain after acceleration. The energy resolution can also be represented in terms of E d /E s and d as will be shown later. 74 Figure 3.11: Photoionization experimental setup 3.6.1.1 TOFMS design The ionization region in the TOF assembly was 1.25 cm. and the acceleration region was 1.88 cm. The initial design for the TOF-MS was developed by Dr. Kin Wong [174] which is further optimized to reduced the fringe field, maximize counts with improved resolution and for better control towards steering the cluster beam. The source backing plate had a round hole of 5 mm whereas the middle and the grounded plate had a 9 mm hole covered by a copper mesh with 100 lines per inch [Model. 02199C-AB from SPI Supplies] (The details for the mesh attachment is provided in the Appendix A). This is because the clusters passing through the first hole are neutrals and would not be affected by the fringe fields appearing at the edges. The mesh in the middle and grounded plates are important since the clusters get ionized before they pass through these plates. The fringe fields diverges the ionized beam and a good fraction of them does not reach the detector. In Figure 3.12 we present the simulation (done by Dr. Anthony Liang) from Simion (ver.7) where we have introduced mesh in the middle and end plates and we can see that the convergence is much improved relative, as the circle drawn around it gives a representation of the divergence 75 without the mesh. The mesh also helps in improving the resolution as the clusters on ionization experience more uniform spatial distribution of the electric field. Note, in principle the mesh could be applied to all the plates, but that reduces the intensity by blocking the clusters as transmission is ≈ t n where t is the transmission coefficient for each mesh and n is the number of mesh in the beam path. For the copper mesh used t=0.85 and thus with n=2, the effective transmission was 0.72. Figure 3.12: Simion simulation for beam spread for TOF plates without mesh (left) and with mesh covering the middle and the grounded plate (right) An additional set of four perpendicular plates was used at the exit of the three plate assembly. This gives an extra control on steering the beam such that the most intense part of the beam can make up to the detector. A typically voltage of few to tens of volts had to be applied for best result. The beam was pretty sensitive to it and one on correct tuning could improve the beam intensity by orders of magnitude. 3.6.1.2 Optical system An optical parametric oscillator (OPO) laser has been used to produce the variable wavelength in the range 210 - 2300 nm, though for our experiment we focused on the UV wavelength range 210 - 250 nm. 76 The laser beam exits the OPO from one of the three available ports depending on the required wavelength. Since we focused mostly around the lower wavelength range, we had to use only a single port which emits 210 - 419.9 nm. The laser beam is highly divergent and has been a serious concern as it traverses a path of ~ 2 m from OPO to the chamber where it interacts with the cluster beam. A couple of 1 inch square cylindrical lenses of 1m focal length (P/N SCX-25.4- 508.6-UV from CVI Mellis griot) has been placed at a distance of ~0.5 m and ~1.5 m from the OPO exit. They compensate quite nicely the divergence along the vertical axis and creates a laser beam of dimension ~ 0.5 cm at the entry to the chamber. The divergence along the horizontal axis is minimal and spans over a length of ~ 1 cm outside the chamber (no additional converging lens had to be put along this direction of the laser beam). Three square right-angled prisms have been used along the beam path of 1" cathetus (customed with Altos Ref # 45158/9451)as the laser beam is wide smaller size prism could not be used. The beam has a flat energy distribution in the central region at the entry to the chamber. A collimator from OWIS with 4 movable lamella (P/N SP 60)are used to manipulate and select the position of the laser beam entering the chamber such that the energy distribution for the collimated beam is uniform along its dimension. The beam entering the chamber is ≈ 2x2 mm in dimension. A UVFS wedged window (P/N W2-IF-0525-UV-193-0) lets the beam enter the chamber, with the wedge preventing back reflection. The cluster beam in the interaction chamber is ≈ 5 mm diameter and the cluster beam dimension is chosen so as to ionize the clusters from the central part of the beam. This plays a critical role in severalaspects: (i) the cluster intensity is uniform in the central region of the cluster beam and can have fluctuations around the edges (ii) the ionized clusters remain more towards the central part of the mesh which covers the 9 mm diameter hole at the center of the middle and the grounded plates. This enables the ionized beam to stay away from any edges that 77 can possibly cause fringe effects. The cluster intensity is maximized by shifting the lamellae (iii) an increase in beam divergence is noticed with wavelength, however due to proper focusing and collimation the beam dimension interacting with the cluster beam could be kept constant over the wavelength range explored. The energy density in the laser beam varies widely as a function of wavelength. In order to keep the energy flux constant, a smoothly varying circular attenuator is used (Thor labs, Item #NDC-100C-2M). The region closer to the edge of the disc has been used as around outer rim the OD variation is slower. Prior to this final tuning an initial alignment is performed using the alignment stick setup. Here a motorized manipulator is used with a custom built alignment setup attached towards the lower end. As shown in Figure 3.13 the setup has two perpendicular surfaces. Figure 3.13: Schematic of the laser alignment design 78 One of those side has a 3 mm diameter hole which lets the cluster beam to pass through. The optimized position is determined by maximizing the intensity measured using an quadrupole mass spectrometer/ analyzer (QMA). As a next step a cross horizontal slit of 2.8 mm width (i.e. slightly smaller than the cluster hole) is used carved in the same setup with its central line at the same height as the center of the hole. The intensity of the laser beam entering the chamber is measured using an energy meter that is mounted inside the vacuum chamber (Model: QE-12 from Gentec). This ensures that all of the laser beam entering the chamber is interacting with the cluster beam. 3.6.1.3 Data collection and Analysis The cluster ions are ionized at every wavelength at an interval of 1 nm. The OPO has the capability to produce wavelengths at an interval of 0.1 nm, however that much precision seems unnecessary. During the data collection procedure minor adjustment of a pair of BBO crystals is necessary for beam intensity optimization. Towards the deeper UV region a few microdivision (1/256 degrees) shift in crystal angle has huge impact in the beam intensity. The OPO crystals produce wavelengths in the range 420 - 2300 nm. The positions of the OPO crystals are pretty robust and thus in general does not need tuning without technicians. Also, whenever the OPO crystals are tuned, the wavelengths should be correspondingly checked, since the phase matching between the OPO crystals is the key to produce the continuously varying wavelengths (for more details chek Ekspla manual). As is shown in Figure 3.14 the OPO wavelength and its frequency doubled counterpart are tested using a BLUE-wave (Item #BW-UV) miniature spectrometer from StellerNet Inc. 79 Figure 3.14: 520 nm output from OPO and frequency doubled UV wavelength 260 nm During data collection the energy pulses are recorded over all time and a running average is done. The standard deviation was recorded to be ~5% which is equivalent to about 1 μJ energy flux. Thus the effect of fluctuation in the beam intensity is negligible. Typically, the data collection time at each wavelength is within the range ≈40-90 mins depending on the cluster beam intensity and laser wavelength. After every individual wavelength a data set has been collected at a fixed normalizing wavelength. This takes care of any fluctuation in the cluster beam intensity due to changes in the source conditions. The cluster ions leaving TOFMS are detected by a custom-built channeltron detector (Detector Technology) located ~1.3 m downstream [175]. The assembly is equipped with a conversion dynode capable of operation at up to 20 kV, which is important for efficient detection of heavy ions. The signal is analyzed by a multi-channel scaler (MCS-32) software from Ortecwhich produces a time resolved spectra of the clusters reaching the channeltron, with a best resolution offered by the electronics is 100 ns. The total flight time for the clusters between ionization and detection is given by [171], 80   D d s d s T T T s U T qdE qsE U U       , 0 0 (3.10) where U 0 is the initial energy of the neutral cluster, {(E s ,E d ),(s,d),(T s ,T d )} are the electric field, width and the time spent by the clusters in the ionizing region (with s tag) and acceleration region (with d tag) respectively. For simplifying the formula we can use two approximations: (1) the drift time for the cluster once it gets out of the accelerating region and gets towards the detector is orders of magnitude higher than the time spent by the cluster within the accelerating region. Thus the drift time can be approximated as the total time of flight and (2) the initial energy of the clusters is much smaller than the energy gained due to the strong electrostatic field. Thus the initial energy, U 0 can be approximated to zero. With these approximations the flight time can be estimated as, U m D T D 2  (3.11) We have efficiently used this formula to estimate the flight time for all the clusters using only a couple of clusters as the reference which simplified the analysis to a great extent. A typical TOF spectrum is provided in Figure 3.15. As we will notice the peaks are well separated for small clusters whereas for larger clusters we start seeing overlap. This is due to the spread in the peaks which is governed by the space and energy resolution as described earlier. As shown in Ref. [171] the individual space and mass resolution are given by,                 0 2 1 0 0 2 1 0 2 1 0 0 0 2 0 0 1 1 16 16 s d k k k k k U U M s s k M t s  (3.12) and the overall resolution is a combination of both these effects given by 81   M M M s s 1 1 1 ,   (3.13) As can be observed both M s and M θ are functions of E d /E s , s 0 and d. The requirements for optimal space and energy resolution are opposite. Thus the best overall resolution is a compromise. s 0 and d had beendetermine d ≈0.6 cm and 1.9 cm during design for providing the best resolution over the range of interest but cannot be tuned during experiment.s 0 , which is ideally have the distance between the first two TOF plates can be tuned a bit by steering the laser beam within the ionization region. We optimized E d /Es to be ≈2.4 (V d ≈ 2100 V, V s ≈ 300 V, d ≈ 1.9 cm, s 0 ≈ 0.65 cm) in order to control the resolution, by which we could resolve clusters typically up to around size 100. Figure 3.15: Typical TOF mass spectra at 90K or Al n clusters To obtain the intensity for individual clusters we deconvoluted the mass spectra from Figure 3.15 using multi-gaussian peak fitting as shown in Figure 3.16. On careful analysis we found that within the investigated size regime each cluster can overlap with at most the second nearest 82 neighbor peak. Thus in order to find the intensity I n for M n , we fit the cluster peaks within the size range M n-2 till M n+2 using,          2 2 2 n n i c b h i i i e a S  (3.14) where I n is calculated by integrating over the central Gaussian,         n n c b h n e a 2  . This method plays a critical role in the intensity determination, mostly for the larger clusters within the size range studied where the overlaps are higher. The yield values, Y(E) are obtained by proper normalization as Y(E)= I n ’(λ)/D(λ) where D(λ) is the laser fluence and I n ’(λ) is the normalized Al n + counting rate. The normalization corrects for any possible beam intensity drifts/fluctuations by referring the ion rate to that measured at the wavelength of 216 nm after each collection interval. The fitted curve is shown in Figure 3.14 shows that using this method very accurate estimation of peak intensity of individual clusters can be performed. I n has been used for different types of analysis as will be explained in the respective chapters. The background in the mass spectra is negligible as has been tested separately and hence now included in the equation. The background is tested in two different ways: (1) if the beam is blocked then the counts acquired in the mass spectra is negligible over the time window where we are focusing ensuring the background counts from minute oil in the chamber is negligible and (2) with the beam open the counts in the region which lies after the real cluster counts (as can be seen within the region with log-normal distribution) are negligible. This method plays a critical role in the intensity determination, mostly for the larger clusters within the size range studied where the overlaps are higher. 83 Figure 3.16: Multi-gaussian peak fitting to the TOF mass spectra. 3.6.2 Electron impact ionization The electron impact ionization coupled with the quadrupole mass spectrometer is an alternative method for mass detection. By this method we can obtain high mass resolution of clusters in the range 1 to 8700 amu. However the data collection is much slower as it records the intensity of one mass point at a time over a time scale in the order of a msec. So here we have a compromise between the resolution and data collections peed. For experiments where the relative intensity of the cluster is important or if the source condition can fluctuate over time, this might not be ideal. However, there are several advantages of using electron impact ionization: (1) the electrons produced by a simple electron gun can carry energy of tens to hundreds to eV. A photon with energy of 10 eV corresponds to 124 nm which is in deep UV region, and for higher energy one needs to go to even deeper UV. These can be produced only at special facilities with advanced light source. (2) the electron gun used by us produces very diffused electrons gas 84 which ionizes ~1 in 10 4 -10 6 clusters and we can guarantee single electron impact ionization. The laser pulses on the contrary carry a large number of photons and unless properly attenuated can easily lead to multiphoton ionization. The systematic study of these two different ionization method have been quite intriguing and is presented in Chapter 7. Figure 3.17: Electron impact ionization setup While using electron impact ionization, based on the requirement of the experiment either: (1) all the voltages in TOFMS are either kept OFF while the plates still remain in the beam path or (2) the whole assembly is removed from the beam path to avoid the intensity loss due to the mesh in TOF plates. As shown in Figure 3.17, 1.4 m downstream from the skimmer the cluster beam enters the ionization region of an Extrel MAX-9000 quadrupole mass spectrometer (Extrel axial electron-impact molecular beam ionizer with a Spectrum Solutions ionizer control power supply FPS-1). At the exit of the mass filter the cluster ions are detected by the aforementioned custom-built channeltron detector (Detector Technology) with a conversion dynode capable of operation at up to 20 kV, which is important for efficient detection of heavy ions. The electron impact ionization has been especially useful to study multiple 85 ionization in clusters. Let us take the example of Ag n clusters to explain the details of the mass spectrum and its analysis. 3.6.2.1 Analysis The mass spectrometer covers the range from approximately 5 Ag  to 80 Ag  and the source operating parameters were optimized for maximal signal within this size range. In actuality the source produces particles of a much wider size distribution [151, 176]: a separate time-of-flight measurement using pulsed laser ionization detected clusters of over 600 atoms (≥3 nm diameter) for the same source conditions. A quadrupole mass spectrum is shown in Figure 3.18. The shape of the overall envelope can be tuned substantially by the source and detector settings. The pulse counting rate for a cluster of size n is a product of several parameters [177]: ( ) e n n e e n n n S E i FT d    , (3.15) where ( ) e n e E  is this cluster’s ionization cross section at electron energy E e , i e is the ionizing current, ρ n is the number density of neutral target clusters in the ionization volume, F is the spatial overlap factor between the electron and the neutral beams, T n is the transmission efficiency of the ion optics and the rods of the mass analyzer, and d n is the detection efficiency of the channeltron. We have checked that the detector signal increases linearly with the ionizing current, confirming that ionization proceeds via single-impact processes and that space charge in the ionization region does not cause a substantial distortion of S n . For conciseness the above expression omits contributions from ionization-induced fragmentation. In fact, interestingly, in the present experiment fragmentation effects do not appear prominent: the shape of the mass spectrum envelope does not change with increasing 86 electron energy, and the spectrum in Figure 3.18 is smooth without marked odd-even and magic- number features that would typically derive from impact-induced evaporation [13,178]. This suggests that the electron ionization process does not deposit a significant amount of internal excitation energy into the originally cold nanoclusters. Figure 3.18: Mass spectra of silver clusters produced by the sputtering/aggregation source and detected by electron-impact ionization mass spectrometry. The mass spectra are shown for several different electron impact energies (30-70 eV). Note the gradual appearance of Ag n 2+ and Ag n 3+ ions at fractional cluster size channels. (Recall that the mass spectrometer sorts particles according to their charge-to-mass ratio, hence doubly- and triply-charged cations appear at the corresponding fraction of their actual size n.) Insert: an enlarged portion of the 70 eV mass spectrum to show the doubly and triply ionized clusters (solid line: original spectrum, dotted lines: Gaussian peak fitting). Ticks on the ordinate axis denote zero levels for the respective spectra. An inspection of the mass spectra in Figure 3.18 reveals that with increasing electron impact energy, a robust population of doubly-charged Ag n 2+ ions appears in the mass spectrum, and at higher impact energies the presence of Ag n 3+ becomes visible. The range of sizes for 87 which ionization yield measurements were carried out was bound from above by the mass spectrometer resolution and from below by the decreasing beam intensity. Odd-sized Ag n 2+ ions were used for analysis because they don’t overlap with the singly-charged cations, and analogously for Ag n 3+ . Cluster signal intensities were calculated as follows. For a chosen Ag n 2+ or Ag n 3+ peak, the peaks of its immediate more intense neighbors were fitted to Gaussian profiles, and the wings of these Gaussians were subtracted from the counts in the area of interest. The remaining signal was then integrated to obtain the intensity S n 2+ or S n 3+ . 88 Chapter 4 High T c pairing in Al n nanoclusters The ion yield curves of most of the clusters shows a monotonic post-threshold rise for all temperatures and can be used to derive the temperature dependent cluster ionization thresholds [179]. However, we noticed for a few clusters in the range a prominent "blister" appears near the ionization threshold with decrease in the cluster temperature. The most prominent example is observed in the photoionization spectrum of the magic closed shell cluster Al 66 with 198 valence electrons [180]. Among the incomplete shell clusters Al 37 which has 1 electron removed from the closed shell Al 37 - (112 electrons) [150, 149] and hence has slight Jahn- Teller deformation shows a strong temperature dependence. Al 44 and Al 68 which has relatively more deformation show a weaker but detectable temperature variation [181]. We will first provide a theoretical argument to justify the possibility of pairing for these specific clusters [135,144] followed by a discussion of the observed spectral feature. In order to quantify the bulge-like feature in the Y(ħω) curve we use two possible procedures: (i) estimation of the area under the bulge of the yield curve and (ii) differentiate the Y(ħω) curve and identify the emergence of large peak with decrease in temperature. The relative difference between the maxima and the subsequent minima of the peak serves as a measure of the bulge of the yield curve. The evolution of Y(ħω) and dY/dE with temperature for the four different sizes shows the same qualitative feature and are suggestive of a transition around T c ~100 K. In Figure 4.1 and Figure 4.2 we show the plots for Al 66 , similar results for the incomplete shell clusters are presented later. 89 4.1 Closed shell clusters Al 66 is a spherical closed shell cluster with high symmetry and angular momentum degeneracy of the HOS and LUS. Based on jellium model calculation [8,42], the HOS and LUS are 1j (l=7, degeneracy, G H : 2(2l+1) = 30) and 2h (l=5, degeneracy, G L =22) respectively [150], G H and G L are the degeneracy of HOS and LUS respectively. This implies that the effective degeneracy of the cluster is significantly high (G H +G L =52). From the photoelectron spectroscopy measurements we can find the energy difference between HOS and LUS is ~0.3 eV [149,150], which along with simultaneous increase of degeneracy results in strong coupling between the electrons, which results in a significant increase of Δ and the transition temperature, T c . Thus, under the circumstance where we have extremely high degeneracy in the HOS, even if the pairing gap Δ does not exceed the shell spacing we can expect a pairing effect. First, let us discuss about the appearance of blister at low temperature in the yield curve of closed shell Al 66 which is the most remarkable observation for our experiment as we present in Figure 4.1and Figure 4.2 (Selective important sections of the complete code for the data analysis is provided in Appendix B). Primarily this hump like feature for Al 66 has not been observed in any neighboring cluster. It is worth mentioning that this kind of structure is sometimes observed during shell opening as is observed in some other nanoclusters with shell structure, like Cs and Cs n O [182,183] but never in closed shell clusters. Moreover, the humps are never reported to have any temperature dependence. In order to confirm this we measured photoyield for Cu n clusters at 60 K and 215 K. No notable structure is observed in any magic copper clusters near threshold. Some open shell clusters showed some structures, which does not have any obvious temperature dependence as we show in Figure 4.3 [181]. These are in strong contrast to Al 66 which can be categorized under a new genre. 90 Figure 4.1: Photoionization of aluminium nanoclusters.(a) Yield plots for Al n (n=64,…,67) obtained at T=65 K. Short vertical bars denote the cluster ionization threshold energies. A strong bulge-like feature appears close to the threshold for n=66. The adjacent clusters show no such feature. The sharp drop in the ionization energy from Al 66 to Al 67 reflects the fact that the latter is a closed-shell “magic”-size cluster. Different color dots correspond to data duplicated over several experimental runs. In both panels the different yield curves are shown shifted with respect to each other for clarity. (b) Evolution of the Al 66 spectral feature. Its growth with decreasing temperature can be seen by comparing the thick experimental yield curve (a spline average of the data points, such as shown in the first panel) with the dashed interpolating line. 4.2 Incomplete shell clusters Theoretical assumption predicted that the best case scenario for pairing among incomplete shell cluster is for a cluster with a pair of electrons removed from the closed shell. Al 37 forms an excellent example with 111 delocalized electron, i.e. 1 electron removed from the outer 3p shell. This should result in a small deviation from sphericity and splitting of the HOS and LUS levels. The effective degeneracy of Al 37 - is 32 and on splitting the HOS and LUS splits into 2 and 7 sublevels respectively. The splitting of the energy level is given by m l l m l Ur E E ) 0 ( 2   , 91 where HOS H l E E E   ) 0 ( which can be approximated to E F and U is the deformation parameter [135]. m l r is given by a combination of the angular momentum l and its projection m as         1 1 2 1 2 3 2 3 1 2        l l m l l r m l . With mild shape deformation of the cluster the splitting of the degenerate energy levels is minimal, but this partial occupancy/ vacancy of LUS/ HOS strongly affects the position of the chemical potential and the energy spectrum also gets significantly reordered which results in a considerable increase of Δ and T c . Figure 4.2: Temperature dependence of the Al 66 spectrum and the density of states.(a) Derivatives of the near-threshold portion of the photoionization yield plots from Fig. 1(b). As discussed in the text, dY/dE represents a measure of the electronic density of states. The intensity of the first peak, which derives from the “bulge” in the Al 66 spectrum, grows with decreasing temperature, implying a rise in the density of states near threshold. The plots are normalized to the amplitude height of the minimum following the derivative peak. (b) To quantify the intensity variation of the peak in (a), we plot its amplitude as a function of cluster temperature. (c) Another measure of the magnitude of the bulge: its area relative to the dashed straight line in Fig. 1(b). It is noteworthy that the behavior of the plots in panels (b) and (c) matches, both suggesting that a transition takes place as the temperature approaches ≈100 K. 92 Figure 4.3: Photoionization yield curvesfor copper nanoclusters (a) and their derivatives (b), illustrated here for two representative sizes, show no temperature dependence similar to that of the detected spectral feature in Al 66 (Figure 4.2). This confirms that the latter case represents a distinctive electronic transition It is noteworthy that Al 37 contains an odd number of electrons in its HOS. Thus in the superconducting phase of this cluster, all but one electron will pair. The unpaired electron should not interfere with the spectroscopic feature of the ion yield curve that is discussed below. This can be qualitatively anticipated since the unpaired electron will sit above the energy gap, but nevertheless the gap should open up and have its spectroscopic manifestation. The other two clusters identified for displaying temperature dependence in the yield curve are Al 68 and Al 44 . Al 68 has 6 electrons added to the LUS and is the first neutral cluster with even number of electrons following Al 66 . Al 44 on the other hand has 6 electrons removed from the closest closed shell Al 46 . PES spectra show that δE for Al 46 is ~ 0.1 eV [149,150]. Thus after bunching the energy gap between multiplets would reduce significantly. As a result in both these clusters there is good possibility for T c enhancement. 93 In Figure 4.4 we show the raw data corresponding to temperature variation for all the three different size Al n clusters (n = 37, 44, 68), where we can see the appearance of a prominent bulge with temperature drop. The individual sizes are treated carefully later. Figure 4.4: Raw data for the photoyield spectrum of incomplete shell clusters Al n (n=37, 44, 68). In each individual columns the temperatures at which the yield curves are collected are 230 K, 170 K, 120 K, 90 K and 65 K (from top to bottom respectively). Al 37 and Al 44 , precedes their closed shell counterparts Al 37 - and Al 46 by 1 and 6 electrons respectively. Thus the appearance of bulge in their ion yield curve at low temperature has no correspondence with any shell opening effects and can be unequivocally categorized under this newly observed novel phenomenon, similar to Al 66 .The propagation of bulge and its derivative with temperature for Al 37 are shown in Figure 4.5 and Figure 4.6 respectively. The same for Al 44 are shown in Figure 4.7 and Figure 4.8. Al 68 has 6 electrons added to the nearest closed shell cluster Al 66 . PES spectra of Al 66 - has shown prominent signature of shell opening demonstrated by the appearance of a peak at low energy, however Al 67 - has no such feature [149,150]. The PES, which is a representation of the density of states measurement, is analogous to the derivative of the yield. The absence of peak in the Al 67 - PES and the monotonicity of yield curve for Al 67 [181] shows that signature of shell opening at Al 66 - is washed out at these subsequent sizes. Hence the appearance of the hump at Al 68 and its’ temperature dependence also represents the novel feature. This is plotted in Figure 94 4.9 and Figure 4.10. In this connection, it should be mentioned that the appearance of the peak in DOS due to photoemission from 3s and 3p shells and its propagation with size is not observed beyond n≈10 [184]. 4.3 Discussion The transition is relatively sharp in Al 66 and Al 37 whereas it is broader for Al 44 and Al 68 . This is not surprising since the later clusters should have more deformation as they are further away from the nearest magic cluster. This should increase the energy gap between the split energy levels. It would be interesting to explore lower temperature, where Δ might increase and we observe a cleaner phase transition even for Al 44 and Al 68 . Figure 4.5: (Left) Evolution of the Al 37 spectral feature. The growth of the bulge with decreasing temperature can be seen by comparing the thick experimental yield curve (a spline average of the data points, such as shown in the first panel) with the dashed interpolating line. The temperatures are 230 K, 170 K, 120 K, 90 K and 65 K from top to bottom curves respectively. (Right) The plot of the shaded area suggesting that a transition takes place as the temperature approaches ≈100 K 95 Figure 4.6: (Left) Derivatives of the near-threshold portion of the photoionization yield plots from Figure 4.5. As discussed in the text, dY/dE represents a measure of the electronic density of states. The intensity of the first peak, which derives from the “bulge” in the Al 37 spectrum, grows with decreasing temperature, implying a rise in the density of states near threshold. The plots are normalized to the amplitude height of the minimum following the derivative peak. (Right) To quantify the intensity variation of the peak, we plot its amplitude as a function of cluster temperature which shows a transition analogous to that presented in Figure 4.5 happening around 100 K. Figure 4.7: (Left) Evolution of the Al 44 spectral feature. Illustration is analogous to Al 37 (please check caption of Figure 4.5: (Left) Evolution of the Al 37 spectral feature. The growth of the bulge with decreasing temperature can be seen by comparing the thick experimental yield curve (a spline average of the data points, such as shown in the first panel) with the dashed interpolating line. The temperatures are 230 K, 170 K, 120 K, 90 K and 65 K from top to bottom curves respectively. (Right) The plot of the shaded area suggesting that a transition takes place as the temperature approaches ≈100 K). The fluctuation is broader during transition which could be an effect of splitting of the HOS and LUS energy levels for this cluster. Al 44 has 6 vacancies in the HOS. 96 Figure 4.8: (Left) Derivatives of the near-threshold portion of the photoionization yield plots from Figure 4.7. For illustration refer to Figure 4.6. The short vertical bars refers to the position of minima used for normalization. (Right) To quantify the intensity variation of the peak, we plot its amplitude as a function of cluster temperature. The similar transition observed as in Figure 4.7 refers to a broad transition occurring at a temperature ranging from 100-150 K. Figure 4.9: (Left) Evolution of the Al 68 spectral feature. For illustration refer to Figure 4.5. (Right) the plot of the shaded area suggesting that a transition takes place in the temperature range ≈100 - 150 K. 97 Figure 4.10: (Left) Derivatives of the near-threshold portion of the photoionization yield plots from Figure 4.9 with 7, 9 and 11 point smoothing. The plots are normalized with respect to the nearest minima.For further illustration refer to Figure 4.6. (Right) The amplitudes from the 11pt smoothing plot is shown in the right plot where a transition in the range 100-150 K has been observed. Thus the effect reported in Figure 4.1(b) and Figure 4.2 for Al 66 and Figure 4.4 till Figure 4.10 for the Jahn-Teller distorted clusters appears to be a new observation, and merits careful attention and exploration. We suggest that it is consistent with being a manifestation of the electron pairing phenomenon, as supported by several factors: - An actual temperature onset is clearly observed. (The gradual decrease in the intensity of the bulge above the transition is consistent with the behavior of pairing fluctuations expected in a finite system [185].) - The onset lies within the temperature range matching theoretical predictions for aluminium cluster superconductivity. According to theory, the value of T c depends strongly on the occupied-to-unoccupied shell spacing (HOMO-LUMO gap) and the value of the material’selectron-phonon coupling constant . Calculations in [186] give T c ~100 K for closed- shell Al clusters with a coupling constant =0.4, as in crystalline aluminium, and a shell gap of 98 ~0.1 eV. Photoelectron spectroscopy 21 reveals that the intershell distance in Al 66 is a bit larger, approximately 0.35 eV (taking the distance between the half-maximum points on the facing slopes of the topmost peaks). However, theoretical studies of Al n clusters 26 suggest that n=66 (and possibly some of the other aforementioned sizes) has amorphous-like structure, and therefore the value of λ used also should correspond to amorphous aluminium. The latter has a T c of 6 K (see, for example, Ref. [84]) which significantly exceeds that for usual crystalline Al, 1.2 K. Correspondingly, the value of λ here is much larger 28 . With the use of McMillan relations (see, e.g., Ref. [58]) for T c and for λ, one can estimate that for amorphous Al the coupling is indeed quite strong: λ≈1-1.2. Note also that Al 66 has the most spherical shape among its peers and therefore the most pronounced shell system [149] and the highest level degeneracy, which matches the optimal scenario for a pairing mechanism. - The effect appears only in a few of the clusters studied. This agrees with the expectation that pairing can take place only in systems with a propitious combination of electronic degeneracy, shell energies, and coupling strength. While it is conceivable that thermal structure fluctuations could themselves somehow cause a bulge in the electronic spectrum, the observed onset temperature lies much below the aluminium clusters’ pre-melting and melting points 30 of 300-900 K. It would be unexpected for purely structural fluctuations in several clusters of different sizes to conspire to produce a single similar-looking feature in their electronic spectra. Although such a possibility merits bearing in mind, the above considerations strongly suggest that we are dealing not with a structural but rather with an electronic transition. Searching for the origin of such an intriguing spectral feature we explore all the essential terms that contributes to provide the photoionization yield. Y(E) as a function of photon energy 99 can be given by,             E d D f M E Y     , where ε is the electron energy, M is the dipole transition matrix element, f is the Fermi-Dirac occupation function, and D is theDOS. Here, the energy of the vacuum level is set to zero. The derivative dY/dE is therefore proportional to the density of states D(ε) multiplied by the Fermi-Dirac function representing the particle statistics f(ε) [182,183]. Thus, it could be inferred that the appearance of the peak in the dY/dE curve shown in Figure 4.2 is a representation of the electronic DOS and its change with temperature. Below T c at the onset of pairing the electrons in the HOS get compressed towards the closely spaced lower shells [135,149,150] (as from the usual BCS theory for bulk superconductors [58,99]). The shift of the energy level downward towards the more tightly bound levels results in an extra pair breaking energy necessary to move an electron to the continuum. The excitation energy for the clusters below T c takes the form 2 2 ~      , where ξ is the excitation energy of the electrons in the normal state referred to the chemical potential. The order parameter Δ(T) and chemical potential, µ(T) are both functions of temperature which is reflected in the spectral feature. Being a finite system the conservation of electrons needs to be satisfied, thus shifting the position of the µ(T) from the Fermi level. Especially, around T c while the gap is opening up the increase of bulge in the photoemission curve is most prominent. It has been observed in infinite 1D and 2D systems the fluctuations fade the phase transitions. In clusters which could be thought of as zero dimensional systems due to their small size, the situation is much more complex. Now, the dimensionality of a system is determined by the relation between the size of the system relative to the characteristic length. For a superconducting state the coherence length plays the role of the scale factor. While for conventional superconductor the coherence length is ~10 2 -10 3 nm, high T c cuprates are known to 100 have relatively small coherence length of the order of ~ 1.5 nm. For the high T c - nanoclusters the coherence length is of the order of ~ 1 – 1.5 nm and is thus comparable to the cluster size Al n (n=37,44,66,68). Hence, the clusters in this respect behaves rather like a quasi-3D system and not a 0D and a finite phase transition is observed for the specific sizes [144]. The transition as we reported is distinct but broader relative to bulk superconductors due to fluctuations in the order parameter [135]. It is curious to note that bulk high-T c cuprates develop cluster-like metallic regions under certain doping conditions and it has been remarked that these formations may be relevant for superconductivity [187]. Even though the bulk high T c superconductors differs from nanoclusters in the key fact that they lack a characteristic discrete spectrum, it would be interesting to explore if the superconducting transition also has a measurable influence on the photoelectric effect of these systems. While photoelectron spectroscopy has been pervasive in research on high T c materials, study of photoemission has been rather scarce. Interestingly, however below T c marked changes in the near-threshold photoelectron yield has been observed in Bi 2 Sr 2 CaCu 2 O 8+δ [146,147]. This strong alteration in the total photocurrent is an effect ascribed to the two-dimensional nature of the CuO 2 planes. However, in the superconducting phase the photoyield spectra obtain some distinct feature which can be detected as peaks in the derivative, dY/dE. The peaks could be attributed to the superconducting DOS analogous to the “bulge” observed in Al clusters. A temperature dependent study of the total photoyield of a cluster beam with constant flux would be quite interesting. Unfortunately, the flux of the cluster beam itself is a function of temperature and such an absolute comparison is the beyond the scope of present measurement. 101 4.4 Conclusion The photoionization yield spectra of a few selected closed and incomplete shell aluminum cluster presents a special feature which increases considerably on lowering temperature. The appearance of the "bulge" at low temperature is suggestive of an increase in density of states near threshold due to electron pairing. However, the transition which is prominent around 100K is not sharp as expected for finite systems. The onset of this feature occurs considerably above this temperature and is broadened due to the fluctuation in the order parameter. This transition is consistent with the theoretical prediction for aluminum cluster superconductivity [135,144]. Metallic nanoclusters are not only the smallest known superconductor with about 10 2 -10 3 delocalized electrons in the system, but has a potential for demonstrating electron pairing at a temperature considerably higher than any known bulk high T c superconductors.Cuprates have ruled the field of high T c -superconductors in 1987 [188,189]. However recently there has been an increased interest in several new categories of superconducting systems: ruthenates, MgBr 2 , borocarbides, nitrides, fullerides etc. This is obviously the most flourishing era in the field of high T c - superconductors. The inclusion of metallic nanoclusters into the group could help in understanding the pairing in the nanoscopic range. It would be curious to extend our investigation to lower temperature where Δ should be larger and provide a cleaner picture for the transition. It would also be interesting to explore other potentially superconducting materials like Zn, Cd, Ga and In which has shown prominent shell structure. Metal nanoclusters would form a novel family of high-T c superconductors that could serve as building blocks for next generation materials and networks [186]. 102 4.5 Possible application of nanocluster based networks Accurate size-selective measurements on individual free clusters are essential for identifying and characterizing those particles which exhibit the “giant strengthening” of pairing. Looking ahead, materials and electronics applications require assembling such superconducting size-selected nanoclusters into arrays, films and compounds. Consider, for example, a chain or network made up of identical nanoclusters with discrete shell-ordered energy spectra, connected by tunneling barriers. Recent theory predicts that not only is such a chain capable of supporting a tunneling current, but also that it will actually enable a 2-3 order of magnitude increase in Josephson current transport compared to conventional systems [186]. Thus the use of high-T c nanoclusters could combine an orders-of-magnitude increase in superconducting current capacity with an orders-of-magnitude increase in the operating temperature. The techniques of size-selective cluster deposition haveadvanced very significantly over the last decade (see, e.g., the publications in [190]). Nevertheless, their application towards preparation of superconducting structures based on, and sensitive to, clusters of a specific chosen size faces a number of practical challenges. First of all, depositing clusterswith precise size selectivity and sufficiently gently to prevent fragmentation, coalescence, and distortion of their electronic spectrum remains a cutting- edge research problem [69, 191]. Secondly, in order to be mass analyzed and electrostatically slowed down for “soft-landing” the clusters have to be ionized, hence the substrate needs to be able to support charge transfer in order to prevent the buildup of repulsive charge. Thirdly, at present such techniques have a relatively low deposition rate. Therefore conventional techniques 103 such as magnetometry are not sensitive enough, and experimental approaches with high sensitivity to superconductivity in low-coverage samples need to be deployed. The challenging project leading towards the first possible development of size-selected nanoclusters as building blocks for superconducting materials, networks, and circuits has currently been conceived and is in pursuit at Prof. Vitaly Kresin's lab. 104 Chapter 5 Temperature dependent appearance energy for Al n clusters Appearance energies of clusters produce useful information about their electronic and geometric properties [192,193,194,195]. Al n clusters being a group-III representative suffer from stronger symmetry breaking perturbations due to axially symmetric distortion, spin-orbit coupling and crystal-field splitting. This was clearly demonstrated in the anomalous shell closings for sizes that could not be explained by the spherical jellium model [35]. Previous measurements have mostly been focused at room temperature [35,36]. Recent developments in the field of nanoclusters predicting exotic features like high temperature electron pairing [135,143] has aroused interest in investigating the temperature dependence of appearance energies in certain class of nanoclusters. Here we will restrict ourselves to the thermal shift of appearance energies for the small Al n clusters and the spectroscopic manifestation of pairing has been discussed elsewhere [180,181]. Photoionization is a complex phenomenon which involves thermal vibrations of ions, the initial and final state of ionized electron, plasmon excitations and photodissociation in clusters. The thermal shift involves interplay between structural and electronic degrees of freedom. Our operational temperature range is low enough such that there should not be any effect of pre- melting or melting of clusters [143] and the measured shift should be of electronic origin. Controlling the temperature of clusters has always been challenging, thus most of the temperature related studies of clusters has been theoretic [196,197]. With our developed technique we can precisely control the temperature of clusters over a wide range. This enables us to measure the thermal shift and estimate the temperature coefficient for the clusters. The 105 classical conducting sphere model for the clusters extrapolates the small clusters’ appearance energies to bulk work function. To our knowledge this is the first experimentally obtained thermal shift measurement for Al clusters and hence in bulk. Direct measurement of temperature dependent work function in bulk Al have been difficult since it is a reactive metal and any surface contamination could mask the thermal effect completely [198]. The precision necessary for direct measurement is beyond the scope of contemporary experiment or numerical ab initio studies [198–201]. From semi-empirical theories [201–204] which incorporate the thermal expansion effects, the predicted temperature dependence is found to be quite small which is also expected since the Fermi temperature greatly exceeds the ambient temperature. Experiments on alkali nanoparticles in gas phase produced temperature dependence of work function to high accuracy [148,205]. By optimizing the photon energies to the range where only elimination of electrons from the pure metallic clusters is possible, we ensure that there is no contamination whatsoever. Once ionized the clusters reach the detector within microsecond under ultrahigh vacuum condition. The photoyield curves Y(E) (a measure of the properly normalized Al n + ion intensity as a function of photon energies) studied for aluminum clusters, Al n (n=32 to 95) have a smooth threshold rise that could be consistently approximated by a quadratic Fowler law, originally derived for bulk surfaces at low temperature [206]. Our study extends to the post threshold region up to ~1 eV above the threshold. While most of the clusters show a monotonic dependence over this energy range some magic clusters or clusters with small Jahn-Teller distortion have interesting features in the post threshold region [181], however these does not have any direct impact in the thermal shift. The thresholds are rigorously determined at 5 different temperatures within the range 65 K - 230 K and are used to derive the averaged 106 temperature coefficient <dφ/dT> N,T , where the averaging is done over all cluster sizes and temperatures. Figure 5.1: Linearity plot for Al 67 5.1 Appearance energy measurement In Figure 5.2 we present a typical time of flight mass spectra at low and high temperature obtained during the same run. At low photon energy we see some outstanding prominent peaks which correspond to clusters with low ionization thresholds. For high photon energy all peak amplitudes are relatively same since the photoionization cross sections become comparable. Besides, we see that the flight time of the clusters marginally reduces with increase in the internal temperature. This is likely to be an effect of the increase of pressure inside the thermalizing tube which occurs with the rise of temperature. Using the specially designed setup for explicitly measuring the pressure inside the thermalizing tube and aggregation chamber (shown in Figure 3.10 (a)) we have monitored the pressure change carefully. For e.g. at a gas load of 80 sccm of Ar and 280 sccm of He gas, the pressure inside the chamber/ tube drops from 800/600 mTorr at 300K to 600/400 mTorr at 120 K respectively. Thus the pressure inside the 107 tube increases relative to the source chamber pressure which does not have any noticeable change with tube temperature. As a result the gas assisted cluster beam progresses with a faster flow towards the detector resulting in a small but noticeable change in the flight time of the clusters. Figure 5.2: The blue and red lines are at cold and hot T respectively. Look at the decrease in TOF with T rise. This has been observed for all days and for different wavelengths. Figure 5.3: Similar plot at a few wavelengths along with multi Gaussian fit for Al x showing the deconvolution There is no unified theory to estimate the shape of photoyield curve, Y(E) in the near threshold region. Previous measurements of ionization threshold for aluminum clusters were limited to ad hoc linear fitting of the yield points with photon energies in the near threshold region [35,36]. 108 We traced the photoyield up to ~1 eV from the threshold region which provides a clearer understanding of its evolution. Focusing only around the region where an intensity rise is observed could mask the real threshold estimation due to the thermal tail. The yield curve could be best approximated by the quadratic fit which is the low temperature approximation of the Fowler function. Y∞(hν-Φ 0 ) 2 (5.1) where Φ 0 is the bulk work function and hν is the photon energy. It has previously been shown for alkali nanoparticles that finite temperature Fowler function provides the most accurate result for work function [207]. Since the Fermi temperature for Al (13.7 x 10 4 K) is much higher than alkalis, the temperature range explored between 65 K and 230 K (T<<T F ), are found to consistently follow the quadratic fit. The appearance energies measured using this method should correspond to the vertical ionization potential which is the energy difference between the neutral and the charged cluster without relaxation, though for the size range explored the difference should be relatively small as the change in cluster geometry on ionization should be minimal [197]. 5.2 Results and discussions 5.2.1 Thermal shift of clusters’ appearance energy For determining the thermal effect on small clusters we performed an exhaustive photoionization measurement at 5 different temperatures within the range 65 K to 230 K. Each data point of the photoyield curve plotted in Figure 5.4 is an average of 4-5 experimental runs and each run lasted for approximately 24 hours. 109 This extensive data collection was important to increase the accuracy of the measured thermal shift. Figure 5.4 shows a representative plot of appearance energy variation with temperature for Al x . Since the temperature dependence can be assumed to be linear within the range explored [203,208], the dashed line is fitted to estimate the average temperature coefficient, <dφ/dT> x . The comprehensive table of data for measured appearance energies at all individual temperatures is provided in Appendix C. Figure 5.4: Temperature dependence of appearance energy for Al n (n=48-63) The thermal shift as observed for most clusters is <1% of the appearance energy. Thus we calculated an universal appearance energy value for each clusters by averaging over all the temperatures, φ T as is plotted in Figure 5.5. The thermal shift is estimated by (Δφ T ) x ≈ <dφ/dT> x ∙ΔT, where ΔT≈165K (230K~65K) and is represented by a bar on the top of (φ T ) x . 110 Figure 5.5: IP with temperature shift represented by the bar Figure 5.6: Thermal shift in appearance energies The temperature shifts for the smaller clusters are relatively larger than the bigger clusters as can be seen from Figure 5.6. As have been explored for bulk metals, the origin of drop in work function is the thermal expansion in solids with increase in temperature which results in decrease of electron density and a change in the Fermi energy. Thus the work needed to 111 remove an electron from the metal surface goes down [148,203,204]. For small clusters, it has been observed by ab initio studies that its’ thermal expansion is larger than that for bulk. While for alkalis this effect is seen in clusters down till 50 K, in small Al clusters the significant increase in volume have been found at temperatures higher than ≈300 K [209]. Since the maximum temperature that we could explore is only ≈230 K and the observed shifts for the clusters are comparable, we calculated an universal low temperature coefficient <dφ/dT> N ≈ - 7.75∙10 -5 eV/K, by averaging over all cluster sizes in the range Al 32 to Al 95 . The negative value of the temperature coefficient implies that the appearance energy drops with temperature rise. The measured value falls well within the window of calculated temperature coefficient for bulk Al, which is ≈ -2∙10 -5 eV/K to ≈-8.3∙10 -5 eV/K based on the applied model [203,204]. This is also consistent with other room temperature measurements on Al clusters and the fact that metallic behavior have been observed for sizes starting from ~30-40.The experiments in support of this fact are the gap closure in the photoelectron spectroscopy measurement following the Kubo criterion [210] or the polarizability measurement where the sharp oscillations predominant in small clusters is found to saturate to the smooth polarizability decay matching with the jellium model estimation [211]. Data for exact thermal coefficient on small clusters are available only for Al 6 and Al 7 where ab initio molecular dynamics studies in the temperature range 300 K - 800 K have produced a thermal shift of approximately 0.1 eV [209, 212]. This is equivalent to a temperature coefficient of -20∙10 -5 eV/K and is consistent with expected coefficient increase for small clusters especially at high temperatures [209, 213]. 5.2.2 Temperature dependence of bulk work function The thermal shift in the appearance energy measured for small Al clusters can be used to derive accurate information about the temperature dependence of bulk work function. It has been 112 studied that the appearance energies of small conductive spheres with radius R differs from the bulk work function WF of an infinite, flat polycrystalline solid as, AE=WF+αe 2 /R (5.2) The radius of the cluster R can be represented in terms of the Wigner-Seitz radius (r s,Al =2.98a 0 [214]) of Al and the number of electrons, n e as R=r s n e 1/3 . The bulk WF contains information about the highest occupied molecular orbital or the Fermi levels with respect to vacuum and the second term is the electrostatic finite size correction for the cluster. In Figure 5.7 we plot the appearance energies of the clusters with n e -1/3 . The straight line is fitted for the clusters with size greater than ~40 as that is estimated as the onset of metal like property from different experiments as discussed in Refs. [210,215]. The fitted line extrapolates to provide the bulk work function of 4.35 eV at 65 K. The result is excellent match with the accepted bulk work function for Al of 4.28 eV estimated at room temperature [203]. The bulk work function obtained in a similar way at 230 K provides a value of 4.34 eV. Thus drop in the work function is 13.7 meV. From the calculated value of <dφ/dT.> ≈ -8.3∙10 -5 eV/K [204] we find the estimated shift for a temperature rise of 165 K, starting from 65 K should be ~10 meV. Thus the match is excellent within the accuracy of measurement. The theoretically estimated value of α is derived from the electrostatics consideration and is estimated to be 3/8 [216] or ½ [217].Though the experimentally obtained value for a wide range of clusters shows variation within 0.2 to 0.4 [218]. The average result of α from our measurement obtained by averaging over all the temperature range is well within the bound of observed values. 5.2.3 Anomalous shell structure Trivalent Al is a nearly free electron metal and thus the electronic structure of its clusters are expected to match the jellium model [150]. We observed shell closings for Al n clusters at 113 n=46,56,66,78 and is consistent with the theoretical prediction based on spherical jellium model (SJM). From photoelectron spectroscopy measurement, the density of states spectra have produced isolated low energy peaks for the corresponding anions of the above mentioned shell closing sizes, thus presenting an alternative proof for shell closing for the neutral clusters [149,150]. In Al 66 this feature is most prominent marked by a drop in appearance energy by ~0.2 eV for Al 67 . This is corroborated by numerical simulation via density functional theory approach to produce a distorted structure for Al 66 with a spherical outer shape [149]. Figure 5.7: WF extrapolation for AE vs n -1/3 plot at 65 K and 230 K However, no clear evidence of shell closings are observed for n e = 106 (Al 35 - ), 112 (Al 37 - ) , 156 (Al 52 ), 166 (Al 55 - ), 220 (Al 73 - ), where n e is the number of electrons in the cluster, showing disagreement with the SJM model [150]. Similar disagreement with the shell closings have also been observed for small size clusters [35,149,150]. Interestingly, unexpected shell or sub-shell openings have been observed at sizes, n=37,39,43,55,57,69 which could be due to pronounced perturbations demonstrated by +3 ionic core on a jellium background. Density functional theory 114 calculations [149,219,220] have shown that the cold clusters attain crystalline, faceted shapes which destroys the electron shell structure. The hot, liquid clusters on the other hand should be spherical and could be better candidates to follow the SJM model [149,221]. It is important to consider the fact that Al being a group-III cluster, many of the predicted shell closings are for anionic clusters. For example, Al 35 - and Al 37 - are expected to have shell closing based on SJM. PES of anionic clusters show shell openings for Al 36 - and Al 38 - which are consistent with the fact that Al 35 - and Al 37 - are closed shells. From the present study of photoionization spectra for neutral clusters, we see a sharp dip in appearance energies for Al 37 and Al 39 which cannot be directly correlated to shell openings based on SJM. Also, DOS of Al 39 - does not have any open shell features observed from the PES [149]. The post threshold photoyield curve for Al 37 has an interesting temperature dependence which could be correlated to a novel feature, and is discussed in detail elsewhere [181]. The derivative of the photoyield curve, can reproduce the density of states as has been confirmed by direct comparison of dY(E)/dE for monovalent Cu clusters with the PES spectra [149,180,181,222]. However, if we now look at the photoyield curve of Al 67 which follows the closed shell Al 66 cluster, we can obviously identify the dip in the appearance energy, but cannot regain the extra peak from the derivative of photoyield. PES of Al 66 - on the contrary shows the obvious peak appearing due to the extra electron of the anion added to the new shell. In Al 67 there are two additional electrons added to the new shell, which possibly results in closure of the HOMO-LUMO gap. The dips at the shell openings as can be seen from Figure 5.5 are noticeably larger than that for alkalis [5,223] consistent with that predicted by theoretical model [196,224,5]. This implies that the gap opening following the major shell closings for Al are typically larger than 115 alkalis at low temperature. This prominent difference at cold clusters should smoothen out with temperature rise as we can see from numerical simulation for alkali clusters [196]. 5.3 Conclusion Here we discussed the wide range control of aluminum cluster temperature and its appearance energy measurement. Photoyield data obtained using low photon flux ensures minimal fragmentation for larger clusters. Quadratic Fowler fit provides the best estimate for the rise in the yield around threshold. It would be interesting to explore how the threshold behavior changes as the cluster temperature is increased close to melting or pre-melting points [225]. Interestingly deviation from spherical shell model is observed in appearance of shell or sub-shell openings at anomalous sizes and missing SJM shell closings. We propose a significant perturbation on the jellium background due to lattice crystal field effect for this group-III cluster is the probable cause behind such inconsistencies. Our detailed study on the thermal shift of appearance energy produced a drop of few meV for most clusters, but relatively larger for small ones. The drop in appearance energies for the small clusters is directly related to the decrease in bulk work function with the temperature rise. Thus the linear extrapolation based on the conducting spherical droplet model (CSD) for the measured appearance energies produces a very accurate value of the bulk work function and its temperature dependence. The average thermal coefficient measured for clusters of size greater than ≈30 is a very good match with the theoretical result for the temperature coefficient of bulk work function [201–203]. This is consistent with the fact that several previous studies [149,215] have observed metal like property in clusters of size larger than ~40. While the CSD turns out to be a great success, one interesting exercise could be to apply the sophisticated effective 116 coordination number (ECN) based model which takes into account cluster structure [226,227] in describing the size evolution for these clusters. This model has worked nicely for Cr and Mn clusters [228,229] and would be curious to explore for Al clusters which has unconventional size dependence of appearance energies [149]. 117 Chapter 6 Temperature dependent appearance energy of Cu n clusters Photoionization spectroscopy has been extensively used to study the appearance energies of polyatomic metal clusters [133,228,229] and work function of bulk metals [203]. This technique pioneers in determining the electronic structure and shape of atoms, molecules, clusters and bulk metals. The analysis of the yield curve has mostly been focused towards the rising trend around the threshold [52,182,183,228,229]. When it comes to determining the band gap and probing the deeper energy levels inside the clusters, photoelectron spectroscopy (PES) has its complete monopoly. Here we demonstrate for the first time that both the near and post threshold feature of the yield curve can be efficiently used to derive the density of states distribution inside the neutral copper clusters and its direct correspondence with the photoelectron spectra of anionic copper clusters [210,230]. Previously, this method had been used for estimating the variation of density of states close to the Fermi level in bulk high T c superconductors [146,147]. We use our DOS results for the Cu n clusters to estimate the HOMO- LUMO gap, which in conjugation with the size dependence of appearance energy predicts the onset of metallicity [210]. From Koopman’s approximation the HOMO-LUMO gap is a property of neutral clusters [32], and the DOS derived here lets us compare the results with that of anionic clusters [231–233]. Tuning cluster temperature in a beam has always been challenging and thus most of the study related to the temperature dependent size evolutionary pattern of the clusters have been theoretic [196,197]. We have used an enhanced method by which the temperature of the clusters can be controlled over a wide range precisely. The variation of the near threshold photoyield 118 helps in determining the temperature dependence of cluster appearance energy. From the derivative of the total yield curve we can derive the dependence of the excitation band gap at low and high temperatures. Previous results of the temperature dependence of bulk Cu work function demonstrated wide range of discrepancy with the theoretical data [202,204]. One hindrance in these measurements is the possible oxidation of the polycrystalline metal surface which gets in the way of the precision necessary for accuracy of the results [198]. Interestingly, gas phase nanoparticles came to the rescue as that can be probed in situ under high vacuum with minimum possibility for contamination. This method had been adapted for highly reactive alkali clusters with great success [148]. In the present investigation the clusters studied are much smaller, thus we get an opportunity to compare its temperature dependence with theoretic value for bulk work function. Figure 6.1: Cluster ion yield (integrated yield in the range from Cu 40 to Cu 48 ) as a function of laser fluence at =216 nm. The linearity of the plot confirms that the ion signals derive from single-photon ionization. Photoionization data discussed in this paper were acquired at a fluence of ≈500 µJ/cm 2 . 6.1 Results The time of flight mass spectra obtained at 60 K for some representative wavelengths are shown in Figure 6.2. For high photon energies we notice a smooth profile indicating that the 119 ionization cross-section of the clusters at high photon energies are comparable. However, for low photon energies distinct odd-even effects among clusters are observed, most prominent for size smaller than Cu 48 . Between Cu 49 and Cu 61 the oscillations ceases and is almost indiscernible for size greater than Cu 63 . Interestingly, Cu 49 and Cu 61 have remarkably low appearance energy relative to its neighbors. Figure 6.2: Spectra at 216 nm and 245 nm. At 245 nm the cluster peaks sticking out corresponds to the ones with low appearance energies The yield curves are derived from the photoionization spectrum using the standard method as described in previous chapters (see Chapter 4). The yield curves obtained at two 120 different temperatures 60 K and 215 K are plotted in Figure 6.3 which serves as the parent curve for further analysis. Deriving the appearance energies from the yield plots has always been debatable due to the lack of a general theory of cluster photoionization. Some of the common methods that have been used are linear and exponential extrapolations close to the threshold, error function fits and displaced oscillator models [205,207,182,214,223]. Applying the finite temperature Fowler law [148,206,207,234] the work functions for alkali nanoparticles have previously been estimated [148] and Indium clusters to high accuracy [234]. Since the temperature of Cu clusters in this study is much lower than its Fermi temperature, we have found that the low temperature estimate of the Fowlar law approximates the threshold rise quite consistently as was discussed in detail in Chapter 5. The photoyield curve Y(E) is further efficiently used for deriving the DOS of the clusters using similar analysis as discussed for AL clusters in Chapter 4. In Figure 6.4 we have plotted the derived DOS at 60 K and 215 K. The DOS distribution for Cu 49 and Cu 61 shows the most striking features with an anomalous shell opening. With these plots we have superposed the PES obtained for Cu 48 - and Cu 60 - with the Fermi level normalized [231]. The HOMO-LUMO band gapturns out to be a pretty good match, which enables us to explore the origin of appearance energy variation and its relation with the density of states distribution. 121 Figure 6.3:Yield spectra of Cu clusters marking the sharp drop in IP at 49 and 61 represented as step like feature 6.2 Discussions 6.2.1 Shell structure, density of states and metallicity The yield plots obtained for the clusters at 60 K and 215 K have been used to derive the appearance energy and the density of states of the neutral clusters simultaneously. The appearance energy demonstrates the minimum energy necessary to remove an electron from the 122 cluster surface whereas the density of states distribution provides the number of available states within the energy interval. Thus the size dependence of the appearance energy trails the position of the highest molecular orbital. In Figure 6.6 we have plotted the appearance of the clusters from Cu 24 till Cu 87 . We observe significant odd-even oscillations in the appearance energies which on comparing with the DOS distribution from Figure 6.4 can be correlated with opening of large band gap for the odd neutral clusters. The data points in the yield curve extend up to ≈1eV above the threshold. We have superposed the dY/dE plot with the DOS for anionic clusters obtained by PES with properly scaled Fermi energies, which shows a nice coincidence [231] as shown in Figure 6.5. Figure 6.4: DOS from derivative of Yield plot for size 49-71. The low energy peak at 49 marks the shell opening. The next opening is at 61 and closes at 63.Following this size no further significant opening is observed marking the onset of metallicity 123 From the previous PES measurements, it has been reported that within this energy depth from the Fermi level, the electrons show consistent s-band characteristics as the 4s derived orbital extends till a depth of 2-5 eV [231,235] below the Fermi level. Clemenger – Nilsson model [46] or detailed ab initio calculations [236,237] have done a fair job in predicting the electron shell structure for the alkali clusters. However for Cu and also previously reported for Ag [231,238,239] there are additional structures that cannot be completely explained within this framework. Figure 6.5: Photoionization yield curvesfor copper nanoclusters (a) and their derivatives (b), illustrated here for two representative sizes. This confirms that the latter case represents a distinctive electronic transition. The additional green line is obtained from [240] which shows nice superposition with our data, thus confirming the validity of our method Strong oscillations in the appearance energy is observed till Cu 48 , associated with the opening of band gaps is a possible result of strong s-d hybridization which changes the free electron nature of the 4s contributed electron towards more directional bonding (Cu electronic configuration: [Ar]3d 10 4s 1 ). Thus even though PES of the uppermost part of the s-band for Cu showed noticeable similarity with that of alkali Na cluster, there are intrinsic differences between 124 the alkalis and noble metal clusters. This raises a very curious question: are the small Cu clusters metallic? Metallicity in clusters is a very interesting field of study. An insulator to metal transition in bulk is denoted by an overlap between the conduction and valence band with a finite DOS at Fermi level. The common transition mechanisms are Bloch-Wilson, Mott-Anderson and Mott-Hubbard transitions [241–243]. Clusters on the other hand due to their finite size always have discrete DOS at Fermi level and consequently a finite energy level spacing δ. Thus, by the contemporary definition MIT in clusters occur when δ is smaller than the Kubo gap, given by k B T≈(4/3)E F /N. where E F is the Fermi energy and N the number of electrons in the valence band [139]. If we look at monovalent alkali metal clusters like Na or K, we see that their outermost shell is completely delocalized and even small clusters have a band gap smaller than the Kubo gap, however for Cu it is quite intriguing as we will discuss below. Figure 6.6: Appearance energies at low and high temperature for Cu n clusters For Cu clusters we observe that most of the odd sized clusters till Cu 48 have a band gap higher than the Kubo gap as is shown in Figure 6.7. The opening of new band at the magic clusters is well understood and is a consequence of electronic shell closing; however a band gap 125 inevery other cluster indicates that every odd electron is occupying a split sub-shell. These gaps are larger than the Kubo gap, thus making the clusters non-metallic and are due to strong perturbation on the free electrons by the deeper d-band electrons. For Cu 49 the AE drops by about 0.5 eV which is consistent with the opening of a HOMO-LUMO gap of approximately that magnitude in the DOS which rather is a typical band gap for semiconductors [244]. From these data and from the simulated results for HOMO-LUMO measurement [245] we can infer that the appearance energy dips are associated with the electron emission from the low lying energy peaks of the DOS spectra. Figure 6.7: The appearance energies of Cu n clusters referred to the Kubo gap [210] 126 Figure 6.8: Kubo gap with HOMO-LUMO gap may be as Fig.6.7a and 6.7b. Beyond Cu 63 we see stable gap closure [245] The dip for Cu 49 is not expected as consequence of shell closing underellipsoidal Clemenger model [46]. However there is an interesting coincidence, as for this size the topmost 1g shell is exactly half filled. Cu clusters have the electronic configuration and the appearance energy variations quite similar to Ag clusters, which has been worked on in gory details. We will analyze some of the simulated structures of Ag n to understand this interesting feature. Using modified dynamic lattice searching method (DLS) it has been found that small Ag clusters till Ag 48 acquire an amorphous structure, whereas from size 49 till 61 it is like an icosahedra and for larger clusters it attains a decahedral geometry [246]. The opening of the band gap at size 49 could be due to phase transition in the cluster geometry where the shape changes from amorphous to icosahedral like. Following this size we observe that the peak appearing at low energy strengthens and migrates to higher energy which is a similar effect as observed during binding energy increase for anionic clusters. The migration ceases at Cu 52 after a rise in appearance energy by ~ 0.3 eV (similar to the binding energy rise) and the peak strengthens. The deeper band which lies ~ 0.5 eV from the top gets pushed down and isolates from the topmost 127 peak. This is observed with the flattening of the yield plot towards high photon energy following the step. For Cu 55 [15,169] which is a perfect icosahedron the cluster has a strong peak in the DOS isolated from deeper band. On addition of more atoms the deeper band starts merging with this isolated peak. Cu 58 is a magic cluster and is consistent with the fact that we see a dip in the appearance energy or appearance of a disjoint peak for Cu 59 . We see another gap opening for Cu 61 , which quickly closes and for clusters bigger than Cu 63 no further shell opening has been observed. This marks the onset of metallic behavior for the clusters. For comparison we have plotted the band gap referred to the Kubo gap for all sizes in Figure 6.7. This conclusion for metallicity is consistent with the observation of plasmons in small ultracold Cu or Ag clusters inside a He droplet matrix [247,248] or in surface deposited Ag clusters [249]. Furthermore, the polarizability measurement [250] and closure of the HOMO-LUMO gap observed from numerical simulation around this size range also supports the metallic transition [251]. 6.2.2 Temperature dependence of appearance energy and density of states The measurement of temperature dependence of the vertical appearance energy for small clusters and also for bulk metals has always been challenging. While for small clusters controlling its temperature have been difficult, for bulk metals there have always been the problem of contamination incorporating inaccuracies in the measurement. Here we measured the photoionization yield at 60 K and 215 K that have been used to obtain the appearance energies and density of states for the clusters. The comprehensive table of data for measured appearance energies at all individual temperatures is provided in Appendix D. Almost no difference could be observed in the density of states distribution at these temperatures since; unlike alkalis Cu has a much stiffer lattice [231]. The temperature under investigation is much smaller than the Fermi temperature and also the pre-melting or melting point which is ≈1000 K [252–254]. Thus, the 128 observed drop of AE with temperature should be an electronic effect. For smaller clustersthe shifts are relatively larger and are plotted as a bar on the top of appearance energies inFigure 6.6. Averaging over all cluster sizes, the temperature coefficient <dφ/dT> is obtained to be ≈-4.9∙10 -5 eV/K. This is in good agreement with the theoretically derived value for bulk Cu,obtained by taking the thermal expansion and the atomic vibrations into account at room temperature. The best estimation for <dφ/dT> is found to be between -4∙10 -5 to -6∙10 -5 eV/K and is a perfect match with the experimental result [201,202,204]. Previous experiments of bulk work function have reported orders of magnitude temperature difference with the theoretical results measured at low temperature [255]. Since the measurement is carried out for small clusters in gas phase produced in high vacuum, it is free from any kind of contamination or oxidation effects that has been the major source of error with the experimental data for bulk work function. Based on the electrostatic model, the AE of the small clusters with a finite radius R exceeds the bulk work function by αe 2 /R. The value of α have been derived to be 3/8 [216] or ½ [217, 238,256] though experimentally it is found to lie within 0.2-0.4 for a wide range of elements [218].Starting from the accepted value for polycrystalline surface work function of 4.65 eV [203] and using α ≈ 0.2, we find the straight line following the spherical droplet model approximates the AE values for the small clusters quite consistently as is shown in Figure 6.9. The difference in the scaling factor relative to electrostatic model could be due to strong AE oscillations for the small clusters and also the periodic AE rise that is observed between consecutive magic numbers as is visible between Cu 58 and Cu 92 (beyond Cu 87 , see [238]) from the mass spectra at low energy plotted in Figure 6.7(a) and also from the AE variation in Figure 6.6. 129 Figure 6.9: IP with n -1/3 based on conducting droplet model. α≈0.2, WF≈4.65 eV (polycrystalline value) 6.3 Conclusion In this study we have discussed how the photoionization yield plot obtained for neutral Cu clusters can be used to derive the appearance energy of the clusters’ and as well as its density of states. The DOS observed by this simple method is in good agreement with the result from the PES previously obtained for anionic Cu clusters [231,232,235]. Simultaneous study of AE and DOS for the noble Cu clusters helps us understand the anomalous shell structure. However, since the appearance energies of the neutral clusters are much higher we could only observe the DOS within 1eV from the Fermi surface. It would be curious to explore the deeper bands using higher energy advanced light source. Alternatively the yield curve of the anionic clusters could also be monitored in order to explore the deeper bands with moderate energy UV photons (e.g. compare the PES of neutral and anionic Hg clusters [257]). In Cu 49 a sharp drop in the AE is observed associated with the appearance of a low energy peak and is a possible signature of geometric phase transition to icosahedral like structure [246]. The peak strengthens as atoms are added in 130 order to form perfect icosahedra at Cu 55 [15,169]. For higher sizes, we have demonstrated the gap closure referring to the Kubo criterion which marks the onset of metallicity in the Cu clusters. These observations are in much contrast to the monovalent alkalis where metallic property have been reported in clusters even with size less than 10. It would be an interesting experiment to study the plasmon resonances in size selected Cu clusters in order to observe the onset of collective oscillations for electrons. The temperature dependence of the DOS within 60 K to 215 K is almost indiscernible, though the appearance energy data could be used to measure the average temperature coefficient for the clusters. The data is in good agreement to the theoretical value obtained for bulk Cu at room temperature whereas previous direct measurement using photoemission from bulk surface produced considerable mismatch [201,204,208]. The electrostatic conducting droplet model [258,259, 260,261] has been used to determine the scaling factor using the appearance energies of the small clusters starting from the accepted polycrystalline bulk WF of Cu [204]. Thus, this can be conceived as an useful tool in order to study the temperature dependence of appearance energies and work function as the clusters in the gas phase produced under high vacuum can be directly probed and is free of any kind of contamination which masks the thermal effect. This method has previously been used for alkali nanoparticles which directly measures the temperature dependence of bulk WF [148]. This can thus be used to study the temperature dependence of several reactive metals for which surface oxidation has been a major disadvantage and could provide a new wealth of data for the temperature related studies. Furthermore, our study has been restricted towards intermediate temperature. It would be curious to explore the high temperatures close to the melting point where strong size dependence of the melting transition is observed [262], if the photoyield spectra would also experience some 131 noticeable variations. Also, ultralow temperatures would be curious to explore since the anomalous shell structures observed in the cold Cu clusters becomes most prominent at low temperature. 132 Chapter 7 Ionization of cold aluminium clusters: Electron impact vs multiphotoionization In the previous chapters we have extensively focused on the near and post threshold behavior of photoionization yield curves obtained via electron impact or photoionization on several clusters. Since our focus has been to explore the near-threshold behavior we optimized the photon flux in order to allow only single photon ionization. We used the electron impact ionization for studying the higher ionization states in silver clusters. For this purpose we had to use the electrons with energy in the order of tens of eV, but we restricted ourselves to single electron impact [256]. In this chapter we will present a comparative study of multiple photon absorption by cold clusters at ≈90 K as opposed to high energy electron impact. The neutral precursors have been found to respond quite differently to these two different modes of ionization. There have been dedicated experiments for cluster photoabsorption cross-section measurements [263]. We have found a simple technique where by finding the photon flux corresponding to the transition from single to multiple photon absorption we could deduce the photoabsorption cross section for the clusters of individual sizes.The result is in good agreement with the off resonance surface plasmon values calculated from the aluminum dielectric functions [264]. 7.1 Ionization modes In this Section we will analyze the difference in the mass spectra response as the aluminum clusters absorb multiple photons vs. a single electron impact. For direct comparison 133 the electron energy is adjusted to the value which is the cumulative sum of the total energy of the absorbed photons. 7.1.1 Photoionization at low and high fluence The cluster is ionized at a fixed wavelength of 220 nm emitted from the OPO laser with the fluence varying within the range 3∙10 2 – 1.5∙10 5 μJ/cm 2 . In Figure 7.1 we show that shift in the peak of the mass spectra with the increase in fluence. The peak in the abundance spectra at ~Al 500 for the single photon ionization regime, shifts heavily towards mass less than ~Al 10 as the fluence is increased to above 5∙10 4 μJ/cm 2 . The peak position is marked by an arrow in the plot. Figure 7.1Top: Singly photoionized spectra. No strong fragmentation Middle: Multiple photon absorption. Heavy fragmentation. Avalanche towards smaller clusters Bottom: Mass distribution shifts towards even smaller clusters In order to see how the cluster intensity changes for individual sizes we plot the intensity of a few size selected clusters, Al x (x=38,67,210,900) intensity with the laser flux density as 134 shown in Figure 7.2. An initial rise in intensity is noticed for all sizes before it starts to saturate which is a result of competition between the clusters of the size being focused fragmenting itself as opposed to the ones produced to fragmentation from bigger clusters. It can be written as, Al x * =Al x -Al <x +Al >x (7.1) where Al x * is the observed intensity, Al x is the cluster intensity with single photoionization and no contamination, Al <x are the number of parent clusters Al x that fragmented into smaller clusters and Al >x are the number of clusters Al x born from larger clusters as a result of fragmentation. Figure 7.2: Intensity of ionized clusters with photon energy flux density In order to get a clearer picture we have plotted the intensity of individual clusters with laser fluence. The peak shifts towards higher energy flux density with decrease in cluster size as is consistent with eq. (7.1). In order to analyze the feature of photofragmentation in these clusters as a result of multiphoton absorption it is necessary to build up a statistics to understand the absorption mechanism. Ideally the absorption of photons follows a Poisson distribution [265]. 135 One reason for a possible failure is due to the spatial variation of the mean number of photons across the beam.We have taken special care to keep the beam uniform so that the photon absorption follows the expected statistics in a similar way as discussed in Section 3.6.1.2. For low fluence the laser beam is kept wide and only the central part of the beam is used to ionize the clusters. The intensity is uniform across the central region, which is confirmed by checking the intensity at several parts of the beam by partially blocking the rest. For high fluence the laser beam was focused into a narrow region and properly attenuated to attain the required fluence. The photoabsorption followed by evaporation of the clusters due to warming up is explained in the following section. Figure 7.3: Arrows mark the point of inflexion in the cluster intensity with energy flux plot 136 7.1.2 Photoabsorption and Photofragmentation Following the Poisson statistics the probability of a cluster to pick up k photons is given by,   z k k e k z z P   ! (7.2) where z is the average number of collisions leading to the pick-up of k photons. z can be written as σΦ; σ is the cluster ionization cross section that is assumed to stay constant for all the k consecutive pick-up events and Φ is the laser fluence. In the multiphoton absorption process we sum over the events of all possible absorptions as,         1 1 k z k e z P (7.3) In Figure 7.4 the plots corresponding to different pick up events have been plotted individually. We see as the cluster ensemble picks up varying number of photons, k in an absorption event, the effective probability distribution tends to saturate. This analytical derivation is consistent with the results from Figure 7.2 where the saturation of smaller clusters at higher fluence is a manifestation of the their smaller ionization cross section. Now let us consider the effect of photofragmentation which plays a major role in understanding the intensity distribution for the varying size clusters. Maximal ion intensity is observed at higher fluences for smaller clusters as the evaporation chains of large clusters, extend towards smaller daughters as the fluence increases.We can represent the avalanche decay process as, 137 ... * 2 * 2 * 1 * 1                   h N N h N N h N Al Al Al Al Al Al Al (7.4) Figure 7.4: Theoretical Poisson distribution resembling pick up event of photons by clusters Thermal evaporation rate constants of free nanoclusters are derived using theory of detailed balance [13]. Here we will consider only monomer evaporation which the major decay channel as the probability to evaporate via monomer loss is orders of magnitude higher than that for the dimer evaporation. The rate of evaporation is given by,   m N b m N T k D m N m N e N E k ~ 3 2    (7.5) where   N m N E k is the rate constant for monomer evaporation from Al N cluster, m N D dissociation energy for an atom from Al surface, ω m can be approximated to be a constant and is a slowly varying function of cluster temperature.   C D E T T m N N N m N 2 ~   where T N (E N ) is cluster 138 temperature upon absorption of n photons of energy hν and initial temperature T 0 and heat capacity C.   C nh T E T N N    0 (7.6) Using eq. (7.5) we have estimated the monomer decay rate in Figure 7.5. Thus the lifetime, τ of a hot cluster can easily be obtained from this plot as τ=1/k. The cluster temperature has been obtained from eq. (7.6) as we show for two specific sizes, Al 67 and Al 210 . In Figure 7.6 we have plotted zone boundary based on the number of photons that a cluster of certain size should absorb in order to either start evaporating in the time scale of the laser pulse and alternatively when will the cluster not evaporate at all during the flight time. The as intermediate zone where the clusters will evaporate along the flight path. Thus when a beam of certain fluence excites an ensemble of clusters of all possible size the evaporative event occurs quite differently. This explains the varying peak positions as observed in Figure 7.3. Figure 7.5: Monomer decay rate for Al 67 (left) and Al 210 (right) 139 Figure 7.6: For the spanned space above the red line the evaporation triggers within the laser pulse width of ns and for the space below the blue line the evaporation triggers within beam flight time of ≈100 μs From these study we can claim that he peak in the intensity spectra for a cluster of certain particular size is a convolution of the cluster evaporation and photon absorption. Interestingly we never observed higher ionization state in these clusters. It has previously been reported that higher ionized states fragments could only be observed in fs laser pulse excitation [266]. 7.1.3 Electron impact ionization The electron density in the ionizer is low enough in order to unsure single collision events. However we have the privilege to increase the energy of the electrons higher than 100 eV. We find the cluster intensity to increase with electron energy as it is directly proportional to collision cross section as is shown in Figure 7.7. 140 Figure 7.7: Normalized cluster intensity with impact electron energy Prominent double and triple ionized peaks appears with increase in electron energy but no significant fragmentation with shift in mass spectrum is observed as we plotted in Figure 7.8. Figure 7.8: Mass spectrum of cluster with varying energy of impact electrons (all events are single electron absorption) 141 Full energy loss of one 90 eV electron would be equivalent to 16 photons with λ=220 nm. Sequential absorption of 16 photons would correspond to 1.6*10 4 µJ/cm 2 fluence for Al 67 and would show heavy fragmentation but not multiply-ionized peaks. Therefore electron ionization is more efficient at direct ionization energy transfer to the electron system. 7.2 Determination of photoabsorption cross section We have used the results from Figure 7.9 in conjugation with eq. (7.3) to estimate the extinction cross section of the cold aluminum clusters. We can see from eq. (7.3) that the onset of saturation can be marked by the relation, 1 ~   using which we determine σ~7.5 Ǻ 2 at 220 nm. The geometrical cross-section is σ geometrical ≈ 110 Ǻ 2 . Figure 7.9: Linearity of cluster intensity with energy flex of electrons marking single absorption event The linear region may be approximated as   1  z P k and after saturation it grows as   1  z P k The plot in Figure 7.10 shows the extinction cross-section per unit volume, C ext (sum of absorption and scattering cross sections) calculated in [264] on the basis of Drude 142 parameters for Al with size correction.The a=10 Ǻ curve corresponds to Al 250 and at 220 nm gives C ext /V=0.77x10 6 cm -1 . The extinction efficiency,Q ext is given by C ext /(πa 2 ) which we may assume to be constant for the clusters of similar size. Thus,   2 250 67 250 , 67 , a a C C ext ext  (7.7) Byproper scaling with the number of electrons in the cluster, we derive for Al 67 C ext ~ 8.5 Ǻ 2 .The Al 67 absorption cross section determined experimentally at hν=5.64 eVis therefore comparable to the value estimated from the Drude (plasmonic) extinction cross section. Figure 7.10: Theoretical plot of absorption cross-section from Drude theory for different size Al nanoparticles [264] 143 7.3 Conclusion In the low laser flux density regime indicating a single photon absorption process, the cold aluminum clusters intensity increases linearly for low photon flux which gradually saturates marking the onset of multiphoton absorption. By locating the saturation transition one can estimate the photoabsorption cross section. For Al N it was found to be in good agreement with extrapolation from Al Drude constants. In the high laser flux density regime with multiphoton absorption the ion intensity of the smaller clusters peaks at increasing laser fluences for decreasing cluster sizes. These ions derive from the intense evaporation chains of large clusters, which grow longer as the fluence increases.A secondary rising tail develops at larger fluence. Its origin is unclear at present. Electron impact ionization of cold Al N clusters with rising electron impact energy shows a signature of abundant production of multiply-charged clusters, but no significant increase in fragmentation. Electron impact transfers sufficient energy to the cluster electrons to cause multiple ionization, but does not cause massive heating and evaporation. Conversely, sequential absorption of photons from ns laser pulses heats up clusters to trigger heavy evaporation but the intervals between photon arrival are slow enough so as not to generate multiple ionization . 144 Chapter 8 Energetics of Quantum dots Advances in the field of nanotechnology have led to the development of nanoscale quantum dots (QD) in which electrons are spatially confined in all three dimensions, within scales ranging from a few up to several hundreds of nanometers. If the motion of the electrons is effectively restricted to a two-dimensional (2D) plane, as occurs, e.g., in semiconductor inversion layers and disks of electrons sandwiched within semiconductor pillars [267], this represents a 2D QD. This happens when the effective Bohr radius,  2 * 2 * e m a B   (8.1) (e<0 and m * are the electron charge and effective mass, and ε is the dielectric constant of the semiconductor material) is commensurate with the layer thickness. In this case we are dealing with strong transverse quantization and the system turns essentially into a flat 2D "artificial atom." For example, for InGaAs dots described in [267], the disk diameter is a few hundred nm, its thickness is about 10 nm, and nm a B 10 *  [268]. In the lateral (soft) direction the electron island is confined within a potential well V created by external electrostatic gates and/or by the sidewalls, and this potential can commonly be approximated as quadratic. In this respect the QD artificial atoms differ from real atoms, in which the electron cloud resides within the Coulomb potential of the nucleus rather than within a parabolic bowl. They are actually more reminiscent of J. J. Thompson's "plum pudding" picture. Another important distinction lies in the fact that the dot's potential does not need to be central: by a suitable design of the electrode or pillar shape, the confinement can be made 145 asymmetric. An important case is when the potential is elliptical, i.e., with different oscillator curvatures along is two principal axes. In understanding the behavior of QD electrons, quantities of interest include the shape and depth of the mean-field potential, the shape and size of the electron cloud, the total energy, the chemical potential, etc. Evaluating these quantities as a function of the number of electrons in the dot must be done in a self-consistent manner, accounting for the electron-electron interaction and screening. Of the vast number of calculations in the literature the majority employ numerical methods such as exact diagonalization for few-electron dots, and Hartree-Fock, density- functional or related treatments for larger ones. It is certainly worthwhile to develop a reliable analytical approach: this allows one to map the system's behavior over a large parameter space and to trace the interdependence between different parameters. This is the goal of our present work [269]. It is well known that a many-electron system within a smooth potential well can be accurately described by the semiclassical statistical approach [the Thomas-Fermi (TF) theory]. This treatment yields reliable values of physical quantities which are averaged over the quantum oscillations. It is a local approximation, stating that the maximum kinetic energy of the electron gas, treated semiclassically, cannot exceed the local depth of the potential well (as measured relative to chemical potential):            r V r e m r p F    * 2 2 (8.2) Here p F is the local value of the Fermi momentum, the chemical potential of the electron system is denoted byμ. (μ=0 for a neutral isolated atom, but not in general), and  is the electrostatic potential generated by the electron cloud. In 2D, n p F 2 2 2    wheren is the number of electrons 146 per unit area. We can also consider electrons confined within a 3D dot, for completeness, in which case   3 2 2 2 2 3 n p F    where nis the number of electrons per unit volume. An important observation made in [270] is that the relative magnitude of the quantum 2 F p (kinetic energy) term contains the factor, R a B * , where R is the radius of the dot's electron cloud. This means that for sufficiently extended two- or three-dimensional QD systems (cf. e.g., the InGaAs dots mentioned above), where the above factor is 1   , it can be adequate to retain only the following from eq. (8.2)         r V r e   (8.3) which can be recognized as the classical equation of force equilibrium. The electrostatic potential is given by     ' ' ' * r d r r r n e r D          (8.4) where   * e and D=2,3 for a 2D or 3D dot, respectively. Quantum deviations from eq. 8.3 show up only in the region near the dot edge, r~R, because here the density profile is changing rapidly and the kinetic energy term in the TF differential equation     0 ) ( 4 2 2 * 2 2              r V r en m r p F     [obtained from (8.2) by differentiating] becomes large. Treatments of the full TF equation including edge effects for such systems as axially symmetric 2D quantum dots, planar metallic surfaces, metal clusters and 3D parabolically confined electrons can be found, e.g., in [271, 272]. Here we make use of the classical limit to derive analytical formulas for those parameters of harmonically confined QD which are well represented by this formalism: the size and shape of 147 the electron cloud and the energetics of the system. In particular, we emphasize the case of asymmetric dots, for which a full analytical solution of the TF equation is unavailable, but a solution in the classical limit can be written out in closed form. Our focus, in Section (8.1), is on 2D systems: elliptically shaped transversely confined quantum dots. For completeness, in Section (8.2) we outline a similar analysis for 3D ellipsoidally confined QD. 8.1 Shape of 2D asymmetric quantum dot The electrons confined inside an elliptic quantum dot are assumed to be spanned over a space which is larger than the effective Bohrradius * B a . Thus the electrons behave asclassical particles and we can assume its local kinetic energy to bezero. Under this approximation we will find the number densitydistribution of the delocalized electron gas under the influence ofa confining asymmetric oscillator potential,   2 2 2 1 y x r V      (8.5) where 1  and 2  are the oscillator strengths along the x and y axes respectivelyand governs the curvature of the oscillator potential. We will write     1 and     2 such that the constant volume constraint 2 2 1     is satisfied for all deformations δ. Theelectrons at equilibrium under the influence of an externalpotential V and the interaction with other electrons in theneighborhood, will have the maximum total energy at any point givenby eq. (8.3) with V=0at the center of the dot. This condition alsoimplicitly assumes that the vertical ground state energy for theelectrons is zero. The number density of electrons for the 2D system is given by eq. (8.4) with D = 2. We assume   0  r n  outside theboundary of the ellipse whose semi-major and semi-minor radius 148 area andb respectively. The eq. (8.4) for D = 2 in cartesian co-ordinates with proper limits is given below,         a b dy dx r r y x n e y x 0 0 * ' ' ' ' , ' ,    (8.6) Fromthe condition of energy equilibrium the potential   y x,  canalso be written as,   2 2 2 1 * , y x y x e        (8.7) Using (9.6) and (9.7) we could express thenumber density of electrons in terms of μand the externallyapplied parameters   2 1 ,   as,   2 2 2 1 0 0 * ' ' ' ' , ' y x dy dx r r y x n e a b            (8.8) The solution to the above equation can be directly obtained byanalogy with the classical electrostatic solution for the 2D chargedistribution inside a flattened ellipsoid that produces an in planepotential to counterbalance the force from V [270,273]. This approach has been previously adapted to find theelectron distribution on the surface of bulk helium [274], following which we obtain the electronic distributioninside the dot can be written as,       2 2 2 2 ' ' 1 2 3 ' , ' y b x a ab F y x n     (8.9) whereF is the proportionality constant and {a,b} defines theouter boundary for the ellipse. We use the normalization condition     N dy dx y x n ' ' ' , ' , to obtain F=N (See Appendix E.1). Thus the number density distribution insidethe dot can be written as, 149       2 2 2 2 0 1 , y b x a n y x n    (8.10) where ab N n  2 3 0  . V(x,y) and the corresponding n(x,y) has been plotted in Figure 8.1 for δ=0.5and N=20, while all other parameters are taken from experimentallymeasured quantities [268,275]. Now we will connect derive the exact shape of the elliptic dot by solving for the parameters {a,b} in terms of the applied potential terms {μ,γ 1 ,γ 2 }. For this we again draw an analogy withthe electrostatic problem [273] where the potential inside aflattened ellipsoid has been obtained in two different ways to yieldthe following identity,                                                  2 2 0 2 2 2 2 0 0 2 2 2 2 ' ' 1 2 ' ' 1 ' ' ' b a d b y a x ab b y a x r r dy dx a b   (8.11) Comparing this relation with (8.8) we can write theparameters {μ,γ 1 ,γ 2 } as follows,                                        0 2 2 2 2 * 2 0 2 2 2 2 * 1 0 2 2 2 * 4 3 4 3 4 3                  b a b d e N b a a d e N b a d e N (8.12) These sets of equations can be solved as we will show in Appendix E.2 to yield the following sets of results, 150             3 3 2 * 2 2 3 2 * 1 1 2 * 8 3 8 3 4 3 F a e N F a e N F a e N          (8.13) where       2 1 2 1 ' ; 1 ; 2 1 , 2 1 k F F  ,       2 1 2 2 ' ; 2 ; 2 3 , 2 1 k F F  ,       2 1 2 3 ' ; 2 ; 2 3 , 2 3 k F F  and 1 2 F are the Gauss Hypergeometric functions [276]. k', is theeccentricity of the ellipse and is defined as k'=√(1-k 2 ); k(δ)=b/a. Using the ratio of the last two eqs. from (9.13) we obtain, ) 3 ( ) 2 ( 2 1 F F    (8.14) This equation can be solved to obtain the eccentricity of theellipse under the influence of the confining potential V. Onreplacing it in either of the last two eqs. (8.13) we can solve for the exact value of the a and hence b for anellipse which contains N electrons. Limiting case: For      2 1 (say) we reduce to the limiting case ofan axially symmetric dot where the radius of the quantum dotconfining N electrons is found to reduce to the correct limit     8 3 2 * e N as was also obtained byShikin et. al. [270]. We can call the ratio of the oscillator strength     2 1 (say) as the tuning parameter,where 1 0    . Using eq. (8.14), the variation of k with δ has been plotted in Figure 8.1. Along with the absolute value of a andb for an ellipse confined inside an InGaAs dot with N =20 and Ω 0 =3 meV has been plotted. The parameter value has been collected from the experimentally obtained parameters [268,275]. By varying δ we can form the quantum dot of any shape like a circular dot for δ=1or a 1D nanowire for δ→0 since we 151 have assumed 1  a b which implies 1 2    as the dot is more strongly confined along y- coordinate. Figure 8.1: Electron distribution inside an asymmetric QD. (Top left) Variation of ratio of semi-minor to semi-major axes, k(δ)=b/a with deformation δ.The red line acts as a guide to show that the curve k(δ) converges to 1 in the circular dotlimit i.e. for δ=1 where the external applied potential is symmetric. (Top right) The semi-minor and semi major axis for a quantum dot confined in InGaAs with N = 20; ω 0 = 3 meV. (Bottom left) The shape of the deformed external potential well V(x; y) for δ=0.5 (in units of J to be changed to meV ) (Bottom right) Normalized number density distributioninside the dot for δ=0.5 may be changed to exact number density units for InGaAs and N = 20 i.e. same as that for V(x,y) 8.1.1 Internal energy of the quantum dot The total internal energy of the electrons inside the dot is due toits interaction with the external potential and also with otherneighboring electrons inside the dot. Classically the energy of theelectrons is purely electrostatic as they are assumed to be at rest.The energy expression can be written as,          dxdy y x n y x e y x V dE p , , 2 1 , *    (8.15) 152 Substituting V,   y x e , *  from eq. (8.7) and theelectron density from eq. (8.10) into the above eq. we get,     dxdy b y a x n y x E p 2 2 2 2 0 2 2 2 1 1 2 1           (8.16) With the proper change of variables to polar coordinates       sin , cos r b y r a x   we reducethe integral to that over an unit circle,                 drd r b r a r r N E p      2 , 1 0 . 0 2 2 2 2 2 2 2 1 2 sin cos 1 4 3 (8.17) The results for the important integrals used to solve the above eq.are tabulated in Appendix E.3. The total energy onevaluating turns out to be,                    5 2 2 2 2 1 b a N E p    (8.18) The contributionto the total energy from the interaction with the externalpotential,   1 p E and that from electron-electron interaction,   2 p E are given by,                                                            3 2 2 1 6 1 3 2 2 1 3 1 4 * 5 2 ) 2 ( 3 2 2 6 1 3 2 2 1 3 1 4 * 5 2 ) 1 ( 5 1 2 8 9 5 2 8 9 F k F F F F e N E F k F F F e N E p p       (8.19) Using the virial theorem we have     1 2 2 p p E E  . Using thiswe can simplify the total energy to the following form, 153         ' 3 20 9 6 1 3 2 2 1 3 1 4 * 5 k K F F e N E p                     (8.20) K is the complete elliptic integrals of the first kind. Limiting case: For the symmetric situation we have      2 1 , a=b=R;k'=0. TheQD takes a purely circular shape with the total internal energy     3 1 5 4 * 2 9 3 . 0 N e E p    . From eq. (8.20) we can find that the total internal energy decreases as we increase the deformation in the quantum dot. The change in the total energy along with the contributing terms   1 p E and   2 p E with deformation of the clusters for a fixed number of electrons inside it are plotted in Figure 8.2. With increase in deformation we find that n 0 which is proportional to k -1 goes to ∞ at the center of the dot. What this essentially means is that all the electrons getsclumped at the center of the dot and the density of electrons decays very fast as we go away from the origin (for a 3D QD on the contrary the electron density always stays constant within the boundary [277]. So in the limiting case for k → 0, we hit the case for which we have a huge charge Ne * δ"(0) where δ" is the Dirac delta at theorigin. As we find from eq. (8.20) the classical energy of the dot reduces to zero, which implies that as the system gets confined to a one-dimensional nanowire the quantum size effects starts playing a significant role and cannot be correctly explained only within the realm of classical electrostatics. The symmetric distribution thus contains the maximum energy which decreases with shape deformation of the elliptic QD. In the large δ limit circular dot tunes to a nanowire which has theleast energy among all the different elliptic QDs but for correct analysis of the system with recent, modifications of the structures at the microscopic level hasenabled in the formation of multiple electron puddles 154 separated by abarrier rather than a single electrons pool in the sandwichedsemiconductor layer confining quantum dots [278,279]. This is also commonly known as multiple quantum dotmolecules [280]. We note that interestingly enough thetotal internal energy for this system of say nN electrons wherethe electrons are equally distributed in n quantum dots decreasesby a factor of n 2/3 . This implies that the multiplequantum dot systems are more stable to hold a certain number ofelectrons relative to electrons confined in a single dot with thesame confining potential. Figure 8.2: This Fig. might be separates in as most of it comes after the next section The internal energy, chemical potential, capacitive energy, ionization potentialand electron affinityof deformedInGaAs quantum dots with N = 20. (Top left)Total internal energy (E p ) along with the two contributing terms to the total energy: theinteraction energy with the external potential,   1 p ee E V  and the electron-electron interactionenergy   2 p ext E V  . For all δ,     1 2 2 p p E E  . (Top right) Chemical potential change with thedeformation parameter δ (Bottom left) Capacitive energy change with the deformationparameter δ (Bottom right) Ionization potential and the electron affinity of the deformedquantum dots plotted with the deformation parameter δ. An analogous plot showing the change in the energy with thevariation of number of electrons inside the dot, for dots withdifferent deformations are shown in Figure 8.3 155 Figure 8.3: Internal energy (top) and chemical potential (bottom) plotted with the numberof electrons in the quantum dot confined in InGaAs and ω 0 = 3meV. For δ'=1.5 andδ'=2.0 (δ'=1/δ) (legends to be changed accordingly). The internal energy plots (top) are offset vertically by 100meV and 200meV respectively and the chemical potential are offsetby 10meV and 20meV for clarity. 8.1.2 Physical properties of the quantum dot The chemical potential of a quantum dot with total number of electrons N and deformation δ can be defined as the difference in total energy between two dots with same deformationscontaining N and N-1electrons respectively,           , 1 , ,    N E N E N (8.21) The addition energy of an electron inside the dot (also known ascapacitive energy) with N electrons, can simply be obtained from the difference in the electro-chemical potential, µ as,              , , 1 , 2 * N N N C e    (8.22) 156 The variation of electrochemical potential µ and addition energywith deformations for a particular N, i.e. fixed number ofelectrons inside the dot is shown in Figure 8.2. Correspondingly, the variations of µ with N for different deformations are shown in Figure 8.3. We notice that the electrochemical potential and addition energy also decreases like the internal energy with the increase in deformation of the dot.Also interestingly, the charge density at the center of the dot which isproportional to k -1 increases with the deformation δ. Thus the change incharge density distribution becomes sharper along the short axis. The addition energy is of particular interest from experimental point of view. The addition energy for circular and rectangular quantum dots has been studied experimentally and modeled numerically using spin density functional theory (SDFT) by Austing et. al. [275,267,268]. Subsequent numerical study has also been performed by diagonalizing N- electron Hamiltonian for circular and ellipsoidal quantum dots [281,282]. Our analytical calculation shows an excellent match with the experimental and numerical calculations as shown in Figure 8.4 and Figure 8.5. Figure 8.4 presents a comparative study for the circular and ellipsoidal quantum dots with our analytical results. The experimental data along with LSDA calculation results presented here are for circular dots [268,275]. A separate set of resultsobtained by diagonalizing N- electron Hamiltonian for the dotswith varying shapes are also presented along with [281,282]. In Figure 8.5 we present the experimental data for rectangular quantum dots [268,275]. Our analytical results for elliptic dots shows close correspondence with the rectangular dots. The analogy is based on the fact that therectangular dots with rounded off corners can be approximated be ellipsoids where β, the ratio of the long to short edge length is compared with the ones with same δ which represents theratio of the axes length for an ellipse. For the system with multiple quantum dots, we have discussed above that the total energy of the dots decreases if the electrons 157 are distributed among multiple dots. But the addition energy of anelectron in the multi-dot system would be larger relative to asingle dot carrying the total numberof electrons. This is because the addition energy to the system of non-interacting puddle of electrons would correspond to that for each individual pool ofelectrons rather than that for the total number of electrons. So adding an electron to this system is more difficult. Figure 8.4: Capacitive energy plotted from the classically derived formula eq. (9.22) for several{ω x ,ω y }and confined in InGaAs quantum dot. The results are compared with the valuesobtained numerically by diagonalizing N-electron Hamiltonian [267,281]. For acircular quantum dot the results are obtained experimentally [279,283]. Also LSDA calculations data obtained for the same parameter set [268,283] is plotted along with Finally, we define the ionization potential I(N) and the electronaffinity A(N) of the dots and their connection with the capacitiveenergy as follows,                              , , , , 1 , , , , 1 , 2 * N A N I N C e N E N E N A N E N E N I         (8.23) 158 for the deformed elliptic QDobtained at tuning parameters δ. Both of them show a sharperdecay with initial deformation from circular geometry. With increase in deformation the change in µ becomes smoother. Figure 8.5: Capacitive energy for quantum dots in InGaAs are plotted with varying number of electrons and varying deformation parameter δ'(δ' corresponds to the deformation parameter from paper). The plots for µ corresponding to δ'=1, 1.375, 1.5, 2; δ=1/(δ') 2 areoffset vertically by 1,3,5 and 7 meV respectively for all corresponding sets of data. Theresults are compared with the experimentally obtained data for circular dot [267,279,281] and rectangular dot [268,283]. 8.1.3 Conclusion In summary, we have developed the complete classical solution forthe electronic distribution in two dimension. The fundamental properties like the internal energy and the electrochemical potential are derived for arbitrary deformations. Subsequently we obtained the results for addition energy of electrons, ionization potential and electron affinity for the elliptical dots. The results show excellent match with that obtained via experiments or numerical simulations. The interesting gradual decrease in the internal energy, electrochemical potential and the addition energy with the increase in deformation of the dot is studied and explained. 159 The techniques developed can be potentially used to study the intrinsic properties inside highly deformed dots ( i.e. for k→0) which forms quasi-one dimensional electron gas also known as quantum wires [284]. Consequently we could study the spatial distributionof particles with a mixture of different charge to mass ratio for different configurations which have only been studied numerically. Another interesting study would be to understand thedistribution of electrons inside the a polygonal quantum dot [285]. 8.2 Shape of asymmetric 3D ellipsoidal quantum dot The electrons inside a three dimensional (3D) quantum dot areconfined by a 3D harmonic oscillator potential which can bemodeled as,   2 3 2 2 2 1 z y x r V        (8.24) where {γ 1 ,γ 2 ,γ 3 } are the oscillator strengths along the {x,y,z} axes respectively and governs the curvature of the oscillator potential. The dimensionof the dot is assumed to be larger than the effective Bohr radius * B a . With this assumption the electrons behave as classical particles and under the influence of such an external potential distribute inside the outer boundary of the dot with a constant density n and vanishes outside the boundary. The result is confirmed by the fact that the 3D self consistent potential due to the constant charge distribution balances the force due to the oscillator potential [273]. The electrons which behave as classical particles can be assumed to have zero local kinetic energy. The electrons at hydrostatic equilibrium under the influence of an external potential V and the interaction with other electrons in the neighborhood, will have the maximum total energy at any point given by eq. (8.3), where r  is now the three dimensional radial vector with V=0 at 160 the center of the dot. The self consistent potential inside the outer boundary of the ellipsoidal dot can begiven as,                                   2 2 2 0 2 2 2 2 2 2 * 1 4 3 , , c b a d c z b y a x Ne z y x (8.25) where N is the total number of electrons within the dot. We will assume azimuthal symmetry in the problem such that γ 1 =γ 2 =γand a=b=R. Under this assumption theabove expression reduces to,                             2 2 0 2 2 2 2 2 * 1 4 3 , , c R d c z R y x Ne z y x (8.26) Expressing   z y x , ,  in terms of µ and the external parameters, from (8.3) and (8.24) we can write(8.25) as,         2 3 2 2 2 2 0 2 2 2 2 2 2 * 1 4 3 z y x c R d c z R y x e N                            (8.27) Comparing the two sides from the above expression (8.27) we can write the parameters {µ,γ,γ 3 } as follows,                                  0 3 2 2 2 * 3 0 2 2 2 2 * 0 2 2 2 * 4 3 4 3 4 3             c R d e N c R d e N c R d e N (8.28) 161 These sets of equations can be solved as we will show in Appendix (E.4) to yield the following sets of results,             3 3 2 * 3 2 3 2 * 1 2 * 2 2 2 3 F R e N F R e N F R e N       (8.29) where       2 1 2 1 ' ; 2 3 ; 2 1 , 2 1 k F F  ,       2 1 2 2 ' ; 2 5 ; 2 3 , 2 1 k F F  ,       2 1 2 3 ' ; 2 5 ; 2 3 , 2 3 k F F  and 1 2 F are theGauss Hypergeometric functions [276]. k', is the eccentricity of the ellipse (k'=√(1-k 2 );k(δ)=b/a) and we assumed R>c. Using the ratio of the last two eqs. from(8.29) we obtain, ) 3 ( ) 2 ( 3 F F    (8.30) By assumption the electrons are less strongly bound in the horizontal plane relative to the vertical direction and hence γ=γ 3 . This follows from the fact that R>c, implying that the electrons are confined more strongly along the vertical direction. We will define 3     as the ratio between the confining potential along the horizontal and vertical direction which will vary within the range 0<δ<1. The shape of the spheroid (governed by its' eccentricity k') is solely defined by the parameter δ irrespective of the actual number of electrons confined within the dot. This was also shown tobe the case for planar pancake like 2D quantum dots [271]. The dependence of k' on the parameter δ is shown in Figure 8.6. Subsequently, we find the change of the long andshort axes i.e. R and c respectively with δ for the electrons confined in the 3D potential within GaAs semiconductor. For this in order to keep the volume of the dot constant weimplement the 162 condition γ 2 γ 3 =1. Thus 3 2 0 3 1 0 ,          , and the corresponding semi-long and short axes, R and c respectively are plotted in Figure 8.6. Figure 8.6: (Left) Variation of ratio of semi-minor to semi-major axes, k = c/R with δ. The red line acts as a guide to show that the curve k(δ) converges to 1 for the symmetric spherical dot where δ = 1 and the external applied potential is symmetric. (Right) The semi minor and semi major axes, corresponding to the vertical and horizontal direction respectively for a quantum dot confined in GaAs with N = 100; ω 0 = 30meV 8.2.1 Internal energy of the quantum dot The total internal energy of the electrons inside the dot is due to its interaction with the external potential and also with other neighboring electrons inside the dot. Classically the energy of the electrons is purely electrostatic as they are assumed to be at rest. The energy expression can be written as,          dxdydz z y x n z y x e z y x V dE p , , , , 2 1 , , *    (8.31) 163 Substituting   z y x e V , , , *  (from eq. (8.3) and (8.24)) for the constant electron density distribution inside the dot into the above eq. we get,       ndxdy z y x E p      2 3 2 2 2 1    (8.32) where the integrated volume is over a spheroid,   1 2 2 2 2 2    c z R y x . With proper change invariables we reduce the integral to one over a sphere 1  r  as is detailed in the Appendix (E.5). The total energy for the spheroidal quantum dot turns out to be,                    5 2 2 2 3 2 c R N E p    (8.33) The contribution to the total energy from the interaction with the external potential,   1 p E and that from electron-electron interaction,   2 p E are given by,                                                            3 2 2 1 6 1 3 2 3 3 1 4 * 5 ) 2 ( 3 2 2 6 1 3 2 3 3 1 4 * 5 ) 1 ( 2 5 1 3 32 2 5 2 32 F k F F F F e N E F k F F F e N E p p     (8.34) Using the virial theorem we have     1 2 2 p p E E  . Thus, we can simplify the total energy to the following form,         1 6 1 3 2 3 3 1 4 * 5 4 4 9 F F F e N E p                    (8.35) 164 For the 3D isotropic quantum dot the total energy reduces to the correct limit     3 1 5 4 * 2 9 . 0 N e E p   . This is similar to what we have for an uniform distribution of electrons inside a sphere with a constant positively charged background given as R Q 10 9 2 , where Q is the total charge and R is the radius of the dot. The total energy of the 3D quantum dot is higher than a 2D dot, confining same number of electrons and held by the same confining potential by a factor of   3 1 2 6  . The total energy E p and the individual contributing terms   1 p E and   2 p E are plotted in Figure 8.7 with variation in the deformation of the dot for a fixed number of electrons confined within the dot. Figure 8.7: The internal energy, chemical potential, capacitive energy, ionization potential and electron affinity of deformed quantum dots confined in GaAs and N = 100; ω 0 = 30meV (Top left) Total internal energy (E p ) along with the two contributing terms to the total energy: the interaction energy with the external potential (E p (1) ) and the electron electron interaction energy (E p (2) ). For all δ, E p (2) = 2 E p (1) . (Top right) Chemical potential change with the deformationparameter δ (Bottom left) Capacitive energy change with the deformation parameter δ (Bottom right) Ionization potential and the electron affinity of the deformed quantum dots plotted with the deformation parameter δ. 165 The energy for the symmetric 3d dot is plotted in Figure 8.8 and is compared with the results obtained numerically by relativistic coupled cluster method (FSCCSD) [286]. Figure 8.8: Internal energy calculated classically for a three dimensional symmetric quantum dot confined in GaAs semiconductor. The relativistic coupled cluster method (FSCCSD) [286] used to derive the energy for these 3D dots has been added for comparison. The classical result can be extended to large QDs for deriving the internal energy of the dots. The parameters are taken as the same as that used in the paper for GaAs. The results show an excellent match which improves with the increase in size of the dot i.e. as the system becomes more classical. We prove this by plotting the relative difference between the internal energy calculated by the two afore mentioned methods in Figure 8.9. The difference reduces to less than 5% for N=10 and <3% for N=40. Simultaneously in Figure 8.10 we compare the results obtained for a deformed quantum dot. 166 Figure 8.9: The relative difference between the energy calculated using FSCCSD [286] and the Classical energy calculation. The difference reduces considerably as the number of electrons confined inside the dot increases and the system becomes more classical. In the classical regime the contribution from the kinetic energy becomes insignificant Here the confining potential along the vertical axis is kept constant and that in the horizontal plane is gradually relaxed. We find that as the system is more loosely confined, the total internal energy ofthe system decreases. This is because the external potential wellbecomes flatter and also the electrons are less loosely bound and are distributed over a larger space. Interestingly, enough comparison of our results with that obtained via spin density functional theory [287,288] shows a considerable mismatch. We also estimated the differences for various N of each deformed dot and plotted in Figure 8.11. We find that the difference reduces with increase in size of the dot, but is relatively much larger compared with a symmetric dot of the same size. The reason for the difference is primarily due to the kinetic energy contribution tothe dot. We find that the discrepancy increases with the 167 deformation of the dot. Within our assumption we see that as the deformation increases the spheroid bulges out and the free electrons are more relaxed due to the weak confinement, hence increasing the average kinetic energy. This also suggests with the increase in deformations, the quantum confinement effect becomes more important for estimationof the correct internal energy for the system. Figure 8.10: Internal energy of the deformed three dimensional quantum dot. The dot isdeformed anisotropically by relaxing the oscillator strength in the xy horizontal plane.As the oscillator is relaxed the total energy of the dot is reduced. The energy is obtainednumerically using spin DFT [287] and shows a considerable mismatch with the classicalresult. The mismatch is due to the kinetic energy contribution that becomes more dominantfor a deformed quantum dot relative to isotropic 3d quantum dot and increases with thedeformation 8.2.2 Physical properties of the quantum dot Chemical potential is an intrinsic property of a physical system and is defined as the rate of change in its internal energy with the number of particles. For a 3D quantum dot with 168 arbitrary deformation δ and containing total number of electrons N can be defined as [287,289], and defined by the same eq. as for a 2D as eq. (8.21). The difference in the chemical potential of quantum dots with N+1 and N electrons gives the addition energy of an electron inside the dot (also known as capacitive energy) with N electrons [287,289] as shown in eq. (8.22). Figure 8.11: The relative difference in the internal energy between the classical and spinDFT results decreases for large N, but is much higher than that obtained for an isotropicQD as shown in 4. Among the three different cases studied you may also note that the relativemismatch is largest for the most deformed dot, which is due to increased contributionfrom the kinetic energy [287] The chemical potential, and the capacitive energy for a 3D quantum dot with varying deformation δ is plotted in Figure. 8.7. Here we have chosen N=100 electrons inside a GaAs semiconductor layer in order to ensure that electrons behave classically even in the deformed states. 169 Figure 8.12 Capacitive energy (commonly known as addition energy) is plotted for a symmetric three dimensional quantum dot for different strength of confining potential where     4 3 * 0 2 * 2 legend a e    . The classical results are compared with that obtained using Hartree-Fock approximation [290]. Several numerical methods have been established in order to studythese properties of quantum dots, which we will like to compare withour analytic classical results. The change in addition energy with increase in size of the quantum dot is plotted in Figure 8.12 and Figure 8.13 for spherical and deformed quantum dots respectively. We notice that with the increase in confining potential strength, γ, theaddition energy reduces. So the electrons are easily sucked into the deeper oscillator potential. The addition energy converges to almost a fixed value in the classical limit. The results are compared with that obtained by Hartree-Fock approximation in Figure 8.12 for a symmetric dot [290] and in Figure 8.13 by spin DFT for a deformeddot [287]. We may note that the addition addition energy show an excellent match in both cases. This implies that the addition energy of the electrons inside a dot is independent of the quantum confinement effect which is otherwise important for determination of the internal energy of small deformed dots. This correspondence between classical and numerically obtained quantal 170 solutions for addition energy of the dots was also observed for a 2D quantum dot and in other similar studies [288].We will now discuss two other experimental observable along with the addition energy. They are ionization potential I(N) and the electron affinity A(N) of the dots. These observables areimplicitly connected with the addition energy of the dots as is shown in eq. (8.23). These are plotted in Figure 8.7 along with the internalenergy and addition energy for the same sized quantum dots. Figure 8.13 Capacitive energy for the deformed quantum dots. The results from the classical and spin DFT calculations [287] are plotted. Note that even though the internal energies showed a considerable mismatch for these small size dots (see Figure 8.11 and Figure 8.12) there is no effect on the addition energy estimates showing that it is not strongly dependent on the kinetic energy contribution of the total internal energy 8.3 Conclusion The shape of the 3D quantum dot obtained by placing the electrons inside a general anisotropic harmonic oscillator potential has been derived. The distribution is independent of the actual size of thedot and is only a function of the relative strength of the confining potential 171 along the horizontal and vertical directions. The total internal energy of the dot is derived for a general spheroidal quantum dot. We see that classically the energy contribution from the interaction with electrons is twice that ofthe contribution from the interaction with the external potentialfor all deformations. We found that the total energy calculated for the isotropic quantum dot is in excellent match with the results obtained by four-component relativistic coupled cluster method [286]. However, for the deformed quantum dots the totalenergy obtained by spin DFT [287] is higher than the classical result. This implies that the contribution to the total energy from the kinetic energy for small dots is larger with the increase in deformation. Interestingly, even though the total internal energy of the deformed dots has noticeable contribution from the kinetic energy, the addition energies for both isotropic and spheroidal quantum dots arein very good agreement with the relativistic coupled cluster method/ Hartree- Fock approximation [290] and spin DFTrespectively [287]. From this we can say that whatever, be the contribution to the total energy from the energy subparts, the total internal energy for a system can be well approximation from theclassically obtained internal energy of any sizes. 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B 84, 075131 (2011). 187 Appendix Appendix A Mesh attachment to the TOF plate We have used a fine Cu mesh of 1" x 1" dimension with wire width of 19.8μm. This ensures an open area of 85%. This lets most of the cluster beam pass through the TOF plates and the holes never get clogged (since there are two following mesh plate the effective transmission is ~73%). Some dry air is blown on the mesh after several runs to take off some little deposition that might develop on the mesh over time. The square wire grid is used to cover the TOF plate hole of 9mm diameter. The most critical part here is to maintain a perfectly flat surface without any wrinkles as that introduces fringe fields in the region. We developed a "double ring method" for having maximal control during mesh bonding. The picture for a mesh bonded TOF plate is provided in Fig.XX. The steps towards bonding is described below: 1. We used the silver-filled conductive adhesive from McMaster-Carr (Product # 7661A11) and mixed the resin and hardener into a nice jelly. The silver colored jelly was spread uniformly inside the inner ring such that it forms a little hump around the ring. 2. We place the mesh symmetrically around the central hole and this makes a nice contact between the silver jelly and the mesh. A rubber o-ring is then placed on the outer ring and covered with an aluminum block. This applies a mild pressure on the outer part of the mesh which enables its central region which goes over the hole to stay perfectly flat. A little touch of 188 the jelly is applied towards the outer edge of the mesh such that they stay in contact with the plate and does not sag. 3. The whole assembly is then placed on a heating plate at ~50-70 C temperature and left overnight for curing. The following day the o-ring is removed and the TOF plate with bonded mesh is ready to use. 189 Appendix B: Algorithm for data analysis for plotting the Photoyield curves. This is a sketch of the programs that has been used for analysis: Step 1: The following code finds the time of flight of different clusters from the accelerating region to the detector: clc clear all close all Startin=27; %Starting size of clusters-1 Endin=114; %End size of clusters+1 Note: The maximum number of clusters that can be fitted is 8 with Gaussians but we could contrain to 4 or 5 %for increasing accuracy % Cluster properties m_Al=26.982*(10^(-3)); N_Av=6.023*(10^23); %Mass for the cluster range of interest Start_clustersize=Startin; %Cluster size of the smallest peak fitted -1 End_clustersize=Endin; %Cluster size of the largest peak fitted +1 N=[Start_clustersize:1:End_clustersize];%The time range should correspond to this set of cluster size, found from the previous fitting program m=m_Al/N_Av*N; t_67=84.5*(10^(-6)); t_87=96.1*(10^(-6));%84.6*(10^(-6)); t_47=70.8*(10^(-6));%78*(10^(-6)); i=find(N==67); k=(t_87-t_47)/(sqrt(m(1,i+20))-sqrt(m(1,i-20))); del_T=k*(sqrt(m(1,i))-sqrt(m)); T=(t_67-del_T)*(10^6); save ('TOF_73K_Mar06_2014.mat', 'T'); % If using the Lasing delay it needs to be found using Dr. Anthony Liang's TOF optimization program (may be provided on request) % T=Lasing_delay+sqrt(2*m).*(((u1-sqrt(u0))./(q*Es))+((u2-u1)./(q*Ed))+L./(2*u2)); % T=10^6*T; % % t_67=t_d+L*(m_67/(2*Vs*Ls/Ls_ion+2*Vd) 190 Step 2: The following matlab code finds the raw intensity contribution at the normalizing wavelength 216 nm, which was repeated after every individual wavelengths for Al 66 , similar analysis can be performed for any arbitrary size. clc clear all close all %load('TOF_73K_Mar06_2014.mat'); % This is used to load the TOF for all the clusters in the range T=[81.3 81.9 82.6 83.2 83.8 84.5 85.1 85.7 86.3]; Startin=62; %Starting size of clusters-1 Endin=70; %End size of clusters+1 Note: The maximum number of clusters that can be fitted is 8 with Gaussians but we could contrain to 4 or 5 %for increasing accuracy TOF_range_min=54.5; %87.0; % %Here we enter the minimum time range for the the cluster sizes to be fitted TOF_range_max=109.9; %89.4; % % %Here we enter the maximum time range for the the cluster sizes to be fitted Numberoffittinggaussians=5; %The number of Gaussians that is to be fitted Note: Max allowable is 8 but may be restrict to 4 or 5 or %check for accuracy. There should be no %contribution to any of the Gaussians except %from one o fthe side peaks. Norm_wave=216; %Here write the normalizing wavelength N_data_normwave=21; %Here write the total number of normalization data points Min_channel=545; %Minimum channel number where we want to calculate the norm data from Max_channel=1099; %Maximum channel number up to where we want to calculate the norm data Nofchannels=9999; %Enter here the total number of channels over which the data is collected Strength_min=0.001; % The min strength of the average data values per point to fit a gaussian %Loading data Normalizing_wavelength=Norm_wave; %Here write the normalizing wavelength Numberofdata_Norm=N_data_normwave; %Here write the total number of normalization data points %Here the normalization data is getting loaded and Norm evaluated Norm=zeros(1,Numberofdata_Norm); x_norm=zeros(Numberofdata_Norm,Nofchannels); x1=zeros(Numberofdata_Norm,Nofchannels); for i=1:Numberofdata_Norm if i<10 str=strcat(['Al_0',num2str(i),'_',num2str(Normalizing_wavelength),'.Asc']); load(str); str_norm=strcat(['Al_0',num2str(i),'_',num2str(Normalizing_wavelength),'(:,2)']); x_norm(i,:)=eval(str_norm); str_norm1=strcat(['Al_0',num2str(i),'_',num2str(Normalizing_wavelength),'(:,1)']); x1(i,:)=eval(str_norm1); else str=strcat(['Al_',num2str(i),'_',num2str(Normalizing_wavelength),'.Asc']); load(str); str_norm=strcat(['Al_',num2str(i),'_',num2str(Normalizing_wavelength),'(:,2)']); x_norm(i,:)=eval(str_norm); 191 str_norm1=strcat(['Al_',num2str(i),'_',num2str(Normalizing_wavelength),'(:,1)']); x1(i,:)=eval(str_norm1); end end Norm_data=zeros(Numberofdata_Norm,length(1:Endin-Startin-5)); %-1-4 is done in order to find the I_fitrange correctly coeffvals1=zeros(length(1:Endin-Startin-5),Numberofdata_Norm,3); for i_range=1:Endin-Startin-5 I_fitrange=0; I_fitrange=find(x1(1,:)>(T(i_range)+T(i_range+1))/2 & x1(1,:)<(T(i_range+5)+T(i_range+6))/2); TOF_range_min=x1(1,min(I_fitrange)); TOF_range_max=x1(1,max(I_fitrange)); TOF_range_min_channels=10*TOF_range_min+1; TOF_range_max_channels=10*TOF_range_max+1; %x1_fit=0; x1_fit=[TOF_range_min:0.1:TOF_range_max]; %Time range of the TOF spectra to be fitted collected by MCS %I_fit=zeros(Numberofdata_Norm,length(x1_fit)); x2_fit=zeros(Numberofdata_Norm,length(x1_fit)); I_fit=find(x1(1,:)>TOF_range_min-0.02 & x1(1,:)<TOF_range_max+0.02); for i=1:Numberofdata_Norm x2_fit(i,:)=x_norm(i,I_fit(1):1:I_fit(1)+length(I_fit)-1)-sum(x_norm(i,9900:9999))/(9999-9900); x2_fit(i,:)=(x2_fit(i,:)+abs(x2_fit(i,:)))/2; end %Here we transpose the matrix since the cftool can only fit the column %matrices and also initialize IP data calculation x1_fit=x1_fit'; x2_fit=x2_fit'; %Here I calculate the starting parameters for Gaussian start_at=zeros(Numberofdata_Norm,3*Numberoffittinggaussians); %Here we find the peak positions Amp=zeros(Numberofdata_Norm,Numberoffittinggaussians); for i=1:Numberofdata_Norm count_amp=0; for j=i_range+1:i_range+Numberoffittinggaussians count_amp=count_amp+1; %I_fit=0; I_fit = find(x1_fit>(T(j-1)+T(j))/2 & x1_fit<(T(j)+T(j+1))/2); %x2_var=0; x2_var=x2_fit(I_fit(1):1:I_fit(1)+length(I_fit)-1,i); Amp(i,count_amp)=max(x2_var); end end for i=1:Numberofdata_Norm 192 for j=1:Numberoffittinggaussians start_at(i,j)=Amp(i,j)*0.75; %for a cefficient start_at(i,j+5)=T(i_range+j); %for b cefficient start_at(i,j+10)=0.2; %for c cefficient end end coeffvals=zeros(Numberofdata_Norm,3*Numberoffittinggaussians); goodness=zeros(Numberofdata_Norm,5); strength_check=zeros(Numberofdata_Norm,1); for i=1:Numberofdata_Norm str_fittype=strcat(['a1*exp(-((x-b1)/c1)^2) + a2*exp(-((x-b2)/c2)^2) + a3*exp(-((x-b3)/c3)^2) + a4*exp(-((x- b4)/c4)^2) + a5*exp(-((x-b5)/c5)^2)']); f=fittype(str_fittype); options=fitoptions(str_fittype); strength_check(i,1)=sum(x2_fit(:,i))/length(x1_fit); if strength_check(i,1)>Strength_min options.Startpoint=start_at(i,:); options.Lower=[0.1*ones(1,Numberoffittinggaussians) start_at(i,6:10)-0.2 0.1*ones(1,Numberoffittinggaussians)]; options.Upper=[Inf(1,Numberoffittinggaussians) start_at(i,6:10)+0.2 0.4*ones(1,Numberoffittinggaussians)]; [fitobject,gof] = fit(x1_fit,x2_fit(:,i),f,options); %[fitobject,gof] = fit(x1_fit,x2_fit(:,i),f); goodness(i,:)=[gof.sse gof.rsquare gof.dfe gof.adjrsquare gof.rmse]; f1=fit(x1_fit,x2_fit(:,i),f,options); %figure;plot(f1,x1_fit,x2_fit(:,i)); coeffvals(i,:) = coeffvalues(f1); else goodness(i,:)=[0 0 0 0 0]; coeffvals(i,:) = Strength_min*ones(1,3*Numberoffittinggaussians); end %pause a=coeffvals(i,3); b=coeffvals(i,8); c=coeffvals(i,13); coeffvals1(i_range,i,1) = a; coeffvals1(i_range,i,2) = b; coeffvals1(i_range,i,3) = c; end Norm_data(:,i_range)=coeffvals(:,3).*coeffvals(:,3+10)*sqrt(pi); %Since I am fitting 5 gaussians and taking the mid gaussian value %pause end save ('Norm_fullrange1_gauss5_73K_Mar06_2014_Al66.mat', 'Norm_data', 'Norm_wave', 'coeffvals', 'goodness'); 193 Step 3: The following matlab code finds the raw intensity contribution at every wavelength where the data is collected for Al 66 , similar analysis can be performed for any arbitrary size. clc clear all close all %load('TOF_73K_Mar06_2014.mat'); T=[81.9 82.6 83.2 83.8 84.5 85.1 85.7]; %Input parameters (this code assumes there is no data taken at normalization wavelength, if taken add a section from the previous code and % some other related changes) Startin=63; %Starting size of clusters-1 Endin=69; %End size of clusters+1 Note: The maximum number of clusters that can be fitted is 8 with Gaussians but we could contrain to 4 or 5 %for increasing accuracy Data_wave=[210 220 212 214 211 213 228 215 232 225 236 219 224 230 217 227 222 231 223 238 233 218 234 221 237 243 245 247 241 226 229 240 244 235 239]; %Here enter the complete data set taken for IP measurement except the data at normalization wavelength TOF_range_min=54.5; %Here we enter the minimum time range for the the cluster sizes to be fitted TOF_range_max=109.9; %Here we enter the maximum time range for the the cluster sizes to be fitted Numberoffittinggaussians=5; %The number of Gaussians that is to be fitted Note: Max allowable is 8 but may be restrict to 4 or 5 or %check for accuracy. There should be no %contribution to any of the Gaussians except %from one o fthe side peaks. Nofchannels=9999; %Enter here the total number of channels over which the data is collected Strength_min=0.001; % The min strength of the average data values per point to fit a gaussian %Loading data Data_set=Data_wave; %The complete data set taken for IP measurement except the data at normalization wavelength Numberofdata=length(Data_set); %Here the data set is getting loaded for i=1:length(Data_set) str=strcat(['Al_01_',num2str(Data_set(i)),'.Asc']); load(str); end x1=zeros(Numberofdata,Nofchannels); %The data collection period x2=zeros(Numberofdata,Nofchannels); %The data collection period for i=1:length(Data_set) str1=strcat(['Al_01_',num2str(Data_set(i)),'(:,1)']); str2=strcat(['Al_01_',num2str(Data_set(i)),'(:,2)']); x1(i,:)=eval(str1); x2(i,:)=eval(str2); end %Here we select the range for fitting in microsec 194 IP_data=zeros(Numberofdata,length(1:Endin-Startin-1-4)); coeffvals1=zeros(length(1:Endin-Startin-1-4),Numberofdata,3); for i_range=1:Endin-Startin-5 %I_fitrange=0; I_fitrange=find(x1(1,:)>(T(i_range)+T(i_range+1))/2 & x1(1,:)<(T(i_range+5)+T(i_range+6))/2); TOF_range_min=x1(1,min(I_fitrange)); TOF_range_max=x1(1,max(I_fitrange)); TOF_range_min_channels=10*TOF_range_min+1; TOF_range_max_channels=10*TOF_range_max+1; %x1_fit=0; x1_fit=[TOF_range_min:0.1:TOF_range_max]; %Time range of the TOF spectra to be fitted collected by MCS %I_fit=zeros(Numberofdata,length(x1_fit)); x2_fit=zeros(Numberofdata,length(x1_fit)); I_fit=find(x1(1,:)>TOF_range_min-0.02 & x1(1,:)<TOF_range_max+0.02); for i=1:Numberofdata x2_fit(i,:)=x2(i,I_fit(1):1:I_fit(1)+length(I_fit)-1)-sum(x2(i,9900:9999))/(9999-9900); x2_fit(i,:)=(x2_fit(i,:)+abs(x2_fit(i,:)))/2; end %Here we transpose the matrix since the cftool can only fit the column %matrices and also initialize IP data calculation % IP_data=zeros(Numberofdata,Numberoffittinggaussians); x1_fit=x1_fit'; x2_fit=x2_fit'; %Here I calculate the starting parameters for Gaussian start_at=zeros(Numberofdata,3*Numberoffittinggaussians); %Here we find the peak positions Amp=zeros(Numberofdata,Numberoffittinggaussians); for i=1:Numberofdata count_amp=0; for j=i_range+1:i_range+Numberoffittinggaussians count_amp=count_amp+1; %I_fit=0; I_fit = find(x1_fit>(T(j-1)+T(j))/2 & x1_fit<(T(j)+T(j+1))/2); %x2_var=0; x2_var=x2_fit(I_fit(1):1:I_fit(1)+length(I_fit)-1,i); Amp(i,count_amp)=max(x2_var); end end for i=1:Numberofdata for j=1:Numberoffittinggaussians start_at(i,j)=Amp(i,j)*0.75; start_at(i,j+5)=T(i_range+j); start_at(i,j+10)=0.2; 195 end end coeffvals=zeros(Numberofdata,3*Numberoffittinggaussians); goodness=zeros(Numberofdata,5); strength_check=zeros(Numberofdata,1); for i=1:Numberofdata str_fittype=strcat(['a1*exp(-((x-b1)/c1)^2) + a2*exp(-((x-b2)/c2)^2) + a3*exp(-((x-b3)/c3)^2) + a4*exp(-((x- b4)/c4)^2) + a5*exp(-((x-b5)/c5)^2)']); f=fittype(str_fittype); options=fitoptions(str_fittype); strength_check(i,1)=sum(x2_fit(:,i))/length(x1_fit); if strength_check(i,1)>Strength_min options.Startpoint=start_at(i,:); options.Lower=[0.1*ones(1,Numberoffittinggaussians) start_at(i,6:10)-0.2 0.1*ones(1,Numberoffittinggaussians)]; options.Upper=[Inf(1,Numberoffittinggaussians) start_at(i,6:10)+0.2 0.4*ones(1,Numberoffittinggaussians)]; [fitobject,gof] = fit(x1_fit,x2_fit(:,i),f,options); %[fitobject,gof] = fit(x1_fit,x2_fit(:,i),f); goodness(i,:)=[gof.sse gof.rsquare gof.dfe gof.adjrsquare gof.rmse]; f1=fit(x1_fit,x2_fit(:,i),f,options); %figure;plot(f1,x1_fit,x2_fit(:,i)); %pause; coeffvals(i,:) = coeffvalues(f1); else goodness(i,:)=[0 0 0 0 0]; coeffvals(i,:) = Strength_min*ones(1,3*Numberoffittinggaussians); end %pause a=coeffvals(i,3); b=coeffvals(i,8); c=coeffvals(i,13); coeffvals1(i_range,i,1) = a; coeffvals1(i_range,i,2) = b; coeffvals1(i_range,i,3) = c; end IP_data(:,i_range)=coeffvals(:,3).*coeffvals(:,3+10)*sqrt(pi); %Since I am fitting 5 gaussians and taking the mid gaussian value end %Here we have the energy data set energy_data=1240./Data_set; save ('IP_fullrange1_gauss5_73K_Mar06_2014_Al66.mat', 'IP_data', 'energy_data', 'coeffvals', 'goodness'); 196 Step 4: The following code normalizes the raw data with respect to all the parameters and calculates the data point for the photoyield curve, clc clear all close all load ('IP_fullrange1_gauss5_73K_Mar06_2014_Al66.mat'); load ('Norm_fullrange1_gauss5_73K_Mar06_2014_Al66.mat'); Startin=63; %71; % %Starting size of clusters-1 Endin=69; %76; %End size of clusters+1 Note: The maximum number of clusters that can be fitted is 8 with Gaussians but we could contrain to 4 or 5 %for increasing accuracy Intensity_normalization=[50.6 49.8 48.3 48.8 46.9 46.1 45.4 44.7 44.1 44.7 43.1 42.8 44.2 42.5 41.9 43.2 40.5 41.2 41.1 41.5 41.0]./50; %for 170K [62.4 59.6 58.1 59.1 55.9 56.9 59.1 57.5 61.6 56.4 55.6 55.2 54.6 54.5]./60; %Here enter the data of the normalizing intensity in order should also include the data wavelength %if matches with the norm wavelength Norm_timebase=[5000*ones(1,4) 6000*ones(1,4) 7000*ones(1,4) 8000*ones(1,9)]; %Here you enter the norm data timebase in order. %Note: The norm data section at the bottom needs to be chaecked for proper ordering N_data_normwave=21; %Here write the total number of normalization data points Data_Intensity_normalization=[41.2 51.6 36.5 43.1 40.9 39.8 51.2 43.5 50.9 53.4 52.2 45.2 52.0 51.0 50.2 52.8 48.6 50.9 50.1 51.9 ... 52.4 46.8 51.9 52.6 53.9 53.9 53.8 53.7 50.4 49.8 52.2 51.3 50.2 52.9 53.2]./50; %Here we enter the data for Laser intensity normalization and the intensity for the %data wavelength if matches with the norm wavelength should go at last Data_timebase=1000*[6 13 6 7 7 8 20 8 25 20 33 12 20 25 9 20 15 26 20 38 29 11 31 16 38 45 47 55 41 21 25 40 45 32 38]; %Here we enter the timebase of the norm with the time for the normalizing %wavelength at the end Data_wave=[210 220 212 214 211 213 228 215 232 225 236 219 224 230 217 227 222 231 223 238 233 218 234 221 237 243 245 247 241 ... 226 229 240 244 235 239]; %Here enter the complete data set taken for IP measurement except the data at normalization wavelength Data_set=Data_wave; %The complete data set taken for IP measurement except the data at normalization wavelength Numberofdata=length(Data_set); Data_Intensity_Laser_normalization=Data_Intensity_normalization; Norm_Laser_normalized=zeros(N_data_normwave,3); for i=1:N_data_normwave Norm_Laser_normalized(i,:)=Norm_data(i,:)./(Intensity_normalization(1,i)*Norm_timebase(1,i)); end Norm_corrected=zeros(Numberofdata,3); Norm_corrected(1,:)=(Norm_Laser_normalized(1,:)+((26*Norm_Laser_normalized(1,:)+6*Norm_Laser_normalize d(2,:))/32))/2; 197 Norm_corrected(2,:)=(((26*Norm_Laser_normalized(1,:)+6*Norm_Laser_normalized(2,:))/32)+((13*Norm_Laser_ normalized(1,:)+19*Norm_Laser_normalized(2,:))/32))/2; Norm_corrected(3,:)=(((13*Norm_Laser_normalized(1,:)+19*Norm_Laser_normalized(2,:))/32)+((7*Norm_Laser_ normalized(1,:)+25*Norm_Laser_normalized(2,:))/32))/2; Norm_corrected(4,:)=(Norm_Laser_normalized(2,:)+((7*Norm_Laser_normalized(1,:)+25*Norm_Laser_normalize d(2,:))/32))/2; Norm_corrected(5,:)=(Norm_Laser_normalized(2,:)+((36*Norm_Laser_normalized(2,:)+7*Norm_Laser_normalize d(3,:))/43))/2; Norm_corrected(6,:)=(((36*Norm_Laser_normalized(2,:)+7*Norm_Laser_normalized(3,:))/43)+((28*Norm_Laser_ normalized(2,:)+15*Norm_Laser_normalized(3,:))/43))/2; Norm_corrected(7,:)=(((8*Norm_Laser_normalized(2,:)+35*Norm_Laser_normalized(3,:))/43)+((28*Norm_Laser_ normalized(2,:)+15*Norm_Laser_normalized(3,:))/43))/2; Norm_corrected(8,:)=(Norm_Laser_normalized(3,:)+((8*Norm_Laser_normalized(2,:)+35*Norm_Laser_normalize d(3,:))/43))/2; Norm_corrected(9,:)=(Norm_Laser_normalized(3,:)+((20*Norm_Laser_normalized(3,:)+25*Norm_Laser_normaliz ed(4,:))/45))/2; Norm_corrected(10,:)=(Norm_Laser_normalized(4,:)+((20*Norm_Laser_normalized(3,:)+25*Norm_Laser_normali zed(4,:))/45))/2; Norm_corrected(11,:)=(Norm_Laser_normalized(4,:)+((12*Norm_Laser_normalized(4,:)+33*Norm_Laser_normali zed(5,:))/45))/2; Norm_corrected(12,:)=(Norm_Laser_normalized(5,:)+((12*Norm_Laser_normalized(4,:)+33*Norm_Laser_normali zed(5,:))/45))/2; Norm_corrected(13,:)=(Norm_Laser_normalized(5,:)+((25*Norm_Laser_normalized(5,:)+20*Norm_Laser_normali zed(6,:))/45))/2; Norm_corrected(14,:)=(Norm_Laser_normalized(6,:)+((25*Norm_Laser_normalized(5,:)+20*Norm_Laser_normali zed(6,:))/45))/2; Norm_corrected(15,:)=(Norm_Laser_normalized(6,:)+((35*Norm_Laser_normalized(6,:)+9*Norm_Laser_normaliz ed(7,:))/44))/2; Norm_corrected(16,:)=(((15*Norm_Laser_normalized(6,:)+29*Norm_Laser_normalized(7,:))/44)+((35*Norm_Lase r_normalized(6,:)+9*Norm_Laser_normalized(7,:))/44))/2; Norm_corrected(17,:)=(Norm_Laser_normalized(7,:)+((15*Norm_Laser_normalized(6,:)+29*Norm_Laser_normali zed(7,:))/44))/2; Norm_corrected(18,:)=(Norm_Laser_normalized(7,:)+((20*Norm_Laser_normalized(7,:)+26*Norm_Laser_normali zed(8,:))/46))/2; Norm_corrected(19,:)=(Norm_Laser_normalized(8,:)+((20*Norm_Laser_normalized(7,:)+26*Norm_Laser_normali zed(8,:))/46))/2; Norm_corrected(20,:)=(Norm_Laser_normalized(8,:)+Norm_Laser_normalized(9,:))/2; 198 Norm_corrected(21,:)=(Norm_Laser_normalized(9,:)+((11*Norm_Laser_normalized(9,:)+29*Norm_Laser_normali zed(10,:))/40))/2; Norm_corrected(22,:)=(Norm_Laser_normalized(10,:)+((11*Norm_Laser_normalized(9,:)+29*Norm_Laser_norma lized(10,:))/40))/2; Norm_corrected(23,:)=(Norm_Laser_normalized(10,:)+((16*Norm_Laser_normalized(10,:)+31*Norm_Laser_norm alized(11,:))/47))/2; Norm_corrected(24,:)=(Norm_Laser_normalized(11,:)+((16*Norm_Laser_normalized(10,:)+31*Norm_Laser_norm alized(11,:))/47))/2; Norm_corrected(25,:)=(Norm_Laser_normalized(11,:)+Norm_Laser_normalized(12,:))/2; Norm_corrected(26,:)=(Norm_Laser_normalized(12,:)+Norm_Laser_normalized(13,:))/2; Norm_corrected(27,:)=(Norm_Laser_normalized(13,:)+Norm_Laser_normalized(14,:))/2; Norm_corrected(28,:)=(Norm_Laser_normalized(14,:)+Norm_Laser_normalized(15,:))/2; Norm_corrected(29,:)=(Norm_Laser_normalized(15,:)+Norm_Laser_normalized(16,:))/2; Norm_corrected(30,:)=(Norm_Laser_normalized(16,:)+((25*Norm_Laser_normalized(16,:)+21*Norm_Laser_norm alized(17,:))/46))/2; Norm_corrected(31,:)=(Norm_Laser_normalized(17,:)+((25*Norm_Laser_normalized(16,:)+21*Norm_Laser_norm alized(17,:))/46))/2; Norm_corrected(32,:)=(Norm_Laser_normalized(17,:)+Norm_Laser_normalized(18,:))/2; Norm_corrected(33,:)=(Norm_Laser_normalized(18,:)+Norm_Laser_normalized(19,:))/2; Norm_corrected(34,:)=(Norm_Laser_normalized(19,:)+Norm_Laser_normalized(20,:))/2; Norm_corrected(35,:)=(Norm_Laser_normalized(20,:)+Norm_Laser_normalized(21,:))/2; Norm_corrected_3peak=zeros(Numberofdata,1); Norm_corrected_3peak(:,1)=(Norm_corrected(:,1)+Norm_corrected(:,2)+Norm_corrected(:,3))/3; IP_data3=zeros(Numberofdata,length(1:Endin-Startin-5)); for i=1:Numberofdata IP_data3(i,:)=IP_data(i,:)./(Norm_corrected_3peak(i,:)*Data_timebase(1,i)*Data_Intensity_Laser_normalization(1,i )); end IP_data_check=zeros(Numberofdata,length(1:Endin-Startin-5)); for i=1:Numberofdata IP_data_check(i,:)=IP_data(i,:)./(Data_timebase(1,i)*Data_Intensity_Laser_normalization(1,i)); end IP_data3_73K_Mar06_14=IP_data3; energy_data_73K_Mar06_14=1240./Data_wave; save ('IP_calculate1_gauss5_73K_Mar06_2014_Al66.mat', 'IP_data3_73K_Mar06_14', 'energy_data_73K_Mar06_14'); 199 Step 5: The following code plots the photoyield and also its derivatives along with the error estimated for individual points: clear all %close all clc load Data_compare_new_all1.mat %70K Numberofdata_73K_Mar06_14=length(energy_data_73K_Mar06_14); IP_data_73K_Mar06_14=zeros(Numberofdata_73K_Mar06_14,84); IP_data_73K_Mar06_14(:,1:82)=IP_data3_73K_Mar06_14(:,1:82); IP_data_73K_Mar06_14(:,83)=energy_data_73K_Mar06_14'; IP_data_73K_Mar06_14(:,84)=-energy_data_73K_Mar06_14'; IP_data_order_73K_Mar06_14=sortrows(IP_data_73K_Mar06_14,83); energy_data_order_73K_Mar06_14=IP_data_order_73K_Mar06_14(:,83); Numberofdata_65K_Mar13_14=length(energy_data_65K_Mar13_14); IP_data_65K_Mar13_14=zeros(Numberofdata_65K_Mar13_14,84); IP_data_65K_Mar13_14(:,1:82)=IP_data3_65K_Mar13_14(:,1:82); IP_data_65K_Mar13_14(:,83)=energy_data_65K_Mar13_14'; IP_data_65K_Mar13_14(:,84)=-energy_data_65K_Mar13_14'; IP_data_order_65K_Mar13_14=sortrows(IP_data_65K_Mar13_14,83); energy_data_order_65K_Mar13_14=IP_data_order_65K_Mar13_14(:,83); Numberofdata_65K_Mar18_14=length(energy_data_65K_Mar18_14); IP_data_65K_Mar18_14=zeros(Numberofdata_65K_Mar18_14,84); IP_data_65K_Mar18_14(:,1:82)=IP_data3_65K_Mar18_14(:,1:82); IP_data_65K_Mar18_14(:,83)=energy_data_65K_Mar18_14'; IP_data_65K_Mar18_14(:,84)=-energy_data_65K_Mar18_14'; IP_data_order_65K_Mar18_14=sortrows(IP_data_65K_Mar18_14,83); energy_data_order_65K_Mar18_14=IP_data_order_65K_Mar18_14(:,83); %90K Numberofdata_89K_Dec02_13=length(energy_data_89K_Dec02_13); IP_data_89K_Dec02_13=zeros(Numberofdata_89K_Dec02_13,84); IP_data_89K_Dec02_13(:,1:82)=IP_data3_89K_Dec02_13(:,1:82); IP_data_89K_Dec02_13(:,83)=energy_data_89K_Dec02_13'; IP_data_89K_Dec02_13(:,84)=-energy_data_89K_Dec02_13'; IP_data_order_89K_Dec02_13=sortrows(IP_data_89K_Dec02_13,83); energy_data_order_89K_Dec02_13=IP_data_order_89K_Dec02_13(:,83); Numberofdata_90K_Dec17_13=length(energy_data_90K_Dec17_13); IP_data_90K_Dec17_13=zeros(Numberofdata_90K_Dec17_13,84); IP_data_90K_Dec17_13(:,1:82)=IP_data3_90K_Dec17_13(:,1:82); 200 IP_data_90K_Dec17_13(:,83)=energy_data_90K_Dec17_13'; IP_data_90K_Dec17_13(:,84)=-energy_data_90K_Dec17_13'; IP_data_order_90K_Dec17_13=sortrows(IP_data_90K_Dec17_13,83); energy_data_order_90K_Dec17_13=IP_data_order_90K_Dec17_13(:,83); Numberofdata_90K_Jan03_14=length(energy_data_90K_Jan03_14); IP_data_90K_Jan03_14=zeros(Numberofdata_90K_Jan03_14,84); IP_data_90K_Jan03_14(:,1:82)=IP_data3_90K_Jan03_14(:,1:82); IP_data_90K_Jan03_14(:,83)=energy_data_90K_Jan03_14'; IP_data_90K_Jan03_14(:,84)=-energy_data_90K_Jan03_14'; IP_data_order_90K_Jan03_14=sortrows(IP_data_90K_Jan03_14,83); energy_data_order_90K_Jan03_14=IP_data_order_90K_Jan03_14(:,83); Numberofdata_90K_Feb07_14=length(energy_data_90K_Feb07_14); IP_data_90K_Feb07_14=zeros(Numberofdata_90K_Feb07_14,84); IP_data_90K_Feb07_14(:,1:82)=IP_data3_90K_Feb07_14(:,1:82); IP_data_90K_Feb07_14(:,83)=energy_data_90K_Feb07_14'; IP_data_90K_Feb07_14(:,84)=-energy_data_90K_Feb07_14'; IP_data_order_90K_Feb07_14=sortrows(IP_data_90K_Feb07_14,83); energy_data_order_90K_Feb07_14=IP_data_order_90K_Feb07_14(:,83); %120K Numberofdata_121K_Jan22_14=length(energy_data_121K_Jan22_14); IP_data_121K_Jan22_14=zeros(Numberofdata_121K_Jan22_14,84); IP_data_121K_Jan22_14(:,1:82)=IP_data3_121K_Jan22_14(:,1:82); IP_data_121K_Jan22_14(:,83)=energy_data_121K_Jan22_14'; IP_data_121K_Jan22_14(:,84)=-energy_data_121K_Jan22_14'; IP_data_order_121K_Jan22_14=sortrows(IP_data_121K_Jan22_14,83); energy_data_order_121K_Jan22_14=IP_data_order_121K_Jan22_14(:,83); Numberofdata_121K_Feb07_14=length(energy_data_121K_Feb07_14); IP_data_121K_Feb07_14=zeros(Numberofdata_121K_Feb07_14,84); IP_data_121K_Feb07_14(:,1:82)=IP_data3_121K_Feb07_14(:,1:82); IP_data_121K_Feb07_14(:,83)=energy_data_121K_Feb07_14'; IP_data_121K_Feb07_14(:,84)=-energy_data_121K_Feb07_14'; IP_data_order_121K_Feb07_14=sortrows(IP_data_121K_Feb07_14,83); energy_data_order_121K_Feb07_14=IP_data_order_121K_Feb07_14(:,83); Numberofdata_120K_Feb22_14=length(energy_data_120K_Feb22_14); IP_data_120K_Feb22_14=zeros(Numberofdata_120K_Feb22_14,84); IP_data_120K_Feb22_14(:,1:82)=IP_data3_120K_Feb22_14(:,1:82); IP_data_120K_Feb22_14(:,83)=energy_data_120K_Feb22_14'; IP_data_120K_Feb22_14(:,84)=-energy_data_120K_Feb22_14'; IP_data_order_120K_Feb22_14=sortrows(IP_data_120K_Feb22_14,83); energy_data_order_120K_Feb22_14=IP_data_order_120K_Feb22_14(:,83); Numberofdata_120K_Apr18_14=length(energy_data_120K_Apr18_14); IP_data_120K_Apr18_14=zeros(Numberofdata_120K_Apr18_14,84); IP_data_120K_Apr18_14(:,1:82)=IP_data3_120K_Apr18_14(:,1:82); 201 IP_data_120K_Apr18_14(:,83)=energy_data_120K_Apr18_14'; IP_data_120K_Apr18_14(:,84)=-energy_data_120K_Apr18_14'; IP_data_order_120K_Apr18_14=sortrows(IP_data_120K_Apr18_14,83); energy_data_order_120K_Apr18_14=IP_data_order_120K_Apr18_14(:,83); Numberofdata_120K_Apr26_14=length(energy_data_120K_Apr26_14); IP_data_120K_Apr26_14=zeros(Numberofdata_120K_Apr26_14,84); IP_data_120K_Apr26_14(:,1:82)=IP_data3_120K_Apr26_14(:,1:82); IP_data_120K_Apr26_14(:,83)=energy_data_120K_Apr26_14'; IP_data_120K_Apr26_14(:,84)=-energy_data_120K_Apr26_14'; IP_data_order_120K_Apr26_14=sortrows(IP_data_120K_Apr26_14,83); energy_data_order_120K_Apr26_14=IP_data_order_120K_Apr26_14(:,83); %145K Numberofdata_145K_Apr18_14=length(energy_data_145K_Apr18_14); IP_data_145K_Apr18_14=zeros(Numberofdata_145K_Apr18_14,84); IP_data_145K_Apr18_14(:,1:82)=IP_data3_145K_Apr18_14(:,1:82); IP_data_145K_Apr18_14(:,83)=energy_data_145K_Apr18_14'; IP_data_145K_Apr18_14(:,84)=-energy_data_145K_Apr18_14'; IP_data_order_145K_Apr18_14=sortrows(IP_data_145K_Apr18_14,83); energy_data_order_145K_Apr18_14=IP_data_order_145K_Apr18_14(:,83); %170K Numberofdata_170K_Dec09_13=length(energy_data_170K_Dec09_13); IP_data_170K_Dec09_13=zeros(Numberofdata_170K_Dec09_13,84); IP_data_170K_Dec09_13(:,1:82)=IP_data3_170K_Dec09_13(:,1:82); IP_data_170K_Dec09_13(:,83)=energy_data_170K_Dec09_13'; IP_data_170K_Dec09_13(:,84)=-energy_data_170K_Dec09_13'; IP_data_order_170K_Dec09_13=sortrows(IP_data_170K_Dec09_13,83); energy_data_order_170K_Dec09_13=IP_data_order_170K_Dec09_13(:,83); Numberofdata_170K_Jan03_14=length(energy_data_170K_Jan03_14); IP_data_170K_Jan03_14=zeros(Numberofdata_170K_Jan03_14,84); IP_data_170K_Jan03_14(:,1:82)=IP_data3_170K_Jan03_14(:,1:82); IP_data_170K_Jan03_14(:,83)=energy_data_170K_Jan03_14'; IP_data_170K_Jan03_14(:,84)=-energy_data_170K_Jan03_14'; IP_data_order_170K_Jan03_14=sortrows(IP_data_170K_Jan03_14,83); energy_data_order_170K_Jan03_14=IP_data_order_170K_Jan03_14(:,83); Numberofdata_170K_Jan31_14=length(energy_data_170K_Jan31_14); IP_data_170K_Jan31_14=zeros(Numberofdata_170K_Jan31_14,84); IP_data_170K_Jan31_14(:,1:82)=IP_data3_170K_Jan31_14(:,1:82); IP_data_170K_Jan31_14(:,83)=energy_data_170K_Jan31_14'; IP_data_170K_Jan31_14(:,84)=-energy_data_170K_Jan31_14'; IP_data_order_170K_Jan31_14=sortrows(IP_data_170K_Jan31_14,83); energy_data_order_170K_Jan31_14=IP_data_order_170K_Jan31_14(:,83); Numberofdata_170K_Feb22_14=length(energy_data_170K_Feb22_14); 202 IP_data_170K_Feb22_14=zeros(Numberofdata_170K_Feb22_14,84); IP_data_170K_Feb22_14(:,1:82)=IP_data3_170K_Feb22_14(:,1:82); IP_data_170K_Feb22_14(:,83)=energy_data_170K_Feb22_14'; IP_data_170K_Feb22_14(:,84)=-energy_data_170K_Feb22_14'; IP_data_order_170K_Feb22_14=sortrows(IP_data_170K_Feb22_14,83); energy_data_order_170K_Feb22_14=IP_data_order_170K_Feb22_14(:,83); Numberofdata_170K_Apr26_14=length(energy_data_170K_Apr26_14); IP_data_170K_Apr26_14=zeros(Numberofdata_170K_Apr26_14,84); IP_data_170K_Apr26_14(:,1:82)=IP_data3_170K_Apr26_14(:,1:82); IP_data_170K_Apr26_14(:,83)=energy_data_170K_Apr26_14'; IP_data_170K_Apr26_14(:,84)=-energy_data_170K_Apr26_14'; IP_data_order_170K_Apr26_14=sortrows(IP_data_170K_Apr26_14,83); energy_data_order_170K_Apr26_14=IP_data_order_170K_Apr26_14(:,83); %230K Numberofdata_230K_Jan22_14=length(energy_data_230K_Jan22_14); IP_data_230K_Jan22_14=zeros(Numberofdata_230K_Jan22_14,84); IP_data_230K_Jan22_14(:,1:82)=IP_data3_230K_Jan22_14(:,1:82); IP_data_230K_Jan22_14(:,83)=energy_data_230K_Jan22_14'; IP_data_230K_Jan22_14(:,84)=-energy_data_230K_Jan22_14'; IP_data_order_230K_Jan22_14=sortrows(IP_data_230K_Jan22_14,83); energy_data_order_230K_Jan22_14=IP_data_order_230K_Jan22_14(:,83); Numberofdata_230K_Jan31_14=length(energy_data_230K_Jan31_14); IP_data_230K_Jan31_14=zeros(Numberofdata_230K_Jan31_14,84); IP_data_230K_Jan31_14(:,1:82)=IP_data3_230K_Jan31_14(:,1:82); IP_data_230K_Jan31_14(:,83)=energy_data_230K_Jan31_14'; IP_data_230K_Jan31_14(:,84)=-energy_data_230K_Jan31_14'; IP_data_order_230K_Jan31_14=sortrows(IP_data_230K_Jan31_14,83); energy_data_order_230K_Jan31_14=IP_data_order_230K_Jan31_14(:,83); Numberofdata_230K_Mar22_14=length(energy_data_230K_Mar22_14); IP_data_230K_Mar22_14=zeros(Numberofdata_230K_Mar22_14,84); IP_data_230K_Mar22_14(:,1:82)=IP_data3_230K_Mar22_14(:,1:82); IP_data_230K_Mar22_14(:,83)=energy_data_230K_Mar22_14'; IP_data_230K_Mar22_14(:,84)=-energy_data_230K_Mar22_14'; IP_data_order_230K_Mar22_14=sortrows(IP_data_230K_Mar22_14,83); energy_data_order_230K_Mar22_14=IP_data_order_230K_Mar22_14(:,83); Numberofdata_230K_Mar28_14=length(energy_data_230K_Mar28_14); IP_data_230K_Mar28_14=zeros(Numberofdata_230K_Mar28_14,84); IP_data_230K_Mar28_14(:,1:82)=IP_data3_230K_Mar28_14(:,1:82); IP_data_230K_Mar28_14(:,83)=energy_data_230K_Mar28_14'; IP_data_230K_Mar28_14(:,84)=-energy_data_230K_Mar28_14'; IP_data_order_230K_Mar28_14=sortrows(IP_data_230K_Mar28_14,83); energy_data_order_230K_Mar28_14=IP_data_order_230K_Mar28_14(:,83); color_data = [1 0 0; 0 1 0; 203 0 0 1; 1 0 1; 0 0 0 0 1 1]; FS_leg = 1; % Ratio1=[1.58 2.08 1.85 1.26 1.4]; % Ratio2=[1.0 1.3 1.16 1 1.18]; % Ratio=Ratio1./Ratio2; Ratio=ones(1,5); j=0; for i=15 %[8 10 14 37 39] %[7:1:15 25:1:31 35:1:41] j=j+1; if i<10 n=3; m=7; s=1; else n=1; m=5; s=1; end en_log=[5.06:0.01:5.91]; en_log1=[5.23:0.01:5.91]; en_log2=[5.06:0.01:5.22]; x1=en_log; lin_log=[35:1:65];%[15:1:35];%[30:1:62];%[32:1:67];%for 37 [5:1:50]; %for Al66, 30 to 55 in [5.06:0.01:5.91] Al68 30 to 55 %Mar06_73K p = csaps(energy_data_order_73K_Mar06_14,smooth(IP_data_order_73K_Mar06_14(:,i),n)); y_fit_73K_Mar06_14=ppval(p,en_log); p_der=fnder(p,1); y_der_73K_Mar06_14=ppval(p_der,en_log); I1=find(energy_data_order_73K_Mar06_14>5.34 & energy_data_order_73K_Mar06_14<5.6); p = csaps(energy_data_order_73K_Mar06_14([1:min(I1) max(I1):length(energy_data_order_73K_Mar06_14)]),smooth(IP_data_order_73K_Mar06_14([1:min(I1) max(I1):length(energy_data_order_73K_Mar06_14)],i),n)); y1_fit_73K_Mar06_14=ppval(p,en_log); %65K_Mar13 p = csaps(energy_data_order_65K_Mar13_14,smooth(IP_data_order_65K_Mar13_14(:,i),n)); y_fit_65K_Mar13_14=ppval(p,en_log); p_der=fnder(p,1); y_der_65K_Mar13_14=ppval(p_der,en_log); I1=find(energy_data_order_65K_Mar13_14>5.34 & energy_data_order_65K_Mar13_14<5.6); 204 p = csaps(energy_data_order_65K_Mar13_14([1:min(I1) max(I1):length(energy_data_order_65K_Mar13_14)]),smooth(IP_data_order_65K_Mar13_14([1:min(I1) max(I1):length(energy_data_order_65K_Mar13_14)],i),n)); y1_fit_65K_Mar13_14=ppval(p,en_log); %65K Mar18 p = csaps(energy_data_order_65K_Mar18_14,smooth(IP_data_order_65K_Mar18_14(:,i),n)); y_fit_65K_Mar18_14=ppval(p,en_log); p_der=fnder(p); y_der_65K_Mar18_14=ppval(p_der,en_log); I1=find(energy_data_order_65K_Mar18_14>5.34 & energy_data_order_65K_Mar18_14<5.65); p = csaps(energy_data_order_65K_Mar18_14([1:min(I1) max(I1):length(energy_data_order_65K_Mar18_14)]),smooth(IP_data_order_65K_Mar18_14([1:min(I1) max(I1):length(energy_data_order_65K_Mar18_14)],i),n)); y1_fit_65K_Mar18_14=ppval(p,en_log); %65K master p_65_lin=polyfit(x1(1,lin_log),(y_fit_73K_Mar06_14(1,lin_log)+y_fit_65K_Mar13_14(1,lin_log)+y_fit_65K_Mar 18_14(1,lin_log))/3,1); y_lin_65K=p_65_lin(1)*x1(1,lin_log)+p_65_lin(2); %90K Dec02 13 p = csaps(energy_data_order_89K_Dec02_13,smooth(IP_data_order_89K_Dec02_13(:,i)/Ratio(1,j),n)); y_fit_89K_Dec02_13=ppval(p,en_log); p_der=fnder(p); y_der_89K_Dec02_13=ppval(p_der,en_log); %90K Dec17 13 p = csaps(energy_data_order_90K_Dec17_13,smooth(IP_data_order_90K_Dec17_13(:,i),n)); y_fit_90K_Dec17_13=ppval(p,en_log); p_der=fnder(p); y_der_90K_Dec17_13=ppval(p_der,en_log); y_fit1_90K_Dec17_13=ppval(p,en_log1); y_der1_90K_Dec17_13=ppval(p_der,en_log1); y_fit2_90K_Dec17_13=ppval(p,en_log2); y_der2_90K_Dec17_13=ppval(p_der,en_log2); %90K Jan03 14 p = csaps(energy_data_order_90K_Jan03_14,smooth(IP_data_order_90K_Jan03_14(:,i),n)); y_fit_90K_Jan03_14=ppval(p,en_log); p_der=fnder(p); y_der_90K_Jan03_14=ppval(p_der,en_log); y_fit1_90K_Jan03_14=ppval(p,en_log1); y_der1_90K_Jan03_14=ppval(p_der,en_log1); 205 y_fit2_90K_Jan03_14=ppval(p,en_log2); y_der2_90K_Jan03_14=ppval(p_der,en_log2); %90 Feb07 14 p = csaps(energy_data_order_90K_Feb07_14,smooth(IP_data_order_90K_Feb07_14(:,i),n)); y_fit_90K_Feb07_14=ppval(p,en_log); p_der=fnder(p); y_der_90K_Feb07_14=ppval(p_der,en_log); y_fit1_90K_Feb07_14=ppval(p,en_log1); y_der1_90K_Feb07_14=ppval(p_der,en_log1); %90K master % p_90_lin=polyfit(x1(1,lin_log),(y_fit_90K_Dec17_13(1,lin_log)+y_fit_90K_Jan03_14(1,lin_log)+y_fit_90K_Feb0 7_14(1,lin_log))/3,1); % y_lin_90K=p_90_lin(1)*x1(1,lin_log)+p_90_lin(2); y_90K_st=[((smooth(y_fit2_90K_Dec17_13,s)+smooth(y_fit2_90K_Jan03_14,s))/2)' ((smooth(y_fit1_90K_Dec17_13,s)+smooth(y_fit1_90K_Jan03_14,s)+smooth(y_fit1_90K_Feb07_14,s))/3)']; p_90_lin=polyfit(x1(1,lin_log),y_90K_st(1,lin_log),1); y_lin_90K=p_90_lin(1)*x1(1,lin_log)+p_90_lin(2); p_90K_st = csaps(x1,y_90K_st); y_fit_st_90K=ppval(p_90K_st,en_log); p_der_90K_st=fnder(p_90K_st); y_der_90K_st=ppval(p_der_90K_st,en_log); y_90K_st_der=[((smooth(y_der2_90K_Dec17_13,s)+smooth(y_der2_90K_Jan03_14,s))/2)' ((smooth(y_der1_90K_Dec17_13,s)+smooth(y_der1_90K_Jan03_14,s)+smooth(y_der1_90K_Feb07_14,s))/3)']; %120K Jan22 14 p = csaps(energy_data_order_121K_Jan22_14,smooth(IP_data_order_121K_Jan22_14(:,i),n)); y_fit_121K_Jan22_14=ppval(p,en_log); p_der=fnder(p); y_der_121K_Jan22_14=ppval(p_der,en_log); y_fit1_121K_Jan22_14=ppval(p,en_log1); y_der1_121K_Jan22_14=ppval(p_der,en_log1); % 120K Feb07 14 p = csaps(energy_data_order_121K_Feb07_14,smooth(IP_data_order_121K_Feb07_14(:,i),n)); y_fit_121K_Feb07_14=ppval(p,en_log); p_der=fnder(p); y_der_121K_Feb07_14=ppval(p_der,en_log); y_fit1_121K_Feb07_14=ppval(p,en_log1); y_der1_121K_Feb07_14=ppval(p_der,en_log1); %120K Feb 22 14 206 p = csaps(energy_data_order_120K_Feb22_14,smooth(IP_data_order_120K_Feb22_14(:,i),n)); y_fit_120K_Feb22_14=ppval(p,en_log); p_der=fnder(p); y_der_120K_Feb22_14=ppval(p_der,en_log); y_fit1_120K_Feb22_14=ppval(p,en_log1); y_der1_120K_Feb22_14=ppval(p_der,en_log1); %120K Apr 18 14 p = csaps(energy_data_order_120K_Apr18_14,smooth(IP_data_order_120K_Apr18_14(:,i),n)); y_fit_120K_Apr18_14=ppval(p,en_log); p_der=fnder(p); y_der_120K_Apr18_14=ppval(p_der,en_log); y_fit1_120K_Apr18_14=ppval(p,en_log1); y_der1_120K_Apr18_14=ppval(p_der,en_log1); y_fit2_120K_Apr18_14=ppval(p,en_log2); y_der2_120K_Apr18_14=ppval(p_der,en_log2); %120K Apr 26 14 p = csaps(energy_data_order_120K_Apr26_14,smooth(IP_data_order_120K_Apr26_14(:,i),n)); y_fit_120K_Apr26_14=ppval(p,en_log); p_der=fnder(p); y_der_120K_Apr26_14=ppval(p_der,en_log); y_fit1_120K_Apr26_14=ppval(p,en_log1); y_der1_120K_Apr26_14=ppval(p_der,en_log1); y_fit2_120K_Apr26_14=ppval(p,en_log2); y_der2_120K_Apr26_14=ppval(p_der,en_log2); %120K master % p_120_lin=polyfit(x1(1,lin_log),(y_fit_121K_Jan22_14(1,lin_log)+y_fit_121K_Feb07_14(1,lin_log)+ ... % y_fit_120K_Feb22_14(1,lin_log)+y_fit_120K_Apr18_14(1,lin_log)+y_fit_120K_Apr26_14(1,lin_log))/5,1); % y_lin_120K=p_120_lin(1)*x1(1,lin_log)+p_120_lin(2); y_120K_st=[((smooth(y_fit2_120K_Apr18_14,s)+smooth(y_fit2_120K_Apr26_14,s))/2)' ... ((smooth(y_fit1_121K_Jan22_14,s)+smooth(y_fit1_121K_Feb07_14,s)+smooth(y_fit1_120K_Feb22_14,s)+ ... smooth(y_fit1_120K_Apr18_14,s)+smooth(y_fit1_120K_Apr26_14,s))/5)']; p_120_lin=polyfit(x1(1,lin_log),y_120K_st(1,lin_log),1); y_lin_120K=p_120_lin(1)*x1(1,lin_log)+p_120_lin(2); p_120K_st = csaps(x1,y_120K_st); y_fit_st_120K=ppval(p_120K_st,en_log); p_der_120K_st=fnder(p_120K_st); y_der_120K_st=ppval(p_der_120K_st,en_log); 207 y_120K_st_der=[((smooth(y_der2_120K_Apr18_14,s)+smooth(y_der2_120K_Apr26_14,s))/2)' ... ((smooth(y_der1_121K_Jan22_14,s)+smooth(y_der1_121K_Feb07_14,s)+smooth(y_der1_120K_Feb22_14,s)+ ... smooth(y_der1_120K_Apr18_14,s)+smooth(y_der1_120K_Apr26_14,s))/5)']; %145K Apr 18 14 p = csaps(energy_data_order_145K_Apr18_14,smooth(IP_data_order_145K_Apr18_14(:,i),n)); y_fit_145K_Apr18_14=ppval(p,en_log); p_der=fnder(p); y_der_145K_Apr18_14=ppval(p_der,en_log); y_fit1_145K_Apr18_14=ppval(p,en_log1); y_der1_145K_Apr18_14=ppval(p_der,en_log1); y_fit2_145K_Apr18_14=ppval(p,en_log2); y_der2_145K_Apr18_14=ppval(p_der,en_log2); %145K master p_145_lin=polyfit(x1(1,lin_log),(y_fit_145K_Apr18_14(1,lin_log))/1,1); y_lin_145K=p_145_lin(1)*x1(1,lin_log)+p_145_lin(2); %170K Dec09 13 p = csaps(energy_data_order_170K_Dec09_13,smooth(IP_data_order_170K_Dec09_13(:,i),n)); y_fit_170K_Dec09_13=ppval(p,en_log); p_der=fnder(p); y_der_170K_Dec09_13=ppval(p_der,en_log); y_fit1_170K_Dec09_13=ppval(p,en_log1); y_der1_170K_Dec09_13=ppval(p_der,en_log1); y_fit2_170K_Dec09_13=ppval(p,en_log2); y_der2_170K_Dec09_13=ppval(p_der,en_log2); % 170K Jan03 14 p = csaps(energy_data_order_170K_Jan03_14,smooth(IP_data_order_170K_Jan03_14(:,i),n)); y_fit_170K_Jan03_14=ppval(p,en_log); p_der=fnder(p); y_der_170K_Jan03_14=ppval(p_der,en_log); y_fit1_170K_Jan03_14=ppval(p,en_log1); y_der1_170K_Jan03_14=ppval(p_der,en_log1); y_fit2_170K_Jan03_14=ppval(p,en_log2); y_der2_170K_Jan03_14=ppval(p_der,en_log2); %170K Jan 31 14 p = csaps(energy_data_order_170K_Jan31_14,smooth(IP_data_order_170K_Jan31_14(:,i),n)); y_fit_170K_Jan31_14=ppval(p,en_log); 208 p_der=fnder(p); y_der_170K_Jan31_14=ppval(p_der,en_log); y_fit1_170K_Jan31_14=ppval(p,en_log1); y_der1_170K_Jan31_14=ppval(p_der,en_log1); %170K Feb22 14 p = csaps(energy_data_order_170K_Feb22_14,smooth(IP_data_order_170K_Feb22_14(:,i),n)); y_fit_170K_Feb22_14=ppval(p,en_log); p_der=fnder(p); y_der_170K_Feb22_14=ppval(p_der,en_log); y_fit1_170K_Feb22_14=ppval(p,en_log1); y_der1_170K_Feb22_14=ppval(p_der,en_log1); %170K Apr 26 14 p = csaps(energy_data_order_170K_Apr26_14,smooth(IP_data_order_170K_Apr26_14(:,i),n)); y_fit_170K_Apr26_14=ppval(p,en_log); p_der=fnder(p); y_der_170K_Apr26_14=ppval(p_der,en_log); y_fit1_170K_Apr26_14=ppval(p,en_log1); y_der1_170K_Apr26_14=ppval(p_der,en_log1); y_fit2_170K_Apr26_14=ppval(p,en_log2); y_der2_170K_Apr26_14=ppval(p_der,en_log2); %170 Master % p_170_lin=polyfit(x1(1,lin_log),(y_fit_170K_Dec09_13(1,lin_log)+y_fit_170K_Jan03_14(1,lin_log)+ ... % y_fit_170K_Jan31_14(1,lin_log)+y_fit_170K_Feb22_14(1,lin_log)+y_fit_170K_Apr26_14(1,lin_log))/5,1); % y_lin_170K=p_170_lin(1)*x1(1,lin_log)+p_170_lin(2); y_170K_st=[((smooth(y_fit2_170K_Dec09_13,s)+smooth(y_fit2_170K_Jan03_14,s)+smooth(y_fit2_170K_Apr26_ 14,s))/3)' ((smooth(y_fit1_170K_Dec09_13,s)+... smooth(y_fit1_170K_Jan03_14,s)+smooth(y_fit1_170K_Jan31_14,s)+smooth(y_fit1_170K_Feb22_14,s)+smooth(y _fit1_170K_Apr26_14,s))/5)']; p_170_lin=polyfit(x1(1,lin_log),y_170K_st(1,lin_log),1); y_lin_170K=p_170_lin(1)*x1(1,lin_log)+p_170_lin(2); p_170K_st = csaps(x1,y_170K_st); y_fit_st_170K=ppval(p_170K_st,en_log); p_der_170K_st=fnder(p_170K_st); y_der_170K_st=ppval(p_der_170K_st,en_log); y_170K_st_der=[((smooth(y_der2_170K_Dec09_13,s)+smooth(y_der2_170K_Jan03_14,s)+smooth(y_der2_170K_ Apr26_14,s))/3)' ((smooth(y_der1_170K_Dec09_13,s)+... smooth(y_der1_170K_Jan03_14,s)+smooth(y_der1_170K_Jan31_14,s)+smooth(y_der1_170K_Feb22_14,s)+smoot h(y_der1_170K_Apr26_14,s))/5)']; 209 %230K Mar22 p = csaps(energy_data_order_230K_Mar22_14,smooth(IP_data_order_230K_Mar22_14(:,i),n)); y_fit_Mar22_230K=ppval(p,en_log); p_der=fnder(p); y_der_Mar22_230K=ppval(p_der,en_log); y_fit1_230K_Mar22_14=ppval(p,en_log1); y_der1_230K_Mar22_14=ppval(p_der,en_log1); y_fit2_230K_Mar22_14=ppval(p,en_log2); y_der2_230K_Mar22_14=ppval(p_der,en_log2); I1=find(energy_data_order_230K_Mar22_14>5.34 & energy_data_order_230K_Mar22_14<5.6); %p = csaps(energy_data_order_230K_Mar22_14([1:min(I1) max(I1):length(energy_data_order_230K_Mar22_14)]),smooth(IP_data_order_230K_Mar22_14([1:min(I1) max(I1):length(energy_data_order_230K_Mar22_14)],i),n)); %y1_fit_230K_Mar22_14=ppval(p,en_log); p = polyfit(energy_data_order_230K_Mar22_14([1:min(I1) max(I1):length(energy_data_order_230K_Mar22_14)]),smooth(IP_data_order_230K_Mar22_14([1:min(I1) max(I1):length(energy_data_order_230K_Mar22_14)],i)),8); y1_fit_230K_Mar22_14=p(1)*x1.^8+p(2)*x1.^7+p(3)*x1.^6+p(4)*x1.^5+p(5)*x1.^4+p(6)*x1.^3+p(7)*x1.^2+p(8) *x1.^1+p(9); %230K Mar28 p = csaps(energy_data_order_230K_Mar28_14,smooth(IP_data_order_230K_Mar28_14(:,i),n)); y_fit_Mar28_230K=ppval(p,en_log); p_der=fnder(p); y_der_Mar28_230K=ppval(p_der,en_log); y_fit1_230K_Mar28_14=ppval(p,en_log1); y_der1_230K_Mar28_14=ppval(p_der,en_log1); y_fit2_230K_Mar28_14=ppval(p,en_log2); y_der2_230K_Mar28_14=ppval(p_der,en_log2); I1=find(energy_data_order_230K_Mar28_14>5.34 & energy_data_order_230K_Mar28_14<5.65); p = csaps(energy_data_order_230K_Mar28_14([1:min(I1) max(I1):length(energy_data_order_230K_Mar28_14)]),smooth(IP_data_order_230K_Mar28_14([1:min(I1) max(I1):length(energy_data_order_230K_Mar28_14)],i),n)); y1_fit_230K_Mar28_14=ppval(p,en_log); %230K Jan22 p = csaps(energy_data_order_230K_Jan22_14,smooth(IP_data_order_230K_Jan22_14(:,i),n)); y_fit_Jan22_230K=ppval(p,en_log); p_der=fnder(p); y_der_Jan22_230K=ppval(p_der,en_log); 210 y_fit1_230K_Jan22_14=ppval(p,en_log1); y_der1_230K_Jan22_14=ppval(p_der,en_log1); I1=find(energy_data_order_230K_Jan22_14>5.34 & energy_data_order_230K_Jan22_14<5.6); p = csaps(energy_data_order_230K_Jan22_14([1:min(I1) max(I1):length(energy_data_order_230K_Jan22_14)]),smooth(IP_data_order_230K_Jan22_14([1:min(I1) max(I1):length(energy_data_order_230K_Jan22_14)],i),n)); y1_fit_230K_Jan22_14=ppval(p,en_log); %230K Jan31 p = csaps(energy_data_order_230K_Jan31_14,smooth(IP_data_order_230K_Jan31_14(:,i),n)); y_fit_Jan31_230K=ppval(p,en_log); p_der=fnder(p); y_der_Jan31_230K=ppval(p_der,en_log); y_fit1_230K_Jan31_14=ppval(p,en_log1); y_der1_230K_Jan31_14=ppval(p_der,en_log1); I1=find(energy_data_order_230K_Jan31_14>5.34 & energy_data_order_230K_Jan31_14<5.65); p = csaps(energy_data_order_230K_Jan31_14([1:min(I1) max(I1):length(energy_data_order_230K_Jan31_14)]),smooth(IP_data_order_230K_Jan31_14([1:min(I1) max(I1):length(energy_data_order_230K_Jan31_14)],i),n)); y1_fit_230K_Jan31_14=ppval(p,en_log); %230 Master % p_230_lin=polyfit(x1(1,lin_log),(y_fit_Mar22_230K(1,lin_log)+y_fit_Mar28_230K(1,lin_log)+ ... % y_fit_Jan22_230K(1,lin_log)+y1_fit_230K_Jan31_14(1,lin_log))/4,1); % y_lin_230K=p_230_lin(1)*x1(1,lin_log)+p_230_lin(2); y_230K_st=[((smooth(y_fit2_230K_Mar22_14,s)+smooth(y_fit2_230K_Mar28_14,s))/2)' ((smooth(y_fit1_230K_Mar22_14,s)+... smooth(y_fit1_230K_Mar28_14,s)+smooth(y_fit1_230K_Jan22_14,s)+smooth(y_fit1_230K_Jan31_14,s))/4)']; p_230_lin=polyfit(x1(1,lin_log),y_230K_st(1,lin_log),1); y_lin_230K=p_230_lin(1)*x1(1,lin_log)+p_230_lin(2); p_230K_st = csaps(x1,y_230K_st); y_fit_st_230K=ppval(p_230K_st,en_log); p_der_230K_st=fnder(p_230K_st); y_der_230K_st=ppval(p_der_230K_st,en_log); y_230K_st_der=[((smooth(y_der2_230K_Mar22_14,s)+smooth(y_der2_230K_Mar28_14,s))/2)' ((smooth(y_der1_230K_Mar22_14,s)+... smooth(y_der1_230K_Mar28_14,s)+smooth(y_der1_230K_Jan22_14,s)+smooth(y_der1_230K_Jan31_14,s))/4)']; if i>10 % error analysis %for Al66 start here % 5-pt smoothing edges % err65_lin_log=find(en_log>5.31 & en_log<5.52); % err90_lin_log=find(en_log>5.19 & en_log<5.44); % err120_lin_log=find(en_log>5.31 & en_log<5.49); 211 % err170_lin_log=find(en_log>5.29 & en_log<5.53); % err230_lin_log=find(en_log>5.28 & en_log<5.47); %7-pt smoothing edges % err65_lin_log=find(en_log>5.21 & en_log<5.52); % err90_lin_log=find(en_log>5.21 & en_log<5.52); % err120_lin_log=find(en_log>5.31 & en_log<5.5); % err170_lin_log=find(en_log>5.29 & en_log<5.52); % err230_lin_log=find(en_log>5.29 & en_log<5.47); %for Al66 ends here %for Al68 start here %7-pt smoothing edges % err65_lin_log=find(en_log>5.28 & en_log<5.56); % err90_lin_log=find(en_log>5.21 & en_log<5.55); % err120_lin_log=find(en_log>5.2 & en_log<5.56); % err170_lin_log=find(en_log>5.21 & en_log<5.52); % err230_lin_log=find(en_log>5.23 & en_log<5.3); %for Al68 ends here % for Al44 starts here err65_lin_log=find(en_log>5.37 & en_log<5.68); err90_lin_log=find(en_log>5.32 & en_log<5.64); err120_lin_log=find(en_log>5.26 & en_log<5.54); err170_lin_log=find(en_log>5.34 & en_log<5.6); err230_lin_log=find(en_log>5.27 & en_log<5.6); % for Al44 ends here error_65K=zeros(length(en_log),3); error_65K(:,1)=smooth(y_der_73K_Mar06_14,m); error_65K(:,2)=smooth(y_der_65K_Mar13_14,m); error_65K(:,3)=smooth(y_der_65K_Mar18_14,m); errorbar_65K=std(error_65K'); %sigsq_u_65K=sum(((errorbar_65K(1,38:47))).^2)/length(38:47); sigsq_u_65K=sum(((errorbar_65K(1,err65_lin_log))).^2)/length(err65_lin_log); error_90K=zeros(length(en_log),3); error_90K(:,1)=smooth(y_der_90K_Dec17_13,m); error_90K(:,2)=smooth(y_der_90K_Jan03_14,m); error_90K(:,3)=smooth(y_der_90K_Feb07_14,m); errorbar_90K=std(error_90K'); %sigsq_u_90K=sum(((errorbar_90K(1,39:48))).^2)/length(39:48); sigsq_u_90K=sum(((errorbar_90K(1,err90_lin_log))).^2)/length(err90_lin_log); error_120K=zeros(length(en_log),5); error_120K(:,1)=smooth(y_der_121K_Jan22_14,m); error_120K(:,2)=smooth(y_der_121K_Feb07_14,m); error_120K(:,3)=smooth(y_der_120K_Feb22_14,m); error_120K(:,4)=smooth(y_der_120K_Apr18_14,m); error_120K(:,5)=smooth(y_der_120K_Apr26_14,m); errorbar_120K=std(error_120K'); %sigsq_u_120K=sum(((errorbar_120K(1,37:44))).^2)/length(37:44); 212 sigsq_u_120K=sum(((errorbar_120K(1,err120_lin_log))).^2)/length(err120_lin_log); % error_145K=zeros(length(en_log),1); % error_145K(:,1)=smooth(y_der_145K_Apr18_14); % errorbar_145K=std(error_145K'); error_170K=zeros(length(en_log),5); error_170K(:,1)=smooth(y_der_170K_Dec09_13,m); error_170K(:,2)=smooth(y_der_170K_Jan03_14,m); error_170K(:,3)=smooth(y_der_170K_Jan31_14,m); error_170K(:,4)=smooth(y_der_170K_Feb22_14,m); error_170K(:,5)=smooth(y_der_170K_Apr26_14,m); errorbar_170K=std(error_170K'); %sigsq_u_170K=sum(((errorbar_170K(1,37:48))).^2)/length(37:48); sigsq_u_170K=sum(((errorbar_170K(1,err170_lin_log))).^2)/length(err170_lin_log); error_230K=zeros(length(en_log),4); error_230K(:,1)=smooth(y_der_Mar22_230K,m); error_230K(:,2)=smooth(y_der_Mar28_230K,m); error_230K(:,3)=smooth(y_der_Jan22_230K,m); error_230K(:,4)=smooth(y_der_Jan31_230K,m); errorbar_230K=std(error_230K'); %sigsq_u_230K=sum(((errorbar_230K(1,37:41))).^2)/length(37:41); sigsq_u_230K=sum(((errorbar_230K(1,err230_lin_log))).^2)/length(err230_lin_log); % error data errordata_65K=zeros(length(en_log),3); errordata_65K(:,1)=y_fit_73K_Mar06_14; errordata_65K(:,2)=y_fit_65K_Mar13_14; errordata_65K(:,3)=y_fit_65K_Mar18_14; errorbardata_65K=std(errordata_65K'); sigsq_w_65K=sum(((errorbardata_65K(1,lin_log))./en_log(1,lin_log)).^2)/length(lin_log); errordata_90K=zeros(length(en_log),3); errordata_90K(:,1)=y_fit_90K_Dec17_13; errordata_90K(:,2)=y_fit_90K_Jan03_14; errordata_90K(:,3)=y_fit_90K_Feb07_14; errorbardata_90K=std(errordata_90K'); sigsq_w_90K=sum(((errorbardata_90K(1,lin_log))./en_log(1,lin_log)).^2)/length(lin_log); errordata_120K=zeros(length(en_log),5); errordata_120K(:,1)=y_fit_121K_Jan22_14; errordata_120K(:,2)=y_fit_121K_Feb07_14; errordata_120K(:,3)=y_fit_120K_Feb22_14; errordata_120K(:,4)=y_fit_120K_Apr18_14; errordata_120K(:,5)=y_fit_120K_Apr26_14; errorbardata_120K=std(errordata_120K'); sigsq_w_120K=sum(((errorbardata_120K(1,lin_log))./en_log(1,lin_log)).^2)/length(lin_log); % error_145K=zeros(length(en_log),1); % error_145K(:,1)=smooth(y_der_145K_Apr18_14); % errorbar_145K=std(error_145K'); errordata_170K=zeros(length(en_log),5); errordata_170K(:,1)=y_fit_170K_Dec09_13; 213 errordata_170K(:,2)=y_fit_170K_Jan03_14; errordata_170K(:,3)=y_fit_170K_Jan31_14; errordata_170K(:,4)=y_fit_170K_Feb22_14; errordata_170K(:,5)=y_fit_170K_Apr26_14; errorbardata_170K=std(errordata_170K'); sigsq_w_170K=sum(((errorbardata_170K(1,lin_log))./en_log(1,lin_log)).^2)/length(lin_log); errordata_230K=zeros(length(en_log),4); errordata_230K(:,1)=y_fit_Mar22_230K; errordata_230K(:,2)=y_fit_Mar28_230K; errordata_230K(:,3)=y_fit_Jan22_230K; errordata_230K(:,4)=y_fit_Jan31_230K; errorbardata_230K=std(errordata_230K'); sigsq_w_230K=sum(((errorbardata_230K(1,lin_log))./en_log(1,lin_log)).^2)/length(lin_log); h1=figure; subplot(4,4,1); myplot(energy_data_order_73K_Mar06_14',smooth(IP_data_order_73K_Mar06_14(:,i),n),'.',color_data(1,:),'','Photo n energy (h \nu)','Int','','Northwest',FS_leg); hold on; myplot(en_log',y_fit_73K_Mar06_14,'-',color_data(1,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %hold on; myplot(en_log',y1_fit_73K_Mar06_14,'--',color_data(1,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_65K_Mar13_14',smooth(IP_data_order_65K_Mar13_14(:,i),n),'.',color_data(2,:),'','Photo n energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_65K_Mar13_14,'-',color_data(2,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %hold on; myplot(en_log',y1_fit_65K_Mar13_14,'--',color_data(2,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_65K_Mar18_14',smooth(IP_data_order_65K_Mar18_14(:,i),n),'.',color_data(3,:),'','Photo n energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_65K_Mar18_14,'-',color_data(3,:),'70K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %hold on; myplot(en_log',y1_fit_65K_Mar18_14,'--',color_data(3,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %myplot(x1(1,lin_log)',(y_lin_65K),'-',color_data(5,:),'70K','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',(smooth(y_fit_73K_Mar06_14,s)+smooth(y_fit_65K_Mar13_14,s)+smooth(y_fit_65K_Mar18_14,s ))/3,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %errorbar(en_log',(y_fit_73K_Mar06_14+y_fit_65K_Mar13_14+y_fit_65K_Mar18_14)/3,errorbardata_65K'); xlim([5 6]); ylim([0 2]); subplot(4,4,5); myplot(en_log',smooth(y_der_73K_Mar06_14,m),'--',color_data(1,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); hold on; myplot(en_log',smooth(y_der_65K_Mar13_14,m),'--',color_data(2,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',smooth(y_der_65K_Mar18_14,m),'--',color_data(3,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',(smooth(y_der_73K_Mar06_14,m)+smooth(y_der_65K_Mar13_14,m)+smooth(y_der_65K_Mar18 _14,m))/3,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %errorbar(en_log',(smooth(y_der_73K_Mar06_14,m)+smooth(y_der_65K_Mar13_14,m)+smooth(y_der_65K_Mar 18_14,m))/3,errorbar_65K'); 214 xlim([5 6]); ylim([0 4]); subplot(4,4,2); % myplot(energy_data_order_89K_Dec02_13',smooth(IP_data_order_89K_Dec02_13(:,i)/Ratio(1,j),n),'.',color_data(1 ,:),'','Photon energy (h \nu)','Int','','Northwest',FS_leg); hold on; %myplot(en_log',y_fit_89K_Dec02_13,'-',color_data(1,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_90K_Dec17_13',smooth(IP_data_order_90K_Dec17_13(:,i),n),'.',color_data(2,:),'','Photo n energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_90K_Dec17_13,'-',color_data(2,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_90K_Jan03_14',smooth(IP_data_order_90K_Jan03_14(:,i),n),'.',color_data(3,:),'','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_90K_Jan03_14,'-',color_data(3,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_90K_Feb07_14',smooth(IP_data_order_90K_Feb07_14(:,i),n),'.',color_data(4,:),'','Photo n energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_90K_Feb07_14,'-',color_data(4,:),'90K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %myplot(x1(1,lin_log)',(y_lin_90K),'-',color_data(5,:),'90K','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',(smooth(y_fit_90K_Dec17_13,s)+smooth(y_fit_90K_Jan03_14,s)+smooth(y_fit_90K_Feb07_14,s)) /3,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %errorbar(en_log',(y_fit_90K_Dec17_13+y_fit_90K_Jan03_14+y_fit_90K_Feb07_14)/3,errorbardata_90K'); xlim([5 6]); ylim([0 2]); subplot(4,4,6); %myplot(en_log',smooth(y_der_89K_Dec02_13),'--',color_data(1,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); hold on; myplot(en_log',smooth(y_der_90K_Dec17_13,m),'--',color_data(2,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',smooth(y_der_90K_Jan03_14,m),'--',color_data(3,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',smooth(y_der_90K_Feb07_14,m),'--',color_data(4,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',(smooth(y_der_90K_Dec17_13,m)+smooth(y_der_90K_Jan03_14,m)+smooth(y_der_90K_Feb07_1 4,m))/3,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %errorbar(en_log',(smooth(y_der_90K_Dec17_13,m)+smooth(y_der_90K_Jan03_14,m)+smooth(y_der_90K_Feb0 7_14,m))/3,errorbar_90K'); xlim([5 6]); ylim([0 4]); subplot(4,4,3); myplot(energy_data_order_121K_Jan22_14',smooth(IP_data_order_121K_Jan22_14(:,i),n),'.',color_data(1,:),'','Phot on energy (h \nu)','Int','','Northwest',FS_leg); 215 hold on; myplot(en_log',y_fit_121K_Jan22_14,'-',color_data(1,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_121K_Feb07_14',smooth(IP_data_order_121K_Feb07_14(:,i),n),'.',color_data(2,:),'','Pho ton energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_121K_Feb07_14,'-',color_data(2,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_120K_Feb22_14',smooth(IP_data_order_120K_Feb22_14(:,i),n),'.',color_data(3,:),'','Pho ton energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_120K_Feb22_14,'-',color_data(3,:),'Mar22 14 120K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_120K_Apr18_14',smooth(IP_data_order_120K_Apr18_14(:,i),n),'.',color_data(4,:),'','Pho ton energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_120K_Apr18_14,'-',color_data(4,:),'120K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_120K_Apr26_14',smooth(IP_data_order_120K_Apr26_14(:,i),n),'.',color_data(6,:),'','Pho ton energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_120K_Apr26_14,'-',color_data(6,:),'120K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %myplot(x1(1,lin_log)',(y_lin_120K),'-',color_data(5,:),'120K','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',(smooth(y_fit_121K_Jan22_14,s)+smooth(y_fit_121K_Feb07_14,s)+smooth(y_fit_120K_Feb22_14 ,s)+smooth(y_fit_120K_Apr18_14,s)+smooth(y_fit_120K_Apr26_14,s))/5,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %errorbar(en_log',(y_fit_121K_Jan22_14+y_fit_121K_Feb07_14+y_fit_120K_Feb22_14+y_fit_120K_Apr18_14+ y_fit_120K_Apr26_14)/5,errorbardata_120K'); xlim([5 6]); ylim([0 2]); subplot(4,4,7); myplot(en_log',smooth(y_der_121K_Jan22_14,m),'--',color_data(1,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); hold on; myplot(en_log',smooth(y_der_121K_Feb07_14,m),'--',color_data(2,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',smooth(y_der_120K_Feb22_14,m),'--',color_data(3,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',smooth(y_der_120K_Apr18_14,m),'--',color_data(4,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',smooth(y_der_120K_Apr26_14,m),'--',color_data(6,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',(smooth(y_der_121K_Jan22_14,m)+smooth(y_der_121K_Feb07_14,m)+smooth(y_der_120K_Feb2 2_14,m)+smooth(y_der_120K_Apr18_14,m)+smooth(y_der_120K_Apr26_14,m))/5,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %errorbar(en_log',(smooth(y_der_121K_Jan22_14,m)+smooth(y_der_121K_Feb07_14,m)+smooth(y_der_120K_F eb22_14,m)+smooth(y_der_120K_Apr18_14,m)+smooth(y_der_120K_Apr26_14,m))/5,errorbar_120K'); xlim([5 6]); ylim([0 4]); subplot(4,4,4); myplot(energy_data_order_145K_Apr18_14',smooth(IP_data_order_145K_Apr18_14(:,i),n),'.',color_data(3,:),'','Pho ton energy (h \nu)','Int','','Northwest',FS_leg); hold on;myplot(en_log',y_fit_145K_Apr18_14,'-',color_data(3,:),'145K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); 216 myplot(en_log',(smooth(y_fit_145K_Apr18_14,s))/1,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %myplot(x1(1,lin_log)',(y_lin_145K),'-',color_data(5,:),'145K','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); xlim([5 6]); ylim([0 2]); subplot(4,4,8); myplot(en_log',smooth(y_der_145K_Apr18_14,m),'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %errorbar(en_log',smooth(y_der_145K_Apr18_14),errorbar_145K'); xlim([5 6]); ylim([0 4]); subplot(4,4,11); myplot(energy_data_order_170K_Dec09_13',smooth(IP_data_order_170K_Dec09_13(:,i),n),'.',color_data(1,:),'','Ph oton energy (h \nu)','Int','','Northwest',FS_leg); hold on; myplot(en_log',y_fit_170K_Dec09_13,'-',color_data(1,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_170K_Jan03_14',smooth(IP_data_order_170K_Jan03_14(:,i),n),'.',color_data(2,:),'','Phot on energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_170K_Jan03_14,'-',color_data(2,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_170K_Jan31_14',smooth(IP_data_order_170K_Jan31_14(:,i),n),'.',color_data(3,:),'','Phot on energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_170K_Jan31_14,'-',color_data(3,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_170K_Feb22_14',smooth(IP_data_order_170K_Feb22_14(:,i),n),'.',color_data(4,:),'','Pho ton energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_170K_Feb22_14,'-',color_data(4,:),'170K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_170K_Apr26_14',smooth(IP_data_order_170K_Apr26_14(:,i),n),'.',color_data(6,:),'','Pho ton energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_170K_Apr26_14,'-',color_data(6,:),'170K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %myplot(x1(1,lin_log)',(y_lin_170K),'-',color_data(5,:),'170K','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',(smooth(y_fit_170K_Dec09_13,s)+smooth(y_fit_170K_Jan03_14,s)+smooth(y_fit_170K_Jan31_14 ,s)+smooth(y_fit_170K_Feb22_14,s)+smooth(y_fit_170K_Apr26_14,s))/5,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %errorbar(en_log',(y_fit_170K_Dec09_13+y_fit_170K_Jan03_14+y_fit_170K_Jan31_14+y_fit_170K_Feb22_14+ y_fit_170K_Apr26_14)/5,errorbardata_170K'); xlim([5 6]); ylim([0 2]); subplot(4,4,15); myplot(en_log',smooth(y_der_170K_Dec09_13,m),'--',color_data(1,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); hold on; myplot(en_log',smooth(y_der_170K_Jan03_14,m),'--',color_data(2,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',smooth(y_der_170K_Jan31_14,m),'--',color_data(3,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); 217 myplot(en_log',smooth(y_der_170K_Feb22_14,m),'--',color_data(4,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',smooth(y_der_170K_Apr26_14,m),'--',color_data(6,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',(smooth(y_der_170K_Dec09_13,m)+smooth(y_der_170K_Jan03_14,m)+smooth(y_der_170K_Jan3 1_14,m)+smooth(y_der_170K_Feb22_14,m)+smooth(y_der_170K_Apr26_14,m))/5,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %errorbar(en_log',(smooth(y_der_170K_Dec09_13,m)+smooth(y_der_170K_Jan03_14,m)+smooth(y_der_170K_J an31_14,m)+smooth(y_der_170K_Feb22_14,m)+smooth(y_der_170K_Apr26_14,m))/5,errorbar_170K'); xlim([5 6]); ylim([0 4]); subplot(4,4,12); myplot(energy_data_order_230K_Mar22_14',smooth(IP_data_order_230K_Mar22_14(:,i),n),'.',color_data(1,:),'','Ph oton energy (h \nu)','Int','','Northwest',FS_leg); hold on; myplot(en_log',y_fit_Mar22_230K,'-',color_data(1,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %hold on; myplot(en_log',y1_fit_230K_Mar22_14,'--',color_data(1,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_230K_Mar28_14',smooth(IP_data_order_230K_Mar28_14(:,i),n),'.',color_data(2,:),'','Ph oton energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_Mar28_230K,'-',color_data(2,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %hold on; myplot(en_log',y1_fit_230K_Mar28_14,'--',color_data(2,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_230K_Jan22_14',smooth(IP_data_order_230K_Jan22_14(:,i),n),'.',color_data(3,:),'','Phot on energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_Jan22_230K,'-',color_data(3,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %hold on; myplot(en_log',y1_fit_230K_Jan22_14,'--',color_data(3,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_230K_Jan31_14',smooth(IP_data_order_230K_Jan31_14(:,i),n),'.',color_data(4,:),'','Phot on energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_Jan31_230K,'-',color_data(4,:),'230K, , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %hold on; myplot(en_log',y1_fit_230K_Jan31_14,'--',color_data(4,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %myplot(x1(1,lin_log)',(y_lin_230K),'-',color_data(5,:),'230K','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',(smooth(y_fit_Mar22_230K,s)+smooth(y_fit_Mar28_230K,s)+smooth(y_fit_Jan22_230K,s)+smoot h(y_fit_Jan31_230K,s))/4,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %errorbar(en_log',(y_fit_Mar22_230K+y_fit_Mar28_230K+y_fit_Jan22_230K+y_fit_Jan31_230K)/4,errorbardata_ 230K'); xlim([5 6]); ylim([0 2]); subplot(4,4,16); myplot(en_log',smooth(y_der_Mar22_230K,m),'--',color_data(1,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); 218 hold on; myplot(en_log',smooth(y_der_Mar28_230K,m),'--',color_data(2,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',smooth(y_der_Jan22_230K,m),'--',color_data(3,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',smooth(y_der_Jan31_230K,m),'--',color_data(4,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',(smooth(y_der_Mar22_230K,m)+smooth(y_der_Mar28_230K,m)+smooth(y_der_Jan22_230K,m)+ smooth(y_der_Jan31_230K,m))/4,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %errorbar(en_log',(smooth(y_der_Mar22_230K,m)+smooth(y_der_Mar28_230K,m)+smooth(y_der_Jan22_230K,m )+smooth(y_der_Jan31_230K,m))/4,errorbar_230K'); xlim([5 6]); ylim([0 4]); % print(gcf, '-r300','-dpng', strcat('Al',num2str(i+29))); % saveas(h1,strcat('Al',num2str(i+29),'.fig')); %pause %for Al66 %figure; % T=[65 90 120 145 170 230]; % Top_peak=[2.703 2.541 2.393 2.41 2.332 2.199];%5-ptfitted derivative data[2.785 2.575 2.55 2.509 2.5 2.36]; % Bot_peak_rt=[1.121 1.016 1.37 1.31 1.627 1.923];%5-ptfitted derivative data[.964 .922 1.348 1.317 1.579 1.83]; % % Bot_peak_lt=[.36 .167 .288 .13 .319 .33];%[.32 .46 .28 .33 .34 .24]; % % Top_peak2=[3.274 2.508 3.007 2.684 2.545 2.605]; % peak_diff_rt=Top_peak-Bot_peak_rt; % peak_diff_lt=Top_peak-Bot_peak_lt; % peak_ratio=peak_diff_rt./peak_diff_lt; % % peak_diff_rt2=Top_peak2-Bot_peak_rt; % peak_ratio2=peak_diff_rt2./peak_diff_lt; % figure;plot(T,peak_ratio2/peak_ratio2(6),'g*-'); % % figure;plot(T,peak_diff_rt/peak_diff_rt(6),'*-'); % hold on;plot(T,peak_diff_lt/peak_diff_lt(6),'r*-'); % plot(T,peak_ratio/peak_ratio(6),'g*-'); % % figure;plot(T,peak_diff_rt/peak_diff_rt(1),'*-'); % hold on;plot(T,peak_diff_lt/peak_diff_lt(1),'r*-'); % plot(T,peak_ratio/peak_ratio(1),'g*-'); %for Al68 with 7-pt smoothing % T=[65 90 120 145 170 230]; % Top_peak=[2.927 3.068 2.810 2.582 3.0334 2.148]; % Bot_peak_rt=[1.792 1.565 1.61 2.118 2.19 1.982]; %forAl44 T=[65 90 120 145 170 230]; Top_peak=[3.27 3.76 3.06 2.55 3 3.35]; 219 Bot_peak_rt=[1.9 0 1.99 1.83 2.47 2.08]; peak_diff_rt=Top_peak-Bot_peak_rt; %figure; p_lin=[p_65_lin(1) p_90_lin(1) p_120_lin(1) p_145_lin(1) p_170_lin(1) p_230_lin(1)]; %plot(T,peak_diff_rt./p_lin,'*-'); % error_Al66=[sqrt(.514^2/(p_65_lin(1))^2+.288^2/(p_65_lin(1))^2) sqrt(.55^2/(p_90_lin(1))^2+.567^2/(p_90_lin(1))^2) sqrt(.406^2/(p_120_lin(1))^2+.535^2/(p_120_lin(1))^2) 0 sqrt(.428^2/(p_170_lin(1))^2+.508^2/(p_170_lin(1))^2) sqrt(.329^2/(p_230_lin(1))^2+.541^2/(p_230_lin(1))^2)]; % std_y=[sum(errorbardata_65K(30:55,1))/length([30:55]) sum(errorbardata_90K(30:55,1))/length([30:55]) ... % sum(errorbardata_120K(30:55,1))/length([30:55]) 0 sum(errorbardata_170K(30:55,1))/length([30:55]) ... % sum(errorbardata_230K(30:55,1))/length([30:55])]; err_66_65K=sqrt(2*sigsq_u_65K/(p_65_lin(1))^2+(peak_diff_rt(1)/(p_65_lin(1))^2)^2*sigsq_w_65K); err_66_90K=sqrt(2*sigsq_u_90K/(p_90_lin(1))^2+(peak_diff_rt(2)/(p_90_lin(1))^2)^2*sigsq_w_90K); err_66_120K=sqrt(2*sigsq_u_120K/(p_120_lin(1))^2+(peak_diff_rt(3)/(p_120_lin(1))^2)^2*sigsq_w_120K); err_66_170K=sqrt(2*sigsq_u_170K/(p_170_lin(1))^2+(peak_diff_rt(5)/(p_170_lin(1))^2)^2*sigsq_w_170K); err_66_230K=sqrt(2*sigsq_u_230K/(p_230_lin(1))^2+(peak_diff_rt(6)/(p_230_lin(1))^2)^2*sigsq_w_230K); error_Al66=[err_66_65K err_66_90K err_66_120K 0 err_66_170K err_66_230K]; figure;errorbar(T,peak_diff_rt./p_lin,error_Al66); Al66_Norm3_T=T; Al66_Norm3_Amp=peak_diff_rt./p_lin; Al66_Norm3_err=error_Al66; figure;errorbar(T(1,[1:3 5 6]),Al66_Norm3_Amp(1,[1:3 5 6]),Al66_Norm3_err(1,[1:3 5 6])); ylim([0 1]); save('Al66_ana_Norm3.mat','Al66_Norm3_T','Al66_Norm3_Amp','Al66_Norm3_err'); else % Comment from here for smaller size clusters % error analysis % error analysis % 5-pt smoothing edges (7-pt used for 37) % err65_lin_log=find(en_log>5.21 & en_log<5.4); % err90_lin_log=find(en_log>5.22 & en_log<5.34); % err120_lin_log=find(en_log>5.21 & en_log<5.3); % err170_lin_log=find(en_log>5.23 & en_log<5.42); % err230_lin_log=find(en_log>5.2 & en_log<5.35); % 7-pt smoothing edge for 43 err65_lin_log=find(en_log>5.23 & en_log<5.4); err90_lin_log=find(en_log>5.23 & en_log<5.3); err120_lin_log=find(en_log>5.26 & en_log<5.41); err170_lin_log=find(en_log>5.26 & en_log<5.3); err230_lin_log=find(en_log>5.23 & en_log<5.4); error_65K=zeros(length(en_log),3); error_65K(:,1)=smooth(y_der_73K_Mar06_14,m); error_65K(:,2)=smooth(y_der_65K_Mar13_14,m); 220 error_65K(:,3)=smooth(y_der_65K_Mar18_14,m); errorbar_65K=std(error_65K'); %sigsq_u_65K=sum(((errorbar_65K(1,38:47))).^2)/length(38:47); sigsq_u_65K=sum(((errorbar_65K(1,err65_lin_log))).^2)/length(err65_lin_log); error_90K=zeros(length(en_log),2); error_90K(:,1)=smooth(y_der_90K_Dec17_13,m); error_90K(:,2)=smooth(y_der_90K_Jan03_14,m); %error_90K(:,3)=smooth(y_der1_90K_Feb07_14,m); %errorbar_90K=std(error_90K'); error2_90K=zeros(length(en_log2),2); error2_90K(:,1)=error_90K(1:length(en_log2),1); error2_90K(:,2)=error_90K(1:length(en_log2),2); errorbar2_90K=std(error2_90K'); error1_90K=zeros(length(en_log1),3); error1_90K(:,1)=error_90K(length(en_log2)+1:length(en_log),1); error1_90K(:,2)=error_90K(length(en_log2)+1:length(en_log),2); error1_90K(:,3)=smooth(y_der1_90K_Feb07_14,m); errorbar1_90K=std(error1_90K'); errorbar_90K=([errorbar2_90K errorbar1_90K]); %sigsq_u_90K=sum(((errorbar_90K(1,39:48))).^2)/length(39:48); sigsq_u_90K=sum(((errorbar_90K(1,err90_lin_log))).^2)/length(err90_lin_log); % errorbar_90K=std(error_90K'); %sigsq_u_90K=sum(((errorbar_90K(1,39:48))).^2)/length(39:48); %sigsq_u_90K=sum(((errorbar_90K(1,err90_lin_log))).^2)/length(err90_lin_log); error_120K=zeros(length(en_log),2); error_120K(:,1)=smooth(y_der_120K_Apr18_14,m); error_120K(:,2)=smooth(y_der_120K_Apr26_14,m); error2_120K=zeros(length(en_log2),2); error2_120K(:,1)=error_120K(1:length(en_log2),1); error2_120K(:,2)=error_120K(1:length(en_log2),2); errorbar2_120K=std(error2_120K'); error1_120K=zeros(length(en_log1),5); error1_120K(:,1)=smooth(y_der1_121K_Jan22_14,m); error1_120K(:,2)=smooth(y_der1_121K_Feb07_14,m); error1_120K(:,3)=smooth(y_der1_120K_Feb22_14,m); error1_120K(:,4)=error_120K(length(en_log2)+1:length(en_log),1); error1_120K(:,5)=error_120K(length(en_log2)+1:length(en_log),2); errorbar1_120K=std(error1_120K'); errorbar_120K=([errorbar2_120K errorbar1_120K]); %sigsq_u_120K=sum(((errorbar_120K(1,37:44))).^2)/length(37:44); sigsq_u_120K=sum(((errorbar_120K(1,err120_lin_log))).^2)/length(err120_lin_log); % error_145K=zeros(length(en_log),1); % error_145K(:,1)=smooth(y_der_145K_Apr18_14); % errorbar_145K=std(error_145K'); error_170K=zeros(length(en_log),3); 221 error_170K(:,1)=smooth(y_der_170K_Dec09_13,m); error_170K(:,2)=smooth(y_der_170K_Jan03_14,m); error_170K(:,3)=smooth(y_der_170K_Apr26_14,m); error2_170K=zeros(length(en_log2),3); error2_170K(:,1)=error_170K(1:length(en_log2),1); error2_170K(:,2)=error_170K(1:length(en_log2),2); error2_170K(:,3)=error_170K(1:length(en_log2),3); errorbar2_170K=std(error2_170K'); error1_170K=zeros(length(en_log1),5); error1_170K(:,1)=error_170K(length(en_log2)+1:length(en_log),1); error1_170K(:,2)=error_170K(length(en_log2)+1:length(en_log),2); error1_170K(:,3)=smooth(y_der1_170K_Jan31_14,m); error1_170K(:,4)=smooth(y_der1_170K_Feb22_14,m); error1_170K(:,5)=error_170K(length(en_log2)+1:length(en_log),3); errorbar1_170K=std(error1_170K'); errorbar_170K=([errorbar2_170K errorbar1_170K]); %sigsq_u_170K=sum(((errorbar_170K(1,37:48))).^2)/length(37:48); sigsq_u_170K=sum(((errorbar_170K(1,err170_lin_log))).^2)/length(err170_lin_log); error_230K=zeros(length(en_log),2); error_230K(:,1)=smooth(y_der_Mar22_230K,m); error_230K(:,2)=smooth(y_der_Mar28_230K,m); error2_230K=zeros(length(en_log2),2); error2_230K(:,1)=error_230K(1:length(en_log2),1); error2_230K(:,2)=error_230K(1:length(en_log2),2); errorbar2_230K=std(error2_230K'); error1_230K=zeros(length(en_log1),4); error1_230K(:,1)=error_230K(length(en_log2)+1:length(en_log),1); error1_230K(:,2)=error_230K(length(en_log2)+1:length(en_log),2); error1_230K(:,3)=smooth(y_der1_230K_Jan22_14,m); error1_230K(:,4)=smooth(y_der1_230K_Jan31_14,m); errorbar1_230K=std(error1_230K'); errorbar_230K=([errorbar2_230K errorbar1_230K]); %sigsq_u_230K=sum(((errorbar_230K(1,37:41))).^2)/length(37:41); sigsq_u_230K=sum(((errorbar_230K(1,err230_lin_log))).^2)/length(err230_lin_log); % error data errordata_65K=zeros(length(en_log),3); errordata_65K(:,1)=y_fit_73K_Mar06_14; errordata_65K(:,2)=y_fit_65K_Mar13_14; errordata_65K(:,3)=y_fit_65K_Mar18_14; errorbardata_65K=std(errordata_65K'); sigsq_w_65K=sum(((errorbardata_65K(1,lin_log))./en_log(1,lin_log)).^2)/length(lin_log); errordata_90K=zeros(length(en_log),2); errordata_90K(:,1)=smooth(y_fit_90K_Dec17_13,m); errordata_90K(:,2)=smooth(y_fit_90K_Jan03_14,m); 222 errordata2_90K=zeros(length(en_log2),2); errordata2_90K(:,1)=errordata_90K(1:length(en_log2),1); errordata2_90K(:,2)=errordata_90K(1:length(en_log2),2); errorbardata2_90K=std(errordata2_90K'); errordata1_90K=zeros(length(en_log1),3); errordata1_90K(:,1)=errordata_90K(length(en_log2)+1:length(en_log),1); errordata1_90K(:,2)=errordata_90K(length(en_log2)+1:length(en_log),2); errordata1_90K(:,3)=smooth(y_fit1_90K_Feb07_14,m); errorbardata1_90K=std(errordata1_90K'); errorbardata_90K=([errorbardata2_90K errorbardata1_90K]); %errorbardata_90K=std(errordata_90K'); sigsq_w_90K=sum(((errorbardata_90K(1,lin_log))./en_log(1,lin_log)).^2)/length(lin_log); errordata_120K=zeros(length(en_log),2); errordata_120K(:,1)=smooth(y_fit_120K_Apr18_14,m); errordata_120K(:,2)=smooth(y_fit_120K_Apr26_14,m); errordata2_120K=zeros(length(en_log2),2); errordata2_120K(:,1)=errordata_120K(1:length(en_log2),1); errordata2_120K(:,2)=errordata_120K(1:length(en_log2),2); errorbardata2_120K=std(errordata2_120K'); errordata1_120K=zeros(length(en_log1),5); errordata1_120K(:,1)=smooth(y_fit1_121K_Jan22_14,m); errordata1_120K(:,2)=smooth(y_fit1_121K_Feb07_14,m); errordata1_120K(:,3)=smooth(y_fit1_120K_Feb22_14,m); errordata1_120K(:,4)=errordata_120K(length(en_log2)+1:length(en_log),1); errordata1_120K(:,5)=errordata_120K(length(en_log2)+1:length(en_log),2); errorbardata1_120K=std(errordata1_120K'); errorbardata_120K=([errorbardata2_120K errorbardata1_120K]); %sigsq_u_120K=sum(((errorbar_120K(1,37:44))).^2)/length(37:44); sigsq_w_120K=sum(((errorbardata_120K(1,lin_log))./en_log(1,lin_log)).^2)/length(lin_log); % error_145K=zeros(length(en_log),1); % error_145K(:,1)=smooth(y_der_145K_Apr18_14); % errorbar_145K=std(error_145K'); errordata_170K=zeros(length(en_log),3); errordata_170K(:,1)=smooth(y_fit_170K_Dec09_13,m); errordata_170K(:,2)=smooth(y_fit_170K_Jan03_14,m); errordata_170K(:,3)=smooth(y_fit_170K_Apr26_14,m); errordata2_170K=zeros(length(en_log2),3); errordata2_170K(:,1)=errordata_170K(1:length(en_log2),1); errordata2_170K(:,2)=errordata_170K(1:length(en_log2),2); errordata2_170K(:,3)=errordata_170K(1:length(en_log2),3); errorbardata2_170K=std(errordata2_170K'); errordata1_170K=zeros(length(en_log1),5); errordata1_170K(:,1)=errordata_170K(length(en_log2)+1:length(en_log),1); errordata1_170K(:,2)=errordata_170K(length(en_log2)+1:length(en_log),2); errordata1_170K(:,3)=smooth(y_fit1_170K_Jan31_14,m); 223 errordata1_170K(:,4)=smooth(y_fit1_170K_Feb22_14,m); errordata1_170K(:,5)=errordata_170K(length(en_log2)+1:length(en_log),3); errorbardata1_170K=std(errordata1_170K'); errorbardata_170K=([errorbardata2_170K errorbardata1_170K]); sigsq_w_170K=sum(((errorbardata_170K(1,lin_log))./en_log(1,lin_log)).^2)/length(lin_log); errordata_230K=zeros(length(en_log),2); errordata_230K(:,1)=smooth(y_fit_Mar22_230K,m); errordata_230K(:,2)=smooth(y_fit_Mar28_230K,m); errordata2_230K=zeros(length(en_log2),2); errordata2_230K(:,1)=errordata_230K(1:length(en_log2),1); errordata2_230K(:,2)=errordata_230K(1:length(en_log2),2); errorbardata2_230K=std(errordata2_230K'); errordata1_230K=zeros(length(en_log1),4); errordata1_230K(:,1)=errordata_230K(length(en_log2)+1:length(en_log),1); errordata1_230K(:,2)=errordata_230K(length(en_log2)+1:length(en_log),2); errordata1_230K(:,3)=smooth(y_fit1_230K_Jan22_14,m); errordata1_230K(:,4)=smooth(y_fit1_230K_Jan31_14,m); errorbardata1_230K=std(errordata1_230K'); errorbardata_230K=([errorbardata2_230K errorbardata1_230K]); sigsq_w_230K=sum(((errorbardata_230K(1,lin_log))./en_log(1,lin_log)).^2)/length(lin_log); h1=figure; subplot(4,4,1); myplot(energy_data_order_73K_Mar06_14',smooth(IP_data_order_73K_Mar06_14(:,i),n),'.',color_data(1,:),'','Photo n energy (h \nu)','Int','','Northwest',FS_leg); hold on; myplot(en_log',y_fit_73K_Mar06_14,'-',color_data(1,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %hold on; myplot(en_log',y1_fit_73K_Mar06_14,'--',color_data(1,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_65K_Mar13_14',smooth(IP_data_order_65K_Mar13_14(:,i),n),'.',color_data(2,:),'','Photo n energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_65K_Mar13_14,'-',color_data(2,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %hold on; myplot(en_log',y1_fit_65K_Mar13_14,'--',color_data(2,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_65K_Mar18_14',smooth(IP_data_order_65K_Mar18_14(:,i),n),'.',color_data(3,:),'','Photo n energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_65K_Mar18_14,'-',color_data(3,:),'70K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %hold on; myplot(en_log',y1_fit_65K_Mar18_14,'--',color_data(3,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',(smooth(y_fit_73K_Mar06_14,s)+smooth(y_fit_65K_Mar13_14,s)+smooth(y_fit_65K_Mar18_14,s ))/3,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %errorbar(en_log',(y_fit_73K_Mar06_14+y_fit_65K_Mar13_14+y_fit_65K_Mar18_14)/3,errorbardata_65K'); myplot(x1(1,lin_log)',(y_lin_65K),'-',color_data(5,:),'70K','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); xlim([5 6]); ylim([0 2]); subplot(4,4,5); myplot(en_log',smooth(y_der_73K_Mar06_14,m),'--',color_data(1,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); 224 hold on; myplot(en_log',smooth(y_der_65K_Mar13_14,m),'--',color_data(2,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',smooth(y_der_65K_Mar18_14,m),'--',color_data(3,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',(smooth(y_der_73K_Mar06_14,m)+smooth(y_der_65K_Mar13_14,m)+smooth(y_der_65K_Mar18 _14,m))/3,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %errorbar(en_log',(smooth(y_der_73K_Mar06_14,m)+smooth(y_der_65K_Mar13_14,m)+smooth(y_der_65K_Mar 18_14,m))/3,errorbar_65K); xlim([5 6]); ylim([0 4]); subplot(4,4,2); % myplot(energy_data_order_89K_Dec02_13',smooth(IP_data_order_89K_Dec02_13(:,i)/Ratio(1,j),n),'.',color_data(1 ,:),'','Photon energy (h \nu)','Int','','Northwest',FS_leg); hold on; %myplot(en_log',y_fit_89K_Dec02_13,'-',color_data(1,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_90K_Dec17_13',smooth(IP_data_order_90K_Dec17_13(:,i),n),'.',color_data(2,:),'','Photo n energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_90K_Dec17_13,'-',color_data(2,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_90K_Jan03_14',smooth(IP_data_order_90K_Jan03_14(:,i),n),'.',color_data(3,:),'','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_90K_Jan03_14,'-',color_data(3,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_90K_Feb07_14',smooth(IP_data_order_90K_Feb07_14(:,i),n),'.',color_data(4,:),'','Photo n energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log1',y_fit1_90K_Feb07_14,'-',color_data(4,:),'90K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(x1(1,lin_log)',(y_lin_90K),'-',color_data(5,:),'90K','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log',(smooth(y_fit_90K_Dec17_13,s)+smooth(y_fit_90K_Jan03_14,s)+smooth(y_fit_90K_Feb07_14,s)) /3,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % errorbar(en_log',(y_fit_90K_Dec17_13+y_fit_90K_Jan03_14+y_fit_90K_Feb07_14)/3,errorbardata_90K'); % errorbar(en_log',y_90K_st,errorbardata_90K'); % myplot(en_log1',(smooth(y_fit1_90K_Dec17_13,s)+smooth(y_fit1_90K_Jan03_14,s)+smooth(y_fit1_90K_Feb07_ 14,s))/3,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log2',(smooth(y_fit2_90K_Dec17_13,s)+smooth(y_fit2_90K_Jan03_14,s))/2,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(x1,y_90K_st,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); xlim([5 6]); ylim([0 2]); subplot(4,4,6); 225 %myplot(en_log',smooth(y_der_89K_Dec02_13),'--',color_data(1,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); hold on; myplot(en_log',smooth(y_der_90K_Dec17_13,m),'--',color_data(2,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',smooth(y_der_90K_Jan03_14,m),'--',color_data(3,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log1',smooth(y_der1_90K_Feb07_14,m),'--',color_data(4,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log',(smooth(y_der_90K_Dec17_13,m)+smooth(y_der_90K_Jan03_14,m))/2,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % errorbar(en_log',(smooth(y_der_90K_Dec17_13,m)+smooth(y_der_90K_Jan03_14,m))/2,errorbar_90K); % myplot(en_log1',(smooth(y_der1_90K_Dec17_13,m)+smooth(y_der1_90K_Jan03_14,m)+smooth(y_der1_90K_Feb 07_14,m))/3,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log2',(smooth(y_der2_90K_Dec17_13,m)+smooth(y_der2_90K_Jan03_14,m))/2,'- ',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(x1,y_der_90K_st,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(x1,smooth(y_90K_st_der,m),'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % errorbar(en_log',smooth(y_90K_st_der,m),errorbar_90K); xlim([5 6]); ylim([0 4]); subplot(4,4,3); myplot(energy_data_order_121K_Jan22_14',smooth(IP_data_order_121K_Jan22_14(:,i),n),'.',color_data(1,:),'','Phot on energy (h \nu)','Int','','Northwest',FS_leg); hold on; myplot(en_log1',y_fit1_121K_Jan22_14,'-',color_data(1,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_121K_Feb07_14',smooth(IP_data_order_121K_Feb07_14(:,i),n),'.',color_data(2,:),'','Pho ton energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log1',y_fit1_121K_Feb07_14,'-',color_data(2,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_120K_Feb22_14',smooth(IP_data_order_120K_Feb22_14(:,i),n),'.',color_data(3,:),'','Pho ton energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log1',y_fit1_120K_Feb22_14,'-',color_data(3,:),'Mar22 14 120K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); hold on;myplot(energy_data_order_120K_Apr18_14',smooth(IP_data_order_120K_Apr18_14(:,i),n),'.',color_data(4,:),'',' Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_120K_Apr18_14,'-',color_data(4,:),'120K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_120K_Apr26_14',smooth(IP_data_order_120K_Apr26_14(:,i),n),'.',color_data(6,:),'','Pho ton energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_120K_Apr26_14,'-',color_data(6,:),'120K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); 226 myplot(x1(1,lin_log)',(y_lin_120K),'-',color_data(5,:),'120K','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log',(smooth(y_fit_120K_Apr18_14,s)+smooth(y_fit_120K_Apr26_14,s))/2,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % errorbar(en_log',(y_fit_121K_Jan22_14+y_fit_121K_Feb07_14+y_fit_120K_Feb22_14+y_fit_120K_Apr18_14+y_ fit_120K_Apr26_14)/5,errorbardata_120K');xlim([5 6]); % errorbar(en_log',y_120K_st,errorbardata_120K'); % myplot(en_log1',(smooth(y_fit1_121K_Jan22_14,s)+smooth(y_fit1_121K_Feb07_14,s)+smooth(y_fit1_120K_Feb2 2_14,s)+smooth(y_fit1_120K_Apr18_14,s)+smooth(y_fit1_120K_Apr26_14,s))/5,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log2',(smooth(y_fit2_120K_Apr18_14,s)+smooth(y_fit2_120K_Apr26_14,s))/2,'- ',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(x1,y_fit_st_120K,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(x1,y_120K_st,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); ylim([0 2]); subplot(4,4,7); myplot(en_log1',smooth(y_der1_121K_Jan22_14,m),'--',color_data(1,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); hold on; myplot(en_log1',smooth(y_der1_121K_Feb07_14,m),'--',color_data(2,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log1',smooth(y_der1_120K_Feb22_14,m),'--',color_data(3,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',smooth(y_der_120K_Apr18_14,m),'--',color_data(4,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',smooth(y_der_120K_Apr26_14,m),'--',color_data(6,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log',(smooth(y_der_120K_Apr18_14,m)+smooth(y_der_120K_Apr26_14,m))/2,'- ',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %errorbar(en_log',(smooth(y_der_120K_Apr18_14,m)+smooth(y_der_120K_Apr26_14,m))/2,errorbar_120K); % myplot(en_log1',(smooth(y_der1_121K_Jan22_14,m)+smooth(y_der1_121K_Feb07_14,m)+smooth(y_der1_120K_ Feb22_14,m)+smooth(y_der1_120K_Apr18_14,m)+smooth(y_der1_120K_Apr26_14,m))/5,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log2',(smooth(y_der2_120K_Apr18_14,m)+smooth(y_der2_120K_Apr26_14,m))/2,'- ',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %myplot(x1,y_der_120K_st,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(x1,smooth(y_120K_st_der,m),'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % errorbar(en_log',smooth(y_120K_st_der,m),errorbar_120K); xlim([5 6]); ylim([0 4]); 227 subplot(4,4,4); myplot(energy_data_order_145K_Apr18_14',smooth(IP_data_order_145K_Apr18_14(:,i),n),'.',color_data(3,:),'','Pho ton energy (h \nu)','Int','','Northwest',FS_leg); hold on;myplot(en_log',y_fit_145K_Apr18_14,'-',color_data(3,:),'145K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(x1(1,lin_log)',(y_lin_145K),'-',color_data(5,:),'145K','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); xlim([5 6]); ylim([0 2]); subplot(4,4,8); myplot(en_log',smooth(y_der_145K_Apr18_14,m),'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); xlim([5 6]); ylim([0 4]); subplot(4,4,11); myplot(energy_data_order_170K_Dec09_13',smooth(IP_data_order_170K_Dec09_13(:,i),n),'.',color_data(1,:),'','Ph oton energy (h \nu)','Int','','Northwest',FS_leg); hold on; myplot(en_log',y_fit_170K_Dec09_13,'-',color_data(1,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_170K_Jan03_14',smooth(IP_data_order_170K_Jan03_14(:,i),n),'.',color_data(2,:),'','Phot on energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_170K_Jan03_14,'-',color_data(2,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_170K_Jan31_14',smooth(IP_data_order_170K_Jan31_14(:,i),n),'.',color_data(3,:),'','Phot on energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log1',y_fit1_170K_Jan31_14,'-',color_data(3,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_170K_Feb22_14',smooth(IP_data_order_170K_Feb22_14(:,i),n),'.',color_data(4,:),'','Pho ton energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log1',y_fit1_170K_Feb22_14,'-',color_data(4,:),'170K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_170K_Apr26_14',smooth(IP_data_order_170K_Apr26_14(:,i),n),'.',color_data(6,:),'','Pho ton energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_170K_Apr26_14,'-',color_data(6,:),'170K , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); % errorbar(en_log',y_170K_st,errorbardata_170K'); % myplot(en_log',(smooth(y_fit_170K_Dec09_13,s)+smooth(y_fit_170K_Jan03_14,s)+smooth(y_fit_170K_Apr26_1 4,s))/3,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % errorbar(en_log',(y_fit_170K_Dec09_13+y_fit_170K_Jan03_14+y_fit_170K_Jan31_14+y_fit_170K_Feb22_14+y_ fit_170K_Apr26_14)/5,errorbardata_170K'); myplot(x1(1,lin_log)',(y_lin_170K),'-',color_data(5,:),'170K','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log1',(smooth(y_fit1_170K_Dec09_13,s)+smooth(y_fit1_170K_Jan03_14,s)+smooth(y_fit1_170K_Jan3 1_14,s)+smooth(y_fit1_170K_Feb22_14,s)+smooth(y_fit1_170K_Apr26_14,s))/5,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); 228 % myplot(en_log2',(smooth(y_fit2_170K_Dec09_13,s)+smooth(y_fit2_170K_Jan03_14,s)+smooth(y_fit2_170K_Apr 26_14,s))/3,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %myplot(x1,y_fit_st_170K,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(x1,y_170K_st,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); xlim([5 6]); ylim([0 2]); subplot(4,4,15); myplot(en_log',smooth(y_der_170K_Dec09_13,m),'--',color_data(1,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); hold on; myplot(en_log',smooth(y_der_170K_Jan03_14,m),'--',color_data(2,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log1',smooth(y_der1_170K_Jan31_14,m),'--',color_data(3,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log1',smooth(y_der1_170K_Feb22_14,m),'--',color_data(4,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log',smooth(y_der_170K_Apr26_14,m),'--',color_data(6,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log',(smooth(y_der_170K_Dec09_13,m)+smooth(y_der_170K_Jan03_14,m)+smooth(y_der_170K_Apr 26_14,m))/3,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %errorbar(en_log',(smooth(y_der_170K_Dec09_13,m)+smooth(y_der_170K_Jan03_14,m)+smooth(y_der_170K_A pr26_14,m))/3,errorbar_170K); % myplot(en_log1',(smooth(y_der1_170K_Dec09_13,m)+smooth(y_der1_170K_Jan03_14,m)+smooth(y_der1_170K_ Jan31_14,m)+smooth(y_der1_170K_Feb22_14,m)+smooth(y_der1_170K_Apr26_14,m))/5,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log2',(smooth(y_der2_170K_Dec09_13,m)+smooth(y_der2_170K_Jan03_14,m)+smooth(y_der2_170K_ Apr26_14,m))/3,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %myplot(x1,y_der_170K_st,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(x1,smooth(y_170K_st_der,m),'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % errorbar(en_log',smooth(y_170K_st_der,m),errorbar_170K); xlim([5 6]); ylim([0 4]); subplot(4,4,12); myplot(energy_data_order_230K_Mar22_14',smooth(IP_data_order_230K_Mar22_14(:,i),n),'.',color_data(1,:),'','Ph oton energy (h \nu)','Int','','Northwest',FS_leg); hold on; myplot(en_log',y_fit_Mar22_230K,'-',color_data(1,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %hold on; myplot(en_log',y1_fit_230K_Mar22_14,'--',color_data(1,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); 229 myplot(energy_data_order_230K_Mar28_14',smooth(IP_data_order_230K_Mar28_14(:,i),n),'.',color_data(2,:),'','Ph oton energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log',y_fit_Mar28_230K,'-',color_data(2,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); %hold on; myplot(en_log',y1_fit_230K_Mar28_14,'--',color_data(2,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_230K_Jan22_14',smooth(IP_data_order_230K_Jan22_14(:,i),n),'.',color_data(3,:),'','Phot on energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log1',y_fit1_230K_Jan22_14,'-',color_data(3,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); % %hold on; myplot(en_log',y1_fit_230K_Jan22_14,'--',color_data(3,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); myplot(energy_data_order_230K_Jan31_14',smooth(IP_data_order_230K_Jan31_14(:,i),n),'.',color_data(4,:),'','Phot on energy (h \nu)','Int','','Northwest',FS_leg); myplot(en_log1',y_fit1_230K_Jan31_14,'-',color_data(4,:),'230K, , Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); % %hold on; myplot(en_log',y1_fit_230K_Jan31_14,'--',color_data(4,:),'Mar22 14 230K, Al_{37}, Yield plot','Photon energy (h \nu)','Int','','Northwest',FS_leg); % errorbar(en_log',y_230K_st,errorbardata_230K'); % myplot(en_log',(smooth(y_fit_Mar22_230K,s)+smooth(y_fit_Mar28_230K,s)+smooth(y_fit_Jan22_230K,s)+smoot h(y_fit_Jan31_230K,s))/4,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % errorbar(en_log',(y_fit_Mar22_230K+y_fit_Mar28_230K+y_fit_Jan22_230K+y_fit_Jan31_230K)/4,errorbardata_2 30K'); myplot(x1(1,lin_log)',(y_lin_230K),'-',color_data(5,:),'230K','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log1',(smooth(y_fit1_230K_Mar22_14,s)+smooth(y_fit1_230K_Mar28_14,s)+smooth(y_fit1_230K_Jan 22_14,s)+smooth(y_fit1_230K_Jan31_14,s))/4,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log2',(smooth(y_fit2_230K_Mar22_14,s)+smooth(y_fit2_230K_Mar28_14,s))/2,'- ',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %myplot(x1,y_fit_st_230K,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(x1,y_230K_st,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); xlim([5 6]); ylim([0 2]); subplot(4,4,16); myplot(en_log',smooth(y_der_Mar22_230K,m),'--',color_data(1,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); hold on; % myplot(en_log2',smooth(y_der2_230K_Mar22_14,m),'--',color_data(1,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log1',smooth(y_der1_230K_Mar22_14,m),'--',color_data(1,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); hold on; 230 myplot(en_log',smooth(y_der_Mar28_230K,m),'--',color_data(2,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log1',smooth(y_der1_230K_Jan22_14,m),'--',color_data(3,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(en_log1',smooth(y_der1_230K_Jan31_14,m),'--',color_data(4,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log',(smooth(y_der_Mar22_230K,m)+smooth(y_der_Mar28_230K,m))/2,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % errorbar(en_log',(smooth(y_der_Mar22_230K,m)+smooth(y_der_Mar28_230K,m))/2,errorbar_230K); % myplot(en_log1',(smooth(y_der1_230K_Mar22_14,m)+smooth(y_der1_230K_Mar28_14,m)+smooth(y_der1_230K _Jan22_14,m)+smooth(y_der1_230K_Jan31_14,m))/4,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % myplot(en_log2',(smooth(y_der2_230K_Mar22_14,m)+smooth(y_der2_230K_Mar28_14,m))/2,'- ',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); %myplot(x1,y_der_230K_st,'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); myplot(x1,smooth(y_230K_st_der,m),'-',color_data(5,:),'Yield plot derivative','Photon energy (h \nu)','D(I(h \nu))','','Northwest',FS_leg); % errorbar(en_log',smooth(y_230K_st_der,m),errorbar_230K); xlim([5 6]); ylim([0 4]); % for 37 T=[65 90 120 145 170 230]; Top_peak=[2.799 2.179 1.896 1.868 1.376 1.801];%[2.837 (2.459+2.049)/2 (2.589+1.946)/2 1.932 (1.39+1.28)/2 (1.946+1.733)/2];%7pt[2.799 2.256 1.774 1.792 1.379 1.779]; Bot_peak_rt=[.611 .445 .904 .669 .646 1.12];%[.544 .337 .848 .593 .582 1.094];%7pt[.611 .387 .936 .705 .36 .585]; % Bot_peak_lt=[.36 .167 .288 .13 .319 .33];%[.32 .46 .28 .33 .34 .24]; % Top_peak2=[3.274 2.508 3.007 2.684 2.545 2.605]; %for 43 T=[65 90 120 145 170 230]; Top_peak=[1.129 .907 .918 1.008 .789 .954];%[2.837 (2.459+2.049)/2 (2.589+1.946)/2 1.932 (1.39+1.28)/2 (1.946+1.733)/2];%7pt[2.799 2.256 1.774 1.792 1.379 1.779]; Bot_peak_rt=[.656 .699 .781 .767 .689 .454]; peak_diff_rt=Top_peak-Bot_peak_rt; p_lin=[p_65_lin(1) p_90_lin(1) p_120_lin(1) p_145_lin(1) p_170_lin(1) p_230_lin(1)]; %plot(T,peak_diff_rt./p_lin,'*-'); % error_Al66=[sqrt(.514^2/(p_65_lin(1))^2+.288^2/(p_65_lin(1))^2) sqrt(.55^2/(p_90_lin(1))^2+.567^2/(p_90_lin(1))^2) sqrt(.406^2/(p_120_lin(1))^2+.535^2/(p_120_lin(1))^2) 0 sqrt(.428^2/(p_170_lin(1))^2+.508^2/(p_170_lin(1))^2) sqrt(.329^2/(p_230_lin(1))^2+.541^2/(p_230_lin(1))^2)]; % std_y=[sum(errorbardata_65K(30:55,1))/length([30:55]) sum(errorbardata_90K(30:55,1))/length([30:55]) ... % sum(errorbardata_120K(30:55,1))/length([30:55]) 0 sum(errorbardata_170K(30:55,1))/length([30:55]) ... % sum(errorbardata_230K(30:55,1))/length([30:55])]; err_66_65K=sqrt(2*sigsq_u_65K/(p_65_lin(1))^2+(peak_diff_rt(1)/(p_65_lin(1))^2)^2*sigsq_w_65K); err_66_90K=sqrt(2*sigsq_u_90K/(p_90_lin(1))^2+(peak_diff_rt(2)/(p_90_lin(1))^2)^2*sigsq_w_90K); err_66_120K=sqrt(2*sigsq_u_120K/(p_120_lin(1))^2+(peak_diff_rt(3)/(p_120_lin(1))^2)^2*sigsq_w_120K); 231 err_66_170K=sqrt(2*sigsq_u_170K/(p_170_lin(1))^2+(peak_diff_rt(5)/(p_170_lin(1))^2)^2*sigsq_w_170K); err_66_230K=sqrt(2*sigsq_u_230K/(p_230_lin(1))^2+(peak_diff_rt(6)/(p_230_lin(1))^2)^2*sigsq_w_230K); error_Al66=[err_66_65K err_66_90K err_66_120K 0 err_66_170K err_66_230K]; figure;errorbar(T,peak_diff_rt./p_lin,error_Al66); % figure;plot(T,peak_diff_rt./p_lin); % % ylim([0.5 1.8]); % figure;errorbar(T,peak_diff_rt,error_Al66); % ylim([0.5 2.5]); % Uncomment till here for smaller size clusters end end % T_log=[65 90 120 170 230]; % % Amp_log_37_65=(2.3+2.2+1.1)/3; % Amp_log_37_90=(1.2+1.8+1.4+1.8)/4; % Amp_log_37_120=(3.2+2.4+1.3)/3; % Amp_log_37_170=(0.2+.8+.5+.4)/4; % Amp_log_37_230=(.8+.7+.3+.2)/4; % % Amp_log_37=[Amp_log_37_65 Amp_log_37_90 Amp_log_37_120 Amp_log_37_170 Amp_log_37_230]; % % Amp_37_65=[2.3 2.2 1.1]; % Amp_37_90=[2 1.8 1.4 1.8]; % Amp_37_120=[3.2 2.4 1.3]; % Amp_37_170=[.2 .8 .5 .4]; % Amp_37_230=[.8 .7 .3 .2]; % std_37_65=std(Amp_37_65); % std_37_90=std(Amp_37_90); % std_37_120=std(Amp_37_120); % std_37_170=std(Amp_37_170); % std_37_230=std(Amp_37_230); % % % Amp_37_error=[std_37_65 std_37_90 std_37_120 std_37_170 std_37_230]; % % Amp_log_39_65=(.3+.6+.8)/3; % Amp_log_39_90=(1+.2+1+.4)/4; % Amp_log_39_120=4/3; % Amp_log_39_170=(0.3+.1+.2+.7)/4; % Amp_log_39_230=(.4+.3+.5+1.2)/4; % % Amp_log_39=[Amp_log_39_65 Amp_log_39_90 Amp_log_39_120 Amp_log_39_170 Amp_log_39_230]; % % Amp_39_65=[.3 .6 .8]; % Amp_39_90=[1 .2 1 .4]; % Amp_39_120=[3.7 4.3 4]; % Amp_39_170=[.3 .1 .2 .7]; % Amp_39_230=[.4 .3 .5 1.2]; % std_39_65=std(Amp_39_65); 232 % std_39_90=std(Amp_39_90); % std_39_120=std(Amp_39_120); % std_39_170=std(Amp_39_170); % std_39_230=std(Amp_39_230); % % % Amp_39_error=[std_39_65 std_39_90 std_39_120 std_39_170 std_39_230]; % % % Amp_log_43_65=(.8+.5+.2)/3; % Amp_log_43_90=(.4+.2+.4+.6)/4; % Amp_log_43_120=(.9+.4+.3)/3; % Amp_log_43_170=(.5+.2+.25+.2)/4; % Amp_log_43_230=(.1+.4+.3+.6)/4; % % Amp_log_43=[Amp_log_43_65 Amp_log_43_90 Amp_log_43_120 Amp_log_43_170 Amp_log_43_230]; % % Amp_43_65=[.8 .5 .2]; % Amp_43_90=[.4 .2 .4 .6]; % Amp_43_120=[.9 .4 .3]; % Amp_43_170=[.5 .2 .25 .2]; % Amp_43_230=[.1 .4 .3 .6]; % std_43_65=std(Amp_43_65); % std_43_90=std(Amp_43_90); % std_43_120=std(Amp_43_120); % std_43_170=std(Amp_43_170); % std_43_230=std(Amp_43_230); % % % Amp_43_error=[std_43_65 std_43_90 std_43_120 std_43_170 std_43_230]; % % % Amp_log_66_65=(1.2+.4+.5)/3; % Amp_log_66_90=(1.0+.9+.4+.3)/4; % Amp_log_66_120=(.1+.05+.05)/3; % Amp_log_66_170=(0.01+.02+.4+.3)/4; % Amp_log_66_230=(.8+.7+.01+.2)/4; % % Amp_log_66=[Amp_log_66_65 Amp_log_66_90 Amp_log_66_120 Amp_log_66_170 Amp_log_66_230]; % % Amp_66_65=[1.2 .4 .5]; % Amp_66_90=[1.0 .9 .4 .3]; % Amp_66_120=[.1 .05 .05]; % Amp_66_170=[.01 .02 .4 .3]; % Amp_66_230=[.8 .7 .01 .2]; % std_66_65=std(Amp_66_65); % std_66_90=std(Amp_66_90); % std_66_120=std(Amp_66_120); % std_66_170=std(Amp_66_170); % std_66_230=std(Amp_66_230); % % % Amp_66_error=[std_66_65 std_66_90 std_66_120 std_66_170 std_66_230]; % % % Amp_log_68_65=(1+.7+.3)/3; 233 % Amp_log_68_90=(2.0+1+.6+.3)/4; % Amp_log_68_120=(1.6+.6+.7)/3; % Amp_log_68_170=(0.1+.2+.2)/4; % Amp_log_68_230=(.2+.3+.1)/4; % % Amp_log_68=[Amp_log_68_65 Amp_log_68_90 Amp_log_68_120 Amp_log_68_170 Amp_log_68_230]; % % Amp_68_65=[1 .7 .3]; % Amp_68_90=[2.0 1 .6 .3]; % Amp_68_120=[1.6 .6 .7]; % Amp_68_170=[.1 .2 .2]; % Amp_68_230=[.2 .3 .1]; % std_68_65=std(Amp_68_65); % std_68_90=std(Amp_68_90); % std_68_120=std(Amp_68_120); % std_68_170=std(Amp_68_170); % std_68_230=std(Amp_68_230); % % % Amp_68_error=[std_68_65 std_68_90 std_68_120 std_68_170 std_68_230]; % % h2=figure;subplot(2,3,1);errorbar(T_log,Amp_log_37,Amp_37_error,'*-');ylim([0 4]);grid minor;title('Al_{37}');xlabel('T(K)');ylabel('Amp(arb.units)'); % hold on; % subplot(2,3,2);errorbar(T_log,Amp_log_39,Amp_39_error,'*-');ylim([0 4]);grid minor;title('Al_{39}');xlabel('T(K)');ylabel('Amp(arb.units)'); % subplot(2,3,3);errorbar(T_log,Amp_log_43,Amp_43_error,'*-');ylim([0 4]);grid minor;title('Al_{43}');xlabel('T(K)');ylabel('Amp(arb.units)'); % subplot(2,3,4);errorbar(T_log,Amp_log_66,Amp_66_error,'*-');ylim([0 4]);grid minor;title('Al_{66}');xlabel('T(K)');ylabel('Amp(arb.units)'); % subplot(2,3,5);errorbar(T_log,Amp_log_68,Amp_68_error,'*-');ylim([0 4]);grid minor;title('Al_{68}');xlabel('T(K)');ylabel('Amp(arb.units)'); % % print(gcf, '-r300','-dpng', 'Amp shift with T'); % saveas(h2,'Amp shift with T.fig'); % % % Pos % % Pos_log_37_65=(5.2+5.23+5.15)/3; % Pos_log_37_90=(5.17+5.21+5.21+5.2)/4; % Pos_log_37_120=(5.2+5.2+5.2)/3; % % % Pos_log_37=[Pos_log_37_65 Pos_log_37_90 Pos_log_37_120 0 0]; % % Pos_37_65=[5.2 5.23 5.15]; % Pos_37_90=[5.17 5.21 5.21 5.2]; % Pos_37_120=[5.2 5.2 5.2]; % % std_37_65=std(Pos_37_65); % std_37_90=std(Pos_37_90); % std_37_120=std(Pos_37_120); % % Pos_37_error=[std_37_65 std_37_90 std_37_120 0 0]; % % Pos_log_39_65=(5.19+5.21+5.2)/3; 234 % Pos_log_39_90=(5.2+5.18+5.17+5.2)/4; % Pos_log_39_120=(5.2+5.2+5.2)/3; % % Pos_log_39=[Pos_log_39_65 Pos_log_39_90 Pos_log_39_120 0 0]; % % Pos_39_65=[5.19 5.21 5.2]; % Pos_39_90=[5.2 5.18 5.17 5.2]; % Pos_39_120=[5.2 5.2 5.2]; % % std_39_65=std(Pos_39_65); % std_39_90=std(Pos_39_90); % std_39_120=std(Pos_39_120); % % Pos_39_error=[std_39_65 std_39_90 std_39_120 0 0]; % % % Pos_log_43_65=(5.21+5.24+5.21)/3; % Pos_log_43_90=(5.21+5.23+5.2+5.2)/4; % Pos_log_43_120=(5.2+5.2+5.2)/3; % % Pos_log_43=[Pos_log_43_65 Pos_log_43_90 Pos_log_43_120 0 0]; % % Pos_43_65=[5.21 5.24 5.21]; % Pos_43_90=[5.21 5.23 5.2 5.2]; % Pos_43_120=[5.2 5.2 5.2]; % % std_43_65=std(Pos_43_65); % std_43_90=std(Pos_43_90); % std_43_120=std(Pos_43_120); % % Pos_43_error=[std_43_65 std_43_90 std_43_120 0 0]; % % % Pos_log_66_65=(5.41+5.4+5.46)/3; % Pos_log_66_90=(5.43+5.48+5.46+5.45)/4; % % % Pos_log_66=[Pos_log_66_65 Pos_log_66_90 0 0 0]; % % Pos_66_65=[5.41 5.4 5.46]; % Pos_66_90=[5.43 5.48 5.46 5.45]; % % std_66_65=std(Pos_66_65); % std_66_90=std(Pos_66_90); % % Pos_66_error=[std_66_65 std_66_90 0 0 0]; % % % Pos_log_68_65=(5.46+5.47+5.46)/3; % Pos_log_68_90=(5.46+5.53+5.48+5.44)/4; % Pos_log_68_120=(5.43+5.42+5.41)/3; % % Pos_log_68=[Pos_log_68_65 Pos_log_68_90 Pos_log_68_120 0 0]; % % Pos_68_65=[5.46 5.47 5.46]; % Pos_68_90=[5.46 5.53 5.48 5.44]; 235 % Pos_68_120=[5.43 5.42 5.41]; % % std_68_65=std(Pos_68_65); % std_68_90=std(Pos_68_90); % std_68_120=std(Pos_68_120); % % Pos_68_error=[std_68_65 std_68_90 std_68_120 0 0]; % % h2=figure;subplot(2,3,1);errorbar(T_log,Pos_log_37,Pos_37_error,'*-');ylim([4.5 6]);grid minor;title('Al_{37}');xlabel('T(K)');ylabel('Pos(eV)'); % hold on; % subplot(2,3,2);errorbar(T_log,Pos_log_39,Pos_39_error,'*-');ylim([4.5 6]);grid minor;title('Al_{39}');xlabel('T(K)');ylabel('Pos(eV)'); % subplot(2,3,3);errorbar(T_log,Pos_log_43,Pos_43_error,'*-');ylim([4.5 6]);grid minor;title('Al_{43}');xlabel('T(K)');ylabel('Pos(eV)'); % subplot(2,3,4);errorbar(T_log,Pos_log_66,Pos_66_error,'*-');ylim([4.5 6]);grid minor;title('Al_{66}');xlabel('T(K)');ylabel('Pos(eV)'); % subplot(2,3,5);errorbar(T_log,Pos_log_68,Pos_68_error,'*-');ylim([4.5 6]);grid minor;title('Al_{68}');xlabel('T(K)');ylabel('Pos(eV)'); % % print(gcf, '-r300','-dpng', 'Pos shift with T'); % saveas(h2,'Pos shift with T.fig'); % % % Wid % % Wid_log_37_65=(.2+.19+.27)/3; % Wid_log_37_90=(.17+.18+.17+.18)/4; % Wid_log_37_120=(.12+.16+.2)/3; % % % Wid_log_37=[Wid_log_37_65 Wid_log_37_90 Wid_log_37_120 0 0]; % % Wid_37_65=[.2 .19 .27]; % Wid_37_90=[.17 .18 .17 .18]; % Wid_37_120=[.12 .16 .2]; % % std_37_65=std(Wid_37_65); % std_37_90=std(Wid_37_90); % std_37_120=std(Wid_37_120); % % Wid_37_error=[std_37_65 std_37_90 std_37_120 0 0]; % % Wid_log_39_65=(.15+.3+.28)/3; % Wid_log_39_90=(.13+.17+.2+.12)/4; % Wid_log_39_120=(.12+.13+.14)/3; % % Wid_log_39=[Wid_log_39_65 Wid_log_39_90 Wid_log_39_120 0 0]; % % Wid_39_65=[.15 .3 .28]; % Wid_39_90=[.13 .17 .2 .12]; % Wid_39_120=[.12 .13 .14]; % % std_39_65=std(Wid_39_65); % std_39_90=std(Wid_39_90); % std_39_120=std(Wid_39_120); % 236 % Wid_39_error=[std_39_65 std_39_90 std_39_120 0 0]; % % % Wid_log_43_65=(.19+.25+.28)/3; % Wid_log_43_90=(.18+.18+.23+.18)/4; % Wid_log_43_120=(.2+.3+.3)/3; % % Wid_log_43=[Wid_log_43_65 Wid_log_43_90 Wid_log_43_120 0 0]; % % Wid_43_65=[.19 .25 .28]; % Wid_43_90=[.18 .18 .23 .18]; % Wid_43_120=[.2 .3 .3]; % % std_43_65=std(Wid_43_65); % std_43_90=std(Wid_43_90); % std_43_120=std(Wid_43_120); % % Wid_43_error=[std_43_65 std_43_90 std_43_120 0 0]; % % % Wid_log_66_65=(.1+.08+.1)/3; % Wid_log_66_90=(.1+.11+.1+.07)/4; % % % Wid_log_66=[Wid_log_66_65 Wid_log_66_90 0 0 0]; % % Wid_66_65=[.1 .08 .1]; % Wid_66_90=[.1 .11 .1 .07]; % % std_66_65=std(Wid_66_65); % std_66_90=std(Wid_66_90); % % Wid_66_error=[std_66_65 std_66_90 0 0 0]; % % % Wid_log_68_65=(.1+.1+.11)/3; % Wid_log_68_90=(.14+.2+.1+.05)/4; % Wid_log_68_120=(.13+.15+.12)/3; % % Wid_log_68=[Wid_log_68_65 Wid_log_68_90 Wid_log_68_120 0 0]; % % Wid_68_65=[.1 .1 .11]; % Wid_68_90=[.14 .2 .1 .05]; % Wid_68_120=[.13 .15 .12]; % % std_68_65=std(Wid_68_65); % std_68_90=std(Wid_68_90); % std_68_120=std(Wid_68_120); % % Wid_68_error=[std_68_65 std_68_90 std_68_120 0 0]; % % h2=figure;subplot(2,3,1);errorbar(T_log,Wid_log_37,Wid_37_error,'*-');ylim([0 .4]);grid minor;title('Al_{37}');xlabel('T(K)');ylabel('Wid(eV)'); % hold on; % subplot(2,3,2);errorbar(T_log,Wid_log_39,Wid_39_error,'*-');ylim([0 .4]);grid minor;title('Al_{39}');xlabel('T(K)');ylabel('Wid(eV)'); 237 % subplot(2,3,3);errorbar(T_log,Wid_log_43,Wid_43_error,'*-');ylim([0 .4]);grid minor;title('Al_{43}');xlabel('T(K)');ylabel('Wid(eV)'); % subplot(2,3,4);errorbar(T_log,Wid_log_66,Wid_66_error,'*-');ylim([0 .4]);grid minor;title('Al_{66}');xlabel('T(K)');ylabel('Wid(eV)'); % subplot(2,3,5);errorbar(T_log,Wid_log_68,Wid_68_error,'*-');ylim([0 .4]);grid minor;title('Al_{68}');xlabel('T(K)');ylabel('Wid(eV)'); % % print(gcf, '-r300','-dpng', 'Wid shift with T'); % saveas(h2,'Wid shift with T.fig'); % comment from here for Amp compare plots % Al64 % T=[65 90 120 145 170 230]; % Top_peak=[1.96 2.41 2.28 2.84 1.99 2.21]; % Bot_peak_rt=[1.54 1.33 1.48 .25 1.62 1.02]; % peak_diff_rt=Top_peak-Bot_peak_rt; % % figure; % p_lin=[p_65_lin(1) p_90_lin(1) p_120_lin(1) p_145_lin(1) p_170_lin(1) p_230_lin(1)]; % plot(T,peak_diff_rt./p_lin,'o-'); % % % Al68 % % T=[65 90 120 145 170 230]; % Top_peak=[2.89 3.06 2.48 2.84 3.03 2.14]; % Bot_peak_rt=[1.78 1.61 1.61 1.79 2.14 2.02]; % peak_diff_rt=Top_peak-Bot_peak_rt; % % figure; % p_lin=[p_65_lin(1) p_90_lin(1) p_120_lin(1) p_145_lin(1) p_170_lin(1) p_230_lin(1)]; % plot(T,peak_diff_rt./p_lin,'s-'); % % % for Al37 % % T=[65 90 120 145 170 230]; % Top_peak=[2.837 2.392 1.866 1.852 1.347 1.708]; % Bot_peak_rt=[.544 .226 .905 .667 .289 .502]; % peak_diff_rt=Top_peak-Bot_peak_rt; % % figure; % p_lin=[p_65_lin(1) p_90_lin(1) p_120_lin(1) p_145_lin(1) p_170_lin(1) p_230_lin(1)]; % plot(T,peak_diff_rt./p_lin,'o-'); % % error_Al37=[sqrt(.958^2/(p_65_lin(1))^2+.412^2/(p_65_lin(1))^2) sqrt(.429^2/(p_90_lin(1))^2+.029^2/(p_90_lin(1))^2) sqrt(.062^2/(p_120_lin(1))^2+.047^2/(p_120_lin(1))^2) 0 sqrt(.073^2/(p_170_lin(1))^2+.226^2/(p_170_lin(1))^2) sqrt(.414^2/(p_230_lin(1))^2+.142^2/(p_230_lin(1))^2)]; % % figure;errorbar(T,peak_diff_rt./p_lin,error_Al37); % % %for Al66 % figure; % T=[65 90 120 145 170 230]; % Top_peak=[2.785 2.575 2.55 2.509 2.5 2.362]; % Bot_peak_rt=[.964 .922 1.348 1.317 1.579 1.857]; % Bot_peak_lt=[.36 .167 .288 .13 .319 .33];%[.32 .46 .28 .33 .34 .24]; 238 % Top_peak2=[3.274 2.508 3.007 2.684 2.545 2.605]; % peak_diff_rt=Top_peak-Bot_peak_rt; % peak_diff_lt=Top_peak-Bot_peak_lt; % peak_ratio=peak_diff_rt./peak_diff_lt; % % peak_diff_rt2=Top_peak2-Bot_peak_rt; % peak_ratio2=peak_diff_rt2./peak_diff_lt; % % figure;plot(T,peak_ratio2/peak_ratio2(6),'g*-'); % % figure;plot(T,peak_diff_rt/peak_diff_rt(6),'*-'); % hold on;plot(T,peak_diff_lt/peak_diff_lt(6),'r*-'); % plot(T,peak_ratio/peak_ratio(6),'g*-'); % % figure;plot(T,peak_diff_rt/peak_diff_rt(1),'*-'); % hold on;plot(T,peak_diff_lt/peak_diff_lt(1),'r*-'); % plot(T,peak_ratio/peak_ratio(1),'g*-'); % % figure; % p_lin=[p_65_lin(1) p_90_lin(1) p_120_lin(1) p_145_lin(1) p_170_lin(1) p_230_lin(1)]; % plot(T,peak_diff_rt./p_lin,'*-'); % % error_Al66=[sqrt(.514^2/(p_65_lin(1))^2+.288^2/(p_65_lin(1))^2) sqrt(.55^2/(p_90_lin(1))^2+.567^2/(p_90_lin(1))^2) sqrt(.406^2/(p_120_lin(1))^2+.535^2/(p_120_lin(1))^2) 0 sqrt(.428^2/(p_170_lin(1))^2+.508^2/(p_170_lin(1))^2) sqrt(.329^2/(p_230_lin(1))^2+.541^2/(p_230_lin(1))^2)]; % % figure;errorbar(T,peak_diff_rt./p_lin,error_Al66); % Uncomment till here for Amp ana plots % peak_diff=Top_peak-Bot_peak; % subplot(2,3,1);plot(T,peak_diff,'*-'); % title('The diff between peak and bot'); % % subplot(2,3,2);plot(T,peak_diff./peak_diff(1),'*-'); % title('Peak diff/Peak diff(65K)'); % % subplot(2,3,3);plot(T,peak_diff./peak_diff(6),'*-'); % title('Peak diff/Peak diff(230K)'); % % p_lin=[p_65_lin(1) p_90_lin(1) p_120_lin(1) p_145_lin(1) p_170_lin(1) p_230_lin(1)]; % subplot(2,3,4);plot(T,p_lin,'*-'); % title('The slope between 5.15eV to 5.5eV'); % % subplot(2,3,5);plot(T,peak_diff./p_lin,'*-'); % title('Peak diff/slope'); % % subplot(2,3,6);plot(T,peak_diff.*p_lin,'*-'); % title('Peak diff*slope'); % % Pos_peak1=[5.44 5.44 5.42*ones(1,4)]; % Pos_peak2=[5.66 5.6 5.6 5.58 5.58 5.51]; % % figure;plot(T,Pos_peak1,'*-'); % hold on;plot(T,Pos_peak2,'r*-'); 239 Starting from these fundamental algorithm and simple modifications we have analysed the data for most of the results plotted from Chapter 4 - 6. 240 Appendix C Appearance energy of Al clusters measured in units of eV. Al cluster AE 65K stdv 65K AE 90K stdv 90K AE 120K stdv 120K AE 170K stdv 170K AE 230K stdv 230K 32 5.57 0.03 5.57 0.04 5.48 0.12 5.55 0.07 5.43 0.2 33 5.44 0.05 5.43 0.17 5.49 0.08 5.36 0.21 5.35 0.26 34 5.35 0.17 5.51 0.11 5.39 0.18 5.7 0.73 5.07 0.7 35 5.48 0.04 5.24 0.33 5.44 0.07 4.77 1.11 5.39 0.16 36 5.53 0.09 5.52 0.09 5.49 0.05 5.51 0.12 5.3 0.34 37 5.03 0.02 4.99 0.03 5.01 0.05 4.79 0.4 4.97 0.07 38 5.39 0.04 5.36 0.04 5.35 0.02 5.37 0.06 5.36 0.07 39 4.95 0.06 4.87 0.08 5.02 0.05 4.89 0.16 5.05 0.11 40 5.35 0.05 5.31 0.06 5.33 0.05 5.33 0.03 5.35 0.08 41 5.32 0.01 5.29 0.03 5.3 0.04 5.3 0.02 5.29 0.04 42 5.35 0.01 5.3 0.06 5.32 0.06 5.33 0.06 5.28 0.16 43 5.08 0.04 5.07 0.05 5.1 0.06 5.05 0.09 5.11 0.05 44 5.24 0.03 5.25 0.03 5.23 0.01 5.22 0.02 5.24 0.05 45 5.21 0.01 5.19 0.07 5.19 0.03 5.18 0.02 5.17 0.03 46 5.27 0 5.25 0.02 5.22 0.05 5.23 0.03 5.27 0.03 47 5.18 0.02 4.82 0.49 5.12 0.08 5.19 0.06 5.24 0.04 48 5.25 0 5.23 0.02 5.25 0.04 5.24 0.04 5.25 0.03 49 5.31 0.01 5.3 0.03 5.3 0.01 5.25 0.06 5.28 0.03 50 5.32 0.02 5.3 0 5.29 0.03 5.3 0.03 5.28 0.03 51 5.27 0.01 5.2 0.05 5.25 0.01 5.24 0.04 5.26 0.03 52 5.34 0.01 5.28 0.01 5.31 0.02 5.2 0.11 5.26 0.04 53 5.27 0.02 5.19 0.1 5.27 0.08 5.23 0.1 5.27 0.08 54 5.3 0.02 5.23 0.02 5.29 0.01 5.25 0.04 5.27 0.04 55 5.25 0.02 5.2 0.04 5.22 0.04 5.18 0.04 5.2 0.03 56 5.2 0.01 5.21 0.01 5.21 0.01 5.2 0.03 5.21 0.02 57 5.18 0.02 5.15 0.04 5.18 0.02 5.1 0.07 5.15 0.06 58 5.17 0 5.16 0.05 5.18 0.01 5.17 0.03 5.18 0.02 59 5.15 0.01 5.17 0.01 5.14 0.03 5.17 0.02 5.14 0.02 60 5.2 0 5.2 0.01 5.21 0.01 5.2 0.01 5.2 0 61 5.17 0 5.16 0.03 5.17 0.02 5.15 0.01 5.16 0.03 62 5.27 0.01 5.25 0.01 5.26 0.02 5.24 0.08 5.21 0.15 63 5.21 0.04 5.18 0.05 5.21 0.04 5.17 0.05 5.2 0.06 64 5.25 0.01 5.24 0.04 5.26 0.01 5.22 0.04 5.24 0.03 65 5.16 0.01 5.18 0.03 5.19 0.01 5.18 0.03 5.18 0.03 66 5.27 0.01 5.24 0.02 5.27 0.01 5.25 0.03 5.25 0.01 67 5.05 0.03 5.05 0.02 5.06 0.03 5.05 0.04 5.04 0.04 241 68 5.21 0.01 5.18 0 5.2 0.01 5.19 0.02 5.13 0.03 69 5.17 0.03 5.16 0.02 5.18 0.01 5.16 0.01 5.15 0.05 70 5.27 0.01 5.22 0.02 5.27 0.04 5.16 0.13 5.26 0.03 71 5.26 0.02 5.23 0.01 5.26 0.02 5.23 0.04 5.22 0.05 72 5.25 0.04 5.24 0.04 5.26 0.05 5.18 0.15 5.36 0.24 73 5.22 0.02 5.19 0.03 5.21 0.02 5.2 0.04 5.19 0.03 74 5.21 0.03 5.19 0.03 5.18 0.04 5.15 0.05 5.16 0.03 75 5.14 0 5.11 0.02 5.13 0.02 5.1 0.04 5.1 0.03 76 5.13 0 5.11 0.02 5.11 0 5.11 0.03 5.09 0.01 77 5.1 0.01 5.08 0.01 5.1 0.02 5.09 0.04 5.1 0.02 78 5.1 0 5.1 0.01 5.1 0.01 5.1 0.03 5.1 0.02 79 5.07 0.01 5.07 0.01 5.06 0.01 5.06 0.02 5.07 0.01 80 5.08 0.02 5.06 0.03 5.07 0.03 5.08 0.03 5.08 0.02 81 5.06 0.03 5.1 0.04 5.03 0.04 5.06 0.07 5.07 0.04 82 5.08 0.01 5.06 0.01 5.09 0.03 5.09 0.03 5.09 0.02 83 5.09 0.02 5.06 0.01 5.08 0.02 5.07 0.02 5.07 0.01 84 5.1 0.03 5.11 0.01 5.1 0.03 5.1 0.02 5.08 0.02 85 5.09 0.01 5.07 0.02 5.08 0 5.08 0.03 5.07 0.04 86 5.06 0.02 5.09 0.01 5.09 0.02 5.07 0.02 5.06 0.06 87 5.07 0.01 5.05 0.02 5.07 0.02 5.06 0.03 5.06 0.01 88 5.1 0.04 5.08 0.02 5.12 0.03 5.08 0.02 5.08 0.07 89 5.07 0 5.02 0.08 5.07 0.02 5.07 0.05 5.06 0.02 90 5.08 0.02 5.08 0.06 5.08 0.02 5.08 0.01 5.05 0.02 91 5.06 0.01 5.06 0.01 5.04 0.03 5.03 0.02 5.05 0.02 92 5.11 0.02 5.07 0.04 5.1 0.02 5.09 0.02 5.04 0.09 93 5.05 0.03 5.08 0.03 5.09 0.02 5.08 0.05 5.06 0.06 94 5.07 0.04 4.99 0.19 5.1 0.01 5.09 0.04 5.07 0.03 95 5.06 0.02 5.05 0.03 5.09 0.03 5.08 0.03 5.07 0.02 242 Appendix D The appearance energies of Cu clusters: N (Custer size Cu n ) 61 K (eV) 215K (eV) 24 5.4 5.3 25 5.11 5 26 5.4 5.39 27 5.13 5.09 28 5.31 5.28 29 5.05 5.03 30 5.44 5.42 31 5.15 5.04 32 5.25 5.16 33 5.1 5.06 34 5.4 5.37 35 5.04 4.97 36 5.34 5.3 37 5.18 5.1 38 5.37 5.32 39 5.26 5.23 40 5.42 5.45 41 4.95 4.97 42 5.13 5.17 43 5.13 5.11 44 5.15 5.15 45 5.17 5.13 46 5.22 5.21 47 5.16 5.14 48 5.1 5.13 49 4.77 4.85 50 5.02 5.03 51 5.06 5.07 52 5.15 5.14 53 5.22 5.22 54 5.26 5.26 55 5.24 5.25 56 5.25 5.26 57 5.3 5.3 58 5.28 5.29 59 5.15 5.15 60 5.21 5.23 243 61 4.94 4.94 62 4.94 4.95 63 4.95 4.94 64 5.01 5.05 65 5.05 5.08 66 5.05 5.09 67 5.04 5.05 68 5.12 5.15 69 5.07 5.09 70 5.12 5.15 71 5.07 5.1 72 5.08 5.1 73 5.07 5.1 74 5.11 5.11 75 5.11 5.13 76 5.13 5.15 77 5.15 5.17 78 5.14 5.11 79 5.12 5.13 80 5.14 5.15 81 5.15 5.17 82 5.1 5.05 83 5.17 5.21 84 5.18 5.15 85 5.2 5.18 86 5.2 5.18 87 5.22 5.2 88 5.25 5.26 89 5.24 5.25 90 5.29 5.3 91 5.3 5.31 92 4.9 4.92 93 5 5.02 94 4.98 5 95 5 5.05 96 5.01 5.01 97 5.02 5.05 98 5 5.02 244 Appendix E E.1: Calculation of proportionality constant F for n(x,y) From the normalization condition which depends on the fact that thetotal number of electrons inside the dot is N we obtain thefollowing condition,         N dy dx b y a x ab F b a             ' ' ' ' 1 2 3 , 0 , 0 2 2 2 2  (E.1) We redefine our variables as y b y x a x ~ ' ; ~ '   . In terms of the new variables (E.1) reduces to,       N y d x d y x F     ~ ~ ~ ~ 1 2 3 1 , 1 0 , 0 2 2  (E.2) The above equation (E.2) can be recognized as the simpleintegration over an unit circle, 1 ~ ~ 2 2   y x where in terms of the circular coordinates (E.2)can bewritten as,       N d r d r r F       ~ ~ ~ ~ 1 2 3 2 , 1 0 , 0 2 (E.3) which yields F=N. E.2: Solution for {µ , γ 1 , γ 2 } For evaluating the integrals from (8.12) we use theidentities [276],                             0 1 2 1 0 Re Re 1 ; ; , , 1                F b x x x (E.4) 245 Where B(x',y') are Beta functions and     z F z F ; ; , ; ; , 1 2        are theGauss Hypergeometric function. Let us write the eqs. from (8.12) as       2 2 2 1 2 1 0 2 4 ~ 3 , 4 ~ 3 , 4 ~ 3 I e N I e N I e N       . The parametric integral I 0 can be solved as follows,         2 1 0 2 1 2 2 1 0 2 2 2 2 3 0 2 2 2 2 2 2 2 0 2 2 0 1 1 1 1                                                 x x k x dx a a a k a a d a a a k a a a d b a d I             (E.5) where we assumed   1 ,    a b k b a and 2 a x   .The above eq. (E.5) can be compared with (E.4)to yield 1 , 2 1 , 2 1       . On usingthese parameters and comparing with (8.12) we get,                  2 1 , 2 1 ' ; 1 ; 2 1 , 2 1 1 2 1 2 0 B k F a I (E.6) where   a b k k k    , 1 ' 2 2 . This I 0 expression can befurther simplified by the use of yet another identity [276],            2 ' ; 1 ; 2 1 , 2 1 2 ' k F k K  (E.7) Thus we have,       2 ' ; 1 ; 2 1 , 2 1 1 2 ' k F k K  . Withall these algebra we solved the first eq. of (8.12) toobtain,       2 ' 2 1 , 2 1 4 ~ 3 2 k K B a e N        (E.8) 246 The second integral I 1 can be solved as follows,               2 1 0 2 1 2 2 3 3 0 2 1 2 1 2 2 3 3 0 2 2 2 2 2 3 2 6 0 2 3 2 1 1 1 1 1 1                                          x x k x dx a x x k x d a a a a k a a a d b a d I          (E.9) with similar analogy as above we get, 2 , 2 1 , 2 1       . With these parameters the integral evaluates to,                  2 1 , 2 3 ' ; 2 ; 2 3 , 2 1 1 2 1 2 3 1 B k F a I (E.10) Finally we solve the last integral I 2 as follows,               2 1 0 2 3 2 2 1 3 0 3 2 5 0 2 2 2 3 2 2 6 2 2 0 3 2 2 2 1 1 1 1 1                                          x x k x dx a x x k x d a a a a k a a a d b a d I          (E.11) The parameters that we get on similar analogy, 2 , 2 1 , 2 3       .With these parameter the integral evaluated to,                  2 1 , 2 3 ' ; 2 ; 2 3 , 2 3 1 2 1 2 3 2 B k F a I (E.12) 247 E.3: Integrals used for deriving E p     15 2 1 3 1 1 sin cos 1 0 2 3 1 0 2 2 0 2 2 0 2           dr r r dr r r d d        (E.13) E.4 :Solution for { µ , γ 1 , γ 2 } For evaluating the integrals from eq.(8.28) we use theidentities as already mentioned in eq. (E.4) [276]. Let us write the eqs. (8.28)as             2 2 * 2 1 2 * 1 0 2 * 4 3 , 4 3 , 4 3 I e N I e N I e N       . The parametric integral I 0 can be solved as follows,                                                       0 2 1 2 1 0 2 2 2 3 0 2 2 2 2 2 0 2 2 0 1 1 1 1 x k x dx R R k R R d R k R R R d c R d I          (E.14) where we assumed   1 ,    R c k c R and 2 R x   .The above eq. (E.14) can be compared with (E.4) to yield 2 3 , 1 , 2 1       . On usingthese parameters and comparing with (8.28) we get,                  1 , 2 1 ' ; 2 3 ; 2 1 , 2 1 1 2 1 2 0 B k F R I (E.15) 248 where   R c k k k    , 1 ' 2 2 . Withall these algebra we solved the first eq. of (8.12) toobtain,          2 1 2 2 * ' ; 2 3 ; 2 1 , 2 1 2 3 k F N R e  (E.16) The second integral I 1 can be solved as follows,                                      0 2 1 2 2 3 0 2 2 2 2 4 0 2 2 2 1 1 1 1 x k x dx R R k R R R d c R d I       (E.17) with similar analogy as above we get, 2 5 , 1 , 2 1       . With these parameters the integral evaluates to,                  1 , 2 3 ' ; 2 5 ; 2 3 , 2 1 1 2 1 2 3 1 B k F R I (E.18) Finally we solve the last integral I 2 as follows,                                      0 2 3 2 1 3 0 3 2 2 3 2 2 0 3 2 2 2 1 1 1 x k x dx R R k R R R d c R d I       (E.19) The parameters that we get on similar analogy, 2 5 , 1 , 2 3       .With these parameter the integral evaluated to,                  1 , 2 3 ' ; 2 5 ; 2 3 , 2 3 1 2 1 2 3 2 B k F R I (E.19) 249 E.5: Integrals used for deriving E p The integration over the spheroid is transformed to that over anunit sphere, by doing the following transformation of variablesx=Rx',y=Ry',z=cz'. The volume elements are transformed as dxdydz = J dx' dy' dz' where J=a 2 c is the Jacobian for transformingthe integral over an ellipsoid   1 2 2 2 2 2    c z R y x to that over an unit spherex’ 2 +y' 2 +z' 2 =1. For evaluating the other integrals, we first transform to the{x',y',z'} coordinate system and subsequently to spherical polarsystem to provide the following important results, 3 2 1 2 4 1 2 15 4 15 4 2 2 2 2 2 2 2 2 2 2 c R dxdydz z c R dxdydz x c z R y x c z R y x             (E.20)

Abstract (if available)

Abstract A unique property of size-resolved metal nanocluster particles is their “superatom”-like electronic shell structure. The shell levels are highly degenerate, and it has been predicted that this can enable exceptionally strong superconducting-type electron pair correlations in certain clusters composed of just tens to hundreds of atoms. In our experiment we observed a spectroscopic signature of such an effect. A bulge-like feature appears in the photoionization yield curve of a few free cold closed-shell (magic clusters) or near closed shell (clusters with slight Jahn-Teller distortion) aluminum clusters and shows a rapid rise as the temperature approaches ≈100 K. The novel behavior, previously not reported for clusters, implies an increase in the effective density of states close to Fermi level and the closing of energy spacing between highest occupied (HOS) and lowest unoccupied shells (LUS). It is consistent with a pairing transition and suggests a high-temperature superconducting state with Tc≳100 K. Unlike bulk superconductors the phase transition has a noticeable broadening due to quantum fluctuations. Our results highlight the promise of metal nanoclusters as high-Tc building blocks for materials and networks. ❧ Photoionization yield (Y(E)) curves obtained for small aluminum and copper clusters show a quadratic rise around the threshold which is consistent with the theoretical Fowler prediction for the bulk surfaces at low temperature. The data can be fitted to this model to derive the appearance energies (AEs) of the clusters. Sharp drop in AEs have been observed in certain specific sizes that can be related to electronic shell closings. Self-consistent shell model and ellipsoidal Clemenger-Nilsson model have been quite successful in predicting the shell structure for alkali clusters. The clusters under survey are much more complicated as they undergo strong perturbation to the free electron behavior. Careful analysis of these features helps in understanding the electronic and geometric structures of these clusters. ❧ A controlled variation of the cluster temperature within the range 60K to 230K has been performed to derive accurate value for the thermal coefficient of appearance energies.The size dependent evolution of AEs is consistent with the electrostatic spherical droplet model and extrapolates to bulk work function (WF). The thermal coefficient is a good match with the theoretically derived value for WF. The AEs measured using this method is free from any contamination which has been the main source of error for the temperature dependent measurements from bulk surfaces. ❧ The photoionization yield curves have been used for a long time in determining the AEs for clusters and WF of bulk metals. We have shown for the first time that the derivative of Y(E) curves gives a very good measure for the density of states (DOS) of the clusters. The energy gap, δ, appearing between the highest occupied shell (HOMO) and the lowest unoccupied shell (LUMO) have been referred to the Kubo gap to observe the metallic transition in copper clusters. ❧ We also performed a comparative study of the two ionization mechanisms by bombarding the cold aluminum clusters at ≈90K with photons from a nanosecond tunable laser or by electrons from an electron-impact ionizer. By monitoring the ionization yield of the same cluster beam under the action of these two different probes, we identified some interesting differences. Whereas multiphoton ionization produces singly-charged cations accompanied by copious fragmentation, the impact of electrons with the same total energy as n photons is substantially “softer”. Specifically, electrons produce multiply charged cations when their total impact energy exceeds the corresponding ionization threshold, but do not give rise to massive shifts in the abundance spectra that are characteristic of extensive fragmentation. Therefore, it appears that the ionizing electrons do not deposit significant excess internal energy in the cluster. ❧ Furthermore, saturation curves corresponding to the transition from single-photon to multiple successive absorption events allowed us to estimate absolute cluster photoabsorption cross sections at these frequencies. They were found to be in good agreement with off-resonance surface plasmon values derived from the aluminum dielectric function. ❧ Theoretical analysis has been carried out of another prototype finite Fermi system where the electrons are confined within an anisotropic harmonic potential background. The two-dimensional (2D) electron “puddle” represents quantum dot geometry whereas the three-dimensional (3D) counterpart represents a metal cluster. The systems have the relaxation to attain any general shape with azimuthal symmetry. A classical electrostatic approach lets us derive the electronic distributions, which are in good agreement with the numerical density functional theory (DFT) calculations and the classical limit solution for the Thomas-Fermi artificial atoms. We also calculated the physical observables like the chemical potential, the electron affinity, the ionization potential and the capacitance (more familiar as “addition energy” to the experimentalists) for the electronic distributions. The experimentally measured value of addition energy for the 2D electronic systems can be reproduced with excellent accuracy by our simple results.

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Creator Halder, Avik (author)

Core Title Temperature-dependent photoionization and electron pairing in metal nanoclusters

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 02/25/2015

Defense Date 11/06/2014

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

Tag appearance energies,densities of states,electron impact ionization,electronic shell structure,metal nanoclusters,OAI-PMH Harvest,photoionization and photoemission spectroscopy,photoionization yields,quantum dots,superconducting pairing

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Kresin, Vitaly V. (committee chair), Cronin, Stephen B. (committee member), El-Naggar, Moh (committee member), Nakano, Aiichiro (committee member), Vilesov, Andrey F. (committee member)

Creator Email ahalder@usc.edu,avikpapan@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-538019

Unique identifier UC11297720

Identifier etd-HalderAvik-3211.pdf (filename),usctheses-c3-538019 (legacy record id)

Legacy Identifier etd-HalderAvik-3211.pdf

Dmrecord 538019

Document Type Dissertation

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Rights Halder, Avik

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

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

Repository Name University of Southern California Digital Library

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