PHYS07200604007 Manas Kumar Dala - Homi Bhabha National ...

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Experimental Techniques 26 the transition probability can be written as, w ∝ 2π¯h |〈ψ f|r|ψ i 〉| 2 δ(E f − E i − hν) = 2π¯h m fiδ(E f − E i − hν) (2.2) where m fi is the square of the transition matrix element. To discuss about the transition matrix element, one has to take certain assumptions of the wave functions contained in it. In the simplest approximation one can take a one electron view for the initial and final state wave functions. The final state is in addition with a free electron having kinetic energy E kin . The initial state wave function is then written as a product of the orbital φ k from which the electron is excited and the wave function of the remaining electrons ψi,R(N k −1), assuming that system under consideration has N electrons, where the index k indicates that the electron k has been left out and R refers to the remaining. Now one can write, ψ i (N) = Cφ i,k ψ k i,R(N − 1) (2.3) where C is the operator that antisymmetrizes the wave function properly. Similarly the final state can be written as a product of the wave function of the photoemitted electron φ f,Ekin and that of the remaining N −1 electrons ψf,R k (N −1), ψ f (N) = Cφ f,Ekin ψf,R k (N − 1) (2.4) So the transition matrix element can be written as, 〈ψ f |r|ψ i 〉 = 〈φ f,Ekin |r|φ i,k 〉〈ψf,R(N k − 1)|ψi,R(N k − 1)〉 (2.5) Thus the matrix element is the product of one electon matrix element and the (N − 1) electron overlap integral. Let us assume that the remaining orbitals (called the passive orbitals) are the same in the final state as they were in the initial state, meaning ψf,R(N k − 1) = ψi,R(N k − 1), which renders the overlap integral unity and the transition matrix element is just the one-electron matrix element. Under this assumption [7] the PES experiment measures the negative Hartree-Fock orbital energy of the orbital k meaning, which is sometimes called Koopman’s binding energy. E B,k ≃ − ǫ k , (2.6)
Experimental Techniques 27 Let us now assume that the final state with N − 1 electrons has s excited states with the wave function ψf,s k and energy E s (N − 1). The transition matrix element can be calculated by summing over all the possible excited final states yielding, 〈ψ f |r|ψ i 〉 = 〈φ f,Ekin |r|φ i,k 〉 ∑ s c s (2.7) with c s = 〈ψf,s k (N − 1)|ψk i,R (N − 1)〉 (2.8) Here |c s | 2 is the probability that removal of electron from orbital φ k of the N electron ground state leaves the system in the excited state s of the N − 1 electron system. For strongly correlated systems many of the c s ’s are nonzero. In terms of PE spectrum this means that for s = k one has the so-called main line and for the other non-zero c s additional so-called satellite lines occur. If the correlations are weak, then ψ k f,s(N − 1)≃ψ k i,R(N − 1) (2.9) meaning that c s 2 ≃ 1 for s = k and c s 2 ≃ 0 for s ≠ k, i.e one has only one peak. Now the photocurrent I detected in a PE experiment as, I∝ ∑ |〈φ f,Ekin |r|φ i,k 〉| 2∑ f,i,k s |c s | 2 δ(E f,kin + E s (N − 1) − E 0 (N) − hν, (2.10) where E 0 (N) is the ground state energy of the N-electron system. The photo current thus consists of the lines created by photoionizing the various orbitals k where each line can be accompanied by satellites according to the number of excited states s created in the photoexcitation of that particular orbital k. The above discussion is very convenient for atoms and molecules [4]. However, it is not always easy to apply for solids. For solids, a slightly different formalism has been developed [3, 8, 9] and hence the above equation can be rewritten in the following form I∝ ∑ f,i,k|〈φ f,Ekin |r|φ i,k 〉| 2 A(k, E) (2.11) where A(k, E) is the so-called spectral function for the wave number k and energy E. The spectral function can be related to the single particle Green’s function by
Page 1 and 2: SPECTROSCOPIC INVESTIGATIONS OF THE
Page 3 and 4: iii To My Parents
Page 5 and 6: Maharana, P. Khare, D. K. Basa, K.
Page 7 and 8: List of Publications/Preprints [1]
Page 9 and 10: Synopsis The perovskite type mangan
Page 11 and 12: to the t 2g and e g spin-up states
Page 13 and 14: Bibliography [1] R. von Helmolt, J.
Page 15 and 16: Contents CERTIFICATE Acknowledgemen
Page 17 and 18: List of Figures 1.1 Temperature dep
Page 19 and 20: 2.2 Schematic view of the energetic
Page 21 and 22: 3.4 O K edge x-ray absorption spect
Page 23 and 24: Chapter 1 Introduction 1
Page 25 and 26: Introduction 3 The temperature and
Page 27 and 28: Introduction 5 Figure 1.2: Schemati
Page 29 and 30: Introduction 7 Figure 1.4: A schema
Page 31 and 32: Introduction 9 Figure 1.6: Schemati
Page 33 and 34: Introduction 11 Figure 1.8: Differe
Page 35 and 36: Introduction 13 Figure 1.10: A sche
Page 37 and 38: Introduction 15 Figure 1.12: Schema
Page 39 and 40: Introduction 17 Figure 1.13: Phase
Page 41 and 42: Bibliography [1] R. von Helmolt, J.
Page 43 and 44: Introduction 21 [31] Damien Saurel,
Page 45 and 46: Chapter 2 Experimental Techniques 2
Page 47: Experimental Techniques 25 Figure 2
Page 51 and 52: Experimental Techniques 29 and G(k,
Page 53 and 54: Experimental Techniques 31 Figure 2
Page 55 and 56: Experimental Techniques 33 point wi
Page 57 and 58: Experimental Techniques 35 negative
Page 59 and 60: Experimental Techniques 37 ∆E = E
Page 63 and 64: Experimental Techniques 41 ( ) dσ
Page 65 and 66: Experimental Techniques 43 The expe
Page 67 and 68: Experimental Techniques 45 are weld
Page 77 and 78: Bibliography [1] H. Hertz, Ann. Phy
Page 79 and 80: Experimental Techniques 57 [35] K.
Page 81 and 82: Chapter 3 Electronic Structure of P
Page 83 and 84: Electronic Structure of Pr 0.67 Ca
Page 95 and 96: Bibliography [1] I. G. Deac, J. F.
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Chapter 4 Electronic Structure of P
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Electronic Structure of Pr 1−x Ca
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Electronic Structure of Ca 0.86 Pr
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Bibliography [1] C. Zener, Phys. Re
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Summary and Conclusions 109 In this
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