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MINISTRY OF EDUCATION AND TRAINING VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY CU SY THANG STUDY OF THE THERMODYNAMIC PARAMETERS AND CUMULANTS OF SOME MATERIALS BY ANHARMONIC XAFS METHOD SUMMARY OF MATERIAL SCIENCE DOCTORAL THESIS Ha Noi - 2020 MINISTRY OF EDUCATION AND TRAINING VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY CU SY THANG STUDY OF THE THERMODYNAMIC PARAMETERS AND CUMULANTS OF SOME MATERIALS BY ANHARMONIC XAFS METHOD SUMMARY OF MATERIAL SCIENCE DOCTORAL THESIS Major: Electronic material Code: 44 01 23 Supervisor: Prof Dr Sc Nguyen Van Hung Dr Le Quang Huy Ha Noi - 2020 INTRODUCTION The X-ray Absorption Fine Structure (XAFS) spectroscopy technique is a modern and high precision method to be used in the structural determination of materials. In general, this method is used to fit the theoretical and experimental spectra to extract data or parameters from the XAFS spectra. Anharmonic Correlated Einstein Model (ACEM) [9] is one of the efficiency theoretical methods [7] used to study thermodynamic parameters of XAFS spectra. Anharmonic effective interaction potential in ACEM has been built. In which Morse potential is assumed to describe the single-pair atomic interaction. By using this effective potential, the ACEM has not only overcome the limitations of using single bond potential [8] but also simplified the many-body system problem back to the simple one-dimensional system problem with the contribution of many-body effects through consideration of the interaction of neighboring atoms. Many previous studies [10-25] showed that the numerical results of the ACEM were good agreement with the experimental data as well as those obtained values by other methods for several different structural materials, However, most of the studies focused on the thermodynamic parameters, especially the cumulants of the XAFS spectra without concerning the anharmonic contribution of the second cumulant as well as the anharmonic contribution to XAFS phase and amplitude components. Debye-Waller factor or the second-order cumulant is an important thermodynamic parameter that characterizes the decrease XAFS amplitude. The relationship between the second-order cumulant and other thermodynamic parameters and XAFS phase and amplitude needs to be continuously studied, considered in more detail and comprehension. Therefore, I 1 choose the research topic: "Study of thermodynamic parameters and cumulants of some materials by anharmonic XAFS method" Target of thesis To develop a method that can simplify the determination of the thermodynamic parameters and XAFS spectra which base on the second-order cumulant only. In particular, this method can be applied for both theoretical calculations and experimental measurements in the XAFS method. Subject and Scope of the thesis: Subject: - Thermodynamic parameters, XAFS cumulants, XAFS spectra, and their Fourier transform magnitudes. - Materials: Diamond crystals (Si, Ge), fcc crystals (Cu), hcp crystals (Zn). Scope: - Models and theoretical methods of XAFS: focus on anharmonic correlated Einstein model using anharmonic effective potential in which Morse potential is assumed to describe the singe-pair atomic interaction. Study methods Theoretical method: - Using the anharmonic correlated Einstein model - Using the anharmonic effective interaction potential method Experimental method: Study on documents, the configuration of XAFS spectroscopy experimental station, the scanning process of XAFS spectra as well as data processing methods at Synchrotron Light Research Institute - Thailand. Numerical calculation Program and XAFS spectra data processing and analysis software - Using Matlab 2014 software, Demeter 9.0.25 software 2 Main contents of the thesis - Study and establish the expressions of thermodynamic parameters and the Debye-Waller factor of the XAFS spectra of materials depending on the temperature. - Study and establish a relationship between the anharmonic contribution of the second-order cumulant and the Debye-Waller factor of XAFS spectra of materials depending on temperature. - Study and establish a relationship between the Debye-Waller factor and anharmonic contribution of the XAFS phase and amplitude of materials depending on temperature. - Analyze and evaluate constructed theoretical calculations as well as evaluating the agreement between theoretical results and obtained experimental values from Synchrotron Light Research Institute - Thailand and other theoretical methods or measurements for fcc (Cu) and hcp (Zn) structural materials. CHAPTER OVERVIEW OF XAFS DEBYE-WALLER FACTOR 1.1 Fundamentals of XAFS 1.1.1 Simple physical description of XAFS X-ray absorption fine structure is the final state of interference between photoelectric waves emitted from absorption atoms and scattering waves by neighboring atoms. 1.1.2 XAFS equation (k) S0 N j e j 2 k 2 j 2 R j / ( k ) e kR j f j (k ) sin 2kR j j (k) (1.14) 1.1.3 XAFS Debye-Waller factor 2 - The factor e2 k in (1.14) have formed as e w+i is called XAFS Debye-Waller factor [29] j 3 1.1.4 XAFS cumulants - J.J. Rehr [28,29,31] showed that XAFS Debye-Waller factor is generally complex and has a natural cumulant expansion approach in term of Taylor series from the generalized cumulant expression [33]: (2ik ) n ( n ) (1.22) ik ( r R ) e exp[ ] j n! n1 - With x= rj R0 and lattice expansion a(T ) (rj R0 ) (1) (T ) Set y =x - a and y ; cumulants are written: (1) (T ) (rj R0 ) y R j rj R0 (1) (1.23) (2) (T ) (T) (rj R0 )2 y (3) (T ) ( rj R0 )3 y 1.2 XAFS Debye-Waller factor studies methods 1.2.1 Correlated Einstein Model [1] The correlated Einstein model is one of the ways of calculating or fitting XAFS thermodynamic parameter values. In this model, the vibrational density of states center at single vibrational frequency: j ( ) ( Ej ) 1.2.2 Equation of motion method [3,38] (1.37) R R (T ) coth M ( ) 2 ii j j ii i i j i 1.2.3 Statistical moment method [39-46] (1) (T) x r r0 a(T) a(0) y0 (T) R ui u0 ui2 u02 ui u0 (1.58) (1.59) CHAPTER ANHARMONIC CORRELATED EINSTEIN MODEL IN STUDY OF XAFS THERMODYNAMIC PARAMETERS 2.1 Effective potential in anharmonic correlated Einstein model The generalized expression of anharmonic effective interaction potential using in ACEM: E (x) (x) M i a ,b j a ,b 4 i x R0i Rij (2.3) Basing on the quantum statistical perturbation theory, we can determine Hamiltonian of the system and can extract anharmonic effective interaction potential expressions: E (a) keff a k3 a (2.6) E (y) (k eff a 3k3 a ) y k3 y3 (2.7) E (x) E (a) keff y E ( y) (2.9) 2.2 Morse potential [53] (rij ) D e 2 (rij r0 ) 2e (rij r0 ) Taylor series expansion in approximation up to the third order: ( x) D(1 x x3 ) (2.13) Table 2.2 Morse parameters of copper (Cu) and zinc (Zn) from theoretical calculation. Materials D (eV) (Å-1) r0(Å) c Cu [20,60,61] Cu [62] Zn [20,15,17,22,23,59,63] 0.3429 0.3364 0.1700 1.3588 1.5550 1.7054 2.868 2.8669 2.793 2 2 1/ 2.2.1 Applying Morse potential to calculate the thermodynamic parameters and effective interaction potential in anharmonic correlated Einstein model for fcc and hcp structural materials Figure 2.3. Face centered cubic Crystal [47] Figure 2.4. Hexagonal closepacked Crystal [47] Derive the expression of effective interaction potential which is used in the anharmonic correlated Einstein model, we can get: x x x E (x) (x) 2 ( ) 8 ( ) 8 ( ) (2.28) 5 4 Derive the expressions of effective local force constant, cubic anharmonic parameter as well as Einstein frequency and temperature for fcc and hcp crystals: k 5D k 5D a D k 5D 1 a 5D eff 2 2 eff E E E k k B B k3 D E ( y ) 5D (ay y ) eff 5D 10 3 k3 D E ( y ) 5D (ay 20 y ) (2.31); (2.32,2.34) (2.33) 2.2.2 Applying Morse potential to calculate thermodynamic parameters and interatomic effective potential in anharmonic correlated Einstein model for diamond structural materials [47] Figure 2.5 Diamond structural crystal Derive the expression of interaction effective potential which is used in the anharmonic correlated Einstein model, we can get: 1 3 M E (x) (x) 3 x 3 3M 1 x (x) 3 x 3 x 6 (2.36) Derive the expressions of effective local force constant, cubic anharmonic parameter as well as Einstein frequency and temperature for fcc and hcp crystals: keff D 35 E k D a D a D eff 12 6 E D k 35 D E k k B B 36 (2.39); Morse parameters for Si [25,64] : D=1.83 (eV); =1.56 (Å-1) và r0=2.34 (Å) Morse parameters for Ge[25,64]: D=1.63 (eV); =1.50 (Å-1) và r0=2.44 (Å) 6 (2.40) 2.3 Stillinger-Weber potential [52,65] ( x) ij Wijk (2.41) where the single-pair interaction potential component: 1 rij p rij q rij A B exp a , rij a ij 0, rij a (2.42) The three-body interaction potential component: 1 Wijk exp (rij a) 1 (rik a) 1 cosijk 3 Parameters for Si[52,65]: A=7.049556277; B=0.6022245584; p=4; q=0; a=1.80; =21.0; =1.20; =2.0951Å; =50 kcal/mol. Parameters for Ge[52]: A=7.049556277; B=0.6022245584; p=4; q=0; a=1.80; =31.0; =1.20; =2.181Å; =1.93 kcal/mol. 2.4 Calculating thermodynamic parameters by anharmonic correlated Einstein model 2.4.1 Calculating cumulants by anharmonic correlated Einstein model Atomic vibration is quantized in terms of phonons, anharmonicity is the result of phonon-phonon interaction. So we can express y in term of annihilation and creation operators [68]: y (aˆ aˆ ) with 0 E and aˆ aˆ n The above operators have the following properties: aˆ , aˆ 1, aˆ n n n , aˆ n n n , aˆ aˆ n n n (2.54) Then the averaging procedure can be calculated by statistical physics as [69]: ym Tr ( y m ), m 1, 2,3, Z (2.55) Calculating (2.55) in cases of: + m is even value: ym 1 Tr ( y m ) Tr ( y m ) Z Z0 Z0 7 e n n E n ym n (2.59) We can be received the second-order cumulant: y (2) Z0 e n E n y n (2.60) n + m is odd value: ym e En e En ' n E n ' n ' y m n Z n ,n ' En En ' (2.64) We can receive the first and third-order cumulants. Finally, we can receive expressions of cumulants for fcc (Cu) and hcp (Zn) structural materials: fcc: a ( ) z hcp: (1) 1 z (1) a ( ) 20 1 z (2) ( z 1) ( ) (1 z ) 3( ) (1 10 z z ) (3) 10 (1 z ) 1 z (2) ( z 1) ( ) (1 z ) (3) ( )4 (1 10 z z ) (1 z )2 (2.63, 2.73, 2.80) 2.4.2 Derive expressions of cumulants based on the second cumulant only in anharmonic correlated Einstein model From the expression about the relationship between temperature variable z and mean square relative displacement given by Rabus [8,9]: z ( )2 , replace into (2.63,2.73,2.83) we can receive ( )2 expressions of cumulants based on the Debye-Waller factor or the XAFS second cumulant only for fcc and hcp structural materials: 9 (1) (2) z 3 (2) 20 (2.82) (1) a ( ) 1 z (2) 2 ( z 1) (2) 2 ( ) 3 (1 z ) (3) [3( ) 2(( ) ) ] (3) ( ) (1 10 z z ) 10 2 2 [3( ) 2(( ) ) ] (1 z )2 where ( )2 E 10 D The relationship among cumulants is determined according to: (1) (2.83) (3) ( ) 02 3 8 Figure 2.6. Temperature Figure 2.7. Temperature dependence of the second-order dependence of the second-order cumulant using Stillinger-Weber cumulant using Stillinger-Weber potential in the statistical moment potential in the statistical moment method for Si. method for Ge. Figures 2.6 and 2.7 show a good agreement of the statistical moment method using in the calculation the XAFS second cumulant values for Si and Ge diamond semiconductors, respectively. For Si, the results were compared with the obtained values given by M. Benfatto in the article [70] at 80 K, 300 K, and 500 K. For Ge, the results have an agreement with experimental values given by A.E. Stern in [71] at 300 K, G. Dalba in [72] at some temperatures and with theoretical calculation results given by J.J. Rehr in the article [4] when using the LDA method at 300 K. Moreover, the obtained results are consistent with experimental results of A.Yoshiasa in [73] in some specific temperatures, even the results are calculated from the GGA and hGGA methods given by J.J.Rehr at 300 K [4]. These results published in the article [19]. Numerical results of the second-order cumulant using Morse and Stillinger-Weber potential for Si and Ge crystals by the anharmonic correlated Einstein model were evaluated and compared in the article [18,24,25]. The anharmonic correlated Einstein model using two potentials is consistent with experimental values as well as those obtained from other methods. Therefore, the anharmonic correlated 11 Einstein model can be applied to diamond semiconductors using Morse and Stillinger-Weber potential. 2.5 Quantum effects in low temperature limit and classical approximation in high temperature The obtained thermodynamic parameter formulas from quantum theory can be applied at all temperature values. At the hightemperatures, the formulas include the results of classical approximation theory. At the low-temperature limit, quantum effects express through contributions of zero-point energy. Thermodynamic Quantity T0 T (1) a (1) (1 z ) 3k3k BT / k eff (2) (1 z ) k BT / k eff (3) (3) (1 12 z ) 6k3 (k BT )2 / k eff T T0 z (ln z ) (1 z ) 3k3 / k B r (1) (3) (1) (1 z )2 3(1 z ) 3 (3) (1 12 z ) 2(1 12 z ) 1 T r.T 3z ln (3) z 1 T r.T (3) CHAPTER EXPERIMENTAL MEASUREMENT AND APPLICATION OF ANHARMONIC CORRELATED EINSTEIN MODEL IN STUDY ON XAFS THERMODYNAMIC PARAMETERS FOR HCP AND FCC STRUCTURE MATERIALS 3.1 Synchrotron facility and XAFS spectra experimental station The preparation for experimental samples depending on temperature: 12 Figure 3.5. Experimental station Beamline 8. SLRI Figure 3.7. Experimental XAFS measurement depending on temperature 3.2 Experimental measurements results of the Debye-Waller factors for hcp structure material Experimental values are shown in figure 3.12 and table 3.1. Figure 3.12 XAFS spectrum and Fourier transform magnitudes of Zn at 300 K, 400 K, 500 K, and 600 K Table 3.1 The value of cumulants and thermal expansion coefficients of Zn: Theoreratical calculation (LT) and experimental value (TN) at temperatures. Symbol: MHĐH – Harmonic model T(K) (1)(Å) (1)(Å) 2(Å) 2(Å) 2(Å) (3)(Å) (3)(Å) T T (10-5/K) (10-5/K) LT TN LT MHĐH TN LT TN LT TN 300 0.0139 0.0143 0.0110 0.0109 0.0113 0.0003 0.0003 1.555 1.582 400 0.0182 0.0189 0.0146 0.0143 0.0149 0.0005 0.0006 1.582 618 500 0.022 0.0232 0.0182 0.0177 0.0185 0.0008 0.0009 1.595 1.599 600 0.0270 0.0279 0.0219 0.0211 0.0223 0.0011 0.0012 1.602 1.630 13 3.3 Determining thermodynamic parameters of XAFS from the experimental values of the Debye-Waller factor or the secondorder cumulant by anharmonic correlated Einstein model for hcp structure materials Figure 3.14 Temperature dependence of the first cumulant, total and the harmonic second cumulant and experimental values From the illustration in figure 3.14b, anharmonic correlated Einstein model, and the harmonic correlated model [82] have certain deviations for the second-order cumulant or Debye-Waller factor in the high-temperature range. ACEM is more suitable for experimental values than the harmonic correlated model. Note that the data of the first-order cumulant is derived from the experimental value of the second-order cumulant. Figure 3.15 Temperature dependence of the third cumulant and thermal expansion coefficient of Zn calculated from cumulant experimental values. 14 Figure 3.16. Temperature dependence of cumulants ratio, the ratio between thermal expansion coefficient and cumulants of Zn. Similar to the first -order cumulant, we are also able to determine the third-order cumulant and thermal expansion coefficient of zinc (Zn) at 300 K, 400 K, 500 K, and 600 K. Figure 3.15a and 3.15b showed the results derived from experimental measurements are very agreement with the calculations from the theoretical model. To assess the validity of the theoretical model, we can also check by establishing the ratio among cumulants according to the expression (2.83) and ratio among thermal expansion coefficient and cumulants according to the expression (2.88). Figure 3.16 showed the above relationships. From figure 3.16, the values are derived from experimental values that make these ratios reach the value of ½. These ratios are used as the standard method for assessing cumulant studies [9,56,81,83], as well as for determining temperature when the classical limit can be applied [9]. The theoretical results and the results of these ratios showed that hcp structure materials, specifically Zn, we can use classical correlation Einstein model when the temperature is higher than Einstein temperature (E = 206 K). 3.4 Experimental results of XAFS Debye-Waller factors for fcc structure material 15 Figure 3.17. XAFS spectrum and Fourier transform magnitudes of Cu at 300 K, 400 K, 500 K Figure 3.18. The process of fitting the XAFS spectrum of Cu at temperatures XAFS spectrum at temperature values after merging are fitted to the theoretical spectra by using Artemis software. The R, k variables are in R space [1-3 Å] or k space [3.00-14.023 Å-1] run to the optimal value between theoretical spectra and experimental spectra. 3.5 Determining thermodynamic parameters of XAFS from experimental values of the Debye-Waller factor or the secondorder cumulant by anharmonic correlated Einstein model for fcc (Cu) structure material Figure 3.19. Temperature dependence of the first cumulant, total and harmonic second cumulant and the experimental values. Anharmonic correlated Einstein model and harmonic correlated Einstein model [81] have certain deviations for the second-order 16 cumulant or Debye-Waller factor in high-temperature range (Figure 3.19). The results showed that anharmonic correlated Einstein model is well suited to experimental values as well as obtained results of S. a Beccara, et al. [82] for the first-order cumulant and V. Pirog, et al. [58] for the second-order cumulant. Note that, the data of the firstorder cumulants are derived from the experimental second-order cumulants values. Figure 3.20 Temperature dependence of the third-cumulant and thermal expansion coefficient of Cu calculated from experimental cumulant values Similar to the first-order cumulant, we can also identify the thirdorder cumulant and thermal expansion coefficient of copper (Cu) at 300 K, 400 K, 500 K. Figure 3.20 showed the results derived from experiment values were very consistent with the obtained data of V. Pirog, et al [58] and T. Yokoyama, et al [88] for the third-order cumulant. Figure 3.20b indicated agreement among calculated results from the present method and the experimental results and obtained results from other documents [89] for the thermal expansion coefficient. To evaluate the validity of the theoretical model, we verified by establishing the ratio among cumulants according to expression (2.83) and the ratio among the thermal expansion coefficient and cumulants according to the expression (2.88). Figure 3.21 showed these relationships. 17 Figure 3.21. Temperature dependence of cumulants ratio, ratio between thermal expansion coefficient and cumulants of Cu The values extracted from experiments make these ratios approach the value of ½ (Figure 3.21). These ratios are used as the standard method for assessing cumulant studies [9, 81, 90], as well as for determining temperatures at which classical limits can be applied [9]. The theoretical results and the results of these ratios showed that hcp structure materials, specifically Cu, we can use classical correlation Einstein model [9,81] when the temperature is higher than Einstein temperature (E = 218 K). CHAPTER ANHARMONIC CORRELATED EINSTEIN MODEL IN STUDY OF XAFS PHASE AND AMPLITUDE CONTRIBUTION OF HCP AND FCC STRUCTURE MATERIALS 4.1 Overview of anharmonic XAFS spectra The anharmonic XAFS function is represented by cumulant expansion approach [21,60,90,91]: 2R e (k) (2ik )n ( n ) (4.1) (k ) F (k ) Im ei (k) exp 2ikR kR n! n XAFS amplitude expression [9,90-92]: W(k , T ) 2ki (1) (T) 2k 2 (T ) (4.2) 4i (T) k R 3 4 1 ik (T) k (T) R (k) A ( k , T ) ( k , T ) ( k , T0 ) 2k[R (T)( 18 1 )] k (3) (T) (4.3) R Với (T) (T) (T0 ) (4.4) 4.2 XAFS Debye-Waller factor with contribution anharmonic In the high-temperature range, the Debye-Waller factor includes 2 components: a harmonic and anharmonic contribution component. (T) H2 (T) A2 (T) (4.5) where A2 (T) (T)[ H2 (T) (T0 )]= (T)[ H2 (T) 02 ] (4.6) Replace (4.5) into (4.4) we receive: (T) H2 (T) (T)[ H2 (T) 02 ] 02 (1 (T)[ H2 (T) 02 ] With (T) is called the anharmonic factor of the XAFS second cumulant which depending on temperature and Grüneisen parameter. (T) 2 G V ln E G V với ln V 4.2.1 Determination of Grüneisen parameter G From (2.32, 2.34) we can determine lnE/T (4.9) and lnV/T (4.10). Therefore, we can determine: ( R ) (4.11) ln E G 2 ln V 4(1 ) 4.2.2 Determination of anharmonic factor (T) Determine the change in volume due to thermal expansion V/V and from (4.12) we can determine: (T ) 9 2 3 (T)[1 (T)(1 (T)] (4.14) 4R 4R 4.5 XAFS spectra with contribution anharmonic components The Debye-Waller factor includes two components as expression (4.5). To accurately describe the actual spectra so that the XAFS phase and amplitude in (1.14) need to be added to the anharmonic factors. In detail, the phase component is added to the anharmonic factor: 2 2 k A (T ) FA (k , T ) e 19 The amplitude component is added to the anharmonic factor: A (k , T ) 2k[R (T)( 1 )] k (3) (T) (4.16) R Then, the generalized expression of XAFS become to: (k) S N j f (k ) F (k , T )e 2 k e 2 R / ( k ) sin 2kR (k) (k, T) (1.17) kR j 2 j j j A j j A j 4.6 XAFS anharmonic phase and amplitude components for HCP (Zn) structure material The anharmonic XAFS components increased with increasing temperature and k-wavenumber values (Figure 4.1). Figure 4.1. Temperature dependence of anharmonic amplitude and phase components with the k-wave number of XAFS spectra for hcp (Zn) structural material. These components (phase and amplitude) contribute to anharmonic XAFS spectra show in figure 4.2 in both of the theoretical calculations by the anharmonic correlated Einstein model and experimental values. Figure 4.2 Theoretical and experimental XAFS spectra with hcp structure material (Zn) at temperatures. 20 Figure 4.3 Comparison of Fourier transform magnitudes of theoretical spectra with experimental XAFS results for hcp (Zn) structural material at temperatures. Figure 4.3 showed an agreement between the theoretical results of the model with obtained Fourier transform magnitudes from experimental measurements. In addition, we can see that the magnitude of spectra decreases with the increasing temperature gradually. Note that the anharmonic contribution components to the XAFS phase and amplitude are calculated base on second-order cumulant only. Moreover, by using anharmonic correlated Einstein model, we can reconstruct XAFS spectra and the Fourier transform magnitudes from the obtained experimental second-order cumulant values. This study has shown that the obtained experimental results are consistent with theoretical calculations at a temperature of 300 K, 400 K, 500 K, and 600 K for Zn. 4.7 Anharmonic contribution of XAFS phase and amplitude for FCC (Cu) structural material 4.7.1 Anharmonic contribution to the second-order cumulant and anharmonic factor (T) Figure 4.4 Temperature dependence of anharmonic contribution to the second-order cumulant and anharmonic factor (T) of fcc (Cu) structural material 21 Figure 4.4a and 4.4b showed an agreement between the calculated results from the present method and experimental values for anharmonic contribution to the second-order cumulant and anharmonic factor (T), respectively. These experimental values are extracted from the experimental second-order cumulant results. Anharmonic factor (T) is a new factor given by Nguyen Van Hung, et al. in the article [21]. In addition, the above anharmonic contribution is difficult to measure directly. So when using anharmonic correlated Einstein model, we can calculate and represent these anharmonic components that depending temperature based on theoretical calculations or experimental measurement values of the second-order cumulant. 4.7.2 Anharmonic contribution to XAFS phase and amplitude For anharmonic contribution to XAFS spectra, indicated in the figure (4.3). We can describe two components including anharmonic contribution to amplitude FA (k , T ) and phase shift A (k, T) of XAFS spectra in expressions (4.16) and (4.17). Figure 4.5 Temperature dependence of anharmonic contribution to amplitude and phase with k - wave number for fcc (Cu) structure material Anharmonic contribution to the XAFS spectra increases with increasing temperature and value of k-wavenumber (Figure 4.5). These components contribute to anharmonic of XAFS spectra presented in figure 4.6 regarding on theory of anharmonic correlated 22 Einstein model as well as experimental results. Figure 4.7 illustrates a good agreement of the theoretically calculated results and obtained spectra from measurements through Fourier transform magnitudes. Besides, figure 4.7 also shows that peak heights decreased and their shifts moved to the left when the temperature increased. Figure 4.6 Theoretical and experimental XAFS spectra of fcc (Cu) structure material at temperatures. Figure 4.7. Comparison of Fourier transform magnitudes of the theoretical spectra with experimental XAFS results for hcp (Zn) structural material at temperatures. Here, we note that the anharmonic contribution of XAFS phases and amplitudes are calculated base on the second-order cumulant only. Furthermore, by using anharmonic correlated Einstein model, we can reproduce XAFS spectra and transform their Fourier magnitudes base on the obtained experimental second-order cumulant values. This study showed that the obtained experimental results are consistent with calculations from the theoretical model at temperatures of 300 K, 400 K, and 500 K for Cu. 23 CONCLUSIONS AND RECOMMENDATIONS This thesis contributed an advanced to the progress of complement and upgrading of anharmonic correlated Einstein model a well-applied in both theory and experiment for XAFS spectroscopy. This thesis consisted of the following key findings: 1. Using calculated Morse potential parameters from theory to determine effective interacted potential in anharmonic correlated Einstein model. In this study, Stillinger-Weber potential was used additionally for diamond structure materials (Si, Ge). 2. This thesis described a procedure known as an advanced method that can simplify the calculations for thermodynamic parameters, XAFS spectra, and their Fourier transform magnitudes only through the second-order cumulant. In particular, this advanced anharmonic correlated Einstein model can apply in both theory and experiment in the XAFS field. 3. The advanced anharmonic correlated Einstein model was applied to derive, calculate and evaluate parameters of XAFS such as cumulants, thermal expansion coefficients T, XAFS spectrum, and their Fourier transform magnitudes, anharmonic factor (T), anharmonic contribution to the second-order cumulant, Grüneisen parameter, the ratio among cumulants as well as ratio among coefficient of thermal expansion and cumulants. 4. Measurements of second-order cumulant for fcc (Cu) and hcp (Zn) structure material were conducted. The obtained experimental results were evaluated and compared with theoretical values as well as results from other methods. 5. The calculated theoretical results obtained by using the advanced method were appropriate to experimental values and those obtained from other measurements. The contents of this study published through five scientific articles in which three ones belong to SCI journals. 24 LIST OF WORKS PUBLISHED 1. Nguyen Van Hung, Cu Sy Thang, Nguyen Cong Toan, Ho Khac Hieu (2014), Temperature dependence of Debye-Waller factors of semiconductors, J. Vacuum, (101), pp 63-66. 2. Nguyen Van Hung, Cu Sy Thang, Nguyen Ba Duc, Dinh Quoc Vuong (2017), Advances in theoretical and experimental XAFS studies of thermodynamic properties, anharmonic effects and structural determination of fcc crystals, The European physical Journal B, 90:256. 3. Nguyen Van Hung, Cu Sy Thang, Nguyen Ba Duc, Dinh Quoc Vuong, Tong Sy Tien (2017), Temperature dependence of theoretical and experimental Debye-Waller factor, thermal expansion and XAFS of metallic Zinc, Physica B, 521, pp 198203. Cu Sy Thang, Nguyen Van Hung, Nguyen Bao Trung, Nguyen Cong Toan (2018). A Method for theoretical and experimental studies of thermodynamic parameters and XAFS of HCP crystals, application to metallic Zinc. Proceeding of The 5th Academic conference on natural science for young scientists, master and Ph.D. students from Asian Countries(4-7 October 2017, Da lat, Viet Nam). ISBN: 978-604-913-714-3, pp 58-65. 5. Nguyen Van Hung, Cu Sy Thang, Nguyen Bao Trung, Nguyen Cong Toan (2018). Theoretical and Experimental studies of Debye-Waller factors and XAFS of FCC crystals. Proceeding of Advances in Applied and Engineering Physics-CAEP V. ISBN: 978-604-913-232-2, pp 47-55. ... especially the cumulants of the XAFS? ? spectra without concerning the anharmonic contribution of the second cumulant? ? as well as the anharmonic contribution to XAFS? ? phase and ... CHAPTER ANHARMONIC CORRELATED EINSTEIN MODEL IN STUDY OF XAFS PHASE AND AMPLITUDE CONTRIBUTION OF HCP AND FCC STRUCTURE MATERIALS 4.1 Overview of anharmonic XAFS spectra The anharmonic XAFS? ?... 400 K, 500 K, and 600 K for Zn. 4.7 Anharmonic contribution of XAFS phase and amplitude for FCC (Cu) structural material 4.7.1 Anharmonic contribution to the second-order cumulant and anharmonic factor (T)
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