DSpace at VNU: ANHARMONIC EFFECTIVE POTENTIAL, LOCAL FORCE CONSTANT AND EXAFS OF HCP CRYSTALS: THEORY AND COMPARISON TO EXPERIMENT

November 20, 2008 16:18 WSPC/140-IJMPB 04928 International Journal of Modern Physics B Vol 22, No 29 (2008) 5155–5166 c World Scientific Publishing Company Int J Mod Phys B 2008.22:5155-5166 Downloaded from www.worldscientific.com by CAMBRIDGE UNIVERSITY on 12/21/14 For personal use only ANHARMONIC EFFECTIVE POTENTIAL, LOCAL FORCE CONSTANT AND EXAFS OF HCP CRYSTALS: THEORY AND COMPARISON TO EXPERIMENT NGUYEN VAN HUNG∗ and TONG SY TIEN Department of Physics, University of Science, VNU-Hanoi, 334 Nguyen Trai,Hanoi, Vietnam ∗ hungnv@vnu.edu.vn LE HAI HUNG Institute of Engineering Physics, Hanoi University of Technology, Dai Co Viet, Hanoi, Vietnam RONALD R FRAHM Bergische Universită at-Gesamthochschule Wuppertal, FB: 8-Physik, Gauß Straße 20, 42097 Wuppertal, Germany Received 30 April 2008 Anharmonic effective potential, Extended X-ray Absorption Fine Structure (EXAFS) and its parameters of hcp crystals have been theoretically and experimentally studied Analytical expressions for the anharmonic effective potential, effective local force constant, three first cumulants, a novel anharmonic factor, thermal expansion coefficient and anhamonic contributions to EXAFS amplitude and phase have been derived This anharmonic theory is applied to analyze the EXAFS of Zn and Cd at 77 K and 300 K, measured at HASYLAB (DESY, Germany) Numerical results are found to be in good agreement with experiment, where unnegligible anharmonic effects have been shown in the considered theoretical and experimental quantities Keywords: Effective potentials; anharmonicity; cumulant; EXAFS Introduction EXAFS and its parameters are often measured at low temperatures and wellanalyzed by the harmonic procedure1 because the anharmonic contributions to atomic thermal vibrations can be neglected However, EXAFS may provide apparently different information on structure and on other parameters of the substances at different high temperatures due to anharmonicity.2–25 Analysis of these effects is necessary to evaluate thermal atomic vibration of the substances The formalism for including anharmonic effects in EXAFS is often based on the cumulant expansion approach.3,4 Using this procedure, the anharmonic effects in EXAFS are often 5155 November 20, 2008 16:18 WSPC/140-IJMPB Int J Mod Phys B 2008.22:5155-5166 Downloaded from www.worldscientific.com by CAMBRIDGE UNIVERSITY on 12/21/14 For personal use only 5156 04928 N V Hung et al valuated by the ratio method,3–10 which is based on the comparison of EXAFS data measured at two different temperatures Another way is direct calculation and analysis of EXAFS and its parameters, including anharmonic effects at any temperature For this purpose, an anharmonic factor has been introduced14–16 to take into account the anharmonic contributions to the mean square relative displacement (MSRD) This procedure provides good agreement with experiment, but the expressions for the anharmonic factor and for the phase change of EXAFS due to anharmonicity contain a fitting parameter, and the cumulants were obtained by an extrapolation procedure from the experimental data.14,15 For calculation of anharmonic effects, it is very important to calculate interatomic potentials.17,21,24 A procedure for calculation of Morse pair potential for crystals of cubic structures is available,26 and its parameters have been extracted from experimental EXAFS data.27,28 However, for calculation of thermodynamic properties of materials including anharmonic effects, the pair potential may not be suitable, and an effective interatomic potential17,24 containing the pair potential is necessary This work is devoted to theoretical and experimental study of the anharmonic EXAFS and its parameters of hcp crystals, an interesting structure Our development presented in Sec is the derivation of analytical expressions for the anharmonic interatomic effective potential, effective local force constant, correlated Einstein frequency and temperature, three first cumulants, where the second cumulant is equal to the Debye-Waller factor (DWF), anharmonic contributions to EXAFS amplitude and phase of hcp crystals This model includes only near-neighbor interactions between absorber and backscattering atoms and their immediate (or first shell) neighbors instead of a single-bond model.12 The cumulants contained in the derived expressions can be calculated by several procedures.5–7,12,13,17,18 In this work, the quantum statistical approach with the anharmonic correlated Einstein model,17 has been used This model avoids full lattices dynamical13 or dynamical matrix18 calculations, contributing to the extraction of physical parameters from the EXAFS data,27,28 to the investigation of local force constants of transition metal dopants in a nickel host,22 compared to Mossbauer studies, and to theoretical approaches to the EXAFS.19 Anharmonic factor has been derived again expressed by the second cumulant or DWF, and the fitting parameter is avoided The procedure for calculation of Morse potentials for crystals of cubic structure26 has been generalized to calculation of those for hcp crystals They characterize the interaction of each pair of atoms, and are contained in the effective potentials, in the cumulants and in the other EXAFS parameters The EXAFS and its parameters for hcp crystals Zn and Cd at 77 K and 300 K have been measured at HASYLAB (DESY, Germany), represented in Sec 3, and its physical parameters have been obtained by a fitting procedure Here the experimental data are analyzed and compared to the calculated values Moreover, unnegligible anharmonic effects appearing in the experimental and calculated EXAFS parameters have been evaluated in detail November 20, 2008 16:18 WSPC/140-IJMPB 04928 Anharmonic Effective Potential of HCP Crystals 5157 Formalism Based on cumulant expansion approach, the EXAFS oscillation function is given by21 Int J Mod Phys B 2008.22:5155-5166 Downloaded from www.worldscientific.com by CAMBRIDGE UNIVERSITY on 12/21/14 For personal use only χ(k) = F (k) e−2R/λ(k) Im eiΦ(k) exp 2ikR + kR2 n (2ik)n (n) σ n! , (1) where F (k) is the real atomic backscattering amplitude; Φ is the net phase shift; k and λ are the wave number and the mean free path of the photoelectron, respectively; R = r with r as the instantaneous bond length between absorber and backscattering atoms and σ (n) (n = 1, 2, 3, ) are the cumulants The total mean square relative displacement (MSRD) σtot (T ) or anharmonic DWF at a temperature T is given as the sum of the harmonic σ (T ) and anharmonic σA (T ) contributions 2 σtot (T ) = σ (T ) + σA (T ) , σA = β0 (T )[σ (T ) − σ02 ] , β0 (T ) = 2G V , V (2) where G is Gră uneisen parameter, ∆V /V is the relative volume change due to thermal expansion, and σ02 is zero-point contribution to σ (T ) The anharmonic effective potential in our approach is expressed as a function of ˆ direction, r and r0 being the instantaneous the displacement x = r−r0 along the R and equilibrium distances between absorber and backscattering atoms, respectively Veff (x) ≈ k x2 + k x3 , (3) where k0 is effective local force constant, and k3 is cubic parameter giving the asymmetry due to anharmonicity (Here and in the following, the constant contributions are neglected) A Morse potential is assumed to describe the interatomic interaction, and expanded to the third order around its minimum V (x) = D(e−2αx − 2e−αx ) ∼ = D(−1 + α2 x2 − α3 x3 + · · ·) , (4) where α describes the width of the potential and D is the dissociation energy In the case of relative vibrations of absorber and backscatterer atoms, including the effect of correlation and taking into account only the nearest or first shell neighbor interactions, the effective interatomic potential is given by Veff (x) = V (x) + V i=a,b j=a,b ˆ0 ˆ xR · Rij , (5) where the first term on the right concerns only absorber and backscatterer atoms, the remaining sums extend over the remaining neighbors, and this relation is used for calculation of the effective potential for monatomic hcp crystals based on its atomic structure November 20, 2008 16:18 WSPC/140-IJMPB 5158 04928 N V Hung et al Applying the Morse potential Eq (4) to Eq (5) and comparing it to Eq (3) for the case of correlated atomic vibrations, the effective local force constants are now expressed in terms of the Morse potential parameters as k0 = 5Dα2 (6) Int J Mod Phys B 2008.22:5155-5166 Downloaded from www.worldscientific.com by CAMBRIDGE UNIVERSITY on 12/21/14 For personal use only For calculation of thermodynamic parameters, we use the further definition y = x − a, a = x ,12,17 to write Eq (5) as Veff (y) ∼ (7) = (k0 + 3k3 a)ay + keff y + k3 y , is an effective local force constant, which is in principle different from where keff k0 In accordance with Eq (7), the anharmonic effective potential can be expressed as the sum of the harmonic contribution and a perturbation δV due to the weak anharmonicity Veff (y) ∼ = keff y + δV (y) , (8) where now the force constant is different from the one of Eq (6) for Veff (x), i.e., keff = 5Dα2 − αa 10 = µωE , θE = ωE , kB µ= MA MS MA + M S (9) from which we obtain the correlated Einstein frequency ωE and temperature θE ; kB is Boltzmann constant; µ is reduced mass of absorber and backscatterer with masses MA and MS , respectively; and the perturbation δV due to the weak anharmonicity is given by δV (y) ∼ = 5Dα2 ay − αy 20 (10) Making use of quantum statistical methods,29 the physical quantity is determined by an averaging procedure using canonical partition function Z and statistical density matrix ρ, e.g., Tr(ρ y m ) , m = 1, 2, 3, (11) Z Atomic vibrations are quantized in terms of phonons, and anharmonicity is the result of phonon-phonon interaction, that is why we express y in terms of annihilation and creation operators, a ˆ and a ˆ+ , respectively ym = y ≡ a0 (ˆ a+a ˆ+ ) , a20 = which have the following properties √ [ˆ a, a ˆ+ ] = , a ˆ+ |n = n + 1|n + , a ˆ+ a ˆ|n = n|n , ωE , 2keff a ˆ|n = (12) √ n − 1|n − , (13) November 20, 2008 16:18 WSPC/140-IJMPB 04928 Anharmonic Effective Potential of HCP Crystals 5159 and use the harmonic oscillator state |n as the eigenstate with the eigenvalue En = n ωE , ignoring the zero-point energy for convenience Using the above results for correlated atomic vibrations and the procedure depicted by Eqs (11)–(13), as well as the first-order thermodynamic perturbation theory and considering the anharmonic component Eq (10) of the potential, we derived the cumulants The second cumulant or MSRD is expressed as Int J Mod Phys B 2008.22:5155-5166 Downloaded from www.worldscientific.com by CAMBRIDGE UNIVERSITY on 12/21/14 For personal use only σ2 = y2 = Z0 e−nβ ωE n n|y |n , β = 1/kB T , (14) where the canonical partition function is given by ∞ Z∼ = Z0 = e −nβ ωE zn = = n n=0 , 1−z z = e−θE /T (15) Applying Eqs (13) to calculate the matrix element in Eq (14), we obtain the second cumulant 1+z ωE σ (T ) = σ02 , σ02 = , z = e−θE /T (16) 1−z 10Dα2 Using Eq (11) to evaluate the traces, the remaining odd moments are given by ym = 5Dα2 Z0 n,n e−βEn − e−βEn n|ay − αy |n En − E n 20 n |y m |n , m = 1, (17) Calculating the matrix elements in Eq (17), we obtain the first (m = 1) and the third (m = 3) cumulants as (1) σ (1) (T ) = a(T ) = σ0 (3) σ (3) (T ) = σ0 +z 9α = σ , 1−z 20 (1) σ0 + 10z + z (3) = σ0 [3(σ /σ02 )2 − 2] , (1 − z)2 = 9α σ , 20 (3) σ0 = 3α 2 (σ ) , 10 (18) (19) and the thermal expansion coefficient αT = α0T (1) z| ln(z)|2 da = α0T = α0T R dT (1 − z)2 5Dα kB T [(σ )2 − (σ02 )2 ] , (20) 9kB = , 100DαR (3) where σ0 , σ02 , σ0 are zero-point contributions to σ (1) , σ , σ (3) , respectively, and α0T is the constant value which αT approaches at high temperatures From the above results, we obtain the following cumulant relations 3z(1 + z) ln(1/z) αT rT σ , = (3) (1 − z)(1 + 10z + z ) σ σ (1) σ , = (3) − 4(σ02 /σ )2 σ (21) where the first equation coincides with the one of Ref 12, and the second equation with the result of Ref 17 for the other crystal structures November 20, 2008 16:18 WSPC/140-IJMPB 5160 04928 N V Hung et al To calculate the total MSRD or anharmonic DWF Eq (2), an anharmonic factor has been derived 3α 3α 9α2 (22) β0 (T ) = σ 1+ σ 1+ σ 4R 4R Int J Mod Phys B 2008.22:5155-5166 Downloaded from www.worldscientific.com by CAMBRIDGE UNIVERSITY on 12/21/14 For personal use only Note that all the above derived thermodynamic quantities are expressed by the second cumulant σ , hence leading our work mainly to the calculation of this quantity The anharmonic contribution to the EXAFS phase at a given temperature is the difference between the total phase and the one of the harmonic EXAFS, and it is given by ΦA (T, k) = 2k σ (1) (T ) − 2σA (T ) 1 + R λ(k) − σ (3) (T )k (23) We obtained from Eq (1), taking into account the above results, the temperature dependent K-edge EXAFS function including anharmonic effects as χ(k, T ) = j 2 S02 Nj Fj (k)e−(2k [σ (T )+σA (T )]+2Rj /λ(k)) sin(2kRj +Φj (k)+ΦjA (k, T )) , kRj2 (24) S02 where is the square of the many-body overlap term, Nj is the atomic number of each shell, the remaining parameters were defined above, the mean free path λ is defined by the imaginary part of the complex photoelectron momentum p = k+i/λ, and the sum is over all atomic shells It is obvious that in Eq (24), σA (T ) determines the anharmonic contributions to the amplitude characterizing attenuation, and ΦA (k, T ) the anharmonic contributions to the phase characterizing the phase shift of the EXAFS spectra They are calculated by Eq (2) and Eq (23), respectively At low temperatures, these values approach zero so that our anharmonic theory becomes the harmonic one, but at high temperatures, it approaches the classical limit results including anharmonic effects, so that the present anharmonic procedure contains the harmonic and classical theories as its special cases Discussion of Experimental and Numerical Results EXAFS spectra at 77 K and 300 K for Zn and Cd have been measured at HASYLAB (DESY, Germany) Some results are presented in Fig The EXAFS for Zn Table Calculated and experimental values of Morse parameters D, α, r , of effective local force constant keff , correlated Einstein frequency ωE and temperature θE for Zn and Cd Bond Zn-Zn, Calc Zn-Zn, Expt Cd-Cd, Calc Cd-Cd, Expt D (eV) α (˚ A−1 ) r (˚ A) keff (N/m) ωE (× 1013 Hz) θE (K) 0.1698 0.1685 0.1675 0.1653 1.7054 1.7000 1.9069 1.9053 2.7931 2.7650 3.0419 3.0550 39.5616 39.0105 48.7927 48.0711 2.6917 2.6729 2.2798 2.2628 205.6101 204.1730 174.1425 172.8499 November 20, 2008 16:18 WSPC/140-IJMPB 04928 Anharmonic Effective Potential of HCP Crystals 0.4 30 Fourier transform magnitude 77K 300K Zn 20 χk 10 -10 -20 -30 Int J Mod Phys B 2008.22:5155-5166 Downloaded from www.worldscientific.com by CAMBRIDGE UNIVERSITY on 12/21/14 For personal use only 5161 10 -1 k(Å ) (a) Fig 300 K 12 14 16 Zn 0.3 -1 -1 3Å < k < 13Å 77K x2, 300K 0.2 0.1 0.0 R(Å) (b) (a) Experimental EXAFS of Zn and (b) its Fourier transform magnitudes at 77 K and (a) (b) Fig (a) Calculated Morse potentials and (b) anharmonic effective potentials compared to harmonic components They all agree well with experimental results is attenuated and shifted to the right [Fig 1(a)], its Fourier transform magnitude is attenuated and shifted to the left [Fig 1(b)] when the temperature changes from 77 K to 300 K Similar properties of the measured EXAFS for Cd have also been obtained This temperature dependence of the EXAFS of hcp crystals need to be analyzed and compared to the developed theory In order to perform it, we apply the expressions derived in the previous section to numerical calculations and compare the results to experimental data for Zn and Cd Morse potential parameters of Zn and Cd have been calculated by generalizing the procedure for cubic crystals26 to the one for hcp crystals The experimental Morse potential and other EXAFS parameters have been obtained from the experimental EXAFS data by a fitting procedure Effective local force constants, correlated Einstein frequencies and temperatures have been calculated using the obtained Morse parameters The results are written in Table They are used for calculation of the anharmonic EXAFS November 20, 2008 16:18 WSPC/140-IJMPB Int J Mod Phys B 2008.22:5155-5166 Downloaded from www.worldscientific.com by CAMBRIDGE UNIVERSITY on 12/21/14 For personal use only 5162 04928 N V Hung et al and its parameters Figure shows the calculated Morse potential (a) and anharmonic effective potentials (b) for Zn and Cd compared to experiment and to their harmonic terms The anharmonic effective interatomic potentials are quite different from the Morse pair potentials due to inclusion of the interaction of nearest neighbors of the absorber and backscattering atoms Figure illustrates the temperature dependence of the calculated total MSRD σtot (T ) or anharmonic DWF for Zn and 2 Cd, compared to their harmonic ones σ (T ) of the anharmonic contribution σA (T ) to the MSRD (b) According to Eq (24), σA (T ) is also the anharmonic contribution to the EXAFS amplitude, and it increases as the temperature increases The 2 calculated values of σtot (T ) and σA (T ) agree well with experiment at 77 K and 300 K We see that both σtot (T ) and σ (T ) contain zero-point contribution at low temperatures and σ (T ) is linearly proportional to the temperature at high temperature, satisfying all standard properties of these quantities.21 Based on Eq (22) and the results of Fig 3, the value of σA is about 3.50% at 300 K and 8.30% at 700 K of σ for Zn, but negligible at 77 K Due to this anharmonic contribution, σtot (T ) is slightly not linear at high temperatures The cumulant relations Eqs (21) have been calculated for Zn and Cd, and the results are shown in Fig They agree with experiment at 77 K and 300 K The first relation equals zero at K and approaches the constant value of 0.5 at high temperatures, the second one equals 1.5 at K and also approaches 0.5 at high temperatures, as these properties were shown in theory 12,17 and in experiment7 for the other crystals They are slightly different for different materials only below the Einstein temperature Hence, we can conclude that these cumulant relations have the same properties for all crystals as the standards for evaluation of cumulants in EXAFS technique Figure shows the anharmonic contributions to the EXAFS phase according to Eq (23) at 77 K, 300 K and 500 K This contribution is negligible at 77 K, but valuable at 300 K and 500 K, especially at high (a) (b) Fig (a) Temperature dependence of total MSRD compared to the harmonic one and (b) anharmonic contribution for Zn and Cd They agree well with experiment at 77 K and 300 K 04928 Anharmonic Effective Potential of HCP Crystals Int J Mod Phys B 2008.22:5155-5166 Downloaded from www.worldscientific.com by CAMBRIDGE UNIVERSITY on 12/21/14 For personal use only 5163 contributions EXAFS phaseEq for(17) Zn for at Zn and Cd compared to Fig Temperature dependence of to cumulant relations experiment at 77 K and 300 K satisfying all standard Based on Eq and the results of Fig 3, the value of s about 3.50 % at 300 K and 8.30 % at Zn -1 ΦA (rad.) perature, November 20, 2008 16:18 WSPC/140-IJMPB 1st shell, single scattering -2 77K 300K 500K -3 -4 -5 10 15 20 -1 k(Å ) Fig Calculated k-dependence of anharmonic contributions to EXAFS phase for Zn at 77 K, 300 K and 500 K k values For calculation of anharmonic EXAFS of Zn and Cd, we modified code FEFF1 by adding our anharmonic contributions For XANES, the multiple scattering is important, but for EXAFS, the single scattering is dominant,31 and the main contribution to EXAFS is given by the first shell.7 This is why for testing theory, only the calculated EXAFS for the first shell for single scattering has been used for comparison to experiment The calculated EXAFS and their Fourier transform magnitudes for Zn at 77 K, 300 K and 500 K for the first shell and for single scattering are illustrated in Fig They reflect all properties of temperature dependence of the experimental results (Fig 1) They show significant changes of amplitude and November 20, 2008 16:18 WSPC/140-IJMPB 5164 04928 N V Hung et al and in for the other crystals They are Zn, Present theory, 77K 300K 500K single scattering, 1st shell 10 χk3 0.4 Fourier transform magnitude 15 -5 -10 -15 10 15 77K 300K 500K 0.2 0.1 0.0 20 k(Å-1) R(Å) (a) (b) Fig (a) Calculated EXAFS and (b) their Fourier transform magnitudes for the first shell for single scattering of Zn at 77 K, 300 K and 500 K 0.18 0.30 0.16 Fourier transform magnitude Fourier transform magnitude Int J Mod Phys B 2008.22:5155-5166 Downloaded from www.worldscientific.com by CAMBRIDGE UNIVERSITY on 12/21/14 For personal use only Zn, Present theory 1st shell, single scattering 3Å-1 < k < 13.5Å-1 0.3 Zn, 300K 0.14 3Å-1 < k < 13.5Å-1 0.12 Expt Present theory FEFF 0.10 0.08 0.06 0.04 0.02 0.00 Cd, 77K 0.25 3Å-1 < k < 12.85Å-1 0.20 FEFF Present theory exp 0.15 0.10 0.05 0.00 R(Å) (a) 6 R(Å) (b) Fig Fourier transform magnitudes of EXAFS for Zn (a) at 300 K and for Cd at 77 K (b) calculated by present anharmonic theory compared to experiment and to those calculated by FEFF code.1 phase as the temperature increases, e.g., the EXAFS is attenuated and shifted to the right, the peak of its Fourier transform magnitude is attenuated and shifted to the left Figure 7(a) shows a good agreement of our calculated anharmonic EXAFS of Zn at 300 K with experiment and its difference from the one calculated by the harmonic FEFF,1 but at low temperature 77 K, where the anharmonic contributions are negligible, as it was shown in Figs 3(b) and 5, then the Fourier transform magnitudes calculated by the present anharmonic procedure and by the harmonic FEFF code both agree well with experiment [Fig 7(b)] Hence, our anharmonic theory is advantageous and shown to be in agreement with experiment both at low and high temperatures, while the harmonic one can obtain this agreement only at low temperatures where the anharmonicity is negligibly small, but disagreement occurs at high temperatures because the anharmonicity is not included November 20, 2008 16:18 WSPC/140-IJMPB 04928 Anharmonic Effective Potential of HCP Crystals 5165 Int J Mod Phys B 2008.22:5155-5166 Downloaded from www.worldscientific.com by CAMBRIDGE UNIVERSITY on 12/21/14 For personal use only Conclusion In this work, the anharmonic EXAFS and its parameters of hcp crystals have been theoretically and experimentally studied The derived expressions for the first, second and third cumulant, and thermal expansion coefficient satisfy all their fundamental properties in temperature dependence, especially in the high temperature region where anharmonicity is significant, and in the low temperature region where zero point vibration is important Anharmonic contribution to the second cumulant or DWF, characterized by a novel derived anharmonic factor is the one to EXAFS amplitude, and anharmonic contribution to EXAFS phase is characterized mainly by the first, third cumulant and anharmonic contribution to MSRD, e.g., by the anharmonic parameters The present anharmonic theory contains the harmonic model at low temperatures, where quantum theory must be used and the classical limit at high temperatures, where the anharmonic effects are significant, as its special cases Moreover, our expression of the derived EXAFS thermodynamic quantities by the second cumulant is advantageous by decreasing the computation Anharmonic effective potentials including the nearest neighbor interactions of absorber and backscatterer and containing Morse potential parameters characterizing the pair interaction are significantly asymmetric due to anharmonic effects, and are very suitable for calculating anharmonic EXAFS parameters The cumulant relations αT rT σ /σ (3) and σ (1) σ /σ (3) for hcp crystals have the same form and the same properties as for the other crystal structures, so that they can be proposed to be the standards for cumulant evaluation in EXAFS procedure The EXAFS and Fourier transform magnitudes for Zn and Cd, measured at HASYLAB (DESY, Germany) provide the necessary physical parameters for comparison to those of the developed anharmonic theory They show their thermodynamic properties in temperature dependence and anharmonic effects The derived analytical expressions for EXAFS parameters in the present anharmonic theory provide the results satisfying all their fundamental properties, and agree well with experiment at 77 K and 300 K They show unnegligible anharmonic effects in the theoretical and experimental considered quantities, which denotes the importance of including anharmonic contributions in the temperature dependent EXAFS and the efficiency of the effective potential procedure Acknowledgments The authors thank J J Rehr, P Fornasini and A I Frenkel for useful comments One of the authors (N V H.) thanks the BUGH Wuppertal for hospitality and appreciates the partial supports of the basic science research project No 405806 and the special research project of VNU Hanoi QG.08.02 November 20, 2008 16:18 WSPC/140-IJMPB 5166 04928 N V Hung et al Int J Mod Phys B 2008.22:5155-5166 Downloaded from www.worldscientific.com by CAMBRIDGE UNIVERSITY on 12/21/14 For personal use only References J J Rehr, J Mustre de Leon, S I Zabinsky and R C Albers, J Am Chem Soc 113, 5135 (1991) B S Clausen, L Grabæck, H Topsøe, L B Hansen, B Stoltze, J K Nørskov and O H Nielsen, J Catal 141, 368 (1993) E D Crozier, J J Rehr and R Ingalls, X-ray Absorption, eds D C Koningsberger and R Prins (Wiley, New York, 1988) G Bunker, Nucl Instrum Methods 207, 438 (1983) J M Tranquada and R Ingalls, Phys Rev B 28, 3520 (1983) L Wenzel, D Arvanitis, H Rabus, T Lederer, L Baberschke and G Gomelli, Phys Rev Lett 64, 1765 (1990) E A Stern, P Livins and Z Zhang, Phys Rev B 43, 8850 (1991) E Burattini, G Dalba, D Diop, P Fornasini and F Rocca, Jpn J Appl Phys 32, 90 (1993) P Fornasini, F Monti and A Sanson, J Synchrotron Rad 8, 1214 (2001) 10 L Tră oger, T Yokoyama, D Arvanitis, T Lederer, M Tischer and K Baberschke, Phys Rev B 49, 888 (1994) 11 T Yokoyama, T Sasukawa and T Ohta, Jpn J Appl Phys 28, 1905 (1989) 12 A I Frenkel and J J Rehr, Phys Rev B 48, 585 (1993) 13 T Miyanaga and T Fujikawa, J Phys Soc Jpn 63, 1036, 3683 (1994) 14 N V Hung and R Frahm, Physica B 208 & 209, 91 (1995) 15 N V Hung, R Frahm and H Kamitsubo, J Phys Soc Jpn 65, 3571 (1996) 16 N V Hung, J de Physique IV(C2), 279 (1997) 17 N V Hung and J J Rehr, Phys Rev B 56, 43 (1997) 18 T Yokoyama, Phys Rev B 57, 3423 (1998) 19 J J Rehr and R C Albers, Rev Mod Phys 72, 621 (2000) 20 H Katsumata, T Miyanaga, T Yokoyama, T Fujikawa and T Ohta, J Synchrotron Radiat 8, 226 (2001) 21 N V Hung, N B Duc and R R Frahm, J Phys Soc Jpn 72, 1254 (2003) 22 M Daniel, D M Pease, N Van Hung and J I Budnick, Phys Rev B 69, 134414 (2004) 23 P Fornasini, S A Beccara, G Dalba, R Grisenti, A Sanson, M Vaccari and F Rocca, Phys Rev B 70, 134414 (2004) 24 N V Hung and P Fornasini, J Phys Soc Jpn 76, 084601 (2007) 25 F D Vila, J J Rehr, H H Rossner and H J Krappe, Phys Rev B 76, 014301 (2007) 26 L A Girifalco and W G Weizer, Phys Rev 114, 687 (1959) 27 I V Pirog, T I Nedoseikina, I A Zarubin and A T Shuvaev, J Phys.: Condens Matter 14, 1825 (2002) 28 I V Pirog and T I Nedoseikina, Physica B 334, 123 (2003) 29 R P Feynman, Statistical Mechanics (Benjamin, Reading, MA, 1972) 30 J M Ziman, Principles of the Theory of Solids (Cambridge University Press, London, 1972) 31 P Rennert and N V Hung, Physica Status Solidi B 48, 49 (1988)  ... the width of the potential and D is the dissociation energy In the case of relative vibrations of absorber and backscatterer atoms, including the effect of correlation and taking into account... contributions EXAFS phaseEq for(17) Zn for at Zn and Cd compared to Fig Temperature dependence of to cumulant relations experiment at 77 K and 300 K satisfying all standard Based on Eq and the results of. .. The EXAFS for Zn Table Calculated and experimental values of Morse parameters D, α, r , of effective local force constant keff , correlated Einstein frequency ωE and temperature θE for Zn and