Physica B 406 (2011) 456–460 Contents lists available at ScienceDirect Physica B journal homepage: www.elsevier.com/locate/physb Pressure effects in Debye–Waller factors and in EXAFS Nguyen Van Hung a,n, Vu Van Hung b, Ho Khac Hieu a,c, Ronald R Frahm d a University of Science,VNU Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam National University of Civil Engineering, 55 Giai Phong, Hai Ba Trung, Hanoi, Vietnam d Bergische Universită at-Gesamthochschule Wuppertal, FB: 8-Physik, Gauß Straße 20, 42097 Wuppertal, Germany b c a r t i c l e i n f o a b s t r a c t Article history: Received 15 July 2010 Received in revised form October 2010 Accepted November 2010 Anharmonic correlated Einstein model (ACEM) and statistical moment method (SMM) have been developed to derive analytical expressions for pressure dependence of the lattice bond length, effective spring constant, correlated Einstein frequency and temperature, Debye–Waller factors (DWF) or second cumulant, first and third cumulants in Extended X-ray Absorption Fine Structure (EXAFS) at a given temperature Numerical results for pressure-dependent DWF of Kr and Cu agree well with experiment and other theoretical values Simulated EXAFS of Cu and its Fourier transform magnitude using our calculated pressure-induced change in the 1st shell are found to be in a reasonable agreement with those using X-ray diffraction (XRD) experimental results & 2010 Elsevier B.V All rights reserved Keywords: EXAFS Pressure dependence effects Debye–Waller factors and cumulants Introduction The anharmonic EXAFS providing information on structure and thermodynamic parameters of substances has been analyzed by means of cumulant expansion approach [1,2] In this formulation, an EXAFS oscillation function w(k) is given by [3] ( " #) X ð2ikÞn FðkÞ wkị ẳ e2R=lkị Im eifkị exp 2ikR ỵ sNị , ð1Þ n! kR n where k and l are the wave number and mean free path of emitted photoelectrons, respectively, F(k) is the real atomic backscattering amplitude, fðkÞ the net phase shift and snị n ẳ 1,2,3, .ị are the cumulants EXAFS is sensitive to pressure [4,5], which can cause certain changes of cumulants including DWF leading to uncertainties in physical information taken from EXAFS The pressure dependence of the DWF or EXAFS second cumulant has been measured at the wiggler beamlines of Stanford Synchrotron Radiation Laboratory (SSRL, USA) for Cu [6], and at the Laboratoire Pour I’Utisation du Rayonnement Electromagne´tique (LURE) (Orsay, France) for Kr [7,8] Such pressure effects have been calculated by correlated Debye model [6], as well as by MonteCarlo (MC) simulation [7] and by Loubeyre’s model [8] to interpret experimental results Recently the ACEM and SMM have been used for calculation of temperature dependence of EXAFS cumulants of crystals at zero n Corresponding author E-mail address: hungnv@vnu.edu.vn (N Van Hung) 0921-4526/$ - see front matter & 2010 Elsevier B.V All rights reserved doi:10.1016/j.physb.2010.11.012 pressure [9] The purpose of this work is to develop the ACEM and SMM for calculation and analysis of pressure effects in DWF and in EXAFS of crystals at a given temperature Our new development presented in Section is establishing and solving the equation of state to obtain the pressure-dependent lattice bond length, and then is the derivation of the analytical expressions for pressure dependence of the effective spring constant, correlated Einstein frequency and temperature, DWF or second cumulant, first and third cumulants in EXAFS Morse and Lennard-Jones (L-J) potentials have been used to characterize the interaction between each pair of atoms Numerical calculations (Section 3) using the developed ACEM and SMM show similar results for the pressure-dependent DWF of Cu (fcc), crystal Kr (fcc) and semiconductor Si (dia), where the calculated values for Kr and Cu agree well with experiment and other theoretical results [6–8] Since the experimental data for pressure dependence of EXAFS of Cu is not yet available, we compare the pressure-dependent EXAFS of Cu and its Fourier transform magnitude simulated using the FEFF code [10] and our calculated pressure-induced change in the 1st shell to those using XRD results [11] They show a reasonable agreement Formalism Using the expressions for free energy and pressure [12] we obtain the equation of state describing the pressure versus volume relation of crystal lattice in the form ! @U0 @K _o Pv ẳ a , zẳ ỵ yzcoth z , y ẳ kB T, 2ị @a 2K @a 2y N Van Hung et al / Physica B 406 (2011) 456–460 a2 a2 a4 Dm ẳ m ỵ2ịm ỵ 4ịm ỵ 6ị Amixỵ ỵ6Amixỵiy8 Table Lennard-Jones potential parameters m, n, e and s for Cu, Kr and Si [13,14] Crystals m n ˚ s (A) e=kB ðK Þ Cu Kr Si 5.5 6.63 6.0 9.0 16.42 12.0 2.5487 3.993 2.2950 4125.7 237.16 3269.1 where P denotes the hydrostatic pressure and v the atomic volume v ¼V/N of a crystal having volume V and N atoms, kB is Boltzmann constant The frequency o and the nearest neighbor distance a(P,T) depend on pressure P and temperature T Hence, solving this equation we can obtain the pressure- and temperature-dependent nearest neighbor distance a(P,T) However, for our numerical calculations it is convenient to determine first the value a(P,0) at pressure P and temperature T¼0 K In this case Eq (2) is reduced to ! @U0 _o0 @K Pv ẳ a , 3ị ỵ @a 4K @a where o0 is the value of o at zero temperature We can take the pair interaction energy in crystals as the power law using the Morse potential  à jrị ẳ D e2arr0 ị 2earr0 ị , 4ị where a describes the width of potential, D the dissociation energy and r the instantaneous bond length between the two immediate neighboring atoms with r0 being its equilibrium value; or using the (L-J) potential sm sn ! e n jrị ẳ Àm , ð5Þ mÀn r r where e describes the dissociation energy, and r has the same definition as for Morse potential in Eq (4), but with s being its equilibrium value The parameters e, s, n and m are determined by fitting experimental data (e.g., cohesive energy and elastic modulus), and the results for some crystals [13,14] are shown in Table Using (L-J) potential Eq (5), it is straightforward to get the interaction energy U0, the quantity K and the parameter g in a crystal [15] as sn sm ! e mAn nAm 6ị U0 ẳ nm a a Kaị ẳ sn sm X @2 j ẳ Cn ÀCm ¼ M o20 2 i @uib a a containing the coefficients i enm h a2 Cn ¼ n ỵ 2ịAnixỵ An ỵ ; 2a nmị i enm h a2 mỵ 2ịAmixỵ Am ỵ ; Cm ẳ 2a nmị 7ị a2 18m þ 2Þðm þ 4ÞAnixþ þ9ðm þ 2ÞAm þ a2 Pv ẳ c1 and gaị ẳ 12a4 nmị Dn sn a sm ! ÀDm , a containing the coefficients a2 a2 a4 Dn ẳ n ỵ2ịn ỵ 4ịn ỵ6ị Anixỵ ỵ 6Anixỵ 8iy a2 18n ỵ2ịn þ 4ÞAnixþ þ9ðn þ 2ÞAn þ ; sm r a2 c2 sn r c3 s=aịn c4 s=aịm ỵ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , c5 ðs=aÞn Àc6 ðs=aÞm ð11Þ containing the coefficients enm enm ; c2 ẳ Am ; 6nmị 6nmị r h i _ enm a2 p n ỵ2ị n ỵ2ịAnixỵ An ỵ ; c3 ẳ a M 2ðnÀmÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih i _ enm a2 pffiffiffiffiffi ðm ỵ2ịAmixỵ Am ỵ ; c4 ẳ a M 2nmị c1 ẳ An a2 c5 ẳ n ỵ 2ịAnixỵ An ỵ ; a2 c6 ẳ m þ 2ÞAmixþ ÀAm þ : ð12Þ Solving Eq (11) using the given parameters n, m, e and s (Table 1) we get the nearest neighbor distance a(P,0) Substituting the obtained a(P,0) in Eqs (7), (9), we find the quantities K(P,0), g(P,0) at pressure P and temperature T¼0 K In SMM the nearest neighbor distance a(P,T) between the two immediate atoms at pressure P and temperature T can be dened as aP,T ị ẳ aP,0ị ỵ y0 P,T ị: ð13Þ Hence, in order to calculate a(P,T) of Eq (13) using the obtained a(P,0) we substitute the obtained K(P,0), g(P,0) in the following expression for the thermally induced lattice expansion [12,16] s 2gP,0ịy AP,Tị, 14ị y0 P,Tị ẳ 3K P,0ị where AP,Tị ẳ a1 ỵ a1 ẳ ỵ a3 ẳ a4 ẳ g2 P,0ịy2 K P,0ị a2 ỵ g3 P,0ịy3 K P,0ị a3 ỵ g4 ðP,0Þy4 K ðP,0Þ a4 , Z 13 47 23 , a2 ẳ ỵ Zỵ Z þ Z , 6 ! 25 121 50 16 ỵ Zỵ Z þ Z þ Z , 3 ð15Þ 43 93 169 83 22 ỵ Zỵ Z ỵ Z ỵ Z ỵ Z 3 3 _oP,0ị zẳ , 2y ð8Þ ð10Þ Here M is the atomic mass, An , Am , Anix , Amix are the structural sums of the given crystal From Eqs (3), (5)–(7) we obtain the equation of state for crystals at zero temperature as Z ¼ zcothðzÞ, rffiffiffiffiffiffiffiffiffiffiffiffiffiffi KðP,0Þ : oðP,0Þ ¼ M To a good approximation the parallel mean square relative displacement (MSRD) corresponds to the second cumulant s2 ¼ enm 457 (h ) ! ! ! i2 R :ð ui À u0 Þ ẳ u2i ỵ u20 ui u0 , ui u0 % ui hu0 i, 2 2 ui % u0 : ð16Þ ð9Þ Here, u0 and ui are the atomic displacements of the zeroth and the ! ith sites from their equilibrium positions, R is the unit vector at the zeroth site pointing towards the ith site, and the brackets /S denote the thermal average The first two terms on the right-hand side are the uncorrelated mean square displacement (MSD), while the third term is the parallel displacement correlation function 458 N Van Hung et al / Physica B 406 (2011) 456–460 In SMM using the expression of the second-order moment [12] we obtain the MSD 2 y ui ẳ ui ỵ yA1 ỵ Z1ị, K ! 2g2 Z 1ỵ 1ỵ Z ỵ 1ị : A1 ẳ K K 17ị s1ị P,Tị ẳ s1ị VPịị The ACEM [17] has been used in analysis of temperaturedependent EXAFS data [3,17,19–23] In this model the cumulants have been calculated using the effective atomic interaction potential X m ^ ij , ^ 12 UR jeff xị % keff x ỵ k3 x3 ỵ ẳ jxị ỵ j 20ị R Mi iaj where x equals either rÀr0 for Morse potential or r–s for (L-J) potential, keff is effective spring constant and k3 is the cubic anharmonicity parameter This model has been defined as an oscillation of a single bond pair of atoms with masses M1 and M2 (e.g., absorber and backscatterer) in a given system Their oscillation is influenced by their neighbors In the center-of-mass frame of this bond, the model is defined further by the second equation, here m ẳ M1 M2 =M1 ỵ M2 ị is reduced mass, R^ is the bond unit vector, the sum i is over absorber (i¼ 1) and backscattering atom (i¼ 2), and the sum j is over all their nearest neighbors, excluding absorber and backscatterer themselves whose contributions are described by the term j(x) Our development in this work is the derivation of pressure dependence of the effective spring constant keff, the correlated Einstein frequency oE and temperature yE and the cumulants ă starting from the denition of Gruneisen parameter @ ln oE : @ ln V ð21Þ This parameter gG depends on volume which, in turn, depends on pressure With a slight modification of the formula found in Refs [18,24], the simplest parameterization of the volume dependence for crystals [6] can be written as gG VPịị VPị ẳ gG V0 ị V0 ẳ const, ỵ zVPị,Tị , 1zVPị,Tị s1ị VPịị ẳ f1 s0 VPịịị, 27ị 18ị From Eqs (14), (16)–(18) we derive the second cumulant or DWF of crystals at pressure P and temperature T in SMM as 3 4g2 P,0ịy Z 2y Z ỵ 1ị þ þ Z: ð19Þ s2 ðP,TÞ % K P,0ị KP,0ị gG ẳ Making use of quantum statistical methods [25] the expressions for the 1st cumulant sð1Þ and the 3rd cumulant sð3Þ have been derived as the functions of pressure P at a given temperature T ð22Þ s3ị P,Tị ẳ s3ị VPịị 3s2 VPị,Tịị2 2s20 VPịịị2 s20 VPịịị2 s3ị VPịị ẳ f3 s0 VPịịị, , ð28Þ ð3Þ where sð1Þ ðVðPÞÞ,s0 ðVðPÞÞ and s0 ðVðPÞÞ are pressure-dependent zeropoint contributions to the 1st, 2nd and 3rd cumulants, respectively They are described by the functions of the zero-point contribution to the 2nd cumulant s20 ðVðPÞÞ with explicit form depending on crystal structure and on our using Morse or (L-J) potential The pressure-dependent EXAFS can be calculated using Eq (1) with the above obtained pressure-dependent DWF and cumulants Numerical results and discussions Now we apply the expressions derived in previous section to numerical calculations for metal Cu, crystal Kr and semiconductor Si The (L-J) potential parameters of Cu, Kr and Si are given in Table Morse potential parameters have the values D ¼0.3429 eV, a ¼1.3588 A˚ À for Cu taken from Ref [26], and D ¼0.9956 eV, a ¼1.3621 A˚ À for Si calculated by our generalizing the method for the cubic crystals [26] to diamond structure These Morse potential parameters were obtained using experimental values of the energy of sublimation, the compressibility and the lattice constant The pressure dependence of DWF, i.e., the MSRD or the second cumulant s2 (Fig 1) and the MSD u2 (Fig 2) for crystal Kr at 300 K has been calculated by the ACEM and by the SMM using (L-J) potential Both methods provide similar results where the values for s2 (Fig 1) calculated by the present theory agree well with experiment [7,8] and with the other theoretical values calculated by the MC simulation [7] and by the Loubeyre’s model (private communication) [8] Our calculated results for MSD (Fig 2) agree well with those calculated by the MC simulation [7] Fig illustrates the pressure dependence s2 ðPÞÀs2 ð0Þ for Cu at 300 K calculated by the ACEM using Morse potential and by the SMM using (L-J) potential The calculated results of both methods for where the subscript indicates zero pressure From Eqs (21), (22) we derive pressure dependence of the correlated Einstein frequency oE and temperature yE as & !' VðPÞ _oE ðVðPÞÞ oE VPịị ẳ oE V0 ịexp gG V0 ị , , yE VPịị ẳ V0 kB 23ị VPị a3 P,Tị , ẳ V0 a 0,0ị 24ị where a(P,T) is calculated by Eq (13) Using these results, we obtain the pressure dependence of the effective spring constant as keff VPịị ẳ mo2E VPịị: 25ị From Eqs (23)(25) we derive the EXAFS DWF of crystals as a function of pressure P at a given temperature T ỵ zVPị,Tị , 1zVPị,Tị _oE VPịị ,zVPị,Tị ẳ exp yE VPịị=T : s20 VPịị ẳ 2keff VPịị s2 P,Tị ẳ s20 ðVðPÞÞ ð26Þ Fig Pressure dependence of MSRD or s2 of Kr calculated by ACEM and SMM compared to experiment [7,8], to MC simulation [7] and Loubeyre’s [8] results N Van Hung et al / Physica B 406 (2011) 456–460 Fig Pressure dependence of MSD u2 of Kr calculated by ACEM and by SMM using (L-J) potential compared to MC simulation [7] 459 Fig Pressure dependence s2 ðPÞÀs2 ð0Þ of Si calculated by ACEM using Morse potential and by SMM using (L-J) potential -0.005 Cu -0.01 Δr (Å) -0.015 -0.02 -0.025 -0.03 -0.035 -0.04 -0.045 10 P (GPa) Fig Pressure dependence s2 ðPÞÀs2 ð0Þ of Cu calculated by ACEM using Morse potential and by SMM using (L-J) potential compared to Debye [6] and model [6] results DWF are also similar and found to be in a good agreement with those calculated by the correlated Debye model [6] and by the model for calculation of the moments of the nearest neighbor distance distribution from an expansion to third order of the potential energy [6] Fig shows the pressure dependence s2 ðPÞÀs2 ð0Þ for Si at 300 K calculated by the ACEM using Morse potential and by the SMM using (L-J) potential They have the form similar to those for Cu but with lower values Fig demonstrates the pressure dependence of our calculated pressure induced change in the 1st shell Dr for Cu Fig illustrates our calculated pressure dependence sð3Þ ðPÞÀsð3Þ ð0Þ for the 3rd cumulant of Cu at 300 K compared to the results calculated by the methods P3 and P4 [6] Since the experimental pressure-dependent EXAFS data of Cu is not yet available, we compared our by FEFF [10] code simulated EXAFS spectrum of Cu (Fig 7) for the 1st shell in single scattering and its Fourier transform magnitude (Fig 8) at 10 GPa using our calculated pressure-induced change in the 1st shell Dr¼0.041 A˚ (Fig 4) to those using Dr ¼0.05 A˚ taken from XRD [11] and to the Fig Pressure dependence of calculated pressure-induced change in the 1st shell Dr ẳ rPịr0ịof Cu Fig Pressure dependence sð3Þ ðPÞÀsð3Þ ð0Þ calculated by ACEM using Morse and (L-J) potential compared to P3 and P4 method results [6] 460 N Van Hung et al / Physica B 406 (2011) 456–460 the pressure effects in cumulants including DWF and in EXAFS has been developed Both methods provide similar results for the pressure dependence of DWF Our development is establishing and solving equation of state to get the pressure dependence of the lattice bond length, and then is the derivation of the analytical expressions of pressure dependence for the 2nd cumulant or DWF, as well as for the 1st and 3rd cumulants, the effective spring constant, the correlated Einstein frequency and temperature The good agreement of our calculated results of the pressuredependent DWF for Kr and Cu with experiment and other theoretical values; the reasonable agreement of our simulated EXAFS spectrum of Cu and its Fourier transform magnitude with those using experimental XRD data; as well as the significant increase in the height and the shifting of simulated EXAFS amplitude of Cu and its Fourier transform magnitude under pressure increase denote a further step of applying the ACEM and SMM to reproduction of the pressure-dependent experimental DWF and to description of the pressure effects in EXAFS Fig Pressure dependence of EXAFS of Cu simulated by FEFF [10] using our calculated 2nd, 3rd cumulant, Dr (Fig 5) and Dr of XRD [11] Acknowledgements The authors thank J.J Rehr, P Fornasini and M Vaccari for useful comments This work is supported by the research project QGTD.10.02 of VNU Hanoi and in part by the research project no 103.01.09.09 of NAFOSTED; one of the authors (V.V.H.) acknowledges the partial support of the research project no 103.01.2609 of NAFOSTED References Fig Fourier transform magnitudes of pressure-dependent EXAFS of Cu of Fig results at P¼0 The EXAFS amplitude and the peak height of its Fourier transform magnitude increase, the phase of EXAFS is shifted to the right, and the position of its Fourier transform magnitude is shifted to the left as the pressure increases Figs and show a reasonable agreement of our calculated results with those obtained from the XRD A small discrepancy in Dr of 0.009 A˚ may be tentatively attributed to neglecting many-body effects in the Morse and (L-J) potentials used in our theory while XRD is threedimensional technique Conclusions In this work a formalism based on our further development of the ACEM and SMM using Morse and (L-J) potentials for investigation of [1] E.D Crozier, J.J Rehr, R Ingalls, D.C Koningsberger, R Prins (Eds.), Wiley, New York, , 1988 chapter [2] G Bunker, Nucl Instrum Methods 207 (1983) 437 [3] N.V Hung, N.B Duc, R.R Frahm, J Phys Soc Jpn 72 (2003) 1254 [4] R Ingalls, G.A Garcia, E.A Stern, Phys Rev Lett 40 (1978) 334 [5] R Ingalls, E.D Crozier, J.E Whitemore, A.J Seary, J.M Tranquada, J Appl Phys 51 (1980) 3158 [6] J Freund, R Ingalls, E.D Crozier, Phys Rev B 39 (1989) 12537 [7] A Di Cicco, A Filipponi, J.P Itie, A Polian, Phys Rev B 54 (1996) 9086 [8] A Polian, J.P Itie, E Daartyge, A Fontaine, G Tourillon, Phys Rev B 39 (1989) 3369 [9] V.V Hung, H.K Hieu, K Masuda-Jindo, Comut Mater Sci 49 (2010) S2417 [10] J.J Rehr, J Mustre de Leon, S.I Zabinsky, R.C Albers, J Am Chem Soc 113 (1991) 5135 [11] American Institute of Physics Handbook, 3rd ed., AIP, New York, 1972 pp 4–100 [12] N Tang, V.V Hung, Phys Status Solidi B 149 (1988) 511 [13] K Masuda-Jindo, V.V Hung, P.D Tam, Phys Rev B 67 (2003) 094301 [14] S Erkoc, Phys Rep 278 (2)(1997) 81 [15] V.V Hung, N.T Hoa, Jaichan Lee, AJSTD 23 (1 and 2)(2006) 27 [16] V.V Hung, K Masuda-Jindo, J Phys Soc Jpn 69 (2000) 2067 [17] N.V Hung, J.J Rehr, Phys Rev B 56 (1997) 43 [18] J Ramakrishnan, R Boehler, G.H Higgins, G.C Kennedy, J Geophys Res 83 (1978) 3535 [19] I.V Pirog, T.I Nedoseikina, A.I Zarubin, A.T Shuvaev, J Phys.: Condens Matter 14 (2002) 1825 [20] F.D Vila, J.J Rehr, H.H Rossner, H.J Krappe, Phys Rev B 76 (2007) 014301 [21] M Daniel, D.M Pease, N Van Hung, J.I Budnick, Phys Rev B 69 (2004) 134414 [22] N.V Hung, P Fornasini, J Phys Soc Jpn 76 (2007) 084601 [23] N.V Hung, T.S Tien, L.H Hung, R.R Frahm, Int J Mod Phys B 22 (2008) 5155 [24] U Walzer, W Ullmann, V.L Pan’kov, Phys Earth Planet Inter 18 (1979) [25] R.P Feynman, Statistical Mechanics, Benjamin, Reading, MA, 1972 [26] L.A Girifalco, V.G Weizer, Phys Rev 114 (1959) 687 ... code simulated EXAFS spectrum of Cu (Fig 7) for the 1st shell in single scattering and its Fourier transform magnitude (Fig 8) at 10 GPa using our calculated pressure- induced change in the 1st... development in this work is the derivation of pressure dependence of the effective spring constant keff, the correlated Einstein frequency oE and temperature yE and the cumulants ă starting from... the quantities K(P,0), g(P,0) at pressure P and temperature T¼0 K In SMM the nearest neighbor distance a(P,T) between the two immediate atoms at pressure P and temperature T can be defined as aðP,T