ARTICLE IN PRESS Physica B 405 (2010) 2519–2525 Contents lists available at ScienceDirect Physica B journal homepage: www.elsevier.com/locate/physb Anharmonic correlated Debye model Debye–Waller factors Nguyen Van Hung a,n, Nguyen Bao Trung a, Barbara Kirchner b a b Department of Physics, University of Science, VNU-Hanoi, 334 Nguyen Trai, Hanoi, Vietnam Wilhelm-Ostwald Institute for Physical and Theoretical Chemistry, University of Leipzig, Linne´str 2, 04103 Leipzig, Germany a r t i c l e in fo abstract Article history: Received 16 September 2009 Received in revised form March 2010 Accepted March 2010 Anharmonic correlated Debye model has been derived for calculation and analysis of extended X-ray absorption fine structure (EXAFS) cumulants using an anharmonic effective potential that takes into account all nearest neighbor interactions of absorber and backscattering atoms Dispersion relation has been considered, and anharmonic effects are included based on cumulant expansion Analytical expressions for the first, second, third, and fourth EXAFS cumulants have been derived They contain anharmonic effective potential parameters Numerical calculations have been applied to Cu, where Morse potential is used to characterize the interaction between each pair of atoms The results are found to be in good agreement with experiment & 2010 Elsevier B.V All rights reserved Keywords: EXAFS cumulants Anharmonic effective potential Correlated Debye model Introduction Extended X-ray absorption fine structure (EXAFS) has been developed into a powerful probe of atomic structure and thermal effects of substances [1–25,30,31,33,34], where the Debye–Waller factor e À W(T) accounts for the effects of atomic thermal vibrations [17,19,24] The dominant term W(T)¼2k2s2(T) depends on the mean square relative displacement (MSRD) s2(T) of a given bond between absorber and backscatterer atoms [1–25] This Debye– Waller factor damps the EXAFS spectra with respect to increasing temperature T and wave number k (or energy) Anharmonicity in atomic interaction potential yields additional contributions, which if ignored can lead to non-negligible errors in structural and thermodynamic parameters [4–21] obtained from EXAFS spectra The formalism for including anharmonic effects in EXAFS is often based on the cumulant expansion [5,6], where the even cumulants contribute to the amplitude and the odd ones to the phase of EXAFS spectra [4,6] Moreover, for a two-atomic molecule, the EAXFS cumulants can be expressed as a function of the force constant of one-dimensional bare interaction potential [10,17,30] For many-atomic systems, the EXAFS cumulants can be connected by the same analytical expressions to the force constants of a one-dimensional potential [12,17] Many efforts have been made to calculate the cumulants for including anharmonic effects or phonon–phonon interactions in the temperature dependent EXAFS At high temperatures the classical approach can work well [8,9,11,13–16] But it cannot be n Corresponding author E-mail address: hungnv@vnu.edu.vn (N.V Hung) 0921-4526/$ - see front matter & 2010 Elsevier B.V All rights reserved doi:10.1016/j.physb.2010.03.013 valid at low temperatures due to zero-point vibration [12,17] Several approaches have been derived to calculate the EXAFS cumulants such as anharmonic correlated Einstein single-bond (SB) model [10], full lattice dynamical (FLD) approach [12], anharmonic correlated Einstein (ACE) model [17], path integral calculation [18], force constant (FC) model [19,31], and dynamic matrix (DM) calculation [24] They have achieved excellent steps in studying EXAFS, including anharmonic contributions or anharmonic EXAFS, but certain limitations still remain The fourth cumulant is important as an anharmonic contribution to EXAFS amplitude [6,9,20] and has been measured for some crystals [20], so that its analytical calculation with comparison to experiment is necessary Our new development in this work is to derive an anharmonic correlated Debye (ACD) model for calculation and analysis of EXAFS cumulants considering dispersion relation and using an anharmonic interatomic effective potential that takes into account all nearest neighbor interactions of absorber and backscattering atoms instead of a Morse pair potential used in the FLD approach [12] or a SB model [10], for approximating EXAFS cumulants up to the fourth order instead of those of the SB [10], ACE [17], and DM [24] model using cubic anharmonicity, including only the first three cumulants without dispersion, or the FC method [19,31] providing information mainly on the MSRD Section presents the derivation of analytical expressions for the correlated Debye frequency and temperature, dispersion relation, and four leading EXAFS cumulants containing anharmonic effective potential parameters and more information taken from integration over the phonon wave numbers varied in the Brillouin zone (BZ) Section illustrates numerical calculations applied to Cu, where Morse potential is used to characterize ARTICLE IN PRESS 2520 N.V Hung et al / Physica B 405 (2010) 2519–2525 interaction between each pair of atoms The influence of the perpendicular MSRD on the calculated cumulants has been discussed The results are compared to different measured values [3,7,20,23] and to other theories [12,17,24,31], showing a good agreement of our results with experiment Formalism 2.1 EXAFS cumulants, anharmonic effective potential, and dispersion relation According to cumulant expansion the anharmonic EXAFS function is given by [21] ( " #) X ð2ikÞn ðnÞ wðkÞ $ Im eiFðkÞ exp 2ikR ỵ s Tị , 1ị n! nẳ1 where F is the net phase shift, R¼/rS with r as the instantaneous bond length between absorber and backscatterer atoms, and s(n) (n¼1, 2, 3, 4, y) are the cumulants To determine the cumulants it is necessary to specify the interatomic potential and force constant [10,17,21,22,25] Let us consider an anharmonic interatomic effective potential expanded to the 4th order: Veff ðxÞ % keff x2 ỵ k3eff x3 ỵ k4eff x4 , ð2Þ where keff is the effective local force constant, k3eff and k4eff are effective anharmonic parameters giving an asymmetry of the anharmonic effective potential, and x is the deviation of the instantaneous bond length between two immediate neighboring atoms from its equilibrium The effective potential equation (2) is defined based on an assumption in the center-of-mass frame of single-bond pair of absorber with mass M1 and backscatterer with mass M2 [17] as ! XX m ^ ^ M1 M2 V xR12 URij , m ẳ , 3ị Veff xị ẳ Vxị ỵ M M ỵ M2 i i ẳ 1,2 j a i where the sum i is over absorbing and backscattering atoms, the ^ ij the sum j is over their neighbors, Mi is the ith atom mass, and R unit vector along the bond between ith and jth atoms The first term on the right concerns only absorber and backscatterer, and the second one describes the lattice contributions to pair interaction and depends on crystal structure type Hence, this effective pair potential describes not only pair interaction of absorbing and backscattering atoms themselves, but also an effect of their neighbors on such interaction This is the difference of our effective potential from the single-pair potential [12], which concerns only each pair of immediate neighboring atoms like V(x) without the 2nd term in the right-hand side of Eq (3) Comparing Eq (3) to Eq (2) we can determine effective parameters keff, k3eff , and k4eff of the anharmonic effective potential In this paper we generalize the pair model of Eq (3) to that of a linear chain with the same effective potential, in order to account for the effects of dispersion Then based on Ref [26] for the case of vibration between absorber and backscattering atoms and using the interatomic effective potential, the dispersion relation is expressed as r k qa p 4ị oqị ẳ eff sin , q r , M a where q is the phonon wave number, M the mass of composite atoms, and a the lattice constant At the bounds of the first BZ of the linear chain, q¼ p/a, the frequency is maximum so that we get the correlated Debye frequency oD and temperature yD in the following form: rffiffiffiffiffiffiffi k _o oD ¼ eff , yD ẳ D , M kB 5ị where kB is the Boltzmann constant This Debye frequency is calculated using our effective spring constant keff for vibration of atoms in crystal lattice as some treatments of the formula oD ¼cqD, which is derived based on vibration of the system of N oscillators with frequencies varied from to the maximal Debye frequency oD, and propagated with sound speed c in the crystal having volume V, where qD is defined by the number density N/V 2.2 Anharmonic correlated Debye model cumulants We describe the parameter x in terms of the displacement of nth atom un of the one-dimensional chain as xn ¼ un ỵ un , 6ị where the displacements un are related to phonon displacement operators Aq [27] by rffiffiffiffiffiffiffiffiffiffiffi _ X eiqan ỵ pAq , Aq ẳ Aq , Aq ,Aq0 ẳ 0: 7ị un ẳ 2NM q oðqÞ Applying un from Eq (7) to Eq (6) the parameter xn is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X _ iqan eiqa À1 : e f ðqÞAq , f qị ẳ 8ị xn ẳ o qị 2NM q In order to include anharmonic effects, Hamiltonian of the system is written as the summation of harmonic and anharmonic components, H0 and Ha, respectively: H ẳ H0 ỵ H a , H a ẳ Hc ỵ H q , 9ị where Ha consists of cubic Hc and quartic Hq term If the anharmonic contribution to the anharmonic interatomic effective potential consists of the cubic term, then it can be expressed as X Vðq1 ,q2 ,q3 ÞAq1 Aq2 Aq3 , ð10Þ Hc ¼ k3eff x3 ¼ q1 ,q2 ,q3 or in the following form using Eq (8) for the displacement of nth atom: X un ỵ un ị3 Hc ẳ k3eff n ¼ k3eff X q1 ,q2 ,q3 ! X eiðq1 þ q2 þ q3 Þan f ðq1 Þf ðq2 Þf ðq3 ÞAq1 Aq2 Aq3 : ð11Þ n Comparing Eq (11) to Eq (10) and denoting X X iqna DðqÞ ¼ e , Dð0Þ ¼ ei0na ¼ N, N n n ð12Þ N being the atomic number we obtain V q1 ,q2 ,q3 ị ẳ k3eff Dq1 ỵq2 ỵ q3 Þf ðq1 Þf ðq2 Þf ðq3 Þ: ð13Þ Using Eq (8) and Eq (12), Eq (13) is transformed to V q1 ,q2 ,q3 ị ẳ ! 3=2 X eiq1 a À1 eiq2 a À1 eiq3 a _ p eiq1 ỵ q2 ỵ q3 ịan : k3eff 2NM oðq1 Þoðq2 Þoðq3 Þ n ð14Þ The 1st cumulant or net thermal expansion has been calculated with the aid of many-body perturbation approach ARTICLE IN PRESS N.V Hung et al / Physica B 405 (2010) 2519–2525 [28] using the expression P f ðqÞ Aq SðbÞ s1ị ẳ /xS ẳ q , Sbị Sbị ẳ given by 15ị Z Z b X 1ịn b dt1 dtn T ẵHa t1 Þ Á Á Á Ha ðtn Þ, n! 0 n¼0 ¼ etH0 Ha eÀtH0 , _a ¼À 2pkeff Ha tị Z p=a ỵ zqị dq, 1zqị oqị 25ị 17ị s3ị ẳ /x3 S3/x2 S/xS ỵ2/xS3 % /x3 SÀ3/x2 S/xS: ð26Þ f ðqÞ q X V ðq1 ,q2 ,q3 Þ Z b q1 ,q2 ,q3 D h iE dt T A^ q ð0ÞA^ q1 ðtÞA^ q2 ðtÞA^ q3 ðtÞ : ð18Þ Using Wick theorem for T-product in the integral, the harmonic phonon Green function [28] D h iE G0q,q0 tị ẳ T A^ q tịA^ q0 ð0Þ , G0q,q0 ðtÞ À_oðqÞt _oqịt ẫ ẩ ỵ nq e , 19ị ẳ dq,q0 nq ỵ1 e the symmetry properties of V(q1, q2, q3) [27], properties of function dq, À q0 , the phonon density /nq S ¼ Z p=a qa1 þ zðqÞ dq sin 1ÀzðqÞ which contains only effective local force constant keff of the anharmonic effective potential The 3rd cumulant has been calculated using the following expression: we obtain hxi ¼ À _a s2 ¼ /x2 S ẳ q p Mkeff 16ị considering backscattering only from the first shell Substituting into Eq (15) the relations [28] Z D h iE ^ ðtÞ , /Aq S ẳ 0, Aq Sbị ẳ dt T Aq H X 2521 , ZðqÞÀ1 Zqị ẳ expb_oqịị, b ẳ 1=kB T, The calculation of /x S is analogous to the one of /xS above, i.e P q1 ,q2 ,q3 f ðq1 Þf ðq2 Þf ðq3 Þ Aq1 Aq2 Aq3 SðbÞ o /x3 S ẳ : 27ị /SbịS0 Using S(b) from Eq (16), limiting only to cubic anharmonic term, the Wick theorem for T-product, and the symmetry properties of V(q1, q2, q3) [27] we calculated /x3S of Eq (27) The product 3/x2S/xS has been calculated using /x2Sfrom Eq (24) and /xS from Eq (21) Substituting the obtained /x3S and 3/x2S/xS in Eq (26) and applying the relation for phonon momentum conservation in the first BZ we obtain the 3rd cumulant as 3_2 k3eff X oq1 ịoq2 ịoq1 ỵq2 ị 2N2 k3eff q1 ,q2 oq1 ịỵ oq2 ị ỵ oq1 ỵ q2 ị s3ị ẳ ( 20ị ) oq1 ị ỵ oq2 ị eb_ẵoq1 ị ỵ oq2 ị eb_oq1 ỵ q2 ị : oq1 ị ỵ oq2 ịoq1 ỵ q2 ị eb_oq1 ị eb_oq1 ị eb_oq1 ỵ q2 ị 1ỵ6 as well as o(q) from Eq (4), f (q) from Eq (8), D(0) from Eq (12), and the phonon momentum conservation in the first BZ we transform Eq (18) to À ÁÀ Á 3_k3eff X eiqa eiqa 1 ỵ Zqị hxi ẳ 2NMkeff q oqị 1Zqị X 3_k3eff qa 1ỵ Zqị : 21ị ¼ qffiffiffiffiffiffiffiffiffiffiffiffi sin 1ÀZðqÞ N Mk3 q eff ð28Þ For large N, the summation over q can be replaced by the corresponding integral, so that the 3rd cumulant is given by Z Z ðp=aÞÀq1 3_2 k3eff a2 p=a sð3Þ ¼ dq dq2 Fðq1 ,q2 Þ, ð29Þ 4p2 k3eff Àp=d where For large N, the summation over q can be replaced by the corresponding integral, so that the 1st cumulant is estimated as Z p=a qa1 ỵ Zqị 3a_k3eff s1ị ẳ /xS ẳ q sin dq 1Zqị p Mk3 Fq1 ,q2 ị ẳ ( Z p=a oqị ỵZqị dq, 1Zqị oq1 ị ỵ oq2 ị eb_ẵoq1 ị ỵ oq2 ị eb_oq1 ỵ q2 ị , oq1 ị ỵ oq2 ịoq1 ỵ q2 ị eb_oq1 ị eb_oq2 ị eb_oq1 ỵ q2 ị 30ị 22ị which contains the effective local force constant keff and the effective anharmonic cubic parameter k3eff of the anharmonic effective potential Now we calculate the second cumulant, which is equal to the Debye–Waller factor (DWF) s2: s2ị ẳ s2 ẳ /x2 S/xS2 % /x2 S: ) ỵ6 eff 3a_k3eff ẳ 2pk2eff oq1 ịoq2 ịoq1 ỵ q2 ị oq1 ị ỵ oq2 ị þ oðq1 Àq2 Þ which contains the effective local force constant keff and cubic anharmonic parameter k3eff For calculation of the 4th cumulant the Hamiltonian of the system includes anharmonic contributions up to the 4th order, so that similar to Eq (10) we obtain X Vðq1 ,q2 ,q3 ÞAq1 Aq2 Aq3 Ha ẳ k3eff x3 ỵk4eff x4 ẳ q1 ,q2 ,q3 ð23Þ Using Eq (4) for o(q), Eq (8) for xn and f(q), Eq (12) for D(q) and D(0), Eq (19) for G0q,q0 (t), and Eq (20) for /nqS we calculate /x2S to obtain the 2nd cumulant: * + X qa1ỵ Zqị X _ xn ỵ xn ị2 ẳ q sin : 24ị /x2 S ẳ 1ÀZðqÞ n N Mkeff q For large N, the summation over q can be replaced by the corresponding integral, so that the 2nd cumulant or DWF is ỵ X V ðq1 ,q2 ,q3 ,q4 ÞAq1 Aq2 Aq3 Aq4 , ð31Þ q1 ,q2 ,q3 ,q4 V ðq1 ,q2 ,q3 ,q4 ị ẳ k4eff _ 2NM ! 2 X iq1 ỵ q2 ỵ q3 ỵ q4 ịan e n À ÁÀ ÁÀ ÁÀ Á eiq1 a À1 eiq2 a À1 eiq3 a À1 eiq4 a À1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :  oðq1 Þoðq2 Þoðq3 Þoðq4 Þ ð32Þ ARTICLE IN PRESS 2522 N.V Hung et al / Physica B 405 (2010) 2519–2525 The 4th cumulant has been calculated based on the following expression: s4ị ẳ /x4 S4/x3 S/xS3/x2 S2 ỵ 12/x2 S/xS2 À6/xS4 % /x4 SÀ3/x2 S2 : ð33Þ Calculation of /x4S is analogous to that of /xS above Using /x2Sfrom Eq (24) we obtain 3/x2S2 Substituting these values in Eq (33) and using the phonon momentum conservation in the first BZ, the 4th cumulant equation (33) is obtained as s4ị ẳ 3_3 k4eff X Dq1 ỵ q2 ỵ q3 þ q4 Þoðq1 Þoðq2 Þoðq3 Þoðq4 Þ oðq1 Þ þ oq2 ị ỵ oq3 ị ỵ oq4 ị 2N k4eff q1 ,q ,q3 ,q4 > < Zðq1 ịZq2 ịZq3 ịZq4 ị 1ỵ8 > Zq1 ị1ịZq2 ị1ịZq3 ị1ịZq4 ị1ị : ) oq3 ị ỵ oq4 ị : oq1 ị ỵ oq2 ịoq3 ịoq4 ị 34ị For large N, the summation over q can be replaced by the corresponding integral, so that the 4th cumulant Eq (34) is given by Âdq3 Z p=a dq1 Z p=aÀq1 dq2 Z p=aq1 ỵ q2 ị p=a oq1 ịoq2 ịoq3 ịoq4 ị oq1 ị ỵ oq2 ị ỵ oq3 ị ỵ oq4 ị > < Zq1 ịZq2 ịZq3 ịZq4 ị 1ỵ8 > Zq1 ị1ịZq2 ị1ịZq3 ị1ịZq4 ị1ị : ) oq3 ị ỵ oq4 ị , oq1 ị þ oðq2 ÞÀoðq3 ÞÀoðq4 Þ ð35Þ Â which contains the effective local force constant keff and the anharmonic parameter k4eff of the anharmonic effective potential Table Expressions of cumulants in LT (T-0) and HT (T-N) limits s À s(4) T-0 T-N 6_k3eff q1 ỵ zị p Mk3eff 3k3eff kB T k2eff 2_ q1 ỵ zị kB T keff 3_2 k3eff P oðq1 Þoðq2 Þoðq3 Þ ð1 þ Z3 Þ 2N k3eff q1 ,q2 ,q3 oðq1 ị ỵ oq2 ịỵ oq3 ị 6k3eff kB T ị À k3eff p Mkeff s(3) to simplify the expressions for the cumulants In the LT limit Z(q)b1, so that all temperature dependent terms approach zero, and the LT limit cumulants approach constant values, e.g., their zero-point contributions These results are presented in Table 1, where Z p=a sinðqaÞ=ð2Þ dq, zẳ a Zqị 6ẵoq1 ị ỵ oq2 ị Zq3 ịZq1 ịZq2 ị , ẵoq1 ị ỵ oq2 ị2 o2 ðq3 Þ Zðq1 ÞZðq2 ÞZðq3 Þ È É oðq1 ị ỵ oq2 ị ỵ oq3 ị Zq ịZq ÞZðq ÞÀZðq Þ oðq1 Þ þ oðq2 Þ þ oðq3 ÞÀoðq4 Þ 7 ẩ ẫ oq3 ị ỵ oq4 ị ỵ Zq1 ịZq2 ịZq3 ịZq4 ị oq1 ị ỵ oq2 ịoq3 ịoq4 ị Z4 ẳ : P oq1 Þoðq2 Þoðq3 Þoðq4 Þ Zðq1 ÞZðq2 ÞZðq3 ÞZðq4 Þ q1 ,q2 ,q3 ,q4 oq1 ị ỵ oq2 ị ỵ oq3 ị ỵ oq4 ị 37ị Note that (Table 1) at high temperatures the 1st and 2nd cumulants are proportional to the temperature T, the 3rd cumulant to T2, and the 4th cumulant to T3 as the standard characters for these quantities as was mentioned in Ref [9] At low temperatures they approach their zero-point contributions, which also involve contributions of q-values from the 1st BZ Moreover, our quartic anharmonicity provides the 4th cumulant contributing to the EXAFS amplitude Now we apply the expressions derived in previous sections to numerical calculations for Cu A Morse potential is assumed to describe the interaction between each pair of atoms, expanded to the 4th order around its minimum: 38ị Vxị ẳ D ỵ a2 x2 a3 x3 ỵ a4 x4 þ Á Á Á , 12 Zðq1 ÞZðq2 ÞÀZðq3 ÞZðq4 ị ỵ6 Zq1 ị1ịZq2 ị1ịZq3 ị1ịZq4 ị1ị s(1) 36ị Numerical results and discussion oq1 ị ỵ oq2 ị ỵ oq3 ị oq1 ị ỵ oq2 ị ỵ oq3 ịoq4 ị Cumulant Zqị % ỵ b_oqị Zq1 ịZq2 ịZq3 ịZq4 ị ỵ6 Zq1 ị1ịZq2 ị1ịZq3 ị1ịZq4 ị1ị 9_3 a3 k4eff 4p3 k4eff It is useful to consider the high-temperature (HT) limit, where the classical approach is applicable, and the low-temperature (LT) limit, where the quantum theory must be used [17] In the HT limit we use the approximation Z3 ẳ oq1 ị ỵ oq2 ị ỵ oq3 ị oq1 ị ỵ oq2 ị ỵ oq3 ịoq4 Þ sð4Þ ¼ 2.3 High- and low-temperature limits where a describes the width of the potential, and D the dissociation À1 energy For Cu we have D ¼ 0:3429 eV and a ¼ 1:3588 A˚ [29], which were obtained using experimental values for the energy of sublimation, the compressibility, and the lattice constant These values have been used for calculation of local force constant kS, anharmonic parameters k3S, k4S, correlated Debye frequency oDS, and temperature yDS for the anharmonic effective (S¼eff) and pair (S¼p) potentials Some results are presented in Table compared to those for the anharmonic effective potential calculated using the À1 measured Morse parameters [20] D¼0.33 eV and a ¼ 1:38 A˚ Table Calculated values of kS, oDS, yDS of Cu for effective (S¼ eff) and pair (S¼ p) potentials using the Morse parameters from Refs [20] (experimental values) and [29] 3_3 k4eff P 24k4eff ðkB T Þ oðq1 Þoðq2 Þoðq3 Þoðq4 Þ ỵ Z4 ị q1 ,q2 ,q3 ,q4 oq1 ịỵ oq2 ị ỵ oq3 ị ỵ oq4 ị N k4eff k4eff S kS(N/m) oDS(  1013 Hz) eff[29] eff[20] p [29] 50.7181 50.3450 20.2872 4.3717 4.3556 2.7649 yDS (K) 333.9399 332.7094 211.2021 ARTICLE IN PRESS N.V Hung et al / Physica B 405 (2010) 2519–2525 They show a significant strengthening of all parameters, as well as of correlated Debye frequency and temperature for the effective potentials compared with those for the pair potential Note that the value kp ¼ 20.3 N/m for the pair potential is significantly smaller than 27.9 N/m needed to approximate the observed phonon spectra with a single parameter [17] This is indicative of the limitations of a pair potential [17] and SB [10] model, and the possible importance of next-neighbor interactions Moreover, the Debye temperature yD ¼333 K calculated using our effective potential is closer to 315 K determined by fitting the observed heat capacity with the Debye formula at the point where the heat capacity is about half the Dulong–Petit value [32], than the value 211 K calculated using the pair potential The calculated potential parameters have been used for calculation of the anharmonic effective, harmonic effective, and pair potentials (Fig 1a) and dispersion relations (Fig 1b) The anharmonic effective potential and dispersion relation calculated using the measured Morse parameters [20] coincide with those calculated by the present theory; hence they are not shown in Fig The maximal frequencies calculated using the effective local force constant keff (Fig 1b) equal the Debye frequencies presented in Table The Morse parameters from Ref [29] have been applied further to calculation of cumulants for Cu using the present theory Based on the calculated 1st cumulant we deduced the temperature dependence of interatomic distance R(T) of the 1st shell (Fig 2a), which is found to be in reasonable agreement with experiment [20], but quite different from the pair potential result In order to receive a better agreement, our result has to be added to the contribution of the perpendicular MSRD [23,33], whose accurate independent evaluation is however not easy There is no way of obtaining the perpendicular MSRD solely from EXAFS; it requires comparison with the three-dimensional XRD technique Recent path-integral Monte Carlo (PIMC) calculations have been performed on Cu [34] Here the 2nd cumulant and the parallel MSRD have been independently evaluated from the set of configurations generated by PIMC; the agreement between the two values suggests that the contribution of the perpendicular MSRD to the 2nd cumulant is negligible, at least for Cu Temperature dependence of the calculated 2nd cumulant s2(T) (Fig 2b) is compared to the results of the FLD approach [12] at 300 K and of the FC model [31] The present theoretical results for s2(T) agree well with experimental values measured at different 0.35 Anharmonic effective Harmonic effective Anharmonic pair Cu 4.5 Cu Frequency ω (1013 Hz) 0.3 Potential V (eV) 2523 0.25 0.2 0.15 0.1 0.05 3.5 2.5 1.5 0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 π/a -π/a 0 Effective potential Pair potential -1 -0.5 0.5 q (Å-1) x (Å) Fig (a) Calculated anharmonic effective potential compared to harmonic effective and pair potential, and (b) dispersion relations calculated using effective and pair potentials 0.02 Cu 2.64 Present theory Expt., Ref Expt., Ref Expt., Ref 20 Expt., Ref 23 FLD, Ref 12 FC, Ref 31 0.018 Present theory Expt., Ref 20 Pair potential 0.016 0.014 2.62 σ (Å2) Interatomic distance R (Å) 2.66 2.6 0.012 Cu 0.01 0.008 2.58 0.006 2.56 0.004 2.54 0.002 100 200 300 400 T (K) 500 600 700 100 200 300 400 T (K) 500 600 700 Fig Temperature dependence of (a) interatomic distance R(T) deduced from calculated 1st cumulant compared to experiment [20] and to the one using pair potential and (b) calculated 2nd cumulant s2(T) compared to experiment [3,7,20,23] and to those using FLD [12]and FC [31] model ARTICLE IN PRESS 2524 N.V Hung et al / Physica B 405 (2010) 2519–2525 Cu Present theory Expt., Ref 20 σ (4) (10-5 Å4) σ (3) (10-4 Å3) Cu 3.5 Present theory Expt., Ref Expt., Ref 20 Expt., Ref 23 2.5 1.5 1 0.5 0 100 200 300 400 500 600 700 100 200 T (K) 300 400 500 600 700 T (K) Fig Temperature dependence of our calculated (a) 3rd cumulant s(3)(T) compared to experiment [7,20,23]and (b) 4th cumulant s(4)(T) compared to experiment [20] temperatures [3,7,20,23] Fig presents good agreement of temperature dependence of the calculated 3rd cumulant s(3)(T) with experiment [7,20,23] (a) and of the calculated 4th cumulant s(4)(T) (b) with experiment [20] The temperature dependences of s2(T) and s(3)(T) calculated by the ACE model [17] and DM procedure [24] coincide with those calculated by the present theory, so that they are not shown in Figs 2b and 3a, respectively Temperature dependences of the 2nd, 3rd, and 4th cumulants calculated using the pair potential are quite different from those calculated using the present theory and from experiment as shown in Fig 2a for R(T) with a discrepancy of about 50% at 300 K, so that they are not shown in the respective figures, and we present only the case for R(T) in Fig 2a as an example Note that the calculated EXAFS cumulants illustrated in the above figures satisfy their highand low-temperature properties shown in Table Conclusions In this work an anharmonic correlated Debye model has been derived for calculation and analysis of four leading EXAFS cumulants considering dispersion relation to get more information of phonons from the first BZ Although our model is one-dimensional, three-dimensional interactions have been taken into account using an anharmonic effective potential that includes all nearest neighbor interactions of absorber and backscattering atoms By projecting the interactions along the bond direction as in Eq (3), the purely one-dimensional model is recovered Hence, we have extended this effective pair interaction model to a onedimensional chain to partly account for dispersion effects Here Morse potential is used only for characterizing interaction between each pair of atoms Derived expressions for dispersion relation, correlated Debye frequency, and temperature using our effective spring constant keff, and four temperature dependent EXAFS cumulants satisfy all their fundamental properties All obtained expressions for the cumulants contain keff, but the 1st and 3rd cumulants appear due to parameter k3eff, and the 4th cumulant due to parameter k4eff of the anharmonic effective potential so that the 1st, 3rd, and 4th cumulants are apparently anharmonic effects This theory has been applied to fcc crystals, but it also can be extended to binary alloys by including both acoustic and optical frequencies The good agreement of numerical results for Cu with experiment shows the advantages and efficiency of the present theory and of using the anharmonic effective potential procedure in EXAFS data analysis Acknowledgments The authors thank J.J Rehr, P Fornasini, and R.R Frahm for useful comments One of the authors (N.V.H.) appreciates the DFG Projects KI 768/5-1, KI 768/5-2 under SPP 1191 and the WilhelmOstwald-Institute for Physical and Theoretical Chemistry, University of Leipzig for the supports and hospitality during his stay here Two of the authors (N.V.H., N.B.T.) acknowledge the supports of the Research Project No 103.01.09.09 of NAFOSTED References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] G Beni, P.M Platzman, Phys Rev B 14 (1976) 1514 E Sevillano, H Meuth, J.J Rehr, Phys Rev B 20 (1979) 4908 R.B Greegor, F.W Lytle, Phys Rev B 20 (1979) 4902 J.M Tranquada, R Ingalls, Phys Rev B 28 (1983) 3520 G Buncker, Nucl Instrum Methods Phys Res 207 (1983) 437 E.D Crozier, J.J Rehr, R Ingalls, D.C Koningsberger, R Prins (Eds.), Wiley, New York, 1988 T Yokoyama, T Sasukawa, T Ohta, Jpn J Appl Phys 28 (1989) 1905 L Wenzel, D Arvanitis, H Rabus, T Lederer, K Baberschke, G Comelli, Phys Rev Lett 64 (1990) 1765 E.A Stern, P Livins, Zhe Zhang, Phys Rev B 43 (1991) 8850 A.I Frenkel, J.J Rehr, Phys Rev B 48 (1993) 585 ă L Troger, T Yokoyama, D Arvanitis, T Lederer, M Tischer, K Baberschke, Phys Rev B 49 (1994) 888 T Miyanaga, T Fujikawa, J Phys Soc Jpn 63 (1994) 1036, 3683 N.V Hung, R Frahm, Physica B 208 & 209 (1995) 91 N.V Hung, in: in: Proceedings of the Fourth International Conference on Synchrotron Radiation Sources, Second Forum on Synchrotron Radiation, Kyongju, Korea, 1995 420–424 N.V Hung, R Frahm, H Kamitsubo, J Phys Soc Jpn 65 (1996) 3571 N.V Hung, J Phys IV C2 (1997) 279 N.V Hung, J.J Rehr, Phys Rev B 56 (1997) 43 T Yokoyama, Phys Rev B 57 (1998) 3423 A.V Poiarkova, J.J Rehr, Phys Rev B 59 (1999) 948 I.V Pirog, T.I Nedoseikina, A.I Zarubin, A.T Shuvaev, J Phys.: Condens Matter 14 (2002) 1825 N.V Hung, N.B Duc, R.R Frahm, J Phys Soc Jpn 72 (2003) 1254 M Daniel, D.M Pease, N Van Hung, J.I Budnick, Phys Rev B 69 (2004) 134414 G Dalba, P Fornasini, R Grisenti, J Purans, Phys Rev Lett 82 (1999) 4240 F.D Vila, J.J Rehr, H.H Rossner, H.J Krappe, Phys Rev B 76 (2007) 014301 N.V Hung, L.H Hung, T.S Tien, R.R Frahm, Int J Mod Phys B 22 (2008) 5155 C Kittel, in: Introduction to Solid State Physics, 6th ed., John-Wiley & Sons, Inc., New York, 1986 A.A Maradudin, in: G.K Horton, A.A Maradudin (Eds.), Dynamical Properties of Solids, vol 1, Amsterdam, North Holland, 1974, p ARTICLE IN PRESS N.V Hung et al / Physica B 405 (2010) 2519–2525 [28] [29] [30] [31] G.D Mahan, in: Many-Particle Physics, 2nd ed., Plenum, New York, 1990 L.A Girifalco, W.G Weizer, Phys Rev 114 (1959) 687 T Yokoyama, J Synchrotron Radiat (1999) 323 R.M Niclow, G Gilat, H.G Smith, L.J Raubenheimer, M.K Wilkinson, Phys Rev 164 (1967) 922 2525 [32] N.W Ashroft and N.D Mermin, Solid State Physics (Holt, Rinehart and Winston, New York, 1976) [33] P Fornasini, S a Beccara, G Dalba, R Grisenti, A Samson, M Vaccari, Phys Rev B 70 (2004) 74301 [34] S a Beccara, P Fornasini, Phys Rev B 77 (2008) 172304 ... composite atoms, and a the lattice constant At the bounds of the first BZ of the linear chain, q¼ p/a, the frequency is maximum so that we get the correlated Debye frequency oD and temperature yD... This Debye frequency is calculated using our effective spring constant keff for vibration of atoms in crystal lattice as some treatments of the formula oD ¼cqD, which is derived based on vibration... energy of sublimation, the compressibility, and the lattice constant These values have been used for calculation of local force constant kS, anharmonic parameters k3S, k4S, correlated Debye frequency