DSpace at VNU: Anharmonic effective potential, correlation effects, and EXAFS cumulants calculated from a morse interaction potential for fcc metals

Journal of the Physical Society of Japan Vol 76, No 8, August, 2007, 084601 #2007 The Physical Society of Japan Anharmonic Effective Potential, Correlation Effects, and EXAFS Cumulants Calculated from a Morse Interaction Potential for fcc Metals Nguyen V AN H UNGà and Paolo F ORNASINI1 Department of Physics, University of Science, VNU Hanoi, 334 Nguyen Trai, Hanoi, Vietnam Dipartimento di Fisica, Universita` di Trento, Via Sommarive 14, I-38050 Povo (Trento), Italy (Received March 2, 2007; accepted June 13, 2007; published August 10, 2007) Anharmonic effective pair potentials and effective local force constants have been studied for fcc metals, assuming an interaction Morse potential and taking into account the influence of nearest neighbours of absorber and backscatterer atoms Analytical expressions for the first three extended X-ray absorption fine structure (EXAFS) cumulants, as well as for the atomic mean square displacements, have been derived as a function of the Morse parameters Numerical results for copper and nickel are compared with experimental data A good agreement is found for the second cumulant Non-negligible discrepancies are instead found for the first and third cumulants, which are tentatively attributed to the central nature of the Morse potential, which neglects many-body effects KEYWORDS: anharmonicity, EXAFS, Debye–Waller factor DOI: 10.1143/JPSJ.76.084601 Introduction Extended X-ray absorption fine structure (EXAFS) has developed into a powerful probe of local atomic structure and thermal effects of substances.1–12) To take into account thermal vibrations, the EXAFS function for a given scattering path is expressed in terms of a canonical average of all distance-dependent factors:4) & ( 2ikr À2r=ðkÞ )' e 2iðkÞ e ðkÞ / = f ðk; ; RÞe ; ð1Þ r2 where r is the instantaneous distance and R an average distance The canonical average of eq (1) is equivalent to a real space average:13) ( 2ikr À2r=ðkÞ ) Z e e Pr; ịe2ikr dr 2ị ẳ r2 where Pr; ị ẳ rịe2r= =r 3ị is an effective distribution of distances, ðrÞ being the real distribution.14) In case of moderate disorder, the integral on the right-hand side of eq (2) can be conveniently parametrized in terms of a few leading cumulants14) of the real distribution, typically the first three:  ð1Þ ,  ð2Þ , and  ð3Þ The cumulants of the real distribution have been elsewhere11,15) indicated by Cià (i ¼ 1; 2; 3; ), with a different meaning of the first cumulant C1à is the average value of the distribution of distances,11,15)  ð1Þ is the thermal expansion with respect to the equilibrium distance.6,8) In principle, C1à and  ð1Þ should have the same temperature dependence, but differ in absolute values The third cumulant  ð3Þ  C3à measures the asymmetry of the distribution The second cumulant   C2à is the variance of the distribution, and corresponds, to a good accuracy, to the parallel mean square relative displacement (MSRD) D  E ^ Á ðub À ua Þ2  ẳ hu2k i ẳ R 4ị E-mail: hungnv@vnu.edu.vn where ub and ua are the instantaneous displacements of ^ is backscatterer and absorber atoms, respectively, and R 2) the unit vector along the equilibrium distance Since EXAFS is sensitive to the relative motion, the MSRD is the sum of the uncorrelated mean square displacements (MSDs) u2 of the two atoms minus the displacement correlation function (DCF) CR For a two-atomic molecule, the EXAFS cumulants can be expressed as a function of the force constant of the onedimensional bare interaction potential.6,16) For many-atomic systems, the EXAFS cumulants can be connected by the same analytical expressions to the force constants of a onedimensional effective pair potential The relationship between EXAFS cumulants and effective potential on the one hand, and physical properties of many-atomic systems on the other, is still a matter of debate, in particular with reference to the very meaning of the effective potential13,17) and its possible dependence on temperature.18,19) The interest towards this subject has been enhanced by recent temperature dependent EXAFS studies, which have questioned the equivalence of the first and third cumulants for measuring the thermal expansion of interatomic bonds.11,20) A method for calculating the EXAFS cumulants from the force constants of the crystal potential, based on first principles finite temperature many-body perturbation theory, has been proposed by Fujikawa and Miyanaga.21) The method has been thoroughly applied mainly to one-dimensional systems,22) and some attempts have been done also for fcc crystals.23) Promising results on three-dimensional systems have been obtained by path-integral techniques, based on the use of effective potentials24) or on Monte Carlo sampling.25) A simpler phenomenological approach was developed by Hung and Rehr8) (henceforth cited as HR), who derived an anharmonic correlated model for the effective potential taking into account the interaction of absorber and backscatterer atoms with their nearest neighbours via a Morse potential HR expressed the second and third EXAFS cumulants, as well as the coefficient of thermal expansion, as a function of the Morse potential parameters for the case 084601-1 J Phys Soc Jpn., Vol 76, No N VAN HUNG and P F ORNASINI of an fcc metal, and compared the theoretical results with the then available experimental data for copper The HR approach, focussing on a local cluster, is suited for treating amorphous and crystalline systems on the same ground New refined experimental data have recently been published on EXAFS cumulants11) and on diffraction thermal factors26) of copper, stimulating a more accurate check of theoretical results This paper is a natural extension of HR Analytical expressions of the anharmonic effective pair potentials and local force constants have been obtained for the first coordination shell of copper and nickel, whence the espressions of EXAFS cumulants have been calculated Besides, a calculation has been attempted also for the singleatom effective potential and the uncorrelated MSD Theoretical results have been compared with recent experimental data from both EXAFS and diffraction experiments, allowing for a careful evaluation of strengths and limitations of the HR procedure.8) The HR approach represents some sort of a ‘‘theoretical laboratory’’, where the connection between the three-dimensional properties of solids and the one-dimensional effective pair potentials can be attempted on the basis of a simple starting model and subsequently ameliorated by a progressive refinement of approximations The comparison with experimental results performed in this work represents a check of the first step, and gives suggestions for further steps In §2 the theoretical approach of HR8) is presented with some detail for the case of fcc metals, to make it more accessible to non specialists, and extended to the uncorrelated MSDs In §3 the EXAFS cumulants and the MSDs calculated as a function of temperature from the Morse potential parameters are presented and compared with experimental data The possible origin of discrepancies is discussed Section is dedicated to conclusions Theory For perfect monatomic crystals, the parallel MSRD is given by  ¼ 2u2 À CR : Here the MSD has been defined as     ^ ị2 ẳ ub R ^ Þ2 ; u2 ¼ ðua Á R ð5Þ ð6Þ where ua ¼ ub are the instantaneous displacements of ^ is absorber and backscatterer atoms, respectively, and R the unit vector connecting the two atoms, so that the DCF is given by   ^ Þðub Á R ^ ị ẳ 2u2  : CR ẳ ðua Á R ð7Þ The anharmonic effective potential can be expressed as a ^0 function of the displacement x ¼ r À r0 along the R direction, r and r0 being the instantaneous and equilibrium distances between absorber and backscatterer atoms, respectively: Veff xị k0 x2 ỵ k3 x3 ; ð8Þ where k0 is an effective local force constant and k3 is a cubic parameter giving the asymmetry due to anharmonicity (Here and in the following, the constant contributions are neglected.) For the calculation of thermodynamical parameters, we use the further definitions a ¼ hr À r0 i and y ¼ x À a,6) to rewrite eq (8) as Veff yị k0 ỵ 3k3 aịay þ keff y2 þ k3 y3 ; ð9Þ where keff is in principle different from k0 Making use of quantum statistical methods,6,27) the physical quantities are determined by an averaging procedure using the statistical density matrix  and the canonical partition function Z, e.g., hym i ¼ Trym ị; Z m ẳ 1; 2; 3; ð10Þ 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^ and a^y , respectively y  a0 a^ ỵ a^y ị; a20 ẳ h" !E =2keff ; ð11Þ and use the harmonic oscillator state jni as the eigenstate with the eigenvalue En ¼ nh" !E , ignoring the zeropoint energy for convenience (here !E is the Einstein frequency) A Morse potential is assumed to describe the interatomic interaction, and expanded to the third order around its minimum Vxị ẳ D e2x 2ex D ỵ 2 x2 3 x3 ; ð12Þ where  describes the width of the potential and D is the dissociation energy The above expressions will be used in the next subsections, where the presence or absence of an apex s will label the quantities concerning the MSD and the MSRD, respectively 2.1 EXAFS cumulants and mean square relative displacement The case of relative vibrations of absorber and backscatterer atoms, including the effect of correlation, has been treated in HR.8) According to eq (2) of HR, taking into account only the nearest neighbour interactions, the effective pair potential is given by  X X 1 ^ ^ Veff xị ẳ Vxị ỵ x R Rij V i¼a;b j6¼a;b       x x x ẳ Vxị ỵ 2V ỵ 8V ỵ 8V ; 13ị 4 where the first term on the right concerns only absorber and backscatterer atoms, the remaining sums extend over the remaining neighbours, and the second equality is for monatomic fcc crystals Applying the Morse potential (12) to eq (13) and comparing with eq (8), for the case of correlated atomic vibrations the force constants are now expressed in terms of the Morse parameters as k0 ¼ 5D2 ; k3 ¼ À D3 : ð14Þ In accordance with eq (9), the potential (13) can be expressed as 084601-2 J Phys Soc Jpn., Vol 76, No N VAN HUNG and P F ORNASINI   a y Veff ðyÞ ’ 5D2 a À 20   a y2 D3 y3 ; ỵ D2 10 ð15Þ where the force constant is different from that of the potential Veff ðxÞ:   a ¼ !2E ; k3 ¼ À D3 : 16ị keff ẳ 5D 10 Here !E is the correlated Einstein frequency, to which it corresponds a correlated Einstein temperature E ¼ h" !E =kB , and  is the reduced mass Note that the third order coefficient has here the correct value 3/4 instead of 5/4 as reported in HR Using the above results for the correlated atomic vibrations and the procedure depicted by eqs (10) and (11), as well as the first order thermodynamic perturbation theory considering the anharmonic component in the potential (15), we derived the EXAFS cumulants The second cumulant (parallel MSRD) is expressed as  Tị ẳ 02 1ỵz ; 1z z ẳ exph" !E =kB Tị; 17ị where 02 ẳ h" !E =10D2 is the zero-point contribution to the MSRD The first cumulant is 9  ;  ð1Þ Tị ẳ 20 3 2 3 ị À 2ð02 Þ2 10 ð19Þ Ã 9D3  2 ð Þ À ð02 Þ2 ; 4RkB T tot ẳ  ỵ  02 ị an anharmonic factor has been derived  ! 92 3 3 Tị ẳ  1ỵ   1ỵ : 4R 4R s Veff yị 8D2 ay ỵ 4D2 y2 ; 22ị 2.2 Mean square displacement and correlation function Taking into account only the influence of the N nearest atomic neighbors, a single-atom effective potential28) can be obtained N X s ^0 ÁR ^ jÞ ðxÞ ¼ Vðx R Veff j¼1     x x ẳ Vxị ỵ Vxị ỵ 4V ỵ 4V ; ð23Þ 2 ^ j are the unit vectors of neighbouring atoms where R with respect to the equilibrium position of the central atom, and the second equality is valid for monatomic fcc crystals ð26Þ s with the same force constants (24) as the potential Veff xị: keff ẳ k0 Based on the unperturbed statistical density matrix 0 and the unperturbed canonical partition function Z0 X Z0 ¼ Tr0 ị ẳ expnh" !sE ị n ẳ X zns ẳ 27ị ; zs where ẳ 1=kB T and zs ẳ exph" !sE =kB Tị, we determined the MSD 1 X Ành" !s E hnjy2 jni u2 ẳ hy2 i Tr0 y2 ị ẳ e Z Z n X h" !s ỵ zs ẳ 2a20 zs ị ỵ nịzns ẳ sE 2keff zs n ẳ 21ị 24ị m and kB being the atomic mass and the Boltzmann constant, respectively In accordance with eq (9), the potential (23) can be expressed as ð20Þ where R is the bond length To calculate the total MSRD including anharmonic contributions10) k3s ¼ 0: The fact that k3 ¼ shows that taking into account the influence of all nearest neighbours compensates the anharmonic component present in each separate term of the potential (23) The Einstein frequency !sE , and the corresponding temperature, can the be expressed as pffiffiffiffiffiffiffiffiffiffiffiffi !sE ¼ 2 2D=m; Es ẳ h" !sE =kB ; 25ị nẳ0 and the thermal expansion coecient T ẳ k0s ẳ m!sE ị2 ẳ 8D2 ; ð18Þ directly proportional to the second cumulant The third cumulants is  3ị Tị ẳ Applying the Morse potential (12) to eq (23) and comparing with eq (8), for the case of single atomic vibrations we obtain the effective force constant and the cubic parameter h" !sE þ zs þ zs ¼ u20 ; 16D2 zs zs 28ị where u20 ẳ h" !sE =16D2 is the zero-point contribution to the MSD 2.3 Low and high temperature limits It is useful to consider the high-temperature (HT) limit, where the classical approach is applicable, and the lowtemperature (LT) limit, where the quantum theory must be used.8) In the HT limit we use the approximation zðzs Þ ’ À h" !E ð!sE Þ=kB T ð29Þ to simplify the expressions for the thermodynamic parameters In the LT limit, zðzs Þ ! 0, so that we can neglect zðz2s Þ and higher power terms These results are written in Table I Table I Expressions of u2 ,  , CR for fcc crystals in the LT and HT limits 084601-3 Expression T!0 T !1 u2 u20 ỵ 2zs ị kB T=8D2 2 02 þ 2zÞ kB T=5D2 CR 2u20 ð1 þ 2zs Þ 02 ỵ 2zị kB T=20D2 J Phys Soc Jpn., Vol 76, No N VAN HUNG and P F ORNASINI Note that from Table I the functions u2 ,  , CR are linearly proportional to the temperature at high-temperatures and contain the zero-point contributions at low-temperatures, satisfying all standard properties of these quantities.29) Table II Parameters of the Morse potentials for copper and nickel, from Lincoln et al.,31Þ Girifalco and Weizer,30Þ Pirog et al.32Þ Cu 2.4 Comparison with single-bond model For comparison, we apply the Morse potential eq (12) to the anharmonic correlated single bond model of ref VSB xị ẳ SB k x ỵ k3SB x3 Ni Ref D (eV)  ˚ À1 ) (A D2 ˚ 2) (eV/A D3 ˚ 3) (eV/A 31 0.33 1.31 0.57 0.74 30 0.34 1.38 0.63 0.89 32 30 0.33 0.42 1.36 1.42 0.63 0.85 0.83 1.20 32 0.41 1.39 0.79 1.1 ð30Þ to obtain the anharmonic correlated single bond potential VSB yị ẳ D2 3aịay ỵ kSB y2 ỵ k3SB y3 1.2 k3SB ẳ D3 : 32ị Since a ( 1, the comparison of eq (32) with eq (16) shows that the effective force constant is non-negligibly strenghtened if the interaction with nearest neighbours is included, while the cubic constant is slightly reduced The effective pair potential cannot be confused with the bare interaction potential By the same procedure used for the effective potential, we derive the first three cumulants: The second cumulant (parallel MSRD) is SB ðTÞ ẳ 02 ịSB ỵ zSB ; zSB zSB ẳ exph" !SB E =kB Tị; 33ị where 02 ịSB ẳ h" !SB E =4D is the zero-point contribution First and third cumulants are 3  ; SB 3ị Tị ẳ  3 Þ2SB À 2ð02 Þ2SB SB ð1Þ SB ðTÞ ¼ ð34Þ 35ị and the thermal expansion coecient is T ẳ V (eV) and the local force constants kSB ¼ 2D2 ð1 À 3aÞ; à 3D3  2 ð ÞSB À ð02 Þ2SB : RkB T Cu ð31Þ ð36Þ Results and Discussion The expressions derived in §2 have been used to evaluate the force constants of the effective pair and single-particle potentials of copper and nickel The first three EXAFS cumulants and the diffraction MSDs have then been calculated We used the D and  parameters of the Morse potential evaluated by Girifalco and Weizer30) from the experimental values of vaporisation energy, lattice constants 0.8 0.4 -0.6 -0.3 x (Å) 0.3 0.6 Fig Comparison of calculated effective single atom potential (dashed line), effective pair potential (continuous line), and single bond pair potential (dotted line) for Cu and compressibility of Cu and Ni (Table II) The calculated values of the force constants and Einstein angular frequencies are listed in Table III for the single atom and pair effective potentials [eqs (24) and (16), respectively] The values calculated for the single-bond potential [eq (32)] are also shown for comparison In Fig 1, the single-bond, effective single atom and effective pair potentials of copper are shown The single atom potential is, to first order, harmonic Let us compare the pair potentials: the force constant keff is much larger and the modulus of the cubic parameter k3 smaller for the effective potential with respect to the single-bond potential; the effect of the interaction with nearest neighbours on the absorber-backscatterer pair is a strenghtening of the spring constant and reduction of its anharmonicity 3.1 Mean square relative displacement The second cumulants (MSRDs) of copper and nickel calculated from the Morse potential parameters are shown in Table III Calculated and experimental values of force constants and Einstein angular frequencies for the single atom effective, pair effective and single bond potentials, for copper and nickel Single atom (eff.) s keff Cu Calc Exp Exp Ni (10 rad/s) keff (N/m) !E (1013 rad/s) k3 ˚ 3) (eV/A (N/m) !SB E (1013 rad/s) k3SB ˚ 3) (eV/A 81.15 2.77 50.72 3.10 À0:64 20.29 1.96 À0:86 51.27 51.26 3.08 3.08 À1:3 À1:37 67.91 3.73 À0:90 27.16 2.36 À1:20 62.50 3.43 À1:6 11Þ 108.66 32Þ Pair (SB) (N/m) 32Þ Calc Exp Pair (eff.) !sE 13 3.34 084601-4 SB keff J Phys Soc Jpn., Vol 76, No N VAN HUNG and P F ORNASINI Cu 10-4 Cu 0.02 10-4 Third cumulant (Å ) MSRD (Å ) 0.01 Ni 0.02 100 10-4 Ni 10-4 0.01 0 200 400 100 600 200 400 600 T (K) T (K) Fig Temperature dependence of the MSRD tot ðTÞ (continuos lines) for Cu (upper panel) and Ni (lower panel) calculated from eq (21); the dashed line is the harmonic part, while the dash-dotted line is calculated from the single-bond potential The experimental data are from ref 32 (crossed squares) for Cu and Ni, from ref 11 (full circles) for Cu and from ref 24 (open circles) for Ni The dotted line is the theoretical calculation for Ni, from ref 23 Fig The dashed lines represent the harmonic expression eq (17), while the continuous lines include the anharmonic contribution [eq (21)] The difference is of the order of 3% at 600 K, and can be considered negligible with respect to the typical accuracy of experimental data The MSRDs calculated from the single-bond potentials (dash-dotted lines) are instead much larger, due to the smaller value of the force constant keff The calculated MSRDs are compared with recently published data from EXAFS experiments: Fornasini et al.11) for copper from to 500 K (full circles), Pirog et al.32) for both copper and nickel from 300 to 630 K (crossed squares), Yokoyama24) for nickel from 40 to 300 K (open circles) The agreement is particularly good for copper, where the discrepancy between theory and experiments is comparable with the discrepancy between the experiments In the case of nickel, a discrepancy is observed not only between theoretical calculations, but also between experimental values It is anyway worth noting that the calculations by Katsumata et al.23) for nickel are based on significantly different values of the Morse parameters than the present calculations These results, besides stressing the difference between single-bond and effective potential, show that a relatively simple model based on the interaction with nearest-neighbours via a Morse potential can in principle reproduce the amplitude of relative displacements of absorber and backscatterer atoms as a function of temperature in fcc metals A reverse procedure was used by Pirog et al.,32) to evaluate the parameters of the Morse potentials from the experimental values of the first three EXAFS cumulants, based on the relations of HR in the high-temperature classical approximation The values of Morse parameters obtained by Pirog et al are shown in Table II One can see that the values of the product D2 of Pirog et al and of Fig Temperature dependence of the third cumulant  ð3Þ ðTÞ (continuos lines) for Cu (upper panel) and Ni (lower panel) calculated from eq (19); the dash-dotted line is calculated from the single-bond potential The experimental data are from ref 32 (crossed squares) for Cu and Ni, from ref 11 (full circles) for Cu and from ref 24 (open circles) for Ni The dotted line is the theoretical calculation for Ni, from ref 23 Girifalco and Weizer agree for copper, but are slightly different for nickel, and this explains the slight discrepancy between theory and experiment for the MSRD of nickel The difference of D2 values for nickel is however of the same order of magnitude of the difference between the values of Girifalco and Weizer30) and of Lincoln et al.31) for copper It is thus difficult to assess wether the slight discrepancy between theory and experiment found for nickel is due to the approximation of the HR model or to inadequacy of the starting Morse parameters 3.2 Third EXAFS cumulant and thermal expansion The third cumulants evaluated from the Morse potential parameters for copper and nickel are shown in Fig and compared with experimental results The inadequacy of the single-bond model (dash-dotted line) with respect to the effective potential model in reproducing experimental data is evident: the single-bond third cumulant is much higher than both the experimental and the effective potential third cumulants The influence of neighbouring atoms reduces the asymmetry of the distance distribution, as a joint effect of reduction of the cubic parameter k3 and of the MSRD  A non negligible residual discrepancy can however be noticed also between the experimental values and the values calculated from the effective pair potential (continuous line) For copper, theoretical values underestimate the experimental ones by about 30% For nickel, a much better agreement is found, in particular for the low-temperature values One possible cause of the discrepancy between theory and experiment could be the inadequacy of a purely central potential in accounting for the asymmetry of the effective potential Actually, in a recent paper25) dedicated to the calculation of the EXAFS cumulants of the first shell of copper by the path-integral Monte Carlo technique, a good 084601-5 N VAN HUNG and P F ORNASINI 0.02 0.06 Cu (a) 0.04 MSD (Å ) Absolute expansion (Å) J Phys Soc Jpn., Vol 76, No 0.02 Relative expansion (Å) 0.01 (b) 200 T (K) 400 600 0.04 Fig Temperature dependence of the MSD u2 for copper calculated from Morse potential parameters through eq (28) (continuous line) and measured by Day et al.26) (diamonds) and by Martin and O’Connor35) (open squares) The dashed line is the MSD reconstructed from the MSRD on the hypothesis of validity of the Debye model (see text) 0.02 0 200 400 600 T (K) Fig (a) Temperature dependence of the calculated first cumulant  ð1Þ ðTÞ of the first shell of copper, calculated from the effective potential (continuos line) and from the single-bond potential (dash-dotted line) (b) Variation  ð1Þ ðTÞ of the calculated first cumulants with respect to the zero kelvin value, compared with the experimental EXAFS data: from ref 32 (first cumulant, crossed squares) and from ref 11 (first cumulant, circles, and thermal expansion from third cumulant, diamonds), as well as with the vlues of thermal expansion from Bragg diffraction (dashed line) agreement with the experimental values of the third cumulant could be obtained only when a many-body interaction potential was used Single-bond and effective pair potentials also differ in the evaluation of the first cumulant  ð1Þ , corresponding to the thermal expansion with respect to the equilibrium distance (Fig 4, top panel) The thermal expansion of the distance between a pair of isolated atoms interacting via a Morse potential is strongly reduced when the atomic pair is embedded within a crystal Different are also the zero-point thermal expansions The comparison with experimental results for copper is done in the lower panel of Fig An EXAFS study of copper in the temperature range from to 500 K has been recently published by Fornasini et al.;11) the data analysis, performed by the ratio method, led to relative values of the rst cumulant C1 Tị ẳ C1à ðTÞ À C1à ð4 KÞ, where C1à is the average value of the distribution of distances (circles in Fig 4) In the same paper, the thermal expansion was also evaluated according to the Frenkel and Rehr6) formula, a ¼ À3k3 C2à =k0 , using the quadratic and cubic constants k0 and k3 calculated from the second and third experimental cumulants Since no zero-point thermal expansion could be measured by the relative values C1à ðTÞ of the first cumulant obtained through the ratio method, relative values  ð1Þ (continuous line) and a (diamonds) have been plotted in Fig 4, lower panel, in order to facilitate comparisons Pirog et al.32) have plotted the interatomic distances measured by EXAFS in the temperature interval from 290 to 570 K; the values have been vertically shifted so as to match, on the average, the C1à data of Fornasini et al Also shown in Fig is the first-shell thermal expansion Rc measured by Bragg diffraction (dashed line) It is well known11,33,34) that the average distance measured by the first EXAFS cumulant C1à ¼ hjrb À ji is larger than the distance between average atomic positions measured by Bragg diffraction, Rc ¼ jhrb i À hra ij, owing to the effect of thermal vibrations perpendicular to the bond direction Correspondingly, the thermal expansion C1à (circles) is larger than the thermal expansion Rc (dashed line) It has also been experimentally verified11,20) that the thermal expansion due solely to the shape of the effective potential, and measured by6) a ¼ À3k3 C2à =k0 (diamonds) does not correspond to the first cumulant thermal expansion (circles) The discrepancy has been tentatively explained in terms of temperature dependence of the minimum position of the effective potential, but a thorough investigation on the subject is still lacking The first cumulant  ð1Þ , which has been calculated from the Morse potential parameters through eq (18), only depends on the shape of the effective potential and should then directly compared with the parameter a of ref One should then expect a correspondence between the theoretical values  ð1Þ (continuous line) and the experimental values a (diamonds) The non negligible discrepancy between  ð1Þ and a confirms the limitations of the model based on the central Morse potential 3.3 Mean square displacement The uncorrelated single-atom MSD calculated for copper from the Morse potential parameters [eq (28)] is compared in Fig with recent experimental values obtained by diraction of Moăssbauer gamma rays.26,35) The theoretical model underestimates the MSD u2 by about 30%, due to an overestimate of the force constant k0s According to eq (5), since the MSRD  is well reproduced, the model underestimates also the DCF CR The inefficiency in reproducing the MSDs (Fig 5), in spite of the ability in reproducing the MSRDs (Fig 2), can be explained as follows The model here used, based on the simple projection of the nearest-neighbour interaction potentials along an interatomic direction, is quite ineffective in accounting for the long-wavelengths acoustic phonons, which can strongly affect the uncorrelated MSD but have negligible influence on the MSRDs 084601-6 J Phys Soc Jpn., Vol 76, No N VAN HUNG and P F ORNASINI An approximate method for obtaining the MSD values of a metal from the experimental MSRD values can be introduced, in the hypothesis that the temperature dependence of both quantities can be described by suitable Debye models, uncorrelated and correlated, respectively, having the same Debye temperature.2,3) The ratio u2 = can be calculated from the Debye models and used to recover the values of u2 from the values of  The ratio is in principle temperature dependent, but, as a first approximation, one can use its high-temperature asymptotic value As an example, for the case of copper we used the Debye asymptotic value, u2 = ’ 5=6, to evaluate the MSD u2 (dashed line in Fig 5) from the MSRD  calculated from the Morse potential parameters (continuous line in Fig 2) This procedure based on the Debye ratio can be interesting for the following reasons: a) It allows a better evaluation of the MSD from the Morse potential parameters; b) It allows the approximate evaluation of the MSD from the experimental values of MSRD (it is generally easier to get accurate experimental values of the MSRD from EXAFS than of the MSD from diffraction) Conclusions In this work we explored the effectiveness of anharmonic effective pair and single-atom potentials, evaluated from the parameters of a Morse interaction potential and taking into account nearest-neighbour interactions, in reproducing the first three EXAFS cumulants and the uncorrelated MSD of fcc metals copper and nickel The main conclusions can be summarized as follows a) The effective pair potential is significantly stronger and less asymmetric (keff larger and k3 smaller) than the single-bond potential, and much better reproduces the experimental EXAFS cumulants b) A very good agreement is found between theory and experiment for the second EXAFS cumulant (parallel MSRD) of copper; less good is the agreement for nickel, where however a non negligible discrepancy is present also between the available experimental data c) The theoretical values underestimate the experimental values of the third EXAFS cumulant; the discrepancy, larger for copper than for nickel, has been tentatively attributed to the radial nature of the Morse potential, which neglects many-body effects d) The theoretical values underestimate also the experimental first cumulant; the discrepancy is due to: d1) the inadequacy of the third cumulant ( ð1Þ and  ð3Þ in HR model are strictly connected); d2) the intrinsic difference between the experimental values C1à and the theoretical values  ð1Þ , due to vibrations perpendicular to the bond, which are neglected by the present formulation of the HR model e) The theoretical values underestimate the uncorrelated MSD, probably due to the difficulty of the model in properly reproducing the effects of long-wavelendth acoustic modes The good agreement found for the second EXAFS cumulant suggests that further improvements of the HR model could lead to a better reproduction also of the third cumulant and of thermal expansion Acknowledgments The authors thanks J J Rehr and A I Frenkel for helpful comments One of the authors (N.V.H.) appreciates the partial supports by projects Nos 058 06 and QG.05.04 The authors are grateful to the Institute of Pure and Applied Physics for financial support in publication 1) C A Ashley and S Doniach: Phys Rev B 11 (1975) 1279 2) G Beni and P M Platzman: Phys Rev B 14 (1976) 1514 3) E Sevillano, H Meuth, and J J Rehr: Phys Rev B 20 (1979) 4908 4) J M Tranquada and R Ingalls: Phys Rev B 28 (1983) 3520 5) T Yokoyama, T Satsukawa, and T Ohta: Jpn J Appl Phys 28 (1989) 1905 6) A I Frenkel and J J Rehr: Phys Rev B 48 (1993) 585 7) L Troăger, T Yokoyama, D Arvanitis, T Lederer, M Tischer, and K Baberschke: Phys Rev B 49 (1994) 888 8) N Van Hung and J J Rehr: Phys Rev B 56 (1997) 43 9) G Dalba and P Fornasini: J Synchrotron Radiat (1997) 243 10) N Van Hung, N B Duc, and R R Frahm: J Phys Soc Jpn 72 (2003) 1254 11) P Fornasini, S a Beccara, G Dalba, R Grisenti, A Sanson, M Vaccari, and F Rocca: Phys Rev B 70 (2004) 174301 12) M Daniel, D M Pease, N Van Hung, and J L Budnick: Phys Rev B 69 (2004) 134414 13) M Vaccari and P Fornasini: Phys Rev B 72 (2005) 092301 14) G Bunker: Nucl Instrum Methods Phys Res 207 (1983) 437 15) G Dalba, P Fornasini, and F Rocca: Phys Rev B 47 (1993) 8502 16) T Yokoyama: J Synchrotron Radiat (1999) 323 17) A Filipponi: J Phys.: Condens Matter 13 (2001) R1 18) J Munstre de Leon, S D Conradson, I Batistic´, A R Bishop, I D Raistrick, M C Aronson, and F H Garzon: Phys Rev B 45 (1992) 2447 19) G Dalba, P Fornasini, R Gotter, and F Rocca: Phys Rev B 52 (1995) 149 20) G Dalba, P Fornasini, R Grisenti, and J Purans: Phys Rev Lett 82 (1999) 4240 21) T Fujikawa and T Miyanaga: J Phys Soc Jpn 62 (1993) 4108 22) T Miyanaga and T Fujikawa: J Phys Soc Jpn 63 (1994) 1036 23) H Katsumata, T Miyanaga, T Yokoyama, T Fujikawa, and T Ohta: J Synchrotron Radiat (2001) 226 24) T Yokoyama: Phys Rev B 57 (1998) 3423 25) S a Beccara, G Dalba, P Fornasini, R Grisenti, F Pederiva, A Sanson, D Diop, and F Rocca: Phys Rev B 68 (2003) 140301 26) J T Day, J G Mullen, and R Shukla: Phys Rev B 52 (1995) 168 27) R P Feynman: Statistical Mechanics (Benjamin, Reading, MA, 1972) 28) B T M Willis and A W Pryor: Thermal Vibrations in Crystallography (Cambridge University Press, Cambridge, U.K., 1975) 29) J M Ziman: Principles of the Theory of Solids (Cambridge University Press, Cambridge, U.K., 1972) 2nd ed 30) L A Girifalco and W G Weizer: Phys Rev 114 (1959) 687 31) R C Lincoln, K M Koliwad, and P B Ghate: Phys Rev 157 (1967) 463 32) I V Pirog, T I Nedoseikina, I A Zarubin, and A T Shuvaev: J Phys.: Condens Matter 14 (2002) 1825 33) W R Busing and H A Levy: Acta Crystallogr 17 (1964) 142 34) P Fornasini, G Dalba, R Grisenti, J Purans, M Vaccari, F Rocca, and A Sanson: Nucl Instrum Methods Phys Res., Sect B 246 (2006) 180 35) C J Martin and D A O’Connor: J Phys C 10 (1977) 3521 084601-7 ... copper and nickel calculated from the Morse potential parameters are shown in Table III Calculated and experimental values of force constants and Einstein angular frequencies for the single atom... potentials of copper and nickel The first three EXAFS cumulants and the diffraction MSDs have then been calculated We used the D and  parameters of the Morse potential evaluated by Girifalco and. .. MSD and the MSRD, respectively 2.1 EXAFS cumulants and mean square relative displacement The case of relative vibrations of absorber and backscatterer atoms, including the effect of correlation,