Journal of the Physical Society of Japan 83, 024802 (2014) http://dx.doi.org/10.7566/JPSJ.83.024802 Pressure-Dependent Anharmonic Correlated Einstein Model Extended X-ray Absorption Fine Structure Debye–Waller Factors Nguyen Van Hung+ Department of Physics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam (Received May 17, 2013; accepted November 13, 2013; published online January 8, 2014) A pressure-dependent anharmonic correlated Einstein model is derived for extended X-ray absorption fine structure (EXAFS) Debye–Waller factors (DWFs), which are presented in terms of cumulant expansion up to the third order The model is based on quantum thermodynamic perturbation theory and includes anharmonic effects based on empirical potentials Explicit analytical expressions of the pressure-dependent changes in the interatomic distance, anharmonic effective potential, thermodynamic parameters, first, second, and third EXAFS cumulants, and thermal expansion coefficient have been derived This model avoids the use of extensive full lattice dynamical calculations, yet provides good and reasonable agreement of numerical results for Cu with experimental results of X-ray diffraction (XRD) analysis and pressure-dependent EXAFS Significant pressure effects are shown by the decrease in the pressure-induced changes in the interatomic distance, EXAFS cumulants and thermal expansion coefficient, as well as by the increase in the pressure-induced changes in the interatomic effective potential, effective spring constant, correlated Einstein frequency, and temperature Introduction Thermal vibrations and disorder in extended X-ray absorption fine structure (EXAFS) give rise to Debye– Waller factors (DWFs), which are presented here in terms of cumulant expansion1) to include anharmonic effects EXAFS is sensitive to pressure,2–4) which can cause changes of information on cumulants and in the structural and thermodynamic parameters of substances taken from EXAFS and makes it possible for some physical effects to occur Such effects can be seen in experimental results of EXAFS for the Br K edge in NaBr,4) where EXAFS is shifted and a Bragg peak appears under pressure EXAFS has been proven to be a valuable tool in various branches of solid state physics, but few attempts have so far been made to use it in high-pressure physics.5) Some efforts have been exerted to study pressure effects in EXAFS and its cumulants.2–10) A pressure-dependent EXAFS experiment on Cu has been performed5) to examine Cu as a calibrant for EXAFS Here, its parameters such as the second cumulant ðPÞ or mean-square-relative displacement (MSRD) and the third cumulant ð3Þ ðPÞ describing the EXAFS phase shift due to anharmonicity, etc., as functions of pressure have been extracted using the expressions of cumulant expansion and Rehr’s correlated Debye model, where the ratio method and parameter fitting are used for data analysis.5) Moreover, MSRD has also been measured,6,7) as well as calculated by Monte-Carlo simulation,6) and by using Loubeyre’s model8) for Kr Interatomic distance as a function of pressure for simple metals has been theoretically studied.9) The pressureinduced interatomic distance change of Cu was measured by XRD analysis and the results are presented in the American Institute of Physics Handbook.10) For further development toward the use of EXAFS as a valuable tool in highpressure physics it is necessary to develop procedures for the description and analysis of the pressure dependences of physical quantities obtained from EXAFS Several methods including anharmonic correlated Einstein model (ACEM)11) have been successfully applied to temperature-dependent EXAFS.11–18) Hence, further developing a model for study- ing high-pressure EXAFS and its parameters may be useful This work is a next step in the approach of HR (Ref 11) to derive a pressure-dependent ACEM for approximating the pressure-dependent interatomic distance, thermodynamic parameters, and cumulants in high-pressure EXAFS In Sect 2, the pressure-dependent anharmonic effective potential is developed and used for deriving an equation whose solution provides the pressure-induced interatomic distance change ÁrðPÞ by a simple means On the basis of quantum thermodynamic perturbation theory, the explicit analytical expressions for pressure-dependent first, second, and third EXAFS cumulants, and the thermal expansion coefficient at a given temperature have been derived The Morse potential is assumed to describe the interaction between each pair of atoms In high-pressure physics, the pressure-induced changes in physical quantities, i.e., the difference of a quantity at pressure P 6¼ from its value at P ¼ are usually measured,5,10) so that in this work we also calculate and discuss such pressure-induced changes in the considered quantities, as well as in the ratios of pressure-induced changes of EXAFS cumulants Numerical results (Sect 3) for Cu, being one of the intensively studied materials,5) are compared to the results of the XRD experiment analysis,10) and pressure-dependent EXAFS,5) which show good and reasonable agreement The physical properties except for ÁrðPÞ depend on ÁrðPÞ Since ÁrðPÞ can be obtained by the XRD method, two calculations are compared in this paper, i.e., using ÁrðPÞ calculated on the basis of the present theory, and using ÁrðPÞ from the experiment.10) Moreover, the pressure effects that appeared in all the considered quantities have been discussed in detail to show more information shown in this high-pressure EXAFS theory Pressure-Dependent Anharmonic Correlated Einstein Model We consider a pressure-dependent anharmonic correlated Einstein model characterized by an anharmonic effective potential corresponding to the case, in which under pressure P the interatomic distance is changed by rPị: 024802-1 â2014 The Physical Society of Japan J Phys Soc Jpn 83, 024802 (2014) N Van Hung keff rPịị2 ỵ k3 rPịị3 ỵ k4 rPịị4 ; rPị ẳ rPị r0ị; 1ị $ Veff Pị ẳ where ÁrðPÞ is the pressure-induced change in interatomic distance, keff is the effective spring constant, k3 and k4 are cubic and quartic anharmonic parameters, respectively, giving an asymmetry of the potential Equation (1) is our definition of anharmonic effective potential for the pressure dependence used in this work, which is different from that for the temperature dependence11) by using ÁrðPÞ instead of x being the deviation in the instantaneous bond length between the two atoms from equilibrium, as well as by including the pressure dependence of the effective spring constant shown below To take into account all nearest neighbor interactions for including three-dimensional interactions in the present derived pressure-dependent ACEM, the pressure-dependent anharmonic interatomic effective potential Eq (1) is defined on the basis of an assumption in the center-of-mass frame of a single-bond pair of an absorber with a mass M1 and a backscatterer with a mass M2 as XX ^ ^ V ÁrðPÞR12 Á Rij ; Veff Pị ẳ VPị ỵ Mi iẳ1;2 j6ẳi M1 M2 ; 2ị M1 ỵ M2 where the sum i is over absorbing and backscattering atoms, ^ ij is the unit vector the sum j is over their nearest neighbors, R along the bond between the ith and jth atoms, and ® is the reduced mass of M1 and M2 The first term on the right concerns only absorbing and backscattering atoms, and the second one describes the lattice contributions to pair interactions and depends on the crystal structure type observed in the potential parameters keff , k3 , and k4 obtained by comparing Eqs (2) to (1) for each considered crystal structure To derive the pressure-induced interatomic distance change ÁrðPÞ, we consider the anharmonic interatomic effective potential given by Eqs (1) and (2) under pressure P, which has the definition P ẳ @F=@VịT , where F is the free energy At finite temperatures and under pressure, the effective potential will be defined to correspond to the appropriate free energy Hence, from the above definition of P, the change of F under pressure P of the system corresponding to the volume change V is given by ẳ 3ị If the system has N elementary cells, each of which has volume vc , then the free-energy change under pressure P corresponding to one elementary cell is equal to frPịị ẳ F VPị ¼ ÀPvc : N Vð0Þ ð4Þ When a material under pressure P is compressed in one direction, it usually tends to expand in the other two directions perpendicular to the compression direction This phenomenon is called the Poisson effect described by the Poisson ratio £.19,20) The Poisson ratio is the ratio or the fraction (or percent) of expansion to the fraction (or percent) of compression and it is given by 5ị From this ratio, we obtain x y z ẳ ¼ : x y z ð6Þ Suppose that x, y, and z are three coordinates of a sample at P ¼ 0, then under pressure P > these coordinates change into x À Áx, y À Áx, and z À Áz, respectively Hence, using Eqs (5) and (6) (Áz=z ( 1), the following ratio is obtained: ÁVðPÞ ðx À xịy yịz ỵ zị xyz ẳ V0ị xyz z $ : 7ị ẳ 2 ị z Atomic displacement is shown by the change of an atomic bond, which can be along the axis, along which the effective potential is directed This treats the axial compression of the atomic bond caused by atomic displacement This axial compression can also be applied to the other atomic bonds along the other axes leading to the uniform compression of the system Suppose that z is the compression direction along the interatomic distance r so that Eq (7) is generalized into ÁVðPÞ $ rPị : ẳ 2 ị V0ị rPị ẳ FPị ẳ Veff Pị ẳ PVPị: x=x y=y ẳ : Áz=z Áz=z ð8Þ Substituting ÁVðPÞ=Vð0Þ from Eq (8) into Eq (4), we obtain frPịị ẳ 2 ịPvc rPị : rð0Þ ð9Þ Since the anharmonic effective potential given in Eq (1) corresponds to the free-energy change under pressure P, as described by Eq (4) or (9), from Eqs (1) and (9), we obtain the following equation: k4 rPịị3 ỵ k3 rPịị2 ỵ ỵ 2 ịPvc ẳ 0; r0ị keff ðÁrðPÞÞ ð10Þ whose solution provides the pressure-induced interatomic distance change rPị and the pressure-dependent interatomic distance rPị ẳ r0ị ỵ rPị Furthermore, on the basis of the denition of the Grỹneisen parameter G ẳ @ln !E ị=@ln Vị and the relation5) G Pị G 0ị ẳ ẳ const; VðPÞ Vð0Þ ð11Þ the pressure-dependent correlated Einstein frequency !E ðPÞ and temperature E ðPÞ, as well as the effective spring constant keff ðPÞ, have been derived and are given by VPị !E Pị ẳ !E 0ị exp G ð0Þ À ; ð12Þ Vð0Þ ð13Þ E ðPÞ ¼ h" !E ðPÞ=kB ; keff ðPÞ ¼ !2E ðPÞ: Making use of quantum thermodynamic perturbation theory,21) as well as of Eqs (1), (2), (12), and (13), the explicit analytical expressions for the three first EXAFS cumulants as functions of the pressure P at a given 024802-2 ©2014 The Physical Society of Japan J Phys Soc Jpn 83, 024802 (2014) N Van Hung Table I Formulas of ð1Þ , , ð3Þ , and P in the HP (P ! 1) limit Quantities ð1Þ ðP; TÞ ðP; TÞ ð3Þ ðP; TÞ P ðP; Tị P!1 ỵ 2zị01ị Pị ỵ 2zị02 Pị ỵ 12zị03ị Pị 4z 0P Pị02 Pịị2 =T temperature T have been derived They are given by, for the rst cumulant, 1ị P; Tị ẳ 01ị Pị ỵ zP; Tị 3k3 ẳ P; Tị; À zðP; TÞ keff ðPÞ 3k3 ðPÞ; keff Pị for the second cumulant or MSRD, ỵ zP; Tị ; P; Tị ẳ 02 Pị zP; Tị h" !E Pị 02 Pị ẳ ; 2keff Pị E Pị zP; Tị ẳ exp À ; T and for the third cumulant, 0ð1Þ Pị ẳ 14ị 15ị Fig Pressure dependence of pressure-induced interatomic distance change ÁrðPÞ of Cu calculated using the present theory compared with that obtained in the experiments (XRD-AIP, XRD-DEW).10) 3ị P; Tị ẳ 03ị Pịẵ3 P; TÞ=02 ðPÞÞ2 À 2; 2k3 ð ðPÞÞ2 : ð16Þ keff ðPÞ Using the obtained first cumulant or net thermal expansion Eq (14), the expression of the pressure-dependent thermal expansion coefficient has been derived and is given by 0ð3Þ Pị ẳ P; Tịị2 02 Pịị2 ; T2 3k3 0P Pị ẳ : rPịkB Moreover, the above considered physical properties except for ÁrðPÞ depend on ÁrðPÞ Since ÁrðPÞ can be obtained by the XRD method, two calculations are compared in this paper (Sect 3), i.e., using ÁrðPÞ calculated by Eq (10) and using ÁrðPÞ from the experiment.10) P P; Tị ẳ 0P Pị 17ị The descriptions of Eqs (14)–(17) in terms of the second cumulant are useful because all the cumulants and thermal expansion coefficients can be deduced when the second cumulant or MSRD is measured or calculated This can lead to significant reductions of the numerical calculations or measurements of the pressure-dependent EXAFS cumulants and thermal expansion coefficient by focusing only on the calculations or measurements of the second cumulant in highpressure EXAFS It is useful to consider the behaviors of the above derived pressure-dependent cumulants and thermal expansion coefficients in the low-pressure (LP) and high-pressure (HP) limits In the LP (P ! 0) limit, they approach those of P ¼ 0.11) In the HP (P ! 1) limit, z ! 0, so that the term z2 and higher powers can be neglected This leads to the approximation of Eqs (14)–(17) The obtained results in the HP limit are presented in Table I Hence, in the HP limit the pressure dependence is defined mainly on the basis of the pressure dependences of 0ð1Þ ðPÞ, 02 ðPÞ, and 0ð3Þ ðPÞ for the EXAFS cumulants and of zðPÞ, 02 ðPÞ, and 0P ðPÞ for the thermal expansion coefficient Note that, in the LP limit (P ! 0) the expressions of the EXAFS cumulants and thermal expansion coefficient, as well as of the effective spring constant, correlated Einstein frequency, and temperature of the present pressure-dependent ACEM, approach those of the temperature-dependent ACEM.11) Numerical Results and Discussion Now, we apply expressions derived in the previous section to numerical calculations for Cu, where Morse potential22) is used to describe the interaction between each pair of atoms Its parameters22) D ẳ 0:3429 eV, and ẳ 1:3588 ạ1 were obtained using the experimental values of the energy of sublimation, compressibility, and lattice constant They have been used for the calculation of the pressure-induced changes in the interatomic distance, anharmonic interatomic effective potential, thermodynamic parameters, EXAFS cumulants, and thermal expansion coefficient of Cu at the given temperature T ¼ 300 K, a temperature at which the measurements are often performed,3,10) as well as using Poisson’s ratio ¼ 0:355,19) zero-pressure Grüneisen parameter22) G 0ị ẳ 2:108, and r0ị ẳ 2:554 .5) The pressure-induced interatomic distance change ÁrðPÞ of Cu calculated by the present theory using Eq (10) (Fig 1) shows good agreement with those of the XRD (AIP, DEW) experiments.10) The obtained ÁrðPÞ values are used for calculating the pressure dependence of the pressure-induced changes in all the quantities considered in this work using the present theory The results are illustrated firstly for the effective spring constant Ákeff ðPÞ (Fig 2) calculated using the second equation of Eq (13), for the anharmonic interatomic effective potential ÁVeff ðPÞ (Fig 3) calculated using Eq (1) including keff ðPÞ, for the correlated Einstein temperature ÁE ðPÞ (Fig 4) calculated using the first equation of Eq (13), and for the first cumulant Á ð1Þ ðPÞ (Fig 5) calculated using Eq (14) Figure shows the pressure-induced change of the second cumulant or MSRD Á Pị calculated using Eq (15) and 024802-3 â2014 The Physical Society of Japan J Phys Soc Jpn 83, 024802 (2014) N Van Hung Fig Pressure dependence of pressure-induced effective spring constant change Ákeff ðPÞ of Cu calculated using the present theory compared with that calculated using ÁrðPÞ obtained from the experiments (XRD-AIP, XRDDEW)10) at different pressures Fig Pressure dependence of pressure-induced first cumulant change Á ð1Þ ðPÞ of Cu calculated using the present theory compared with that calculated using ÁrðPÞ obtained from the experiments (XRD-AIP, XRDDEW)10) at different pressures Fig Pressure dependence of pressure-induced anharmonic interatomic effective potential change ÁVeff ðPÞ of Cu calculated using the present theory compared with that calculated using ÁrðPÞ obtained from the experiments (XRD-AIP, XRD-DEW)10) at different pressures Fig Pressure dependence of pressure-induced second cumulant change Á ðPÞ of Cu calculated using the present theory compared with the experimental results extracted from EXAFS using Rehr’s correlated Debye model,5) and the method (A1),5) as well as with that calculated using ÁrðPÞ obtained from the experiments (XRD-AIP, XRD-DEW)10) at different pressures Fig Pressure dependence of pressure-induced correlated Einstein temperature change ÁE ðPÞ of Cu calculated using the present theory compared with that calculated using ÁrðPÞ obtained from the experiments (XRD-AIP, XRD-DEW)10) at different pressures experimental data taken from the results of the method (A1) and Rehr’s correlated Debye model in Ref Similarly, Fig shows the pressure-induced change of the third cumulant Á ð3Þ ðPÞ calculated using Eq (16) and experimental data taken from the results of the methods (P3) and (P4) in Ref The method (A1) is one of the four methods used in Ref to obtain Á ðPÞ from the amplitude ratio of the high- Fig Pressure dependence of pressure-induced third cumulant change Á ð3Þ ðPÞ of Cu calculated using the present theory compared with the experimental results extracted from EXAFS using the methods (P3)5) and (P4),5) as well as with that calculated using ÁrðPÞ obtained from the experiments (XRD-AIP, XRD-DEW)10) at different pressures pressure EXAFS datasets The methods (P3) and (P4) are two methods of fitting the phase difference of the high-pressure EXAFS datasets The parameter sets are (ÁR, Á ð3Þ ) for (P3) and (ÁE, ÁR, Á ð3Þ ) for (P4), where R ẳ 1ị , and E is 024802-4 â2014 The Physical Society of Japan J Phys Soc Jpn 83, 024802 (2014) N Van Hung Fig Pressure dependence of pressure-induced thermal expansion coefficient change Á P ðPÞ of Cu calculated using the present theory compared with that calculated using ÁrðPÞ from the experiments (XRD-AIP, XRD-DEW)10) at different pressures the photoelectron zero-energy difference between different datasets Both figures show reasonable agreement between the results of the present theory and the experimental data, where the lines of the calculated results are compared with the average curve of the fluctuating experimental values On the basis of Eq (14), the pressure-dependent first and second cumulants (Figs and 6) are proportional to each other by the factor 3k3 =keff ðPÞ However, the effective spring constant keff ðPÞ is pressure-dependent (Fig 2) thus, illustrating both these cumulants can be useful for consideration of the effect of the pressure on this proportionality Furthermore, Fig shows the pressure dependence of the pressureinduced change in the thermal expansion coefficient Á P ðPÞ of Cu calculated using the present theory by Eq (17) It approaches a constant value at very high pressures The increases in ÁVeff ðPÞ, Ákeff ðPÞ, Á!E ðPÞ, and ÁE ðPÞ, as well as the decreases in Á ð1Þ ðPÞ, Á ðPÞ, Á ð3Þ ðPÞ, and Á P ðPÞ under pressure denote significant pressure effects of the anharmonic effective potential, thermodynamic parameters, and EXAFS cumulants The decrease in the pressureinduced change of the second cumulant under pressure calculated using the present theory has also been reflected in the experimental results for Kr.6,7) The decrease in the pressure-induced interatomic distance change (Fig 1) under pressure denotes that the material becomes denser as the atoms get closer together under pressure This leads to the enhancement in the degree of interatomic interaction shown by the increase in the pressure-induced change of effective spring constant Ákeff ðPÞ (Fig 2) and anharmonic effective potential ÁVeff ðPÞ (Fig 3) under pressure From the first equation in Eq (13), the pressure-dependent correlated Einstein frequency is proportional to the correlated Einstein temperature by the factor kB =h" , i.e., !E ¼ kB E =h" , so that they can be determined from each other On the basis of this proportionality one can also get from the increase in the correlated Einstein temperature (Fig 4) making the material harder, the increase in the correlated Einstein frequency making the atomic vibrations faster under pressure Consequently, the pressure dependence in the pressure-induced changes in the calculated quantities shown in the above figures affect EXAFS, causing it to shift, as observed in the experimental results for the Br K edge in NaBr.4) Hence, the Fig Pressure dependences of cumulant ratios Á ð1Þ Á =Á ð3Þ and Á P ÁrTÁ =Á ð3Þ of Cu calculated using the present theory present theory can provide more information on the pressure effects, that occur in high-pressure EXAFS It is seen from Figs 2–8 that the results of the pressureinduced changes in the considered EXAFS parameters calculated using ÁrðPÞ from the experiments (XRD-AIP, XRD-DEW)10) at different pressures are found to be in good agreement with those calculated using ÁrðPÞ obtained by Eq (10) for all the considered quantities, as well as with the experimental second cumulant obtained using Rehr’s correlated Debye model; they are also in reasonable agreement with the high-pressure EXAFS experimental data5) for the second cumulant Á ðPÞ obtained by (A1) and for the third cumulant Á ð3Þ ðPÞ obtained by (P3) and (P4) In the temperature-dependent EXAFS (P ẳ 0) the cumulant ratios 1ị = ð3Þ and T rT = ð3Þ are often considered as standards for cumulant calculations,11,15,23) because ð1Þ ðTÞ, ð3Þ ðTÞ, and T ðTÞ contain the anharmonic potential parameter k3 , where these ratios approach a constant value, i.e., 1/2, at high temperatures for both classical23) and quantum11,15) theories In high-pressure EXAFS, the important quantities are the pressure-induced changes of pressure-dependent rðPÞ, ð1Þ ðPÞ, ðPÞ, ð3Þ ðPÞ, and P ðPÞ at a given temperature Since, in the present theory, they also contain the anharmonic potential parameter k3 , it is crucial to consider the ratios of the pressure-induced changes in these quantities to look for some characteristics indicating the relationship between anharmonicity and pressure dependence in high-pressure EXAFS Figure shows the pressure dependences of the cumulant ratios Á ð1Þ Á =Á ð3Þ and Á P ÁrTÁ =Á ð3Þ calculated using the present theory The ratios increase markedly at low pressures and then slowly as pressure increases; they approach constant values at very high pressures We are hoping that these properties of the ratios of the pressure-induced changes in the cumulants, thermal expansion coefficient, and interatomic distance will also be considered as standards for pressure-dependent cumulant calculations similarly to those of the cumulant ratios in temperature-dependent EXAFS Conclusions The present derived pressure-dependent ACEM has led to a method of calculating and analyzing the pressure-induced changes in the interatomic distance, anharmonic effective potential, thermodynamic parameters, and three first cumulants in high-pressure EXAFS 024802-5 ©2014 The Physical Society of Japan J Phys Soc Jpn 83, 024802 (2014) N Van Hung The behaviors of the EXAFS cumulants and thermal expansion coefficient in the LP and HP limits have been considered It is shown that the expressions of these quantities, as well as of the effective spring constant, correlated Einstein frequency, and temperature approach those of the temperature-dependent ACEM, when the pressure approaches zero (P ! 0) Significant pressure effects have been shown by this pressure-dependent ACEM, namely, under pressure the material becomes denser as the interatomic distance gets shorter, as the atoms get closer together and vibrate faster, and as the material becomes harder This leads to the enhancement in the degree of interatomic interaction shown by the increases in anharmonic effective potential, and effective spring constant, as well as to the changes in the thermodynamic properties of the material characterized by the thermal expansion coefficient and Debye–Waller factors presented in terms of cumulants All these pressure effects affect high-pressure EXAFS profiles This model avoids the use of extensive full lattice dynamical calculations, yet provides good and reasonable agreement of the calculated results for Cu with the experimental results of XRD analysis and high-pressure EXAFS All the above results illustrate the advantage, simplicity, and efficiency of the present theory in highpressure EXAFS data analysis Acknowledgements The author thanks J J Rehr and P Fornasini for helpful comments This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.01-2012.03 The author is grateful to the Physical Society of Japan for financial support in publication + hungnv@vnu.edu.vn 1) E D Crozier, J J Rehr, and R Ingalls, in X-ray Absorption, ed D C Koningsberger and R Prins (Wiley, New York, 1988) Chap 2) R Ingalls, G A Garcia, and E A Stern, Phys Rev Lett 40, 334 (1978) 3) A Yoshiasa, T Nagai, O Ohtaka, O Kamishima, and O 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ðPÞ (Fig 3) under pressure From the first equation in Eq (13), the pressure-dependent correlated Einstein frequency is proportional to the correlated Einstein. .. XRD-DEW)10) at different pressures Fig Pressure dependence of pressure-induced correlated Einstein temperature change ÁE ðPÞ of Cu calculated using the present theory compared with that calculated