Advances in Materials Science and Engineering

Volume 2018, Article ID 3263170, 9 pages

https://doi.org/10.1155/2018/3263170

## Thermodynamic Properties and Anharmonic Effects in XAFS Based on Anharmonic Correlated Debye Model Debye–Waller Factors

^{1}Department of Physics, Tan Trao University, Km 6, Trung Mon, Yen Son, Tuyen Quang, Vietnam^{2}Institute of Research and Development, Duy Tan University, 03 Quang Trung, Da Nang, Vietnam^{3}School of Engineering Physics, Hanoi University of Science and Technology, 1 Dai Co Viet, Hanoi, Vietnam^{4}Quang Ninh Education & Training Department, Cam Pha School, Nguyen Van Cu, Ha Long, Quang Ninh, Vietnam^{5}Department of Basic Sciences, University of Fire Fighting & Prevention, 243 Khuat Duy Tien, Thanh Xuan, Hanoi, Vietnam

Correspondence should be addressed to Nguyen Van Hung; nv.ude.unv@vngnuh

Received 28 June 2017; Revised 14 January 2018; Accepted 7 May 2018; Published 29 July 2018

Academic Editor: Francesco Ruffino

Copyright © 2018 Nguyen Ba Duc et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Thermodynamic properties and anharmonic effects in X-ray absorption fine structure (XAFS) have been studied based on the anharmonic correlated Debye model Debye–Waller factors presented in terms of cumulant expansion. The derived analytical expressions of three first XAFS cumulants involve more information on phonon-phonon interactions taken from integration over the first Brillouin zone. Many-body effects are taken into account in the present one-dimensional model based on the first shell near neighbor contributions to the vibrations between absorber and backscatterer atoms. Morse potential is assumed to describe single-pair atomic interaction included in the derived anharmonic interatomic effective potential. The present theory can be applied to any crystal structure including complex systems. Numerical results for Cu and Ni are found to be in good agreement with experiment and with those of the other theories.

#### 1. Introduction

X-ray absorption fine structure (XAFS) has developed into a powerful probe of atomic structure, thermodynamic properties, and anharmonic effects in atomic vibration of substances [1–28]. Thermal atomic vibrations and disorder give rise to Debye–Waller factors (DWFs) in XAFS describing these properties of the considered material. The formalism for including anharmonic effects in XAFS is often based on cumulant expansion [1] where the even cumulants contribute to the amplitude and the odd ones to the phase of XAFS spectra. Hence, the accurate DWFs presented in terms of cumulant expansion are crucial to quantitative treatment of anharmonic XAFS. Consequently, the lack of the precise DWFs or cumulants has been one of the biggest limitations to accurate structural determinations, thermodynamic properties, and anharmonic effects in atomic vibration of materials taken from XAFS experiments. Therefore, a reliable and effective method for treatment of thermal and structural disorders based on DWFs still represents an open problem, whose solution is expected to increase the amount and accuracy of information obtainable from XAFS.

Many efforts have been made to derive procedures for the calculation and analysis of cumulants describing the thermodynamic properties and anharmonic effects or phonon-phonon interactions in temperature-dependent XAFS [2–28] using the classical approach [3–6] and quantum theory [7–28]. Classical theories have the advantages shown by simplicity and possibility of application to high temperatures where anharmonicity is dominant, and quantum methods can be applied to both high and low temperatures where quantum effects are evident. The anharmonic correlated Einstein model (ACEM) [9] has been derived for the calculation and analysis of DWFs presented in terms of cumulant expansion up to the third order. Its simplicity and efficiency in XAFS studies is demonstrated by calculating and analyzing XAFS cumulants of fcc [9–13], hcp [14], and crystals and in studying pressure-dependent XAFS [15]. The statistical moment method [16, 17] has been applied to calculate the mean square relative displacement (MSRD) including anharmonic contributions of some crystals. The derived path-integral effective potential (PIEP) method [18] has the advantage for calculating XAFS DWFs presented in terms of cumulant expansion up to the fourth order based on quantum theory including three dimension, correlation, anharmonicity, and many-body effects. The other efforts in XAFS cumulant studies have been shown, for example, by the path-integral Monte Carlo (PIMC) calculations [19], the force constant method (FCM) [20], and the local density approximation (LDA) [21] based on the density functional theory calculations of dynamic matrix.

The purpose of this work is XAFS study of the thermodynamic properties and anharmonic effects of materials based on the anharmonic correlated Debye model (ACDM) DWFs presented in terms of cumulant expansion up to the third order. In Section 2, the analytical expressions for the dispersion relation, correlated Debye frequency and temperature, and three first XAFS cumulants have been derived. They involve more information of phonon-phonon interactions taken from integration over the first Brillouin zone (BZ). The many-body effects are taken into account in the present one-dimensional model based on the first shell near neighbor contributions to the vibrations between absorber and backscatterer atoms. Morse potential is assumed to describe the single-pair atomic interaction included in the derived anharmonic interatomic effective potential. The thermodynamic properties and anharmonic effects in atomic vibration have been analyzed based on the obtained temperature-dependent XAFS cumulants where the anharmonicity causing thermal expansion and the MSRD component perpendicular to bond direction have been discussed in detail. This theory can be applied to any crystal structure including complex systems. Numerical results for Cu and Ni presented in Section 3 are compared to a large number of experimental data [13, 19–26] and to those calculated using several well-known methods such as ACEM [9], SMM [16], PIEP [18], PIMC [19], FCM [20], and LDA [21] at lowand high temperatures to show the advantage of the present theory. The conclusions on the obtained results are presented in Section 4 of the paper.

#### 2. Formalism

##### 2.1. Anharmonic Effective Potential

To determine XAFS cumulants, it is necessary to specify the interatomic interaction potential and force constant [3–27]. Let us consider an anharmonic interatomic effective potential expanded to the third order as follows:where is the effective local force constant, is the cubic effective parameter giving an asymmetry of the anharmonic effective potential, and is deviation of the instantaneous bond length between two immediate neighboring atoms from its equilibrium value .

The anharmonic effective potential (2) is defined based on an assumption in the center-of-mass frame of single bond pair of absorber and backscatterer atoms:where the first term on the right concerns only absorber and backscatterer with the masses and , respectively, and the second one includes the contributions of their immediate near neighbors to the oscillation between absorber and backscatterer atoms. By projecting such interactions along the bond direction as in (2), the purely one-dimensional model is recovered. Hence, we have extended this effective pair-interaction model to a one-dimensional chain to partly account for dispersion effects of the crystals. It is the difference of the present anharmonic effective potential from the single-bond (SB) [7] and single-pair (SP) [8] potentials, which concern only each pair of immediate neighboring atoms given by without the second one on the right of (2).

Note that the lattice contributions to the oscillation between absorber and backscatterer atoms illustrated by the second term of (2) can be obtained using the *first shell near neighbors contributions approach* (FSNNCA), which has been successfully applied to bcc crystals [28]. Hence, based on the first shell near neighbor contributions to the vibration between absorber and backscatterer, the many-body effects have been taken into account in the present one-dimensional model.

A Morse potential is assumed to describe the atomic pair-interaction contained in the effective potential (2) and expanded to the third order around its minimum:where describes the width of the potential and is dissociation energy.

Applying Morse potential (3) to (2) and comparing the results to (1), the values of of the anharmonic effective potential presented in terms of Morse potential parameters are determined.

##### 2.2. XAFS Cumulants Based on ACDM

In order to include anharmonic effects in the present ACDM, the Hamiltonian of system is written in the summation of the harmonic and cubic anharmonic components, and , respectively:where containing the local force constant, is used for derivation of second cumulant, and the term containing the cubic anharmonic parameter is used for derivation of the first and third cumulants of the materials.

Derivation of the present ACDM is performed using the *many-body perturbation approach* (MBPA) [29] based on the dualism of an elementary particle in quantum theory, that is, its corpuscular and wave property. Then, we can describe the system in the present ACDM involving all different frequencies up to the Debye frequency as a system consisting of many bodies, that is, many phonons, each of which corresponds to a wave having frequency and wave number varied in the first BZ. Moreover, based on the FSNNCA only backscattering from the first shell of absorber and backscatterer atoms is taken into consideration. This reduces and simplifies the derivations of the analytical expressions of the considered XAFS cumulants.

For this purpose, the displacement in the parameter in terms of the displacement of atom of the one-dimensional chain described byis related to the phonon displacement operators [30] in the form

Then, (5) has resulted aswhere is the atomic number, is reduced mass, and is the lattice constant.

The frequency contained in (7) and then in all cumulant expressions derived for the oscillation between absorber and backscatterer atoms in XAFS process under the interactions of these atoms with their first shell near neighbors describes the dispersion relation. Using the obtained local force constant, it has resulted as

At the bounds of the first BZ of the linear chain, , the frequency is maximum so that from (8), we obtain the correlated Debye frequency and temperature in the formwhere is Boltzmann constant.

Based on the above results, the cubic component of Hamiltonian is expressed asor in the following form using (6) for the displacement of atom

Comparing (11) to (10) and indicatingwith as the atomic number, we obtain

Using (6) and (12), (13) changes into

In the MBPA [28], the value is calculated using the expressionwhich takes backscattering only from the first shell.

Substituting into (15) the relations [29]we obtain

Using the Wick theorem for T-product in the integral, the harmonic phonon Green function [29]the symmetric properties of [30], properties of function , the phonon densityas well as from (8), from (7), from (12) and the phonon momentum conservation in the first BZ, we change (18) into the following:

Using this expression, the first cumulant describing the net thermal expansion or lattice disorder in XAFS theory has resulted aswhere is the second cumulant describing the mean square relative displacement (MSRD) and has the following form:where using (8) for , (7) for and , (12) for and , (19) for , and (20) for we calculate

The third cumulant is the mean cubic relative displacement (MCRD) describing the asymmetry of the pair distribution function in XAFS theory and has resulted aswhere the calculation of is analogous to the one of above, that is,where using from (17) with limiting only the cubic anharmonic term, the Wick theorem for T-product, and the symmetric properties of [30], we calculated . The product has been calculated using from (24) and from (21).

In the above expressions for the cumulants in the present ACDM, are zero-point energy contributions to the first, second, and third cumulant, respectively, and these cumulant expressions have been obtained for the case of large phonon numbers, when the summation over is replaced by the corresponding integral in the first BZ. Moreover, we have used the phonon momentum conservation in the first BZ [30] to describe the value of by and for the first and third cumulant. This leads to reducing the integrations for these cumulants given by (22) and (25), respectively.

Note that the present theory is valid for any crystal structure including complex systems; therefore, the other developments for the monatomic [27] and bcc [28] crystals are only its special cases.

##### 2.3. High- and Low-Temperature Limits of XAFS Cumulants

It is useful to consider the high-temperature (HT) limit, where the classical approach [5, 6] is applicable, and the low-temperature (LT) limit, where the quantum theory must be used [9]. In the HT limit, we use the approximationto simplify the expressions for the cumulants. In the LT limit, , so that all temperature-dependent terms approach zero and the cumulants approach constant values, for example, their zero-point contributions. These results are written in Table 1.where