Investigation of temperature and pressure effects on thermodynamic parameters of intermetallic alloy in EXAFS

Abstract This work advances the anharmonic correlated Einstein model to investigate how pressure and temperature affect the Debye–Waller factor and thermodynamic parameters of an intermetallic alloy. By using the anharmonic correlated Einstein model in extended X-ray absorption fine structure (EXAFS) theory, analytical expressions are derived for the effective spring constant, EXAFS cumulants, the anharmonic factor, and the thermal expansion coefficient at high temperature and pressure. The results show that both the anharmonicity of the thermal atomic vibration and the effects of pressure are essential for the thermodynamic parameters and the EXAFS Debye–Waller factor, as well as showing that the anharmonic contributions are important in the EXAFS spectrum. Numerical calculations of these thermodynamic quantities are performed for pure Cu and Ag and for CuAg72 alloy, and the results are consistent with those obtained experimentally and from other theories.


PUBLIC INTEREST STATEMENT
The extended X-ray absorption fine structure (EXAFS) spectroscopy is one of the powerful techniques for investigating structures of crystalline. The formalism for including anharmonic effects in EXAFS is often written through cumulant expansion approach. There are many methods that have been developed to study the temperature dependence of EXAFS cumulants. However, no theoretical calculations have been done to predict the dependence of cumulants and thermodynamic parameters on temperature with the effects of pressure in EXAFS spectra. It requires the more accurate interatomic interaction form for metallic systems such as the manybody embedded-atom potentials. The purpose of this work is to investigate the dependence of cumulants and thermodynamic parameters on temperature with effects of pressure in EXAFS spectra of crystals. Anharmonic correlated Einstein model has been applied for compound of copper and silver with 72% ratio through the interatomic potential which has been derived by the Morse effective potential.

Introduction
Extended X-ray absorption fine structure (EXAFS) spectroscopy has developed into a powerful probe of atomic structure and the high-temperature thermodynamics of substances due to anharmonicity (Iwasawa et al., 2017;Rehr, 2000). Numerous methods have been developed to investigate how temperature affects the EXAFS cumulants, such as path-integral effective-potential theory (Yokoyama, 1999), the statistical moment method (V. v. Hung et al., 2010), the ratio method (Bunker, 1983), the Debye model (Beni & Platzman, 1976), the Einstein model (Frenkel & Rehr, 1993), and the anharmonic correlated Einstein model (ACEM) (N. v. Hung & Rehr, 1997). Several groups have applied ACEM theory to EXAFS to study how the thermodynamic properties depend on temperature with the effect of the material doping ratio (DR) (Duc et al., 2017;Hung et al., 2015;Kraut & Stern, 2000;Nafi et al., 2013). However, no reports to date have discussed how the thermodynamic parameters and the Debye-Waller factor (DWF) depend on temperature and pressure for Cu, Ag, and their intermetallic alloy CuAg72. A CuAg alloy contains the elements Cu and Ag, with the Ag atoms referred to as the substitution atoms and the Cu atoms referred to as the host atoms. CuAg72 has a ratio of 72% Ag and 28% Cu (±1%) and is also known as CuSil or UNS P0772 (note: CuSil should not be confused with Cusil-ABA, which has the composition 63.0% Ag, 35.25% Cu, and 1.75% Ti). It is an eutectic alloy and is used primarily for vacuum brazing (Nafi et al., 2013).
Herein, we use EXAFS theory (N. v. Hung & Rehr, 1997) to investigate how the DWF depends on temperature at high pressure. We also investigate thermodynamic parameters such as (i) the effective spring constant, (ii) the thermal expansion coefficient, (iii) the anharmonic factor, and (iv) the Einstein frequency and temperature and how they depend on temperature at ambient pressure for CuAg72.

Formalism
EXAFS is usually derived by using the cumulant-expansion approach, which contains the second cumulant σ 2 corresponding to the parallel mean-square relative displacement (MSRD) (Bunker, 1983). The second cumulant-often called the DWF-is an important factor in EXAFS analysis because the thermal lattice vibrations and high pressure strongly influence the EXAFS amplitudes through the function e À2σ 2 k 2 (Duc et al., 2018;Rehr, 2000). For simplicity, the temperature and pressure dependences of the DWF are denoted as σ 2 ðTÞ and σ 2 ðpÞ, respectively. One way to investigate how temperature and pressure affect the EXAFS cumulant is to combine ACEM with EXAFS (Hung, 2014;N. v. Hung & Rehr, 1997), which gives results that are consistent with experiments. ACEM uses the anharmonic effective interaction potential under pressure p, and the interatomic distance x is supplemented by δrðpÞ to give V E x; p ð Þ% 1 2 k eff ðxδrðpÞÞ 2 þ k 3eff ðxδrðpÞÞ 3 þ k 4eff ðxδrðpÞÞ 4 þ ::: where k eff is the effective spring constant, k 3eff and k 4eff are the effective cubic and quartic anharmonic parameters, respectively, which cause the asymmetry in the pair-distribution function, δrðpÞ ¼ rðpÞ À rð0Þ is the pressure-induced change in the interatomic distance, x = r−r 0 is the instantaneous bond length between atoms from the equilibrium location, r is the spontaneous bond length between absorbing and backscattering atoms, and r 0 is the equilibrium value of r. ACEM is determined by the vibration of single pairs of atoms, with M 1 and M 2 being the masses of the absorber and backscattering atoms, respectively. The oscillations of the absorber and backscattering atoms depend on their neighbors, so the interaction potential in Equation (1) is written in the form of an anharmonic effective interaction potential under ambient pressure, namely In Equation (2), V(x, p) is the interaction potential between absorbing and backscattering atoms, the sum i is over absorber (i = 1) and backscattering (i = 2) atoms, and the sum j is over all nearest neighbors whose contributions are described by the term V(x, p) excluding the absorber and backscattering atoms themselves. Furthermore, M i is the atomic mass of atom i, μ is the reduced atomic mass, namely μ ¼ M 1 M 2 =ðM 1 þ M 2 Þ, andR is the unit vector for the bond between atoms i and j. Therefore, this effective pair potential describes not only the interaction between absorber and backscattering atoms but also how the nearest-neighbor atoms affect such interactions, which is the difference between the effective potential used in this work and the single-pair potential (Tranquada & Ingalls, 1997) or single-bond potential (Frenkel & Rehr, 1993), which consider only each pair of immediate-neighbor atoms [i.e., only V(x)] without considering the remaining terms on the right-hand side of Equation (2). The atomic vibration is calculated using a quantum statistical approach with an approximate anharmonic vibration in which the system Hamiltonian includes a harmonic term H 0 with respect to the equilibrium at a given temperature plus an anharmonic perturbation, namely Here, the interaction potential V E ðaÞ and anharmonic perturbation δV E ðaÞ are which is an anharmonic potential of Morse pairs and is appropriate for approximating the structure of cubic crystals. The Morse anharmonic potential is where DðeVÞ ¼ ÀV r 0 ð Þ is the dissociation energy and α 12 (Å −1 ) is the width of the potential. We expand Equation (5) in x to obtain the third-order term that describes approximately the cubic structure of doped crystals. When considering only crystals with orderly doping, we also assume that the lattice is not corrupted, and we designate Cu as the host atom with indicator 1 and Ag as the substituted atom with indicator 2. ACEM uses the Morse anharmonic pair potential to describe the pair interaction between atoms, namely V E p ð Þ ¼ D 12 ðe À2α 12 xδrðpÞ À 2e Àα 12 xδrðpÞ Þ % D 12 À1 þ α 2 12 ðxδrðpÞÞ 2 À α 3 12 ðxδrðpÞÞ 3 þ ::: For simplicity, we approximate the parameters of the Morse potential in Equation (6) at a given temperature by where c 1 ; c 2 are the doping ratios (%) of the intermetallic alloys. We calculate the sums in the second term of Equation (2) and compare the results with the terms of Equations (1) and (6) to obtain the effective force constant k eff ðx; pÞ ¼ μω 2 E ðx; pÞ ¼ 5D 12 α 2 12 of the effective anharmonic potential. At ambient pressure, the effective force constant is k 0 eff ¼ 23D 12 α 2 12 =4.
To derive analytical expressions for the cumulants, we use the perturbation theory (Duc et al., 2018;Hung, 2014). Atomic vibrations are quantized as phonons, and the phonon-phonon interaction leads to anharmonicity, with the phonon vibration frequency taking the form where a 0 is the lattice constant at temperature T, and q is the phonon wave number. The correlated Einstein frequency and temperature at ambient pressure are, respectively Using quantum thermodynamic perturbation theory (Hung et al., 2015) and Equations (1), (2), and (8), we obtain the first three EXAFS cumulants as functions of the ambient pressure and temperature. For the first cumulant or net thermal expansion, we have : We also obtain the thermal expansion coefficient α T and the anharmonic factor βðT; pÞ as functions of ambient pressure and temperature as The second cumulant σ 2 contributes to the anharmonic EXAFS amplitude, while σ ð1Þ and σ ð3Þ contribute to the phase shift of the EXAFS due to anharmonicity. Note that σ ð1Þ , σ ð3Þ , and αðp; TÞ contain the anharmonicity parameter k 3eff and exist only when this parameter is included, which is why σ ð1Þ , σ ð3Þ , and αðp; TÞ must be considered when calculating the anharmonic effects in EXAFS. Under ambient pressure, the factor β is proportional to the temperature and inversely proportional to the shell radius, which is consistent with the anharmonicity obtained in experimental research into catalysis (Clausen et al., 1993), and R is considered as the particle radius. In Equations (9)-(13), zðp; TÞ ¼ exp Àθ 0 E =T À Á is the heat and pressure function, which describes how the cumulants, the thermal expansion coefficient, and the anharmonic factor depend on the absolute temperature T and pressure applied to the intermetallic alloy.

Results and discussion
For Cu-Cu and Ag-Ag pure metals and the alloy CuAg72, Table 1 gives the calculated and experimental (Benassi, 2018) parameters of the Morse potential, D 12 and α 12 , respectively. Substituting the parameters D 12 and α 12 from Table 1 (2001) and Okube et al. (2003). (1) and (9)-(13) gives the effective anharmonic potential V E ðx; pÞ as a function of the departure x from equilibrium bond length and ambient pressure (see Figure 1). The cumulants σ ðnÞ ðnÞ depend on the absolute temperature T and are influenced by pressure up to 14 GPa (see Figures 2-5). Figure 6 shows the thermal expansion coefficient αðT; pÞ as a function of absolute temperature T and pressure, and Figure 7 shows the anharmonic factor βðT; pÞ.  (2001) and Okube et al. (2003), indicating that the coefficients k eff , k 3eff , and k 0 eff calculated by using the ACEM (given in Table 2) are in reasonable agreement with measurements and the calculations of Okube et al. for CuAg72.  Figure 2 shows our calculation of the second cumulant or DWF as a function of the DR at 300 K and an ambient pressure of 14 GPa for the crystalline alloy CuAg72. These results illustrate that for DRs of 0-50% and 50-100%, the DWF varies linearly with the DR (with different slopes in each range). For DR = 100% (i.e., where the Ag content is 0% and the Cu content is 100%), the calculated values are in good agreement with experimental values determined at 300 K (see symbols *, □) (Hung et al., 2002;N. v. Hung & Duc, 1999, 2000. However, there are breakpoints in the lines at 50% DR, which means that we do not have ordered atoms at DR = 50%. Thus, the Cu-Ag alloy does not form an ordered phase at the molar composition of 1:1 (i.e., the alloy CuAg50 does not exist), and this result is consistent with the findings of Kraut and Stern (2000). As the ambient pressure increases, the DWF decreases: with 0% Ag, 100% Cu, and 101 kPa (i.e., normal atmospheric pressure), we have DWF = 0.2330 Å 2 (for 14 GPa, DWF = 0.2241 Å 2 ); with 100% Ag, 0% Cu, and 101 kPa, we have DWF = 0.1796 Å 2 (for 14 GPa, DWF = 0.1718 Å 2 ). At the breakpoints, we have DWF = 0.2005 Å 2 and 0.1928 Å 2 at 101 kPa and 14 GPa, respectively. Thus, increasing the ambient pressure decreases the EXAFS amplitude by reducing the atomic MSRD that characterizes the EXAFS DWF. Figure 3 shows the first cumulant σ ð1Þ as a function of temperature at the pressure of 14 GPa for Cu, Ag, and CuAg72. At approximately the zero point with 101 kPa and 14 GPa ambient pressure, we have σ ð1Þ ¼ 0:0027 Å and σ ð1Þ ¼ 0:0047 Å, respectively; at 700 K, we have σ ð1Þ ¼ 0:0184 Å and σ ð1Þ ¼ 0:0201 Å, respectively. Thus, as the pressure increases, the first cumulant also increases, but at low temperature, it deviates more, meaning that the pressure causing the net thermal expansion is more pronounced at low temperatures. Figure 4 shows the second cumulant or DWF as a function of absolute temperature with the effects of ambient pressure for Cu-Cu, Ag-Ag, and CuAg72 and compares these results with experimental results (Hung et al., 2002;N. v. Hung & Duc, 1999). The calculated values for the first cumulant (Figure 3) and the DWF (Figure 4) for different DRs at the pressure of 14 GPa are proportional to temperature from around 100 K and above. Consider the change in the second cumulant (DWF) for different temperatures. At approximately 0 K, the DWF increases from σ ð2Þ = 0.0026 Å 2 to σ ð2Þ = 0.0046 Å 2 as the pressure increases from normal atmospheric pressure up to 14 GPa. At 700 K, the DWF increases from σ ð2Þ = 0.018 Å 2 to σ ð2Þ = 0.0197 Å 2 as the pressure increases from normal atmospheric pressure up to 14 GPa. At low temperatures, the DWF changes more than it does at high temperatures because the change in ambient pressure from 101 kPa to 14 GPa causes a greater MSRD of the atoms (or second cumulant σ ð2Þ ). Furthermore, Figure 4 shows that from room temperature upward (approximately 300 K), the DWF remains almost constant as the ambient pressure increases, so the ambient pressure has a stronger effect at low temperatures.    (Hung et al., 2002;N. v. Hung & Duc, 1999, 2000 at normal atmospheric pressure. At 0 K for both 101 kPa and 14 GPa, the third cumulant σ ð3Þ ≈0, but as the temperature increases, σ ð3Þ for CuAg72 decreases with pressure: at 700 K, we have σ ð3Þ = 0.0026 Å 3 at 101 kPa and 0.0023 Å 3 at 14 GPa. Thus, high ambient pressure reduces the asymmetry of the atomic interaction potential at higher temperatures.

Inserting the thermodynamic parameters from Tables 1 and 2 into Equations
The results shown in Figures 3-5 for CuAg72 at all pressures are very similar to the results for Cu-Cu, demonstrating the consistency between theoretical and experimental results. The calculated first three cumulants contain zero-point contributions at low temperatures, resulting from an asymmetry of the atomic interaction potential due to anharmonicity even at high pressure, which is consistent with established theories (Duc et al., 2018;Frenkel & Rehr, 1993;Rehr, 2000;N. v. Hung & Rehr, 1997). Figure 6 shows the thermal expansion coefficient α T for Cu-Cu, Ag-Ag, and CuAg72 as a function of absolute temperature with the effects of ambient pressure. The calculated results for Cu-Cu are consistent with experimental results (Hung et al., 2002) at normal atmospheric pressure; however, the result for CuAg72 is deflected from 70 to 400 K −1 when the ambient pressure is 14 GPa, which shows that because of the effect at high pressure, the thermal expansion coefficient α T for CuAg72 is reduced significantly in the room-temperature range but changes very little when the temperature exceeds 700 K.
The graph of α T has the form of the specific heat C V , thus reflecting the fundamental principle of solid-state theory, which states that thermal expansion results from anharmonic effects and is proportional to the specific heat C V (Duc et al., 2018). Our calculated values of α T approach the constant value α 0 T at high temperatures and vanish exponentially with θ E =T at low temperatures, which is consistent with the results of previous research (N. v. Hung & Duc, 1999, 2000.  Figure 7 shows the anharmonic factor βðTÞ as a function of absolute temperature and pressure for CuAg72. For both normal and high pressure (14 GPa), βðTÞ is negligibly small at low temperature and increases strongly when the temperature exceeds 100 K. At normal atmospheric pressure, we have θ E = 176 K for Ag, θ E = 236 K for Cu, and θ E = 207 K for CuAg72. At high pressure, we have θ 0 E = 279 K for Ag, θ 0 E = 364 K for Cu, and θ 0 E = 333 K for CuAg72. The results shown in Figure 7 are consistent with experimental results (Hung et al., 2002), which demonstrates that our calculations for CuAg72 are appropriate for normal atmospheric pressure. At temperatures above 100 K with  increasing pressure, the anharmonic factor β 0 ðTÞ is less than at normal pressure βðTÞ; in other words, β 0 ðTÞ= 0.3125βðTÞ at 100 K, β 0 ðTÞ= 0.7439βðTÞ at 300 K, and β 0 ðTÞ = 0.898βðTÞ at 700 K. Thus, the anharmonic factor describes the temperature dependence of the anharmonic EXAFS theory under the influence of high ambient pressure.

Conclusions
In this work, based on quantum statistical theory and by applying the effective ACEM to EXAFS spectra, we derive analytical expressions for the temperature dependence of the cumulants and thermodynamic parameters of crystalline Cu, Ag, and their alloy CuAg72 under the influence of pressures up to 14 GPa. The expressions for the second cumulant or DWF, the thermodynamic parameters, the effective force constant, and the correlated Einstein frequency and temperature for Cu, Ag, and CuAg72 agree with the known properties for these quantities. The expressions for calculations involving orderly doped crystals have forms similar to those for pure crystals. Figures 1-7 show the cumulants and thermodynamic parameters for doped crystals as functions of absolute temperature and pressure. The calculated results are consistent with experimental results and other studies of Cu and Ag, and the results for CuAg72 are coherent. Thus, the method developed herein, which is based on applying the ACEM to EXAFS, is appropriate for calculating and analyzing the cumulant and thermodynamic properties of intermetallic alloys. Author details Nguyen Ba Duc 1 E-mail: ducnb@daihoctantrao.edu.vn 1 Faculty of Physics, Tan Trao University, Tuyen Quang, Viet Nam.

Citation information
Cite this article as: Investigation of temperature and pressure effects on thermodynamic parameters of intermetallic alloy in EXAFS, Nguyen Ba Duc, Cogent Engineering (2020), 7: 1759184.