Thermodynamic parameters depend on temperature with the influence of doping ratio of the crystal structure metals in extended X-Ray absorption fine structure

Recieved: 02/11/2018 Accepted: 10/12/2018 The effects of the doping ratio and temperature on the cumulants and thermodynamic parameters of crystal structure metals and their alloys was investigated using the anharmonic correlated Einstein model, in extended X-ray absorption fine structure (EXAFS) spectra. We derived analytical expressions for the EXAFS cumulants, correlated Einstein frequency, Einstein temperature, and effective spring constant. We have considered parameters of the effective Morse potential and the Debye-Waller factor depend on temperature and the effects of the doping ratio of face-centered-cubic (fcc) crystals of copper (Cu-Cu), silver (Ag-Ag), and hexagonal-close-packed (hcp) crystal of zinc (Zn-Zn), and their alloys of Cu-Ag and Cu-Zn. The derived anharmonic effective potential includes the contributions of all the nearest neighbors of the absorbing and scattering atoms. This accounts for three-dimensional interactions and the parameters of the Morse potential, to describe single-pair atomic interactions. The numerical results of the EXAFS cumulants, thermodynamic parameters, and anharmonic effective potential agree reasonably with experiments and other theories.


Introduction
Extended X-ray absorption fine structure spectra has developed into a powerful probe of atomic structures and the thermal effects of substances [1,5,[8][9][10][11][12][13][14][15]. The dependence of the thermodynamic properties and cumulants of the lattice crystals of a substance on the temperature with influence doping ratio (DR) was studied using this technique. The thermodynamic parameters and the EXAFS cumulants for pure cubic crystals, such as crystals of copper (Cu) doped with silver (Ag) (Cu-Ag), which depend on DR and temperature, have been derived using the anharmonic correlated Einstein model (ACEM) in EXAFS theory [6,8,10]. However, the effect of the doping ratio and temperature on the thermodynamic parameters and cumulants of the EXAFS for copper doped with zinc (Cu-Zn), copper doped with silver at a level not above 50%, is yet to be determined.
In this study, we use anharmonic effective potential from EXAFS theory [8,10,15] to formulate thermodynamic parameters, such as the effective force constants, expressions of cumulants, thermal expansion coefficient, correlated Einstein frequency, and correlated Einstein temperature, these parameters are contained in the anharmonic EXAFS spectra. The Cu-Ag and Cu-Zn doped crystals contain pure Cu, Ag, and Zn atoms. The Ag and Zn atoms are referred to as the substitute atoms and the Cu atoms are referred to as the host atoms. The expression CuAg72 indicates a ratio of 72% Ag and 28% Cu atoms in the alloy, and CuZn45 indicates 45% Zn and 55% Cu in the alloy. Numerical calculations have been conducted for doped crystals to determine the thermodynamic effects and how they depend on the DR and temperature of the crystals. The results of the calculations are in good agreement with experimental values and those of other studies [2][3][4][5][6][7][8][9][10][11]13,16,17].

Formalism
The anharmonic EXAFS function, including the anharmonic contributions of atomic vibration, is often expressed as [1,10,15] where R r   with r is the instantaneous bond length between absorbing and scattering atoms at temperature T and 0 r is its equilibrium value, 2 0 S is the intrinsic loss factor due to many electron effects, N is the atomic number of a shell, ( ) F k is the atomic backscattering amplitude, k and  are the wave number and mean free path of the photoelectron, and ( ) k  is the total phase shift of the photoelectron.
In the ACEM [10,15], interaction between absorbing and scattering atoms with contributions from atomic neighbors is characterized by an effective potential. To describe the asymmetric components of the interactive potential, the cumulants   n    1, 2, 3, 4,... n  are used. To determine the cumulants, it is necessary to specify the interatomic potential and force constant.
Consider a high-order expanded anharmonic interatomic effective potential, expanded up to fourth order, namely where eff k is an effective spring constant that includes the total contribution of the neighboring atoms, and 3eff k and 4eff k are effective anharmonicity parameters that specify the asymmetry of the anharmonic effective potential, 0 x r r   is net deviation. The effective potential, given by Eq. 2, is defined based on the assumption of an orderly centerof-mass frame for a single-bond pair of an absorber and a bacskcatterer [7,10,15]. For monatomic crystals, the masses of the absorber and backscatterer are the same, so the effective potential is given by where V(x) includes only absorber and backscatter atoms, i is the sum of the absorber ( 1 i  ) and backscatter ( 2 i  ) atoms, and j is the sum of all their near neighbors, excluding the absorber and backscatterer themselves, whose contributions are described by the term V(x),  is the reduced atomic mass, R is the unit bond-length vector. Therefore, this effective pair potential describes not only the pair interaction of the absorber and backscatter atoms but also how their near-neighbor atoms affect such interactions. This is the difference between the effective potential of this study and the single-pair potential [7] and single-bond potential [1], which consider only each pair of immediate neighboring atoms, i.e., only V(x), without the remaining terms on the right-hand side of Eq. 3. The atomic vibration is calculated based on a quantum statistical procedure with an approximate quasi-harmonic vibration, in which the Hamiltonian of the system is written as a harmonic term with respect to the equilibrium at a given temperature, plus an anharmonic perturbation: where D12 is the dissociation energy,  describes the width of the potential. For simplicity, we approximate the parameters of the Morse potential in Eq. 5 at a certain temperature by where 1 2 , c c are the DR (%) of the alloy and To derive analytical formulas for the cumulants, we use perturbation theory [15]. The atomic vibration is quantized as phonons, considering the phonon-phonon interactions to account for anharmonicity effects, with correlated Einstein frequency and correlated Einstein temperature: Where B k is the Boltzmann constant, we obtain the cumulants up to third order: dependence of the linear thermal expansion coefficient on the absolute temperature T with efects the DR of the doped metals: and the anharmonic factor as 12 12 12 12 12

Results and discussion
The calculated and experimental [4] parameter values of the Morse potential, 12

D and 12
 , for the pure metals and their alloy crystals are given in Table I.  Table II.

TABLE II. Anharmonic effective parameter values
Substituting the values of the thermodynamic parameters from Tables I and II into Eqs. 2, 9-13, we obtain expressions for the anharmonic effective potential ( ) V x , which depends on T, and the cumulants ( ) ( ) n n  , which depend on the DR and T.

FIG. 1. Comparison between present theory and experimental values of anharmonic effective Morse potential FIG. 2. Dependence of cumulants on doping ratio (DR) CuAg50.
In Figure 1, we compare the calculated anharmonic effective Morse potential (solid lines) and experimental data (dotted lines) from H.Ö. Pamuk and T.Halicioğlu [4], for Cu (blue curve with symbol ○), Ag (red curve with symbol Δ), and Zn (black curve with symbol □).
The calculated curves of the Morse potential align closely with the experimental curves, indicating that the calculated data for the coefficients keff, k3eff, and k4eff, from the ACEM, are in good agreement with the measured experimental values. Figure 2 shows how the first three calculated cumulants depend on the DR at a given temperature (300 K), for the compound Cu-Ag.
The graphs of (1) Figure 3 shows the temperature dependence of the calculated first cumulant, or net thermal expansion (1)  for Cu, Ag, CuAg72 (the alloy with 28% Cu atoms and 72% Ag atoms, referred to as CuSil or UNS P07720 [16]), and CuZn45 (the alloy with 55% Cu atoms and 45% Zn atoms referred to as the brass [17], a yellow alloy of copper and zinc). Figure 4 illustrates the temperature dependence of the calculated second cumulant or DWF ( 2)  , for Cu-Cu, Ag-Ag, Zn-Zn, and their alloys CuAg72 and CuNi45, and comparison with the experimental values [8,12]. There good agreement at low temperatures and small differences at high temperatures, and the measured results between the results for CuAg72 and CuNi45 with Cu values are reasonable. Calculated values for the first cumulant (Fig. 3), and the DWF (Fig. 4) with the effects of the DRs, are proportional to the temperature at high temperatures. At low temperatures there are very small, and contain zero-point contributions, which are a result of an asymmetry of the atomic interaction potential of these crystals due to anharmonicity. Figure 5 shows the temperature dependence of the calculated third cumulant (3)  , for Cu-Cu, Ag-Ag, Zn-Zn, and their alloys CuAg72 and CuZn45. The calculated results are in good agreement with the experimental values [8,12].
The curves in Figures 3, 4, and 5 for CuZn45 and CuAg72 are very similar to the Cu-Cu curve, illustrating the fit between theoretical and experimental results. The calculated first three cumulants contain zero-point contributions at low temperatures are in agreement with established theory. Furthermore, the calculations and graphs demonstrate that the alloys of two Cu-Zn elements with Zn content less than or equal 45% enhances the durability and ductility of copper alloys, when the Zn content exceeds 50% in the Cu-Zn alloy, it becomes hard and brittle. Alloy CuZn45 is often used as heat sinks, ducts and stamping parts because of its high viscosity [17]. Also, CuAg72 is an eutectic alloy, primarily used for vacuum brazing [16].  [12].