Determination Morse potential parameters by the theoretical method in EXAFS

: A new method for estimating Morse potential's effective parameters has developed using the sublimation energy, the compressibility, and the lattice constant in expanded X-ray absorption fine structure spectra. Application the received parameters of Morse potential to calculate the mean square relative displacement, elastic constant, anharmonic effective potential, correlated function, and local force constant for diamond-type structure crystals such as silicon and germanium crystals, and face-centered cubic crystals as copper, silver. Numerical results agree well with the experimental values and other theories.


Introduction
In Expanded X-ray Absorption Fine Structure (EXAFS) spectra with the anharmonic effects, the anharmonic Morse potential [1] is suitable for describing the interaction and oscillations of atoms in the crystals [2][3][4][5][6][7]. In the EXAFS theory, photoelectrons are emitted by the absorber scattered by surrounded vibrating atoms [1,2]. This thermal oscillation of atoms contributes to the EXAFS spectra, especially the anharmonic EXAFS [2][3][4][5][6][7], which is affected by these spectra's physical information. In the EXAFS spectrum analysis, the parameters of Morse potential is usually extracted from the experiment. Still, experimental data are not available in many cases, so a theory is necessary to deduce Morse potential parameters. The only calculation has been carried out for cubic crystals [8]. The results have been used actively for calculations EXAFS thermodynamic parameters [6] and reasonable with those extracted from EXAFS data [9] using anharmonic correlated Einstein model [8]. Therefore, the requirement for calculation of the anharmonic Morse interaction potential due to thermal disorder for other structures is essential.
The purpose of this study is to expand a method for calculating the Morse potential parameters using the energy of sublimation, the compressibility, and the lattice constant with the effect of the disorder of temperature. Use the Morse potential parameters received to calculate the mean square relative displacement (MSRD), mean square displacement (MSD), elastic constant, anharmonic interatomic effective potential, and effective local force constant for diamond-type (DIA) structure crystals such as silicium (Si), germanium (Ge), SiGe semiconductor, and face-centered cubic (fcc) crystals as copper (Cu), silver (Ag) and CuAg alloy. Numerical results are suitable for the experimental values and other theories [10], [11], [17][18][19][20]. 2

Formalism
The ε(rij) potential of atoms i and j separated by a distance rij is given in by the Morse function: where 1/α describes the width of the potential, D is the dissociation energy (ε(r0) = -D); r0 is the equilibrium distance of the two atoms.
To obtain the potential energy of a large crystal whose atoms are at rest, it is necessary to sum Eq. (1) over the entire crystal. It is quickly done by selecting an atom in the lattice as origin, calculating its interaction with all others in the crystal, and then multiplying by N/2, where N is the total number of atoms in a crystal. Therefore, the potential E is given by: ( Here rj is the distance from the origin atom to the jth atom. It is beneficial to describe the following quantities: where mj, nj, lj are position coordinates of atoms in the lattice. Substitute the Eq. (3) into Eq. (2), the potential energy can be rewritten as: The first and second derivatives of the potential energy of Eq. (4) concerning a, we have: is related to the compressibility [10]. That is, where E0(a0) is the energy of sublimation at zero pressure and temperature, and the compressibility is given by [8] where V0 is the volume at T = 0 and 0 κ is compressibility at zero temperature and pressure. The (13) Solving the system of Eq. (12, 13) we obtain α and r0. Using α and Eq. (4) to solve Eq. (7) we receive D. The Morse potential parameters D, α depends on the compressibility 0 κ , the energy of sublimation E0 and the lattice constant a. These values of all crystals are available already [12].
Next, we apply the above expressions to claculate the equation of state and elastic constants. It is possible to calculate the state equation from the potential energy E. If we assumed that the Debye model could express the thermal section of the free energy, then the Helmholtz energy is given by where B k is Boltzmann constant, θD is Debye temperature.
Using Eqs. (14,15) we derive the equation of state as where G  is the Grüneisen parameter, V is the volume. 4 After transformations, the Eq. (16) is resulted as The equation of state (17) Hence, the derived elastic constants contain the Morse potential parameters.
Next, apply to calculate of anharmonic interatomic effective potential and local force constant in EXAFS theory. The expression for the anharmonic EXAFS function [2] is described by where A(k) is scattering amplitude of atoms, φ(K) is the total phase shift of photoelectron, k and λ are wave number and mean free path of the photoelectron, respectively. The σ (n) are the cumulants, they describe asymmetric of anharmonic Morse potential, and they appear due to 5 being average of the function e -2ikr , in which expanded of the asymmetric terms in a Taylor series around value  = <r>, with r is the instantaneous bond length between absorber and backscatter atoms at T temperature.
For describing anharmonic EXAFS, effective anharmonic potential [6] of the system is derived which in the current theory is expanded up to the third -order and given by   Applying Morse potential given by Eq. (1) expanded up to 4 th order around its minimum point From Eqs. (26)-(28), we obtain the anharmonic effective potential Eeff, effective local force constant keff, anharmonic parameters k3eff for lattice crystals presented in terms of our calculated Morse potential parameters D and α.
In the Eq. (25), σ (n) are cumulants, in which second cumulant σ 2 (T) or Debye-Waller factor (DWF) or Mean Square Relative Displacement (MSRD) of the bond between two nearest atoms [6]. During the diffraction of neutrons or X-ray absorption, the DWF has a similar u 2 (T) form. In the EXAFS spectra, DWF is considered to correlated averages over the relative displacement of  Similarly, for the anharmonic Debye model, u 2 (T) have been determined as: where a is the lattice constant, ω(q) and q are the frequency and phonon wavenumber, M is the mass of composite atoms.

Numerical results and discussion
To receive the Morse potential parameters, we need to calculate the parameter c in Eq. (10). The space lattice of the diamond is the fcc. The primordial basis has two identical atoms at 0 0 0, ¼ ¼ ¼ connected with each point of the fcc lattice. Thus, the conventional unit cube contains eight atoms so that we obtain the value c = 1/4 for this structure.
Apply the above derived expressions, we calculate thermal parameters for DIA structure crystals (Si, Ge and SiGe) using the energy of sublimation [10], the compressibility [14] and the lattice constants [11], as well as calculate for fcc structure crystals (Cu, Ag, and CuxAgy, x = 72, 50; y = 28,50).
The numerical results of the Morse potential parameters shown in Tables 1, 2, 5. The theory values of D, α fit well with the measured experiment [10], [18][19][20]. The elastic constants ci, effective spring force constants keff and effective spring cubic parameters k3eff calculated by Morse potential parameters for Si, Ge, Cu, Ag and their alloys are presented in Tables 3, 4 and compared to the experimental values [11], [18][19][20]. Table 5 shown Morse potential parameters, spring force constants and cubic parameters under pressure effects for Cu50Ag50 up to 14GPa.  The computed results for the state equation illustrated in Figure 1 for Si crystal, Figure 2 for Ge crystal, and compared with the experimental ones (dashed line) [10] represented by an 8 extrapolation procedure of the measured data, the graphs shown they agree well, especially at low pressure.

Figures 3 and 4 illustrate good agreement of the anharmonic interatomic effective potentials for
Si, Ge and SiGe semiconductor calculated using the present theory (solid line), and the experiment values obtained from Morse potential parameters of J. C. Slater (solid line and symbol ) [10], and simultaneously shows strong asymmetry of these potentials due to the anharmonic contributions in atomic vibrations of these DIA structure crystals illustrate by their anharmonic shifting from the harmonic terms (dashed line). Figures 5, 6 showns the temperature and pressure dependence of mean square relative displacement σ2(T) and mean square displacement u2(T) for Si and Ge crystals. They show linear proportional to the temperature T at high temperatures, and the classical limit is applicable.
At low temperatures, the curves for Si and Ge contain zero-point energy contributions -a   [10], [18], [19], [20], and those for CuxAgy (x=72, 50, y=28, 50) agree well with other theories [17]. Thus, it is possible to deduce that the calculation results of the present method for Morse potential parameters are reasonable. Reasonable agreement between our calculated results and the experimental data show the efficiency of the present procedure. The calculation of potential atomic parameters is essential for estimating and analyzing physical effects in the EXAFS technique. It can solve the problems involving any deformation and of atom interaction in the diamond structure crystals.