CALCULATION MORSE POTENTIAL PARAMETERS UNDER TEMPERATURE AND PRESSURE EFFECTS IN EXPANDED X-RAY ABSORPTION FINE STRUCTURE SPECTRA

Recieved: 8/9/2020 Accepted: 10/12/2020 A new method for estimating the effective parameters of Morse potential under thermal disorder and pressure effects for materials has crystals structure developed by using the energy of sublimation, the compressibility, and the lattice constant. Use the Morse potential parameters received to calculate the mean square relative displacement, spring constants, anharmonic interatomic effective potential, and local force constant for silicic and germanium semiconductor crystals, are the materials have diamond structure crystals. The received results suitable for the experimental values and other theories.


Introduction
In 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, photoelectron emitted from an 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's potential value. 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, the application for diamond (DIA) structure crystals. The obtained results applied to the equation of state, mean square relative displacement, spring constants, the effective anharmonic potential, and local force constant. Numerical calculations have carried out for Silicium (Si), Germanium (Ge), and SiGe semiconductor, which are suitable for the experimental values [10], [11].

Formalism
The ε(r ij ) potential of atoms i and j separated by a distance r ij is given in by the Morse function: where 1/α describes the width of the potential, D is the dissociation energy (ε(r 0 ) = -D); r 0 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 r j is the distance from the origin atom to the jth atom. It is beneficial to describe the following quantities: where m j , n j , l j are position coordinates of atoms in the lattice. Substitute the Eq.
The first and second derivatives of the potential energy of Eq. (4) concerning a, we have: where E 0 (a 0 ) is the energy of sublimation at zero pressure and temperature, and the compressibility is given by [8] 0 0 a a 2 2 0 2 where V 0 is the volume at T = 0 and 0 κ is compressibility at zero temperature and pressure.
Solving the system of Eq. (12, 13) we obtain α and r 0 . Using α and Eq. (4) to solve Eq. (7) we receive D. The obtained Morse potential parameters D, α depends on the compressibility 0 κ , the energy of sublimation E 0 and the lattice constant a. These values of all crystals are available already [12].
Next, apply the above expressions to 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 [8] where B k is Boltzmann constant, θ D is Debye temperature.
Using Eqs. (14,15) we derive the equation of state as where G  is the Grüneisen parameter, V is the volume.
After transformations, the Eq. (16) is resulted as The equation of state (17) contains the obtained Morse potential parameters, c is a constant and has value according to the structure of the crystal.
Elastic properties of a crystal described by an elastic tensor contained in the motion equation of the crystal. The non-vanishing elements of the elastic tensor defined as elastic constants. They are given for DIA structure crystals by [13]  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) is the cumulants to describe asymmetric of anharmonic Morse potential, and they appear due to 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 Here k eff is the effective local force constant, For DIA structure crystals, the anharmonic interatomic effective potential Eq. (24) has the form 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 E eff , effective local force constant k eff , anharmonic parameters k 3 , k 4 for DIA crystals presented in terms of our calculated Morse potential parameters D and α.

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 fee 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 to numerical calculations 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, the values of θ D and D [10, 15,16].
The numerical results of Morse potential parameters showed in Table 1. The theory values of α fit well with the measured experiment [10]. The elastic constants calculated by Morse potential parameters for Si and Ge are presented in Table 2 and compared to the experimental values [11].   Figure  1 for Si crystal, Figure 2 for Ge crystal, and compared with the experimental ones (dashed line)   Table 1 and compared to experiment obtained from Morse potential parameters of J. C. Slater (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 shows 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 quantum effect. Simultaneously, the values of σ2(T) are greater than the values of u2(T). The calculated results of σ2(T), u2(T) for the Si, Ge crystals fitting well with the experimental values [10]. Thus, it is possible to deduce that the calculation results of the present method for diamond structure crystals such as Si, Ge crystals are reasonable.

Conclusions
In this work, a calculation method of Morse potential parameters and application for diamond structure crystals have been developed based on the calculation of volume and number of an atom in each basic cell and the sublimation energy, compressibility, and lattice constant. The results applied to the mean square relative displacement, mean square displacement, the state equation, the elastic constants, anharmonic interatomic effective potential, and local force constant in EXAFS theory. The derived expressions have programmed for the calculation of the above physical quantities. 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.