The ε(r_{ij}) potential of atoms i and j separated by a distance r_{ij} is given in by the Morse function:

$$ \varepsilon \left({\mathrm{r}}_{\mathrm{ik}}\right)=\mathrm{D}\left\{{\mathrm{e}}^{-2\alpha \left({\mathrm{r}}_{\mathrm{ij}}-{\mathrm{r}}_{\mathrm{o}}\right)}-2{\mathrm{e}}^{-\alpha \left({\mathrm{r}}_{\mathrm{ij}}-{\mathrm{r}}_{\mathrm{o}}\right)}\right\}, $$

(1)

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:

$$ E=\frac{1}{2} ND\sum \limits_j\left\{{e}^{-2\alpha \left({r}_j-{r}_o\right)}-2{e}^{-\alpha \left({r}_j-{r}_o\right)}\right\}. $$

(2)

Here *r*_{j} is the distance from the origin atom to the jth atom. It is beneficial to describe the following quantities:

$$ {r}_j={\left[{m}_j^2+{n}_j^2+{l}_j^2\right]}^{1/2}a={M}_ja, $$

(3)

where m_{j}, n_{j}, l_{j} are position coordinates of atoms in the lattice. Substitute the Eq. (3) into Eq. (2), the potential energy can be rewritten as:

$$ E(a)=\frac{1}{2}{NDe}^{\alpha {r}_0}\left[{e}^{\alpha {r}_0}\sum \limits_j{e}^{-2\alpha {aM}_j}-2\sum \limits_j{e}^{-\alpha {aM}_j}\right]. $$

(4)

The first and second derivatives of the potential energy of Eq. (4) concerning *a*, we have:

$$ \frac{dE}{da}=-\alpha {NDe}^{\alpha {r}_0}\left[{e}^{\alpha {r}_0}\sum \limits_j{M}_j{e}^{-2\alpha {aM}_j}+\sum \limits_j{M}_j{e}^{-\alpha {aM}_j}\right], $$

(5)

$$ \frac{d^2E}{da^2}={\alpha}^2{NDe}^{\alpha {r}_0}\left[2{e}^{\alpha {r}_0}\sum \limits_j{M}_j^2{e}^{-2\alpha {aM}_j}-\sum \limits_j{M}_j^2{e}^{-\alpha {aM}_j}\right]. $$

(6)

At absolute zero *T* = 0, a_{0} is the value of *a* for which the lattice is in equilibrium, then *E*(*a*_{0}) gives the energy of cohesion, \( {\left[\frac{dE}{da}\right]}_{a_0}=0 \), and \( {\left[\frac{d^2E}{da^2}\right]}_{a_0} \)is related to the compressibility [15]. That is,

$$ dE\left({a}_0\right)={E}_0\left({a}_0\right), $$

(7)

where *E*_{0}(*a*_{0}) is the energy of sublimation at zero pressure and temperature,

$$ {\left(\frac{dE}{da}\right)}_{a_0}=0, $$

(8)

and the compressibility is given by [8]

$$ \frac{1}{\upkappa_0}={V}_0{\left(\frac{d^2{E}_0}{dV^2}\right)}_{a_0}={V}_0{\left(\frac{d^2E}{dV^2}\right)}_{a_0}, $$

(9)

where *V*_{0} is the volume at *T* = 0 and κ_{0} is compressibility at zero temperature and pressure. The volume per atom *V*/*N* is related to the lattice constant *a* by

$$ \frac{V}{N}={ca}^3. $$

(10)

Substituting Eq. (10) into Eq. (9) the compressibility is formulated by

$$ \frac{1}{\upkappa_0}=\frac{1}{9{cNa}_0}{\left(\frac{d^2E}{da^2}\right)}_{a={a}_0}. $$

(11)

Using Eq. (5) to solve Eq. (8), we obtain

$$ {e}^{\alpha {r}_0}=\frac{\sum \limits_j{M}_j{e}^{-\alpha {aM}_j}}{\sum \limits_j{M}_j{e}^{-2\alpha {aM}_j}}. $$

(12)

From Eqs. (4, 6, 7, 11), we derive the relation

$$ \frac{e^{\alpha {r}_0}\sum \limits_j{e}^{-2\alpha {aM}_j}-2\sum \limits_j{e}^{-\alpha {aM}_j}}{4{\alpha}^2{e}^{\alpha {r}_0}\sum \limits_j{M}_j^2{e}^{-2\alpha {aM}_j}-2{\alpha}^2\sum \limits_j{M}_j^2{e}^{-\alpha {aM}_j}}=\frac{E_0{\upkappa}_0}{9{cNa}_0}. $$

(13)

Solving the system of Eq. (12, 13), we obtain *α* and *r*_{0}. Using α and Eq. (4) to solve Eq. (7), we have *D*. The interatomic Morse potential parameters *D*, *α* depend on the compressibility κ_{0}, the energy of sublimation *E*_{0}, and the lattice constant *a*. These values of all crystals are available already [16].

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 [8]

$$ F=E+3{Nk}_BT\ln \left(1-{e}^{-{\theta}_D/T}\right)-{Nk}_B TD\left({\uptheta}_D/T\right), $$

(14)

$$ D\left(\frac{\uptheta_D}{T}\right)=3{\left(\frac{T}{\uptheta_D}\right)}^3\underset{0}{\overset{\theta_D/T}{\int }}\frac{x^3}{e^x-1} dx, $$

(15)

where *k*_{B} is Boltzmann constant, and θ_{D} is Debye temperature.

Using Eqs. (14, 15), we derive the equation of state as

$$ P=-{\left(\frac{\partial F}{\partial V}\right)}_T=\frac{1}{3{ca}^2}\frac{dE}{da}+\frac{3{\gamma}_G RT}{V}D\left(\frac{\uptheta_D}{T}\right), $$

(16)

where *γ*_{G} is the Grüneisen parameter, and *V* is the volume.

After transformations, the Eq. (16) is resulted as

$$ P=\frac{\left[{NDe}^{\alpha {r}_0}\alpha \sum \limits_j{M}_j{e}^{-\alpha {a}_0{M}_j{\left(1-x\right)}^{1/3}}\right]}{3{ca}_0^2{\left(1-x\right)}^{2/3}}-{NDe}^{2\alpha {r}_0}\alpha \sum \limits_j{M}_j{e}^{-2\alpha {a}_0{M}_j{\left(1-x\right)}^{1/3}}+\frac{3{\gamma}_G RT}{V_0\left(1-x\right)}D\left(\frac{\uptheta_D}{T}\right), $$

(17)

$$ x=\frac{V_0-V}{V_0},\kern0.48em {V}_0={ca}_0^3,\kern0.74em R={Nk}_B,\kern0.62em N=6.02\times {10}^{23}. $$

(18)

The equation of state (17) contains the obtained interatomic Morse potential parameters; *c* is a constant and has value according to the structure of the crystal.

An elastic tensor describes the elastic properties of a crystal in the crystal’s motion equation. The non-vanishing components of the elastic tensor are defined as elastic constants. They are given for crystals of lattice structure by [17]:

$$ {c}_{11}={c}_{22}=\sqrt{2}{r}_0\left[10{\Psi}^{{\prime\prime}}\left({r}_0^2\right)+16{\Psi}^{{\prime\prime}}\left(2{r}_0^2\right)+81{\Psi}^{{\prime\prime}}\left(3{r}_0^2\right)\cdots \right]-\frac{{\left\{\sqrt{\frac{2}{3}}\left[-2{\Psi}^{{\prime\prime}}\left({r}_0^2\right)+16{\Psi}^{{\prime\prime}}\left(2{r}_0^2\right)-40{\Psi}^{{\prime\prime}}\left(3{r}_0^2\right)\cdots \right]\right\}}^2}{\sqrt{2}{r}_0^{-1}\left[4{\Psi}^{{\prime\prime}}\left({r}_0^2\right)+16{\Psi}^{{\prime\prime}}\left(2{r}_0^2\right)+12{r}_0^{-1}{\Psi}^{\prime}\left(2{r}_0^2\right)\cdots \right]}, $$

(19)

$$ {c}_{12}=\frac{\sqrt{2}{r}_0\left[10{\Psi}^{{\prime\prime}}\left({r}_0^2\right)+16{\Psi}^{{\prime\prime}}\left(2{r}_0^2\right)+81{\Psi}^{{\prime\prime}}\left(3{r}_0^2\right)\cdots \right]}{3}+\frac{{\left\{\sqrt{\frac{2}{3}}\left[-2{\Psi}^{{\prime\prime}}\left({r}_0^2\right)+16{\Psi}^{{\prime\prime}}\left(2{r}_0^2\right)-40{\Psi}^{{\prime\prime}}\left(3{r}_0^2\right)\cdots \right]\right\}}^2}{\sqrt{2}{r}_0^{-1}\left[4{\Psi}^{{\prime\prime}}\left({r}_0^2\right)+16{\Psi}^{{\prime\prime}}\left(2{r}_0^2\right)+12{r}_0^{-1}{\Psi}^{\prime}\left(2{r}_0^2\right)\cdots \right]}, $$

(20)

$$ \kern0.24em {c}_{33}=\frac{\sqrt{2}}{3}{r}_0\left[32{\Psi}^{{\prime\prime}}\left({r}_0^2\right)+32{\Psi}^{{\prime\prime}}\left(2{r}_0^2\right)+\frac{512}{3}{\Psi}^{{\prime\prime}}\left(3{r}_0^2\right)+\cdots \right], $$

(21)

$$ {c}_{13}={c}_{23}=\sqrt{2}{r}_0\left[8{\Psi}^{{\prime\prime}}\left({r}_0^2\right)+32{\Psi}^{{\prime\prime}}\left(2{r}_0^2\right)+112{\Psi}^{{\prime\prime}}\left(3{r}_0^2\right)+\cdots \right], $$

(22)

$$ {\Psi}^{\prime }(r)=-2 D\alpha \left[{e}^{-2\alpha \left(r-{r}_0\right)}-{e}^{-\alpha \left(r-{r}_0\right)}\right]\frac{1}{r}, $$

(23)

$$ {\Psi}^{{\prime\prime} }(r)=D{\alpha}^2\left[2{e}^{-2\alpha \left(r-{r}_0\right)}-\frac{1}{2}{e}^{-\alpha \left(r-{r}_0\right)}\right]\frac{1}{r^2}+ D\alpha \left[{e}^{-2\alpha \left(r-{r}_0\right)}-{e}^{-\alpha \left(r-{r}_0\right)}\right]\frac{1}{2{r}^3}. $$

(24)

Hence, the derived elastic constants contain the interatomic 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

$$ \chi (k)=A(k)\frac{\exp \left[-2\Re /\lambda (k)\right]}{k\Re^2}\operatorname{Im}\left\{{e}^{i\phi (k)}\exp \left[2 ik\mathit{\Re}+\sum \limits_n\frac{{\left(2 ik\right)}^n}{n!}{\sigma}^{(n)}\right]\right\}, $$

(25)

where *A*(*k*) is scattering amplitude of atoms, φ(*K*) is the total phase shift of photoelectron, and *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 interatomic Morse potential, due to the average of the function *e*^{−2ikr}, *ℜ* = < *r*>, and *r* is the instantaneous bond length between absorber and backscatter atoms at *T* temperature.

For describing anharmonic EXAFS, effective anharmonic potential [9] of the system is derived which in the current theory is expanded up to the third order and given by

$$ {\mathrm{E}}_{\mathrm{eff}}\left(\mathrm{x}\right)=\frac{1}{2}{\mathrm{k}}_{\mathrm{eff}}{\mathrm{x}}^2+{\mathrm{k}}_{3\mathrm{eff}}{\mathrm{x}}^3+\dots +=\mathrm{E}\left(\mathrm{x}\right)+\sum \limits_{\mathrm{j}{}^1\mathrm{i}}\mathrm{E}\left(\frac{\upmu}{{\mathrm{M}}_{\mathrm{i}}}\mathrm{x}{\hat{\mathrm{R}}}_{12}.{\hat{\mathrm{R}}}_{\mathrm{i}\mathrm{j}}\right),\kern1em \mu =\frac{M_1{M}_2}{M_1+{M}_2};\kern1em \hat{\Re}=\frac{\Re }{\mid R\mid }. $$

(26)

Here, *k*_{eff} is the effective local force constant, and *k*_{3eff} is the cubic parameter characterizing the asymmetry in the pair interatomic Morse potential, and *x* is the deviation of instantaneous bond length between the two atoms from equilibrium. The correlated model defined as the oscillation of a pair of particles with *M*_{1} and *M*_{2} mass. Their vibration influenced by their neighbors atoms given by the sum in Eq. (24), where the sum i is over absorber (*i* = 1) and backscatterer (*i* = 2), and the sum *j* is over all their near neighbors, excluding the absorber and backscatterer themselves whose contributions are described by the term E(x). The advantage of this model is a calculation based on including the contributions of the nearest neighbors of absorber and backscatter atoms in EXAFS. The anharmonic interatomic effective potential Eq. (26) has the form

$$ {E}_{eff}(x)={E}_x(x)+2{E}_x\left(-\frac{x}{2}\right)+8{E}_x\left(-\frac{x}{4}\right)+8{E}_x\left(\frac{x}{4}\right). $$

(27)

Applying interatomic Morse potential given by Eq. (1) expanded up to 4th order around its minimum point

$$ {E}_{eff}(x)=D\left({e}^{-2\alpha x}-2{e}^{-\alpha x}\right)\approx D\left(-1+{\alpha}^2{x}^2-{\alpha}^3{x}^3+\frac{7}{12}{\alpha}^4{x}^4\dots \right). $$

(28)

From Eqs. (26)–(28), we obtain the anharmonic effective potential *E*_{eff}, effective local force constant *k*_{eff}, anharmonic parameters *k*_{3eff} for lattice crystals presented in terms of our calculated interatomic Morse potential parameters *D* and *α*.

In Eq. (25), *σ*^{(n)} is cumulants, in which *σ*^{2}(*T*) is the Debye-Waller factor (DWF) or MSRD [9]. In the diffraction or X-ray absorption, the DWF has a form similar *u*^{2}(*T*). In the EXAFS spectrum, DWF is regarded as to correlated averages over the relative displacement of σ^{2}(*T*) for a pair of atoms, while neutron diffraction allude to the MSD *u*^{2}(*T*) of an atom [18]. From *σ*^{2}(*T*) and *u*^{2}(*T*), the correlated function *C*_{R}(*T*) to describe the effects of correlation in the vibration of atoms can be deduced. Using the anharmonic correlated Debye model (ACDM), the MSRD σ^{2}(*T*) has the form [19]:

$$ {\sigma}^2(T)=\frac{\mathrm{\hslash}a}{10\pi \mathrm{D}{\alpha}^2}\underset{0}{\overset{\pi /a}{\int }}{\omega}_A(q)\frac{1+{z}_A(q)}{1-{z}_A(q)} dq, $$

(29)

$$ z(q)={e}^{-\left(\beta \mathrm{\hslash}{\omega}_A(q)\right)},\kern2em {\omega}_A(q)=2\sqrt{\frac{10\mathrm{D}{\alpha}^2}{M}}\left|\sin \left( qa/2\right)\right|,\kern2em \left|q\right|\le \uppi /\mathrm{a}. $$

(30)

Similarly, for the anharmonic Debye model, *u*^{2}(*T*) have been determined as:

$$ {u}^2(T)=\frac{\mathrm{\hslash}a}{16\pi \mathrm{D}{\alpha}^2}\underset{0}{\overset{\pi /a}{\int }}{\omega}_D(q)\frac{1+{z}_D(q)}{1-{z}_D(q)} dq, $$

(31)

$$ {z}_D(q)={e}^{-\left(\beta \mathrm{\hslash}{\omega}_D(q)\right)},\kern2em {\omega}_D(q)=2\sqrt{\frac{8\mathrm{D}{\alpha}^2}{M}}\left|\sin \left( qa/2\right)\right|,\kern2em \left|q\right|\le \uppi /\mathrm{a},\kern1em $$

(32)

where *a* is the lattice constant, ω(q) and *q* are the frequency and phonon wavenumber, and *M* is the mass of composite atoms.