An Approach to Manufacture Small Multiplexer with Dense Field-Effect Transistors

To solve our aim, we determine and analyzed spatio-temporal distribution of concentration of dopant in the considered heterostructure. We determine the distribution by solving the second Fick's law in the following form [1, 20-23].

$\frac{\partial \,C\,\left( x,y,z,t \right)}{\partial \,t}=\frac{\partial \,}{\partial \,x}\left[ D\frac{\partial \,C\,\left( x,y,z,t \right)}{\partial \,x} \right]+\frac{\partial \,}{\partial \,y}\left[ D\frac{\partial \,C\,\left( x,y,z,t \right)}{\partial \,y} \right]+$

$\frac{\partial \,}{\partial \,z}\left[ D\frac{\partial \,C\,\left( x,y,z,t \right)}{\partial \,z} \right]+\Omega \frac{\partial }{\partial \,x}\left[ \frac{{{D}_{S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{C\,\left( x,y,W,t \right)\,d\,W} \right]+$

$\Omega \frac{\partial }{\partial \,x}\left[ \frac{{{D}_{S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{C\,\left( x,y,W,t \right)\,d\,W} \right]+$

$\Omega \frac{\partial }{\partial \,y}\left[ \frac{{{D}_{S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{C\,\left( x,y,W,t \right)\,d\,W} \right]+$  (1)

$\frac{\partial \,}{\partial \,z}\left[ D\frac{\partial \,C\,\left( x,y,z,t \right)}{\partial \,z} \right]+\frac{\partial }{\partial \,\,x}\left[ \frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,x} \right]+$

$\frac{\partial }{\partial \,\,y}\left[ \frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,y} \right]+\frac{\partial }{\partial \,\,z}\left[ \frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,z} \right]$

with boundary and initial conditions

${{\left. \frac{\partial \,C\,\left( x,y,z,t \right)}{\partial \,x} \right|}_{x=0}}=0$, ${{\left. \frac{\partial \,C\,\left( x,y,z,t \right)}{\partial \,x} \right|}_{x={{L}_{x}}}}=0$,

${{\left. \frac{\partial \,C\,\left( x,y,z,t \right)}{\partial \,y} \right|}_{y=0}}=0$, ${{\left. \frac{\partial \,C\,\left( x,y,z,t \right)}{\partial \,y} \right|}_{x={{L}_{y}}}}=0$,  (2)

${{\left. \frac{\partial \,C\,\left( x,y,z,t \right)}{\partial \,z} \right|}_{z=0}}=0$, ${{\left. \frac{\partial \,C\,\left( x,y,z,t \right)}{\partial \,z} \right|}_{x={{L}_{z}}}}=0$,

$C(x, y, z, 0)=f_{C}(x, y, z)$.

Here $C(x, y, z, t)$ is the spatio-temporal distribution of concentration of dopant; $\Omega$ is the atomic volume of dopant; $\nabla_{s}$ is the symbol of surficial gradient; $\int\limits_{0}^{{{L}_{z}}}{C\,\left( x,y,z,t \right)\,d\,z}$ is the surficial concentration of dopant on interface between layers of heterostructure (in this situation we assume, that Z-axis is perpendicular to interface between layers of heterostructure); $\mu_{1}(x,y,z,t)$ and $\mu_{2}(x,y,z,t)$ are the chemical potential due to the presence of mismatch-induced stress and porosity of material; D and D_S are the coefficients of volumetric and surficial diffusions. Values of dopant diffusions coefficients depends on properties of materials of heterostructure, speed of heating and cooling of materials during annealing and spatio-temporal distribution of concentration of dopant. Dependences of dopant diffusions coefficients on parameters could be approximated by the following relations [24-26].

$\left\{ \begin{align}  & {{D}_{C}}={{D}_{L}}\left( x,y,z,T \right)\,\,\left[ 1+\xi \frac{{{C}^{\gamma }}\left( x,y,z,t \right)}{{{P}^{\gamma }}\left( x,y,z,T \right)} \right]\,\,\times  \\ & \times \,\,\left[ 1+{{\varsigma }_{1}}\frac{V\,\left( x,y,z,t \right)}{{{V}^{*}}}+{{\varsigma }_{2}}\frac{{{V}^{2}}\left( x,y,z,t \right)}{{{\left( {{V}^{*}} \right)}^{2}}} \right] \\ & {{D}_{S}}={{D}_{S\,L}}\left( x,y,z,T \right)\,\left[ 1+{{\xi }_{S}}\frac{{{C}^{\gamma }}\left( x,y,z,t \right)}{{{P}^{\gamma }}\left( x,y,z,T \right)} \right]\,\,\times  \\ & \times \,\,\left[ 1+{{\varsigma }_{1}}\frac{V\,\left( x,y,z,t \right)}{{{V}^{*}}}+{{\varsigma }_{2}}\frac{{{V}^{2}}\left( x,y,z,t \right)}{{{\left( {{V}^{*}} \right)}^{2}}} \right] \\\end{align} \right.$ (3)

Here D_L (x,y,z,T) and D_LS (x,y,z,T) are the spatial (due to accounting all layers of heterostruicture) and temperature (due to Arrhenius law) dependences of dopant diffusion coefficients; T is the temperature of annealing; $P (x,y,z,T)$ is the limit of solubility of dopant; parameter $\gamma$ depends on properties of materials and could be integer in the following interval $\gamma \in$ [1,3] [24]; $V (x,y,z,t)$ is the spatio-temporal distribution of concentration of radiation vacancies; V^* is the equilibrium distribution of vacancies. Concentrational dependence of dopant diffusion coefficient has been described in details in Ref. [24]. Spatio-temporal distributions of concentration of point radiation defects have been determined by solving the following system of equations [20-23, 25, 26].

$\frac{\partial \,\,I\,\left( x,y,z,t \right)}{\partial \,\,t}=\frac{\partial }{\partial \,\,x}\left[ {{D}_{I}}\left( x,y,z,T \right)\frac{\partial \,I\,\left( x,y,z,t \right)}{\partial \,x} \right]+$

$\frac{\partial }{\partial \,\,y}\left[ {{D}_{I}}\left( x,y,z,T \right)\frac{\partial \,I\,\left( x,y,z,t \right)}{\partial \,y} \right]-{{k}_{I,I}}\left( x,y,z,T \right)\,\,{{I}^{2}}\left( x,y,z,t \right)+$

$\frac{\partial }{\partial \,\,z}\left[ {{D}_{I}}\left( x,y,z,T \right)\frac{\partial \,I\,\left( x,y,z,t \right)}{\partial \,\,z} \right]-{{k}_{I,V}}\left( x,y,z,T \right)\,I\,\left( x,y,z,t \right)\,\,\times $

$I\,\left( x,y,z,t \right)\,V\,\left( x,y,z,t \right)+\frac{\partial }{\partial \,x}\left[ {{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{I\,\left( x,y,W,t \right)\,d\,W}\,\times  \right.$

$\left. \frac{{{D}_{IS}}}{k\,T} \right]\,\,\Omega +\Omega \frac{\partial }{\partial \,y}\left[ \frac{{{D}_{IS}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{I\,\left( x,y,W,t \right)\,d\,W} \right]+$

$\frac{\partial }{\partial \,x}\left[ \frac{{{D}_{I\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,x} \right]+\frac{\partial }{\partial \,y}\left[ \frac{{{D}_{I\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,y} \right]+$

$\frac{\partial }{\partial \,z}\left[ \frac{{{D}_{I\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,z} \right]$  (4)

$\frac{\partial \,V\,\left( x,y,z,t \right)}{\partial \,\,t}=\frac{\partial }{\partial \,\,x}\left[ {{D}_{V}}\left( x,y,z,T \right)\frac{\partial \,V\,\left( x,y,z,t \right)}{\partial \,x} \right]+$

$\frac{\partial }{\partial \,\,y}\left[ {{D}_{V}}\left( x,y,z,T \right)\frac{\partial \,V\,\left( x,y,z,t \right)}{\partial \,y} \right]-{{k}_{V,V}}\left( x,y,z,T \right)\,\,{{V}^{2}}\left( x,y,z,t \right)+$

$\frac{\partial }{\partial \,\,z}\left[ {{D}_{V}}\left( x,y,z,T \right)\frac{\partial \,V\,\left( x,y,z,t \right)}{\partial \,\,z} \right]-{{k}_{I,V}}\left( x,y,z,T \right)\,I\,\left( x,y,z,t \right)\,\,\times $

$I\,\left( x,y,z,t \right)\,V\,\left( x,y,z,t \right)+\frac{\partial }{\partial \,x}\left[ {{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{V\,\left( x,y,W,t \right)\,d\,W}\,\times  \right.$

$\left. \frac{{{D}_{VS}}}{k\,T} \right]\,\,\Omega +\Omega \frac{\partial }{\partial \,y}\left[ \frac{{{D}_{VS}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{V\,\left( x,y,W,t \right)\,d\,W} \right]+$

$\frac{\partial }{\partial \,x}\left[ \frac{{{D}_{V\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,x} \right]+\frac{\partial }{\partial \,y}\left[ \frac{{{D}_{V\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,y} \right]+$

$\frac{\partial }{\partial \,z}\left[ \frac{{{D}_{V\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,z} \right]$

with boundary and initial conditions

${{\left. \frac{\partial \,\,I\,\left( x,y,z,t \right)}{\partial \,x} \right|}_{x=0}}=0$, ${{\left. \frac{\partial \,\,I\,\left( x,y,z,t \right)}{\partial \,x} \right|}_{x={{L}_{x}}}}=0$,

${{\left. \frac{\partial \,\,I\,\left( x,y,z,t \right)}{\partial \,y} \right|}_{y=0}}=0$, ${{\left. \frac{\partial \,\,I\,\left( x,y,z,t \right)}{\partial \,y} \right|}_{y={{L}_{y}}}}=0$,

${{\left. \frac{\partial \,\,I\,\left( x,y,z,t \right)}{\partial \,z} \right|}_{z=0}}=0$, ${{\left. \frac{\partial \,\,I\,\left( x,y,z,t \right)}{\partial \,z} \right|}_{z={{L}_{z}}}}=0$,

${{\left. \frac{\partial \,\,V\,\left( x,y,z,t \right)}{\partial \,x} \right|}_{x=0}}=0$, ${{\left. \frac{\partial \,\,V\,\left( x,y,z,t \right)}{\partial \,x} \right|}_{x={{L}_{x}}}}=0$,

${{\left. \frac{\partial \,\,V\,\left( x,y,z,t \right)}{\partial \,y} \right|}_{y=0}}=0$, ${{\left. \frac{\partial \,\,V\,\left( x,y,z,t \right)}{\partial \,y} \right|}_{y={{L}_{y}}}}=0$, (5)

${{\left. \frac{\partial \,\,V\,\left( x,y,z,t \right)}{\partial \,z} \right|}_{z=0}}=0$, ${{\left. \frac{\partial \,\,V\,\left( x,y,z,t \right)}{\partial \,z} \right|}_{z={{L}_{z}}}}=0$,

$I(x, y, z, 0)=f_{I}(x, y, z), V(x, y, z, 0)=f_{V}(x, y, z)$

$V\,\left( {{x}_{1}}+{{V}_{n}}t,{{y}_{1}}+{{V}_{n}}t,{{z}_{1}}+{{V}_{n}}t,t \right)={{V}_{\infty }}\left( 1+\frac{2\,\ell \,\omega }{k\,T\,\sqrt{x_{1}^{2}+y_{1}^{2}+z_{1}^{2}}} \right)$.

Here $I (x,y,z,t)$ is the spatio-temporal distribution of concentration of radiation interstitials; I^* is the equilibrium distribution of interstitials; $D_I(x,y,z,T)$, $D_V(x,y,z,T)$, $D_{IS}(x,y,z,T)$, $D_{VS}(x,y,z,T)$ are the coefficients of volumetric and surficial diffusions of interstitials and vacancies, respectively; terms $V^2(x,y,z,t)$ and $I^2(x,y,z,t)$ correspond to generation of divacancies and diinterstitials, respectively (see, for example, [26] and appropriate references in this book); $k_{I,V}(x,y,z,T)$, $k{I,I}(x,y,z,T)$ and $k_{V,V}(x,y,z,T)$ are the parameters of recombination of point radiation defects and generation of their complexes; k is the Boltzmann constant; $\omega^{*}=a^{3}$, a is the interatomic distance; $\ell$ is the specific surface energy. To account porosity of buffer layers we assume, that porous are approximately cylindrical with average values $\mathrm{r}=\sqrt{\mathrm{x}_{1}^{2}+\mathrm{y}_{1}^{2}}$ and z₁ before annealing [23]. With time small pores decomposing on vacancies. The vacancies absorbing by larger pores [27]. With time large pores became larger due to absorbing the vacancies and became more spherical [27]. Distribution of concentration of vacancies in heterostructure, existing due to porosity, could be determined by summing on all pores, i.e.

$V\left( x,y,z,t \right)=\sum\limits_{i=0}^{l}{\sum\limits_{j=0}^{m}{\sum\limits_{k=0}^{n}{{{V}_{p}}\left( x+i\,\alpha ,y+j\,\beta ,\,z+k\,\chi ,t \right)}}}$,

$R=\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}$ (6)

Here $\alpha, \beta$ and $\chi$ are the average distances between centers of pores in directions x, y and z; l, m and n are the quantity of pores inappropriate directions.

Spatio-temporal distributions of divacancies $\Phi_{V} (x,y,z,t)$ and diinterstitials $\Phi_{I} (x,y,z,t)$ could be determined by solving the following system of equations [25, 26].

$\frac{\partial \,\,{{\Phi }_{I}}\left( x,y,z,t \right)}{\partial \,\,t}=\frac{\partial }{\partial \,x}\left[ {{D}_{{{\Phi }_{I}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{I}}\left( x,y,z,t \right)}{\partial \,x} \right]+$

$\frac{\partial }{\partial \,y}\left[ {{D}_{{{\Phi }_{I}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{I}}\left( x,y,z,t \right)}{\partial \,y} \right]+\frac{\partial }{\partial \,z}\left[ {{D}_{{{\Phi }_{I}}}}\left( x,y,z,T \right)\,\,{{\times }^{^{^{^{^{{}}}}}}} \right.$

$\left. \frac{\partial \,{{\Phi }_{I}}\left( x,y,z,t \right)}{\partial \,z} \right]+\Omega \frac{\partial }{\partial \,x}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{{{\Phi }_{I}}\left( x,y,W,t \right)\,d\,W} \right]+$

$\Omega \frac{\partial }{\partial \,y}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{{{\Phi }_{I}}\left( x,y,W,t \right)\,d\,W} \right]+$

$\frac{\partial }{\partial \,x}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,x} \right]+\frac{\partial }{\partial \,y}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,y} \right]+$

$\frac{\partial }{\partial \,z}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,z} \right]+{{k}_{I,I}}\left( x,y,z,T \right)\,{{I}^{2}}\left( x,y,z,t \right)+$

$\,{{k}_{I}}\left( x,y,z,T \right)\,I\,\left( x,y,z,t \right)$  (7)

$\frac{\partial \,\,{{\Phi }_{V}}\left( x,y,z,t \right)}{\partial \,\,t}=\frac{\partial }{\partial \,x}\left[ {{D}_{{{\Phi }_{V}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{V}}\left( x,y,z,t \right)}{\partial \,x} \right]+$

$\frac{\partial }{\partial \,y}\left[ {{D}_{{{\Phi }_{V}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{V}}\left( x,y,z,t \right)}{\partial \,y} \right]+\frac{\partial }{\partial \,z}\left[ {{D}_{{{\Phi }_{V}}}}\left( x,y,z,T \right)\,\,{{\times }^{^{^{^{^{{}}}}}}} \right.$

$\left.  \frac{\partial \,{{\Phi }_{V}}\left( x,y,z,t \right)}{\partial \,z} \right]+\frac{\partial }{\partial \,z}\left[ {{D}_{{{\Phi }_{V}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{V}}\left( x,y,z,t \right)}{\partial \,z} \right]+$

$\Omega \frac{\partial }{\partial \,x}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{{{\Phi }_{V}}\left( x,y,W,t \right)\,d\,W} \right]+$

$\Omega \frac{\partial }{\partial \,y}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{{{\Phi }_{V}}\left( x,y,W,t \right)\,d\,W} \right]+$

$\frac{\partial }{\partial \,x}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,x} \right]+\frac{\partial }{\partial \,y}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,y} \right]+$

$\frac{\partial }{\partial \,z}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,z} \right]+{{k}_{V,V}}\left( x,y,z,T \right)\,{{V}^{2}}\left( x,y,z,t \right)+$

$\,{{k}_{V}}\left( x,y,z,T \right)\,V\,\left( x,y,z,t \right)$

with boundary and initial conditions

${{\left. \frac{\partial \,\,{{\Phi }_{I}}\,\left( x,y,z,t \right)}{\partial \,x} \right|}_{x=0}}=0$, ${{\left. \frac{\partial \,\,I\,\left( x,y,z,t \right)}{\partial \,x} \right|}_{x={{L}_{x}}}}=0$,

${{\left. \frac{\partial \,\,I\,\left( x,y,z,t \right)}{\partial \,y} \right|}_{y=0}}=0$, ${{\left. \frac{\partial \,\,I\,\left( x,y,z,t \right)}{\partial \,y} \right|}_{y={{L}_{y}}}}=0$,

${{\left. \frac{\partial \,\,{{\Phi }_{I}}\,\left( x,y,z,t \right)}{\partial \,z} \right|}_{z=0}}=0$, ${{\left. \frac{\partial \,\,I\,\left( x,y,z,t \right)}{\partial \,z} \right|}_{z={{L}_{z}}}}=0$,

${{\left. \frac{\partial \,\,{{\Phi }_{V}}\,\left( x,y,z,t \right)}{\partial \,x} \right|}_{x=0}}=0$, ${{\left. \frac{\partial \,\,V\,\left( x,y,z,t \right)}{\partial \,x} \right|}_{x={{L}_{x}}}}=0$,

${{\left. \frac{\partial \,\,V\,\left( x,y,z,t \right)}{\partial \,y} \right|}_{y=0}}=0$, ${{\left. \frac{\partial \,\,V\,\left( x,y,z,t \right)}{\partial \,y} \right|}_{y={{L}_{y}}}}=0$, (8)

${{\left. \frac{\partial \,\,V\,\left( x,y,z,t \right)}{\partial \,z} \right|}_{z=0}}=0$, ${{\left. \frac{\partial \,\,{{\Phi }_{V}}\,\left( x,y,z,t \right)}{\partial \,z} \right|}_{z={{L}_{z}}}}=0$,

$\Phi_{\mathrm{I}}(\mathrm{x}, \mathrm{y}, z, 0)=\mathrm{f}_{\Phi \mathrm{I}}\left(\mathrm{x}, \mathrm{y}, \mathrm{z}\right), \Phi_{\mathrm{V}}(\mathrm{x}, \mathrm{y}, \mathrm{z}, 0)=\mathrm{f}_{\Phi \mathrm{V}}\left(\mathrm, \mathrm{y}, \mathrm{z}\right)$

Here $\mathrm{D}_{\Phi \mathrm{I}}(\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{T}), \mathrm{D}_{\Phi \mathrm{V}}(\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{T}), \mathrm{D}_{\Phi \mathrm{IS}}(\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{T})$ and $\mathrm{D}_{\Phi \mathrm{VS}}(\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{T})$ are the coefficients of volumetric and surficial diffusions of complexes of radiation defects; k_I(x,y,z,T) and k_V(x,y, z,T) are the parameters of decay of complexes of radiation defects.

Chemical potential $\mu_{1}$ in Eq. (1) could be determine by the following relation [20].

$\mu_{1}=\mathrm{E}(z) \Omega \sigma_{\mathrm{ij}}\left[\mathrm{u}_{\mathrm{ij}}(\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{t})+\mathrm{u}_{\mathrm{ji}}(\mathrm{x}, \mathrm{y}, \mathrm{z}, \mathrm{t})\right] / 2$ (9)

where, E(z) is the Young modulus, $\sigma_{\text {ij }}$ is the stress tensor; ${{u}_{ij}}=\frac{1}{2}\left( \frac{\partial \,{{u}_{i}}}{\partial \,{{x}_{j}}}+\frac{\partial \,{{u}_{j}}}{\partial \,{{x}_{i}}} \right)$ is the deformation tensor; u_i, u_j are the components u_x(x,y,z,t), u_y(x,y,z,t) and u_z(x,y,z,t) of the displacement vector $\vec{u}\,\left( x,y,z,t \right)$; x_i, x_j are the coordinate x, y, z. The Eq. (3) could be transform to the following form

$\mu \,\,\left( x,y,z,t \right)=E\,\left( z \right)\frac{\Omega }{2}\,\left[ \frac{\partial \,{{u}_{i}}\left( x,y,z,t \right)}{\partial \,{{x}_{j}}}+\frac{\partial \,{{u}_{j}}\left( x,y,z,t \right)}{\partial \,{{x}_{i}}} \right]\,\,\,\times $

$ \,\,\left\{ \frac{1}{2}\left[ \frac{\partial \,{{u}_{i}}\left( x,y,z,t \right)}{\partial \,{{x}_{j}}}+\frac{\partial \,{{u}_{j}}\left( x,y,z,t \right)}{\partial \,{{x}_{i}}} \right]-{{\varepsilon }_{0}}{{\delta }_{ij}}+\frac{\sigma \,\left( z \right)\,{{\delta }_{ij}}}{1-2\,\sigma \,\left( z \right)} \right.\times $

$ \,\,\left. \left[ \frac{\partial \,{{u}_{k}}\left( x,y,z,t \right)}{\partial \,{{x}_{k}}}-3\,{{\varepsilon }_{0}} \right]-K\,\left( z \right)\,\beta \,\left( z \right)\,\left[ T\,\left( x,y,z,t \right)-{{T}_{0}} \right]\,\,{{\delta }_{ij}} \right\}$  (10)

where, $\sigma$ is Poisson coefficient; $\varepsilon_{0}=\left(a_{s}-a_{E L}\right) / a_{E L}$ is the mismatch parameter; $a_{s}, \quad a_{E L}$ are lattice distances of the substrate and the epitaxial layer; K is the modulus of uniform compression; $\beta$ is the coefficient of thermal expansion; T_r is the equilibrium temperature, which coincide (for our case) with room temperature. Components of displacement vector could be obtained by solution of the following equations [21].

$\left\{ \begin{align}  & \rho \,\,\left( z \right)\frac{{{\partial }^{2}}\,{{u}_{x}}\left( x,y,z,t \right)}{\partial \,{{t}^{2}}}= \\ & \,\,\,\,\,=\frac{\partial \,{{\sigma }_{xx}}\left( x,y,z,t \right)}{\partial \,x}+\frac{\partial \,{{\sigma }_{xy}}\left( x,y,z,t \right)}{\partial \,y}+\frac{\partial \,{{\sigma }_{xz}}\left( x,y,z,t \right)}{\partial \,z} \\ & \rho \,\,\left( z \right)\frac{{{\partial }^{2}}\,{{u}_{y}}\left( x,y,z,t \right)}{\partial \,{{t}^{2}}}= \\ & \,\,\,\,\,=\frac{\partial \,{{\sigma }_{yx}}\left( x,y,z,t \right)}{\partial \,x}+\frac{\partial \,{{\sigma }_{yy}}\left( x,y,z,t \right)}{\partial \,y}+\frac{\partial \,{{\sigma }_{yz}}\left( x,y,z,t \right)}{\partial \,z} \\ & \rho \,\,\left( z \right)\frac{{{\partial }^{2}}\,{{u}_{z}}\left( x,y,z,t \right)}{\partial \,{{t}^{2}}}= \\ & \,\,\,\,\,=\frac{\partial \,{{\sigma }_{zx}}\left( x,y,z,t \right)}{\partial \,x}+\frac{\partial \,{{\sigma }_{zy}}\left( x,y,z,t \right)}{\partial \,y}+\frac{\partial \,{{\sigma }_{zz}}\left( x,y,z,t \right)}{\partial \,z}, \\\end{align} \right.$  (11)

where, ${{\sigma }_{ij}}=\left[ \frac{\partial \,{{u}_{i}}\left( x,y,z,t \right)}{\partial \,{{x}_{j}}}-\frac{{{\delta }_{ij}}}{3}\frac{\partial \,{{u}_{k}}\left( x,y,z,t \right)}{\partial \,{{x}_{k}}}+\frac{\partial \,{{u}_{j}}\left( x,y,z,t \right)}{\partial \,{{x}_{i}}} \right]\,\,\times $$\frac{E\,\left( z \right)}{2\,\left[ 1+\sigma \,\left( z \right) \right]}+K\,\left( z \right)\,\,{{\delta }_{ij}}\frac{\partial \,{{u}_{k}}\left( x,y,z,t \right)}{\partial \,{{x}_{k}}}-\,\,\left[ T\left( x,y,z,t \right)-{{T}_{r}} \right]\,\,\times  \,\beta \,\left( z \right)\,K\,\left( z \right)$, $\rho(z)$ is the density of materials of heterostructure, $\delta_{i j}$ Is the Kronecker symbol. With account the relation for $\sigma_{i j}$ last system of equation could be written as

$\rho \,\left( z \right)\frac{{{\partial }^{2}}\,{{u}_{x}}\left( x,y,z,t \right)}{\partial \,{{t}^{2}}}=\left\{ K\,\left( z \right)+\frac{5E\,\left( z \right)}{6\,\left[ 1+\sigma \,\left( z \right) \right]} \right\}\frac{{{\partial }^{2}}{{u}_{x}}\left( x,y,z,t \right)}{\partial \,{{x}^{2}}}+$

$\left\{ K\,\left( z \right)-\frac{E\,\left( z \right)}{3\,\left[ 1+\sigma \,\left( z \right) \right]} \right\}\frac{{{\partial }^{2}}{{u}_{y}}\left( x,y,z,t \right)}{\partial \,x\,\partial \,y}+\frac{E\,\left( z \right)}{2\,\left[ 1+\sigma \,\left( z \right) \right]}\times $

$ \left[ \frac{{{\partial }^{2}}{{u}_{y}}\left( x,y,z,t \right)}{\partial \,{{y}^{2}}}+\frac{{{\partial }^{2}}{{u}_{z}}\left( x,y,z,t \right)}{\partial \,{{z}^{2}}} \right]+\left[ K\,\left( z \right)+\frac{E\,\left( z \right)}{3\left[ 1+\sigma \,\left( z \right) \right]} \right]\times $

$ \frac{{{\partial }^{2}}{{u}_{z}}\left( x,y,z,t \right)}{\partial \,x\,\partial \,z}-K\,\left( z \right)\,\beta \,\left( z \right)\frac{\partial \,T\,\left( x,y,z,t \right)}{\partial \,x}$

$\rho \,\left( z \right)\frac{{{\partial }^{2}}\,{{u}_{y}}\left( x,y,z,t \right)}{\partial \,{{t}^{2}}}=\left[ \frac{{{\partial }^{2}}{{u}_{y}}\left( x,y,z,t \right)}{\partial \,{{x}^{2}}}+\frac{{{\partial }^{2}}{{u}_{x}}\left( x,y,z,t \right)}{\partial \,x\,\partial \,y} \right]\,\,\times $

$\frac{E\,\left( z \right)}{2\,\left[ 1+\sigma \,\left( z \right) \right]}-\,K\,\left( z \right)\,\beta \,\left( z \right)\frac{\partial \,T\,\left( x,y,z,t \right)}{\partial \,y}+\frac{\partial }{\partial \,z}\left\{ \frac{E\,\left( z \right)}{2\,\left[ 1+\sigma \,\left( z \right) \right]}\times  \right.$

$\left.  \,\,\left[ \frac{\partial \,{{u}_{y}}\left( x,y,z,t \right)}{\partial \,z}+\frac{\partial \,{{u}_{z}}\left( x,y,z,t \right)}{\partial \,y} \right] \right\}+\frac{{{\partial }^{2}}{{u}_{y}}\left( x,y,z,t \right)}{\partial \,{{y}^{2}}}\times $

$\frac{{{\partial }^{2}}{{u}_{y}}\left( x,y,z,t \right)}{\partial \,{{y}^{2}}}\left\{ \frac{5E\,\left( z \right)}{12\,\left[ 1+\sigma \,\left( z \right) \right]}+K\,\left( z \right) \right\}+\frac{{{\partial }^{2}}{{u}_{y}}\left( x,y,z,t \right)}{\partial \,y\,\partial \,z}\times $

$ \left\{ K\,\left( z \right)-\frac{E\,\left( z \right)}{6\,\left[ 1+\sigma \,\left( z \right) \right]} \right\}+K\,\left( z \right)\frac{{{\partial }^{2}}{{u}_{y}}\left( x,y,z,t \right)}{\partial \,x\,\partial \,y}$ (12)

$\rho \,\left( z \right)\frac{{{\partial }^{2}}\,{{u}_{z}}\left( x,y,z,t \right)}{\partial \,{{t}^{2}}}=\frac{E\,\left( z \right)}{2\,\left[ 1+\sigma \,\left( z \right) \right]}\left[ \frac{{{\partial }^{2}}{{u}_{z}}\left( x,y,z,t \right)}{\partial \,{{x}^{2}}}+ \right.$

$\left. \frac{{{\partial }^{2}}{{u}_{z}}\left( x,y,z,t \right)}{\partial \,{{y}^{2}}}+\frac{{{\partial }^{2}}{{u}_{x}}\left( x,y,z,t \right)}{\partial \,x\,\partial \,z}+\frac{{{\partial }^{2}}{{u}_{y}}\left( x,y,z,t \right)}{\partial \,y\,\partial \,z} \right]+$

$\frac{\partial }{\partial \,z}\left\{ K\,\left( z \right)\,\left[ \frac{\partial \,{{u}_{x}}\left( x,y,z,t \right)}{\partial \,x}+\frac{\partial \,{{u}_{y}}\left( x,y,z,t \right)}{\partial \,y}+\frac{\partial \,{{u}_{x}}\left( x,y,z,t \right)}{\partial \,z} \right] \right\}+$

$\frac{1}{6}\frac{\partial }{\partial \,z}\left\{ \left[ 6\frac{\partial \,{{u}_{z}}\left( x,y,z,t \right)}{\partial \,z}-\frac{\partial \,{{u}_{x}}\left( x,y,z,t \right)}{\partial \,x}-\frac{\partial \,{{u}_{y}}\left( x,y,z,t \right)}{\partial \,y}- \right. \right.$

$\left. \left. -\frac{\partial \,{{u}_{z}}\left( x,y,z,t \right)}{\partial \,z} \right]\frac{E\,\left( z \right)}{1+\sigma \,\left( z \right)} \right\}-K\,\left( z \right)\,\beta \,\left( z \right)\frac{\partial \,T\,\left( x,y,z,t \right)}{\partial \,z}$.

Conditions for the system of Eq. (8) could be written in the form

$\frac{\partial \,\,\vec{u}\left( 0,y,z,t \right)}{\partial \,x}=0$; $\frac{\partial \,\vec{u}\,\left( {{L}_{x}},y,z,t \right)}{\partial \,x}=0$; $\frac{\partial \,\,\vec{u}\,\left( x,0,z,t \right)}{\partial \,y}=0$;

$\frac{\partial \,\,\vec{u}\,\left( x,{{L}_{y}},z,t \right)}{\partial \,y}=0$; $\frac{\partial \,\,\vec{u}\,\left( x,y,0,t \right)}{\partial \,z}=0$; $\frac{\partial \,\,\vec{u}\,\left( x,y,{{L}_{z}},t \right)}{\partial \,z}=0; \vec{u}\,\left( x,y,z,0 \right)={{\vec{u}}_{0}}$; $\vec{u}\,\left( x,y,z,\infty  \right)={{\vec{u}}_{0}}$ (13)

We determine spatio-temporal distributions of concentrations of dopant and radiation defects by solving the Eqns. (1), (4) and (7) framework standard method of averaging of function corrections [28]. Previously we transform the Eqns. (1), (4) and (7) to the following form with account initial distributions of the considered concentrations.

$\frac{\partial \,C\left( x,y,z,t \right)}{\partial \,t}=\frac{\partial \,}{\partial \,x}\left[ D\frac{\partial \,C\,\left( x,y,z,t \right)}{\partial \,x} \right]+\frac{\partial \,}{\partial \,y}\left[ D\frac{\partial \,C\,\left( x,y,z,t \right)}{\partial \,y} \right]+$

$\frac{\partial \,}{\partial \,z}\left[ D\frac{\partial \,C\,\left( x,y,z,t \right)}{\partial \,z} \right]+\frac{\partial }{\partial \,\,x}\left[ \frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,x} \right]+$  

$\frac{\partial }{\partial \,\,y}\left[ \frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,y} \right]+\frac{\partial }{\partial \,\,z}\left[ \frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,z} \right]+$

$\Omega \frac{\partial }{\partial \,x}\left[ \frac{{{D}_{S}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{C\,\left( x,y,W,t \right)\,d\,W} \right]+$

$\Omega \frac{\partial }{\partial \,y}\left[ \frac{{{D}_{S}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{C\,\left( x,y,W,t \right)\,d\,W} \right]$ (1a)

$\frac{\partial \,\,I\,\left( x,y,z,t \right)}{\partial \,\,t}=\frac{\partial }{\partial \,\,x}\left[ {{D}_{I}}\left( x,y,z,T \right)\frac{\partial \,I\,\left( x,y,z,t \right)}{\partial \,x} \right]+$

$\frac{\partial }{\partial \,\,y}\left[ {{D}_{I}}\left( x,y,z,T \right)\frac{\partial \,I\,\left( x,y,z,t \right)}{\partial \,y} \right]+\frac{\partial }{\partial \,\,z}\left[ {{D}_{I}}\left( x,y,z,T \right)\,\,{{\times }^{^{^{^{{}}}}}} \right.$

$\left. \frac{\partial \,I\,\left( x,y,z,t \right)}{\partial \,\,z} \right]+\Omega \frac{\partial }{\partial \,x}\left[ \frac{{{D}_{IS}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{I\,\left( x,y,W,t \right)\,d\,W} \right]+$

$\Omega \frac{\partial }{\partial \,y}\left[ \frac{{{D}_{IS}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{I\,\left( x,y,W,t \right)\,\,d\,\,W} \right]-{{I}^{2}}\left( x,y,z,t \right)\,\,\times $

$ \,\,{{k}_{I,I}}\left( x,y,z,T \right)-{{k}_{I,V}}\left( x,y,z,T \right)\,I\,\left( x,y,z,t \right)\,\,V\,\left( x,y,z,t \right)+{{f}_{I}}\,\left( x,y,z \right)\,\delta \,\left( t \right)$  (4a)

$\frac{\partial \,\,V\,\left( x,y,z,t \right)}{\partial \,\,t}=\frac{\partial }{\partial \,\,x}\left[ {{D}_{V}}\left( x,y,z,T \right)\frac{\partial \,V\,\left( x,y,z,t \right)}{\partial \,x} \right]+$

$\frac{\partial }{\partial \,\,y}\left[ {{D}_{V}}\left( x,y,z,T \right)\frac{\partial \,V\,\left( x,y,z,t \right)}{\partial \,y} \right]+\frac{\partial }{\partial \,\,z}\left[ {{D}_{V}}\left( x,y,z,T \right)\,\,{{\times }^{^{^{^{{}}}}}} \right.$

$\left.  \frac{\partial \,V\,\left( x,y,z,t \right)}{\partial \,\,z} \right]+\Omega \frac{\partial }{\partial \,x}\left[ \frac{{{D}_{VS}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{V\,\left( x,y,W,t \right)\,d\,W} \right]+$

$\Omega \frac{\partial }{\partial \,y}\left[ \frac{{{D}_{IS}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{I\,\left( x,y,W,t \right)\,d\,W} \right]-{{I}^{2}}\left( x,y,z,t \right)\,\,\times $

$\,\,{{k}_{I,I}}\left( x,y,z,T \right)-{{k}_{I,V}}\left( x,y,z,T \right)\,I\,\left( x,y,z,t \right)\,V\,\left( x,y,z,t \right)+{{f}_{V}}\,\left( x,y,z \right)\,\,\delta \left( t \right)$

$\frac{\partial \,\,{{\Phi }_{I}}\left( x,y,z,t \right)}{\partial \,\,t}=\frac{\partial }{\partial \,x}\left[ {{D}_{{{\Phi }_{I}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{I}}\left( x,y,z,t \right)}{\partial \,x} \right]+$

$\frac{\partial }{\partial \,y}\left[ {{D}_{{{\Phi }_{I}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{I}}\left( x,y,z,t \right)}{\partial \,y} \right]+\frac{\partial }{\partial \,z}\left[ {{D}_{{{\Phi }_{I}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{I}}\left( x,y,z,t \right)}{\partial \,z} \right]+$

$\Omega \frac{\partial }{\partial \,x}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{{{\Phi }_{I}}\left( x,y,W,t \right)\,d\,W} \right]+$

$\Omega \frac{\partial }{\partial \,y}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{{{\Phi }_{I}}\left( x,y,W,t \right)\,d\,W} \right]+$

$\frac{\partial }{\partial \,x}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,x} \right]+\frac{\partial }{\partial \,y}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,y} \right]+$

$\frac{\partial }{\partial \,z}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,z} \right]+\,{{k}_{I}}\left( x,y,z,T \right)\,\,I\,\left( x,y,z,t \right)+{{k}_{I,I}}\left( x,y,z,T \right)\,{{I}^{2}}\left( x,y,z,t \right)+{{f}_{{{\Phi }_{I}}}}\,\left( x,y,z \right)\,\delta \left( t \right)$  (7a)

$\frac{\partial \,\,{{\Phi }_{V}}\left( x,y,z,t \right)}{\partial \,\,t}=\frac{\partial }{\partial \,x}\left[ {{D}_{\Phi {}_{V}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{V}}\left( x,y,z,t \right)}{\partial \,x} \right]+\frac{\partial \,\,{{\Phi }_{V}}\left( x,y,z,t \right)}{\partial \,\,t}=\frac{\partial }{\partial \,x}\left[ {{D}_{\Phi {}_{V}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{V}}\left( x,y,z,t \right)}{\partial \,x} \right]+$

$\frac{\partial }{\partial \,z}\left[ {{D}_{{{\Phi }_{V}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{V}}\left( x,y,z,t \right)}{\partial \,z} \right]+\Omega \frac{\partial }{\partial \,x}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{{{\Phi }_{V}}\left( x,y,W,t \right)\,d\,W} \right]+$

$\Omega \frac{\partial }{\partial \,y}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{{{\Phi }_{I}}\left( x,y,W,t \right)\,d\,W} \right]+\frac{\partial }{\partial \,x}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,x} \right]+\frac{\partial }{\partial \,y}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,y} \right]+$

$\frac{\partial }{\partial \,z}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,z} \right]+\,{{k}_{I}}\left( x,y,z,T \right)\,I\,\left( x,y,z,t \right)+{{k}_{V,V}}\left( x,y,z,T \right)\,{{V}^{2}}\left( x,y,z,t \right)+{{f}_{{{\Phi }_{V}}}}\,\left( x,y,z \right)\,\delta \left( t \right)$.

Farther we replace concentrations of dopant and radiation defects in right sides of Eqns. (1a), (4a) and (7a) on their not yet known average values $\alpha_{1} \rho$. In this situation we obtain equations for the first-order approximations of the required concentrations in the following form

$\frac{\partial \,{{C}_{1}}\left( x,y,z,t \right)}{\partial \,t}={{\alpha }_{1C}}\Omega \frac{\partial }{\partial \,x}\left[ z\frac{{{D}_{S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right) \right]+{{\alpha }_{1C}}\Omega \frac{\partial }{\partial \,y}\left[ z\frac{{{D}_{S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right) \right]+{{f}_{C}}\,\left( x,y,z \right)\,\delta \,\left( t \right)+$

$\frac{\partial }{\partial \,\,x}\left[ \frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,x} \right]+\frac{\partial }{\partial \,\,y}\left[ \frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,y} \right]+\frac{\partial }{\partial \,\,z}\left[ \frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,z} \right]$  (1b)

$\frac{\partial \,\,{{I}_{1}}\left( x,y,z,t \right)}{\partial \,\,t}={{\alpha }_{1I}}z\,\Omega \frac{\partial }{\partial \,x}\left[ \frac{{{D}_{IS}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right) \right]+{{\alpha }_{1I}}\Omega \frac{\partial }{\partial \,y}\left[ z\frac{{{D}_{IS}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right) \right]+\frac{\partial }{\partial \,x}\left[ \frac{{{D}_{I\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,x} \right]+$

$\frac{\partial }{\partial \,y}\left[ \frac{{{D}_{I\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,y} \right]+\frac{\partial }{\partial \,z}\left[ \frac{{{D}_{I\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,z} \right]+{{f}_{I}}\,\left( x,y,z \right)\,\delta \left( t \right)-\alpha _{1I}^{2}{{k}_{I,I}}\left( x,y,z,T \right)-{{\alpha }_{1I}}{{\alpha }_{1V}}{{k}_{I,V}}\left( x,y,z,T \right)$

$\frac{\partial \,\,{{V}_{1}}\left( x,y,z,t \right)}{\partial \,\,t}={{\alpha }_{1V}}z\,\Omega \frac{\partial }{\partial \,x}\left[ \frac{{{D}_{VS}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right) \right]+$

${{\alpha }_{1V}}\Omega \frac{\partial }{\partial \,y}\left[ z\frac{{{D}_{VS}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right) \right]+\frac{\partial }{\partial \,x}\left[ \frac{{{D}_{V\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,x} \right]+$

$\frac{\partial }{\partial \,y}\left[ \frac{{{D}_{V\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,y} \right]+\frac{\partial }{\partial \,z}\left[ \frac{{{D}_{V\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,z} \right]+$

${{f}_{V}}\,\left( x,y,z \right)\,\delta \left( t \right)-\alpha _{1V}^{2}{{k}_{V,V}}\left( x,y,z,T \right)-{{\alpha }_{1I}}{{\alpha }_{1V}}{{k}_{I,V}}\left( x,y,z,T \right)$  (1b)

$\frac{\partial \,\,{{\Phi }_{1I}}\left( x,y,z,t \right)}{\partial \,\,t}={{\alpha }_{1{{\Phi }_{I}}}}z\,\Omega \frac{\partial }{\partial \,x}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right) \right]+$

$\frac{\partial }{\partial \,x}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,x} \right]+{{\alpha }_{1{{\Phi }_{I}}}}\frac{\partial }{\partial \,y}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right) \right]\,\,\times $

$ z\,\Omega +\frac{\partial }{\partial \,y}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,y} \right]+\frac{\partial }{\partial \,z}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,z} \right]+{{k}_{I}}\left( x,y,z,T \right)\,\,I\,\left( x,y,z,t \right)+{{k}_{I,I}}\left( x,y,z,T \right)\,{{I}^{2}}\left( x,y,z,t \right)+{{f}_{{{\Phi }_{I}}}}\,\left( x,y,z \right)\,\delta \left( t \right)$  (7b)

$\frac{\partial \,\,{{\Phi }_{1V}}\left( x,y,z,t \right)}{\partial \,\,t}={{\alpha }_{1{{\Phi }_{V}}}}z\,\Omega \frac{\partial }{\partial \,x}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right) \right]+$

$\frac{\partial }{\partial \,x}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,x} \right]+\,\,z\,\Omega \frac{\partial }{\partial \,y}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right) \right]\,\,\times $

$ {{\alpha }_{1{{\Phi }_{V}}}}+\frac{\partial }{\partial \,y}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,y} \right]+\frac{\partial }{\partial \,z}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,z} \right]+$

${{k}_{V}}\left( x,y,z,T \right)\,V\,\left( x,y,z,t \right)+{{k}_{V,V}}\left( x,y,z,T \right)\,{{V}^{2}}\left( x,y,z,t \right)+{{f}_{{{\Phi }_{V}}}}\,\left( x,y,z \right)\,\delta \left( t \right)$.

Integration of the left and right sides of the Eqns. (1b), (4b) and (7b) on time gives us possibility to obtain relations for above approximation in the final form

${{C}_{1}}\left( x,y,z,t \right)={{\alpha }_{1C}}\Omega \frac{\partial }{\partial \,x}\int\limits_{0}^{t}{{{D}_{S\,L}}\left( x,y,z,T \right)\,\left[ 1+{{\varsigma }_{1}}\frac{V\,\left( x,y,z,\tau  \right)}{{{V}^{*}}}+ \right.}$

$\left. {{\varsigma }_{2}}\frac{{{V}^{2}}\left( x,y,z,\tau  \right)}{{{\left( {{V}^{*}} \right)}^{2}}} \right]\,{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,\tau  \right)\,\left[ 1+\frac{{{\xi }_{S}}\alpha _{1C}^{\gamma }}{{{P}^{\gamma }}\left( x,y,z,T \right)} \right]\,\,\times $

$ \frac{z}{k\,T}d\,\tau +{{\alpha }_{1C}}\Omega \frac{\partial }{\partial \,y}\int\limits_{0}^{t}{{{D}_{S\,L}}\left( x,y,z,T \right)\,\left[ 1+\frac{{{\xi }_{S}}\alpha _{1C}^{\gamma }}{{{P}^{\gamma }}\left( x,y,z,T \right)} \right]\,\,\times }\,{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,\tau  \right)\,\left[ 1+{{\varsigma }_{1}}\frac{V\,\left( x,y,z,\tau  \right)}{{{V}^{*}}}+{{\varsigma }_{2}}\frac{{{V}^{2}}\left( x,y,z,\tau  \right)}{{{\left( {{V}^{*}} \right)}^{2}}} \right]\,\,\times $

$ \frac{z}{k\,T}d\,\tau +\frac{\partial }{\partial \,\,x}\int\limits_{0}^{t}{\frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,\,x}\,d\,\tau }+\frac{\partial }{\partial \,\,y}\int\limits_{0}^{t}{\frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}}\times $

$\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,\,y}\,d\,\tau +\frac{\partial }{\partial \,\,z}\int\limits_{0}^{t}{\frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,\,z}\,d\,\tau }+{{f}_{C}}\,\left( x,y,z \right)$   (1c)

${{I}_{1}}\left( x,y,z,t \right)={{\alpha }_{1I}}z\,\Omega \frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{IS}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,\tau  \right)\,d\,\tau }+{{\alpha }_{1I}}z\,\Omega \,\,\times $

$ \frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\frac{{{D}_{IS}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,\tau  \right)\,\,d\,\tau }+\frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{I\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,x}d\,\tau }+$

$\frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\frac{{{D}_{I\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,y}d\,\tau }+\frac{\partial }{\partial \,z}\int\limits_{0}^{t}{\frac{{{D}_{I\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,z}d\,\tau }-$

$\alpha _{1I}^{2}\int\limits_{0}^{t}{{{k}_{I,I}}\left( x,y,z,T \right)\,d\,\tau }-{{\alpha }_{1I}}{{\alpha }_{1V}}\int\limits_{0}^{t}{{{k}_{I,V}}\left( x,y,z,T \right)\,\,d\,\tau }+{{f}_{I}}\,\left( x,y,z \right)$   (1c)

${{V}_{1}}\left( x,y,z,t \right)={{\alpha }_{1V}}z\,\Omega \frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{IS}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,\tau  \right)\,d\,\tau }+{{\alpha }_{1V}}z\,\Omega \,\,\times $

$ \frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\frac{{{D}_{IS}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,\tau  \right)\,\,d\,\tau }+\frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{V\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,x}d\,\tau }+$

$\frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\frac{{{D}_{V\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,y}d\,\tau }+\frac{\partial }{\partial \,z}\int\limits_{0}^{t}{\frac{{{D}_{V\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,z}d\,\tau }-$

$-\alpha _{1V}^{2}\int\limits_{0}^{t}{{{k}_{V,V}}\left( x,y,z,T \right)\,d\,\tau }-{{\alpha }_{1I}}{{\alpha }_{1V}}\int\limits_{0}^{t}{{{k}_{I,V}}\left( x,y,z,T \right)\,\,d\,\tau }+{{f}_{V}}\,\left( x,y,z \right)$

${{\Phi }_{1I}}\left( x,y,z,t \right)={{\alpha }_{1{{\Phi }_{I}}}}z\,\Omega \frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{I}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,\tau  \right)\,\,d\,\tau }+{{\alpha }_{1{{\Phi }_{I}}}}z\,\,\Omega \frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{I}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,\tau  \right)\,\,d\,\tau }+{{f}_{{{\Phi }_{I}}}}\,\left( x,y,z \right)+$

$+\frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,x}d\,\tau }+\frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,y}d\,\tau }+$

$+\frac{\partial }{\partial \,z}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,z}d\,\tau }+\int\limits_{0}^{t}{{{k}_{I}}\left( x,y,z,T \right)\,I\,\left( x,y,z,\tau  \right)\,d\,\tau }+\int\limits_{0}^{t}{{{k}_{I,I}}\left( x,y,z,T \right)\,{{I}^{2}}\left( x,y,z,\tau  \right)\,d\,\tau }$   (7c)

${{\Phi }_{1V}}\left( x,y,z,t \right)={{\alpha }_{1{{\Phi }_{V}}}}z\,\Omega \frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{V}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,\tau  \right)\,\,d\,\tau }+\,{{\alpha }_{1{{\Phi }_{V}}}}\Omega \,\,\times $$\,\times \,\,z\frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{V}}S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,\tau  \right)\,\,d\,\tau }+{{f}_{{{\Phi }_{V}}}}\,\left( x,y,z \right)+\frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}}\times $

$\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,x}d\,\tau +\frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,y}d\,\tau }+\frac{\partial }{\partial \,z}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,z}d\,\tau }+\int\limits_{0}^{t}{{{k}_{V}}\left( x,y,z,T \right)\,\,V\,\left( x,y,z,\tau  \right)\,\,d\,\tau }+$

$+\int\limits_{0}^{t}{{{k}_{V,V}}\left( x,y,z,T \right)\,{{V}^{2}}\left( x,y,z,\tau  \right)\,d\,\tau }$.

We determine average values of the first-order approximations of concentrations of dopant and radiation defects by the following standard relation [28].

${{\alpha }_{1\rho }}=\frac{1}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}\int\limits_{0}^{\Theta }{\int\limits_{0}^{{{L}_{x}}}{\int\limits_{0}^{{{L}_{y}}}{\int\limits_{0}^{{{L}_{z}}}{{{\rho }_{1}}\left( x,y,z,t \right)\,d\,z\,d\,y\,d\,x\,d\,t}}}}$.  (14)

Substitution of the relations (1c), (4c) and (7c) into relation (9) gives us possibility to obtain required average values in the following form

${{\alpha }_{1C}}=\frac{1}{{{L}_{x}}{{L}_{y}}{{L}_{z}}}\int\limits_{0}^{{{L}_{x}}}{\int\limits_{0}^{{{L}_{y}}}{\int\limits_{0}^{{{L}_{z}}}{{{f}_{C}}\left( x,y,z \right)\,d\,z\,d\,y\,d\,x}}}$, ${{\alpha }_{1V}}=\frac{1}{{{S}_{IV\,00}}}\times $

$ \,\,\left[ \frac{\Theta }{{{\alpha }_{1I}}}\int\limits_{0}^{{{L}_{x}}}{\int\limits_{0}^{{{L}_{y}}}{\int\limits_{0}^{{{L}_{z}}}{{{f}_{I}}\,\left( x,y,z \right)\,\,d\,z\,d\,y\,d\,x}}}-{{\alpha }_{1I}}{{S}_{II\,00}}-\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}} \right]$,  (15)

${{\alpha }_{1I}}=\sqrt{\frac{{{\left( {{a}_{3}}+A \right)}^{2}}}{4\,a_{4}^{2}}-4\,\,\left( B+\frac{\Theta \,{{a}_{3}}B+{{\Theta }^{2}}{{L}_{x}}{{L}_{y}}{{L}_{z}}{{a}_{1}}}{{{a}_{4}}} \right)}-\frac{{{a}_{3}}+A}{4\,{{a}_{4}}}$,

where,

${{S}_{\rho \rho \,\,ij}}=\int\limits_{0}^{\Theta }{\int\limits_{0}^{{{L}_{x}}}{\int\limits_{0}^{{{L}_{y}}}{\int\limits_{0}^{{{L}_{z}}}{{{k}_{\rho ,\rho }}\left( x,y,z,T \right)\,V_{1}^{j}\left( x,y,z,t \right)\,I_{1}^{i}\left( x,y,z,t \right)\,\,d\,z\,d\,y\,d\,x\,\,\times }}}}$

$ \,\,\left( \Theta -t \right)\,d\,t$, ${{a}_{4}}={{S}_{II\,00}}\left( S_{IV\,00}^{2}-{{S}_{II\,00}}{{S}_{VV\,00}} \right)$, ${{a}_{2}}={{S}_{IV00}}\Theta \,L_{x}^{2}\,\times  \,L_{y}^{2}L_{z}^{2}+{{S}_{IV00}}S_{IV00}^{2}\int\limits_{0}^{{{L}_{x}}}{\int\limits_{0}^{{{L}_{y}}}{\int\limits_{0}^{{{L}_{z}}}{{{f}_{V}}\,\left( x,y,z \right)\,d\,z\,d\,y\,d\,x}}}+2{{S}_{VV00}}{{S}_{II00}}\times $

$ \int\limits_{0}^{{{L}_{x}}}{\int\limits_{0}^{{{L}_{y}}}{\int\limits_{0}^{{{L}_{z}}}{{{f}_{I}}\,\left( x,y,z \right)\,d\,z\,d\,y\,d\,x}}}-\Theta \,L_{x}^{2}L_{y}^{2}L_{z}^{2}{{S}_{VV00}}-S_{IV00}^{2}\int\limits_{0}^{{{L}_{x}}}{\int\limits_{0}^{{{L}_{y}}}{\int\limits_{0}^{{{L}_{z}}}{{{f}_{I}}\,\left( x,y,z \right)\,d\,z\,d\,y\,d\,x}}}$, ${{a}_{3}}={{S}_{IV00}}{{S}_{II00}}+S_{IV00}^{2}-$

${{S}_{II00}}{{S}_{VV00}}$, ${{a}_{1}}={{S}_{IV00}}\int\limits_{0}^{{{L}_{x}}}{\int\limits_{0}^{{{L}_{y}}}{\int\limits_{0}^{{{L}_{z}}}{{{f}_{I}}\,\left( x,y,z \right)\,d\,z\,d\,y\,d\,x}}}$, ${{a}_{0}}={{S}_{VV00}}\times  \,{{\left[ \int\limits_{0}^{{{L}_{x}}}{\int\limits_{0}^{{{L}_{y}}}{\int\limits_{0}^{{{L}_{z}}}{{{f}_{I}}\,\left( x,y,z \right)\,d\,z\,d\,y\,d\,x}}} \right]}^{2}}$, $A=\sqrt{8\,y+{{\Theta }^{2}}\frac{a_{3}^{2}}{a_{4}^{2}}-4\,\Theta \,\frac{{{a}_{2}}}{{{a}_{4}}}}$,  (15)

$B=\frac{\Theta \,{{a}_{2}}}{6\,{{a}_{4}}}+\sqrt[3]{\sqrt{{{q}^{2}}+{{p}^{3}}}-q}-\sqrt[3]{\sqrt{{{q}^{2}}+{{p}^{3}}}+q}$, $q=\frac{{{\Theta }^{3}}{{a}_{2}}}{\,24\,a_{4}^{2}}\times  \,\left( 4{{a}_{0}}-\Theta \,{{L}_{x}}{{L}_{y}}{{L}_{z}}\frac{{{a}_{1}}{{a}_{3}}}{{{a}_{4}}} \right)-{{\Theta }^{2}}\frac{{{a}_{0}}}{8a_{4}^{2}}\left( 4\,\Theta \,{{a}_{2}}-{{\Theta }^{2}}\frac{a_{3}^{2}}{{{a}_{4}}} \right)-\frac{{{\Theta }^{3}}a_{2}^{3}}{54\,a_{4}^{3}}-$

$L_{x}^{2}L_{y}^{2}L_{z}^{2}\frac{{{\Theta }^{4}}a_{1}^{2}}{8\,a_{4}^{2}}$, $p={{\Theta }^{2}}\frac{4{{a}_{0}}{{a}_{4}}-\Theta \,{{L}_{x}}{{L}_{y}}{{L}_{z}}{{a}_{1}}{{a}_{3}}}{12\,a_{4}^{2}}-\frac{\Theta \,{{a}_{2}}}{18\,{{a}_{4}}}$,

$\left\{ \begin{align}  & {{\alpha }_{1{{\Phi }_{I}}}}=\frac{{{R}_{I1}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}+\frac{{{S}_{II\,20}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}+ \\ & +\frac{1}{{{L}_{x}}{{L}_{y}}{{L}_{z}}}\int\limits_{0}^{{{L}_{x}}}{\int\limits_{0}^{{{L}_{y}}}{\int\limits_{0}^{{{L}_{z}}}{{{f}_{{{\Phi }_{I}}}}\,\left( x,y,z \right)\,\,d\,z\,d\,y\,d\,x}}} \\ & {{\alpha }_{1{{\Phi }_{V}}}}=\frac{{{R}_{V1}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}+\frac{{{S}_{VV\,20}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}+ \\ & +\frac{1}{{{L}_{x}}{{L}_{y}}{{L}_{z}}}\int\limits_{0}^{{{L}_{x}}}{\int\limits_{0}^{{{L}_{y}}}{\int\limits_{0}^{{{L}_{z}}}{{{f}_{{{\Phi }_{V}}}}\,\left( x,y,z \right)\,\,d\,z\,d\,y\,d\,x}}}, \\\end{align} \right.$   (16)

where, ${{R}_{\rho \,\,i}}=\int\limits_{0}^{\Theta }{\,\left( \Theta -t \right)\int\limits_{0}^{{{L}_{x}}}{\int\limits_{0}^{{{L}_{y}}}{\int\limits_{0}^{{{L}_{z}}}{{{k}_{I}}\left( x,y,z,T \right)\,I_{1}^{i}\left( x,y,z,t \right)\,\,d\,z\,d\,y\,d\,x\,\,d\,t}}}}$.

We determine approximations of the second and higher orders of concentrations of dopant and radiation defects framework standard iterative procedure of method of averaging of function corrections [28]. Framework this procedure to determine approximations of the n-th order of concentrations of dopant and radiation defects we replace the required concentrations in the Eqns. (1c), (4c), (7c) on the following sum $\alpha_{n \rho}+\rho_{n-1}(x, y, z, t)$. The replacement leads to the following transformation of the appropriate equations

$\frac{\partial \,{{C}_{2}}\left( x,y,z,t \right)}{\partial \,t}=\frac{\partial \,}{\partial \,x}\left( \,\left\{ 1+\xi \frac{{{\left[ {{\alpha }_{2C}}+{{C}_{1}}\left( x,y,z,t \right) \right]}^{\gamma }}}{{{P}^{\gamma }}\left( x,y,z,T \right)} \right\}\,\,\times  \right. \,{{D}_{\,L}}\left( x,y,z,T \right)\,\,\left[ 1+{{\varsigma }_{1}}\frac{V\,\left( x,y,z,t \right)}{{{V}^{*}}}+{{\varsigma }_{2}}\frac{{{V}^{2}}\left( x,y,z,t \right)}{{{\left( {{V}^{*}} \right)}^{2}}} \right]\,\,\times $

$\left. \frac{\partial \,{{C}_{1}}\left( x,y,z,t \right)}{\partial \,x} \right)+\frac{\partial \,}{\partial \,y}\left( \,\,\left[ 1+{{\varsigma }_{1}}\frac{V\,\left( x,y,z,t \right)}{{{V}^{*}}}+{{\varsigma }_{2}}\frac{{{V}^{2}}\left( x,y,z,t \right)}{{{\left( {{V}^{*}} \right)}^{2}}} \right]\,\,\,\times  \right. \left.  \,{{D}_{\,L}}\left( x,y,z,T \right)\frac{\partial \,{{C}_{1}}\left( x,y,z,t \right)}{\partial \,y}\left\{ 1+\xi \frac{{{\left[ {{\alpha }_{2C}}+{{C}_{1}}\left( x,y,z,t \right) \right]}^{\gamma }}}{{{P}^{\gamma }}\left( x,y,z,T \right)} \right\} \right)+$

$\frac{\partial \,}{\partial \,z}\left( {{D}_{\,L}}\left( x,y,z,T \right)\,\left[ 1+{{\varsigma }_{1}}\frac{V\,\left( x,y,z,t \right)}{{{V}^{*}}}+{{\varsigma }_{2}}\frac{{{V}^{2}}\left( x,y,z,t \right)}{{{\left( {{V}^{*}} \right)}^{2}}} \right]\,\,\times  \right. \left. \,\frac{\partial \,{{C}_{1}}\left( x,y,z,t \right)}{\partial \,z}\left\{ 1+\xi \frac{{{\left[ {{\alpha }_{2C}}+{{C}_{1}}\left( x,y,z,t \right) \right]}^{\gamma }}}{{{P}^{\gamma }}\left( x,y,z,T \right)} \right\} \right)+$

${{f}_{C}}\,\left( x,y,z \right)\,\delta \left( t \right)+\frac{\partial }{\partial \,\,x}\left[ \frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,x} \right]+\frac{\partial }{\partial \,\,y}\left[ \frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,y} \right]+\frac{\partial }{\partial \,\,z}\left[ \frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,z} \right]+$

$+\Omega \frac{\partial }{\partial \,x}\left\{ \frac{{{D}_{S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2C}}+C\,\left( x,y,W,t \right) \right]\,\,d\,W} \right\}+\Omega \frac{\partial }{\partial \,y}\left\{ \frac{{{D}_{S}}}{k\,T}{{\nabla }_{S}}{{\mu }_{1}}\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2C}}+C\,\left( x,y,W,t \right) \right]\,\,d\,W} \right\}$  (1d)

$\frac{\partial \,\,{{I}_{2}}\left( x,y,z,t \right)}{\partial \,\,t}=\frac{\partial }{\partial \,\,x}\left[ {{D}_{I}}\left( x,y,z,T \right)\frac{\partial \,{{I}_{1}}\left( x,y,z,t \right)}{\partial \,x} \right]+\frac{\partial }{\partial \,\,y}\left[ {{D}_{I}}\left( x,y,z,T \right)\frac{\partial \,{{I}_{1}}\left( x,y,z,t \right)}{\partial \,y} \right]-{{k}_{I,I}}\left( x,y,z,T \right)\,\,\times $

$ \,\,\,{{\left[ {{\alpha }_{1I}}+{{I}_{1}}\left( x,y,z,t \right) \right]}^{2}}+\frac{\partial }{\partial \,\,z}\left[ {{D}_{I}}\left( x,y,z,T \right)\frac{\partial \,{{I}_{1}}\left( x,y,z,t \right)}{\partial \,\,z} \right]-{{k}_{I,V}}\left( x,y,z,T \right)\,\,\left[ {{\alpha }_{1I}}+{{I}_{1}}\left( x,y,z,t \right) \right]\,\left[ {{\alpha }_{1V}}+{{V}_{1}}\left( x,y,z,t \right) \right]+$

$\Omega \frac{\partial }{\partial \,x}\left\{ \frac{{{D}_{IS}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2I}}+{{I}_{1}}\left( x,y,W,t \right) \right]\,\,d\,W} \right\}+\Omega \frac{\partial }{\partial \,y}\left\{ \frac{{{D}_{IS}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2I}}+{{I}_{1}}\left( x,y,W,t \right) \right]\,\,d\,W} \right\}+$

$\frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{I\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,x}d\,\tau }+\frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\frac{{{D}_{I\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,y}d\,\tau }+\frac{\partial }{\partial \,z}\int\limits_{0}^{t}{\frac{{{D}_{I\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,z}d\,\tau }$   (4d)

$\frac{\partial \,\,{{V}_{2}}\left( x,y,z,t \right)}{\partial \,\,t}=\frac{\partial }{\partial \,\,x}\left[ {{D}_{V}}\left( x,y,z,T \right)\frac{\partial \,{{V}_{1}}\left( x,y,z,t \right)}{\partial \,x} \right]+\frac{\partial }{\partial \,\,y}\left[ {{D}_{V}}\left( x,y,z,T \right)\frac{\partial \,{{V}_{1}}\left( x,y,z,t \right)}{\partial \,y} \right]-{{k}_{V,V}}\left( x,y,z,T \right)\,\,\times $

$\times \,\,\,{{\left[ {{\alpha }_{1V}}+{{V}_{1}}\left( x,y,z,t \right) \right]}^{2}}+\frac{\partial }{\partial \,\,z}\left[ {{D}_{V}}\left( x,y,z,T \right)\frac{\partial \,{{V}_{1}}\left( x,y,z,t \right)}{\partial \,\,z} \right]-{{k}_{I,V}}\left( x,y,z,T \right)\,\,\left[ {{\alpha }_{1I}}+{{I}_{1}}\left( x,y,z,t \right) \right]\,\left[ {{\alpha }_{1V}}+{{V}_{1}}\left( x,y,z,t \right) \right]+$

$+\Omega \frac{\partial }{\partial \,x}\left\{ \frac{{{D}_{VS}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2V}}+{{V}_{1}}\left( x,y,W,t \right) \right]\,\,d\,W} \right\}+$

$+\Omega \frac{\partial }{\partial \,y}\left\{ \frac{{{D}_{VS}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2V}}+{{V}_{1}}\left( x,y,W,t \right) \right]\,\,d\,W} \right\}+$

$+\frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{V\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,x}d\,\tau }+\frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\frac{{{D}_{V\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,y}d\,\tau }+$

$\frac{\partial }{\partial \,z}\int\limits_{0}^{t}{\frac{{{D}_{V\,S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,z}d\,\tau }\frac{\partial \,\,{{\Phi }_{2I}}\left( x,y,z,t \right)}{\partial \,\,t}=\frac{\partial }{\partial \,x}\left[ {{D}_{{{\Phi }_{I}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{1I}}\left( x,y,z,t \right)}{\partial \,x} \right]+$

$\frac{\partial }{\partial \,y}\left[ {{D}_{{{\Phi }_{I}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{1I}}\left( x,y,z,t \right)}{\partial \,y} \right]+\frac{\partial }{\partial \,\,z}\left[ {{D}_{{{\Phi }_{I}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{1I}}\left( x,y,z,t \right)}{\partial \,z} \right]+$

$\Omega \frac{\partial }{\partial \,x}\left\{ \frac{{{D}_{{{\Phi }_{I}}S}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2{{\Phi }_{I}}}}+{{\Phi }_{1I}}\left( x,y,W,t \right) \right]\,\,d\,W} \right\}+\Omega \frac{\partial }{\partial \,y}\left\{ \frac{{{D}_{{{\Phi }_{I}}S}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2{{\Phi }_{I}}}}+{{\Phi }_{1I}}\left( x,y,W,t \right) \right]\,\,d\,W} \right\}+$

$+{{k}_{I,I}}\left( x,y,z,T \right)\,{{I}^{2}}\left( x,y,z,t \right)+{{k}_{I}}\left( x,y,z,T \right)\,I\,\left( x,y,z,t \right)+\frac{\partial }{\partial \,x}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,x} \right]+\frac{\partial }{\partial \,y}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,y} \right]+$

$+\frac{\partial }{\partial \,z}\left[ \frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,z} \right]+{{f}_{{{\Phi }_{I}}}}\,\left( x,y,z \right)\,\,\delta \,\left( t \right)$  (7d)

$\frac{\partial \,\,{{\Phi }_{2V}}\left( x,y,z,t \right)}{\partial \,\,t}=\frac{\partial }{\partial \,x}\left[ {{D}_{{{\Phi }_{V}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{1V}}\left( x,y,z,t \right)}{\partial \,x} \right]+\frac{\partial }{\partial \,y}\left[ {{D}_{{{\Phi }_{V}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{1V}}\left( x,y,z,t \right)}{\partial \,y} \right]+$

$\frac{\partial }{\partial \,\,z}\left[ {{D}_{{{\Phi }_{V}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{1V}}\left( x,y,z,t \right)}{\partial \,z} \right]+\Omega \frac{\partial }{\partial \,x}\left\{ \frac{{{D}_{{{\Phi }_{V}}S}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2{{\Phi }_{V}}}}+{{\Phi }_{1V}}\left( x,y,W,t \right) \right]\,\,d\,W} \right\}+$

$\Omega \frac{\partial }{\partial \,y}\left\{ \frac{{{D}_{{{\Phi }_{V}}S}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,t \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2{{\Phi }_{V}}}}+{{\Phi }_{1V}}\left( x,y,W,t \right) \right]\,\,d\,W} \right\}+{{k}_{V,V}}\left( x,y,z,T \right)\,{{V}^{2}}\left( x,y,z,t \right)+{{k}_{V}}\left( x,y,z,T \right)\,V\,\left( x,y,z,t \right)+$

$\frac{\partial }{\partial \,x}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,x} \right]+\frac{\partial }{\partial \,y}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,y} \right]+\frac{\partial }{\partial \,z}\left[ \frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,z} \right]+{{f}_{{{\Phi }_{V}}}}\,\left( x,y,z \right)\,\,\delta \,\left( t \right)$.

Integration of the left and the right sides of Eqns. (1d), (4d) and (7d) gives us possibility to obtain relations for the required concentrations in the final form

${{C}_{2}}\left( x,y,z,t \right)=\frac{\partial \,}{\partial \,x}\int\limits_{0}^{t}{\,\left[ 1+{{\varsigma }_{1}}\frac{V\,\left( x,y,z,\tau  \right)}{{{V}^{*}}}+{{\varsigma }_{2}}\frac{{{V}^{2}}\left( x,y,z,\tau  \right)}{{{\left( {{V}^{*}} \right)}^{2}}} \right]\,\,\times } \,{{D}_{\,L}}\left( x,y,z,T \right)\,\,\left\{ 1+\xi \frac{{{\left[ {{\alpha }_{2C}}+{{C}_{1}}\left( x,y,z,\tau  \right) \right]}^{\gamma }}}{{{P}^{\gamma }}\left( x,y,z,T \right)} \right\}\frac{\partial \,{{C}_{1}}\left( x,y,z,\tau  \right)}{\partial \,x}\,d\,\tau +$

$\frac{\partial \,}{\partial \,y}\int\limits_{0}^{t}{{{D}_{\,L}}\left( x,y,z,T \right)\,\left[ 1+{{\varsigma }_{1}}\frac{V\,\left( x,y,z,\tau  \right)}{{{V}^{*}}}+{{\varsigma }_{2}}\frac{{{V}^{2}}\left( x,y,z,\tau  \right)}{{{\left( {{V}^{*}} \right)}^{2}}} \right]\,\,\times } \frac{\partial \,\,{{C}_{1}}\left( x,y,z,\tau  \right)}{\partial \,y}\left\{ 1+\xi \frac{{{\left[ {{\alpha }_{2C}}+{{C}_{1}}\left( x,y,z,t \right) \right]}^{\gamma }}}{{{P}^{\gamma }}\left( x,y,z,T \right)} \right\}+$

$\frac{\partial \,}{\partial \,z}\int\limits_{0}^{t}{\frac{\partial \,{{C}_{1}}\left( x,y,z,\tau  \right)}{\partial \,z}\left[ 1+{{\varsigma }_{1}}\frac{V\,\left( x,y,z,\tau  \right)}{{{V}^{*}}}+{{\varsigma }_{2}}\frac{{{V}^{2}}\left( x,y,z,\tau  \right)}{{{\left( {{V}^{*}} \right)}^{2}}} \right]\,\,\times } \,\,{{D}_{\,L}}\left( x,y,z,T \right)\,\left\{ 1+\xi \frac{{{\left[ {{\alpha }_{2C}}+{{C}_{1}}\left( x,y,z,\tau  \right) \right]}^{\gamma }}}{{{P}^{\gamma }}\left( x,y,z,T \right)} \right\}\,d\,\tau +{{f}_{C}}\,\left( x,y,z \right)+$

$\Omega \frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{S}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,\tau  \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2C}}+{{C}_{1}}\left( x,y,W,\tau  \right) \right]\,\,d\,W}\,d\,\tau }+\Omega \frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\frac{{{D}_{S}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,\tau  \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2C}}+{{C}_{1}}\left( x,y,W,\tau  \right) \right]\,\,d\,W}\,d\,\tau }+$

$\frac{\partial }{\partial \,\,x}\left[ \frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,x} \right]+\frac{\partial }{\partial \,\,y}\left[ \frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,y} \right]+\frac{\partial }{\partial \,\,z}\left[ \frac{{{D}_{C\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,z} \right]$  (1e)

${{I}_{2}}\left( x,y,z,t \right)=\frac{\partial }{\partial \,\,x}\int\limits_{0}^{t}{{{D}_{I}}\left( x,y,z,T \right)\frac{\partial \,{{I}_{1}}\left( x,y,z,\tau  \right)}{\partial \,x}d\,\tau }+$

$\frac{\partial }{\partial \,\,y}\int\limits_{0}^{t}{{{D}_{I}}\left( x,y,z,T \right)\frac{\partial \,{{I}_{1}}\left( x,y,z,\tau  \right)}{\partial \,y}d\,\tau }+\frac{\partial }{\partial \,\,y}\int\limits_{0}^{t}{{{D}_{I}}\left( x,y,z,T \right)\,\,\times } \frac{\partial \,{{I}_{1}}\left( x,y,z,\tau  \right)}{\partial \,z}d\,\tau -\int\limits_{0}^{t}{{{k}_{I,I}}\left( x,y,z,T \right)\,\,{{\left[ {{\alpha }_{2I}}+{{I}_{1}}\left( x,y,z,\tau  \right) \right]}^{2}}d\,\tau }-$

$\int\limits_{0}^{t}{{{k}_{I,V}}\left( x,y,z,T \right)\,\,\left[ {{\alpha }_{2I}}+{{I}_{1}}\left( x,y,z,\tau  \right) \right]\,\left[ {{\alpha }_{2V}}+{{V}_{1}}\left( x,y,z,\tau  \right) \right]\,\,d\,\tau }+\Omega \frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{IS}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,\tau  \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2I}}+{{I}_{1}}\left( x,y,W,\tau  \right) \right]\,\,d\,W}\,d\,\tau }+$

$\Omega \frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\frac{{{D}_{IS}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,\tau  \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2I}}+{{I}_{1}}\left( x,y,W,\tau  \right) \right]\,\,d\,W}\,d\,\tau }+\frac{\partial }{\partial \,\,x}\left[ \frac{{{D}_{I\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,x} \right]+\frac{\partial }{\partial \,\,y}\left[ \frac{{{D}_{I\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,y} \right]+$

$+\frac{\partial }{\partial \,\,z}\left[ \frac{{{D}_{I\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,z} \right]+{{f}_{I}}\,\left( x,y,z \right)$  (4e)

${{V}_{2}}\left( x,y,z,t \right)=\frac{\partial }{\partial \,\,x}\int\limits_{0}^{t}{{{D}_{V}}\left( x,y,z,T \right)\frac{\partial \,{{V}_{1}}\left( x,y,z,\tau  \right)}{\partial \,x}d\,\tau }+\frac{\partial }{\partial \,\,y}\int\limits_{0}^{t}{{{D}_{V}}\left( x,y,z,T \right)\frac{\partial \,{{V}_{1}}\left( x,y,z,\tau  \right)}{\partial \,y}d\,\tau }+\frac{\partial }{\partial \,\,y}\int\limits_{0}^{t}{{{D}_{V}}\left( x,y,z,T \right)\,\,\times }$

$ \frac{\partial \,{{V}_{1}}\left( x,y,z,\tau  \right)}{\partial \,z}d\,\tau -\int\limits_{0}^{t}{{{k}_{V,V}}\left( x,y,z,T \right)\,\,{{\left[ {{\alpha }_{2V}}+{{V}_{1}}\left( x,y,z,\tau  \right) \right]}^{2}}d\,\tau }-\int\limits_{0}^{t}{{{k}_{I,V}}\left( x,y,z,T \right)\,\,\left[ {{\alpha }_{2I}}+{{I}_{1}}\left( x,y,z,\tau  \right) \right]\,\left[ {{\alpha }_{2V}}+{{V}_{1}}\left( x,y,z,\tau  \right) \right]\,\,d\,\tau }+$

$\Omega \frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{VS}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,\tau  \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2V}}+{{V}_{1}}\left( x,y,W,\tau  \right) \right]\,\,d\,W}\,d\,\tau }+\Omega \frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\frac{{{D}_{VS}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,\tau  \right)\int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2V}}+{{V}_{1}}\left( x,y,W,\tau  \right) \right]\,\,d\,W}\,d\,\tau }+$

$\frac{\partial }{\partial \,\,x}\left[ \frac{{{D}_{V\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,x} \right]+\frac{\partial }{\partial \,\,y}\left[ \frac{{{D}_{V\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,y} \right]+\frac{\partial }{\partial \,\,z}\left[ \frac{{{D}_{V\,S}}}{\bar{V}\,k\,T}\frac{\partial \,\,{{\mu }_{2}}\left( x,y,z,t \right)}{\partial \,\,z} \right]+{{f}_{V}}\,\left( x,y,z \right)$

${{\Phi }_{2I}}\left( x,y,z,t \right)=\frac{\partial }{\partial \,\,x}\int\limits_{0}^{t}{{{D}_{{{\Phi }_{I}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{1I}}\left( x,y,z,\tau  \right)}{\partial \,\,x}d\,\tau }+\frac{\partial }{\partial \,\,y}\int\limits_{0}^{t}{{{D}_{{{\Phi }_{I}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{1I}}\left( x,y,z,\tau  \right)}{\partial \,\,y}d\,\tau }+\frac{\partial }{\partial \,\,z}\int\limits_{0}^{t}{{{D}_{{{\Phi }_{I}}}}\left( x,y,z,T \right)}\,\,\times $

$\frac{\partial \,{{\Phi }_{1I}}\left( x,y,z,\tau  \right)}{\partial \,\,y}d\,\tau +\Omega \frac{\partial }{\partial \,x}\int\limits_{0}^{t}{{{\nabla }_{S}}\mu \,\,\left( x,y,z,\tau  \right)\frac{{{D}_{{{\Phi }_{I}}S}}}{k\,T}}\,\,\times  \int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2{{\Phi }_{I}}}}+{{\Phi }_{1I}}\left( x,y,W,\tau  \right) \right]\,\,d\,W}\,d\,\tau +\frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{I}}S}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,\tau  \right)\,}\times $

$ \,\,\Omega \,\,\int\limits_{0}^{{{L}_{z}}}{\left[ \,{{\alpha }_{2{{\Phi }_{I}}}}+{{\Phi }_{1I}}\left( x,y,W,\tau  \right) \right]\,\,d\,W}\,d\,\tau +\int\limits_{0}^{t}{{{k}_{I,I}}\left( x,y,z,T \right)\,}\,\times  \,\,{{I}^{2}}\left( x,y,z,\tau  \right)\,\,d\,\tau +\frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,x}d\,\tau }+$

$\frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,y}d\,\tau }+\frac{\partial }{\partial \,z}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{I}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,z}d\,\tau }+$

${{f}_{{{\Phi }_{I}}}}\,\left( x,y,z \right)+\int\limits_{0}^{t}{{{k}_{I}}\left( x,y,z,T \right)\,I\,\left( x,y,z,\tau  \right)\,\,d\,\tau }$   (7e)

${{\Phi }_{2V}}\left( x,y,z,t \right)=\frac{\partial }{\partial \,\,x}\int\limits_{0}^{t}{{{D}_{{{\Phi }_{V}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{1V}}\left( x,y,z,\tau  \right)}{\partial \,\,x}d\,\tau }+\frac{\partial }{\partial \,\,y}\int\limits_{0}^{t}{{{D}_{{{\Phi }_{V}}}}\left( x,y,z,T \right)\frac{\partial \,{{\Phi }_{1V}}\left( x,y,z,\tau  \right)}{\partial \,\,y}d\,\tau }+\frac{\partial }{\partial \,\,z}\int\limits_{0}^{t}{{{D}_{{{\Phi }_{V}}}}\left( x,y,z,T \right)}\,\,\times $

$ \frac{\partial \,{{\Phi }_{1V}}\left( x,y,z,\tau  \right)}{\partial \,\,y}d\,\tau +\Omega \frac{\partial }{\partial \,x}\int\limits_{0}^{t}{{{\nabla }_{S}}\mu \,\,\left( x,y,z,\tau  \right)\frac{{{D}_{{{\Phi }_{V}}S}}}{k\,T}}\,\,\times  \int\limits_{0}^{{{L}_{z}}}{\left[ {{\alpha }_{2{{\Phi }_{V}}}}+{{\Phi }_{1V}}\left( x,y,W,\tau  \right) \right]\,\,d\,W}\,d\,\tau +\frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{V}}S}}}{k\,T}{{\nabla }_{S}}\mu \,\,\left( x,y,z,\tau  \right)\,}\times $

$\,\,\Omega \,\,\int\limits_{0}^{{{L}_{z}}}{\left[ \,{{\alpha }_{2{{\Phi }_{V}}}}+{{\Phi }_{1V}}\left( x,y,W,\tau  \right) \right]\,\,d\,W}\,d\,\tau +\int\limits_{0}^{t}{{{k}_{V,V}}\left( x,y,z,T \right)\,}\,\times  \,\,{{V}^{2}}\left( x,y,z,\tau  \right)\,\,d\,\tau +\frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,x}d\,\tau }+$

$\frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,y}d\,\tau }+\frac{\partial }{\partial \,z}\int\limits_{0}^{t}{\frac{{{D}_{{{\Phi }_{V}}S}}}{\bar{V}\,k\,T}\frac{\partial \,{{\mu }_{2}}\left( x,y,z,\tau  \right)}{\partial \,z}d\,\tau }+{{f}_{{{\Phi }_{V}}}}\,\left( x,y,z \right)+\int\limits_{0}^{t}{{{k}_{V}}\left( x,y,z,T \right)\,V\,\left( x,y,z,\tau  \right)\,\,d\,\tau }$.

Average values of the second-order approximations of required approximations by using the following standard relation [28].

${{\alpha }_{2\rho }}=\frac{1}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}\times  \int\limits_{0}^{\Theta }{\int\limits_{0}^{{{L}_{x}}}{\int\limits_{0}^{{{L}_{y}}}{\int\limits_{0}^{{{L}_{z}}}{\left[ {{\rho }_{2}}\left( x,y,z,t \right)-{{\rho }_{1}}\left( x,y,z,t \right) \right]\,\,d\,z\,d\,y\,d\,x\,d\,t}}}}$.  (17)

Substitution of the relations (1e), (4e), (7e) into relation (10) gives us possibility to obtain relations for required average values $\alpha_{2 \rho}$_.

$\alpha_{2 C}=0, \alpha_{2 \Phi l} =0, \alpha_{2 \Phi V}=0$, ${{\alpha }_{2V}}=\sqrt{\frac{{{\left( {{b}_{3}}+E \right)}^{2}}}{4\,b_{4}^{2}}-4\,\,\left( F+\frac{\Theta \,{{a}_{3}}F+{{\Theta }^{2}}{{L}_{x}}{{L}_{y}}{{L}_{z}}{{b}_{1}}}{{{b}_{4}}} \right)}-\frac{{{b}_{3}}+E}{4\,{{b}_{4}}}$,

${{\alpha }_{2I}}=\left[ {{C}_{V}}-\alpha _{2V}^{2}{{S}_{VV00}}-{{\alpha }_{2V}}\left( 2\,{{S}_{VV01}}+{{S}_{I\,V10}}+\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}} \right)\,\,-{{S}_{V\,V02}}- \right.$

${\left. -{{S}_{I\,V11}} \right]}/{\left( {{S}_{IV01}}+{{\alpha }_{2V}}{{S}_{IV00}} \right)}\;$,   (18)

where,

${{b}_{4}}=\frac{1}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}S_{IV00}^{2}{{S}_{V\,V00}}-\frac{1}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}S_{VV00}^{2}{{S}_{II00}}$, ${{b}_{3}}=-\,\left( \Theta {{L}_{x}}\times  \right.$

$\left.  \,\,{{L}_{y}}{{L}_{z}}+2{{S}_{VV01}}+{{S}_{IV10}} \right)\frac{{{S}_{II00}}{{S}_{VV00}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}+\frac{{{S}_{IV00}}{{S}_{VV00}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}\left( {{S}_{IV01}}+{{L}_{x}}{{L}_{y}}{{L}_{z}} \right.\times \left. \,\,\Theta +{{S}_{IV01}}+2{{S}_{II10}} \right)+\frac{S_{IV00}^{2}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}\left( 2{{S}_{VV01}}+\Theta \,{{L}_{x}}{{L}_{y}}{{L}_{z}}+{{S}_{IV10}} \right)-$

$-\frac{S_{IV00}^{2}{{S}_{IV10}}}{{{\Theta }^{3}}L_{x}^{3}L_{y}^{3}L_{z}^{3}}$, ${{b}_{2}}=\frac{{{S}_{II00}}{{S}_{VV00}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}\left( {{S}_{VV02}}+{{S}_{IV11}}+{{C}_{V}} \right)-$   (19)

${{\left( {{S}_{IV10}}-2{{S}_{VV01}}+\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}} \right)}^{2}}+\frac{{{S}_{IV01}}{{S}_{VV00}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}\left( 2{{S}_{II10}}+{{S}_{IV01}}+ \right.\left. \Theta {{L}_{x}}{{L}_{y}}{{L}_{z}} \right)+\frac{{{S}_{IV00}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}\left( {{S}_{IV01}}+2{{S}_{II10}}+2{{S}_{IV01}}+\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}} \right)\,\left( \Theta \times  \right.$

$\left.  {{L}_{x}}{{L}_{y}}{{L}_{z}}+2{{S}_{VV01}}+{{S}_{IV10}} \right)-S_{IV00}^{2}\frac{{{C}_{V}}-{{S}_{VV02}}-{{S}_{IV11}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}-2\frac{{{S}_{IV10}}}{\Theta {{L}_{x}}}\times  \frac{{{S}_{IV00}}}{{{L}_{y}}{{L}_{z}}}{{S}_{IV01}}+\frac{{{C}_{I}}S_{IV00}^{2}}{{{\Theta }^{2}}L_{x}^{2}L_{y}^{2}L_{z}^{2}}$, ${{b}_{1}}={{S}_{II00}}\frac{{{S}_{IV11}}+{{S}_{VV02}}+{{C}_{V}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}\left( \Theta \,{{L}_{x}} \right.\times $

$\left.  {{L}_{y}}{{L}_{z}}+2{{S}_{VV01}}+{{S}_{IV10}} \right)+{{S}_{IV01}}\frac{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}+2{{S}_{II10}}+{{S}_{IV01}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}\left( {{S}_{IV10}}+ \right.\left. 2{{S}_{VV01}}+\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}} \right)-\frac{{{S}_{IV10}}S_{IV01}^{2}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}-\frac{{{S}_{IV00}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}\left( 3{{S}_{IV01}}+2{{S}_{II10}}+ \right.$

$\left. \Theta {{L}_{x}}{{L}_{y}}{{L}_{z}} \right)\,\left( {{C}_{V}}-{{S}_{VV02}}-{{S}_{IV11}} \right)+2\,{{C}_{I}}{{S}_{IV00}}{{S}_{IV01}}$, ${{b}_{0}}={{S}_{II00}}\times  \frac{{{\left( {{S}_{IV00}}+{{S}_{VV02}} \right)}^{2}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}-\frac{{{C}_{V}}-{{S}_{VV02}}-{{S}_{IV11}}}{{{L}_{x}}{{L}_{y}}{{L}_{z}}\Theta }\left( \Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}+2\,{{S}_{II10}}+{{S}_{IV01}} \right)\,\,\times $

$ \,\,{{S}_{IV01}}+2{{C}_{I}}S_{IV01}^{2}-{{S}_{IV01}}\frac{{{C}_{V}}-{{S}_{VV02}}-{{S}_{IV11}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}\left( \Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}+2\,{{S}_{II10}}+ \right.\left. {{S}_{IV01}} \right)$, ${{C}_{I}}=\frac{{{\alpha }_{1I}}{{\alpha }_{1V}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}{{S}_{IV00}}+\frac{\alpha _{1I}^{2}{{S}_{II00}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}-\frac{{{S}_{II20}}{{S}_{II20}}}{\Theta {{L}_{x}}{{L}_{y}}{{L}_{z}}}-\frac{{{S}_{IV11}}}{\Theta {{L}_{x}}}\times \frac{1}{{{L}_{y}}{{L}_{z}}}$

${{C}_{V}}={{\alpha }_{1I}}{{\alpha }_{1V}}{{S}_{IV00}}+\alpha _{1V}^{2}{{S}_{VV00}}-{{S}_{VV02}}-{{S}_{IV11}}$, $F=\frac{\Theta \,{{a}_{2}}}{6\,{{a}_{4}}}+\sqrt[3]{\sqrt{{{r}^{2}}+{{s}^{3}}}-r}-\sqrt[3]{\sqrt{{{r}^{2}}+{{s}^{3}}}+r}$, $E=\sqrt{8\,y+{{\Theta }^{2}}\frac{a_{3}^{2}}{a_{4}^{2}}-4\,\Theta \,\frac{{{a}_{2}}}{{{a}_{4}}}}$,

$r=\frac{{{\Theta }^{3}}{{b}_{2}}}{\,24\,b_{4}^{2}}\left( 4{{b}_{0}}-\Theta \,{{L}_{x}}{{L}_{y}}{{L}_{z}}\frac{{{b}_{1}}{{b}_{3}}}{{{b}_{4}}} \right)-{{b}_{0}}\frac{{{\Theta }^{2}}}{8b_{4}^{2}}\left( 4\,\Theta \,{{b}_{2}}-{{\Theta }^{2}}\frac{b_{3}^{2}}{{{b}_{4}}} \right)-$

$\frac{{{\Theta }^{3}}b_{2}^{3}}{54\,b_{4}^{3}}-L_{x}^{2}L_{y}^{2}L_{z}^{2}\frac{{{\Theta }^{4}}b_{1}^{2}}{8\,b_{4}^{2}}$, $s={{\Theta }^{2}}\frac{4{{b}_{0}}{{b}_{4}}-\Theta \,{{L}_{x}}{{L}_{y}}{{L}_{z}}{{b}_{1}}{{b}_{3}}}{12\,b_{4}^{2}}-\frac{\Theta \,{{b}_{2}}}{18\,{{b}_{4}}}$.

Farther we determine solutions of Eq. (8), i.e. components of displacement vector. To determine the first-order approximations of the considered components framework method of averaging of function corrections we replace the required functions in the right sides of the equations by their not yet known average values $\alpha_{i}$. The substitution leads to the following result.

$\rho \,\left( z \right)\frac{{{\partial }^{2}}\,{{u}_{1x}}\left( x,y,z,t \right)}{\partial \,{{t}^{2}}}=-K\,\left( z \right)\,\beta \,\left( z \right)\frac{\partial \,T\,\left( x,y,z,t \right)}{\partial \,x}$,

$\rho \,\left( z \right)\frac{{{\partial }^{2}}\,{{u}_{1y}}\left( x,y,z,t \right)}{\partial \,{{t}^{2}}}=-K\,\left( z \right)\,\beta \,\left( z \right)\frac{\partial \,T\,\left( x,y,z,t \right)}{\partial \,y}$,

$\rho \,\left( z \right)\frac{{{\partial }^{2}}\,{{u}_{1z}}\left( x,y,z,t \right)}{\partial \,{{t}^{2}}}=-K\,\left( z \right)\,\beta \,\left( z \right)\frac{\partial \,T\,\left( x,y,z,t \right)}{\partial \,z}$.

Integration of the left and the right sides of the above relations on time t leads to the following result.

${{u}_{1x}}\left( x,y,z,t \right)={{u}_{0x}}+K\,\left( z \right)\,\frac{\beta \,\left( z \right)}{\rho \,\left( z \right)}\frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{\,T\,\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}-K\,\left( z \right)\,\frac{\beta \,\left( z \right)}{\rho \,\left( z \right)}\frac{\partial }{\partial \,x}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{\,T\,\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}$,

${{u}_{1y}}\left( x,y,z,t \right)={{u}_{0y}}+K\,\left( z \right)\,\frac{\beta \,\left( z \right)}{\rho \,\left( z \right)}\frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{\,T\,\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}-K\,\left( z \right)\,\frac{\beta \,\left( z \right)}{\rho \,\left( z \right)}\frac{\partial }{\partial \,y}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{\,T\,\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}$,  (19)

${{u}_{1z}}\left( x,y,z,t \right)={{u}_{0z}}+K\,\left( z \right)\,\frac{\beta \,\left( z \right)}{\rho \,\left( z \right)}\frac{\partial }{\partial \,z}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{\,T\,\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}-K\,\left( z \right)\,\frac{\beta \,\left( z \right)}{\rho \,\left( z \right)}\frac{\partial }{\partial \,z}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{\,T\,\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}$.

Approximations of the second and higher orders of components of displacement vector could be determined by using standard replacement of the required components on the following sums $\alpha_{i}+u_{i}(x, y, z, t)$ [28]. The replacement leads to the following result.

$\rho \,\left( z \right)\frac{{{\partial }^{2}}\,{{u}_{2x}}\left( x,y,z,t \right)}{\partial \,{{t}^{2}}}=\left\{ K\,\left( z \right)+\frac{5E\,\left( z \right)}{6\,\left[ 1+\sigma \,\left( z \right) \right]} \right\}\frac{{{\partial }^{2}}{{u}_{1x}}\left( x,y,z,t \right)}{\partial \,{{x}^{2}}}+$

$\left\{ K\,\left( z \right)-\frac{E\,\left( z \right)}{3\,\left[ 1+\sigma \,\left( z \right) \right]} \right\}\frac{{{\partial }^{2}}{{u}_{1y}}\left( x,y,z,t \right)}{\partial \,x\,\partial \,y}+\left[ \frac{{{\partial }^{2}}{{u}_{1y}}\left( x,y,z,t \right)}{\partial \,{{y}^{2}}}+ \right.\left. \frac{{{\partial }^{2}}{{u}_{1z}}\left( x,y,z,t \right)}{\partial \,{{z}^{2}}} \right]\frac{E\,\left( z \right)}{2\,\left[ 1+\sigma \,\left( z \right) \right]}-K\,\left( z \right)\,\beta \,\left( z \right)\frac{\partial \,T\,\left( x,y,z,t \right)}{\partial \,x}+$

$+\left\{ K\,\left( z \right)+\frac{E\,\left( z \right)}{3\,\left[ \,1+\sigma \,\left( z \right) \right]} \right\}\frac{{{\partial }^{2}}{{u}_{1z}}\left( x,y,z,t \right)}{\partial \,x\,\partial \,z}$

$\rho \,\left( z \right)\frac{{{\partial }^{2}}\,{{u}_{2y}}\left( x,y,z,t \right)}{\partial \,{{t}^{2}}}=\left[ \frac{{{\partial }^{2}}{{u}_{1y}}\left( x,y,z,t \right)}{\partial \,{{x}^{2}}}+\frac{{{\partial }^{2}}{{u}_{1x}}\left( x,y,z,t \right)}{\partial \,x\,\partial \,y} \right]\,\,\times \frac{E\,\left( z \right)}{2\left[ 1+\sigma \,\left( z \right) \right]}-\,K\,\left( z \right)\,\beta \,\left( z \right)\frac{\partial \,T\,\left( x,y,z,t \right)}{\partial \,y}+\frac{\partial }{\partial \,z}\left\{ \frac{E\,\left( z \right)}{2\,\left[ 1+\sigma \,\left( z \right) \right]}\times  \right.$

$\left.  \left[ \frac{\partial \,{{u}_{1y}}\left( x,y,z,t \right)}{\partial \,z}+\frac{\partial \,{{u}_{1z}}\left( x,y,z,t \right)}{\partial \,y} \right] \right\}+\frac{{{\partial }^{2}}{{u}_{1y}}\left( x,y,z,t \right)}{\partial \,{{y}^{2}}}\times $

$ \left\{ \frac{5E\,\left( z \right)}{12\,\left[ 1+\sigma \,\left( z \right) \right]}+K\,\left( z \right) \right\}+\left\{ K\,\left( z \right)-\frac{E\,\left( z \right)}{6\,\left[ 1+\sigma \,\left( z \right) \right]} \right\}\frac{{{\partial }^{2}}{{u}_{1y}}\left( x,y,z,t \right)}{\partial \,y\,\partial \,z}+K\,\left( z \right)\frac{{{\partial }^{2}}{{u}_{1y}}\left( x,y,z,t \right)}{\partial \,x\,\partial \,y}$  (20)

$\rho \,\left( z \right)\frac{{{\partial }^{2}}{{u}_{2z}}\left( x,y,z,t \right)}{\partial \,{{t}^{2}}}=\left[ \frac{{{\partial }^{2}}{{u}_{1z}}\left( x,y,z,t \right)}{\partial \,{{x}^{2}}}+\frac{{{\partial }^{2}}{{u}_{1z}}\left( x,y,z,t \right)}{\partial \,{{y}^{2}}}+ \right.$

$\left. \frac{{{\partial }^{2}}{{u}_{1x}}\left( x,y,z,t \right)}{\partial \,x\,\partial \,z}+\frac{{{\partial }^{2}}{{u}_{1y}}\left( x,y,z,t \right)}{\partial \,y\,\partial \,z} \right]\frac{E\,\left( z \right)}{2\,\,\left[ 1+\sigma \,\left( z \right) \right]}+\frac{\partial }{\partial \,z}\left\{ K\,\left( z \right)\,\,\times  \right.$

$\left.  \,\,\left[ \frac{\partial \,{{u}_{1x}}\left( x,y,z,t \right)}{\partial \,x}+\frac{\partial \,{{u}_{1y}}\left( x,y,z,t \right)}{\partial \,y}+\frac{\partial \,{{u}_{1x}}\left( x,y,z,t \right)}{\partial \,z} \right]\, \right\}+\frac{\partial }{\partial \,z}\left[ 6\frac{\partial \,{{u}_{1z}}\left( x,y,z,t \right)}{\partial \,z}-\frac{\partial \,{{u}_{1x}}\left( x,y,z,t \right)}{\partial \,x}-\frac{\partial \,{{u}_{1y}}\left( x,y,z,t \right)}{\partial \,y}- \right.$

$\left. \frac{\partial \,{{u}_{1z}}\left( x,y,z,t \right)}{\partial \,z} \right]\frac{E\,\left( z \right)}{6\,\left[ 1+\sigma \,\left( z \right) \right]}-K\,\left( z \right)\,\beta \,\left( z \right)\frac{\partial \,T\,\left( x,y,z,t \right)}{\partial \,z}$.

Integration of the left and right sides of the above relations on time t leads to the following result.

${{u}_{2x}}\left( x,y,z,t \right)=\frac{1}{\rho \,\,\left( z \right)}\frac{{{\partial }^{2}}}{\partial \,{{x}^{2}}}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{{{u}_{1x}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}\,\,\left\{ \frac{5E\,\left( z \right)}{6\,\left[ 1+\sigma \,\left( z \right) \right]}+ \right.$

$\left. ^{^{^{^{^{{}}}}}}K\,\left( z \right) \right\}-\frac{1}{\rho \,\left( z \right)}\frac{{{\partial }^{2}}}{\partial \,x\,\partial \,y}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{{{u}_{1y}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}\left\{ \frac{E\,\left( z \right)}{3\,\,\left[ 1+\sigma \,\left( z \right) \right]}- \right.$

$\left. ^{^{^{^{^{{}}}}}}K\,\left( z \right)\,\, \right\}+\frac{E\,\left( z \right)}{2\,\rho \,\left( z \right)\,\left[ 1+\sigma \,\left( z \right) \right]}\left[ \frac{{{\partial }^{2}}}{\partial \,{{y}^{2}}}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{{{u}_{1y}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}+ \right.$

$\left. \frac{{{\partial }^{2}}}{\partial \,{{z}^{2}}}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{{{u}_{1z}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }} \right]+\frac{{{\partial }^{2}}}{\partial \,x\,\partial \,z}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{{{u}_{1z}}\left( x,y,z,\tau  \right)\,d\,\tau \,d\,\vartheta }}\,\,\times $

$\frac{1}{\rho \,\left( z \right)}\left\{ K\,\left( z \right)\,+\frac{E\,\left( z \right)}{3\,\,\left[ 1+\sigma \,\left( z \right) \right]} \right\}-K\,\left( z \right)\frac{\partial }{\partial \,x}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{T\,\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}\,\,\times  \frac{\beta \,\left( z \right)}{\rho \,\left( z \right)}-\left\{ K\,\left( z \right)+\frac{5E\,\left( z \right)}{6\,\,\left[ 1+\sigma \,\left( z \right) \right]} \right\}\frac{{{\partial }^{2}}}{\partial \,{{x}^{2}}}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1x}}\left( x,y,z,\tau  \right)\,d\,\tau \,d\,\vartheta }}\,\,\times $

$ \frac{1}{\rho \,\left( z \right)}-\left\{ K\,\left( z \right)-\frac{E\,\left( z \right)}{3\,\left[ 1+\sigma \,\left( z \right) \right]} \right\}\frac{{{\partial }^{2}}}{\partial \,x\,\partial \,y}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1y}}\left( x,y,z,\tau  \right)\,d\,\tau \,d\,\vartheta }}\,\,\times $

$ \frac{1}{\rho \,\left( z \right)}-\frac{E\,\left( z \right)}{2\,\rho \,\left( z \right)\,\left[ 1+\sigma \,\left( z \right) \right]}\left[ \frac{{{\partial }^{2}}}{\partial \,{{y}^{2}}}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1y}}\left( x,y,z,\tau  \right)\,d\,\tau \,d\,\vartheta }}+ \right.$

$\left. \frac{{{\partial }^{2}}}{\partial \,{{z}^{2}}}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1z}}\left( x,y,z,\tau  \right)\,d\,\tau \,d\,\vartheta }} \right]-\frac{{{\partial }^{2}}}{\partial \,x\,\partial \,z}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1z}}\left( x,y,z,\tau  \right)\,d\,\tau \,d\,\vartheta }}\,\,\times $

$ \frac{1}{\rho \,\left( z \right)}\left\{ K\,\left( z \right)+\frac{E\,\left( z \right)}{3\,\,\left[ 1+\sigma \,\left( z \right) \right]} \right\}+\frac{\partial }{\partial \,x}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{T\,\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}\,\,\times  {K\,\left( z \right)\,\beta \,\left( z \right)}/{\rho \,\left( z \right)}\;+{{u}_{0x}}$  (21)

${{u}_{2y}}\left( x,y,z,t \right)=\frac{E\,\left( z \right)}{2\,\rho \,\left( z \right)\,\left[ 1+\sigma \,\left( z \right) \right]}\left[ \frac{{{\partial }^{2}}}{\partial \,{{x}^{2}}}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{{{u}_{1x}}\left( x,y,z,\tau  \right)\,d\,\tau \,d\,\vartheta }}+ \right.$

$\left. \frac{{{\partial }^{2}}}{\partial \,x\,\partial \,y}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{{{u}_{1x}}\left( x,y,z,\tau  \right)\,d\,\tau \,d\,\vartheta }} \right]+\frac{{{\partial }^{2}}}{\partial \,x\,\partial \,y}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{{{u}_{1y}}\left( x,y,z,\tau  \right)\,d\,\tau \,d\,\vartheta }}\times $

$\\frac{K\,\left( z \right)}{\rho \,\left( z \right)}+\frac{1}{\rho \,\left( z \right)}\frac{{{\partial }^{2}}}{\partial \,{{y}^{2}}}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{{{u}_{1x}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}\,\left\{ \frac{5E\,\left( z \right)}{12\,\left[ 1+\sigma \,\left( z \right) \right]}+ \right.$

$\left. ^{^{^{^{{}}}}}K\,\left( z \right) \right\}+\frac{1}{2\rho \,\left( z \right)}\frac{\partial }{\partial \,z}\left\{ \frac{E\,\left( z \right)}{1+\sigma \,\left( z \right)}\left[ \frac{\partial }{\partial \,z}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{{{u}_{1y}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}+ \right. \right.$

$\left. \left. \frac{\partial }{\partial \,y}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{{{u}_{1z}}\left( x,y,z,\tau  \right)\,d\,\tau \,d\,\vartheta }} \right] \right\}-\,\,K\,\left( z \right)\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{T\,\,\left( x,y,z,\tau  \right)\,d\,\tau \,d\,\vartheta }}\,\,\times $

$ \,\frac{\beta \,\left( z \right)}{\rho \,\left( z \right)}-\frac{{{\partial }^{2}}}{\partial \,y\,\partial \,z}\int\limits_{0}^{t}{\int\limits_{0}^{\vartheta }{{{u}_{1y}}\left( x,y,z,\tau  \right)\,d\,\tau \,d\,\vartheta }}\,\left\{ \frac{E\,\left( z \right)}{6\,\left[ 1+\sigma \,\left( z \right) \right]}-K\,\left( z \right) \right\}\times $

$ \frac{1}{\rho \,\left( z \right)}-\frac{E\,\left( z \right)}{2\,\rho \,\left( z \right)\,\,\left[ 1+\sigma \,\left( z \right) \right]}\left[ \frac{{{\partial }^{2}}}{\partial \,{{x}^{2}}}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1x}}\left( x,y,z,\tau  \right)\,d\,\tau \,d\,\vartheta }}+ \right.$

$\left. \frac{{{\partial }^{2}}}{\partial \,x\,\partial \,y}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1x}}\left( x,y,z,\tau  \right)\,d\,\tau \,d\,\vartheta }} \right]-K\,\left( z \right)\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{T\,\,\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}\,\,\times $

$\,\frac{\beta \,\left( z \right)}{\rho \,\left( z \right)}-\frac{1}{\rho \,\left( z \right)}\frac{{{\partial }^{2}}}{\partial \,{{y}^{2}}}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1x}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}\,\left\{ \,\frac{5E\,\left( z \right)}{12\,\left[ 1+\sigma \,\left( z \right) \right]}+ \right.$

$\left. ^{^{^{^{^{{}}}}}}K\,\left( z \right) \right\}-\frac{K\,\left( z \right)}{\rho \,\left( z \right)}\frac{{{\partial }^{2}}}{\partial \,x\,\partial \,y}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1y}}\left( x,y,z,\tau  \right)\,d\,\tau \,d\,\vartheta }}-\frac{\partial }{\partial \,z}\left\{ \frac{E\,\left( z \right)}{1+\sigma \,\left( z \right)}\times  \right.$

$ \,\,\left. \left[ \frac{\partial }{\partial \,z}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1y}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}+\frac{\partial }{\partial \,y}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1z}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }} \right] \right\}\times $

$ \frac{1}{2\rho \,\left( z \right)}+\frac{1}{\rho \,\left( z \right)}\frac{{{\partial }^{2}}}{\partial \,y\,\partial \,z}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1y}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}\,\left\{ \frac{E\,\left( z \right)}{6\,\left[ 1+\sigma \,\left( z \right) \right]}- \right.\left. K\,\left( z \right)\, \right\}+{{u}_{0y}}$

${{u}_{z}}\left( x,y,z,t \right)=\frac{E\,\left( z \right)}{2\,\left[ 1+\sigma \,\left( z \right) \right]}\left[ \frac{{{\partial }^{2}}}{\partial \,{{x}^{2}}}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1z}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}+ \right.$

$\frac{{{\partial }^{2}}}{\partial \,{{y}^{2}}}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1z}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}+\frac{{{\partial }^{2}}}{\partial \,x\,\partial \,z}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1x}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}+$

$\left. \frac{{{\partial }^{2}}}{\partial \,y\,\partial \,z}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1y}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }} \right]\frac{1}{\rho \,\left( z \right)}+\frac{1}{\rho \,\left( z \right)}\frac{\partial }{\partial \,z}\left\{ K\,\left( z \right)\,\,{{\times }^{^{^{{}}}}} \right.$

$ \,\,\left[ \frac{\partial }{\partial \,x}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1x}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}+\frac{\partial }{\partial \,y}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1x}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}+ \right.$

$\left. \left. \frac{\partial }{\partial \,z}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1x}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }} \right] \right\}+\frac{1}{6\rho \,\left( z \right)}\frac{\partial }{\partial \,z}\left\{ \frac{E\,\left( z \right)}{1+\sigma \,\left( z \right)}\times  \right.$

$ \,\,\left[ 6\frac{\partial }{\partial \,z}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1z}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}-\frac{\partial }{\partial \,x}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1x}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}- \right.$

$\left. \left. \frac{\partial }{\partial \,y}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1y}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}-\frac{\partial }{\partial \,z}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{{{u}_{1z}}\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }} \right] \right\}-$

$K\,\left( z \right)\frac{\beta \,\left( z \right)}{\rho \,\left( z \right)}\frac{\partial }{\partial \,z}\int\limits_{0}^{\infty }{\int\limits_{0}^{\vartheta }{T\,\left( x,y,z,\tau  \right)\,\,d\,\tau \,d\,\vartheta }}+{{u}_{0z}}$.

Framework this paper we determine concentration of dopant, concentrations of radiation defects and components of displacement vector by using the second-order approximation framework method of averaging of function corrections. This approximation is usually enough good approximation to make qualitative analysis and to obtain some quantitative results. All obtained results have been checked by comparison with results of numerical simulations.