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Photo-thermal-elastic waves of excitation microstretch semiconductor medium under the impact of rotation and initial stress

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Abstract

A novel mathematical-theoretical model is intreduced to study the microstretch properties of elastic semiconductor medium. The theoretical model is obtained during the photo-excitation transport processes in the generalized thermoelasticty theory. The new model can be called Microstretch-photo-thermoelasticity (MPT) theory, the MPT model is studied under the impact of hydrostatic initial stress. The carriers charge field (carrier density or plasma wave) appears due to optical excitation. During this model the interaction between thermal–mechanical-plasma waves is obtained when the medium is in a rotating case. When the medium is linear, isotropic and homogenous, the two dimension (2D) elastic and electronics deformation governing equations are investigated when the microinertia of the medium particles is taken into consideration. The basic physical variables are obtained with the mathematical plane harmonic wave technique to get the general solutions. The complete analytical solutions have been solved completely when some conditions from mechanical-thermal and plasma type are acted at the free surface of the medium. The silicon (Si) and Germanium (Ge) materials are used to make the numerical simulations. The numerical results are displayed graphically and discussed.

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Abbreviations

\(\lambda ,\,\,\mu \quad \quad \;\) :

Lame’s elastic parameters

\(\delta_{n}\) :

The deformation potential difference

\(T\) :

The thermodynamic temperature

\(T_{0} \;\) :

The reference rest temperature

\(\hat{\gamma } = (3\lambda + 2\mu + k)\alpha_{{t_{1} }}\) :

The volume thermal expansion

\(\sigma_{ij}\) :

The stress coefficient tensor

\(\rho \quad \quad\) :

The density of the sample

\(\alpha_{{t_{1} }} ,\alpha_{{t_{2} }}\) :

Coefficients of linear thermal expansion

\({\text{e}}\) :

Cubical dilatation

\(C_{e}\) :

Specific heat of the material at constant strain

\(k\) :

The thermal conductivity

\(D_{E}\) :

The carrier diffusion coefficient

\(\tau\) :

The carrier lifetime

\(E_{g}\) :

The energy gap

\(e_{ij}\) :

Components of strain tensor

\(\Pi ,\Psi\) :

Two scalar functions

\(j_{0}\) :

The microinertia of microelement

\(m_{ij}\) :

Couple stress tensor

\(\alpha_{0} ,\lambda_{0} ,\lambda_{1}\) :

Microstretch elastic constants

\(\tau_{0} ,\nu_{0}\) :

Thermal relaxation times

\(\phi\) :

Rotation inertia vector

\(\phi^{*}\) :

The scalar microstretch

\(\lambda_{i}^{{}}\) :

The first moment (microstress) tensor

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Authors and Affiliations

  1. Department of Mathematics, College of Science and Humanities, Prince Sattam Bin Abdulaziz University, Aflaj, Riyadh, Saudi Arabia

    M. S. Mohamed & A. M. S. Mahdy

  2. Mathematics Department, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt

    Kh. Lotfy

  3. Arab Academy for Science, Technology and Maritime Transport, P.O. Box 1029, Alexandria, Egypt

    A. El-Bary

  4. Department of Mathematics, Faculty of Science, Taibah University, Medina, Saudi Arabia

    Kh. Lotfy

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  1. M. S. Mohamed
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  3. A. El-Bary
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Appendix

Appendix

The basic coefficients of Eq. (38) are:

$$\Theta_{1} = A_{2} \,A_{7} - a_{3} a_{7} - A_{1} - A_{3} - A_{4} - A_{6} - \alpha_{1} + \frac{{a_{1} \,a_{11} - A_{5} }}{{a_{8} }}$$
(53)
$$\begin{gathered} \Theta_{2} = \left( {\frac{{a_{1} \,a_{11} - A_{5} }}{{a_{8} }} - A_{6} - A_{1} - b^{2} - \alpha_{1} - \frac{{A_{5} }}{{a_{8} }}} \right)a_{3} a_{7} - \varepsilon_{4} A_{7} \,a_{4} + \left( {\frac{{(b^{2} + A_{3} + A_{4} + A_{6} + \alpha_{1} )a_{11} }}{{a_{8} }} + \frac{{cA_{7} }}{{a_{8} }}} \right)a_{1} - \hfill \\ \frac{{\omega a_{11} \varepsilon_{1} A_{2} }}{{a_{8} }} - a_{14} A_{9} + (A_{2} A_{4} + A_{2} A_{3} + (b^{2} + \alpha_{1} )A_{2} )A_{7} + ( - A_{1} - A_{3} - A_{4} - \alpha_{1} )A_{6} + ( - A_{3} - A_{1} - \alpha_{1} )A_{4} \hfill \\ ( - A_{1} - \alpha_{1} )A_{3} - \alpha_{1} A_{3} + \varepsilon_{4} \varepsilon_{5} + \frac{{A_{2} A_{5} A_{7} - A_{5} A_{6} + ( - A_{1} - A_{3} - A_{4} - \alpha_{1} )A_{5} - c\omega \varepsilon_{1} }}{{a_{8} }} \hfill \\ \end{gathered}$$
(54)
$$\left. \begin{gathered} \Theta_{3} = - \varepsilon_{4} A_{7} a_{3} a_{4} + ( - (2b^{2} - \alpha_{1} )A_{2} A_{7} + ( - b^{2} - A_{1} - \alpha_{1} )A_{6} - (b^{2} + \alpha_{1} )A_{1} - b^{2} \alpha_{1} + \varepsilon_{4} \varepsilon_{5} )a_{3} )a_{7} \hfill \\ + ( - b^{2} \varepsilon_{4} - A_{3} \varepsilon_{4} - A_{4} \varepsilon_{4} )A_{7} \,a_{4} + ( - A_{6} - A_{4} - \alpha_{1} )A_{9} \,a_{14} + \hfill \\ ((A_{2} A_{3} - ( - b^{2} - \alpha_{1} )A_{2} A_{3} + b^{2} \alpha_{1} A_{2} )A_{7} + (( - A_{3} - A_{1} - \alpha_{1} )A_{4} \hfill \\ + ( - A_{1} - \alpha_{1} )A_{3} - \alpha_{1} A_{1} )A_{6} + (( - A_{1} - \alpha_{1} )A_{3} - \alpha_{1} A_{1} + \varepsilon_{4} \varepsilon_{5} )A_{4} + ( - \alpha_{1} A_{1} + \varepsilon_{4} \varepsilon_{5} )A_{3} + \varepsilon_{4} \varepsilon_{5} A_{1} \hfill \\ + \frac{1}{{a_{8} }}\left( \begin{gathered} \left( {\left( {\left( {2b^{2} + A_{6} + \alpha_{1} } \right)a_{11} + cA_{7} } \right)a_{1} - \omega \varepsilon_{1} A_{2} a_{11} + A_{2} A_{5} A_{7} - A_{5} A_{6} + \left( { - b^{2} - A_{1} - \alpha_{1} } \right)A_{5} - \omega c\varepsilon_{1} } \right) + \left( {\omega a_{11} \varepsilon_{1} \varepsilon_{4} - A_{5} A_{7} \varepsilon_{4} } \right)a_{4} \hfill \\ + (((b^{2} + A_{3} + A_{4} + \alpha_{1} )A_{6} + (b^{2} + A_{3} + \alpha_{1} )A_{4} - ( - b^{2} - \alpha_{1} )A_{3} + b^{2} \alpha_{1} - \varepsilon_{4} \varepsilon_{5} )a_{11} + (b^{2} c + cA_{3} + cA_{4} + c\alpha_{1} )A_{7} )a_{1} \hfill \\ \end{gathered} \right) \hfill \\ - A_{5} A_{9} a_{14} + ( - \omega \varepsilon_{1} A_{2} A_{4} - \omega \varepsilon_{1} A_{2} A_{3} - (b^{2} \omega \varepsilon_{1} + \omega \alpha_{1} \varepsilon_{1} )A_{2} )a_{11} + (A_{2} A_{4} + A_{2} A_{3} - ( - b^{2} - \alpha_{1} )A_{2} )A_{5} A_{7} + ( - A_{4} - A_{3} - A_{1} - \alpha_{1} ) \hfill \\ A_{5} A_{6} + (( - A_{3} - A_{1} - \alpha_{1} )A_{4} + ( - A_{1} - \alpha_{1} )A_{3} - \alpha_{1} A_{1} + \varepsilon_{4} \varepsilon_{5} )A_{5} - c\,\varepsilon_{1} A_{4} \hfill \\ - c\varepsilon_{1} \omega A_{3} - c\varepsilon_{1} \omega A_{1} - c\omega \varepsilon_{1} \alpha_{1} ) \hfill \\ \end{gathered} \right\}$$
(55)
$$\left. \begin{gathered} \Theta_{6} = \frac{1}{{a_{8} }}(((b^{4} \varepsilon_{4} (\omega a_{11} \varepsilon_{1} - A_{5} A_{7} \varepsilon_{4} ) + (((b^{4} a_{11} (A_{6} \alpha_{1} - \varepsilon_{4} \varepsilon_{5} ) + b^{2} c\alpha_{1} A_{7} )a_{1} \hfill \\ - b^{4} \omega \alpha_{1} \varepsilon_{1} A_{2} a_{11} + b^{4} \alpha_{1} A_{2} A_{5} A_{7} - b^{2} \alpha_{1} A_{1} A_{5} A_{6} + b^{2} \varepsilon_{4} \varepsilon_{5} A_{1} A_{5} - b^{2} c\omega \alpha_{1} \varepsilon_{1} A_{1} )a_{3} )a_{7} \hfill \\ + (b^{2} \omega A_{3} A_{4} \varepsilon_{1} \varepsilon_{4} - b^{2} A_{3} A_{4} A_{5} A_{7} \varepsilon_{4} ) + ((b^{2} A_{3} A_{4} A_{6} \alpha_{1} - b^{2} A_{3} A_{4} \varepsilon_{4} \varepsilon_{5} )a_{11} + b^{2} \alpha_{1} cA_{3} A_{4} A_{7} )a_{1} \hfill \\ + ( - c\omega A_{4} \alpha_{1} \varepsilon_{1} - A_{4} A_{5} A_{6} \alpha_{1} + A_{4} A_{5} \varepsilon_{4} \varepsilon_{5} )A_{9} a_{14} - b^{2} \alpha_{1} A_{2} \varepsilon_{1} A_{3} A_{4} a_{11} + b^{2} \alpha_{1} A_{2} A_{3} A_{4} A_{5} A_{7} \hfill \\ - \alpha_{1} A_{1} A_{3} A_{4} A_{5} A_{6} + \varepsilon_{4} \varepsilon_{5} A_{1} A_{3} A_{4} A_{5} - c\omega \varepsilon_{1} \alpha_{1} A_{1} A_{3} A_{4} ) \hfill \\ \end{gathered} \right\}$$
(56)
$$\left. \begin{gathered} \Theta_{4} = (((b^{4} + 2b^{2} \alpha_{1} )A_{2} - 2b^{2} a_{4} \varepsilon_{4} )A_{7} + (( - b^{2} - \alpha_{1} )A_{1} - b^{2} \alpha_{1} )A_{6} \hfill \\ + ( - b^{2} \alpha_{1} + \varepsilon_{4} \varepsilon_{5} )A_{1} + b^{2} \varepsilon_{4} \varepsilon_{5} )a_{3} a_{7} + (( - A_{4} - \alpha_{1} )A_{6} - A_{4} \alpha_{1} + \varepsilon_{4} \varepsilon_{5} )A_{9} a_{14} + \hfill \\ ((((b^{2} + \alpha_{1} )A_{2} - a_{4} \varepsilon_{2} )A_{3} + b^{2} \alpha_{1} A_{2} - b^{2} a_{4} \varepsilon_{4} )A_{4} + (\alpha_{1} b^{2} A_{2} - b^{2} a_{4} \varepsilon_{4} )A_{3} )A_{7} \hfill \\ + ((( - A_{1} - \alpha_{1} )A_{3} - A_{1} \alpha_{1} )A_{4} - A_{1} \alpha_{1} A_{3} )A_{6} + (( - A_{1} \alpha_{1} + \varepsilon_{4} \varepsilon_{5} )A_{3} + \varepsilon_{4} \varepsilon_{5} A_{1} )A_{4} + \varepsilon_{4} \varepsilon_{5} A_{1} A_{3} + \hfill \\ \frac{1}{{a_{8} }}(((((2b^{2} + \alpha_{1} )A_{6} + b^{4} + 2b^{2} \alpha_{1} - \varepsilon_{4} \varepsilon_{5} )a_{11} + (2b^{2} c + c\alpha_{1} )A_{7} )a_{1} + (( - 2b^{2} \omega \varepsilon_{1} - \omega \alpha_{1} \varepsilon_{1} )A_{2} + \hfill \\ \omega a_{4} \varepsilon_{1} \varepsilon_{4} ) + ((2b^{2} + \alpha_{1} )A_{2} - a_{4} \varepsilon_{4} )A_{5} A_{7} + ( - b^{2} - A_{1} - \alpha_{1} )A_{5} A_{6} + (( - b^{2} - \alpha_{1} )A_{1} - b^{2} \alpha_{1} + \varepsilon_{4} \varepsilon_{5} )A_{5} \hfill \\ - c\omega \varepsilon_{1} A_{1} - cb^{2} \omega \varepsilon_{1} - c\omega \alpha_{1} \varepsilon_{1} )a_{3} a_{7} + ((((b^{2} + A_{3} + \alpha_{1} )A_{4} + (b^{2} + \alpha_{1} )A_{3} + b^{2} \alpha_{1} )A_{6} \hfill \\ + ((b^{2} + \alpha_{1} )A_{3} + b^{2} \alpha_{1} - \varepsilon_{4} \varepsilon_{5} )A_{4} + (b^{2} \alpha_{1} - \varepsilon_{4} \varepsilon_{5} )A_{3} ) - b^{2} \varepsilon_{4} \varepsilon_{5} )a_{11} + ((b^{2} c + cA_{3} + \alpha_{1} c)A_{4} \hfill \\ + (b^{2} c + c\alpha_{1} )A_{3} + b^{2} c\alpha_{1} )A_{7} )a_{1} + ( - A_{5} A_{6} + ( - A_{4} - \alpha_{1} )A_{5} - c\omega \varepsilon_{1} )A_{9} a_{14} \hfill \\ + (( - \omega \varepsilon_{1} A_{2} A_{3} + ( - b^{2} \omega \varepsilon_{1} - \alpha_{1} \omega \varepsilon_{1} )A_{2} + \omega \varepsilon_{1} \varepsilon_{4} )A_{4} + (( - \omega b^{2} \varepsilon_{1} - \omega \alpha_{1} \varepsilon_{1} )A_{2} + \omega a_{4} \varepsilon_{1} \varepsilon_{4} )A_{3} \hfill \\ - b^{2} \alpha_{1} \omega \varepsilon_{1} A_{2} + b^{2} a_{4} \omega \varepsilon_{1} \varepsilon_{4} )a_{11} + ((A_{2} A_{3} + (b + \alpha_{1} )A_{2} - a_{4} \varepsilon_{1} )A_{4} + \varepsilon_{1} \varepsilon_{4} + ((b^{2} + \alpha_{1} )A_{2} - a_{4} \varepsilon_{4} )A_{3} \hfill \\ + b^{2} \alpha_{1} A_{2} - b^{2} a_{4} \varepsilon_{4} )A_{5} A_{7} + (( - A_{3} - A_{1} - \alpha_{1} )A_{4} + ( - A_{1} - \alpha_{1} )A_{3} - A_{1} \alpha_{1} )A_{5} A_{6} + \hfill \\ ((( - A_{1} - \alpha_{1} )A_{3} - A_{1} \alpha_{1} + \varepsilon_{4} \varepsilon_{5} )A_{4} + ( - A_{1} \alpha_{1} + \varepsilon_{4} \varepsilon_{5} )A_{3} + \varepsilon_{4} \varepsilon_{5} A_{1} \hfill \\ )A_{5} + ( - c\omega A_{1} \varepsilon_{1} - c\omega A_{3} \varepsilon_{1} - c\omega \alpha_{1} \varepsilon_{1} )A_{4} + ( - c\omega A_{1} \varepsilon_{1} - c\omega \alpha_{1} \varepsilon_{1} )A_{3} - c\omega \alpha_{1} \varepsilon_{1} A_{1} ) \hfill \\ \end{gathered} \right\}$$
(57)
$$\left. \begin{gathered} \Theta_{5} = ( - b^{4} \varepsilon_{4} A_{7} a_{3} a_{4} + (b^{4} A_{2} A_{7} \alpha_{1} - b^{2} A_{1} A_{6} + b^{2} A_{1} \varepsilon_{4} \varepsilon_{5} )a_{3} )a_{7} - b^{2} A_{3} A_{4} A_{7} \varepsilon_{4} a_{4} \hfill \\ + ( - A_{4} A_{6} \alpha_{1} + A_{4} \varepsilon_{4} \varepsilon_{5} )A_{9} a_{14} + b^{2} \alpha_{1} A_{2} A_{3} A_{4} A_{7} - \alpha_{1} A_{1} A_{3} A_{4} A_{6} + \varepsilon_{4} \varepsilon_{5} A_{1} A_{3} A_{4} \hfill \\ + \frac{1}{{a_{8} }}(((2b^{2} \omega a_{11} \varepsilon_{1} \varepsilon_{4} - 2b^{2} A_{5} A_{7} \varepsilon_{4} )a_{3} a_{4} + ((( - ( - b^{4} - 2b^{2} \alpha_{1} )A_{6} + b^{4} \alpha_{1} - 2b^{2} \varepsilon_{4} \varepsilon_{5} )a_{11} \hfill \\ - ( - b^{4} c - 2b^{2} c\alpha_{1} )A_{7} )a_{1} - (b^{4} \omega \varepsilon_{1} + 2b^{2} \alpha_{1} \omega \varepsilon_{1} )A_{2} a_{11} - ( - b^{4} - 2b^{2} \alpha_{1} )A_{2} A_{5} A_{7} \hfill \\ + ( - (b^{2} + \alpha_{1} )A_{1} - b^{2} )A_{5} A_{6} + ( - (b^{2} - \varepsilon_{4} \varepsilon_{5} )A_{1} + b^{2} \varepsilon_{4} \varepsilon_{5} )A_{5} - (b^{2} c\,\omega \,\varepsilon_{1} + c\alpha_{1} \omega \varepsilon_{1} )A_{1} \hfill \\ - b^{2} c\alpha_{1} \omega \varepsilon_{1} )a_{3} )a_{7} + (((b^{2} \omega \varepsilon_{1} \varepsilon_{4} + A_{3} \omega \varepsilon_{1} \varepsilon_{4} )A_{4} + b^{2} \varepsilon_{1} \varepsilon_{4} A_{3} )a_{11} + (( - b^{2} \varepsilon_{4} - A_{3} \varepsilon_{4} )A_{4} \hfill \\ - b^{2} \varepsilon_{4} A_{3} )A_{5} A_{7} )a_{4} + (((( - ( - b^{2} - \alpha_{1} )A_{3} + b^{2} \alpha_{1} )A_{4} + b^{2} \alpha_{1} A_{3} )A_{6} + ( - (b^{2} \alpha_{1} + \varepsilon_{4} \varepsilon_{5} )A_{3} \hfill \\ - b^{2} \varepsilon_{4} \varepsilon_{5} )A_{4} - b^{2} \varepsilon_{4} \varepsilon_{5} A_{3} )a_{11} + (( - (b^{2} c - c\alpha_{1} )A_{3} + b^{2} \alpha_{1} c)A_{4} + b^{2} c\alpha_{1} A_{3} )A_{7} )a_{1} + (( - A_{4} - \alpha_{1} )A_{5} A_{6} \hfill \\ + ( - A_{4} \alpha_{1} + \varepsilon_{4} \varepsilon_{5} )A_{5} - c\omega \varepsilon_{1} A_{4} - c\alpha_{1} \omega \varepsilon_{1} )A_{9} a_{14} + (( - (b^{2} \omega \varepsilon_{1} + \alpha_{1} \omega \varepsilon_{1} )A_{2} A_{3} - b^{2} \alpha_{1} \omega \varepsilon_{1} A_{2} )A_{4} \hfill \\ - b^{2} \varepsilon_{1} \alpha_{1} \omega A_{2} A_{3} )a_{11} + ( - ( - b^{2} - \alpha_{1} )A_{2} A_{3} + b^{2} \alpha_{1} A_{2} )A_{4} + b^{2} \alpha_{1} A_{2} A_{3} )A_{5} A_{7} + ((( - A_{1} - \alpha_{1} )A_{3} \hfill \\ - \alpha_{1} A_{1} )A_{4} - \alpha_{1} A_{1} A_{3} )A_{5} A_{6} + ((( - A_{1} \alpha_{1} + \varepsilon_{4} \varepsilon_{5} )A_{3} + \varepsilon_{4} \varepsilon_{5} A_{1} )A_{4} + \varepsilon_{4} \varepsilon_{5} A_{1} A_{3} )A_{5} + \hfill \\ (( - c\omega \varepsilon_{1} A_{1} - c\omega \alpha_{1} \varepsilon_{1} )A_{3} - c\omega \alpha_{1} \varepsilon_{1} A_{1} )A_{4} - c\omega \alpha_{1} \varepsilon_{1} A_{1} A_{3} ) \hfill \\ \end{gathered} \right\}$$
(58)

On the other hand, the basic coefficients of Eq. (47) are:

$$E_{1} = \frac{{\left( \begin{gathered} (\varepsilon_{1} \omega + A_{7} (a_{8} k_{n}^{2} - A_{5} ))(k_{n}^{2} - b^{2} )\left( {(k_{n}^{2} - A_{6} )(k_{n}^{2} - \alpha_{1} ) - \varepsilon_{4} \varepsilon_{5} } \right) - \hfill \\ \,\,\,\,\,A_{7} (k_{n}^{2} - b^{2} )(k_{n}^{2} - \alpha_{1} )\left[ {\varepsilon_{1} \omega A_{8} - \varepsilon_{4} \varepsilon_{5} + (k_{n}^{2} - A_{6} )(a_{8} k_{n}^{2} - A_{5} )} \right] \hfill \\ \end{gathered} \right)}}{{(k_{n}^{2} - \alpha_{1} )\left( {\varepsilon_{1} \omega A_{9} + (k_{n}^{2} - A_{6} )(a_{8} k_{n}^{2} - A_{5} )} \right) - \varepsilon_{4} \varepsilon_{5} }}$$
(59)
$$E_{2} = \frac{{\left( {(\varepsilon_{1} \omega + A_{7} (a_{8} k_{n}^{2} - A_{5} ))(k_{n}^{2} - b^{2} )(k_{n}^{2} - \alpha_{1} )} \right)}}{{(k_{n}^{2} - \alpha_{1} )\left( {\varepsilon_{1} \omega A_{9} + (k_{n}^{2} - A_{6} )(a_{8} k_{n}^{2} - A_{5} )} \right) - \varepsilon_{4} \varepsilon_{5} }}$$
(60)
$$E_{3} = \frac{{\varepsilon_{4} \left( {(\varepsilon_{1} \omega + A_{7} (a_{8} k_{n}^{2} - A_{5} ))(k_{n}^{2} - b^{2} )} \right)}}{{\varepsilon_{4} \varepsilon_{5} - (k_{n}^{2} - \alpha_{1} )\left( {\varepsilon_{1} \omega A_{9} + (k_{n}^{2} - A_{6} )(a_{8} k_{n}^{2} - A_{5} )} \right)}}$$
(61)
$$E_{4} = \frac{{a_{14} \left( {k_{n}^{2} - A_{4} )} \right)}}{{k_{n}^{4} - (a_{3} a_{7} + A_{3} + A_{7} )k_{n}^{2} + \left( {A_{3} A_{4} + b^{2} a_{3} a_{7} } \right)}}$$
(62)
$$E_{5} = \frac{{a{}_{7}a_{14} \left( {k_{n}^{2} - b^{2} )} \right)}}{{k_{n}^{4} - (a_{3} a_{7} + A_{3} + A_{7} )k_{n}^{2} + \left( {A_{3} A_{4} + b^{2} a_{3} a_{7} } \right)}}$$
(63)

The main parameters of Eqs. (52) can be obtained as:

$$\left. \begin{gathered} s_{1} = \,f_{1} k_{1} ,\quad s_{2} = \,f_{1} k_{2} ,\quad s_{3} = f_{2} + \lambda \,k_{3}^{2} - f_{3} ,\quad s_{4} = f_{2} + \lambda \,k_{4}^{2} - f_{3} , \hfill \\ s_{5} = f_{2} + \lambda \,\,k_{5}^{2} - f_{3} ,\quad s_{6} = f_{2} + \lambda \,\,k_{6}^{2} - f_{3} ,\quad f_{5} = b^{2} \mu .\quad \hfill \\ r_{1} = f_{5} + f_{4} \,\,k_{1}^{2} + k,\quad r_{2} = f_{5} + f_{4} \,k_{2}^{2} + k,\quad r_{3} = \,f_{1} k_{3} ,\quad \hfill \\ r_{4} = \,f_{1} k_{4} ,\quad r_{5} = \,f_{1} k_{5} ,\quad r_{6} = \,f_{1} k_{6} ,f_{1} = - i\,b\,(2\mu + k),\quad \hfill \\ f_{2} = \lambda_{0} - b^{2} \,(\lambda + 2\mu + k),\,f_{3} = \hat{\gamma }(1 + d_{n} ),\,f_{4} = \mu + k.\quad \hfill \\ \end{gathered} \right\}$$
(64)

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Mohamed, M.S., Lotfy, K., El-Bary, A. et al. Photo-thermal-elastic waves of excitation microstretch semiconductor medium under the impact of rotation and initial stress. Opt Quant Electron 54, 241 (2022). https://doi.org/10.1007/s11082-022-03533-x

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