On the use of Hankel transformation in mathematical modeling of catodoluminescence in a homogeneous semiconductor material

The possibility of using the Hankel transform in solving the non–stationary differential heat and mass transfer equation and the subsequent description of the decrease in the intensity of linear radiative recombination of minority charge carriers (or excitons) generated by an electron probe in a homogeneous semiconductor target is considered. The mathematical model of unsteady diffusion in the problem under consideration (when the electron probe is turned off) is described using the partial differential equation
\begin{equation*} \frac{\partial c(x, y, z, t)}{\partial t} = D\Delta c(x, y, z, t) - \frac{c(x, y, z, t)}{\tau } \end{equation*}
with initial condition
\begin{equation*} c(x, y, z, 0) = n(x, y, z). \end{equation*}
Here $c(x, y, z, t)$ is the concentration of minority carriers (or excitons) at a point with coordinates $(x, y, z)$ at time $t$, $ \Delta=\partial^{2}/\partial x^{2} + \partial^{2}/\partial y^{2} + \partial/\partial z^{2} $ — Laplace operator and all the coefficients in the differential equation that characterize the semiconductor, are constant values. Function $n (x, y, z)$ satisfies the stationary differential equation, describing diffusion in the state of quasiequilibrium, with the electron probe turned on, when the number of minority charge carriers (or excitons) generated and recombined in the semiconductor volume per unit time is constant and equal to each other:
\begin{equation*} \Delta n(x, y, z) - \frac{n(x, y, z)}{{\lambda ^2 }} = - \rho(x, y, z). \end{equation*}
The boundary conditions for this equation have the form:
\begin{equation*} D \frac{\partial n(x, y, 0)}{\partial z} = v_s n(x, y, 0), \lim_{z \to +\infty} n(x, y, z)=0, \end{equation*}

\begin{equation*} \lim_{x \to +\infty} n(x, y, z)=0, \lim_{x \to -\infty} n(x, y, z)=0, \lim_{y \to +\infty} n(x, y, z)=0, \lim_{y \to -\infty} n(x, y, z)=0, \end{equation*}
and the function $\rho (x, y, z)$ describes the concentration of carriers, generated in a semiconductor per unit time. Cathodoluminescence intensity for linear radiative recombination
\begin{equation*} I\left( t \right) \cong \int \!\! \int\limits_P \!\!\int c(x, y, z, t) \, dV. \end{equation*}
Here $P$ is the region in which minority charge carriers or excitons are generated, the recombination of which gives cathodoluminescent radiation emerging from the semiconductor volume. Received expression for $I\left( t \right)$, which is more convenient for practical implementation than previously used in this method. In general, the obtained results can be used in planning the experiment, conducting quantitative calculations and processing the results in electron-probe technologies.