L. A. Bakaleinikov

Silicon dioxide modification by an electron beam

The overheating temperature of a microvolume of silicon dioxide produced by bombardment by a high specific-power electron beam has been estimated. Calculations showed that the maximum temperature to which a microvolume of silicon dioxide is overheated can be as high as 1200°C for an electron beam current of 100 nA. The variation in the cathodoluminescence characteristics of amorphous silica with different contents of hydroxyl groups was studied for various electron beam specific-power levels. The impact of a high specific-power electron beam was shown to create additional lattice defects up to the formation of silicon clusters.

Some general properties of the nonlinear collision integral in the Boltzmann equation

A new method for computing matrix elements of the collision integral in the Boltzmann equation makes it possible to consider many problems of the kinetic theory of gases in a new way. Nonlinear kernels of the collision integral are studied and similarity relations, which simplify significantly the problem of constructing of such kernels, are proved.

Matrix elements and kernels of the collision integral in the Boltzmann equation

This article elaborates upon our previous work in which some general properties of the matrix elements and kernels of the gain and loss terms of the collision integral were found. The object of study is the loss term of the collision integral, since related analytical expressions are simple. Formulas to calculate the matrix elements are derived. The kernels of power-law interaction potentials are completely investigated and constructed using analytical and numerical approaches.

Relations between nonlinear kernels of the collision integral

Previously, to solve the Boltzmann equation by the moments method with expansion of the distribution function by spherical Hermit polynomials, a new computational method was suggested which allowed to construct nonlinear matrix elements of the collision integral with very large indices. This made it possible to substantially advance in construction of the distribution function. Limitations to convergence of the distribution function that appear in moment method are eliminated if we come to expansion by spherical harmonics from expansion by spherical Hermit polynomials. In this case, a complex five-fold collision integral is replaced by a set of comparatively simple integral operators, and kernels Gl1,l2l(c, c1, c2) of these operators become the analog of matrix elements. We found the relations between expansions of the distribution function in the reference frames with various velocities of motion along marked axis. Starting from the invariance condition of the collision integral with respect to selection of such reference frames, we derived recurrent relations between the kernels with various indices. These relations allow us to construct any nonlinear kernel Gl1, l2l(c, c1, c2), if the kernel G0,00(c, c1, c2) is known.

The collision integral kernels of the scalar nonlinear Boltzmann equation for pseudopower potentials

Analytic expressions are obtained for the collision integral kernels of the isotropic nonlinear Boltzmann equation for pseudopower interaction potentials. It is shown that for an arbitrary power exponent, the kernels can be expressed in terms of hypergeometric functions. In some cases, the kernels can be expressed in terms of elementary functions.

Construction of kernels of the nonlinear collision integral in the Boltzmann equation using laplace transformation

It is shown that the relation between kernels Ll(v, v1) of the linear collision integral and kernels Gl,0l(v, v1, v2) of the nonlinear collision integral can be reduced to the Laplace transformation. Analytic expressions for nonlinear kernels G0,0+0(v, v1, v2) and G1,0+1(v, v1, v2) are determined for hard spheres and pseudo-Maxwellian molecules.

Analytical and numerical approaches to calculating the escape function for the emission of medium-energy electrons from uniform specimens

A program for the simulation of electron transport by the Monte Carlo method has been developed. This program implies the model of single scattering and dielectric approach (to calculate the characteristics of an inelastic interaction). The escape functions for aluminum, germanium, and gold in the 0.012–20 keV energy range were calculated. The characteristic lengths determining the electron dynamics (the elastic and inelastic mean free paths, isotropization length, and integral path) were calculated using the differential cross sections for both elastic and inelastic interactions for these elements. The analysis of the relations between the characteristic lengths made it possible to determine the applicability range of the analytical expressions for the emission functions obtained in [1]. The comparison of the results obtained analytically and numerically confirmed the conclusion of [1] about the form of the analytical approximation of the emission function for electrons of various energies and showed the validity of the obtained analytical expressions for the escape lengths of electrons.

Calculation of the Linear Kernel of the Collision Integral for the Hard-Sphere Potential

The expansion of a distribution function in spherical harmonics transforms the Boltzmann equation into a system of integro-differential equations with kernels depending only of the magnitudes of velocities. The kernels can be expressed in terms of the sums including the matrix elements (MEs) of the collision integral. The kernels are constructed using new results of ME calculations; analysis of errors is carried out with the help of analytic expressions for kernels, which were derived by Hilbert and Hecke for the hard-sphere model. The concept of generalized matrix elements is introduced and their asymptotic representation is constructed for large values of indices. Analytic expressions for the contribution from MEs with large indices to the kernels are constructed. The high accuracy of the construction of a kernel using MEs is demonstrated.

Depth Profiling of Semiconductor Structures by X-Ray Microanalysis Using the Electron Probe Energy Variation Technique

Depth profiling of semiconductor structures by X-ray microanalysis using the electron probe variation technique followed by mathematical processing of the measurement results is described. Experimental dependences of the relative X-ray intensity on the energy of the electron probe were compared to the Monte Carlo simulation results. Concentration depth profiles were estimated using a priori assumptions, and numerical parameters of the depth distributions were determined from the best fit of calculated curves to the experimental data. The technique was applied to determine Al depth profiles in SiC and GaN samples.

Construction of kernels G l, 0 l of the nonlinear collision integral in the Boltzmann equation for arbitrary l

It was shown in [1] that kernels Ll(v, v1) of linear collision integral and kernels Gl, 0l(v, v1, v2) of nonlinear collision integral are related by the Laplace transform. Here, analytical expressions are derived for nonlinear kernels Gl, 0+l(v, v1, v2) with arbitrary l for models of hard spheres and pseudo-Maxwellian molecules using the Laplace transform method.