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Utilization of Symmetry in Solving the Boltzmann Equation
Author(s) -
Mann E.
Publication year - 1969
Publication title -
physica status solidi (b)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.51
H-Index - 109
eISSN - 1521-3951
pISSN - 0370-1972
DOI - 10.1002/pssb.19690320241
Subject(s) - degenerate energy levels , mathematics , symmetry (geometry) , boltzmann equation , simple (philosophy) , scattering , kernel (algebra) , algebraic number , ordinary differential equation , mathematical analysis , differential equation , physics , quantum mechanics , pure mathematics , geometry , epistemology , philosophy
Boltzmann's integral or integro‐differential equation may be reduced to a finite system of linear algebraic or simple integral equations if the kernel (the transition probability) is degenerate. This happens for elastic scattering of electrons by localized defects if the scattering is described in terms of Wannier functions. It is shown how the transition probability may be represented as a sum of terms transforming according to the irreducible representations of the point group of the problem. As a consequence, the system of linear equations is split into smaller systems. Since the calculation of the electrical current density involves only a few of these representations, only a few systems of smaller order need to be solved. The reduction procedure is described in detail for three examples with cubic, tetragonal, and orthorhombic symmetry.