Generalized Bessel functions in incommensurate structure analysis

Abstract

The analysis of incommensurate structures is computationally more difficult than that of normal ones. This is mainly a result of the structure-factor expression, which involves numerical integrations or infinite series of Bessel functions. Both approaches have been implemented in existing computer programs. Compact analytical expressions are known for special cases only. Recently, a new theory of generalized Bessel functions has been developed. The number of theoretical results and applications is increasing rapidly. Numerical properties and algorithms are being studied. A possible application of the generalized Bessel functions for incommensurate structure analysis is proposed. These functions can be used to derive analytical expressions for structure factors and all partial derivatives for a wide class of incommensurate crystals. The existing programs can be improved by taking into account some interesting numerical and analytical properties of these new functions, like recurrence relations, analytical expressions for derivatives, generating functions and integral representations. A new class of special functions, suitable for dealing with incommensurate structures in a more analv:ical way, is emerging.