Computer simulation of bent perfect crystal diffraction profles

M. Sanchez del Rio, C. Ferrero and V. Mocella
European Synchrotron Radiation Facility
BP 220, 38043 Grenoble-Cedex France

ABSTRACT

Various theoretical methods for calculating diffraction proffles of perfect crystals are available in literature. Although
these methods hold within certain validity ranges due to their inherent approximations, they constitute the current
state-of-the-art of numerical computation of diffraction profles.
In this paper we summarize the theory of Zachariasen for at crystals, the multi-lamellar approximation for
bent crystals and the Penning-Polder approximation for bent Laue crystals. Some examples of their results are
presented. Another method to calculate the diffraction profle consists in solving the Takagi-Taupin equations. The
finite difference method, that provides a numerical solution of these equations, is brie y discussed.
A new method for solving numerically these equations using the finite element method is proposed. This method
is very exible, because it can consider a crystal with an arbitrary shape and cover the case of critical regions (i.e.,
inhomogeneities and deformations) with tne elements. In addition, it can couple naturally the diffraction calculation
with thermal or mechanical crystal deformations. These deformations are generally induced by the x-ray beam (heat
load), the crystal bender (mechanical stress) or are intrinsic to the crystal (inhomogeneities, impurities, dislocations,
etc.). An example of the feasibility of this method is shown.

Keywords: crystal optics, dynamical diffraction, bent or distorted crystals, perfect crystals, finite element method,
finite difference method, Takagi-Taupin equations