Darwin calculations of the Debye-Waller factor

Jin-Seok Chung, Purdue University

Abstract

The effect of temperature on x-ray diffraction from perfect crystals has been investigated with an extension of the Darwin theory of dynamical diffraction. We tried to test the validity of the Debye-Waller model, in which the crystal structure factors are multiplied by a Debye-Waller factor which is derived from the time-averaged distribution of charge induced by thermal vibrations. The Debye-Waller factor was rigorously derived only for kinematical diffraction, which is appropriate for imperfect crystals or for very small domains. The common usage of Debye-Waller factors to describe dynamical diffraction from perfect crystals with thermal vibrations has never been rigorously justified. The extended Darwin theory keeps the original Darwin recursion relations, but allows each plane in the crystal to be separately evaluated. The effective planar displacements as a function of temperature were determined numerically from measured phonon dispersion curves and Bose-Einstein statistics. The application to a realistic model of a macroscopic crystal is made possible by the enormous advances in computational speed available with current computers. A comparison with published experimental data for the temperature dependence of Si (660) showed good agreement with the data. The fit was comparable to the Debye-Waller fit, even though the Darwin theory is essentially parameter-free. When the calculations are extended to various Ge reflections at temperatures spanning 5-900 K, very large disagreements with the Debye-Waller model are found for higher-order reflections and higher temperatures. By changing the values of the atomic form factor, it was possible to use the Darwin theory to investigate the role of finite atomic size in the temperature dependence of x-ray diffraction. The calculations for point-like atoms showed systematically different behavior with temperature. This is the first evidence that atomic size plays any role in the temperature dependence of diffraction. These Darwin calculations effectively describe the diffraction of x-rays from an instantaneous configuration of planar displacements caused by thermal vibrations. The differences with the Debye-Waller model show that the average reflectivity from many different instantaneous configurations is not the same as the reflectivity from the average configuration, at least for dynamical diffraction.

Degree

Ph.D.

Advisors

Durbin, Purdue University.

Subject Area

Condensation

COinS