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Multiple Bragg reflection by a thick mosaic crystal. II. Simplified transport equation solved on a grid
Folkmar Bornemann1, Yun Yvonna Li2, Joachim Wuttke2
1Zentrum Mathematik M3, Technische Universität München, 85747 Garching, Germany.
Summary
Generalized Darwin-Hamilton equations model multiple Bragg reflections in imperfect crystals. Energy conservation simplifies these to the two-ray Darwin-Hamilton equations, with a new numeric solution presented.
Area of Science:
- Crystallography
- Solid State Physics
- X-ray Diffraction
Background:
- The Darwin-Hamilton equations model X-ray diffraction in crystals.
- Describing multiple Bragg reflections in imperfect crystals requires advanced theoretical frameworks.
Purpose of the Study:
- To simplify and solve the generalized Darwin-Hamilton equations for imperfect crystals.
- To investigate the effects of multiple Bragg reflections on X-ray diffraction patterns.
Main Methods:
- Utilizing energy conservation to simplify the generalized Darwin-Hamilton equations.
- Developing a numerical solution using transfer matrices and spectral collocation.
- Comparing results with the conventional two-ray Darwin-Hamilton equations.
Main Results:
- The generalized equations simplify to the two-ray Darwin-Hamilton equations as a first-order approximation.
- The numerical method efficiently solves for multiply reflected rays.
- Demonstrated orientational spread and rocking curve distortion in simulations.
Conclusions:
- The simplified generalized Darwin-Hamilton equations provide a robust model for X-ray diffraction in imperfect crystals.
- The presented numerical method offers an efficient way to analyze complex diffraction phenomena.
- Understanding ray distortion is crucial for interpreting experimental data, especially with limited detector angles.
