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Absorption and weighted path lengths in cylinders and spheres
1Department of Mathematics and Natural Science, Stavanger College, Ullandhaug, Norway. gunnar.thorkildsen@tn.his.no
Summary
This study introduces a Green-function technique to calculate absorption factors and path lengths in cylinders and spheres. The method provides accurate results for numerical computations and analytical solutions in specific angular limits.
Area of Science:
- Solid-state physics
- Crystallography
- Mathematical physics
Background:
- Accurate calculation of absorption and path length is crucial in understanding wave propagation in crystalline materials.
- Existing methods may lack efficiency or analytical tractability for specific geometries.
Purpose of the Study:
- To develop an efficient and accurate method for calculating the normal absorption factor and weighted path length.
- To apply this method to cylindrical and spherical crystal geometries.
- To derive analytical expressions in limiting cases.
Main Methods:
- Utilizing the boundary-value Green-function technique applied to the Takagi equations.
- Expressing calculations as double integrals in angular coordinates.
- Evaluating results in the limits of Bragg angles theta oh = 0° and theta oh = 90°.
Main Results:
- The Green-function technique allows for the computation of normal absorption factor and weighted path length.
- The derived double integrals are well-defined and suitable for numerical implementation.
- Exact analytical expressions were obtained for both cylindrical and spherical geometries at specific Bragg angles.
Conclusions:
- The developed Green-function method offers a robust approach for analyzing wave propagation in extended objects.
- The ease of numerical implementation and availability of analytical solutions enhance its practical applicability.
- This technique provides valuable insights into the behavior of X-rays or other waves in crystalline structures.