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New algorithm for generating all coincidence-site lattices of the cubic crystal system
Kazuaki Kawahara1,2,3, Kazutoshi Inoue3,4,5, Mitsuhiro Saito1
1Institute of Engineering Innovation, University of Tokyo, Yayoi, Bunkyo, Tokyo, 113-8656, Japan.
Summary
This study introduces a linear algebra method to identify all coincidence-site lattice (CSL) orientations in cubic systems. This approach simplifies finding CSL relationships and analyzing grain boundaries without complex number theory.
Area of Science:
- Materials Science
- Crystallography
- Solid State Physics
Background:
- Coincidence-site lattice (CSL) theory is crucial for understanding commensurate crystal orientations and grain boundaries.
- Ranganathan's formula identifies CSL relationships but can lead to duplicates requiring number theoretical analysis.
- Current methods for CSL analysis can be complex and computationally intensive.
Purpose of the Study:
- To develop a simplified method for identifying all CSL orientation relationships in cubic systems.
- To avoid duplication and the need for number theoretical analysis in CSL determination.
- To investigate the relationship between CSL values and the occurrence of symmetric tilt grain boundaries.
Main Methods:
- Utilized linear algebra to establish a necessary condition for CSL orientation relationships.
- Developed a method based on the commensurability of (hkl) and (001) planes.
- Compared the number of solutions from the new condition with those from symmetric tilt grain boundary equations.
Main Results:
- A simple necessary condition using linear algebra was derived to find all CSL orientation relationships in cubic systems.
- The method successfully avoids duplication and the need for number theoretical analysis.
- The study found that the number of CSL orientations without symmetric tilt grain boundaries increases with higher Σ values.
Conclusions:
- The proposed linear algebra approach offers a more straightforward method for determining CSL orientations in cubic materials.
- This simplification aids in the analysis of grain boundary structures and properties.
- The findings contribute to a better understanding of the prevalence of specific grain boundary types as a function of CSL order.
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