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An algorithm for the arithmetic classification of multilattices.

Giuliana Indelicato¹

¹Department of Mathematics, York Center for Complex Systems Analysis, The University of York, Heslington, England, UK. giuliana.indelicato@york.ac.uk

Acta Crystallographica. Section A, Foundations of Crystallography

|December 20, 2012

Summary

A new procedure constructs and classifies monoatomic multilattices in any dimension. This algorithm aids in locating lattice points and determining arithmetic equivalence, offering a robust computational tool.

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Area of Science:

Crystallography
Computational Materials Science
Mathematical Physics

Background:

Monoatomic multilattices are fundamental structures in various scientific domains.
Efficient methods for their classification and point determination are crucial for advancing research.
Existing methods may lack generality or computational efficiency across arbitrary dimensions.

Purpose of the Study:

To develop a general procedure for constructing and classifying monoatomic multilattices.
To create an algorithm for determining the location of points in multilattices with specific symmetries.
To enable the determination of arithmetic equivalence between assigned multilattices.

Main Methods:

The procedure leverages concepts from integral matrix theory.
Key techniques include the reduction of matrices to their Smith normal form.
The approach is designed for implementation in a classification software package.

Main Results:

A systematic procedure for the construction and classification of monoatomic multilattices is established.
The algorithm successfully determines the location of points for multilattices of a given symmetry.
The method provides a definitive way to check for arithmetic equivalence between multilattices.

Conclusions:

The developed procedure offers a comprehensive framework for understanding monoatomic multilattices.
The computational approach based on integral matrix theory is effective and generalizable.
This work facilitates the creation of software for lattice classification and analysis.