On the arithmetic classification of crystal structures

Area of Science:

• Crystallography
• Materials Science
• Mathematical Physics

Background:

• Existing classification of crystal structures relies on space groups and lattice complexes.
• Pitteri & Zanzotto introduced an arithmetic criterion for classifying n-lattices.
• The applicability of Fischer & Koch's lattice complex criterion to complex structures can be challenging.

Purpose of the Study:

• To systematically analyze monoatomic 2-lattices using the arithmetic criterion.
• To compare the arithmetic criterion with the Fischer & Koch lattice complex classification for monoatomic 2-lattices.
• To demonstrate the utility of the arithmetic criterion for classifying more complex crystal structures, such as monoatomic 4-lattices.

Main Methods:

• Systematic analysis of monoatomic 2-lattices.
• Application of the arithmetic criterion for crystal structure classification.
• Examination of elemental allotropes to illustrate the classification of monoatomic 4-lattices.

Main Results:

• Identified 29 distinct arithmetic types of monoatomic 2-lattices.
• Demonstrated that the arithmetic criterion and the Fischer & Koch criterion coincide for monoatomic 2-lattices.
• Showcased the arithmetic criterion's ability to classify complex structures like monoatomic 4-lattices.

Conclusions:

• The arithmetic criterion provides a robust method for classifying crystal structures, particularly when space-group classifications are ambiguous.
• The 29 arithmetic types represent a complete classification of monoatomic 2-lattices.
• The arithmetic criterion offers a valuable group-theoretical framework for advanced crystallographic analysis.