The crystallographic fast Fourier transform. III. Centred lattices

Area of Science:

• Crystallography
• Computational Science
• Mathematical Physics

Background:

• Crystallographic Fast Fourier Transforms (FFT) are crucial for structure determination.
• Evaluating FFT for centered lattices presents computational challenges.
• Existing algorithms may not achieve maximal symmetry reduction for all space groups.

Purpose of the Study:

• To present novel algorithms for evaluating crystallographic FFT for centered lattices.
• To identify space groups where maximal symmetry reduction can be achieved.
• To discuss grid size requirements and their impact on algorithm selection.

Main Methods:

• Development of new algorithms for crystallographic FFT evaluation.
• Application of algorithms to 80 space groups with centring operators.
• Comparison with existing methods for symmetry reduction.

Main Results:

• The presented algorithms are applicable to 80 space groups.
• Maximal symmetry reduction is achieved for 44 space groups by combining new and existing algorithms.
• New algorithms are required for the remaining space groups.

Conclusions:

• The developed algorithms significantly improve FFT evaluation for centered lattices.
• Maximal symmetry reduction is attainable for a substantial subset of space groups.
• Further research is needed for algorithms applicable to all space groups.