Normally supportive sublattices of crystallographic space groups

Area of Science:

• Crystallography
• Group Theory
• Solid State Physics

Background:

• Understanding normal subgroups of crystallographic space groups (csg's) is crucial for various applications.
• Normal subgroups that are themselves csg's are termed `csg-normal', and their point groups must be `cpg-normal' subgroups.

Purpose of the Study:

• To tabulate translational subgroups (sublattices) capable of supporting csg-normal subgroups.
• To identify necessary conditions for sublattices to be `normally supportive'.

Main Methods:

• Identifying two necessary conditions on the relative sublattice basis for supporting csg-normal subgroups.
• Tabulating normally supportive sublattices for each cpg-normal subgroup across 3D space groups' arithmetic crystal classes.

Main Results:

• Defined `normally supportive' sublattices based on point group and cpg-normal subgroup operations.
• Generated symbolic tables of infinite families of normally supportive sublattices.

Conclusions:

• The tabulated sublattices provide a foundation for constructing csg-normal subgroups via group extension.
• The existence of a csg-normal subgroup is not guaranteed by the pair of cpg-normal subgroup and normally supportive sublattice.