Direct methods: the identification of space-group-specific inconsistent three-phase structure invariants.

Area of Science:

• Crystallography
• Solid-state chemistry

Background:

• Crystallographic data analysis can yield multiple phase estimates.
• Space-group symmetry influences the relationships between these estimates.

Purpose of the Study:

• To identify space groups that generate pairs of inconsistent triple phase relationships.
• To catalog the forms of these triple pairs and their phase inconsistencies.

Main Methods:

• Systematic examination of all 230 crystallographic space groups.
• Analysis of triple reflections and their symmetry-related forms.

Main Results:

• Certain space groups permit the generation of two equally probable, yet inconsistent, invariant estimates.
• A known phase shift exists between these inconsistent estimates due to space-group symmetry.
• A complete list of space groups exhibiting this phenomenon is provided.

Conclusions:

• Understanding these space-group-dependent phase inconsistencies is crucial for accurate crystallographic phase determination.
• The findings facilitate the selection of appropriate space groups and the interpretation of crystallographic data.