Special phase invariant formulas of higher order: Expected values.

Area of Science:

• Crystallography
• Mathematical Chemistry
• Data Analysis

Background:

• Crystal structure analysis relies on phase information, which is often difficult to obtain.
• Higher-order phase invariants are crucial for solving complex crystal structures.
• Existing methods for calculating phase invariants can be computationally intensive.

Purpose of the Study:

• To derive new formulas for the cosines of higher-order phase invariants.
• To express these cosines using averages of simple functions of structure factor magnitudes.
• To highlight the utility of these formulas in conjunction with space group symmetries.

Main Methods:

• Derivation of formulas using determinantal joint probability distributions.
• Expressing invariant cosines as expected values.
• Utilizing averages over simple functions of known structure factor magnitudes.

Main Results:

• Development of "special" formulas for higher-order phase invariant cosines.
• Formulas are expressed in terms of readily calculable averages.
• Identification of embedded seminvariants derived from invariants and space group symmetries as a key application.

Conclusions:

• The derived formulas provide a novel approach to calculating higher-order phase invariants.
• These formulas simplify the estimation of invariant cosines from experimental data.
• The integration with space group symmetries enhances the practical application of these crystallographic tools.