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General phase-invariant formulas of higher order: Expected values.
1Laboratory for the Structure of Matter, Naval Research Laboratory, Washington, D.C. 20375.
Summary
New formulas for higher-order phase invariants in crystal structure analysis are derived using probability distributions. These general formulas express invariant cosines using structure factor magnitudes, aiding crystallographic phase determination.
Area of Science:
- Crystallography
- Mathematical Chemistry
- Structural Analysis
Background:
- Crystal structure analysis relies on determining phase invariants.
- Existing methods for phase invariants have limitations.
- Higher-order phase invariants are crucial for complex structures.
Purpose of the Study:
- To derive new, general formulas for the cosines of higher-order phase invariants.
- To express these invariant cosines in terms of measurable structure factor magnitudes.
- To enhance the utility of phase invariants in crystal structure determination.
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
- general
- formulas for higher-order phase invariant cosines.
- Formulas are distinct from previously established
- special
- formulas.
- Results show invariant cosines are functions of structure factor magnitude averages.
Conclusions:
- The derived general formulas offer a new approach to phase invariant calculation.
- Combining these formulas with space group symmetries (embedded seminvariants) maximizes their utility.
- This work advances computational methods in crystal structure analysis.