Ambiguity of structure determination from a minimum of diffraction intensities

Area of Science:

• Crystallography
• Materials Science
• Mathematical Chemistry

Background:

• The determination of crystal structures from diffraction intensities is a fundamental problem in materials science.
• Historically, the ambiguity in crystal structure determination has been recognized but not quantitatively understood.
• Bernstein's theorem provides a method to estimate the number of possible crystal structures from a minimal diffraction intensity dataset.

Purpose of the Study:

• To quantitatively estimate the ambiguity in determining two- and three-dimensional crystal structures from minimal diffraction intensity data.
• To extend the application of Bernstein's theorem to higher dimensions for structure determination.
• To analyze the impact of data resolution on structural ambiguity.

Main Methods:

• Utilizing Bernstein's theorem to analyze the number of possible crystal structures.
• Extending algebraic methods to estimate ambiguity in 2D and 3D crystal structures.
• Evaluating the effect of substituting low-resolution with high-resolution diffraction intensities.

Main Results:

• The ambiguity in crystal structure determination increases exponentially with the size of the structure.
• The number of possible crystal structures grows rapidly with increasing dimensionality.
• Substituting a small number of lower-resolution intensities with higher-resolution ones has minimal impact on ambiguity.

Conclusions:

• Crystal structure determination from minimal diffraction data is inherently ambiguous, with ambiguity escalating rapidly with structural complexity.
• The dimensionality and size of the crystal structure are critical factors influencing the degree of ambiguity.
• Data resolution plays a limited role in mitigating ambiguity when only a few substitutions are made in the minimal dataset.