A Minimal Principle in the Direct Methods of X-Ray Crystallography
The electron density function, ρ(r), in a crystal determines its diffraction pattern, that is, both the magnitudes and phases of its X-ray diffraction maxima, and conversely. If, however, as is always the case, only magnitudes are available from the diffraction experiment, then the density function ρ(r) cannot be recovered. If one invokes prior structural knowledge, usually that the crystal is composed of discrete atoms of known atomic numbers, then the observed magnitudes are, in general, sufficient to determine the positions of the atoms, that is, the crystal structure.
The intensities of a sufficient number of X-ray diffraction maxima determine the structure of a crystal. The available intensities usually exceed the number of parameters needed to describe the structure. From these intensities a set of members ∣EH∣ can be derived, one corresponding to each intensity. However, the elucidation of the crystal structure also requires a knowledge of the complex numbers EH = ∣EH∣ exp (iϕH), the normalized structure factors, of which only the magnitudes ∣EH∣ can be determined from experiment. Thus, a “phase” ϕH, unobtainable from the diffraction experiment, must be assigned to each ∣EH∣, and the problem of determining the phases when only the magnitudes ∣EH∣ are known is called “the phase problem”. Owing to the known atomicity of crystal structures and the redundancy of observed magnitudes ∣EH∣, the phase problem is solvable in principle.
KeywordsConditional Expectation Diffraction Experiment Minimal Principle Electron Density Function Structure Invariant
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