The Phase Problem of X-ray Crystallography: Overview

Abstract

The intensities of a sufficient number of X-ray diffraction maxima determine the structure of a crystal, that is, the positions of the atoms in the unit cell of the crystal. The available intensities usually exceed the number of parameters needed to describe the structure. From these intensities a set of numbers |E _H |can be derived, one corresponding to each intensity. However, the elucidation of the crystal structure also requires a knowledge of the complex numbers E _H =|E _H |exp(iф _H ), the normalized structure factors, of which only the magnitudes |E _H | can be determined from experiment. Thus, a “phase” фH, unobtainable from the diffraction experiment, must be assigned to each |E _H |, and the problem of determining the phases when only the magnitudes |E _H | are known is called the “phase problem”. Owing to the known atomicity of crystal structures and the redundancy of observed magnitudes |E _H |, the phase problem is solvable in principle.

Probabilistic methods have traditionally played a key role in the solution of this problem. They have led, in particular, to the so-called tangent formula which, in turn, has played the central role in the development of methods for the solution of the phase problem. A number of computer programs, stressing different aspects of the central theme, have proven to be particularly effective.

Finally, the phase problem may be formulated as one in constrained global optimization. A method for avoiding the countless local minima in order to arrive at the constrained global minimum leads to the Shake-and-Bake algorithm, a completely automatic solution of the phase problem for structures containing as many as 600 atoms when data are available to atomic resolution.