The Phase Problem of X-ray Crystallography

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 jEHj can be derived, one corresponding to each intensity. However, the elucidation of the crystal structure also requires a knowledge of the complex numbers EH = jEHj exp(i'H), the normalized structure factors, of which only the magnitudes jEHj can be determined from experiment. Thus, a “phase” 'H, unobtainable from the diffraction experiment, must be assigned to each jEHj, and the problem of determining the phases when only the magnitudes jEHj are known is called the “phase problem”. Owing to the known atomicity of crystal structures and the redundancy of observed magnitudes jEHj, 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. 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 1000 atoms when data are available to atomic resolution. In the case that single wavelength anomalous scattering (SAS) data are available, the probabilistic machinery leads to estimates of special linear combinations of the phases, the so-called structure invariants. A method of going from estimates of the structure invariants to the values of the individual phases is described. 164 H.A. Hauptman / The Phase Problem of X-ray Crystallography