Section 15.2.1. Introduction

The intensities of X-ray diffraction spots measured from a crystal give us only the amplitudes of the diffracted waves. To reconstruct a map of the electron density in the crystal, the unmeasured phase information is also required. In fact, the phases are much more important to the appearance of the map than the measured amplitudes. When phases are supplied by an atomic model, therefore, some degree of model bias is inevitable.

The optimal use of model phase information requires an estimate of its reliability, specifically the probability that various values of the phase angle are true. Such a probability distribution can be derived, starting first with the relationship between the structure factor (amplitude and phase) of the model and that of the true crystal structure. The phase probability distribution can then be obtained from this and used, for instance, to provide a figure-of-merit weighting that minimizes the r.m.s. error from the true electron density.

Even with figure-of-merit weighting, model-phased electron density is biased towards the model. The systematic bias component of model-phased map coefficients can be predicted, allowing the derivation of map coefficients that give electron-density maps with reduced model bias. With the help of a few simple assumptions, a correction for bias can also be made when different sources of phase information are combined.

Finally, the refinement of a model against the observed amplitudes allows a certain amount of overfitting of the data, which leads to an extra `refinement bias'. Fortunately, the use of appropriate refinement strategies, including maximum-likelihood targets, can reduce the severity of this problem.