Section 15.2.3.2. Probability distributions for variable coordinate errors

In the Sim distribution, an atom is considered to be either exactly known or completely unknown in its position. These are extreme cases, since there will normally be varying degrees of uncertainty in the positions of various atoms in a model. The treatment can be generalized by allowing a probability distribution of coordinate errors for each atom. In this case, the centroid for the individual atomic contribution to the structure factor will no longer be obtained by multiplying by either zero or one. Averaged over the circle corresponding to possible phase errors, the centroid will generally be reduced in magnitude, as illustrated in Fig. 15.2.3.1. In fact, averaging to obtain the centroid is equivalent to weighting the atomic scattering contribution by the Fourier transform of the coordinate-error probability distribution, . By the convolution theorem, this in turn is equivalent to convoluting the atomic density with the coordinate-error distribution. Intuitively, the atom is smeared over all of its possible positions. The weighting factor, , is thus analogous to the thermal-motion term in the structure-factor expression.

Figure 15.2.3.1| top | pdf | Centroid of the structure-factor contribution from a single atom. The probability of a phase for the contribution is indicated by the thickness of the line.

The variances for the individual atomic contributions will differ in magnitude, but if there are a sufficient number of independent sources of error, we can invoke the central limit theorem again and assume that the probability distribution for the structure factor will be a Gaussian centred on . If the coordinate-error distribution is Gaussian, and if each atom in the model is subject to the same errors, the resulting structure-factor probability distribution is the Luzzati (1952) distribution. In this special case, for all atoms, where D is the Fourier transform of a Gaussian and behaves like the application of an overall B factor.