Multi-cell Method

An alternative method for calculating the diffuse scattering was used for models which are not homogeneous. The total scattered intensity is given exactly by

The most general equation for the diffuse scattering intensity is therefore,

which is simply the difference between the total and Bragg scattering where the symbol denotes an ensemble average in which the atoms in each molecule are displaced from their average position according to some model of motion (obtained from normal mode analysis, molecular dynamics, domain movements, etc). If very long-range motions are to be examined, the electron density consists of a contiguous three-dimensional block (termed a supercell) of unit cells allowing for correlation distances greater than the unit cell dimensions. This method was introduced to analyze the diffuse scattering from tropomyosin crystals [Chacko \& Phillips Jr., 1992][Boylan \& Phillips, 1986]. If the correlations are local to the unit cell, then electron density from a single unit cell is used to calculate the diffuse scattering using equation 4. This single-cell version of the method was used to calculate the diffuse scattering from the low frequency hinge-bending mode in lysozyme [Clarage et al., 1992]. Here the total scattering, , is calculated by Fourier-transforming each unit cell, squaring the result, and then averaging over all unit cells. Similarly, the Bragg scattering, , is calculated by first averaging the electron density over all unit cells and then Fourier-transforming and squaring this average.

A final step in both the convolution and multi-cell methods involves the projection of all calculated intensities intersecting the Ewald sphere (oriented to match the diffraction image orientation) onto the film plane. These projections are compared to the diffraction images to determine the quality of the calculation. Aspects of these methods are being incorporated into a general purpose software package for diffuse scattering analysis called XCADS (X-ray Crystallographic Analysis of Diffuse Scattering).