Assume that the atomic displacements are homogeneous and can be factored into statistically independent components, , (e.g. inter-molecular, intra-molecular, etc.) where is a vector with direction and magnitude characterizing the th component of motion. Then to lowest order, the diffuse scattering from each component can be expressed as a convolution of the ideal structure factors with the Fourier transform of an atomic displacement correlation function. Explicitly [Clarage et al., 1992],
where is the reciprocal space vector with magnitude (). is the intensity calculated as the square of the ideal structure factor using the atomic coordinates (at their average position) assuming no atomic motion. The correlation function , , describes how displacements along the component are correlated for atomic separations, .
The total diffuse scattering is actually given by a power series in . At resolutions of interest in this study (3-4 Å), however, second and higher order terms should contribute only a small fraction compared to the first order term above [Clarage et al., 1992]. Explicit second order calculations carried out for the tRNA data verified that the first-order approximation suffices for tRNA.
The form of the displacement correlation function, and thus that of the halo function, , surrounding Bragg peaks, determines the diffuse scattering distribution in reciprocal space. If the characteristic width of is less than a lattice constant, corresponding to intra-molecular movements, the halos will overlap to yield a smoothly varying ring with modulations proportional to the range over which the movements are correlated. Conversely, if the characteristic width of exceeds a lattice constant, the diffuse scattering consists of halos or streaks clearly associated with the Bragg reflections. A spherically symmetric, exponentially decaying, function, , produces spherical halos. It is clear from the diffraction data that the streaks are not spherically symmetrical about the Bragg positions. A possible re-parameterization of the exponential for anisotropic correlations is
The values of , , and , which are the coupling distances along three orthogonal axes, can now be adjusted separately to match the observed halo shape.
An advantage of this convolution (or Patterson) based formalism is that being analytic, there are only a handful of adjustable parameters, particularly the correlation distances and mean squared displacements. The primary disadvantage is that only homogeneous disorder can be modeled. That is, for each component of the disorder, the entire molecule is subject to the same global for all atoms. In this study, we extend this model to include non-homogeneous disorder of a specific part of the molecule which moves independently of the rest of the structure. The convolution method described above is still used but now only the atoms from a specified portion of the molecule are used in the ideal structure factor calculation.