Helical diffraction. I. The paracrystalline helix and disorder analysis.

In a new approach to helical diffraction a helix generating function is defined, and thence an expression for the autocorrelation function (a.c.f.) for a helix is obtained. The Fourier transform of this a.c.f. gives a new expression for the diffracted intensity, which is shown to be equivalent formally to the classical expression of Cochran, Crick & Vand [Acta Cryst. (1952), 5, 581-586] and A. R. Stokes (unpublished). The new expression allows straightforward examination of the effects of helical disorders on the diffracted intensity. The thermal and paracrystalline effects of disorders with cylindrical symmetry are shown, and examples are given from the diffraction of a model of the actin helix. The general case, disorder with no symmetry, is derived and the effects of axial and radial disorder, separately and together, are computed, again for the model actin helix. Translational disorder is also included, and its effects are explained. The new results are compared with existing accounts of the effects of helical disorders on fibre diffraction.