Analysis of the X-ray diffraction pattern of helical structures

Abstract

The paper deals with the theory of the diffraction pattern of helical structures having the number of units per turn (n) neither integral nor rational. The conventional treatment suffers from the defect that the repeat spacing along the axis of the helix is taken as the standard of reference, and this does not exist, being infinite, when n is irrational. The difficulty is got over in this paper by focussing attention on the 'unit height' (h = resolved component of a unit along the axis) and 'unit twist' (t = fraction of a complete rotation for one unit, = l/n), which vary continuously irrespective of n being rational or irrational. Explicit formulae are obtained in terms of their Bessel indices for the observed layer line-spacings which turn out to be very simply related to the reciprocals of the unit height and the pitch of the helix. A technique of analysing, the observed diffraction pattern for the elements of the helical structure is also given, with examples. The case of a coiled-coil is seen to have the same general features as the simple coil, the layer line-spacing being now related to two pitches, namely, those of the major and the minor helices, and the unit height. The relationship of the diffraction pattern of a helix in its uncoiled and its coiled-coil form is also found to be rather simple, being similar to the multiplet splitting produced by a magnetic field in spectral lines.