X-ray diffraction from a helix of any length that displays cumulative azimuthal disorder

An explicit formula has been derived to describe the attenuation and broadening of cylindrically averaged diffraction intensities from a helix of any given length which possesses cumulative azimuthal disorder. The application limits of an approximate formula, represented by the first term of this formula, are defined. Strategies to estimate the length of fibers, the degree of disorder, and the overlap of adjacent layer lines are outlined. Some features of diffraction patterns from the disordered helical structure of the HbS fiber are interpreted in light of these results. In these patterns, non-zero-order Bessel functions are attenuated and broadened due to azimuthal disorder and finite length. Adjacent layer lines overlap because of the very large axial repeat distance of the HbS fibers. As a result, the contribution of any Bessel function term with n > or = 10 is not discernible in these patterns. Only Bessel terms with n < 6 may be accurately estimated in these patterns, if instrumental broadening is negligible or correctable. The theory presented here may also be used to make a rough estimate of the degree of disorder in F-actin fibers by comparison of X-ray diffraction patterns with serial peak projections calculated assuming various degrees of disorder.