On Variance as a Measure of Line Broadening in Diffractometry III. A Note on Dislocations

According to the Stokes-Wilson approximation, the tails of the line profile resulting from diffraction by a crystal containing a screw dislocation approach zero in accordance with an inverse-cube law. The variance of the line profile, therefore, would increase asymptotically as the logarithm of the range, instead of being independent of the range, as for strains that nowhere exceed a few per cent. The difference arises from the singularity at the dislocation axis, for which the usual models give strains of the order of unity. Very large strains are presumably relieved by some physical mechanism, but the logarithmic law might well apply over the measurable portion of a line profile.The accuracy of the Stokes-Wilson approximation is difficult to assess, and the inverse-cube law has therefore been rederived without serious approximation from the asymptotic form of the Bessel-function expression for the diffraction pattern of a screw dislocation. The logarithmic term is shown to increase as the square of the index of reflection corresponding to the direction of the dislocation axis. Independent confirmation of the inverse-cube law has been obtained by Wilkens in the course of a discussion of the Warren and Averbach analysis of line profiles.