Variance as a Measure of Line Broadening

Abstract

THE measures of line-breadth most widely used in X-ray and electron diffraction are perhaps the half width and the integral breadth. The measure of dispersion most widely used in mathematical treatments of distributions is, however, the variance or mean-square breadth. Its comparative neglect in crystallographic problems has been due to the large variance of the wave-length spread and of the diffraction profile, the ‘tails’ of the lines approaching zero approximately as the square of the distance from the line centroids. In spite of this, however, the variance has been used in discussions of spectrometer¹ and diffractometer^2–4 aberrations. When the line profile is ‘truncated’, as it has to be in practical centroid determination^5–8, the large contribution to the variance from the tails is eliminated, and it becomes a measure of line-broadening worth serious consideration. It has the great advantage over all other measures of breadth that it can be corrected for geometrical aberrations¹ and wave-length spread by simple subtraction, instead of by ‘unfolding’⁹. Tournarie¹⁰ has criticized the use of half width and integral breadth on the ground that they give undue weight to a few isolated points of the diffraction profile, instead of treating all observed points on the same basis, and derived a relation between the crystallite size and the variance of the line profile over a deliberately restricted angular range. Tournarie's result is immediately obtained by considering the asymptotic expressions for the line profile given by Wilson¹¹; in the present communication a unified treatment is outlined which not only recovers Tournarie's particle-size result but also gives analogous expressions applicable to certain types of ‘mistake’ and strain broadening.