The Diffraction of X-Rays by Distorted-Crystal Aggregates III: Remarks on the Interpretation of the Fourier Coefficients

Abstract

The line profile of the hth order reflection from a set of lattice planes is known to be given by a Fourier series with coefficients A(h,m)=N(m)J(h,m). The function N(m) depends on the particle shapes and sizes, and J(h,m) depends on the state of strain Neither can be completely determined from the experimental Fourier coefficients for a single value of h. They have, however, different behaviours for small m, and A(h,m) has an unambiguous interpretation in this region only.: its initial slope gives the mean particle thickness, and its initial curvature sets a lower limit to the mean-square strain. The practical value of this interpretation is subject to two limitations: (1) the initial shape of the A(h,m) curve depends on the `tail' of the line profile, (11) if the crystals are large enough to contain several regions of compression and extension, probability theory shows that J(m) is likely to be approximately proportional to exp {-const.|m|} for m large. Extrapolation to small m could then tend to give fictitiously small values for both particle size and strain The problem of obtaining the greatest possible amount of information from observed line profiles of all orders is discussed Usually only a few orders can be measured, and the information obtainable is less, but in three special cases the Fourier coefficients of two lines can yield all information obtainable from the full set