The determination of crystal size and disorder from X-ray diffraction photographs of polymer fibres. 1. The accuracy of determination of Fourier coefficients of the intensity profile of a reflection

Abstract

Methods which determine the number and disorder of lattice planes in a crystal from the Fourier cosine coefficients of the intensity profile of an X-ray reflection use only the low harmonics and require that the coefficients be normalized so that the zero harmonic is unity. Experimentally, the profiles can only be recorded over a smaller range of scattering angle than required by the theory, and it is necessary to subtract background, which is likely to be estimated with considerable error, before determining the coefficients. It is shown that with polymer fibres this causes serious errors in the normalization, and in the values of those low harmonics used in the size and disorder determination, and prevents reliable values being obtained. Methods which avoid normalization and use only high harmonics are needed. It is shown that disorder may be obtained in such a way, but not size, for which low-order normalized coefficients are essential. A method of extrapolation is described and tested which enables the accurate high harmonics to be used to improve the estimates of the low ones. Whilst this will yield more reliable values of crystal size than are obtainable from existing methods, the accuracy depends entirely on the validity of the extrapolation, which cannot be tested in many cases of interest.