Journal of Applied Crystallography · 1991 · 75 citations · 0 references
Natural SciencesMaterials EngineeringMaterials ScienceMaterials CharacterizationMaterials PhysicsPolymer MaterialsCrystalline MaterialsCrystallographyDisordered SystemsStructure DeterminationX-ray CrystallographyX-ray DiffractionCrystal Size AnalysisCrystallite Size DeterminationDiffraction AnalysisEstimation Of Structural ParametersFiber DiffractionFourier Cosine SeriesRefinementX-ray Reflection
X‑ray reflection intensity profiles encode lattice disorder and crystal size distribution through Fourier cosine coefficients, but existing methods require multiple reflection orders and rely on inaccurate low‑order harmonics, which are rarely available for polymer fibres. The authors develop and test a single‑order method that overcomes these limitations. They model paracrystalline disorder with an assumed crystal‑size distribution, refine parameters by minimizing the difference between calculated and experimental coefficients, and validate the approach using two‑order diffraction patterns and intensity‑profile refinement, though harmonics offered no advantage and sometimes caused instability. The asymmetric length distribution model fits the data within experimental error, yet a single‑order measurement cannot uniquely determine parameters, and the refinement algorithm proved unstable when using harmonics.
The intensity profile of the X-ray reflection from a crystalline material is related to the lattice disorder and the distribution of crystal sizes through its Fourier cosine coefficients. However, existing methods of obtaining these structural parameters from the coefficients require more than one order of reflection and this is seldom available with polymer fibres. They also rely heavily on the low-order harmonics which are those determined with least accuracy. The development and testing of a method which overcomes this weakness and which is suitable for use with a single order is described. The coefficients are calculated for a model with paracrystalline disorder and an assumed distribution of crystal sizes and the parameters describing this model are refined to minimize the discrepancy between the calculated and experimental values of the coefficients. Provided the distribution of lengths is asymmetric this discrepancy is no greater than would be expected from experimental error and so the assumed model cannot be rejected on the evidence available. Since a range of model parameters all gave equally good agreement with experiment, it was not possible with a single order to obtain a well defined set of values. Diffraction patterns displaying two orders had been chosen and results from the second order were consistent with the first, only a narrow range satisfying both simultaneously. The method was further developed by calculating the intensity profile from the harmonics and using this in the refinement. There was no advantage over using harmonics; indeed, on occasions the refinement algorithm was unstable producing unreliable results.