International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 3.2, p. 256
Section 3.2.2.3.2. Strain^{a}Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794–3800, USA |
An individual crystal in a sample may be subject to lattice deformation, either due to forces external to that crystal, or to internal factors such as substitutional disorder and dislocations. A comprehensive treatment of microstructural properties and their effects on powder-diffraction peak shapes is given in Chapter 3.6 of this volume; this section gives a general overview from a phenomenological basis.
The simplest description of strain broadening imagines an ensemble of independent crystallites with different lattice parameters. If the crystallites have an isometric distribution of lattice parameters with a fractional width δa/a, that will be reflected in the range of d-spacings for each reflection. In an angle-dispersive measurement, this will lead to an angular width δ2θ = 2 tan θ δa/a (radians).
Physically plausible mechanisms for a distribution of strains in a powder sample include random inter-grain forces arising during crystallization and the elastic response to internal defects such as dislocations. In practice, these effects often give rise to peak widths which likewise grow as tan θ. However, there is no a priori basis for expecting any particular functional form for that distribution. Frequent practice is to assume a Gaussian, Lorentzian, Voigt, or pseudo-Voigt form for an isotropic strain broadening, with an adjustable parameter representing the FWHM of the strain distribution. The FWHM Γ of the chosen functional form is then for angle-dispersive measurements, for energy-dispersive X-ray measurements and for neutron time-of-flight measurements.