Notes on the order-of-reflection dependence of microstrain broadening

In order to obtain systematic insight into the different manifestations of microstrain broadening in powder diffraction patterns, the consequences of the breakdown of the Stokes–Wilson approximation (negligible strain gradient in a stack of lattice planes) were investigated. To this end, a phenomenological approach for the decay of the variance of the microstrain with increasing correlation distance L, 〈[epsilon]_L²〉, was adopted, as well as a Gaussian microstrain distribution for each L. For the case of an L-independent 〈[epsilon]_L²〉 (i.e. the Stokes–Wilson approximation) the (Gaussian) microstrain distribution directly shows up (is affinely mapped) on the diffraction angle scale as well as on the length of the diffraction vector scale. Furthermore, the integral breadth (on the length of the diffraction vector scale) then increases linearly with the order of reflection or, expressed another way, with the length of the diffraction vector (i.e. the integral breadth increases with tanθ on the diffraction angle 2θ scale). For the case of a decay of 〈[epsilon]_L²〉 with L, deviations from such simple behaviour of the integral breadth occur. In particular, Lorentzian line broadening and nonlinear dependence of the integral breadth on the length of the diffraction vector scale (i.e. non-tanθ dependence on the 2θ scale) are induced. It is argued that the approaches used for the description of microstrain broadening in many procedures (integral breadth and Rietveld refinement) are of limited validity, i.e. they do not warrant general unverified application.