Deformation density variance

The variances and covariances of the deformation density result both from the errors in the observations, and from the uncertainties in the refined parameters,... 

Under general hypotheses, the optimisation of the Bayesian score under the constraints of MaxEnt will require numerical integration of (29), in that no analytical solution exists for the integral. A Taylor expansion of Ao(R) around the maximum of the P(R) function could be used to compute an analytical expression for A and its first and second order derivatives, provided the spread of the A distribution is significantly larger than the one of the P(R) function, as measured by a 2. Unfortunately, for accurate charge density studies this requirement is not always fulfilled for many reflexions the structure factor variance Z2 appearing in Ao is comparable to or even smaller than the experimental error variance o2, because the deformation effects and the associated uncertainty are at the level of the noise. 

All in all, the WPPM approach can provide a simultaneous structure and microstructure refinement, based on physical models of the phases under study, without using any arbitrary profile function. Considering the terms of Equation (26), refinement parameters to be optimized in a least-squares analysis are relatively few, namely, mean (p) and variance (cr) of a suitable distribution of coherent domain sizes, dislocation density (p), effective outer cut-off radius (R ) and character (/e, effective fraction of edge dislocations), twin fault (P), deformation fault (a) and APB (y) probabilities. 

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