>The coefficients Lij in the formula you wrote have no significance.
Our whole science is a so bad approximation to the Universe...
For the representation of an isotropic size effect , you may imagine the mean size being the same in all directions, obtaining a sphere. The same for a mean strain value.
Introducing some anisotropy in mean size and mean strain in the Rietveld method was done in the years 1983-87 by the "naive" view that the mean size M(hkl) in any direction could be approximated by an ellipsoid rather than a sphere, and the same for the mean strain <Z**2>(hkl). See for instance J. Less-Common Metals 129 (1987) 65-76.
Less "naive" representations were applied in the years 1997-98 (so, ten years later). But these less naive representations were not providing any size and strain estimations, the fit was quite better (especially in cases showing stacking faults, with directional effects hardly approximated by ellipsoids) but remained "phenomenological".
The old naive view provided at least (bad) estimations of the directional values of the mean size and strain parameters. Behind that ellipsoidal approximation of the mean size and strain are even more important "details": one has also to define what could be the size distribution and the strain distribution. Simplifying naively, the mean size and strain play on the profile width, and the size and strain distributions play on the profile shape. In the earlier approachs, the size and strain distributions were also naively represented (frequently Cauchy-like for the size distribution, and Gaussian for the strain distribution - but do not confuse the shapes of the size and strain distributions with profile shapes).
Nowadays, people are using flexible profile shapes and seem to be not concerned at all with the exact relation between the profile shape and the size and strain distributions (some profile shapes could correspond to unrealistic size and strain distributions (for instance a negative proportion of crystallites for some given sizes, etc). This looks quite naive to me as well...
Probably in ten or twenty years, more essential improvements will make the current view looking very naive; this is to be expected ;-).
You can find experts in thermal vibration explaining that the ellipsoid representation used by crystallographers is an extremely naive view of the reality, and they are right. But crystallographers continue to calculate these Uij (and there is a table giving Uij restrictions) which in most cases provide a minimal and sufficient representation of thermal vibrations...
With some experience, looking at a powder diffraction pattern, you may visually approximate the mean coherent domain size : small, medium or large. Do you really need the exact size distribution and exact mean size in all crystallographic directions ? Rarely ;-).
Armel
