> 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.
Hello Messieur Le Bail, (and thanks for explaining how to pass from sphere - isotropy to ellipsoid - anisotropy). The naive character doesn't come from the approximation of the crystallite shape by an ellipsoid, but from the approximation of the size effect in powder diffraction by ellipsoid. In powder diffraction it is seen not one, but a (big) number of crystallites more or less randomly oriented. The crystallites in reflection "show" different diameters, not only one. Concerning the mean strain, another confusion. In fact the mean strain gives the peak shift, sometimes reasonably described by an ellipsoid in (hkl) (for example not-textured samples under hydrostatic pressure). But the strain broadening is related on the strain dispersion (you wrote <Z**2> not <Z>) that in first approximation is a symmetrized quartic form and its square root (giving breadth) is never an ellipsoid. Certainly, always one can use ellipsoids as a first approximation for any kind of anisotropy, with the condition to not violate some elementary principles, in particular, here, the invariance to symmetry. It has no relevance to use the thermal ellipsoids as argument. The thermal ellipsoids are a natural consequence of the harmonic vibration of the atoms and no principle is violated, even if, some times, this is a rough approximation because of a high contribution of anharmonicity. > > 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, Not surprisingly, people are mainly interested to obtain a good structure refinement and ignore by-products like strain an size. Doesn't mean that strain and size can not be estimated better. >the fit was quite better > (especially in cases showing stacking faults, with directional effects > hardly approximated by ellipsoids) but remained "phenomenological". The thermodynamics is phenomenological science, have we to consider it a naive or a less naive science? Best wishes a happy Easter, Nicolae Popa > 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... > Armel > >
