Dear Jon,
Jon Wright wrote:
>... to answer to your (too) long questions. May be later, OK?
Going back to this quartics versus ellipsoids peak broadening stuff, maybe I can summarise:
Why should the distribution of lattice parameters (=strain) in a sample match the crystallographic symmetry? If the sample has random, isolated defects then I see it, but if the strains are induced (eg: by grinding) then I'd expect the symmetry to be broken. Suggests to me the symmetry constraints should be optional, and that the peak shape function needs to know about the crystallographic space group and subgroups. Either the program or the user would need to recognise equivalent solutions when the symmetry is broken (like the "star of k" for magnetic structures).
I would like to present my thoughts here, although some points overlap with points mentined previously by others:
One has to think what one looks at: If you look at a single crystal grain, which may show, for whatever reason microstrain broadening (i.e. local distortions, e.g. due to dislocations) which is incompatible with the Laue group/crystal system symmetry. In powder diffraction, however, you look at an ensemble of crystallites, and the line broadening information is averaged. Even if you have a powder of identical crystals, each showing identical line broadening incompatible with Laue symmetry reflections of different width overlap because they have the same d-spacing. By that the line broadening of the crystals incompatible with Laue symmetry cannot be obtained directly from the powder pattern, e.g . by analysing the powder peaks' widths as a function of hkl, as you could do it for a single crystal. You can only analyse averaged widths. Thus you loose the decisive information. However, you may detect hkl dependent changes of the shapes of the reflections, e.g. superlorentzian peaks where, e.g. one broad and two narrow overlap (h00 reflection of a cubic crystals which show strong microstrain along [001] but low microstrain along [100] and [010]). It will be difficult to recognise such effects, unless they are really strong, and it may be even more difficult to interpret that, if you have not specific information about possible sources of the microstrain. The same problem of overlapping reflections of different widths is predicted for certain cases of quartic line broadening when the Laue group symmetry is lower than the crystal system symmetry (E.g. for the Laue class 4/m), as remarked by Stephens (1999). As much as I know, no such case has been reported yet.
To summarise: I think refinement of anisotropic line broadening will be much more stable if constrained by symmetry, such that reflections equivalent by symmetry have the same width.
One example related with that problem was presented at (I think it was a size broadening case, but similar conclusions may be valid for microstrain)
Young, R. A., Sakthivel, A., Bimodal Distributions of Profile-Broadening Effects in Rietveld Refinement, J. Appl. Cryst. 21 (1988) 416
Why would anyone have anything against using an ellipsoid? That same function can be described by the quartic approach, it just has less degrees of freedom.
Here I fully agree, and there ARE cases where theory predicts an ellipsoid (e.g. microstrain-like broadening due to composition variations) which should, however, obey the symmetry restrictions. If such a case is present an ellipsoid model should be used obeying the rule to use a minimum of refined parameters. On the other hand, one might imagine other cases, where you have ellipsoid broadening for the single crystals incompatible with symmetry and being then powder-averaged. However, this will be difficult to be recognised and interpreted (see above).
In short, I don't understand why there is such a strong recommendation to use the quartics instead of ellipsoids or why the symmetry is not optional. I'm still persuing this because I have looked at something with a very small anisotropic broadening which seems to fit better with an ellipsoid which breaks the symmetry compared to a quartic which doesn't!
Thanks for any advice,
Jon
I think it should be recommended
first to use a minimum number of parameters to describe the line broadening, and if possible,
secondly to use models which are mathematically compatible with the physics of the origin of the line broadening.
Thus in some cases an ellipsoid model should be preferred prior the quartic model, because it needs less parameters.
However, I think that are good reasons to keep symmetry restrictions for both the quartic and elipsoid models (see above). But there may be reasons in certain, and probably few cases, where the symmetry restrictions can be lifted, e.g. when you have direction dependent line shapes of the broadening contribution to the peak shapes. But maybe that could also be better modelled by direction dependent shape factors compatible with the crystal symmetry.
Andreas
-- Dr. Andreas Leineweber Max-Planck-Institut fuer Metallforschung Heisenbergstrasse 3 70569 Stuttgart Germany
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