-------- Original Message --------
Subject: Re: Fw: Anisotropic peak broadening with TOPAS
Date: Thu, 30 Oct 2008 08:43:07 +0100
From: Frank Girgsdies <[EMAIL PROTECTED]>
References: <[EMAIL PROTECTED]>

Thanks a lot Peter for this piece of code.
It was really helpful.

First I tried it "as is", but the fit was only
partially successful.
Then I thought about the meaning of the Tan(Th) term,
and that you mentioned it would fit strain broadening.
Thus, I changed "* Tan(Th)" into "/ Cos(Th)" and:
bingo!, the fit worked perfectly well. In fact,
the calculated pattern looks almost exactly like the
one I had obtained before using spherical harmonics.
However, in contrast to the (at least to me) abstract
spherical harmonics function, the Miller indices are
now explicitly included, plus that I have basically
a modified 1/cos(theta) function.
Instead of lor_fwhm = c / Cos(Th) (as applied by the
CS_L macro), I have now lor_fwhm =
(D_spacing^2 * Sqrt(Max(0,mhkl)) / 10000) / Cos(Th),
effectively replacing the overall parameter c with
an hkl-dependent term.
I'm currently trying to make Topas calculate equivalents
of LVol-IB for me. As this calculation seems to need the
convolution features of Topas (because it involves
"translation" of lor_fwhm into Voight Intergal Breadth),
I think I can't do it by hand.
My main problem now is how to make Topas' macros
address one particular hkl peak instead of all peaks,
and then output the result.

Thanks again to Peter and all other contributors!



In response to the question posted this morning by Frank Girgsdies:

Here is a little snatch of topas code for anisotropic broadening in orthorhombic system. (The fitted parameters come from a particular refinement I took this from; the factor of 10^4 is empirically chosen to get parameter values that are generally between unity and several thousand.) The "theory" behind this is given in PWS, Journal of Applied Crystallog. vol. 32, pp 281-289 (1999) and references therein. Generalization to other crystal systems is given in that paper as well.

                prm s400  0.00000`_LIMIT_MIN_0 min 0
                prm s004  1287.33878` min 0
                prm s040  1.56981`_LIMIT_MIN_0 min 0
                prm s220  32.53311`
                prm s202  5410.01715`
                prm s022  792.15586`
prm mhkl = H^4 s400 + K^4 s040 + L^4 s004 + H^2 K^2 s220 + H^2 L^2 s202 + K^2 L^2 s022; lor_fwhm = D_spacing^2 * Tan(Th) * Sqrt(Max(0,mhkl)) / 10000;

Application of this expansion to Lorentzian broadening is not mathematically rigorous, but seems to work OK within the framework of crystallographic refinements (i.e., if you don't try to interpret the fitted broadening parameters in some fundamental way). My experience is based on refinements from synchrotron data, which have mostly Lorentzian shape - you might play with using a Gaussian width as well.

Note that the lor_fwhm command increases the width by that amount, so other terms in the lineshape, such as possibly a Lorentzian term in crystallite size (e.g., CS_L(@, 1000) ) are included as well. This formalism handles all of the strain broadening - even an isotropic component.

At the risk of triggering a lengthy discussion, I would point out that this 4-th order polynomial expansion in Miller indices actually has some basis in elasticity theory (e.g., discussion at the end of the JAC paper cited above and subsequent work, for example by T. Ungar et al.), whereas spherical harmonic expansions really do not.

Good luck with it,

Peter W. Stephens
Professor, Department of Physics and Astronomy
Stony Brook University
Stony Brook, NY 11794-3800
fax 631-632-8176

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