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