matthew.row...@csiro.au said:
> From what I've read of Cagliotti's paper, the V term should always be
> negative; or am I reading it wrong?
That's right. If
FWHM^2 = U.tan^2(T) + V.tan(T) + W
then the W term is just the Full Width at Half-Maximum (FWHM) squared at
zero scattering angle (2T). FWHM^2 is then assumed to decrease linearly
with tan(T) so V is necessarily negative, but at higher angles a quadratic
term (+ve W) produces a rapid increase with tan^2(T).
Cagliotti's formula assumes a minimum in FWHM^2, but if that minimum is
not well defined, U,V,W will be highly correlated and refinement may even
give negative FWHM. In that case you can reasonably constrain V by
assuming the minimum is at a certain angle 2Tm, which may be close to the
monochromator angle for some geometries. So setting the differential of
Cagliotti's equation with respect to tan(T) to zero at that minimum gives:
2U.tan(T) + V =0 at T=Tm or V = -2U.tan(Tm)
this approximate constraint removes the correlation and allows refinement.
Cagliotti's formula simply describes the purely geometrical divergence of
a collimated white neutron beam hitting a monochromator, passing through a
second collimator, then scattered by a powder sample into a collimated
detector. It takes no account of other geometrical effects (eg vertical
divergence) or sample line broadening etc. This geometry is appropriate
for classical neutron powder diffractometers, but not really for X-ray and
other geometries. Still, such a quadratic expression with a well defined
minimum in FWHM, may be a good first approximation in many other cases,
requiring only a few parameters, hence its success. There are many more
ambitious descriptions of FWHM for various scattering geometries and
sample line broadening, usually allowing more parameters to be refined to
produce lower R-factors :-)
Alan
______________________________________________
Dr Alan Hewat, NeutronOptics, Grenoble, FRANCE
<alan.he...@neutronoptics.com> +33.476.98.41.68
http://www.NeutronOptics.com/hewat
______________________________________________
