Back to basics and First Principles....
 
As Alan says, the [use of the Cagliotti function is appropriate for the neutron 
case], "but not really for X-ray and other geometries."
 
My recollection is the Cagliotti function was adapted to the x-ray case when we 
had low resolution x-ray instruments and slow (or no) computers.  Now that we 
have high resolution instruments and fast computers, why does this 
inappropriate function continue to be used?
 
On another note, the world is venturing into the infinitely small realm of 
"nano-particles."  The classical rules for crystallography work very well for 
ordered structures in the macro-world (particles of the order of micron-sizes). 
 However, as the particles become smaller, does one not need to address the 
contribution of the "surface" of the particles?  The volume of the "surface" 
becomes much greater relative to the volume of the "bulk" of the crystal.  
Models today account for "stress" and "strain" in the macro-world.  As the 
relative fraction of the "bulk" becomes smaller, both the physical structure as 
well as the mathematics used to describe the bulk suffer from 
termination-of-series effect, do they not?  Does any of this make sense?  Any 
thoughts?
 
Frank May
St. Louis, Missouri  U.S.A.

________________________________

From: Alan Hewat [mailto:he...@ill.fr]
Sent: Fri 3/20/2009 2:13 AM
To: rietveld_l@ill.fr
Subject: RE: UVW - how to avoid negative widths?



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
______________________________________________




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