Dear All:
Indeed, despite some more advanced approaches to modeling diffraction line
shapes, the good, old Cagliotti function is still in use probably for
historical reasons (as it is the case with many other things in sciences).
However, to be fair to (practically all major) Rietveld programs, the
original Cagliotti function (Gaussian term) was a long time ago amended by a
Lorentzian contribution (linear in FWHM as opposed to a quadratic Gaussian
term). This term immensely helps to accurately model high-resolution
measurements, both x-ray and neutron (see, for instance examples of ESRF and
ISIS data in Size-Strain Line-Broadening Analysis of the Ceria Round-Robin
Sample, Journal of Applied Crystallography 37 (2004) 911-924--article and
data available at http://www.du.edu/~balzar/s-s_rr.htm).
As discussed in this thread, the original Cagliotti function often gets into
trouble because of the square root of a negative number. Bill David
described a much better function. Thus, the general approach described by
Bill and in the article cited above is to refine coefficient of a function
used on a pattern obtained from a suitable standard, such as LaB6, and
then fix them (unfortunately, not always possible for all instruments,
because some instrumental parameters depend on the angle in the same way as
the strain term in the Bragg-Brentano geometry). Moreover, for
high-resolution data it might be helpful to add a Lorentzian FWHM to that
expression and then post it to Wikipedia or perhaps publish a paper... :-)
Davor
-Original Message-
From: simon.billi...@gmail.com
[mailto:simon.billi...@gmail.com] On Behalf Of Simon Billinge
Sent: Sunday, March 22, 2009 2:50 PM
To: rietveld_l
Subject: Cagliotti and Other Issues
Dear Rietvelders
What is the most complete and authoritative source for issues such as
profile function definitions, what is their scientific basis, and when
are they appropriate to use, etc.? I am guessing there is not
one-stop-shop solution (Young's book? GSAS manual? Rietveld list
archive?) but advice on this would be helpful.
I wonder if we should, as a community, put some of this stuff on
wikipedia, or another such place. In other words, distill the
community's collective knowledge in a single place that can be updated
in the future, and also curated for correctness also by the community.
What are people's thoughts on this? Rietveldipedia?
S
2009/3/20 May, Frank frank.l@umsl.edu:
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
