### 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
__

```

### RE: UVW - how to avoid negative widths?

```As I have said before countless time, one should not lose sight of the
objective of Rietveld refinement, that it is to refine a sensible
crystallographic structure. One can reduce R factors in all sorts of
ways by playing with the peak shape functions (even by using lower
symmetry and increasing the number of refinable parameters!) but in the
end what matters is: does the structure make sense? My own experience is
that by judicious use of methods like bond valence calculations, studies
of the bond lengths etc one can rule out unlikely refinements better
than by concentrating on R factors. Many times one can reduce R factors
by playing with the diffraction geometry terms, but with little obvious
improvement of the structural results.

-Original Message-
From: Alan Hewat [mailto:he...@ill.fr]
Sent: 20 March 2009 07:13
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
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
__

```

### RE: UVW - how to avoid negative widths?

```Hi to all!,
in fact, the Cagliti's expression is just a way to show the angular
variation of fwhm, as was mentioned was usef for neutron diffraction and
adopted in XRD, we can also build another dependence such as  FWMH vs
2theta directly and  it is useful to evaluate size and strain, the problem
is that many refinement codes have the FWMH angular dependence in terms of
Caglioti's equation. By the way, how can I get the paper of Young and
Desai?,because I have tried a search in the web,but i did'nt find the
article-
best regards.

Miguel Hesiquio

Miguel Hesiquio-Garduño
Profesor Titular A
Departamento de Ciencia de Materiales
Academia de Ciencias de la Ingeniería
ESFM-IPN. tel 57 29 60 00 ext. 55003, ext. 55011

On Thu, March 19, 2009 5:22 pm, matthew.row...@csiro.au wrote:
From what I've read of Cagliotti's paper, the V term should always be
negative; or am I reading it wrong?

Additionally, there is some good work on the use of the Cagliotti (and
TCHZ) functions in the paper by Young and Desai; it also goes over how to
incorporate sample dependent terms into the expression.

Young, R. A.  Desai, P. 1989, 'Crystallite Size and Microstrain
Indicators in Rietveld Refinement', Archiwum Nauki o Materialach, vol.
10, no. 1-2, pp. 71-90.

Alan Hewat wrote:

This is why I love Topas. All of the the code used in the refinements is
there for you to see!  :)

Cheers

Matthew

Matthew Rowles

CSIRO Minerals
Box 312
Clayton South, Victoria
AUSTRALIA 3169

Ph: +61 3 9545 8892
Fax: +61 3 9562 8919 (site)
Email: matthew.row...@csiro.au

```

### Re: UVW - how to avoid negative widths?

``` According to Caglioti relation, the dimensions of U,V,W are as (angle)^2.

Quick question - does anyone have a trick to stop the Cagliotti formula
going negative? Prodd currently has a habit of bugging out on a
sqrt(negative) and I'm wondering how other folks worked around that, or
if I've got something completely wrong...?

Thanks,

Jon

```

### RE: UVW - how to avoid negative widths?

```Back to basics.  Caglioti Function is instrument function and is often times
used inappropriately.

Just my two cents worth (and these days, 2-cents ain't worth much...)

Frank May
St. Louis, Missouri  U.S.A.

From: Jon Wright [mailto:wri...@esrf.fr]
Sent: Thu 3/19/2009 8:58 AM
Cc: Rietveld Method
Subject: Re: UVW - how to avoid negative widths?

According to Caglioti relation, the dimensions of U,V,W are as (angle)^2.

Quick question - does anyone have a trick to stop the Cagliotti formula
going negative? Prodd currently has a habit of bugging out on a
sqrt(negative) and I'm wondering how other folks worked around that, or
if I've got something completely wrong...?

Thanks,

Jon

```

### Re: UVW - how to avoid negative widths?

```Jon Wright said:
Quick question - does anyone have a trick to stop the Cagliotti formula
going negative?

This can happen if the resolution is relatively flat, so that there is no
well defined minimum. Then the quadratic Cagliotti formula produces large
correlations between U,V,W. The trick is to constrain the minimum to a
fixed angle; eg in the case of a monochromated white beam, this is the
parallel geometry when the scattering angle equals the monochromator
angle. This is an approximation, but better than nothing. You could also
replace the Cagliotti formula by a linear equation if you have access to
the refinement code.

So just differentiate the Cagliotti formula and equate the derivative to
zero to obtain V in terms of U and the fixed angle for minimum half-width.
This works well for classic neutron diffractometers and in general is
probably the best you can do if the refinement programme only allows the
Cagliotti formula.

Alan.
__
Dr Alan Hewat, NeutronOptics, Grenoble, FRANCE
alan.he...@neutronoptics.com +33.476.98.41.68
http://www.NeutronOptics.com/hewat
__

```

### Re: UVW - how to avoid negative widths?

```
Alan Hewat wrote:

Jon Wright said:

Quick question - does anyone have a trick to stop the Cagliotti formula
going negative?

This can happen if the resolution is relatively flat, so that there is no
well defined minimum.

Seems to be the problem - also rather close to zero anyway.

Here I am lucky! As suggested off list - simply return the function to
the form it was in before I edited the code :-)

Thanks a lot for the help

Jon

```

### RE: UVW - how to avoid negative widths?

```Hi,

I sent this to Jon this afternoon and thought that I'd pass the e-mail
on.

Here goes ...

You'll probably find a pre-PRODD CCSL subroutine that I wrote that
goes along the lines of

width^2 = u^2 * ( tan(theta)-tan(theta_m) )^2 + w^2

also theta_m should refine to close to half the monochromator take-off

angle.

Bill

P.S. I also sent him a slightly longer note later ...

Because of the quadrature properties of Gaussians, we get strain terms
like

width^2 = e^2 * tan(theta)^2

size terms go like width^2 = s^2 / cos(theta)^2 = s^2 * (1 +
tan(theta)^2)

so if the monochromator half angle is theta_m so that the pure
instrument term looks like

width^2 = u^2 * ( tan(theta)-tan(theta_m) )^2 + w^2

then instrument  size  strain go like

width^2 = u^2 * ( tan(theta)-tan(theta_m) )^2 + w^2 + e^2 * tan(theta)^2
+ s^2 * (1 + tan(theta)^2)

Obviously you get u, tan(theta_m) and w from a calibration like LaB6 -
you can then plug in (certainly into TOPAS) - then with u0, v0 and w0
fixed at the LaB6 values we get

width^2 = (u0 + f^2) * tan(theta)^2 + v0 *tan(theta) + (w0 + s^2) where
e^2 = f^2-s^2 - and we've got the Gaussian component of the size and
strain directly from the Cagliotti relationship.

-Original Message-
From: Jon Wright [mailto:wri...@esrf.fr]
Sent: 19 March 2009 20:49
To: alan.he...@neutronoptics.com
Cc: Rietveld Method
Subject: Re: UVW - how to avoid negative widths?

Alan Hewat wrote:
Jon Wright said:
Quick question - does anyone have a trick to stop the Cagliotti
formula
going negative?

This can happen if the resolution is relatively flat, so that there is
no
well defined minimum.

Seems to be the problem - also rather close to zero anyway.

Here I am lucky! As suggested off list - simply return the function to
the form it was in before I edited the code :-)

Thanks a lot for the help

Jon

--
Scanned by iCritical.

```

### RE: UVW - how to avoid negative widths?

```From what I've read of Cagliotti's paper, the V term should always be
negative; or am I reading it wrong?

Additionally, there is some good work on the use of the Cagliotti (and TCHZ)
functions in the paper by Young and Desai; it also goes over how to incorporate
sample dependent terms into the expression.

Young, R. A.  Desai, P. 1989, 'Crystallite Size and Microstrain Indicators in
Rietveld Refinement', Archiwum Nauki o Materialach, vol. 10, no. 1-2, pp. 71-90.

Alan Hewat wrote:

This is why I love Topas. All of the the code used in the refinements is there
for you to see!  :)

Cheers

Matthew

Matthew Rowles

CSIRO Minerals
Box 312
Clayton South, Victoria
AUSTRALIA 3169

Ph: +61 3 9545 8892
Fax: +61 3 9562 8919 (site)
Email: matthew.row...@csiro.au

```