### 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: K6H2Nb6O19.8H2O

``` I didn´t find the crystalline phase
K_6 H_2 Nb_6 O_19 .8H_2 O
in ICSD database

There are very many phases containing niobium oxide blocks, with various
other cations and hydration states, so you may not find an exact match.

You can still use ICSD to help understand the structure if you relax the
search criteria. If you search eg for elements K Nb6 H O you get only 10
results, including eg K7(H Nb6O19).(H2O)10 which is already pretty close,
with one more K, one less H, plus some extra water.

You should of course remain sceptical about the precise formula that
people report, especially if they report X-ray structures of hydrogenous
heavy metal oxides :-)

If you can index the pattern determine the unit cell, you can narrow the
possibilities greatly, but again watch out for weak peaks due perhaps to
superstructure, especially for heavy metal oxides.

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

```

### Re: K6H2Nb6O19.8H2O

``` I didn´t find the crystalline phase
K_6 H_2 Nb_6 O_19 .8H_2 O
in ICSD database

K7(H Nb6O19).(H2O)10 which is already pretty close,

I should have given the reference for this and other similar Nb6O19
cluster compounds as:

Solid-state structures and solution behavior of alkali salts of the (Nb6
O19)(8-) Lindqvist ion.
Nyman, M.;Alam, T.M.;Bonhomme, F.;Rodriguez, M.A.;Frazer, C.S.;Welk, M.E.
Journal of Cluster Science (2006) 17, 197-219

If you use Jmol in ICSD to draw these structures you find that they
consist of clusters of 6 Nb-oxide octahedra (Nb6O19) held together with
water hydrogen bonds and K+ or other cations for charge balance.

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

```

### Cagliotti and Other Issues

```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
__

```

### 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

```

### LANSCE Neutron School at the Lujan Neutron Scattering Center - July 7-17, 2009

```LANSCE will host its 6th annual Neutron School at the Lujan Neutron Scattering
Center, this year focused on Application of Neutron Scattering to Study Phase
Transformations, on July 7-17, 2009.

The school will include lectures from some of the preeminent scientists in the
field and hands on experiments on the HIPPO, SMARTS, NPDF, FDS, and LQD
instruments at the Lujan Center.

The school is focused of graduate students beginning there thesis research and
post doctoral researchers.  Industrial applicants will also be considered.

There is no tuition and travel expenses are covered. As an added bonus, we
will take Sunday off for a day of hiking, sightseeing, etc.

School website (http://lansce.lanl.gov/neutronschool). Applications are due
April 13, 2009. On the website is also a link to the poster announcing the
school, we kindly ask you to print and post it appropriately at your
institution.

Please send questions to Sven Vogel (s...@lanl.gov).

Sven Vogel
Chair of the 2009 LANSCE Neutron School

We wish to acknowledge the support of the Department of Energy, National
Science Foundation, and New Mexico State University

```