### 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 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?

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-GarduĂ±o Profesor Titular A Departamento de Ciencia de Materiales Academia de Ciencias de la IngenierĂa 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: if you have access to the refinement code. 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. if you have access to the refinement code. 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 Two neat things about this equation is that it doesn't go negative and 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. if you have access to the refinement code. 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: if you have access to the refinement code. 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