Re: Size Strain in GSAS

[alan coelho]

Title: Message

This is by far the best topic on this list for a long time as opposed to requests for Journal papers which as pointed out by someone else is inappropriate in the first place and illegal in the second.   Nicolae wrote: >(i) but a sum of two Lorentzians is not sharper than the sum of two pVs (Voigts)?   This I know, it should not matter what is used as long as the mapping of the function to a distribution is done accurately. Whether it is lognomal, gamma or something else does does not matter.   Every thing we are talking about is additive meaning that the sum of what ever in 2Th space translates to the sum of what ever distributions. From the resulting distribution you are free to extract what ever parameter you choose.   The idea in Nick Armstrong’s work of obtaining a distribution without knowing its functional form is a powerful one. But the Baysean approach without a functional form results in large errors bars in the distribution, see  http://nvl.nist.gov/pub/nistpubs/jres/109/1/cnt109-1.htm   What we should be looking for are cases where Voigt approximations are not possible. I have only ever seen one case where the actual Sinc type ripples are seen in a pattern. This was a pattern by Bob Cheary of gold columns. Other reports of ripples do exist in the literature (not available to me as I write). We must be careful not to include 2Th independent bumps produced by long narrow Soller slits inserted in the axial plane that limits horizontal divergence.   When sample related ripples are seen then you can throw Voigt based approximations out the window. In the case of the gold columns we fitted three Sinc functions added together. In other words the distribution was really a limited one.   The work of Nick et al is sound and approaches the problems from a different perspective; it does not however negate the need to determine a priori information. The question that is open in my opinion is whether a priori information is more easily incorporated into a least squares process or a Bayesian Maxent approach.   This discussion has reinvigorated my interest and like Bob, whom is now looking for an equation to approximate the log normal disribution, I will resurrect  some code myself that I did a while back which calculates profiles from a arbitrary distribution for the Sinc function. Instead I will include the equation for spherical crystallites and of course a user defined one which can be hkl dependent.   As a hint to those who write such code the calculation of a profile for an arbitraty distribution operates at around 5000 profiles per second as I noticed over the weekend - not much slower that a gaussian Nicolae. Maybe there's no need for a pseuod-Voigt / Lorentzian based approximations after all.     all the best Alan   -----Original Message----- From: Nicolae Popa [mailto:[EMAIL PROTECTED] Sent: Sunday, April 17, 2005 9:00 AM To: rietveld_l@ill.fr Subject: Re: Size Strain in GSAS Alan,   (i) but a sum of two Lorentzians is not sharper than the sum of two pVs (Voigts)?   (ii) We  fitted the exact size profile caused by the lognormal distribution by a pV (for low lognormal dispersion) or by a sum of maximum 3 Lorenzians (for large lognormal dispersion). This is "cheaper"  than the sum of 2 pVs. It involves the calculation of maximum 3 elementary functions with 4 independent parameters (3 breadths + 2 mixing parameters minus 1 constraint = 4)  Sum of two pVs presumes 4 elementary function and 5  independent parameters (2 for one pV + 2 for the second one + a mixing parameter).   Best wishes, Nicolae       A pure peak fitting approach shows that two pV’s (or two Voigts) when added with different FWHMs and integrated intensities but similar peak positions and eta values can almost exactly fit Pearsons II functions that are sharper that Lorentzians. This is not surprising as both profiles comprise 6 parameters.   Thus from my observations two pVs added together can fit a bimodal distributions quite easily. In fact my guess is that two pVs can fit a large range of crystallite size distributions.     all the best alan