Dear colleagues, First of all, thanks to Dr. Ortiz for the advice/explanation.
I would like to point out some discrepancy in the standard formulae for VWDS/SWDS, which are given in various well-respected articles. 1) N.C. Popa, D. Balzar, JAC 35, 338, formulae 7-8: Dv=3mu_4/2mu_3 (7) and Da=4mu_3/3mu_2 (8) 2) P. Scardi, M. Leoni, JAC 39, 24, formulae 1a-1b: <L>s=2mu_3/3mu_2 (1a) and <L>v=3mu_4/4mu_3 (1b) It seems that these definitions are differing merely by a factor of two. I am curious if anybody could clarify this point. Sincerely, Maxim. -----Original Message----- From: alor...@unex.es [mailto:alor...@unex.es] Sent: Friday, December 12, 2008 7:55 PM To: Максим В. Лобанов Subject: RE: calculation of VWDS and SWDS from distributions? Dear Maxim: Thanks for your interest in our paper. The first thing that I should mention is that the our model was formulated for spherical crystallites under the hypothesis of absence of lattice strains, and therefore it can be used under these circunstances. In the case of co-existence of size and strain broadenings, I understand that the model will still provide a first order estimate of the crystallite size distribution (but I have not tested such a case and thus this is only an asumption). With respect to your questions: 1.) If your data show the typical alpha1-alpha2 doublets, then you should fit a pseudo-Voigt (pV) function with Ka1 and Ka2 components to the measured peak. The peak background should also be modeled, for example by a first-order polynomial function. The Cu Ka2 component can be assumed to have the same shape (mixing parameter and width) as the Cu Ka1 component, but with the half of its intensity and shifted toward higher angles according to the Bragg law. Thus, you can model your experimental doublet very well. This how I proceed. 2.) The model correlates the parameters of the pV function to the parameters of the lognormal distribution of spherical crystallites. Of course, in an experiment the value of the mixing parameter of the pV function can change along the pattern. This can be due to various factors: (a) instrumental causes (the variation of the instrumental profile with the 2theta angle); (b) co-existence of strain broadening which depends on d-spacings (this is used in the Warren-Averbach method to resolve sizes and strains); (c) anisotropic crystallite size distributions, that is, non-spherical crystals. In our paper we assumed that the crystallite were spherical to construct our model (see page 3); and (d) all together!!!!. It is hard to say which of these are present in your case. You will always have the first one. In metals and alloys you can also have the second one, but this is rare in ceramics unless they are doped. The third one depends on the synthesis/procesing history of your material. Hope this helps you, angel L. Ortiz > Dear Angel, > > Thank you very much. > This indeed seems a simplest way to extract width of particle size > distribution (basically, formula 22 of your article; using only pV mixing > parameter value). > > There are two questions: > 1) (purely practical) could anybody suggest a good way to proceed in case > of alpha1-alpha2 doublets? Fitting by single peak even at moderately low > angles would yield strong bias (in my test, ~10% in eta vales for 2theta > in the 30-35deg. range). Alpha2-stripping (using standard methods > implemented e.g. in Powder4) inevitably gives artifacts which would not > allow to do a further single-peak fit with the desired accuracy. > Constrained fits are possible, but rather difficult. If this is indeed a > preferred option, can anybody suggest a good software to do such fits? > > 2) as far as I understand, the lognormal/pV model predicts that pV mixing > parameter is independent of 2theta. In case when one observes "really > strong" size broadening, small angle-dependent contribution from > instrumental broadening can be neglected. > So, if such dependence is experimentally observed (for data where > instrumental contribution is small), would this mean that there is > non-negligible strain (or "non-trivial" size distribution...)? Or I simply > misunderstood something? > > Sincerely, > Maxim. > > -----Original Message----- > From: alor...@unex.es [mailto:alor...@unex.es] > Sent: Monday, December 08, 2008 11:03 PM > To: íÁËÓÉÍ ÷. ìÏÂÁÎÏ× > Subject: Re: calculation of VWDS and SWDS from distributions? > > Maxin: > > If you fit a lognormal function to your distribution of sizes, then you > can easily use Eqs.(24) to (27) in my attached paper. Hope this is easier. > > Best regards, > > Angel. L. Ortiz > > > >> Dear colleagues, >> >> I am facing a problem of correlating laser scattering (DLS) and X-ray >> diffraction data. >> For correct comparison, I need either to calculate some model >> distribution from X-ray data (this is feasible assuming lognormal >> distribution - there are ready software solutions for that) or typical >> "X-ray sizes" (Dv and/or >> Da) from given distributions. >> The inverse task (calculating Dv and/or Da from given distributions) >> appears to be very simple, but it seems there is no ready software >> solution, and I need to manually integrate the data. But before doing >> that I would like just to be sure that I use correct formulae. >> I read the paper dealing with that in great detail (JAC, 35, 338 by >> Popa & Balzar, Ref. 1), but it is too mathematical, and I am not >> completely confident if I understood everything correctly. >> >> If we denote distribution (normalized to unity) as p(x), x=particle >> size then, according to Ref.1: >> Dv=3mu_4/2mu_3 (1) >> and >> Da=4mu_3/3mu_2 (2) >> >> Do I understand correctly, that moments mu_i are just: >> >> mu_i=Int[0, Infinity]{x^i*p(x)dx} >> >> Or there are some missing factors somewhere? >> >> Sincerely, >> Maxim. >> >> ------------------------------------------- >> Dr. Maxim Lobanov >> R&D Director >> Huntsman-NMG >> mailto: m_loba...@huntsman-nmg.com >> >> ********************************* >> If you encounter any difficulties >> sending e-mails to the addresses in huntsman-nmg.com domain, this >> could be due to the our spam filter malfunction. >> In case of such an event please send a message to n...@fromru.com >> >> Please note that the old domain nmg.com.ru does not exist anymore - >> please update your address book accordingly >> >> >> >
