Re: LP factor in the Rietveld refinement
Right, but specially for students- beginners we must be much, let say, didactic LP means (Lorentz) * (Polarisation) What is important in Rietveld refinement when a lot of mirrors & monochromators are present is how they change (Polarization) because (Lorentz) is changed by adding factors independent on hkl, then entering in the scaling factor Presuming the same scattering plane for all "scatterers" the polarization factor is: pol = SIN(PSI)**2 + COS(PSI)**2*COS(2*TET1)**2*COS(2*TET2)**2 *.*COS(2*TETm)**2*COS(2*TETb)**2 where TET1, TET2, ., TETm are the Bragg angles at monochromator 1, 2, ,m and where TETb is the Bragg angle at sample (depending on hkl) and where PSI is the angle between polarization vector of the incident beam - IF it is TOTALLY POLARIZED!!! - and the scattering plane; If the incident beam is NOT POLARIZED the averages of both SIN(PSI)**2 and COS(PSI)**2 result in 1/2. If the incident beam is partially polarized one replace for example SIN(PSI)**2 by P0 , consequently COS(PSI)**2 = 1 - P0 and one refine P0 If the geometry is much complicated (different scattering planes for different monochromators) "pol" should be calculated for the given geometry by applying successively the known formula (see a book of electrodynamics, e.g.. Landau) Ej+1 = (Ej X u)Xu and taking at the END: |E(last)|**2 / |E0|**2 (X means vectorial product) where Ej is the electric field vector in the beam scattered j times and u is the unit vector along the scattered beam j+1 Best wishes, Nicolae Popa - Original Message - From: "Leonid Solovyov" To: Sent: Sunday, July 26, 2009 9:05 AM Subject: RE: LP factor in the Rietveld refinement In principle, the LP correction for a multi-bounce monochromator is similar to that for a single-crystal one with the same crystal type and reflection indexes (or diffraction angle). The exact LP value depends, as well, on the crystal perfection (mosaicity) and for supremely precise measurements one might consider refining the LP value as was mentioned by Kurt and Peter. Besides the angular range, the correlation with thermal parameters, and the instrument alignment, one more problem of the LP refinement is the correct choice of the atomic scattering curves in accordance with the oxidation states which might be not quite obvious in general. Leonid *** Leonid A. Solovyov Institute of Chemistry and Chemical Technology 660049, K. Marx 42, Krasnoyarsk , Russia www.icct.ru/eng/content/persons/Sol_LA www.geocities.com/l_solovyov *** --- On Sun, 7/26/09, Peter Y. Zavalij wrote: From: Peter Y. Zavalij Subject: RE: LP factor in the Rietveld refinement To: rietveld_l@ill.fr Date: Sunday, July 26, 2009, 5:03 AM That's right. LP refinement works just fine within TOPAS but angular range as wide as possible is needed. If it is up to 140-150 deg. 2thteta LP does not correlate much with thermal parameters. Refined LP is not exact but very close. Peter Zavalij X-ray Crystallographic Center University of Maryland College Park, MD Office: (301)405-1861 Lab: (301)405-3230 Fax: (301)314-9121 -Original Message- From: Kurt Leinenweber [mailto:ku...@asu.edu] Sent: Saturday, July 25, 2009 8:53 PM To: alor...@unex.es; Leonid Solovyov Cc: rietveld_l@ill.fr Subject: RE: LP factor in the Rietveld refinement Hi all, I haven't actually DONE this, so maybe I shouldn't put my 2 cents in, but can't you refine the polarization factor by using a standard such as Y2O3 and fixing the structure and thermal parameters of the standard while refining the polarization angle? The angle so obtained should agree with what the theory tells you for your diffractometer configuration, but it seems more comforting to verify it by a measurement. - Kurt From: alor...@unex.es [mailto:alor...@unex.es] Sent: Sat 7/25/2009 1:29 PM To: Leonid Solovyov Cc: rietveld_l@ill.fr Subject: Re: LP factor in the Rietveld refinement In this context: What about the LP for a Goebel mirror followed by a 4-bounce or 2-bounce primary monochromator? Best regards angel l. ortiz __ NOD32 4280 (20090726) Information __ This message was checked by NOD32 antivirus system. http://www.eset.com
Re: calculation of VWDS and SWDS from distributions?
Right.
Nevertheless they can be "phenomenological" extended (with some precautions)
to other crystallite shapes like ellipsoids or cylinders -
see JAC(2008)41, 615-627, sections 2 and 4.3.
In principle Dv & Da can be rigorously calculated for much complex shapes
and distributions if the user is able to 'run' through the three steps
described in section 4.2.
but he must be very careful when comparing experimental data with these
calculations.
Best wishes,
Nicolae Popa
- Original Message -
From: "Leonid Solovyov" <[EMAIL PROTECTED]>
To:
Sent: Tuesday, December 09, 2008 6:59 AM
Subject: Re: calculation of VWDS and SWDS from distributions?
Dear Maxim,
The formulae
Dv=3mu_4/2mu_3
Da=4mu_3/3mu_2
are valid for spherical crystallites only.
Accordingly, in the expression
mu_i=Int[0, Infinity]{x^i*p(x)dx}
x should be the radius of spherical crystallite, but not “particle size”.
Best regards,
Leonid
***
Leonid A. Solovyov
Institute of Chemistry and Chemical Technology
660049, K. Marx 42, Krasnoyarsk, Russia
www.icct.ru/eng/content/persons/Sol_LA
www.geocities.com/l_solovyov
***
--- On Mon, 12/8/08, Максим В. Лобанов <[EMAIL PROTECTED]> wrote:
From: Максим В. Лобанов <[EMAIL PROTECTED]>
Subject: calculation of VWDS and SWDS from distributions?
To: "rietveld_l@ill.fr"
Date: Monday, December 8, 2008, 2:04 PM
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: [EMAIL PROTECTED]
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Re: question on size-strain analysis
Privet Leonid, Nice to "see" you too! By contrary, most people are thinking that broad size distributions (resulting in super-Lorentzian profile) is very rare. Nevertheless perhaps you have some time to take a look at JAC(2008)41, 615-627 Best wishes, Nicolae - Original Message - From: "Leonid Solovyov" <[EMAIL PROTECTED]> To: Sent: Tuesday, October 21, 2008 3:23 PM Subject: Re: question on size-strain analysis Hi Nicolae, Nice to hear from you! I couldn't help noting that in reality your emergence in the Rietveld list is similarly rare as the narrow size distribution :-). Instrumental problems are far more regular. Best wishes, Leonid *** Leonid A. Solovyov Institute of Chemistry and Chemical Technology K. Marx av., 42 660049, Krasnoyarsk Russia Phone: +7 3912 495663 Fax: +7 3912 238658 www.icct.ru/eng/content/persons/Sol_LA *** --- On Tue, 10/21/08, Nicolae Popa <[EMAIL PROTECTED]> wrote: From: Nicolae Popa <[EMAIL PROTECTED]> Subject: Re: question on size-strain analysis To: [EMAIL PROTECTED], rietveld_l@ill.fr Date: Tuesday, October 21, 2008, 11:39 AM Hi, Besides strain and instrument also size broadening can be close to a Gaussian if the crystallite size distribution is very narrow. For zero dispersion of size distribution the size peak profile is about 75% Gaussian and 25% Lorentzian (on tails). For details see JAC (2002) 35, 338-346 (self citation) and other ref. cited there (Langford, Louer, Scardi (2000)) Best, Nicolae Popa __ NOD32 3541 (20081021) Information __ This message was checked by NOD32 antivirus system. http://www.eset.com
Re: question on size-strain analysis
Hi, Besides strain and instrument also size broadening can be close to a Gaussian if the crystallite size distribution is very narrow. For zero dispersion of size distribution the size peak profile is about 75% Gaussian and 25% Lorentzian (on tails). For details see JAC (2002) 35, 338-346 (self citation) and other ref. cited there (Langford, Louer, Scardi (2000)) Best, Nicolae Popa - Original Message - From: "Maxim V. Lobanov" <[EMAIL PROTECTED]> To: Cc: "Дмитрий А. Павлов" <[EMAIL PROTECTED]> Sent: Tuesday, October 21, 2008 9:20 AM Subject: question on size-strain analysis Dear colleagues, I have a probably very basic question related to size-strain analysis: we have a pattern of a nanocrystalline oxide, which shows (from Williamson-Hall plot) almost purely size broadening, and shape of reflections is to good accuracy Gaussian. I am curious what type of microstructure this Gaussian behavior can reflect. Maybe it is just something usual, but so far I had only observed predominantly Lorentzian broadening in similar materials... Sincerely, Maxim. --- Dr. Maxim Lobanov R&D Director Huntsman-NMG mailto: [EMAIL PROTECTED] * 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 [EMAIL PROTECTED] Please note that the old domain nmg.com.ru does not exist anymore - please update your address book accordingly __ NOD32 3541 (20081021) Information __ This message was checked by NOD32 antivirus system. http://www.eset.com
Re: question on size-strain analysis
Hi, Besides strain and instrument also size broadening can be close to a Gaussian if the crystallite size distribution is very narrow. For zero dispersion of size distribution the size peak profile is about 75% Gaussian and 25% Lorentzian (on tails). For details see JAC (2002) 35, 338-346 (self citation) and other ref. cited there (Langford, Louer, Scardi (2000)) Best, Nicolae Popa - Original Message - From: "Leonid Solovyov" <[EMAIL PROTECTED]> To: Sent: Tuesday, October 21, 2008 10:04 AM Subject: Re: question on size-strain analysis Dear Maxim, Gaussian peak shape is, normally, related to the instrumental, strain and/or lattice distribution broadening. The Williamson-Hall plot shows the dependency of broadening on the diffraction angle and if this dependency can not be explained by strain then it is most probably instrumental. If you use the Bragg-Brentano geometry a possible reason might be a loose powder, sample misalignment or surface roughness. Regards, Leonid *** Leonid A. Solovyov Institute of Chemistry and Chemical Technology K. Marx av., 42 660049, Krasnoyarsk Russia Phone: +7 3912 495663 Fax:+7 3912 238658 www.icct.ru/eng/content/persons/Sol_LA *** --- On Tue, 10/21/08, Maxim V. Lobanov <[EMAIL PROTECTED]> wrote: From: Maxim V. Lobanov <[EMAIL PROTECTED]> Subject: question on size-strain analysis To: rietveld_l@ill.fr Cc: "Дмитрий А. Павлов" <[EMAIL PROTECTED]> Date: Tuesday, October 21, 2008, 7:20 AM Dear colleagues, I have a probably very basic question related to size-strain analysis: we have a pattern of a nanocrystalline oxide, which shows (from Williamson-Hall plot) almost purely size broadening, and shape of reflections is to good accuracy Gaussian. I am curious what type of microstructure this Gaussian behavior can reflect. Maybe it is just something usual, but so far I had only observed predominantly Lorentzian broadening in similar materials... Sincerely, Maxim. --- Dr. Maxim Lobanov R&D Director Huntsman-NMG mailto: [EMAIL PROTECTED] * 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 [EMAIL PROTECTED] Please note that the old domain nmg.com.ru does not exist anymore - please update your address book accordingly __ NOD32 3541 (20081021) Information __ This message was checked by NOD32 antivirus system. http://www.eset.com
Re: Hyper Lorentzian peak shape
Not necessarily! If (some) peaks are super-Lorentzian the effect could be caused by a broad lognormal distribution of crystallite sizes. Anyway if the peaks are super-Lorentzians the pattern decomposition by using pseudo-Voigt is, in principle, incorrect. Indeed, imagine that after decomposition we wish to clean this profile of instrumental contribution. Presuming the instrumental profile is Voigt (if x-ray laboratory), the Fourier transform of such profile is ~exp(-B*y-C*y**2). On the other hand the Fourier transform of the Lorentzian part of "measured" pseudo-Voigt is ~exp(-A*y), then the Fourier transform of the "deconvoluted" profile for this term should be ~exp[-(A-B)*y + C*y**2]. Certainly A>B (breadth of "measured" peak should be larger than of the instrumental), consequently at small values of y the Fourier transform curve decreases with increasing y, as expected, but when y = (A-B)/C this curve begins to increase, becoming infinite when y=infinite, which is absurd. In fact pseudo-Voigt has been used in pattern decomposition algorithms (including Rietveld) only to mime and to calculate faster the Voigt profile, then with ETA<1 (in TCH pseudo-Voigt this condition is explicitly given). In this case there is no problem with the "deconvolution", that, in fact, is made between two Voigt functions. Best wishes, Nicolae Popa - Original Message - From: "Jon Wright" <[EMAIL PROTECTED]> To: Sent: Thursday, November 22, 2007 12:53 PM Subject: Re: Hyper Lorentzian peak shape If you can fit via a sum of a broader lorentzian and a narrower one it may indicate the sample is a "mixture". Are you sure the sample is microscopically homogenous? Best, Jon Francois Goutenoire schrieb: Dear Rietveld users, I am currently working on Williamson-Hall graph in order to check strain effect on some new phase. I have some hyper-lorentzian peaks eta>1 from a pseudo-Voigt decomposition , then I am not able to get BetaL and BetaG from Winplotr decomposition. Does anyone have an idea ? Francois Goutenoire Laboratoire des Oxydes et des Fluorures, UMR6010. __ NOD32 2675 (20071121) Information __ This message was checked by NOD32 antivirus system. http://www.eset.com
Re: Size Strain in GSAS
Title: Message Alan, But the analytical representation of the profile, even by empirical functions, also helps in the analysis of Size/Strain you don't think? You don't agree with 3 Lorentzians even if they are sharper than two pVs? Probably is a reason that I don't see. Concerning the numerical derivatives probably the providers of popular codes have an opinion. Personally I use quite exclusively the numerical gradients in my personal least square program but I know that many people avoid, if possible. Best wishes, Nicolae Nicolae >To resume, I think (for example) that is better to approximate by a sum of >three Lorentzians (involving 4 profile parameters) than by a sum of two >pVs (involving 5 profile parameters). I couldnt agree more. >Concerning the numerical calculation of the profiles, still I'm not convinced >that is the ideal solution. You have not only the size profile to calculate, >but also at least two convolutions, with the strain and the instrumental >profiles. Moreover, what are doing the codes that use the gradient >calculated analytically? Size distribution is a start and Leonie does seem to have worked in some of the other effects to his credit. Whether all this actually helps in the analysis of Size/Strain does not seem to concern many - so why not experiment. By gradient, I presume you mean the derivatives of the distribution with respect to parameters. This would require a mixture of numerical and analytical derivatives - very simpe using the chain rule. Dont see why the same cant be done in other codes. all the best alan
Re: Size Strain in GSAS
Buna Mateo (where from you know Romanian?), I spiked about the "regular" Rietveld programs that operate in the space of measurement. Sure, FFT is a solution, but I'm not sure that is the best solution. Even fast, the calculation of the profile by Fourier transform is longer than to calculate by elementary functions (don't forget that Voigt can be always approximated by pV-TCH). Moreover, I think problems could be with sampling and truncation. If you have sharp peaks you must go far away in the Fourier space, otherwise the truncation errors become critical; but in this case you have to add many terms. Moreover you have to calculate numerically the Fourier coefficients of the measured instrumental profile that introduce other errors. And also is the problem with the codes that use analytical gradients. Concerning the paper from JAC (2004) I don't know it. It is a least square method that needs not to define the model that follow to fit? Can you send me an electronic reprint? I have not access online and the journal is missing in library. Thanks and best wishes, Nicolae > buna Nicolae, > > > Not only arithmetic, I think is clear that both and c were refined in a > > whole pattern least square fitting. A private program, not a popular > > Rietveld program because no one has inplemented the size profile caused by > > the lognormal distribution. > > not sure no one did.. we're working with that kind of profiles at > least since 2000 (published in 2001 Acta Cryst A57, 204), without the need > for any approximation going through Voigts or Pseudo Voigts. Using FFT and > some math tricks you can compute the "true" profile for a distribution of > crystallites almost in the same time you calculate a Voigt curve, so why > the need to use any approximate function? > I think this agrees with what Alan just pointed out (well 5000 profiles > per second if you do not include any hkl dependent broadening that has > to be calculated for each of them (and perhaps for each subcomponent)... > otherwise the speed reduces.. but yes few ms for each profile is the > current speed for my WPPM code, implementing all this stuff within the > WPPM frame). > > > > But the most important disadvantage is the necessity to choose the > > > exact type of size distribution. For Sample 1 (which, obviously, have > > > certain distribution with certain and c) you got quite different > > > values of and c for lognorm and gamma models, but the values of Dv > > > and Da were nearly the same. Don't you feel that Dv and Da values > > > "contain" more reliable information about and c than those > > > elaborate approximations described in chapter 6? > > > > Well, this is the general feature of the least square method. In the least > > square you must firstly to choose a parametrised model for something that > > you wish to fit. Do you know another posibility with the least square than > > to priory choose the model? Without model is only the deconvolution, and > > even there, if you wish a "stable" solution you must use a deconvolution > > method that requres a "prior, starting model" (I presume you followed the > > disertation of Nick Armstrong on this theme). > > also in this case it has ben shown possible to obtain a distribution > without any prior information on its functional shape (J.Appl.Cryst (2004), > 37, 629) and without taking the maxent treatment into account. > I'm currently using without much problems for the analysis of > nanostructured materials... advantages with respect to maxent are the > speed and the fact that it can coexist with other broadening models (still > not possible with maxent and still have to see a specimen where strain > broadening is absent) and it's able to recover also a polydisperse > distribution if it's present Just need to test it against maxent (if > data would be kindly provided to do so). > For the purists, just redo the calculation starting from different points > and you can evaluate the error in the distribution using a > MonteCarlo-like approach... > > As for the TCH-pV, well, it is no more than a pV with the Scherrer > trend (1/cos) and the differential of Bragg's law (tan term) plugged in. > This means it is ok as long as you consider a Williamson-Hall plot a good > quantitative estimator for size and strain (IMHO). > > Mat > > PS I fully agree with Alan on the continuous request for Journals, but I > bet the other Alan (the deus ex machina of the mailing list) should warn > the members somehow... > > -- > Matteo Leoni > Department of Materials Engineering > and Industrial Technologies > University of Trento > 38050 Mesiano (TN) > ITALY > > > > >
Re: Size Strain in GSAS
Title: Message Alen, you right, as far as the profile corresponding to a given distribution is accurately described, any representation is good. Nevertheless, at comparative accuracies, it is not better to use a representation with smaller numbers of parameters? The profile parameters are functions of the parameters of distribution; in particular for the discussed distributions, lognormal, gamma, the profile parameters are functions of a unique distribution parameter "c". Further, to have a functional formula for profile to be implemented in the Rietveld programs, the dependence on "c" of the profile parameters must be found (somehow, empirical formulae for example). Now, the number of profile parameters you have used in the primary fit becomes critical. If this number is small the dependence of everyone on "c" is more or less smoothed. If the number of these parameters is higher than necessary, the dependence on "c" is rather "ugly" and difficult describe. To resume, I think (for example) that is better to approximate by a sum of three Lorentzians (involving 4 profile parameters) than by a sum of two pVs (involving 5 profile parameters). Concerning the numerical calculation of the profiles, still I'm not convinced that is the ideal solution. You have not only the size profile to calculate, but also at least two convolutions, with the strain and the instrumental profiles. Moreover, what are doing the codes that use the gradient calculated analytically? Best wishes, Nicolae 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. . 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
Re: Size Strain in GSAS
Leonid, The lognormal distribution for particle size is not my modeling (unfortunately), but if you insist, let see once again your equations. = Da + 0.25(DaDv)^0.5 and sigma = (Dv/Da - 1/2)/2 For lognormal distribution first equation becomes: 2=(4/3)(1+c)**2+(1/4)sqrt[2*(1+c)**5] For c=0.05 we obtain: 2=1.87, for c=0.4, 2=3.43 The second equation becomes: sqrt(c)=[(9/8)(1+c)-1/2] For c=0.05, 0.22=0.68, for c=0.4, 0.63=1.75 Well, taking account that the world is not ideal I'm ready to accept that, then I think is time to close our discussion. Best wishes, Nicolae - Original Message - From: "Leonid Solovyov" <[EMAIL PROTECTED]> To: Sent: Sunday, April 17, 2005 2:58 PM > Dear Nicolae, > > I will comment only upon your last statement because the limitations of > your modeling are clear. > > > Well, I don't know where from you taken these formulae > > but I observe that for spheres of equal radius, then zero dispersion, > > you have: > > sigma(D)=5/4, different from zero! > > First of all, for spheres of equal radius and IDEAL definition > of Dv and Da: > sigma = (Dv/Da - 1/2)/2 = (9/8 - 1/2)/2 = 5/16 > Yes it is not zero, but the expressions I derived work only for > 0.05 < c < 0.4 and I derived them not for IDEAL Dv and Da. If you > perform WEIGHTED least-squares fitting of TCH p-V function to a profile > simulated for spherical crystal and added by ~10% background level (to > be closer to real Rietveld refinement) you will obtain the ratio of > Dv/Da~3/4 not 9/8! This ratio is wrong but this is what we have from > WEIGHTED least-squares fitting of TCH p-V to simulated data. In this > case > sigma = (Dv/Da - 1/2)/2 = (3/4 - 1/2)/2 = /8, > different from zero again, sorry, this world is not IDEAL. > > Best wishes, > Leonid > > > > > __ > Do you Yahoo!? > Plan great trips with Yahoo! Travel: Now over 17,000 guides! > http://travel.yahoo.com/p-travelguide >
Re: Size Strain in GSAS
Dear Leonid, See coments below. > > Dear Nicolae, > > This arithmetic is clear, thanks, but since you did not specify this > exact way of calculation in the paper it was not evident. There are > several other ways of deriving , for instance: to calculate Dv from > the inverse integral breadth and then use eq. (12) or (17) etc. Not only arithmetic, I think is clear that both and c were refined in a whole pattern least square fitting. A private program, not a popular Rietveld program because no one has inplemented the size profile caused by the lognormal distribution. > Besides, you did not refine for simulated data in chapter 6 - it > was "fixed". When you apply this formalism to real data you refine both > and c, they may correlate and the result of such correlation is not > apparent. Let's clarify this point. What you call "simulated data in chapter 6" are in fact the exact function PHIbar(x) given in (15b). This can be calculated only by numerical integration and this function, as you can see from (15b), has only one parameter, c. This was the clue to use the parameters and c=sigma(R)**2/**2 in place of the original and sigma(R). It is improper to say that " was fixed", because is contained in the argument x=2*pi*s* of PHIbar(x). > > But the most important disadvantage is the necessity to choose the > exact type of size distribution. For Sample 1 (which, obviously, have > certain distribution with certain and c) you got quite different > values of and c for lognorm and gamma models, but the values of Dv > and Da were nearly the same. Don't you feel that Dv and Da values > "contain" more reliable information about and c than those > elaborate approximations described in chapter 6? Well, this is the general feature of the least square method. In the least square you must firstly to choose a parametrised model for something that you wish to fit. Do you know another posibility with the least square than to priory choose the model? Without model is only the deconvolution, and even there, if you wish a "stable" solution you must use a deconvolution method that requres a "prior, starting model" (I presume you followed the disertation of Nick Armstrong on this theme). Concerning the fact that Dv and Da are the same although the parameters and "c" of the two size distributions are different, it is not surprisingly. By contrary, it should have been very bad that Dv and Da be dependent on the choosen model of distribution. Dv & Da are quantities "seen" in diffraction. In fact the dispute on this subject started from the doubt of one of the participants that the physical model of microstructure determined ONLY from diffraction is unique! And that is essential to search for physical models, etc. (you can follow in archive). "Dv and Da contain more reliable information about and c than those elaborate approximation...". You mix the planes doing comparison between disjunct things. Once the model choosen (lognormal, gamma, etc.), "those elaborate approximations" give the possibility to find the model parameters in an automatic way, by direct refinement, if these are introduced in the whole pattern fiting (Rietveld in particular). These "elaborate approximations" are doing nothing else than to approximate analytically the exact profile that can be only calculated by numerical integration, then time costly in a whole pattern fitting. > > In new version of DDM (see the following message) I included some > estimations of average crystallite diameter and its dispersion > sigma based on empirical approximations derived from fitting TCH-pV > function to simulated profiles for the model of spherical > crystallites with different size distribution dispersions. For > simulated data (which are supplied with the DDM package) these "magic" > expressions: > > = Da + 0.25(DaDv)^0.5 and sigma = (Dv/Da - 1/2)/2 > > allowed reproducing and sigma with less than 10% deviation in > the interval of relative dispersions 0.05 < c < 0.4 for both gamma and > lognorm distributions. Of course, I don't think that these expressions > are perfect and I would be glad to see better estimations. Well, I don't know where from you taken these formulae but I observe that for spheres of equal radius, then zero dispersion, you have: sigma(D)=5/4, different from zero! Best wishes, Nicolae > > Best regards, > Leonid > > > > > __ > Do you Yahoo!? > Plan great trips with Yahoo! Travel: Now over 17,000 guides! > http://travel.yahoo.com/p-travelguide >
Re: Size Strain in GSAS
Bob, A "nice" math. description amenable to RR exists, take a look at JAC(2002) 35, 338-346. "Nice" because the size profile is described by a pV (at "regular" lognormal dispersions) or by a sum of maximum three Lorentzians (at large lognormal dispersions - those 3% that Alan spiked about). The breadths and mixing parameters of pV or of the sum of Lorenzians are calculated analytically from the two parameters of the lognormal distribution, and c. You realize that if approximate the instrumental and the strain profiles by Voigts, it results a sum of three Voigts for the whole profile and for a fast computation, every Voigt can be replaced by our loved TCH pV. Best wishes, Nic > Nic, > Well, I have been tempted from time to time to implement a "log normal" type distribution in on eof the profile functions. A "nice" math description ameanable to RR would help. > Bob >
Re: Size Strain in GSAS
Title: Message 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 pVs (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
Re: Size Strain in GSAS
Title: Message Dear Bob, Perhaps I was not enough clear. Let me be more explicit. It's about one sample of CeO2 (not that from round-robin) that we fitted in 4 ways. (i) by GSAS with TCH-pV (ii) by another pV resulted from gamma distribution of size (iii) by Lorentz - (the limit of any "regular" pV - eta=1) All these 3 variants given bad fits. For example (ii): Rw=0.144, similarly the rest. (In principle if one pV works (TCH for example) any other kind of pV must work.) (iv) by the profile resulted from lognormal distribution of size; this time the fit was reasonably good: Rw=0.047. It resulted c=2.8, that means a "super Lorentzian" profile (I remember that the Lorentzian limit for lognormal is c~0.4 - JAC (2002) 35, 338). Attention, this "super Lorentzian" profile is not constructed as a pV with eta>1. Sure, such samples are rare, or, perhaps, not so rare. A Jap. group (Ida,, Toraya, JAC (2003) 36, 1107) reported "super Lorentzian" on a sample of SiC. They found c=1.37 Best wishes, Nic Popa Nic, This is true for the internal math but the TCH function was assembled to reproduce the true Voigt over the entire range of differing Lorentz and Gauss FWHM values so it works as if the two FWHM components are independent. As for your question, I'm not aware that anyone has actually tried to do the fit both ways on a "super Lorentzian" (eta>1 for old psVoigt) sample to see if a) the fit is the same and b) the eta>1 was an artifact. Any takers to settle this? Bob R.B. Von Dreele IPNS Division Argonne National Laboratory Argonne, IL 60439-4814 -Original Message-From: Nicolae Popa [mailto:[EMAIL PROTECTED] Sent: Thursday, April 14, 2005 9:11 AMTo: rietveld_l@ill.frSubject: Re: Size Strain in GSAS Dear Bob, If I understand well, you say that eta>1 (super Lorenzian) appeared only because eta was free parameter, but if TCH is used super Loreanzians do not occur? Nevertheless, for that curious sample of cerium oxide we tried GSAS (with TCH) and the fit was very bad. Best wishes, Nicolae PS. By the way, TCH also forces FWHM of the Gaussian and Lorenzian components to be equal, but indeed, eta is not free and cannot be greater than 1. Nic, I know about "super Lorentzians". Trouble is that many of those older reports were from Rietveld refinements "pre TCH" and used a formulation of the pseudo-Voigt which forced the FWHM of the Gaussian and Lorentzian components to be equal and allowed the mixing coefficient (eta) to be a free variable (n.b. it is not free in the TCH formulation). Thus, these ought to be discounted in any discussion about the occurence of super Lorentzian effects in real samples. Bob R.B. Von Dreele IPNS Division Argonne National Laboratory Argonne, IL 60439-4814 -Original Message-From: Nicolae Popa [mailto:[EMAIL PROTECTED] Sent: Thursday, April 14, 2005 8:10 AMTo: rietveld_l@ill.frSubject: Re: Size Strain in GSAS Right, is rare, but we have meet once. A cerium oxide sample from a commercial company, c=2.8. I don't know if they did deliberately, probably not, otherwise the hard work to obtain such curiosity is costly and the company risks a bankruptcy. On the other hand superlorenzian profiles were reported from a long time, only were interpreted as coming from bimodal size distributions. And third, you see, people have difficulties to extract size distribution from the Rietveld codes as they are at this moment. Nicolae Popa A word from a "provider" of a Rietveld code (please don't call me a "programmer"). "But if c>0.4 any pV fails" - OK, for what fraction of the universe of "real world" samples is "c">0.4? I suspect, given the general success of the TCH pseudoVoigt function, that it is exceedingly small and only occurs when one works hard to deliberately make a sample like that. R.B. Von Dreele IPNS Division Argonne National Laboratory Argonne, IL 60439-4814 -Original Message-From: Nicolae Popa [mailto:[EMAIL PROTECTED] Sent: Thursday, April 14, 2005 7:14 AMTo: [EMAIL PROTECTED]Cc: rietveld_l@ill.frSubject: Re: Size Strain in GSAS >Dear Nicolae, >Ma
Re: Size Strain in GSAS
Title: Message Dear Bob, If I understand well, you say that eta>1 (super Lorenzian) appeared only because eta was free parameter, but if TCH is used super Loreanzians do not occur? Nevertheless, for that curious sample of cerium oxide we tried GSAS (with TCH) and the fit was very bad. Best wishes, Nicolae PS. By the way, TCH also forces FWHM of the Gaussian and Lorenzian components to be equal, but indeed, eta is not free and cannot be greater than 1. Nic, I know about "super Lorentzians". Trouble is that many of those older reports were from Rietveld refinements "pre TCH" and used a formulation of the pseudo-Voigt which forced the FWHM of the Gaussian and Lorentzian components to be equal and allowed the mixing coefficient (eta) to be a free variable (n.b. it is not free in the TCH formulation). Thus, these ought to be discounted in any discussion about the occurence of super Lorentzian effects in real samples. Bob R.B. Von Dreele IPNS Division Argonne National Laboratory Argonne, IL 60439-4814 -Original Message-From: Nicolae Popa [mailto:[EMAIL PROTECTED] Sent: Thursday, April 14, 2005 8:10 AMTo: rietveld_l@ill.frSubject: Re: Size Strain in GSAS Right, is rare, but we have meet once. A cerium oxide sample from a commercial company, c=2.8. I don't know if they did deliberately, probably not, otherwise the hard work to obtain such curiosity is costly and the company risks a bankruptcy. On the other hand superlorenzian profiles were reported from a long time, only were interpreted as coming from bimodal size distributions. And third, you see, people have difficulties to extract size distribution from the Rietveld codes as they are at this moment. Nicolae Popa A word from a "provider" of a Rietveld code (please don't call me a "programmer"). "But if c>0.4 any pV fails" - OK, for what fraction of the universe of "real world" samples is "c">0.4? I suspect, given the general success of the TCH pseudoVoigt function, that it is exceedingly small and only occurs when one works hard to deliberately make a sample like that. R.B. Von Dreele IPNS Division Argonne National Laboratory Argonne, IL 60439-4814 -Original Message-From: Nicolae Popa [mailto:[EMAIL PROTECTED] Sent: Thursday, April 14, 2005 7:14 AMTo: [EMAIL PROTECTED]Cc: rietveld_l@ill.frSubject: Re: Size Strain in GSAS >Dear Nicolae, >Maybe ya ploho chitayu i ploho soobrazhayu, but even after your>explanation I can't see a way to calculate from the results of>fitting described in chapters 6 & 7 of JAC 35 (2002) 338-346. From such>fitting you obtain only dispersion parameter c. Or I missed something?>Anyway, being "Rietvelders" we still have to deal with TCH-pV function>and we need to extract as much as possible correct information from it.>Hope we shall see more appropriate functions for microstructure>analysis in popular Rietveld programs.>Cheers,>Leonid Dear Leonid, Indeed you missed something. I presume you have the paper. Then, take a look to the formula (15a). This is the size profile for lognormal. There is the function PHI - bar of argument 2*pi*s*. Replace this function PHI - bar from (15a) by the _expression_ (21a) with the argument x=2*pi*s*. You get it? So, not only "c" but also . "We are Rietvelders" means that we must be only "codes drivers", "cheffeurs des codes", "voditeli program"? Have we to accept the "Procust bed" of the Rietveld codes at a given moment? All Rietveld codes are improving in time, isn't it? In particular for the Round_Robin sample TCH-pV works because c=0.18. (Davor explained how Dv and Da are found). But if c>0.4 any pV fails. Best wishes, Nicolae
Re: Size Strain in GSAS
Title: Message Right, is rare, but we have meet once. A cerium oxide sample from a commercial company, c=2.8. I don't know if they did deliberately, probably not, otherwise the hard work to obtain such curiosity is costly and the company risks a bankruptcy. On the other hand superlorenzian profiles were reported from a long time, only were interpreted as coming from bimodal size distributions. And third, you see, people have difficulties to extract size distribution from the Rietveld codes as they are at this moment. Nicolae Popa A word from a "provider" of a Rietveld code (please don't call me a "programmer"). "But if c>0.4 any pV fails" - OK, for what fraction of the universe of "real world" samples is "c">0.4? I suspect, given the general success of the TCH pseudoVoigt function, that it is exceedingly small and only occurs when one works hard to deliberately make a sample like that. R.B. Von Dreele IPNS Division Argonne National Laboratory Argonne, IL 60439-4814 -Original Message-From: Nicolae Popa [mailto:[EMAIL PROTECTED] Sent: Thursday, April 14, 2005 7:14 AMTo: [EMAIL PROTECTED]Cc: rietveld_l@ill.frSubject: Re: Size Strain in GSAS >Dear Nicolae, >Maybe ya ploho chitayu i ploho soobrazhayu, but even after your>explanation I can't see a way to calculate from the results of>fitting described in chapters 6 & 7 of JAC 35 (2002) 338-346. From such>fitting you obtain only dispersion parameter c. Or I missed something?>Anyway, being "Rietvelders" we still have to deal with TCH-pV function>and we need to extract as much as possible correct information from it.>Hope we shall see more appropriate functions for microstructure>analysis in popular Rietveld programs.>Cheers,>Leonid Dear Leonid, Indeed you missed something. I presume you have the paper. Then, take a look to the formula (15a). This is the size profile for lognormal. There is the function PHI - bar of argument 2*pi*s*. Replace this function PHI - bar from (15a) by the _expression_ (21a) with the argument x=2*pi*s*. You get it? So, not only "c" but also . "We are Rietvelders" means that we must be only "codes drivers", "cheffeurs des codes", "voditeli program"? Have we to accept the "Procust bed" of the Rietveld codes at a given moment? All Rietveld codes are improving in time, isn't it? In particular for the Round_Robin sample TCH-pV works because c=0.18. (Davor explained how Dv and Da are found). But if c>0.4 any pV fails. Best wishes, Nicolae
Re: Size Strain in GSAS
>Dear Nicolae, >Maybe ya ploho chitayu i ploho soobrazhayu, but even after your>explanation I can't see a way to calculate from the results of>fitting described in chapters 6 & 7 of JAC 35 (2002) 338-346. From such>fitting you obtain only dispersion parameter c. Or I missed something?>Anyway, being "Rietvelders" we still have to deal with TCH-pV function>and we need to extract as much as possible correct information from it.>Hope we shall see more appropriate functions for microstructure>analysis in popular Rietveld programs.>Cheers,>Leonid Dear Leonid, Indeed you missed something. I presume you have the paper. Then, take a look to the formula (15a). This is the size profile for lognormal. There is the function PHI - bar of argument 2*pi*s*. Replace this function PHI - bar from (15a) by the _expression_ (21a) with the argument x=2*pi*s*. You get it? So, not only "c" but also . "We are Rietvelders" means that we must be only "codes drivers", "cheffeurs des codes", "voditeli program"? Have we to accept the "Procust bed" of the Rietveld codes at a given moment? All Rietveld codes are improving in time, isn't it? In particular for the Round_Robin sample TCH-pV works because c=0.18. (Davor explained how Dv and Da are found). But if c>0.4 any pV fails. Best wishes, Nicolae
Re: Size Strain in GSAS
... > > Yes, profiles can be approximated, but the question is not in > approximating profiles. The primary topic of the discussion is "Size > Strain in GSAS". GSAS and most other Rietveld refinement programs use > TCH-pV profile function which provides the simplest and more or less > correct way for separating microstructural and instrumental broadening > contributions. Unfortunately, the microstructural parameters such as Dv > and Da sizes derived (classically) from TCH-pV deviate significantly > from reality for narrow and broad dispersions. That's why the > TCH-pV-based calculations of Dv, Da or average crystallite diameter > need to be modified and calibrated on, at least, simulated data for > various dispersions. > The formalisms presented in chapters 6 and 7 of JAC 35 (2002) 338-346 > are more complex and they don't give a clear way for separating > microstructural from instrumental effects and, besides, for estimating > the values of Dv, Da or . > > Leonid > Dear Leonid, It is not exact what you say, ty ploho cital. 6 & 7 from JAC 35 (2002) 338-346 gives the size profile - formulae (15a) combined with (21,22) or (20a) combined with (23,24). If you look carefully, these profiles are approximated by pseudo-Voigt or sums of 2 or 3 Lorentzians. These profiles depends of 2 parameters, and c, that are refinable and, once refined, both Dv and Da can be calculated (formulae (12,13) or (17,18)). NOW, if approximate the instrumental function by Voigt, that is possible in very many cases (or sums of Voigts to account for asymmetry for example) and also the strain effect by Gaussian or even Voigt, the resulted profile will be a Voigt (or sum of Voigt), that is used as profile in the whole pattern fitting, this profile including in principle all broadening effects (isotropic). You are claiming that it is not TCH-pseudoVoigt. Right, it is not, and can not be, in general, because for c>0.4 the size profile is no more pseudo-Voigt. The size profile given in that paper cover a much wider range of "c" (for lognormal distribution), including superlorentzians. On the other hand is a trivial matter for a programmer to include this profile in any whole pattern fitting code (Rietveld included). (We did that in a "private" whole pattern fitting program). But certainly not, if the programmer wish to use exclusively TCH and nothing else. Why? I don't know. Note that TCH is an empirical profile that reasonably approximate a Voigt function (not the tails) that contains an empirical constraint: that FWHH of Lorenz and Gauss components are equal one to another and equal with that of the whole psudoVoigt. Best wishes, Nicolae Popa
Re: Size Strain In GSAS
rate that it's important to take into > consideration the metric or geometry of the problem. > > In a mathematical physics context, line profile analysis is simply not > about parameter estimations and curve fitting, but more fundamentally a > problem of mapping a functional space. By functional space I mean that > each point represents a size, shape and/or a dislocation distribution. The > present methods used in line profile analysis, make specific assumptions > about the distributions and/or profile functions, and only represent a > "point" in the functional space. Moreover, using a Bayesian/maximum > entropy reasoning these assumptions may not be physically justified. > This is the basic underlying weakness in the present methods. By stating > this I don't mean to be critical of the present efforts. It is a > statement drawn from a theoretical/mathematical point of view. > > The Bayesian/MCMC and Bayesian/MaxEnt methods have been tested on a > wider verity of simulated data and applied and being used to re-analyze > experimental data (i.e. CEO2 round-robin data). Here I can't stress > enough the need to carryout full and rigorous simulations which take > inot account eh instrumental and noise/background effects etc. In > addition, when/where possible blind test should be carried out. I will > be presenting the two talks: at Denver and Florence where I will be > giving a full description of these methods as applied to developing the > latest NIST Nanocrystallite size SRM1979. (Hope to see you there...) > > About microstrain/dislocations. This si a really hard problem... We are > presently working on applying the Bayesian/MaxEnt methods to determining > microstrain/dislocation distributions. We have number theoretically > approaches for this up our sleeves. A simplified approach as been > presented in [10] (see chapter 5). It has been generalized to include > elastically anisotropic materials by applying the contrast factors > (unpublished). The second approach is computationally time consuming but > involves simulating various microstructures and determining their > probabilities and entropy relative to the experimental data. But this > problem is really hard and progress is slow. A third approaches, > quantifies the probabilities for various models and selects the best > give the experimental data. This includes both size and dislocations > broadening effects (I'm never short on ideas) > > I hope this helps and has addressed some of the queries/questions. Best > Regards, Nick > > > References > [1] Armstrong, N. et al. (2004a), "Bayesian inference of nanoparticle > broadened x-ray line profiles", J. Res. Nat. Inst. Stand. Techn., > 109(1),155-178, > URL:http://nvl.nist.gov/pub/nistpubs/jres/109/1/cnt109-1.htm > [2] Armstrong, N. et al. (2004b) "A Bayesian/Maximum Entropy method for > certification of a > nanocrystallite-size NIST Standard Reference Material", chapter 8 in > "Diffraction > analysis of the microstructure of materials", Springer-Verlag. > [3] Armstrong, N. et al. (2004c), "X-ray diffraction characterisation of > nanoparticle size and shape distributions:--Application to bimodal > distributions", > Proceedings of the Wagga-Wagga Condense Matter Physics & Materials > Science Conference, Janurary 2004. > URL:http://www.aip.org.au/wagga2004/papers.php > [4] Armstrong, N. et al. (2005), "Bayesian analysis of ceria > nanoparticles from line profile data", to be published in Advances in > X-ray Analysis. > [5] Sivia, D.S. (1996), "Data Analysis: A Bayesian tutorial" > [6] Jaynes(1982), Proc. IEEE, 70(9), 939-952 > [7] Johnson & Shore, (1983), IEEE Trans. IT,26(6), 942-943 > [8] Shore & Jhnson (1980), IEEE Trans. IT,26(1), 26-37 > [9] Amari (1985), "Differential-geometrical methods in statistics", > Springer, Berlin > [10] Armstrong, N. PhD Thesis, UTS, Australia > > Nicolae Popa wrote: > > >Hi, > >So, to resume your statements, by using Bayesian/Max.Entr. we can > >distinguish between two distributions that can not be distinguished by > >maximum likelihood (least square)? Hard to swallow, once the restored peak > >profiles are "the same" inside the noise. What other information than the > >peak profile, instrumental profile and statistical noise we have that > >Bayes/Max.ent. can use and the least square cannot? > > > >"prior distributions to be uniform" - if I understand correctly you refer to > >the distributions of "D0" and "sigma" of the lognormal (gamma) distribution > >from which the least square "chooses" the solution, not to the distribution > >itself (logn, gamm). Then, how is this prior distribution for Baye/Max.ent.? > > > >Best, > >Nick Popa > > > > > > > > > >>Hi > >>Sorry for the delay. The Bayesian results showed that the lognormal was > >> > >> > >more probable. Yes, the problem is ill-condition which why you need to
Re: Size Strain In GSAS
Hi, So, to resume your statements, by using Bayesian/Max.Entr. we can distinguish between two distributions that can not be distinguished by maximum likelihood (least square)? Hard to swallow, once the restored peak profiles are "the same" inside the noise. What other information than the peak profile, instrumental profile and statistical noise we have that Bayes/Max.ent. can use and the least square cannot? "prior distributions to be uniform" - if I understand correctly you refer to the distributions of "D0" and "sigma" of the lognormal (gamma) distribution from which the least square "chooses" the solution, not to the distribution itself (logn, gamm). Then, how is this prior distribution for Baye/Max.ent.? Best, Nick Popa > Hi > Sorry for the delay. The Bayesian results showed that the lognormal was more probable. Yes, the problem is ill-condition which why you need to use the Bayesian/Maximum entropy method. This method takes into account the ill-conditioning of the problem. The idea being it determines the most probable solutions from the set of solutions. This solution can be shown to be the most consistent solution or the solution with the least assumptions given the experimental data, noise, instrument effects etc (see Skilling & Bryan 1983; Skilling 1990; Sivia 1996). This is the role of entropy function. There are many mathemaitcal proofs for this (see Jaynes' recent book). The Bayesian analysis maps out the solution/model spaces. > > Also the least squares solution is simple a special case of a class of deconvolution problems. This s well established result. It is not the least ill-posed, since it assumes the prior distributions to be uniform (in a Bayesian case. See Sivia and reference therein). In fact it's likely to be the worst solution since it assumes a most ignorant state knowledge (ie. uniform proir) and doesn't always take into consideration the surrounding information. Moreover, it doesn't account for the underlying physics/mathematics, that the probability distributions/line profiles are positive & additive distributions (Skilling 1990; Sivia 1996). > > Best wishes, Nick > > > Dr Nicholas Armstrong > > Hi, once again, > > Fine, I'm sure you did. And what is the most plausible, lognormal > > or gamma? > > From the tests specific for least square (pattern fitting) they are > > equallyplausible. And take a combination of the type w*Log+(1- > > w)*Gam, that will be > > equally plausible. > > On the other hand, why should believe that the Baesian > > deconvolution (or any > > other sophisticated deconvolution method that can imagine) give the > > answermuch precisely? Both, the least square and deconvolution are > > ill-posed > > problems, but the least square is less ill-posed than the > > deconvolution. At > > least that say the manuals for statistical mathematics. > > > > Best wishes, > > Nicolae Popa > > > > > > > > > > > > > Hi, > > > I pointed out that each model needs to be tested and their > > plausibilitydetermined. This can be achieved by employing Bayesian > > analysis, which > > takes into account the diffraction data and underlying physics. > > > > > > I have carried out exactly same analysis for the round robin CeO2 > > samplefor both size distributions using lognormal and gamma > > distributionfunctions, and similarly for dislocations: screw, edge > > and mixed. The > > plausibility of each model was quantified using Bayesian analysis, > > where the > > probability of each model was determined, from which the model with > > thegreatest probability was selected. This approach takes into > > account the > > assumptions of each model, parameters, uncertainties, instrumental > > andnoise effects etc. See Sivia (1996)Data Analysis: A Bayesian > > Tutorial(Oxford Science Publications). > > > > > > Best wishes, > > > Nick > > > > > > Dr Nicholas Armstrong > > > > > > > > > > > > > > Hi, > > > > But the diffraction alone cannot determine uniquely the physical > > > > model. An > > > > example at hand: the CeO2 pattern from round-robin can be > > equally well > > > > described by two different size distributions, lognormal and gamma > > > > and by > > > > any linear combinations of these two distributions. Is the > > situation> > different with the strain profile caused by different > > types of > > > > dislocations,possible mixed? > > > > > > > > Best wishes,
Re: Size Strain In GSAS
Hi, once again, Fine, I'm sure you did. And what is the most plausible, lognormal or gamma? >From the tests specific for least square (pattern fitting) they are equally plausible. And take a combination of the type w*Log+(1-w)*Gam, that will be equally plausible. On the other hand, why should believe that the Baesian deconvolution (or any other sophisticated deconvolution method that can imagine) give the answer much precisely? Both, the least square and deconvolution are ill-posed problems, but the least square is less ill-posed than the deconvolution. At least that say the manuals for statistical mathematics. Best wishes, Nicolae Popa > Hi, > I pointed out that each model needs to be tested and their plausibility determined. This can be achieved by employing Bayesian analysis, which takes into account the diffraction data and underlying physics. > > I have carried out exactly same analysis for the round robin CeO2 sample for both size distributions using lognormal and gamma distribution functions, and similarly for dislocations: screw, edge and mixed. The plausibility of each model was quantified using Bayesian analysis, where the probability of each model was determined, from which the model with the greatest probability was selected. This approach takes into account the assumptions of each model, parameters, uncertainties, instrumental and noise effects etc. See Sivia (1996)Data Analysis: A Bayesian Tutorial (Oxford Science Publications). > > Best wishes, > Nick > > Dr Nicholas Armstrong > > > > Hi, > > But the diffraction alone cannot determine uniquely the physical > > model. An > > example at hand: the CeO2 pattern from round-robin can be equally well > > described by two different size distributions, lognormal and gamma > > and by > > any linear combinations of these two distributions. Is the situation > > different with the strain profile caused by different types of > > dislocations,possible mixed? > > > > Best wishes, > > Nicolae Popa > > > > > > > > > Best approach is to develop physical models for the line profile > > broadening and test them for their plausibility i.e. model selection. > > > > > > Good luck. > > > > > > Best Regards, Nick > > > > > > > > > Dr Nicholas Armstrong > > > > > > > > > -- > UTS CRICOS Provider Code: 00099F > DISCLAIMER: This email message and any accompanying attachments may contain > confidential information. If you are not the intended recipient, do not > read, use, disseminate, distribute or copy this message or attachments. If > you have received this message in error, please notify the sender immediately > and delete this message. Any views expressed in this message are those of the > individual sender, except where the sender expressly, and with authority, > states them to be the views the University of Technology Sydney. Before > opening any attachments, please check them for viruses and defects. >
Re: Size Strain In GSAS
Hi, But the diffraction alone cannot determine uniquely the physical model. An example at hand: the CeO2 pattern from round-robin can be equally well described by two different size distributions, lognormal and gamma and by any linear combinations of these two distributions. Is the situation different with the strain profile caused by different types of dislocations, possible mixed? Best wishes, Nicolae Popa > Best approach is to develop physical models for the line profile broadening and test them for their plausibility i.e. model selection. > > Good luck. > > Best Regards, Nick > > > Dr Nicholas Armstrong
Re: spherical hamonics
Hi, In Bragg-Brentano geometry the sample symmetry play no role because the scattering vector has a single direction in sample for the whole pattern (the normal to the plate). So in this geometry it is "seen" only the dependence of the pole distribution on the crystal direction (hkl). By forcing sample symmetry mmm it is like you fit a measured line by a+bx+cx+dx in place of a+Bx. The fit works, a,b,c take any value, positive or negative, with the condition that a+b+c=B. But when you calculate a**2+b**2+c** (the texture strength) you are risking to find an enormous value. The program must have an option for Bragg-Brentano to avoid this trouble, ask the authors. Good luck, Nicolae Popa > Hi everyone, > > I tried to use spherical harmonics incorporated in EXPGUI/GSAS for > modelling a preferential orientation in my sample. I used a cylindrical > sample symmetry and it worked fine but I am confused about using such type > of symmetry for a flat sample (Bragg-Brentano goniometer). The rolling > (mmm) symmetry gave me a texture parameter about 400, which I can not > accept. I would appreciate any comments related to this topic. > > One more question: Does GSAS or any other software have any option for > visualization of March-Dollase multi-axis preferential orientation? > > sorry for using your time, > > Yaroslav > > Yaroslav Mudryk > 252 Spedding Hall > Ames Laboratory > Ames, IA 50011, USA > Phone: 1-(515)-233-2041 > >
Re: Size distribution from Rietveld refinement
> actually I implemented the size and strain distributions (both) in my > Rietveld code (Maud) and I demoed it in Praha beginning of September. Thanks Luca, I'm very, very happy to hear that. Really you are moving very fast! I was prepared to come in Prague but, unfortunately, I had to cancel one week before from an unexpected family problem. Keep in touch. Best wishes, Nicolae Popa
Re: Size distribution from Rietveld refinement
> gamma, or whatever we assume it to be. On the former, it is easy to see if
> observed profiles can't be successfully fit ("super-Lorentzian" peak
shapes,
> for instance), which means that the TCH peak shape cannot be used.
However,
> an assumption that physically broadened profiles (size and strain) are
also
> Voigt function is more difficult to prove; if not and one uses the
equations
> described above, a systematic error will be introduced. On the latter, a
Good answer Davor, but why you are avoiding to say that if the size profile
(15a, 21, 22) from JAC(2002)35, 338-346 (used in 3.1 of RR paper) would be
implemented in the Rietveld codes these codes would become much "powerful"
and with a wider application in the size distribution determination?
Nicolae
Re: rietveld refinement
> > It is also true that no development has been done for anisotropy. Not yet! > > Well, if all previous works about trying to take account of size/strain > anisotropy in the Rietveld method are nothing yet, this allows to > close the discussion. Let us wait for really serious developments to > come. You not correctly understood me (I would like to believe that not ill-disposed). I said that no development for size anisotropy has been done including "physical" size distributions (like lognormal, etc.) as were done for the isotropic case. For example: Langford, Louer & Scardi, JAC (2000) 33, 964-974 and Popa & Balzar JAC (2002) 35, 338-346. Concerning previous (phenomenological) works trying to take account of strain/size anisotropy in the Rietveld method, I have myself a contribution: "The (hkl) dependence of diffraction-line broadening caused by strain and size for all Laue groups in Rietveld refinement, N. C. Popa, J. Appl. Cryst. (1998) 31, 176-180." Could I be so stupid to say that such kind of works, including mine, are nothing? Best wishes, Nicolae Popa
Re: rietveld refinement
Title: RE: rietveld refinement Doesn't help with a size distribution, as it only works well for a relatively monodisperse material - but it does work under some circumstances. Pam I disagree, it works also for large dispersion. One example you can find in JAC (2002) 35, 338-346, "Sample 2". It is true that the specific peak profile (that can be "superlorentzian") can not be found in no available Rietveld code. It is also true that no development has been done for anisotropy. Not yet! Best wishes, Nicolae Popa
Re: rietveld refinement
> So I cannot let say that "Significantly different "physical" > size distributions could describe equally well the peak profile". > This is confusing. You may say that : significantly different > crystallite shapes could describe equally well the peak profile > in cubic symmetry. I am not sure that this sentence is > valuable equally for other symmetries when looking at all Sorry, it seems me that rather your sentence is confusing, not mine. In the example with CeO2 the crystallites are quite spherical (one shape) even seen by microscope. But two significantly different distributions of the sphere radius (6a1, 6a2) (lognormal & gamma, respectively) given quite the same column length distribution (6b1, 6b2) and practically the same peak profile. It is no matter here of different crystallite shapes because the shape is unique (sphere). And also the cubic symmetry has no relevance, this should happen for any symmetry (I mean not an unique solution for the sphere radius distribution). (By the way, the sample of CeO2 in discussion is just the sample used in the round-robin paper that you co-authored; in this last paper we used only the lognormal distribution, but doesn't mean that this is the unique solution from powder diffraction). Concerning the different crystallite shapes, this is another storry. I said that even if the cristallites are not spherical, it is not obligatory to observe an anisotropic size broadening effect. Not spherical crystallites is only the necessary condition for size anisotropy effect, but not sufficient. The anisotropic size broadening effect is observable only if the non spherical shape is preferentially orientated with respect to the crystal axes (don't confuse with the texture). It is the case of your nickel hydroxyde in which the plate-like normal is preferentially oriented along the hexagonal c axis. But, if the not spherical crystallite shapes are randomly oriented with respect to the crystal axes (which is possible) the size broadening effect is isotropic and, only from powder diffraction, we can conclude erroneously that the crystallites are spherical. On the other hand, if the anisotropy is observed, the crystallite shape (and the distributions of specific radii) can not be uniquely determined only from powder diffraction. What we can determine is an apparent shape (and column lengths averages). Has any sense, in this case, to search for so called "physical models", or we have to be content with "phenomenological" findings (so much blamed, at least implicitely)? It is only a question, valid also for the strain effect. > So, let us have more fun with a size strain round robin on some > complex sample (or even a size-only round robin not on a > cubic compound ;-). I agree entirely. Best wishes, Nicolae Popa
Re: Unexpected honour
- >>Why "dissident" Armel ? >I am an adept of the open access to the knowledge, >your religion looks different. >Armel >from http://www.dictionary.com: >Disagreeing, as in opinion or belief. >\Dis"si*dent\, a. [L. dissidens, -entis, p. pr. of dissidere to sit apart, >to disagree; dis- + sedere to sit: cf. F. dissident. See ><http://dictionary.reference.com/search?q=sit>Sit.] No agreeing; >dissenting; discordant; different >Our life and manners be dissident from theirs. --Robynson (More's Utopia). >\Dis"si*dent\, n. (Eccl.) One who disagrees or dissents; one who separates >from the established religion. >The dissident, habituated and taught to think of his dissidenc? as a >laudable and necessary opposition to ecclesiastical usurpation. --I. Taylor. Sorry, but I think there is here a mal interpretation (is correct that in English?) of the dictionary. I don't think that the people in SDPD list are thinking the powder diffraction differently than the people in the Rietveld list. Or, if there are differences on some particular subject from one member to other, this can happen also inside the same chat list. An alternative, a diversification, does not mean automatically a disidence. Let us not blurred a word very dear to people like me, rising and living most of the life in a dictatorial regime. Yours, Nicolae Popa
Re: rietveld refinement
> > >The diffraction alone can not decide. Significantly different "physical" > >size distributions could describe equally well the peak profile > >(J.Appl.Cryst. v35 (2002) 338-346 - self citation too). > >Nicolae Popa > > Looking at your figures 6b1 and 6b2, I measure how we > differ on the sense of "significantly different". As you comment > in the text, "The curves 1 and 2 differ in the position of the > maximum by only 2 A and in height of the maximum by > 9.76%". > > I would not call that "significantly different" but "very similar". > > Armel Yes, but the figure 6b represents the COLUMN LENGTH distribution not the CRYSTALLITE RADIUS distribution (in this case of spherical crystallites). The crystallite radius distributions are given in 6a1 and 6a2 (lognormal and gamma, respectively) and they are significantly different, what can be seen also in the table 1: the average radius and the dispersions are completely different. Nevertheless the profile of the diffraction peak is equaly well described. And the column length distribution is quite the same (as discussed in text and as you observed). But when we are speaking about the "physical model" we understand in fact the distribution of the crystallite radius (if spherical). Is that lognormal or gamma? Is the average radius 90(6) or 69(1) Angstroms, is the parameter c (determining the dispersion) 0.18 or 0.39? We can not say only from diffraction that one is more "physical" than other. On the other hand is the column length distribution a full "physical" description of the crystallites, I mean of the shape and radius (radii) distribution? I think not. You can imagine, for example, that the crystallites are even not spherical, but ellipsoidal. It is easy to understand that if the Euler angles representing the orientations of the ellipsoidal principal axes with respect to the crystal axes are UNIFORMLY distributed in their domains of definition, will be NO anisotropy effect. Then we can think the crystallite are spherical with a certain distribution of radius, when in fact they are ellipsoidal with other distributions of (three) radii. But the column length distribution (and the peak profile) is the same. What we see in diffraction is the column lengths (volume & area averaged) and the classics were not full ignoring the shape and radius (radii) distribution(s). Nicolae Popa (Mister, Messieur, Don, Dom, etc.)
Re: rietveld refinement
> methodology, if not that they are "physical" (I believe they are > "physical" in case of size-only effect). The diffraction alone can not decide. Significantly different "physical" size distributions could describe equally well the peak profile (J.Appl.Cryst. v35 (2002) 338-346 - self citation too). Nicolae Popa
Re: Unexpected honour
Don't worry, it is only funny. And now is funny even more (my opinion) Nicolae Popa > Dear Respected Sir, > Have I done anything wrong! I am really scared, because I am not that much sound in english. > I wrote like that because I respect Prof. Armel very much. > > Please forgive me if there were some thing wrong or funny. > > With best regards, > Apu > > > > > /_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/ > Apu Sarkar > Research Fellow > Variable Energy Cyclotron Centre > Kolkata 700 064 > phone: 91-33-2337-1230 (extn. 3190) > Fax: 91-33-2334-6871 > INDIA > /_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/_/ > > > - Original Message - > From: "Shankland, K (Kenneth)" <[EMAIL PROTECTED]> > Date: Thursday, November 18, 2004 1:47 pm > > > > > Apu wrote.. > > > > > Dear Sir Armel > > > > Presumably this long-overdue recognition > > must be part of the centenary celebrations > > of the "Entente Cordiale"? > > > > > > > > > > > > > > >
Re: Anisotropic line broadening in cubic material
Dear Jens, Peter Sthephens is right, try first to see if you have an anisotropic strain effect. But if not, it doesn't mean that you have not a simple size effect, not necessarily staking faults. The size anisotropy model in GSAS is in fact the rod (or plate) model (I wonder why the needles model was not introduced - sin(phi) in place of cos(phi)) and you have to give apriori the "broadening axis". For non cubic is easy to guess because frequently is the n-fold axis (n=2,3,4,6) and the average over equivalents has no effect. But as Peter said the guess is ambiguous for cubic (and not only). Nevertheless you have an approach for size anisotropy that needs no apriori information (except the Laue group), the spherical harmonics approach. For details see the same (J. Appl. Cryst. 31, 176 (1998)). In spite of some skeptical opinions (not clearly argued) the approach is enough robust and you can expect to obtain accurate volume averaged column length as function of direction. Best wishes, Nicolae Popa > > Jens, > > Your effect might be more related to strain than size broadening. You > would have to check widths at various diffraction orders in a given > direction (i.e., 111, 222, 333, etc., vs 200, 400, 600, etc. for an fcc > material). If the widths increase roughly in proportion to diffraction > order, but with a different slope for the two directions, you have > anisotropic strain broadening. > > This was noted by Stokes and Wilson (Proc. Phys. Soc. London 56, 174-181 > (1944)) in cold-worked fcc metals, who had a model as a random distribution > of stresses. N. Popa and I have independently considered the effect more > recently from a phenomenological viewpoint (J. Appl. Cryst. 31, 176 (1998) > and ibid 32, 281 (1999), respectively). And there is a growing literature, > especially from the group of Tamas Ungar, on the effect of specific lattice > defects on strain-broadening in diffraction patterns. > > Regarding your use of the anisotropic size broadening model in GSAS, as you > point out, "broadening axis" for a cubic material is a rather iffy concept. > If my understanding is correct, GSAS does not do the full symmetry > equivalents in that calculation, and so it's a matter of luck how the > calculation will be done. That is, if you list a (111) broadening axis, > and the reflection list contains (111), you'll get one answer, but if you > list (-1 1 1) broadening axis, the (111) reflection will be calculated > differently. > > -Peter > > ~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~^~ > Peter W. Stephens, Professor > Department of Physics & Astronomy > State University of New York > Stony Brook, NY 11794-3800 >
Modesty
CNRS is wrong, but take into account that nobody is a prophet in his own country N. Popa - Original Message - From: "Armel Le Bail" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Monday, August 23, 2004 1:38 PM > > >And he is modest as well :-) > > I have to be modest. According to the CNRS, I am a > second-class researcher... > > Armel >
thick marks shift
Doinitza, Doinitza, scuze fara rost. Cei mai multi folosesc GSAS, dupa cum vezi Bob a mers la sigur. - Original Message - From: "Doinita E Neiner" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Monday, June 28, 2004 11:35 PM Hello, I am so sorry, I have forgotten to specify that i am using GSAS.
Re: GSAS informations
> Dear Prof Popa, > > I had been meaning to implement the quartic form for peak width in a > refinement program for some time, but did not figure out how to generate > the constraints from a general list of symmetry operators. Is there a > simple trick for doing this? I was thinking of just choosing a Dear Jon, Sorry, I had no time and I'll have not at least 5 days to answer to your (too) long questions. May be later, OK? Best wishes Nic. Popa
Re: GSAS informations
> > >(you could be a good boxeur, Armel!), > > Knocked out at round 4 ! Argh ! Some people believe that "fair play" is mainly an Anglo-Saxon apanage (prerogative). Obviously they are wrong. > > Anyway, a sphere was good enough for the previous > size-strain round robin... Hope that the next size-strain > round robin will be more complex, and will succeed > in excluding definitely any ellipsoid from the ring. > > Armel > > Agree with you concerning the complexity of the next SS-RR but, perhaps, we can keep some ellipsoids if they are properly used and in the right place. Best wishes, Nicolae
Re: GSAS informations
> Not violating symmetry restrictions you may either > have the sphere with the terms 11=22=33 and 12=13=23=0 > or something else allowing the 12=13=23 terms to be equal > but different from 0. These two possibilities are all you can do > in cubic symmetry with h,k,l permutable. If I am not wrong. You are. The cross terms have disappear even at orthorhombic (monoclinic has only one). Cubic is an orthorhombic to which a 3-fold axis is added on the big diagonal resulting in 11=22=33. Nicolae
Re: GSAS informations
> > >Presume one of your students makes a fit on a sample having only size > >anisotropy and he is able to determine the six parameters of the ellipsoid. > >But after that he has a funny idea to repeat the fit changing (hkl) into > >equivalents (h'k'l'). He has a chance to obtain once again a good fit, with > >other ellipsoid parameters but with (approximately) the same average size, > >this > >time in other direction > >[lamda/cos(theta)/M(hkl)=lamda/cos(theta)/M(h'k'l')]. Am I wrong? > > But why to presume so soon that people are dumb ? > > You may also presume that the student is not stupid enough > for trying to determine the 6 parameters of the ellipsoid in any case > and that he applies restrictions related to the symmetry, as > recommended in the software manual (the software name was > ARIT)... That manual says that the 6 parameters are obtainable > only in triclinic symmetry, etc. > > I prefer to presume first that people are smart, and may be change > my opinion later. > > I guess that the Lij in GSAS are explained to be symmetry- > restricted as well. > > Armel > By contrary, I presumed a smart student observing immediately that by applying to the ellipsoid the symmetry restrictions he obtains some strange ellipsoids: for orthorhombic the principal axes are always along the crystal axis, for trigonal, tetragonal & hexagonal they are always rotation ellipsoids with 3,4,6 - fold axis as rotation axes. He could ask the master how is the nature so perfect. How know the crystal to grows always along the symmetry axis? But the most wondered will be the student seeing that for cubic crystals the ellipsoid is in fact a sphere. To not risk the next examination probably he will not put this question: how then you searched for size anisotropy in CeO2 with ARIT? Or the symmetry restrictions are optional? Nicolae Popa
Re: GSAS informations
Hi, > It seems that we disagree on the meaning of some > english words. English is not my mother language, so I may be > wrong. Nor mine, so I can be equally wrong (or worse). > I was able to put one word on that definition (thanks for it) in my > previous email : distribution (a size distribution). > > In these earlier works (maybe you define any earlier work as > being "naive" ?) it is not at all the crystallite shape which is > approximated by an ellipsoid. The ellipsoid is there for > modelling the variation of the average size M(hkl) (which is > the mean of the size distribution). If ellipsoid models the crystallite shape is an approximation, good or not good, if models the average size "seen" in powder diffraction as function of direction is a mistake (see next comment). > > So, thanks, I used ellipsoids in 1983-87 for describing some > simple size and strain anisotropy effects in the Rietveld method. > I think that no elementary principle was violated, though Presume one of your students makes a fit on a sample having only size anisotropy and he is able to determine the six parameters of the ellipsoid. But after that he has a funny idea to repeat the fit changing (hkl) into equivalents (h'k'l'). He has a chance to obtain once again a good fit, with other ellipsoid parameters but with (approximately) the same average size, this time in other direction [lamda/cos(theta)/M(hkl)=lamda/cos(theta)/M(h'k'l')]. Am I wrong? If not, how you explain him that averages of the size distribution are the same in different directions once were approximated by an ellipsoid? And what set of ellipsoid parameters you advice to consider, the first or the second one? > particularly stupid. The ellipsoid method was applied to the > recent Size-Strain Round Robin CeO2 sample, giving > results not completely fool (in the sense that not a > lot of anisotropy was found for that cubic sample > showing almost size-effect only, and quasi-isotropy). Not surprisingly. Take a sphere and put two cones at the ends of one diameter. Certainly the finite high of cone means anisotropy. Refine the same CeO2 pattern. Very probably you will find zero for the cone high. Is this funny model equally good? > cubic case showing strong stacking fault effects for HNbO3 > (cubic symmetry). A neutron pattern is available. I would be interested > in a better estimation of the size and strain effects on that sample > (not only a phenomenological fit). Can you provide that better > estimation ? > > Best wishes with HNbO3, > > Armel Le Bail Strong staking faults effect? I would accept your challenge, but I'm not sure that with a knife in place of scissors is possible to do easy tailoring. That doesn't mean the knife is good for nothing. Best wishes and ... il faut pas s'enerver Nicolae Popa
Re: GSAS informations
> Our whole science is a so bad approximation to the Universe... > > For the representation of an isotropic size effect , you may imagine > the mean size being the same in all directions, obtaining a > sphere. The same for a mean strain value. > > Introducing some anisotropy in mean size and mean strain in the > Rietveld method was done in the years 1983-87 by the "naive" view that > the mean size M(hkl) in any direction could be approximated by > an ellipsoid rather than a sphere, and the same for the mean > strain (hkl). See for instance J. Less-Common Metals > 129 (1987) 65-76. Hello Messieur Le Bail, (and thanks for explaining how to pass from sphere - isotropy to ellipsoid - anisotropy). The naive character doesn't come from the approximation of the crystallite shape by an ellipsoid, but from the approximation of the size effect in powder diffraction by ellipsoid. In powder diffraction it is seen not one, but a (big) number of crystallites more or less randomly oriented. The crystallites in reflection "show" different diameters, not only one. Concerning the mean strain, another confusion. In fact the mean strain gives the peak shift, sometimes reasonably described by an ellipsoid in (hkl) (for example not-textured samples under hydrostatic pressure). But the strain broadening is related on the strain dispersion (you wrote not ) that in first approximation is a symmetrized quartic form and its square root (giving breadth) is never an ellipsoid. Certainly, always one can use ellipsoids as a first approximation for any kind of anisotropy, with the condition to not violate some elementary principles, in particular, here, the invariance to symmetry. It has no relevance to use the thermal ellipsoids as argument. The thermal ellipsoids are a natural consequence of the harmonic vibration of the atoms and no principle is violated, even if, some times, this is a rough approximation because of a high contribution of anharmonicity. > > Less "naive" representations were applied in the years 1997-98 > (so, ten years later). But these less naive representations were not > providing any size and strain estimations, Not surprisingly, people are mainly interested to obtain a good structure refinement and ignore by-products like strain an size. Doesn't mean that strain and size can not be estimated better. >the fit was quite better > (especially in cases showing stacking faults, with directional effects > hardly approximated by ellipsoids) but remained "phenomenological". The thermodynamics is phenomenological science, have we to consider it a naive or a less naive science? Best wishes a happy Easter, Nicolae Popa > You can find experts in thermal vibration explaining that the ellipsoid > representation used by crystallographers is an extremely naive view > of the reality, and they are right. But crystallographers continue to > calculate these Uij (and there is a table giving Uij restrictions) > which in most cases provide a minimal and sufficient representation > of thermal vibrations... > Armel > >
Re: GSAS informations
Dear Christophe, The coefficients Lij in the formula you wrote have no significance. This formula is a naive representation of strain anisotropy that falls at the first analysis. It is enough to change the indices hkl into equivalent indices and you obtain other Gamma. As a consequence, in cubic classes for example, the microstrain anisotropy doesn't exist, which is a nonsense. The correct formulae are indeed in Peter Stephens paper (at least for a part of Laue classes) but also in a paper by Popa, J. Appl. Cryst. (1998) 31, 176-180, where the physical significance of coefficients is explicitly stated. Hence, if denote by Eij the components of the microstrain tensor in an orthogonal coordinate system related to crystallite, then the coefficients are some linear combinations (specific to every Laue class) of the averages . Best wishes, Nicolae Popa - Original Message - From: Christophe Chabanier To: [EMAIL PROTECTED] Sent: Wednesday, April 07, 2004 6:45 PM Subject: GSAS informations Hello everybody,i have a question about the GSAS software. Indeed, i would like to know what are exactly the L11, L22, L33L23 parameters. I saw that these parameters represent the anisotropic microstrain in material. Moreover, there is an empirical _expression_ which uses these parameters as following : Gamma(L) = L11*h^2 + L22*k^2 + L33*l^2 + 2*L12*hk + 2*L13*hl + 2*L23*kl I would like to know and understand the physical representation of these parameters and this _expression_.Thanks in advance Christophe ChabanierINRS-Énergie, Matériaux et Télécommunications 1650 Blvd. Lionel Boulet C. P. 1020, Varennes Qc, Canada J3X 1S2Tél: (450) 929 8220Fax: (450) 929 8102Courriel: [EMAIL PROTECTED]
