Program
Method
Microstrain
Size
Anisotropy

Results
Lab-1
Lab-2
Sync-1
Sync-2
Neut-1
Neut-2

Conclusion
References

Results on CeO2
'sharp and broad'

Contribution to the Size/Strain Round Robin
http://www.boulder.nist.gov/div853/balzar/s-s_rr.htm


Armel Le Bail

October 18, 2000


A - Program used

The software used in this study is ARIT available at :

http://sdpd.univ-lemans.fr/arit.html
There are 2 versions allowing to deal with data having a constant step in 1/100° (2q) or in 1/1000°.

B - Method in ARIT

 
ARIT is a Rietveld program for structure refinement which may also apply the “Le Bail method” in order to extract structure factor amplitudes from a powder pattern by iterating the Rietveld decomposition formula.
 
Description of the special profile shapes and size/microstrain approach in the ARIT program can be found in reference [1], although the program was first described in a slightly different form at the XIIIth International Congress of Crystallography, Hamburg, 1984 [2].
Summarizing, profile shapes are described in ARIT by Fourier series, allowing the replace the h = f*g convolution by a simple product in Fourier space : H = F.G. The calculated G part is obtained from the fit of a well crystallized sample representing g (CeO2 “sharp” in this case). The experimental pattern h for an ill-crystallized material is fitted by reusing the previously determined G part, multiplied by the F sample contribution, where Fn = AnS.AnD, the traditional product of the size and microstrain Fourier coefficients.
Currently, ARIT applies a flexible model for the microstrain part and only one model for the size-broadening part which was found to give relative satisfaction (but other size models could be introduced as well). In the following, we use the B. E. Warren formalism [3].
 
Microstrain effect in ARIT
 
A hypothetical Gaussian strain distribution is considered (see Warren, p. 270 [3]) such as :
AnD = <cos 2plZn> = exp(-2p2l2<Z2n>)

In that equation, <Z2n> is modelled in ARIT by a flexible variation law of the distortion versus the distance according to the following equation:

<Z2n> = |n|K<Z21>

The ARIT program refines two microstrain parameters : K and <Z21> (i.e. <Z2n> for n = 1). It is to be noted that if K is refined to K = 2, the calculated microstrain profile shape will be Gaussian, and if K is refined to K = 1, it will be Lorentzian. Other shapes being possible, depending on the final refined value of K.

Size effect in ARIT

The size Fourier coefficient AnS is given in terms of p(i), the fraction of the columns of length i cells by the expression [3]:

where N3 is the mean column length number.

Modelling p(i) allows to define AnS. It would not be difficult to introduce several models in ARIT, however, there is currently only one model proposed which is a continuously decreasing size distribution function defined by :

p(n) = g2exp(-g|n|)

The size Fourier coefficients corresponding to this arbitrary size distribution function is :

AnS = exp(-g|n|) ,

And the average number of unit cells is N3 = 1/g

Practically, fictitious quantities have to be defined like in the Warren book ([3] p. 273). The real distance along the columns of cells perpendicular to the reflecting planes is defined by :

L = n a3

Where a’3 depends on the interval of definition of the reflections [q2q1]:

a3 = l / 2 (sin q2 – sin q1)

Anisotropic size and microstrain effects in ARIT

Details about how this is undertaken in ARIT (ellipsoids describing the <Z21>(hkl) and N3(hkl) values) may be found in [1]
 
The Size/Strain Round Robin indicates that effects in CeO2 are isotropic. We do not see how this can be decided in advance. Thus, the possibility that the size/strain effects could depends on the orientation was also tested by using ARIT, selecting ellipsoids constrained to have the 11=22=33 terms equals but possibly different from the 12=13=23 terms.
 
Comments about the method in ARIT :
 
It should be clear that this is modelling, so that if the experimental case departs from the model, then the result will be unpredictably false. Anyway, this is a method for approaching size/microstrain effects when there is so heavy overlapping that other methods needing isolated profiles cannot be applied. It is thus absurd to apply ARIT to CeO2, since all reflections are isolated. Nevertheless, it is interesting to see how the results will compare to methods which do not model the size/microstrain effects but extract them rigorously, as far as this is possible.

C – Results on CeO2

The ARIT program was used in “Le Bail method” mode (cell constraint, but no structure constraint), which allows to obtain generally bette profile reliability factors than when the Rietveld mode is activated. Producing thermal B factors was of no interest in this Size-Strain Round Robin, and there is no free atomic coordinate to refine here.
 
The data necessary for applying ARIT and the final refinement listings are joined. Comment are given when data were processed in some special way.
 
Needed by the ARIT calculation is that the sharp and broad patterns are treated with the same step and same number of points per reflection.
 
Results presentation :
 
Are given in each case :
The filenames necessary for running ARIT : 
  • filename.dat : contains the intensities and background, which was defined visually/manually; 
  • filename.str : contains the starting parameters describing profile shapes;
  • filename.pr1 : contains the starting structure factor amplitudes;
  • filename.typ : contains the final results;
  • filename.gif : contains the drawing from the .prf file (not given, but can be rebuilt by running ARIT on the joined data) by WinPlotr.
Most files have been renamed with .txt extension for being well downloaded through the Web.
 
For the sharp CeO2 will be given :
 
The RP and RWP values as defined by Hugo Rietveld : background subtracted, and from “peak only” zones in the pattern (however, the range in which were defined the profiles is so large that all points are included). 
 
The refined cell parameter, the zeropoint and the profile parameters are in the filename.typ files (nothing else, since the refinement is made by the Rietveld method).
 
For the broadened CeO2 will be given :
  • The range of definition choosen in common for all the reflections, and the a3 corresponding value in Å.
  • The RP and RWP values as obtained by ARIT from a fit without size/microstrain calculation.
  • The RP and RWP values from a fit with isotropic size/microstrain calculation.
  • The RP and RWP values from a fit with anisotropic size/microstrain calculation.
  • The mean size as obtained from :*= N3a3
  • The K parameter defining the variation law of the distortion versus the distance.
  • The strain parameter defined as <Z21> a32 2) from which one can easily estimate values like <e2L>, by using the K parameter and L as defined above.
  • In the anisotropic calculation, the size and microstrain values will be given for directions orthogonal to 3 planes (111), (200) and (220).
    Standard deviations will be given under parenthesis.
    This value is related to the “surface” size distribution, while another definition exists for a “volume” distribution. According to the model of size distribution retained in these calculation, we have approximately ~ 2.
     
    Be careful, when comparing with results from other methods, that you are effectively comparing the same parameters. Note that there is not only one <e2> value coming with ARIT, but a series of values depending on the distance. More details on how to compare strain from ARIT and from other methods were given page 149 in [4]. Other review papers on microstructure parameters extractions by the Rietveld method are available [5, 6], as well as an online conference [7].

    1- Laboratory x-ray sources : "Common" instrumental setup: University of Le Mans (Armel Le Bail)

      l (CuKa1) = 1.54056 Å, l (CuKa2) = 1.54439 Å, I (CuKa2)/ I ((CuKa1) = 0.48, P = 0.8 

    Comments : for all patterns, definition range for every reflection : 2000 points by steps of 0.01°(2q), corresponding to a3 = 4.419 Å.

    I - "Instrumental standard" 

    - Filenames for running ARIT : lebailsh.dat; lebailsh.str; lebailsh.pr1
    - Results in lebailsh.typ.
     
    RP = 9.3% ; RWP = 11.9% ;       See the fit.

    II - "Broadened sample" 

    A - Fit with ARIT without size/strain effect :
     
    - Filenames for running ARIT : lebailbr.dat; lebailbr.str; lebailbr.pr1
    - Results in lebailbr.typ
     
    RP = 5.2% ; RWP = 7.7% ;         See the fit.
     
    B - Fit with isotropic size/strain effect :
    - Filenames for running ARIT : lebailbrss.dat (rename lebailbr.dat); lebailbrss.str; lebailbrss.pr1
    - Results in lebailbrss.typ
     
    RP = 8.9% ; RWP = 11.7%        See the fit
    Mean size : *= 157(1) Å
    Distortion law : K = 2.96(3)
    Strain parameter for n = 1, <Z21> a32 = 9(2)x10-72)
     
    C - Fit with anisotropic size/strain effect : 
    - Filenames for running ARIT : lebailbrssa.dat (rename lebailbr.dat); lebailbrssa.str; lebailbrssa.pr1
    - Results in lebailbrssa.typ
     
    RP = 8.5% ; RWP = 11.1%        See the fit.
    Mean sizes : *(111) = 170(2) Å;*(200) = 150(2) Å; *(220) = 159(2) Å
    Distortion law : K = 2.57(7)
    Strain parameter for n = 1,<Z21> a32(111) = 1.4(7)x10-52);
    <Z21> a32 (200) = 1.4(7)x10-62); <Z21> a32 (220) = 2(1)x10-62)



    2 - Laboratory x-ray sources : Incident-beam monochromator:
    University of Birmingham (J. Ian Langford)

    l (CuKa1) = 1.54056 Å, P = 0.8 

    Comments : for all patterns, definition range for every reflection : 1000 points by steps of 0.02°(2q), corresponding to a3 = 4.419 Å. The Kalpha-2 contribution was neglected. The 2x3 original powder patterns were gathered in 2x1 by changing the scales and step.

    I - "Instrumental standard" 

    - Filenames for running ARIT : langfsh.dat; langfsh.str; langfsh.pr1
    - Results in langfsh.typ
     
    RP = 22.9% ; RWP = 14.8%  - It may be seen from the drawing .gif file that these high R values are mainly due to bad statistics (all points of the pattern are taken ito account).
     

    II - "Broadened sample" 

    A - Fit with ARIT without size/strain effect :
     
    - Filenames for running ARIT : langfbr.dat; langfbr.str; langfbr.pr1
    - Results in langfbr.typ
     
    RP = 13.2% ; RWP = 9.4% (same comment as above).   See the fit.
     
    B - Fit with isotropic size/strain effect :
    - Filenames for running ARIT : langfbrss.dat (rename langfbr.dat); langfbrss.str; langfbrss.pr1
    - Results in langfbrss.typ
     
    RP = 14.9% ; RWP = 11.1%       See the fit.
     
    Mean size : *= 134(1) Å
    Distortion law : K = 3.06(3)
    Strain parameter for n = 1, <Z21> a32 = 2(1)x10-72)
     
    C - Fit with anisotropic size/strain effect :
    - Filenames for running ARIT : langfbrssa.dat (rename langfbr.dat); langfbrssa.str; langfbrssa.pr1
    - Results in langfbrssa.typ
     
    RP = 14.5% ; RWP = 10.6%       See the fit.
     
    Mean sizes : *(111) = 144(2) Å;*(200) = 123(2) Å; *(220) = 132(2) Å
    Distortion law : K = 2.99(5)
    Strain parameter for n = 1,<Z21> a32(111) = 1.4(7)x10-62);
    <Z21> a32 (200) = 1.5(7)x10-72); <Z21> a32 (220) = 2(1)x10-72)

    3 - Synchrotron x-ray sources : 2nd-generation synchrotron, flat-plate geometry: NSLS X3B1 beamline, Brookhaven National Laboratory
    (Peter W. Stephens)

     l = 0.6998 Å, P = 0 

    Comments : for all patterns, definition range : 3000 points by steps of 0.003°(2q), corresponding to a3 = 4.456 Å. The original powder patterns were rebuilt for having both the same 0.003°(2q) step…
     

    I - "Instrumental standard" 

    - Filenames for running ARIT : stephsh3.dat; stephsh3.str; stephsh3.pr1
    - Results in stephsh3.typ
     
    RP = 28.9% ; RWP = 34.6% (the .gif file shows an horrible fit due to high asymmetry and bad peak positions, sometimes displaced toward large or small angles !).
     

    II - "Broadened sample" 

     
    A - Fit with ARIT without size/strain effect :
     
    - Filenames for running ARIT : stephbr3.dat; stephbr3.str; stephbr3.pr1
    - Results in stephbr3.typ
     
    RP = 4.5% ; RWP = 6.9% (the contrast with the R values of the “sharp” sample is astonishing).         See the fit.
     
    B - Fit with isotropic size/strain effect :
    - Filenames for running ARIT : stephbr3ss.dat (rename stephbr3.dat); stephbr3ss.str; stephbr3ss.pr1
    - Results in stephbr3ss.typ
     
    RP = 8.2% ; RWP = 10.3%      See the fit.
     
    Mean size : *= 141(1) Å
    Distortion law : K = 2.65(2)
    Strain parameter for n = 1, <Z21> a32 = 1.3(2)x10-62)
     
    C - Fit with anisotropic size/strain effect :
    - Filenames for running ARIT : stephbr3ssa.dat (rename stephbr3.dat); stephbr3ssa.str; stephbr3ssa.pr1
    - Results in stephbr3ssa.typ
     
    RP = 8.2% ; RWP = 10.3%       See the fit.
     
    Mean sizes : *(111) = 141(1) Å;*(200) = 141(1) Å; *(220) = 141(1) Å
    Distortion law : K = 2.65(2)
    Strain parameter for n = 1,<Z21> a32(111) = 1.2(2)x10-62);
    <Z21> a32 (200) = 8(1)x10-72); <Z21> a32 (220) = 1.0(2)x10-62)



    4 - Synchrotron x-ray sources : 3rd-generation synchrotron, capillary geometry: ESRF BM16 beamline, Grenoble
    (Olivier Masson and Andy Fitch)

      l = 0.39982 Å, P = 0 

    Comments : for all patterns, definition range : 2600 points by steps of 0.002°(2q), corresponding to a3 = 4.406 Å. The original powder patterns were rebuilt for having both the same 0.002°(2q) step…
     

    I - "Instrumental standard" 

    - Filenames for running ARIT : mash2.dat; mash2.str; mash2.pr1
    - Results in mash2.typ
     
    RP = 14.4% ; RWP = 18.0%      See the fit.
     

    II - "Broadened sample" 

     
    A - Fit with ARIT without size/strain effect :
    - Filenames for running ARIT : mabr2.dat; mabr2.str; mabr2.pr1
    - Results in mabr2.typ
     
    RP = 3.6% ; RWP = 4.9%       See the fit.
     
    B - Fit with isotropic size/strain effect :
    - Filenames for running ARIT : mabr2ss.dat (rename mabr2.dat); mabr2ss.str; mabr2ss.pr1
    - Results in mabr2ss.typ
     
    RP = 7.6% ; RWP = 8.8%       See the fit.
    Mean size : *= 136(1) Å
    Distortion law : K = 2.89(2)
    Strain parameter for n = 1, <Z21> a32 = 6.2(8)x10-72)
     
    C - Fit with anisotropic size/strain effect :
    - Filenames for running ARIT : mabr2ssa.dat (rename mabr2.dat); mabr2ssa.str; mabr2ssa.pr1
    - Results in mabr2ssa.typ
     
    RP = 7.5% ; RWP = 8.7%       See the fit.
     
    Mean sizes : *(111) = 134(1) Å;*(200) = 141(1) Å; *(220) = 138(1) Å
    Distortion law : K = 2.86(2)
    Strain parameter for n = 1,<Z21> a32(111) =6.7(8)x10-72);
    <Z21> a32(200) =6.6(8)x10-72); <Z21> a32 (220) =6.7(8)x10-72)
     



    5 - Neutron sources : ILL D1A diffractometer,
    Grenoble (Alan Hewat)

     l = 1.91 Å 

    Comments : for all patterns, definition range : 500 points by steps of 0.05°(2q), corresponding to a3 = 4.386 Å. 
     

    I - "Instrumental standard" 

    - Filenames for running ARIT : hewsh.dat; hewsh.str; hewsh.pr1
    - Results in hewsh.typ
     
    RP = 9.2% ; RWP = 10.5%       See the fit.
     

    II - "Broadened sample" 

     
    A - Fit with ARIT without size/strain effect :
    - Filenames for running ARIT : hewbr.dat; hewbr.str; hewbr.pr1
    - Results in hewbr.typ
     
    RP = 7.4% ; RWP = 6.7%       See the fit.
     
    B - Fit with isotropic size/strain effect :
    - Filenames for running ARIT : hewbrss.dat (rename hewbr.dat); hewbrss.str; hewbrss.pr1
    - Results in hewbrss.typ
     
    RP = 7.0% ; RWP = 6.3%       See the fit.
    Mean size : *= 143(1) Å
    Distortion law : K = 2.64(2)
    Strain parameter for n = 1, <Z21> a32 = 1.4(5)x10-62)
     
    C - Fit with anisotropic size/strain effect :
    - Filenames for running ARIT : hewbrssa.dat (rename hewbr.dat); hewbrssa.str; hewbrssa.pr1
    - Results in hewbrssa.typ
     
    RP = 6.8% ; RWP = 6.2%       See the fit.
    Mean sizes : *(111) = 147(1) Å;*(200) = 141(1) Å; *(220) = 144(1) Å
    Distortion law : K = 2.86(2)
    Strain parameter for n = 1,<Z21> a32(111) = 1.7(7)x10-62);
    <Z21> a32 (200) =2.5(9)x10-62); <Z21> a32 (220) =2.0(8)x10-62)
     

    6 - Neutron sources : NCNR BT1 diffractometer, 
    NIST-Gaithersburg (Brian Toby)

     l = 1.5905 Å
     

    Comments : for all patterns, definition range : 400 points by steps of 0.05°(2q) for every reflection, corresponding to a3 = 4.876 Å. 
     

    I - "Instrumental standard" 

    - Filenames for running ARIT : tobysh.dat; tobysh.str; tobysh.pr1
    - Results in tobysh.typ
     
    RP = 12.0% ; RWP = 16.1%       See the fit.
     

    II - "Broadened sample" 

     
    A - Fit with ARIT without size/strain effect :
    - Filenames for running ARIT : tobybr.dat; tobybr.str; tobybr.pr1
    - Results in tobybr.typ
     
    RP = 10.8% ; RWP = 9.7%       See the fit.
     
    B - Fit with isotropic size/strain effect :
    - Filenames for running ARIT : tobybrss.dat (rename tobybr.dat); tobybrss.str; tobybrss.pr1
    - Results in tobybrss.typ
     
    RP = 12.5% ; RWP = 10.8%       See the fit.
    Mean size : *= 161(1) Å
    Distortion law : K = 2.45(2)
    Strain parameter for n = 1, <Z21> a32 =2.6(8)x10-62)
     
    C - Fit with anisotropic size/strain effect :
    - Filenames for running ARIT : tobybrssa.dat (rename tobybr.dat); tobybrssa.str; tobybrssa.pr1
    - Results in tobybrssa.typ
    RP = 12.4% ; RWP = 10.7%       See the fit.
    Mean sizes : *(111) = 168(2) Å;*(200) = 157(2) Å; *(220) = 162(2) Å
    Distortion law : K = 2.28(2)
    Strain parameter for n = 1,<Z21> a32(111) =6(2)x10-62);
    <Z21> a32(200) =3(1)x10-62); <Z21> a32 (220) =3(1)x10-62)
     



    General comments and conclusion

    The decrease in the RP and RWP reliability factors observed when going from an isotropic size/microstrain refinement to an anisotropic refinement is quite small, showing that if there is really any anisotropy, it is quite negligible.This is reflected by the small differences in mean size along the directions orthogonal to the (111), (200) and (220) planes. The distortion is so small that it appears negligible. This explains the extremely dispersed values when comparing the results from the 6 difftactometers : multiplying by ten an extremely small full width at half maximum (FWHM) gives still a very small FWHM…). The distortion presents thus very large estimated standard deviations. The CeO2 “broad” case is close to a “size-only effect” situation. The generally much better fit without size/microstrain effects on the “broadened” powder pattern shows that the size effect model in ARIT is certainly not really adapted : the experimental size distribution function is likely to be different from a simply exponentially decreasing function. A more flexible model would have to be introduced and tested. The mean size proposed may well be 50% in error. But ARIT does not pretend more than to give an idea of the size/microstrain magnitudes. Discrepancies between the results from the various data set may essentially come from the problem of finding the background position, and also from the quite different resolution (for instance, neglecting the instrumental g contribution would be almost possible for the synchrotron Masson data, but certainly not for the neutron Hewat data).

    For this high-resolution reason, the “best” result in this series treated by ARIT is very probably that from the 3rd generation synchrotron data (Masson), for which the size/microstrain parameters obtained by using the anisotropic option gives almost the same result as those obtained by applying the isotropic option.From ARIT, the microstructure characteristics of CeO2 “broad” are finally :

    Mean size : *= 136(1) Å

    Distortion law : K = 2.89(2)

    Strain parameter for n = 1, <Z21> a32 =6.2(8)x10-72)

    The corresponding fit :

    These values have now to be compared with the results from other methods. The Warren Fourier analysis method will undoubtedly give THE solution, since there is no serious overlapping here...

    And, well, if there is too much discrepancies, the ARIT program will possibly vanish completely ;-). - but this would be a pity, since the concept behind ARIT can be improved by adding a series of different size/strain models. And it will continue to work when the Warren Fourier analysis method will be impossible to apply : in presence of strong overlapping.


    References

    [1] A new study of the structure of LaNi5D6.7 using a modified Rietveld method for the refinement of neutron powder diffraction data. C. Lartigue, A. Le Bail and A. Percheron-Guegan, Journal of the Less-Common Metals129, 65-76 (1987).
    [2] A. Le Bail, Acta Crystallogr. A40, suppl. c369 (1984).
    [3] B. E. Warren, “X-Ray Diffraction”, Addison-Wesley, 1969, chapter 13.

    [4] Modelling Anisotropic Crystallite Size/Microstrain in Rietveld Analysis”, A. Le Bail, Proceedings of “Accuracy in Powder Diffraction 2”, National Institute of Standard and Technology Special Publication 846 (1992) 142-153.
    [5] New developments in microstructure analysis via Rietveld refinements. A. Le Bail, Advances in X-ray Analysis, Vol 42 (1998) 191-203. Available online.
    [6] Accounting for Size and Microstrain in Whole Powder Pattern Fitting. A. Le Bail, in "Defect and Microstructure Analysis by Diffraction", edited by R.L. Snyder, J. Fiala and H.J. Bunge. Proceedings of SIZE-STRAIN'95, Slovakia, IUCr Monographs on Crystallography 10, Oxford Science Publications, 1999, Chapter 22, 535-555. 
    [7] Advances in Microstructure Analysis by the Rietveld Method, A. Le Bail, Conference given at ESCA 2000, The Sixth International School and Workshop on Crystallography, Structural-Characterization: Amorphous and Nano-Crystalline Materials, 22-27 January 2000, Olympic Village, Ismailia, Egypt.