Method Microstrain Size 
'sharp and broad' Contribution to the Size/Strain
Round Robin
Armel Le Bail October 18, 2000

A
 Program used
The software used in this study is ARIT available at : 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 (CeO_{2}
“sharp” in this case). The experimental pattern h for an illcrystallized
material is fitted by reusing the previously determined G part,
multiplied by the F sample contribution, where F_{n}
= A_{n}^{S}.A_{n}^{D},
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 sizebroadening 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].
A
hypothetical Gaussian strain distribution is considered (see Warren, p.
270 [3]) such as :
In that equation, <Z^{2}_{n}> is modelled in ARIT by a flexible variation law of the distortion versus the distance according to the following equation: <Z^{2}_{n}> = n^{K}<Z^{2}_{1}> The ARIT program refines two microstrain parameters : K and <Z^{2}_{1}> (i.e. <Z^{2}_{n}> 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. The size Fourier coefficient A_{n}^{S} is given in terms of p(i), the fraction of the columns of length i cells by the expression [3]: where N_{3} is the mean column length number. Modelling p(i) allows to define A_{n}^{S}. 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) = g^{2}exp(gn) The size Fourier coefficients corresponding to this arbitrary size distribution function is : And the average number of unit cells is N_{3} = 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 a’_{3} Where a’_{3} depends on the interval of definition of the reflections [q_{2}q_{1}]: a’_{3} = l / 2 (sin q_{2} – sin q_{1}) Anisotropic size and microstrain effects in ARITDetails
about how this is undertaken in ARIT (ellipsoids describing the <Z^{2}_{1}>(hkl)
and N_{3}(hkl) values) may be found in [1].
The Size/Strain Round Robin indicates that effects in CeO_{2} 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 CeO_{2}, 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 CeO_{2}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 SizeStrain 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 :
Most
files have been renamed with .txt extension for being well downloaded through
the Web.
For
the sharp CeO_{2} will be given :
The
R_{P} and R_{WP} 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 CeO_{2} will be given :
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 <e^{2}>
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 xray 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 a’_{3} = 4.419 Å. I  "Instrumental standard"R_{P}
= 9.3% ; R_{WP} = 11.9% ;
See
the fit.
II  "Broadened sample"A
 Fit with ARIT without size/strain effect :
R_{P}
= 5.2% ; R_{WP} = 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 R_{P}
= 8.9% ; R_{WP} = 11.7%
See the fit.
Mean
size : =
157(1) Å
Distortion
law : K = 2.96(3)
Strain
parameter for n = 1, <Z^{2}_{1}> a’_{3}^{2
}=
9(2)x10^{7} (Å^{2})
C
 Fit with anisotropic size/strain effect :

Filenames for running ARIT : lebailbrssa.dat (rename lebailbr.dat);
lebailbrssa.str;
lebailbrssa.pr1
 Results in lebailbrssa.typ R_{P}
= 8.5% ; R_{WP} = 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,<Z^{2}_{1}> a’_{3}^{2}(111)
= 1.4(7)x10^{5} (Å^{2});
<Z^{2}_{1}>
a’_{3}^{2 }(200) = 1.4(7)x10^{6} (Å^{2});
<Z^{2}_{1}> a’_{3}^{2 }(220)
= 2(1)x10^{6} (Å^{2})
2  Laboratory xray
sources : Incidentbeam monochromator:
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 a’_{3} = 4.419 Å. The Kalpha2 contribution was neglected. The 2x3 original powder patterns were gathered in 2x1 by changing the scales and step. I  "Instrumental standard"R_{P}
= 22.9% ; R_{WP} = 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 :
R_{P}
= 13.2% ; R_{WP} = 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 R_{P}
= 14.9% ; R_{WP} = 11.1%
See
the fit.
Mean size : = 134(1) Å Distortion
law : K = 3.06(3)
Strain
parameter for n = 1, <Z^{2}_{1}> a’_{3}^{2
}=
2(1)x10^{7} (Å^{2})
C
 Fit with anisotropic size/strain effect :

Filenames for running ARIT : langfbrssa.dat (rename langfbr.dat);
langfbrssa.str;
langfbrssa.pr1
 Results in langfbrssa.typ R_{P}
= 14.5% ; R_{WP} = 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,<Z^{2}_{1}> a’_{3}^{2}(111)
= 1.4(7)x10^{6} (Å^{2});
<Z^{2}_{1}>
a’_{3}^{2 }(200) = 1.5(7)x10^{7} (Å^{2});
<Z^{2}_{1}> a’_{3}^{2 }(220)
= 2(1)x10^{7} (Å^{2})
3  Synchrotron
xray sources : 2ndgeneration synchrotron, flatplate geometry: NSLS X3B1
beamline, Brookhaven National Laboratory
l = 0.6998 Å, P = 0 Comments
: for all patterns, definition range : 3000 points by steps of 0.003°(2q),
corresponding to a’_{3} = 4.456 Å. The original powder
patterns were rebuilt for having both the same 0.003°(2q)
step…
I  "Instrumental standard"R_{P}
= 28.9% ; R_{WP} = 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 :
R_{P}
= 4.5% ; R_{WP} = 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 R_{P}
= 8.2% ; R_{WP} = 10.3% See
the fit.
Mean size : = 141(1) Å Distortion
law : K = 2.65(2)
Strain
parameter for n = 1, <Z^{2}_{1}> a’_{3}^{2
}=
1.3(2)x10^{6} (Å^{2})
C
 Fit with anisotropic size/strain effect :

Filenames for running ARIT : stephbr3ssa.dat (rename stephbr3.dat);
stephbr3ssa.str;
stephbr3ssa.pr1
 Results in stephbr3ssa.typ R_{P}
= 8.2% ; R_{WP} = 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,<Z^{2}_{1}> a’_{3}^{2}(111)
= 1.2(2)x10^{6} (Å^{2});
<Z^{2}_{1}>
a’_{3}^{2 }(200) = 8(1)x10^{7} (Å^{2});
<Z^{2}_{1}> a’_{3}^{2 }(220)
= 1.0(2)x10^{6} (Å^{2})
4  Synchrotron
xray sources : 3rdgeneration synchrotron, capillary geometry: ESRF BM16
beamline, Grenoble
l = 0.39982 Å, P = 0 Comments
: for all patterns, definition range : 2600 points by steps of 0.002°(2q),
corresponding to a’_{3} = 4.406 Å. The original powder
patterns were rebuilt for having both the same 0.002°(2q)
step…
I  "Instrumental standard"R_{P}
= 14.4% ; R_{WP} = 18.0% See
the fit.
II  "Broadened sample"A
 Fit with ARIT without size/strain effect :
R_{P}
= 3.6% ; R_{WP} = 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 R_{P}
= 7.6% ; R_{WP} = 8.8% See
the fit.
Mean
size : =
136(1) Å
Distortion
law : K = 2.89(2)
Strain
parameter for n = 1, <Z^{2}_{1}> a’_{3}^{2
}=
6.2(8)x10^{7} (Å^{2})
C
 Fit with anisotropic size/strain effect :

Filenames for running ARIT : mabr2ssa.dat (rename mabr2.dat);
mabr2ssa.str;
mabr2ssa.pr1
 Results in mabr2ssa.typ R_{P}
= 7.5% ; R_{WP} = 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,<Z^{2}_{1}> a’_{3}^{2}(111)
=6.7(8)x10^{7} (Å^{2});
<Z^{2}_{1}>
a’_{3}^{2}(200) =6.6(8)x10^{7} (Å^{2});
<Z^{2}_{1}> a’_{3}^{2 }(220)
=6.7(8)x10^{7} (Å^{2})
5  Neutron sources
: ILL D1A diffractometer,
l = 1.91 Å Comments
: for all patterns, definition range : 500 points by steps of 0.05°(2q),
corresponding to a’_{3} = 4.386 Å.
I  "Instrumental standard"R_{P}
= 9.2% ; R_{WP} = 10.5% See
the fit.
II  "Broadened sample"A
 Fit with ARIT without size/strain effect :
R_{P}
= 7.4% ; R_{WP} = 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 R_{P}
= 7.0% ; R_{WP} = 6.3% See
the fit.
Mean
size : =
143(1) Å
Distortion
law : K = 2.64(2)
Strain
parameter for n = 1, <Z^{2}_{1}> a’_{3}^{2
}=
1.4(5)x10^{6} (Å^{2})
C
 Fit with anisotropic size/strain effect :

Filenames for running ARIT : hewbrssa.dat (rename hewbr.dat);
hewbrssa.str;
hewbrssa.pr1
 Results in hewbrssa.typ R_{P}
= 6.8% ; R_{WP} = 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,<Z^{2}_{1}> a’_{3}^{2}(111)
= 1.7(7)x10^{6} (Å^{2});
<Z^{2}_{1}>
a’_{3}^{2 }(200) =2.5(9)x10^{6} (Å^{2});
<Z^{2}_{1}> a’_{3}^{2 }(220)
=2.0(8)x10^{6} (Å^{2})
NISTGaithersburg (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 a’_{3} = 4.876 Å.
I  "Instrumental standard"R_{P}
= 12.0% ; R_{WP} = 16.1% See
the fit.
II  "Broadened sample"A
 Fit with ARIT without size/strain effect :
R_{P}
= 10.8% ; R_{WP} = 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 R_{P}
= 12.5% ; R_{WP} = 10.8% See
the fit.
Mean
size : =
161(1) Å
Distortion
law : K = 2.45(2)
Strain
parameter for n = 1, <Z^{2}_{1}> a’_{3}^{2
}=2.6(8)x10^{6}
(Å^{2})
C
 Fit with anisotropic size/strain effect :

Filenames for running ARIT : tobybrssa.dat (rename tobybr.dat);
tobybrssa.str;
tobybrssa.pr1
 Results in tobybrssa.typ R_{P}
= 12.4% ; R_{WP} = 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,<Z^{2}_{1}> a’_{3}^{2}(111)
=6(2)x10^{6} (Å^{2});
<Z^{2}_{1}>
a’_{3}^{2}(200) =3(1)x10^{6} (Å^{2});
<Z^{2}_{1}> a’_{3}^{2 }(220)
=3(1)x10^{6} (Å^{2})
The decrease in the R_{P} and R_{WP} 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 CeO_{2} “broad” case is close to a “sizeonly 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 highresolution reason, the “best” result in this series treated by ARIT is very probably that from the 3^{rd} 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 CeO_{2} “broad” are finally : Mean size : = 136(1) Å Distortion law : K = 2.89(2) Strain parameter for n = 1, <Z^{2}_{1}> a’_{3}^{2 }=6.2(8)x10^{7} (Å^{2}) 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.
[1] A new study of the
structure of LaNi_{5}D_{6.7} using a modified Rietveld
method for the refinement of neutron powder diffraction data. C. Lartigue,
A. Le Bail and A. PercheronGuegan, Journal of the LessCommon Metals129,
6576 (1987).
[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)
142153.
[5]
New developments in microstructure analysis via Rietveld refinements. A.
Le Bail, Advances in Xray Analysis, Vol 42 (1998) 191203. 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 SIZESTRAIN'95,
Slovakia, IUCr Monographs on Crystallography 10, Oxford Science Publications,
1999, Chapter 22, 535555.
[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,
StructuralCharacterization: Amorphous and NanoCrystalline Materials,
2227 January 2000, Olympic Village, Ismailia, Egypt.

