Method
Microstrain
Size
'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
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.
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.
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 :
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 a’3
Where a’3 depends on the interval of definition of the reflections [q2q1]:
a’3 = l / 2 (sin q2 – sin q1)
Anisotropic size and microstrain effects in ARIT
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.
C – Results on CeO2
- 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.
The range of definition choosen in common for all the reflections, and the a’3 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 :
= N3a’3
The K parameter defining the variation law of the distortion versus the distance.
The strain parameter defined as <Z21> a’32 (Å2) from which one can easily estimate values like <e2L>, by using the K parameter and L as defined above.
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
.
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 a’3 = 4.419 Å.
I - "Instrumental standard"
II - "Broadened sample"
- Results in lebailbrss.typ
=
157(1) Å- Results in lebailbrssa.typ
(111)
= 170(2) Å;
(200)
= 150(2) Å; 
(220)
= 159(2) Å
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 a’3 = 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"
II - "Broadened sample"
- Results in langfbrss.typ
Mean size :
=
134(1) Å- Results in langfbrssa.typ
Mean sizes :
(111)
= 144(2) Å;
(200)
= 123(2) Å; 
(220)
= 132(2) Å3 - 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 a’3 = 4.456 Å. The original powder
patterns were rebuilt for having both the same 0.003°(2q)
step…
I - "Instrumental standard"
II - "Broadened sample"
- Results in stephbr3ss.typ
Mean size :
=
141(1) Å- Results in stephbr3ssa.typ
Mean sizes :
(111)
= 141(1) Å;
(200)
= 141(1) Å; 
(220)
= 141(1) Å
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 a’3 = 4.406 Å. The original powder
patterns were rebuilt for having both the same 0.002°(2q)
step…
I - "Instrumental standard"
II - "Broadened sample"
- Results in mabr2ss.typ
=
136(1) Å- Results in mabr2ssa.typ
Mean sizes :
(111)
= 134(1) Å;
(200)
= 141(1) Å; 
(220)
= 138(1) Å
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 a’3 = 4.386 Å.
I - "Instrumental standard"
II - "Broadened sample"
- Results in hewbrss.typ
=
143(1) Å- Results in hewbrssa.typ
(111)
= 147(1) Å;
(200)
= 141(1) Å; 
(220)
= 144(1) ÅNIST-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 a’3 = 4.876 Å.
I - "Instrumental standard"
II - "Broadened sample"
- Results in tobybrss.typ
=
161(1) Å- Results in tobybrssa.typ
(111)
= 168(2) Å;
(200)
= 157(2) Å; 
(220)
= 162(2) Å
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> a’32 =6.2(8)x10-7 (Å2)
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 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.