Dear Reinhard,
Thank you for your comments and questions.
1) What does "peak shift" really mean in this context? Obviously you
discuss any difference between a "measured" or "refined" angular peak
position and the theoretical Bragg angle calculated from referenced or
refined lattice parameters, plotted in fig. 1. Ok, the reference Bragg
position is clear. It is also known (since Klug & Alexander, Wilson etc.
in the 1950ies) that numerous geometrical effects complicate the peak
shape, apparently "shift" the peak maximum from its theoretical value,
and that these effects are strongly angular dependent (Jenkins,
Schreiner in the 1980ies, textbooks...). Btw this is how the trends in
fig 1 and 3 look like. Of course, if not taken into account correctly,
these effects will bias the lattice parameters systematically.
Basically, the primary question is how accurate we can
model/calculate/correct all these effects in our Rietveld refinement
procedure.
For the "peak-shift":
As described in the Abstract section, the peak-shift is the deviation
in diffraction angle from the theoretical Bragg position.
And we assumed the geomtrical origin with Eq. (1) for the peak-shift.
Then, we found that the "analytical" peak-shift arised in the calculation
when the value of the lattice parameter is not the true one.
As you mentioned, there are the other factors for the peak-shift.
And they must affect to the result.
If you take them into account correctly, the accuracy (and the precision)
of the parameter may get more improved. This is also pointed out by
Norberto in his post on 2017/11/15 7:00.
However, we think that the effect of the peak-shift originating from
the other factors would be much smaller than that from the geometry,
especially the [Ds cos th] term in Eq. (1).
The reason why we think so is that we also conducted the refinements
by changing 2th_min in the 2th-range from 2th_mim to 152 deg.
(The results are not reported in the article.) I recognize this is a rough
approximation to evaluate the effects of the peak shape etc. on the accuracy
of lattice parameters.
Considering the Bragg's equation, the correct evaluation of the peak-shift
in the high 2th region is the key to obtain the lattice parameters
accurately.
The "analytical" peak-shift causes a big deviation with increasing 2th
in the high 2th region as shown in Eq. (2) and Figs. 1c and 1d.
Or, do you rely only on the systematic, geometrically caused
deviations of peak maximum positions when fitted "traditionally" (by
analytical functions) in your Rietveld program? If yes, is your
criterion maybe obsolete in refinements applying good FPA models?
The answer to your question is yes.
If you apply "good" FPA models, the proposed criterion doesn't need.
I suppose that our criterion would be still worth to know the FPA model
is "good" or not, i.e., for cross-checking the goodness of the model.
2) What does "reproducibility of the peak shift" mean? How good a peak
position (or shift from the reference value) can be reproduced by
repetition of the measurement (including preparation), as I would expect
from the usual meaning of the word "reproducibility" in analytical work?
Or do you simply mean "how good your refinement model can fit the peak
positions of the measured pattern"?
As for the "reproducibility":
I mean the latter one, "how good your refinement model can fit
the peak positions of the measured pattern." Do you have a good term for it?
If you mean the latter, we are back
to the discussion above that an incorrect (or oversimplified) peak shape
model in a Rietveld analysis may bias the refined lattice parameters. In
my understanding, such kind of bias is a systematic error, simply caused
by an incorrect model, and the term "reproducibility" sounds a bit
misleading in this context.
You can, of course, go back to the discussion but I think I don't need.
As described the above, I think that the geometry affects the peak-shift
in the high 2th region larger than the peak shape etc.
It must be true that the peak shape etc. affects the peak-shift in
principle,
but the effct seems not to be very big in the high 2th region.
That is because the the peak shape gets asymmetric in the low 2th region
with decreasing 2th.
3) It is clear that systematically wrong (fixed) lattice parameters will
enhance the number of the proposed criterion (fig. 2 e,f). But what will
happen in a practical refinement when the angular range is smaller, Z,
Ts and Ds become more correlated, and maybe additional nonsense profile
parameters are started to refine and compensate each other, e.g. for
more broadened peak profiles?
Yes, the peak profiles get slightly broaden with increasing |Z-Zturue|.
Will the values of this criterion still
give a clear indication of any systematic error, or will they maybe
masked by such correlation effects?
I guess, no. (I have no evidence to have a conclusion.)
To know such information, the other foctors for the peak-shift
should be considered as Norberto and you mentioned. What do you think?
A few cents from a personal view: Of course I agree that the observation
of any systematic misfit in the peak positions in Rietveld refinement is
a valuable information, e.g. for identification of instrumental
misalignment, or pointing to erroneous peak profile description in a FPA
model. However, to get reliable lattice parameters in Rietveld
refinement with a FPA model it is a recommended strategy to admix
silicon SRM640c to a powder sample, fix the lattice parameters of this
internal standard to the theoretical value during the refinement, and
refine/adjust for example sample transparency parameters (a typical
unknown in practice) in the peak profile model, until the profile and
the peak position of the standard peaks are well fitted. So we can
"anchor" the correlated angular effects by the standard peaks. But this
is possible to do without this new criterion, so I'm not convinced of
the necessity of the criterion proposed, at the moment.
You don't need to use the new criterion if you admix SRM.
You don't need to admix SRM if you use the new criterion.
I understand that the "anchor" effect depends on a ratio of amount between
a powder sampe and SRM. So, for me, the new criterion would be better.
Sincerely yours,
Masami
--------
Physonit Inc.
TSUBOTA Masami
6-10 Minami-Horikawa, Kaita, Aki, Hiroshima 736-0044 Japan
Phone/FAX: +81 82 822 0710
E-mail: tsub...@physonit.jp
URL: http://physonit.jp/
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