[Ifeffit] my problem
Anatoly Frenkel
frenkel at bnl.gov
Mon Dec 19 17:25:13 CST 2005
Hi Matt,
Rb-Br/Rb-Cl is a different story. There, the neighbors are different (Br,Cl)
and the central atom is the same (Rb). It is a piece of cake. Here (in
Nb2O5) the neighbors are the same (O), and the central atoms are at multiple
sites. Of course, it would be very cute if they very neatly arranged around
each individual Nb out of 7 (seven) inequivalent Nb atoms so that each
Nb(i)-O shell had a distinct (and degenerate) Nb(i)-O distance, and so that
all 7 Nb(i)-O shells were separated by the distance larger than the
resolution in EXAFS experiment to detect such split.
It may be possible for ABO3 perovskites with a single B site to detect the
split in B-O shell (Nb in KNbO3 displaces toward 111 direction and there is
a distinct group of three oxygens closer to it and another group of 3
oxygens, farther away from it, and that spread can be as large as 0.3 A or
so). But: during the analysis you constrain these shells to have 3 oxygens
in each group, and it is not the case here because the site occupancy is not
known.
Your other comments are too open ended and not exactly interpreting what I
wrote in my original message. Forgive me for not replying.
Anatoly
-----Original Message-----
From: matt.newville at gmail.com [mailto:matt.newville at gmail.com] On Behalf Of
Matt Newville
Sent: Monday, December 19, 2005 2:48 PM
To: anatoly.frenkel at yu.edu; XAFS Analysis using Ifeffit
Subject: Re: [Ifeffit] my problem
Anatoly --
I think I'm siding with Scott on this, but maybe I'm no understanding
all your points. As far as I can tell, Tadij did not say that only N
would be varied in an anlysis -- The question was how to constrain the
total coordination number from a set of scattering paths: Shelly and
Scott answered that, pretty well I think.
Now, I agree that if 7 oxygen paths make up a "first shell" with a
narrow range of distances (though this was not actually clear from the
original question, it does seem likely), there's no way you'll be able
to measure 6 independent coordination numbers. But if there were 2
paths / atom types that made up a first shell, than relative occupancy
can certainly be determined. As a concrete example, determining the
number of Rb-Br and Rb-Cl neighbors in a salt solution is almost
certainly possible. ;). If the two atoms are the same species, they
just need to be sufficiently separated in distance.
On 12/17/05, Anatoly Frenkel <frenkel at bnl.gov> wrote:
> Hi Scott,
>
> Thank you for pointing out at Vince's work on ferrites. It is a nice
> demonstration of the effect of site occupancy by dopants on EXAFS. In
> Vince's APL 68, 2082 (1996), they found the distribution of Zn between A
and
> B sites (octahedral and tetrahedral) by including MS, as you described.
> However, he varied only one parameter in the fit to the MS range - site
> occupancy, keeping all other fixed (p. 2083). As such, it is not a
> non-linear least square fit, but a linear least square fit of a (linear)
> combination of fixed functions, and thus the chi-squared of such fit must
> have a single minimum - either within the range of x, or at the boundary
> (x=0 or 1, as they obtained in most cases).
I don't think it's important whether a linear or non-linear algorithm
is used for the analysis, especially when compared to what was varied
in the analysis (well, for some problems a linear approach is not
possible, but even when a linear approach is possible it's not obvious
that it's better than a non-linear approach). Whether or not there
is a single minimum or not (and, related, what would make different
minima distinguishable) is also more a matter of the problem, and not
the algorithm used to find a solution. For a simple problem with 1
variable for relative intensities of two basis functions, it is very
likely to have one minimum, no matter whether a linear or non-linear
approach is used (as an aside, 1 variable does not always mean one
minimum, but we're usually dealing with well-behaved problems). In
any event, claiming that linear v. non-linear algorithm makes a
difference is a dangerous and/or silly position to take for someone
analyzing XAFS data. We have to use non-linear methods -- do you find
this troubling?
> What Tadej is describing seems more like a general, non-linear least
squares
> fit problem, that has N minima in chi squared of which N-1 are false.
False? What does that mean? Sure, there may be multipe minima in
chi-square. If so, either these can be distinguished because some
have lower chi-squares than others, or that cannot be distinguished
because the chi-squares are close. I assume that by using the word
False that you somehow don't believe statistics or the model used.
In my view, since one is making a model to describe a distribution of
atoms, "False" is a useles concept -- all models are false in the
sense that they are simplifications of reality, but some models are
better than others.
In this example, if 6 relative coordination numbers were used, I would
expect the error bars to blow up, which means local minima in
chi-square would be within the estimated errors, and there would not
be N-1 "False minima", but 1 very broad minima. Perhaps you have a
counter-example?
> It is counter-intuitive to assume that the contributions of EXAFS from
each
> inequivalent Nb can be fixed in the fit except for its fractional
occupancy.
> What about sigma2 of different MS paths?
I don't think Tadij said this. But it does sound like what Harris et
al may have done (ie, if they only had one unknown in their model,
they must have made some assertions abou the other parameters. Does
that not bother you as much??
> ... Thus, you obviously reduce the number of variables but you may exclude
the
> true, physical, minimum of chi squared from your parameter space.
Again, I think I do not understand. What does "True, physical,
minimum of chi squared" mean? You make a model, you do a fit, you
understand the statistical parameters and consider making other
models. In the end you pick the model that fits your data best and
has the most sensible interpretation.
I would consider "True, physical" to mean the complete description of
the distribution of atoms sampled. A typical measurement samples the
local coordination of around 10^9 atoms. If we have good data, we may
be able to fit as many as 10 parameters to describe the first shell of
the partial pair distribution function of these 10^9 atoms. So "True"
and "Physical" are a bit far removed from our ability to see with our
data. We must make model for the distribution and compare these
models to our data. Generally, one picks the model that best matches
the data as the most likely model. It's often called "best fit", but
rarely called "True".
Are you worried about the situation in which there are minima in
chi-square that are clearly distinct (outside the error bars) and for
which the model with significantly lower chi-square is a worse
physical explanation (or disagrees with other measurements)? This
seems to be a common fear, but I find it to be largely unfounded.
Can you (or anyone) give an example of this? There seem to be a lot
of stories about this, but most of the stories I've heard end up being
at least partially due to ignoring (or not even trying to estimate)
the error bars (and often partially due to someone wanting to believe
something that their data
doe not support). Of course, you can certainly pick multiple
models and get different results. Is that a problem??
Since I'm not understanding your concerns with the way analysis is
done with Ifeffit, perhaps you can clarify these concerns.
Thanks,
--Matt
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