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Re: feff polarization
This is the continuation of discussion with Matt on
the EXAFS analysis of polarization dependent data.
With EXAFS one hopes to extract knowledge of pair distribution
fuction P(\vec r) = P(r)* P(\hat r), where I assumed that
angular distribution is independent of radial one.
I will not discuss the analysis of polarization average
data that gives approximate P(r) after Fourier transform.
With polarization dependent data one hopes to
extract the data about Bond Orientation Distribution: P(\hat r)
i.e. probability that bond is oriented exactly along \vec r.
As follows from Eq.4 of Heald/Stern from polarization dependent
data one can extract from EXAFS only convolution of this function
f(\vec \epsilon) = \int (d \hat r) P(\hat r) (\hat r \cdot \vec \epsilon)^2,
where \epsilon is polarization vector.
This function can be called Bond Orientation Probability
(or Bond Orientation Anisotropy, let me know if you have a better
name for that),
since (\hat r \cdot \vec \epsilon)^2 = can be interpreted
for a bond along (\vec r) as a probability to be oriented
along \epsilon. (f(x) + f(y) + f(z)=1 for arbitrary \vec r).
Anyway polarized exafs measures f(\vec \epsilon), but one wants to
say something about P(\hat r): e.g. how probabilty for
a bond to be in some plane relates to the probabilty for a bond
to be perpendicular to that plane. One of the ways to get P would
be to use deconvolution, but it is not always possible:
for cubic systems one cannot get information about P since f=1/3
for all directions of \epsilon. From the other hand for a single
bond on a surface the deconvolution is straighforward if one
assumes that P(\vec \alpha) = \delta(\vec \alpha - \vec \alpha_0).
I.e. bond is directed along the direction where f(\epsilon) is
the largest.
Thus I would say that the main goal of polarized EXAFS should be
getting f(\epsilon) and it's interpretation in terms of P(\hat r).
The interpretation may depend on what else you know about system.
What is the role of FEFF calculations in this process?
For K-edge the standard output seems to be enough, except
one probably wants to exclude degeneracy checker. For L3 edge
the cross term obviously complicates analysis but FEFF
can report c and \delta_0(k). This information should be enough
to perform analysis using Eq.4 of Heald/Stern.
The constant c probably can be used as a fitting parameter also,
but for \delta_0(k) I would take FEFF calculations. Anyways,
\delta_2(k) is already used.
Best wishes
Alex Ankudinov
On Fri, 3 May 2002, Matt Newville wrote:
> Hi Alex,
>
> Thanks for getting back on this. For now I'll just respond to
> the 'regular/generic' polarization (ie, ignoring the exact
> details of the angular dependence of L-edge polarization).
>
> The confusion about polarization dependence is due to the
> nature of EXAFS -- not FEFF. But maybe FEFF's implementation
> could make it easier to work with polarized data.
For K-edge one probably really wants to turn off degeneracy checker.
For L-edge additional phase shift \delta_0(k) is needed.
Ratio c can be estimated by FEFF and from the fit.
>
> I think the issue is whether polarization should be included in
> the path-finder. That is, the path-finder _could_ be (nearly)
> purely geometrical. This has some appeal for a few reasons,
> such as that the paths would be the same for all polarizations.
>
> I think historically that the path-finder included scattering
> for it's filtering criteria. If we stick to EXAFS only (or
> expect that XANES calcs won't rely on very high numbers of
> paths), then the problem of path explosion may be less
> important, and we may be able to get by with a nearly-pure
> geometrical path-finder based only on the cluster geometry and
> tabularized plane-wave criteria.
I agree that if one restricts the number of legs to 4,
the number of paths will be small. I would even start
with single scattering analysis (NLEG 2), since I don't
think the angular bond distribution for distant neighbors
beyond first shell can be reliably determined.
>
> Then there would be no polarization dependence to 'degeneracy'.
> Though more paths would end up with very little amplitude due
> to polarization, it would be more obvious what was happening --
> and have a more highly predictable set of paths.
>
> I don't have a strong opinion one way or the other on this, but
> I do think this is where part of the confusion comes from.
>
> But analyzing polarization average data is not the issue!
> Sometimes you _want_ to use the polarization to better
> illuminate certain directions (say, normal to a surface). If
> that 1 sulfur was in-plane or out-of-plane, you would get very
> different EXAFS!
>
> For L-edges, I'm still confused what Feff is doing. I'll look
> into this a little more, and write a separate message on that,
> but maybe not for a week or so. But for now:
>
> - Which version(s) of Feff have a working MULTIPOLE card?
To use both MULTIPOLE and PRINT 0 2 cards you will need
the latest version of feff8.2. I'll send you one.
>
> - You said you thought that LeFevre's results were more
> reliable than Heald/Stern, but your value for c=|M21|/|M01|
> (= 0.24) is much closer to Heald/Stern's (0.2) than
> LeFevre's (0.4). Any thoughts on this?
I did FEFF8.2 esimate of c for W and got c=0.28.
Heald/Stern neglect difference in phase shift \delta, and their
c'=c*cos(\delta) should be smaller than calculated one.
Also \delta is energy dependent and typically changes by \pi
over EXAFS region.
The LeFevre et all analysis is based on their Eq.3 and known Cd
structure. Thus you can expect tehm to extract \chi_2
and various phase shifts and amplitudes in that formula.
First they had experimental data only to k=7. In this region
multiple scattering may contribute. Also curved wave corrections
are improtant in this region which lead to sin^2 \theta contribution
even for K-edge (Benfatto et al., PRB 39, 1936 (1989).
Also the question is whether the incident radiation was 100% polarized?
I thought that usually it is about 96% from synchrotron.
So really it would be better to do LeFevre et al. analysis for
k=7-20 range where multiple scattering and curved wave corrections
play smaller role.This will also improve Fourier filtering and
will probably settle the question about value of c for Cd.
>
> - You said that this ratio should have very little
> k-dependence. But the plots of scattering amplitudes
> from the feff.dat files shows clear k-dependence. Can
> you explain this? My guess is that the strong k-dependence
> could explain the high value of c that LeFevre got: I see
> much less angular dependence at low k (which would give a
> higher value of c), and their data stops at k=5 or so.
One source of energy dependence is backscattering amplitude
f(\pi), the other is energy dependence in \delta.