[Ifeffit] Re: EXAFS-Question concerning \delta(k)

[Bruce Ravel]

Frank,

I am taking the liberty of forwarding my answer to the Ifeffit mailing
list.  I am not sure that I can answer your question entirely to your
satisfaction and thought it would be useful to offer it up to the list
for more commentary.

> It is well known that there is a difference in the peak maxima of the
> Fourier transformed EXAFS, G(r), and the "real" pair distribution function,
> say F(r). As I have observed in spectra taken on pure metals, the maxima of
> G(r) and the calculated core-shell-distances (calculated out of the crystal
> structure) coincide pretty good. On oxidic samples, this is not the case,
> G(r) is always shifted to lower r-values, about 0.5 \AA, sometimes even
> almost 1 \AA.

I agree that the shift due to the phase might be smaller for a pure
metal than for an oxide, but it isn't 0.  For copper, for instance,
the highest point of the first peak is about 1/4 angstrom lower than
the distance to the first shell.

> It is always said that this difference is due to the phase shift \delta(k)
> in the EXAFS equation, but I have not found a reasonable explanation, WHY
> \delta(k) is influenced in this way! What are the parameters "behind"
> \delta(k)?
>
> Maybe you have a little time to answer my question, because I am really not
> satisfied with the explanations I found in literature.

If I understand your question, there are two parts.  (1) Why is the
shift due to the phase correction always negative (i.e. always to a
smaller value in FT[chi(R)])?  And (2) Why is the shift always in the
neighborhood of 1/2 angstrom?

This is one of those things that I take for granted and rarely think
too hard about.  Here are my stabs at decent answers to your
questions.  I'll try the second part first.

I wnet all the way back to the Lee and Beni paper from 1977 (Phys Rev
B v.15 #6, p. 2862) which shows some plane wave calculations of the
various central atom phase shifts.  (There are lots of reasons to look
at later papers as we now know that curved wave effects are very
important, but for motivating a quick 'n' dirty understanding, it is
useful to think about plane waves.)  These are shown in Fig. 5 on page
2868.  There is a Z dependence -- Rh is very different from Mg in the
plot -- but all of them have essentially the same k dependence.  That
is, the traces in that plot are roughly parallel.

We can approximate those phase shifts as linear -- 2delta = a*k + b.
(They are obviously not linear but the quadratic term isn't huge, and
it's helpful to keep the argument simple.)  With this assumption, we
are Fourier transforming an exafs function with a sine term like this:

    sin( 2kR + a*k + b)
  = sin( (2R+a)*k + b )

When you Fourier transform this, the peak will be shifted away from
the "correct" R value by a/2.  As you can see in the figure in
Lee&Beni, all these functions are downward-sloping.  Thus a is a
negative number, resulting in an inward shift of the peak.  All the
central atoms have about the same slope, so they are all shifted by
about the same amount.  Any variation would have to do with the phase
shift of the scatterer.

So why is the slope of the central atom phase shift always negative
and never positive?  I think I am going to be less helpful for this
question, but maybe someone else can pick up the slack.

When you solve the problem of scattering of partial waves off a hard
sphere, you find that each partial wave has a phase shift of -pi/2
times the angular momentum of the partial wave.  That is, the effect
of the scattering is to *decrease* the phase of the particle scattered
in a way that is dependent on the angular momentum.  An atom isn't a
hard sphere and the calculation is a lot harder than for a hard
sphere, but the essential physics is the same.



Of course, this discussion has been hand-waving.  For a proper
understanding of all this stuff, I recommend the important Rehr-Albers
review of XAS in Rev. Mod. Phys. v.72 #3 (2000) pp 621-654.  Indeed,
anyone who does XAS regularly should probably keep a copy of that
paper on the desk.

HTH,
B



> Thanks!!
>
> Yours,
>
> Frank
>
> ---
> Dipl.-Ing. Frank Haass
> Darmstadt University of Technology
> Institute for Materials Science / Structure Research
> Petersenstrasse 23
> D-64287 Darmstadt
>
> Tel.: +49 (0) 6151 16-2697
> Fax: +49 (0) 6151 16-6023
>
> eMail: haass at tu-darmstadt.de

-- 
 Bruce Ravel  ----------------------------------- ravel at phys.washington.edu
 Code 6134, Building 3, Room 405
 Naval Research Laboratory                          phone: (1) 202 767 2268
 Washington DC 20375, USA                             fax: (1) 202 767 4642

 NRL Synchrotron Radiation Consortium (NRL-SRC)
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 National Synchrotron Light Source
 Brookhaven National Laboratory, Upton, NY 11973

 My homepage:    http://feff.phys.washington.edu/~ravel 
 EXAFS software: http://feff.phys.washington.edu/~ravel/software/exafs/