Re: k

[Matt Newville <newville@cars.uchicago.edu>]

Hi Konstantin,

> The XAFS "photoelectron wave number" k is always determined from the TOTAL
> photoelectron energy: k^2=\hbar\omega-E_c. Whereas the real (deBroglie) wave
> is determined from the KINETIC energy: k^2=T.
> These two k's are not the same since the potential is essentially non-zero
> in the vicinity of atomic cores (otherwise we wouldn't have XAFS phase
> shifts).
>
> Q1: What is the physical meaning of the XAFS k? Is it just a convenient
> something for subsequent Fourier analysis?
> Q2: This artificial "k" (or "p") is included somehow into FEFF calculations.
> Does this  imitation of real k affect the calculations? In which step
> mostly? Extrinsic losses are incorporated into FEFF - as k- or E- dependent?

Well, absorption spectra is measured as a function of x-ray
energy, and there is no obvious way to tell the total energy of
the photoelectron from the experiment.  So one might interpret
*any* definition or assignment of 'k' as "imitation" or
"artificial".  That's probably a little pessimistic for those of
us who use EXAFS on a regular basis :).

Of course, the physical meaning of the EXAFS k is 'the wavenumber
of the scattered photoelectron'.  This is useful for
understanding EXAFS in terms of photoelectron wave interference,
which does lead to the use of the Fourier transform.  This
definition of k is only as good as the single-particle picture of
scattering from which it is derived.  But it seems your questions
are more about how Feff treats the wavenumber in the single
particle picture.

Feff uses its calculation of the total, complex wavenumber of the
photoelectron (p) which is simply related to the total energy
(p*p ~ E), and includes loss terms and self-energies (that is,
including the corrections to the single particle picture).
These values are used in its calculation of the scattering phase
shifts and amplitudes, and ultimately the EXAFS.

When Feff breaks up the calculated EXAFS into a form that
resembles the classic EXAFS equation to report its results, Feff
reports both real and imaginary parts of p, which give the total
wavenumber (momentum, energy) of the photoelectron relative the
top of the muffin tin.  Feff also reports a value for k at each
value of p.  This k is real and relative to the Fermi level
(which it calculates somewhat crudely, and which may not be
completely accurate).  The idea is that the Fermi-level is likely
to be closer to the empirically selected value of 'E_0' for the
experimental data than the top of the muffin-tin would be.  In
that sense, 'k' is close enough to "the right value" for
photoelectron wavenumber to be used, or at least sensibly
corrected in further analysis.  But for Feff, k is really only
used as an index for the calculated EXAFS, and as the nominal
dependent variable for Fourier transforms and so forth.

This does mean that when matching data and calculation, it is
important to keep the different uses of k and p straight:  A
shift of E_0 to have calculation and experiment match should
change Feff's 'k' so that it matches data, but then the
corresponding 'p' value should be used to alter the EXAFS
amplitudes and phases.

I'm not sure I understood all your questions, but hope that
helps,

--Matt Newville