[Ifeffit] Cumulant expansion fittings

[John J. Rehr]

Hi Everyone,

  Instead of using fuzzy descriptions like XAFS is a "one-dimensional"  or
"three-dimensional" probe to describe XAFS, I think it's preferable to define
the physical quantity that XAFS measures and then to characterize that
quantity in terms of its distribution. Following Fornasini's analysis in
PRB70,174301(04), for example, the XAFS measures the average

         <chi_k> = <(feff/kR**2)sin(2kR+Phi) exp(-2R/lam_k)>

which therefore depends on an average over the the *path-length*
distribution R.  Obviously the mean path-length <R> is not precisely the
same as the mean bond-length, since the path-length includes a second-order
term due to motion perpendicular to a bond, Thus the first cumulant is

           sigma^1 = <u_par + u_perp^2/2R>

This is mostly one-dimensional but not entirely. Moreover the
multiple-scattering terms are sensitive to the three-dimensional character
of a system.  For a recent discussion see also F. Vila et al. PRB76,014301(07),
which discusses various theoretical methods for calculating XAFS
Debye-Waller factors and the third cumulant, and also includes some
comparisons with experiment.

  J. Rehr