Re: Debye-Waller factors in FEFF

["John J. Rehr" <jjr@leonardo.phys.washington.edu>]

Dear Feffusers,

  Scott Calvin has raised the important issue of how to treat large
structural disorder in FEFF or other path-by-path XAFS calculations.

  In particular he suggests that:

> The difficulty is that it seems to me that under these 
> circumstances FEFF dramatically overestimates the relative amplitude 
> of focused paths; focusing should be much more disrupted by disorder 
> than either direct scattering paths or multiple scattering paths with 
> other geometries, such as triangles.

   Interestingly, I do think this contention is valid. That is,
for focusing paths, Debye-Waller effects tend to cancel, so that
for the path 0 1 2 1 0, the net sigma**2 is sigma**2_0,2

    The reason is the following; in general the DEBYE-WALLER
factor for a path with N sites, 0, 1, ..., N-1 is exp(-k**2 sigma**2)/2

      sigma**2 = < (delta R)**2 >
               = sum_ij <[u_i . (^R_(i-1,i)+^R_(i+1,i)]
                         [u_j . (^R_(j-1,j)+^R_(j+1,j)]>

where ^R_(ij) are unit  vectors in the direction of bond R_(i,j).
Thus for collinear paths e.g., 0 1 2, there is no second order
contribution from a displacement of the shadowing atom 1, since
(^R_(0,1)+^R_(2,1)=0.  I think this is well known.  Of course
large perpindicular displacements can be important, but come
in at higher order, as do 3rd cumulant effects.

   If one can assume (?) that the structural displacements u_i
and u_j are uncorrelated (unlike those of vibrational motion),
then one can derive a structural Debye-Waller factor contribution
with

      sigma**2 = (16/3)  < u**2 > sum_i  sin**2(theta_i/2),

where theta_i is the angle between the bonds at vertex i (assuming
I haven't missed some factors of 2), and < u**2 > is the  mean
square structural disorder.

    As for  strategies for treating structural Debye-Waller factors,
I would do it in an ad hoc path by path way in the fitting routine.
Alternatively, if one has a model, one could do a configurational
average using FEFF's CFAVERAGE card and a suitabe number of
configurations. I'd  be interested in hearing how others have
treated these factors.

  J. Rehr