[Ifeffit] R dependence of S02

[Scott Calvin]

Hi all,

I've been pondering how much R dependence we should expect to see in  
S02; that is, how much R dependence is shown by intrinsic losses.

I've looked at quite a bit of the literature, not to mention old  
threads from this mailing list. One recent (2002) reference is this  
one by Campbell et al.:

http://prb.aps.org/pdf/PRB/v65/i6/e064107

My understanding is that S02 is intended to account for intrinsic  
losses; that is, those that are determined when the core hole forms.  
Extrinsic losses, such as inelastic scattering of the photoelectron  
and the effect of the core-hole lifetime, are accounted for by a mean  
free path term. The mathematics of how ifeffit implements this are  
here: http://cars9.uchicago.edu/~newville/feffit/feffit.ps

Intrinsic losses are dependent on k for at least two reasons. One is  
that the cross-section of the intrinsic effects themselves depends on  
the energy of the incident x-ray. This is evident if one thinks in  
terms of shake-up and shake-off events. But another reason is that  
shake-up and shake-off events rob some energy from the primary  
photoelectron. At low k, ten or twenty electron volts can alter the  
phase of the primary photoelectron significantly, and thus shake-up  
and shake-off events will tend to cancel each other out. But at high  
k, the energy robbed from the photoelectron is less significant,  
because the EXAFS oscillations are more spread out in energy. Thus,  
shake-up and shake-off events, while still occurring, will not  
suppress the EXAFS amplitude as much at high k. S02 therefore  
gradually rises through most of the EXAFS region to reach a limiting  
value of 1 well above the top of the EXAFS region.

The latter effect--the fact that removing a specified amount of energy  
from the primary photoelectron has less of an effect at high k than at  
low--also implies an R dependence. Low R oscillations are further  
apart in energy than high R oscillations, and thus over a specified k  
range low R oscillations should be less affected. In other words, S02  
should show a modest decrease with increasing R over typical EXAFS  
ranges. Some papers, in particular those with John Rehr as an author,  
confirm that S02 should have an R dependence, but don't discuss the  
implications much.

While ifeffit allows for floating ei, a parameter related to the mean  
free path, as I understand it that will still give a damping of the  
amplitude that is exponential in R. It seems to me that the S02  
dependence on R, in contrast, is likely to be more gentle.

Why is this important?

Several authors, including myself, have analyzed crystallite size and/ 
or morphology by comparing the coordination number of successive  
scattering shells. This is potentially much more accurate than just  
finding the first-shell coordination number, because it is independent  
of any amplitude effects that are independent of R, such as  
normalization errors and many experimental effects. Errors in the mean  
free path are a bit more significant, but the exponential dependence  
of the mean free path gives it a very different shape than effects  
from size and morphology. But an R-dependence of S02 would be  
troubling, as the functional form might look a bit more like a size  
effect.

So one question is this: does anyone have an order-of-magnitude  
estimate of how much R dependence to expect in S02 over the EXAFS  
range? If over a range of 2 to 6 angstroms S02 changed by even a few  
percent, that could have a significant effect on the kind of size  
analyses I mentioned in the preceding paragraph.

Of course, another question is if I've completely blown it anywhere in  
my discussion above; I've just been puzzling this out over the last  
few days!

--Scott Calvin
Faculty at Sarah Lawrence College
Currently on sabbatical at Stanford Synchrotron Radiation Laboratory