[Ifeffit] Distortion of transmission spectra due to particle size

[Scott Calvin]

Hi all,

I'm tracking down a piece of EXAFS lore which I think is incorrect.

I've seen it said that you cannot compensate for the distortion  
introduced by large particle sizes by making the sample thicker.  
Certainly thick samples have their own set of issues (e.g. "thickness  
effects" from harmonics), but I've seen the claim that the mathematics  
of the distortions introduced by nonuniformity means that there is a  
particle-size distortion that is independent of thickness. This claim  
is sometimes accompanied by an equation giving chi_eff/chi_real as a  
function of particle size diameter D and various absorption  
coefficients.

I've eventually traced this equation back to a paper by Lu and Stern  
from 1983, have walked through the derivation, and believe there is a  
flaw in the logic that has led to the erroneous--and widely quoted-- 
conclusion that thickness cannot compensate for particle size.

The paper, for those who want to follow along, is K. Lu and E. A.  
Stern, "Size effect of powdered sample on EXAFS amplitude," Nucl.  
Instrm. and Meth. 212, 475-478 (1983).

They calculate the intensity transmitted by a spherical particle, and  
from there calculate the attenuation in the normalized EXAFS signal  
for a beam passing through that particle.

They then, however, extend this to multiple layers of particles by the  
following argument:

"Finally, the attenuation in N layers is given by (I/I0)^N, where I is  
the transmitted intensity through one layer. Xeff for N layers is then  
the same as for a single layer since N will cancel in the final result."

This is not the case, is it? It seems to me that their analysis  
assumes that the spheres in subsequent layers line up with the spheres  
in previous ones, so that thick spots are always over thick and thin  
spots over thin. It's little wonder, then, that making the sample  
thicker does not improve the uniformity according to that analysis.

I've done a calculation for the effects of uniformity in a somewhat  
different way, and found that it is indeed true that multiple layers  
on particles show less distortion due to nonuniformity that a single  
layer of particles of the same size, just as one would intuitively  
imagine, and in contrast to Lu and Stern.

Do you agree that the extrapolation to multiple layers in the original  
Lu and Stern paper is not correct, or have I misled myself somehow?

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

P.S. None of this should be taken as an endorsement of overly thick  
samples! Harmonics and the like are a concern regardless of the  
uniformity issue.