[Ifeffit] Distortion of transmission spectra due to particle size

[Scott Calvin]

On Nov 22, 2010, at 2:55 PM, Scott Calvin wrote:

> But compare a monolayer of particles with a diameter equal to 0.4  
> absorption lengths with four strips of tape of that kind stacked. Do  
> we really think the distortion due to nonuniformity will be as bad  
> in the latter case as in the first? In practice, I think many  
> transmission samples fall in roughly that regime, so the question  
> isn't just academic.

OK, I've got it straight now. The answer is yes, the distortion from  
nonuniformity is as bad for four strips stacked as for the single  
strip. This is surprising to me, but the mathematics is fairly clear.  
Stacking multiple layers of tape rather than using one thin layer  
improves the signal to noise ratio, but does nothing for uniformity.  
So there's nothing wrong with the arguments in Lu and Stern, Scarrow,  
etc.--it's the notion I had that we use multiple layers of tape to  
improve uniformity that's mistaken.

A bit on how the math works out: for Gaussian distributions of  
thickness, the absorption is attenuated (to first order) by a term  
directly proportional to the variance in the distribution. The  
standard deviation in thickness from point to point in a stack of N  
tapes generally increases as the square root of N (typical statistical  
behavior). This means that the fractional standard deviation goes down  
as the square root of N. In casual conversation, we would usually  
identify a sample with thickness variations of +/-5 % as being
"more uniform" than one with thickness variations of +/- 8%, so it's  
natural to think that a stack of tapes is more uniform than a single  
one. But since the attenuation is proportional to the variance (i.e.  
the square of the standard deviation), it actually increases in  
proportion to N. Since the absorption is also increasing in proportion  
to N, the attenuation remains the same size relative to the  
absorption, and the spectrum is as distorted as ever.

This result doesn't actually depend on having a Gaussian distribution  
of thickness. if each layer has 10% pinholes, for instance, at first  
blush it seems as if two layers should solve most of the problem: the  
fraction of pinholes drops to 1%. But those pinholes are now compared  
to a sample which is twice as thick, on average, and thus create  
nearly as much distortion as before. Add to this that there is now 9%  
of the sample that is half the thickness of the rest, and the  
situation hasn't improved any. I've worked through the math, and the  
cancellation of effects is precise--a two layer sample has the  
identical nonuniformity distortion to a one layer one.

(There is probably a simple and compelling argument as to why this  
distortion is independent of the number of randomly aligned layers for  
ANY thickness distribution, but I haven't yet found it.)

* * *

For me personally, knowing this will cause some changes in the way I  
prepare samples.

First of all, I'm going to move my bias more toward the thin end. My  
samples are generally pretty concentrated, so signal to noise is not a  
big issue. If I'm also not improving uniformity by using more layers  
of tape, there's no reason for me not to keep the total absorption  
down around 1, rather than around 2.

Secondly, I'll approach the notion of eyeballing the assembled stack  
of tapes for uniformity, whether with the naked eye or a microscope,  
with more caution--particularly when teaching new students. The idea  
that a sample which has no evident pinholes is a better sample than  
one that does is not necessarily true, as the example above with the  
single layer exhibiting 10% pinholes as compared to the double layer  
exhibiting 1% demonstrates. Stressing the elimination of visible  
pinholes will tend to bias students toward thicker samples, but not  
necessarily better ones.


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