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X-ray absorption, A

X-rays are absorbed as they pass through materials according to the exponential law:

I = I0e-μt
where I0 is the incident intensity, I is the (reduced) intensity after travelling a distance t through the material and μ is the absorption coefficient (sometimes also referred to as the photoelectric absorption (coefficient) due to photo-electron energy losses being the principal cause). Its value varies with the atomic composition (increasing with atomic number) and X-ray wavelength (increasing with wavelength); the wavelength dependence is approximately cubic overall, but with discontinuities (known as absorption edges) at wavelengths/energies corresponding to specific electronic transitions within the absorbing atoms.

Sample absorption will obviously affect measured diffraction intensities; we might first consider the potential scale of the problem. A value often used, when discussing absorption, is 1/μ since this equals the optimal sample thickness (i.e. t = 1/μ) for a transmission diffraction experiment: in fact you can easily show from the above equation that this thickness will reduce intensities to around 37%. It is quite instructive to look at some typical values of 1/μ, as in the table below:

λ=0.56 Å λ=1.54 Å
perfect vacuum
10−10 mbar vacuum ≈1011 km ≈1010 km
helium 600 m 300 m
air 15 m 1 m
beryllium 23.5 mm 3.6 mm
aluminium 1.4 mm 76 μm
lead 13 μm 4 μm
NaCl 1,110 μm 61 μm
Ca (OH)2 840 μm 48 μm

Just simply looking at these figures, and remembering that these distances will cut down intensities to 37%, you should be able to immediately appreciate some simple practicalities:

In considering the effect of absorption, in the context of powder diffraction, we can envisage three very different situations:

The three situations are depicted below:

Thin Powder Plate in Transmission Thick Powder Plate in Reflection
Thin Glass Capillary (mixed)  
The resultant effects are quite different. In summary they are:
  1. Thin plate: The effect is easier to calculate: the absorption changes with diffraction angle, 2θ, simply because the path length of X-rays through the sample lie between t and t/cos 2θ.
  2. Bragg-Brentano: The effect might not be so obvious how to calculate but the actual result is so simple: since the incident X-ray beam has a finite width there will be many paths of different lengths in and out of the sample. In the example above we see a typical X-ray path for two different angles: at a low "glancing" (θ) angle, the depth below the sample is small; at higher angles the same path length requires deeper penetration into the sample. However the surprising result is that when all the possible path lengths are taken in to account, the net absorption remains constant with θ. This means that the effect of absorption can effectively be ignored in this case since it affects all reflections equally. This is one reason, among others, why this geometry is often chosen.
  3. Capillary: The effect is not easy to calculate but the result is satisfying. As the example above illustrates, some paths appear to be like transmission (a) and others like reflection (b). There is also the absorption through the thin glass wall to consider, though this is often negligible by comparison.

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© Copyright 1997-2006.  Birkbeck College, University of London.
Author(s): Paul Barnes
Simon Jacques
Martin Vickers