Fancy background subtraction: a second look.

Schematic overview of the considerations of the fancy background subtraction.
Schematic overview of the considerations of the fancy background subtraction.

During last week’s visit to BAM in Berlin, I and Ingo played around with some equations. In particular, we were curious if we need to do something special to get the scattering from a sample in a capillary, i.e. a sample between an upstream and a downstream sample cell wall. Long story short: we arrive at a rather ordinary equation after a lengthy derivation.

So we derived the method to extract the sample scattering from the total scattering signal for this case. We discussed this topic before in this post where self-absorption (discussed in this post) was considered, but where the derivation was not completely worked out. To make it easier to derive, we neglect the direction-dependent absorption of the scattered signal(s), but a future derivation may include this term (only for flat walls and phases, as the derivation thereof of round capillary-type cross-sections is ridiculously complex).

Schematic overview of the considerations of the fancy background subtraction.
Figure 1: Schematic overview of the considerations of the fancy background subtraction.

We consider three separate processes as an incident photon beam travels through a given phase: the attenuation of the incident beam by the material before a scattering event, a probability for scattering of this attenuated beam, and the attenuation of the scattered radiation by the remaining material of the phase. Each incident and scattered beam is further attenuated by the preceding and subsequent phases. We also assume that the scattering event does not significantly reduce the intensity of the remaining unscattered beam.

The derivation is provided in a separate document here, together with most of the text in this post. I cannot promise I did the derivation in the most efficient way, there may be shorter ways of doing it, and it most certainly will have been done before in literature. Apropos, it very much resembles a nice exercise for an exam question or class derivation, for those who are into teaching small-angle scattering.

Despite the rather long intermediate equations in the derivation, the final equation is quite short:

P_2 = \frac{1}{D_2} \left\{ \frac{ I_s }{ I_0 \zeta_{\mathrm{1+2}} } - \frac{I_b} { I_0 \zeta_{\mathrm{1}} } \right\}

where D_2 is the thickness of the sample phase (only), I_s the measured intensity of sample + cell, I_b the measured intensity of the cell, I_0 the primary beam flux, and \zeta_{\mathrm{1+2}} and \zeta_{\mathrm{1}} the transmission factors measured for the sample + cell and cell measurements, respectively.

There are interesting aspects when we use this background subtraction equation in practice. Firstly, we find that it is not necessary to determine the sample cell wall thickness D_1. Secondly, both the sample measurement and the background measurement are normalised to the thickness of the sample phase D_2 only. Lastly, it should be noted that this is, of course, only valid if the same sample cell is used for both the background and the sample measurement.

So, nothing shocking here, but it is interesting that it can be derived from more basic considerations.


1 Trackback / Pingback

  1. Thoughts on the displaced volume correction – Looking At Nothing

Leave a Reply

Your email address will not be published.



This site uses Akismet to reduce spam. Learn how your comment data is processed.