Larch Plugin Tutorial

A Useful Plugin: the MBACK algorithm

Here we will walk through the creation of a Larch plugin for the MBACK algorithm by Tsu-Chien Weng, et al.. MBACK is a method of normalizing XAS $μ (E)$ by matching it to tabulated values of atomic cross sections.

To accommodate instrumental and sample-related effects in the background of the measured data, this background function is used:

μ ​ b a c k ​ ​ (E) = [\sum ​ 0 ​ m ​ ​ C ​ i ​ ​ (E - E ​ 0 ​ ​) ​ i ​ ​] + A \cdot e r f c ((E - E ​ e m ​ ​) / ξ)

This includes a Legendre polynomial of order m added to a complementary error function used to approximate the effect of Compton scattering in the discriminator window of an energy-discriminating detector, which is often responsible for a highly non-linear shape to the pre-edge region of the measured $μ (E)$ data.

This normalization function has m+3 parameters -- m Legendre polynomial coefficients, the amplitude (A) and width ( $ξ$ ) of the complementary error function, and a scaling factor (s) for the measured data. The centroid of the error function ( $E e m $ ) is set to the centroid of the emission line associated with the measured absorption edge. This function is then minimized to determine the background function parameters:

\frac{​ 1 ​ ​}{​ n ​ 1 ​ ​ ​} \sum ​ 1 ​ n ​ 1 ​ ​ ​ ​ [μ ​ t a b ​ ​ (E) + μ ​ b a c k ​ ​ (E) + s μ ​ d a t a ​ ​ (E)] ​ 2 ​ ​ + \frac{​ 1 ​ ​}{​ n ​ 2 ​ ​ ​} \sum ​ n ​ 1 ​ ​ ​ N ​ ​ [μ ​ t a b ​ ​ (E) + μ ​ b a c k ​ ​ (E) + s μ ​ d a t a ​ ​ (E)] ​ 2 ​ ​

Here, $μ d a t a (E)$ is the measured spectrum and $μ t a b (E)$ is the tabulated cross section.

:FIXME: katex does not yet support \mathrm or \operatorname, so erfc() looks bad.... Also display math would be nice.

The plugin that implements this is at https://github.com/xraypy/xraylarch/blob/master/plugins/xafs/mback.py . In the next few sections, we will step through the parts of that file.