5.1 Overview

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5.1 Overview

autobk uses a piecewise polynomial, or spline, to approximate $\ensuremath{\mu_{0}(E)}$ .The spline is chosen to optimize the R -components of $\ensuremath{\tilde\chi(R)}$ , the Fourier Transform of $\ensuremath{\chi(k)}$ , below $\ensuremath{R_{\rm bkg}}$ . The stiffness of the spline [12] is controlled by the number of knots , points at which the different polynomial pieces meet, and where there can be a discontinuity in some high order derivative. The number of knots in the background spline is chosen to be the number of independent points in the low-R range of $\ensuremath{\tilde\chi(R)}$ , between R = [0.0, $\ensuremath{R_{\rm bkg}}$ ]. This is simply given by the number of independent points in this region, which is $\begin{displaymath} N_{\rm bkg} = 1 + {{2 \Delta k R_{\rm bkg} }\over {\pi} }.\end{displaymath}$
where $\ensuremath{R_{\rm bkg}}$ is an estimate of the low-R edge of the first peak in the resulting $\ensuremath{\tilde\chi(R)}$ , and $\Delta k$ is the k -range of the data. $\ensuremath{N_{\rm bkg}}$ is the number of degrees of freedom in the data below $\ensuremath{R_{\rm bkg}}$ . Based on ideas of information theory, the knots of the spline are equally spaced in k , which will minimize the spectral leakage of the background into the region above $\ensuremath{R_{\rm bkg}}$ . autobk uses fourth order splines (i.e. , cubic splines) to ensure that no more than one full oscillation of the spline can occur between knots. This means that the highest measurable R value (the so-called Nyquist critical frequency) is $\ensuremath{R_{\rm bkg}}$ , and that all components of the background above $\ensuremath{R_{\rm bkg}}$ comes from spectral leakage due to the finite k -range.

Next: 5.2 Information theory and XAFS Up: 5 Post-Edge Background Function Previous: 5 Post-Edge Background Function

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