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The iterative Kramers-Kronig method

The Kramers-Kronig method finds a pair of f'(E) and f''(E) functions which are a self-consistent Kramers-Kronig transform pair, and which also agree with the data. For centrosymmetric systems with crystallographic weights, tex2html_wrap_inline1526, this method uses the intensity fitting function,
where tex2html_wrap_inline1664 and tex2html_wrap_inline1666, and where the energy dependent instrument correction function tex2html_wrap_inline1668.

To obtain starting values for tex2html_wrap_inline1670, tex2html_wrap_inline1672, b, and m, f' and f'' are estimated using Cromer-Liberman tex2html_wrap_inline1682 and tex2html_wrap_inline1684 values, and Eqn. 15 is fit to the data. Then the functions f' and f'' are obtained by using an iterative procedure: Equating the fitting function to the data, the resulting quadratic equation is solved for f'. This f' is then Kramers-Kronig transformed to obtain a corresponding f''. With these new values for f' and f'', the data is refit to obtain new values of tex2html_wrap_inline1670, tex2html_wrap_inline1672, etc., and in turn a new value for f'. This process usually converges after three or four iterations.

The iterative Kramers-Kronig method has several advantages. First, little knowledge of the crystal structure is required to obtain the f' and f'' functions. Second, the iterative method properly accounts for the diffracted intensity's dependence on the real and imaginary parts of the anomalous amplitude, and also properly accounts for its dependence on the square of the fine structure. Note that the direct method for isolating the fine structure, presented in the previous subsection, neglected the square of the fine structure.