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6. Avoidance of Imprecision in f′(E) Below and Near Edges

Corrections for below-edge behavior applied to the second (intermediate energy) form in Ref. [18] should also be applied to the other two forms as detailed in section 4. Near to but above edges where -ε1 < 1 keV, the integration is improved by using a form similar to the "higher-energy" forms rather than that previously indicated, which was optimum only for E >> -ε1. Well below edges where -ε1 > 1 keV, the integration is improved by using a form similar to the `high-energy' form rather than that previously indicated, which was optimum for E simeq  -ε1. In any of these regions where the functional forms are inappropriate, some improvement in precision can often be gained by using 10-point or higher order Gauss-Legendre integration. This represents improved precision of the integration using a less optimum integral transformation.

Previous below-edge errors are illustrated for the K edge of carbon in carbon in Fig. 2. The edge was apparently displaced and asymmetric (typically with a spurious value for the edge itself) as opposed to the revised form ("o" versus "+" in figure). There were also significant errors in the earlier prescription 12% below the edge. This also occurs with similar magnitude (4 e/atom) for the L-edges of uranium at 17-22 keV in Fig 1. These errors occur at energies well within the range of crystallographic investigation and concern, and are not restricted to regimes of specialized anomalous or other diffraction experiments.

The theoretical-experimental synthesis of Ref. [16] (triangles) is in reasonable agreement with the current revised results well away from the edge, with the exception that aliasing is introduced due to the finite logarithmic grid on which linear interpolation is carried out. This means that the transform is calculated appropriately from the given absorption coefficients to yield f1 values for specific energies, but subsequent interpolation between these points smooths out any finer structure, in addition to generating an interpolation error even in the absence of structure. The earlier synthesis [7] contains a much coarser grid, so direct comparison of the two primarily indicates interpolation limitations. The relativistic correction factor is only about 0.004 (e/atom or eu in crystallographic notation) and makes negligible contribution to the results (whatever form is postulated for the correction), while f0 simeq Z-0.04 eu, and does not affect the comparison. Hence a typical experiment in this regime with graphite or other elemental carbon should relate to a real component of the atomic scattering factor given to high precision by f1. The current revised formalism yields a width (below zero) of 4.6 eV and a mean f1 of circa -0.9 eu over this region, with f1 = -2.4 eu at 0.24 eV from the edge, or f1 = -4.0 eu at 0.07 eV from the edge (or about the edge width).

The independent particle approximation common to all theories discussed herein should yield the revised smooth, symmetric form, but the formalism may be expected to fail near edges or at low energies, where molecular or collective behavior may occur. However, the revised result is in good agreement with Cherenkov radiation studies on the carbon K edge [29], which have derived widths of 5 eV and a mean susceptibility χ′ = 0.0016 or f1 simeq  -0.82 eu, and minima of -2.0 eu > f1,min > -5.1 eu.

Conversely, aliased profiles [16] give minima of 0.3 eu > f1,min > -1.3 eu, non-aliased transforms of a simple edge structure [30] yield f1,min from -5.3 eu to -∞ eu and a width of order 5 eV, while other recent experimental f″ measurements yielded a width of around 14 eV and f1,min simeq  -1.2 eu after integration [31]. All are in disagreement with the Cherenkov studies. In this respect, the importance of direct measurements of the scattering factor (anomalous or atomic) cannot be overestimated. Finally, the result of Ref. [21], which might be expected to include contributing collective behavior, appears to give a shift and singularity in the structure [15].

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