7. Integration Precision Requirements
Several integration procedures were investigated and compared.
Five-point Gauss-Legendre integration yields convergence to 5% in many but
not all cases where the form is appropriate. In some cases (near edges,
particularly with functions not corrected for above or below-edge behavior)
a 30-point integration was found inadequate .
Five-point integration appears adequate for low Z to about
0.01 eu in most regions, while yielding errors of 0.1 eu for
silicon near edges, where 10-point and 16-point results appear identical.
This error rises to above 1 eu for silver (Z = 47) and
5 eu for uranium (Fig. 3), with shifts of
0.3 eu and 1.3 eu, respectively, even midway between edges. The
difference between 10-point and 16-point integration is also significant at
the 0.15 eu level for Z = 92, and is fairly uniform for
all energies. The figure indicates major integration and interpolation error
in the unrevised formalism [17-20,24] of magnitude
1-4 eu across the medium energy range of 4-6 keV. This may also
be compared to Refs. [15,16] (including a
partial frel) correction of perhaps -1.5 eu),
apparently erroneous by up to 5 eu. Although uranium is perhaps an
extreme case, this situation is reflected throughout the medium and higher
range of Z. The 16-point integration is therefore used and commended,
covering a much larger energy regime with higher precision.