NIST Physical Measurement Laboratory X-Ray Form Factor, Attenuation, and Scattering Tables Main Page

[skip navigation] National Institute of Standards and Technology NIST Physical Measurement Laboratory

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 [20]. 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.
Back to Main Page
Table of Contents - 1995
previous page next page