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RTAB data files are available at:
ftp://www-phys.llnl.gov/pub/rayleigh/RTAB
13. Interpolating S-matrix cross sections
From Figure 1, it is evident that direct interpolation in energy on differential cross sections that include the contribution from bound-bound transitions can be problematic. Our experience suggests that the MFASF approximation should serve as a reasonable basis for smoothing the SM values, and that heavy atoms at high energy and back angles are the most challenging cases to consider.In Table 10 we list the ratio of SM/MFASF unpolarized differential scattering cross sections for selected energies and angles for Pb. We display only selected energies, but all energies on our 56-point grid have been checked and the values in Table 10 are representative of the behavior of the SM/MFASF ratio. (A similar table for the SM/RFASF ratio would show variations of a factor of 106.) We observe that the SM/MFASF ratio is reasonably smooth at all energies and angles, and the ratio is very close to one for the bulk of the anomalous scattering region for L-shell and lower energies (<~20 keV for Pb). While the SM/MFASF ratio varies by a factor of about 5 as shown in Table 10, the underlying differential cross sections differ by a factor of 10^7.
Focusing on 90° scattering for Pb, we have interpolated the SM/MFASF ratio using 2-point linear interpolation from the nearest-neighbor grid points of our 56-point energy grid, neglecting the point under consideration. That is, for example, to test the accuracy of interpolating from our 56-point grid at 59.54 keV, we have interpolated the value of the SM/MFASF ratio at 59.54 keV using the ratio values at 57.53 and 66.83 keV. Our assumption is that this will provide a pessimistic estimate of the error for interpolating intermediate ratios as we have purposely ignored data in the table and there are significantly better interpolation methods that could be employed. We find that error in the interpolated ratio is accurate to much less than 1% for energies below the K-shell binding energy (about 88 keV for Pb), and generally about 1% or less at higher energies. The exceptions are 2% errors for 80-90 keV, 3% errors around 250 keV and 1173 keV, and a 9% error at 1408 keV.
We expect generally then, that our energy grid is sufficient to interpolate SM unpolarized differential scattering cross sections by interpolating on the SM/MFASF ratio to an accuracy of about 2% or better for energies less than about 1 MeV for all angles for all atoms. An estimate of the SM value at an intermediate energy can be obtained by scaling the MFASF predictions of FFTAB by the interpolated SM/MFASF ratio. Further efforts to confirmation these expectations should be undertaken in the future.
Table 10. SM/MFASF ratio of unpolarized differential scattering cross sections at selected energies and angles for Pb. While the SM/MFASF ratio varies by about a factor of 5 in this table, the underlying differential cross sections vary by a factor of 107.
return to tables
|
Photon energy (keV) |
dσ SM/dσMFASF |
|||||
|
0° |
10° |
30° |
60° |
90° |
120° |
|
|
0.0543 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
1.000 |
|
0.1833 |
1.000 |
1.001 |
1.001 |
1.000 |
1.001 |
1.001 |
|
0.3924 |
1.000 |
1.000 |
1.000 |
1.000 |
1.001 |
1.000 |
|
1.041 |
1.000 |
1.000 |
1.000 |
0.999 |
0.998 |
0.998 |
|
5.415 |
1.000 |
1.000 |
0.999 |
0.996 |
0.994 |
0.992 |
|
8.048 |
1.000 |
1.000 |
0.999 |
0.994 |
0.992 |
0.989 |
|
17.48 |
1.000 |
1.000 |
0.997 |
0.989 |
0.983 |
0.975 |
|
22.16 |
1.000 |
1.000 |
0.996 |
0.984 |
0.975 |
0.965 |
|
59.54 |
1.000 |
0.999 |
0.991 |
0.958 |
0.939 |
0.916 |
|
145.4 |
1.000 |
0.998 |
0.978 |
0.901 |
0.872 |
0.823 |
|
279.2 |
0.999 |
0.996 |
0.947 |
0.770 |
0.728 |
0.730 |
|
411.8 |
0.999 |
0.994 |
0.919 |
0.696 |
0.744 |
0.783 |
|
779.1 |
0.998 |
0.985 |
0.817 |
0.628 |
0.703 |
0.643 |
|
889.2 |
0.998 |
0.982 |
0.805 |
0.588 |
0.636 |
0.539 |
|
1173.2 |
0.999 |
0.975 |
0.787 |
0.453 |
0.412 |
0.313 |
|
1332.5 |
0.999 |
0.966 |
0.759 |
0.369 |
0.325 |
0.252 |
|
2754.1 |
0.999 |
0.895 |
0.429 |
0.238 |
0.295 |
0.278 |
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Information date: Sep. 2, 2000 lk