RTAB data files are available at:
ftp://www-phys.llnl.gov/pub/rayleigh/RTAB

6. Differential elastic-scattering cross-section tables (tables_SM, tables_MFASF, tables_RFASF, tables_MF, tables_RF, tables_NF)

• tables_SM - numerical S-matrix predictions for inner electrons and modified relativistic form-factor predictions for outer electrons due to Kissel and Pratt (our best predictions - most computationally intensive to produce, limitations as noted below);

• tables_MFASF - modified relativistic form factors with angle-independent anomalous scattering factors (yields predictions close to SM values in many cases, except for back angles at high energies of heavy atoms - much less computationally intensive to produce than SM values - predictions for all Z, E, theta can be easily prepared using the FFTAB code and data in RTAB database - can be easily extended to ions, excited and hollow atoms, inclusion of experimental information such as more realistic photoeffect thresholds, inclusion of environmental effects such as scattering in plasmas or solids);

• tables_RFASF - relativistic form factors with angle-independent anomalous scattering factors (yields predictions identical to MFASF in the forward direction, much poorer predictions for non-forward angles of heavy atoms, especially at 100 keV and higher energies);

• tables_MF - modified relativistic form factors (generally the best of the form-factor-only predictions);

• tables_RF - relativistic form factors (generally yields poorer predictions than those from the modified relativistic form factor, only);

• tables_NF - non-relativistic form factors (Hubbell et al., 1975) (generally yields predictions better than those from relativistic form factors, only).

The numerical S-matrix values are our best predictions, but they have limitations.

• We compute scattering for zero-temperature isolated atoms, neglecting all atomic-environment effects. Interference of the scattering of photons from neighboring atoms is known to make important modifications to photoabsorption near to photoionization thresholds, and thermal diffuse scattering will be important at small momentum transfers. Atom-environment effects in scattering have been observed for solids (see, for example, Gonçalves et al., 1993) and amorphous materials (see, for example, Gonçalves et al., 1994).

• We compute scattering for single-particle transitions in a local potential. This approach only includes a local-density approximation to exchange, neglects correlations, and yields photoionization thresholds that differ noticeably from experiment. Energy scaling procedures have been found to improve predictions near thresholds (Basavaraju et al., 1995) although a more general approach to incorporate experimental information (such as accurate threshold values) is needed. Non-local exchange effects were found to be responsible for the disagreement between our S-matrix predictions and experiment for scattering from Neon (Jung et al., 1998).

Recent studies (Carney et al., 2000a; Carney and Pratt, 2000) suggest that the neglect of correlations might be partially remedied. Correlations are found to matter most for dipole transitions at low energy, where the anomalous scattering approach is valid. This implies that the S-matrix amplitudes might be replaced by better MFASF predictions, using photoeffect cross sections that include correlations.

• Our prescription utilizes modified form-factor approximation to estimate the contribution for outer electrons (defined in this case as electrons wherein h-bar*omega > 300e, e is electron binding energy). We expect that this will be a good approximation for nonrelativistic momentum transfers (h-bar*q << mc), which for back-angle scattering translates to h-bar*omega << ½mc^2. At large momentum transfers, s-states dominate the scattering amplitude with a contribution related to the square of the wavefunction normalization. As a consequence, inner-shell s-states strongly dominate over outer-shells.

  For intermediate and back angles for photon energies greater than about 100 keV, we expect that our MF approximation will be an increasingly poor estimate for the scattering from outer electrons. Under our definition, all electrons for Carbon (eK ~ 300 eV), as an example, are estimated by MF by about 100 keV. In general it is anticipated that the Rayleigh scattering amplitude is very small under these circumstances, and the errors introduced by use of MF are likely masked by the contribution from inner electrons and the nuclear Thomson amplitude.

• Our approximate treatment of partially filled shells is felt to be adequate for scattering from the ground states of atoms in most cases, but significant effects are expected for certain excited states (e.g., scattering from the excited 2p state of Hydrogen), or scattering at sufficiently low energies such that the scattering is dominated by loosely-bound electrons. A proper treatment (Carney et al., 2000b) includes contributions from incoherent elastic scattering and inelastic scattering from nearly degenerate subshells.

• We only consider scattering to lowest (2nd) order in e^2. In some circumstances, such as the scattering from Helium (Lin et al., 1975), higher order contributions can be important.

As noted in the parenthetical thumbnail critique of each approximation in this list, the MFASF values hold special promise for accurate scattering predictions that go beyond that available from our current SM prescription. The major disappointment in our MFASF approach has been the inability to discover, to date, an angular dependence for the anomalous scattering factors that properly handle back angles of high energies for heavy atoms.

We strongly prefer the MFASF values over the RFASF values due to the fact that the modified relativistic form factors appear to correctly predict the forward-angle high-energy scattering limit. [footnote - Under the assumption of angle-independent anomalous scattering factors, f' completely dominates f(q) in the real part of the scattering amplitude by 10° at about 1 MeV for Pb, while g' is still small compared with g(q) in the same circumstances. The SM/MFASF ratio for Pb of unpolarized differential cross sections varies in the range of about 0.8-1 for energies 0-100 keV, expanding to about 0.2-1 for energies 100-3000 keV. In contrast, the SM/RFASF ratio varies in the range of 1-10 for energies 0-100 keV, and 1-106 for energies 100-3000 keV.] As a consequence, g' (the real anomalous factor used with modified relativistic form factor, g) goes to zero at high energies. Under our assumption of angle-independent anomalous scattering factors, MFASF is a much better approximation than RFASF for non-forward angles for heavy atoms, especially at energies of 100 keV and above. If a suitable angle dependence for the anomalous scattering factors could be devised, angle-dependent MFASF and angle-dependent RFASF could give similar predictions.

Individual differential-scattering tables have been generated for a fixed Z and E on a 97-point grid for scattering angles 0-180º. The explicit angle grid is listed for reference in Table 2. The step size of the angle grid starts at 0.01º at forward angle and increases to 2.5º at back angles and is expected to easily support accurate interpolation to intermediate angles.

Individual differential-scattering tables are computed for a single Z on a fixed 56-point grid for photon energies 0.05430-2754.1 keV, and stored as individual logical files (information blocks) within a UFO-structured library in a single file. The explicit energy grid for the tabulations is listed for reference in Table 3. For the most part, these energies have been selected for their experimental interest and are largely common x-ray and gamma-ray energies. Because of the rapid variation in scattering as a function of energy, this energy grid is NOT expected to support accurate interpolation to intermediate energies. Some comments about interpolation of the SM results in energy are made subsequently.

In addition, for each of these approximations, two separate evaluations (stored in separate library files) are provided for the differential scattering cross sections; one file includes only the contribution from the Rayleigh (R) amplitude, and the second file includes the contribution of the summed Rayleigh and nuclear Thomson (R+NT) amplitudes. For example, the S-matrix differential cross sections for neutral Pb are stored in two separate files: '082_cs0sl_sm' includes only the contribution of Rayleigh (R) amplitudes; '082_cs0sl_sm+nt' contains the contribution of Rayleigh (R) and nuclear Thomson (NT) amplitudes.

The value of the unpolarized differential scattering cross section for 59.54-keV photons scattered through 90º by Pb in various approximations is shown in Table 4. To increase one's confidence in utilization of these data files, prospective users are encouraged to find the corresponding values in the data files and compare them with the values listed in the table.

LLNL home page and LLNL Disclaimers
UCRL-MI-137026

Information date: Sep. 2, 2000 lk