Section 4.2.6.2.4. Intercomparison of theories

A discussion of the validity of the non-relativistic dipole approximation for the calculation of forward Rayleigh scattering amplitudes has been given by Roy & Pratt (1982). They compared their relativistic multipole calculations with the relativistic dipole approximation and with the non-relativistic dipole approximation for two elements, silver and lead. They concluded that a relativistic correction to the form factor of order persists in the high-energy limit, and that this constant correction accounts for much of the deviation from the non-relativistic dipole approximation at all energies above threshold. In addition, their results illustrate that cancelling occurs amongst the relativistic, retardation, and higher multipole contributions to the scattering amplitude. This implies that care must be taken in assessing where to terminate the series that describes the multipolarity of the scattering process.

In a later paper, Roy, Kissel & Pratt (1983) discussed the elastic photon scattering for small momentum transfers and the validity of the form-factor theories. In this paper, which compares the relativistic modified form factor with experimental results for lead and a relativistic form factor and the tabulation by Hubbell, Veigele, Briggs, Brown, Cromer & Howerton (1975), it is shown that the modified relativistic form-factor approach gives better agreement with experiment for high momentum transfers ( 104  Å⁻¹) than the non-relativistic, form-factor theories.

Kissel et al. (1980) used the S-matrix technique to calculate the real part of the forward-scattering amplitude for the inert gases at the wavelength of Mo . These values are compared with the predictions of the relativistic dipole theory (RDP) and the relativistic multipole theory (RMP) in Table 4.2.6.2. In most cases, the agreement between the S matrix and the RMP theory is excellent, considering the differences in the methodology of the two sets of calculations. Table 4.2.6.3(a) shows comparisons of the real part of the forward-scattering amplitude calculated for the atoms aluminium, silicon, zinc, germanium, silver, samarium, tantalum and lead using the approach of Kissel et al. (1980) with that of Cromer & Liberman (1970, 1981), with tabulations by Wagenfeld (1975), and with values taken from the tables in this section. Although reasonably satisfactory agreement exists between the relativistic values, large differences exist between the non-relativistic value (Wagenfeld, 1975) and the relativistic values. The major difference between the relativistic values occurs because of differences in estimation of the self-consistent-field term, which is proportional to . The Cromer & Liberman (1970) relativistic dipole value is , whereas the tabulation in this section uses the relativistic multipole value of . This causes a vertical shift of the curve, but does not alter its shape. Should better estimates of the self-energy term be found, the correction is simply that of adding a constant to each value of for each atomic species. There is a significant discrepancy between the Kissel et al. (1980) result for ⁶²Sm and the other theoretical values. This is the only major point of difference, however, and the results are better in accord with the relativistic multipole approach than with the relativistic dipole approach. Note that the relativistic multipole approach does not include the Stibius-Jensen correction, which alters the shape of the curve.

Table 4.2.6.2| top | pdf | Comparison between the S-matrix calculations of Kissel (K) (1977) and the form-factor calculations of Cromer & Liberman (C & L) (1970, 1981, 1983) and Creagh & McAuley (C & M) for the noble gases and several common metals; f′(ω, 0) values are given for two frequently used photon energies Energy (keV) Element RDP (C & L) S matrix (K) RMP (C & M) 17.479 (Mo ) Ne 0.021 0.024 0.026 Ar 0.155 0.170 0.174 Kr −0.652 −0.478 −0.557 Xe −0.684 −0.416 −0.428 22.613 (Ag ) Al 0.032 0.039 0.041 Zn 0.260 0.323 0.324 Ta −0.937 −0.375 −0.383 Pb −1.910 −1.034 −1.162

Table 4.2.6.3| top | pdf | A comparison of the forward-scattering amplitudes computed using different theoretical approaches (a) Real part. KPR (Kissel et al., 1980); C & L (Cromer & Liberman, 1970, 1981); W (Wagenfeld, 1975); and C & M (this data set). Atom Radiation KPR C & L W C & M 1970 1981 13Al Cr 13.320 13.328 13.316 13.376 13.326 Cu 13.209 13.204 13.203 13.235 13.213 Ag 13.039 13.032 13.020 13.078 13.041 14Si Cr   14.333 14.354 14.441 14.365 Cu   14.244 14.242 14.282 14.254 Ag   14.042 14.029 14.071 14.052 30Zn Cr 29.161 29.316 29.314   29.383 Cu 28.369 28.388 28.383   28.451 Ag 30.323 30.260 30.232   30.324 32Ge Cr   31.538 31.538 30.20 31.614 Cu   30.837 30.837 31.92 30.911 Ag   32.228 32.228 32.14 32.302 47Ag Cu 47.075 46.940 46.936   47.131 62Sm Ag 58.307 56.304 56.299   56.676 73Ta Ag 72.625 72.063 71.994   72.617 82Pb Ag 80.966 80.090 80.012   80.832 (b) Imaginary part . KPR (Kissel et al., 1980); C & L (Cromer & Liberman, 1981); W (Wagenfeld, 1975); and C & M (this data set). Atom Radiation KPR C & L W C & M 13Al Cr 0.514 0.522   0.512 Cu 0.243 0.246   0.246 Ag 0.031 0.031   0.031 14Si Cr   0.694 0.70 0.692 Cu   0.330 0.33 0.330 Ag   0.043 0.047 0.043 30Zn Cr 1.370 1.373   1.371 Cu 0.678 0.678   0.678 Ag 0.932 0.938   0.938 32Ge Cr   1.786 1.84 1.784 Cu   0.886 0.87 0.886 Ag   1.190 1.23 1.190 47Ag Cu 4.242 4.282   4.282 62Sm Cu 12.16 12.218   12.218 73Ta Ag 4.403 4.399   4.399 82Pb Ag 6.937 6.929   6.929

In §4.2.6.3.3, some examples are given to illustrate the extent to which predictions of these theories agree with experimental data for .

That there is little to choose between the different theoretical approaches where the calculation of is concerned is illustrated in Table 4.2.6.3(b). In most cases, the agreement between the scattering matrix, relativistic dipole, and relativistic multipole values is within 1%. In contrast, there are some significant differences between the relativistic and the non-relativistic values of . The extent of the discrepancies is greater the higher the atomic number, as one might expect from the assumptions made in the formulation of the non-relativistic model. Some detailed comparisons of theoretical and experimental data for linear attenuation coefficients [proportional to have been given by Creagh & Hubbell (1987) for silicon, and for copper and carbon by Gerward (1982, 1983). These tend to confirm the assertion that, at the 1% level of accuracy, there is little to choose between the various relativistic models for computing scattering cross sections.

Further discussion of this is given in §4.2.6.3.3.