Section 4.3.1.6. Tables of atomic scattering amplitudes for electrons

Tables 4.3.1.1 and 4.3.1.2 list values of in Å for all neutral atoms and most chemically significant ions, respectively. The values have been given by Doyle & Turner (1968) for several cases, denoted by RHF using the relativistic Hartree–Fock atomic potentials of Coulthard (1967). For all other atoms and ions, has been found using the Mott–Bethe formula [equation (4.3.1.15)] for , and the X-ray scattering factors of Table 2.2A of IT IV (1974). Thus all other neutral atoms except hydrogen are based on the relativistic Hartree–Fock wavefunctions of Mann (1968). These are designated by *RHF. For H and for ions below Rb, denoted by HF, is ultimately based on the nonrelativistic Hartree–Fock wavefunctions of Mann (1968). For ions above Rb, denoted by *DS, modified relativistic Dirac–Slater wavefunctions calculated by Cromer & Waber (1974) are used.

For low values of s, the Mott formula becomes less accurate, since tends to zero with s for neutral atoms. Except for the RHF atoms, for s from 0.01 to 0.03 are omitted in Table 4.3.1.1 and for s from 0.04 to 0.11, only two decimal places are given. is then accurate to the figure quoted. For these atoms, was found using the formula given by Ibers (1958): where is the mean-square atomic radius.

For ionized atoms, . The values listed at s = 0 in Table 4.3.1.2 for RHF atoms were calculated by Doyle & Turner (1968) with in equation (4.3.1.13) replaced by , where Here, is the ionic charge. This approach omits the Coulomb field due to the excess or deficiency of charge on the nucleus. With the use of these values, the structure factor for forward scattering by a neutral unit cell containing ions may be found in the conventional way. Similar values are not available for other ions because the atomic potential data are lacking.

For computer applications, numerical approximations to the f(s) of these tables have been given by Doyle & Turner (1968) as sums of Gaussians for the range s = 0 to 2 Å⁻¹. An alternative is to make Gaussian fits to X-ray scattering factors, then use the Mott formula to derive electron scattering factors. As discussed by Peng & Cowley (1988), this practice may lead to problems for small values of s. An additional problem occurs in high-resolution electron-microscopy (HREM) image-simulation programs, where it is usually necessary to have electron scattering factors for the range 0 to 6 Å⁻¹. Fox, O'Keefe & Tabbernor (1989) point out that extrapolation of the Gaussian fits of Doyle & Turner (1968) to values past 2 Å⁻¹ can be highly inaccurate. For the range of s from 2 to 6 Å⁻¹, Fox et al. have used sums of polynomials to make accurate fits to the X-ray scattering factors of Doyle & Turner (1968) for many elements (Section 6.1.1 ), and electron scattering factors can be generated from these data by use of the Mott formula.

Recently, Rez, Rez & Grant (1994) have published new tables of X-ray scattering factors obtained using a multiconfiguration Dirac–Fock code and two parameterizations in terms of four Gaussians, one of higher accuracy over the range of about 2 Å⁻¹ and the other of lower accuracy over the extended range of about 6 Å⁻¹. These authors suggest that electron scattering factors may best be obtained from these X-ray scattering factors by using the Mott formula. They provide a table of values for the electron scattering factor values for zero scattering angle, , for many elements and ions, which may be of value for the calculation of mean inner potentials.