NIST Atomic Form Factors: Concerns with standard conventions

X-Ray Form Factor, Attenuation, and Scattering Tables Main Page

3. Concerns with Standard Conventions

Coherence of cross-sections: In (eq 8), σ_coh is the cross-section for "coherent" or Rayleigh scattering. This is not always coherent - the complex Rayleigh amplitude for adjacent atoms may add in phase or may add with random relative phase. This component represents the elastic scattering contribution to the interaction coefficient. It relates directly to the structure factor F. The structure factor depends on the material under observation and the crystallographic arrangement of atoms, and hence on both the real and imaginary components of the atomic form factor. For an isolated atom or elemental metal, the total elastic scattering of a material is dominated by the real component of the atomic form factor Re(f).

The "incoherent or 'Compton' cross-section" σ_incoh is likewise not always incoherent but represents the inelastic scattering contribution to the total interaction coefficient. This also depends upon the atomic form factor. The atomic photoabsorption cross-section τ_PE or σ_PE is directly related to the imaginary component of the form factor.

Simple addition of cross-sections: Simple addition of cross-sections from scattering and photoabsorption depends on the relative phases of scattered waves being incoherent, and may in some cases be quite inappropriate. In general the amplitudes should be summed including any relative phases. However the simple summation of the cross-sections represents a common and often very good approximation.

Contributions of high-energy terms in the medium-energy x-ray regime: The remaining terms in (eq 8) represent large contributions only for MeV energies and above, and as such are not the concern of the current discussion. They represent the pair production cross-section in the nuclear field (κ_n), the pair production cross-section in the atomic electron field (or triplet cross-section, κ_e), and the photonuclear cross-section σ_p.n.. An excellent review of these cross-sections is given elsewhere [2]. Below MeV energies all interaction coefficients depend directly and implicitly upon the real and imaginary components of the atomic form factor. The graphs below depict the mass attenuation coefficients and the values of the form factors themselves, since it is critical to present not only quantities in use but also the fundamental parameters underlying the used quantities.

Dependence of f′ and f″ on angle: There have been concerns regarding a possible angular dependence (or scattering vector dependence) of the anomalous dispersion (i.e., energy-dependent) components f′ and f″ of the form factor (eq 4) and (eq 7). The current status of this query is well represented by Creagh and McAuley, who summarize that there is no dependence of either quantity upon scattering vector [17]. Hence all angular dependence of the form factor for an isolated atom is contained in f₀.

The justification for the separability of the angular and energy-dependent components as given in (eq 4) is a related issue. If the two dependencies upon angle (in f₀) and energy (in f′) are truly independent, then the components are clearly separable. However, it has been argued that this separation may not be valid for large energies and large momentum scattering vectors [18].

Because of this, some authors define a modified form factor (MFF) (g) and anomalous scattering factors (g′ and g″) [14]. This formalism appears useful for MeV energies, but not relevant for the current discussion (the differences for even up to 500 keV energies are unobservable) [19].

S-matrix and general formalisms: Recent S-matrix computations have predicted new structure in angular dependences of Rayleigh scattering [18-20]. A recent report and review for incoherent scattering factors has summarized much important information in this area [21]. There is no doubt that higher order corrections, particularly relating to the relativistic correction factor, are important and observable in principle. However, it is often not realized that the relativistic formulations of Cromer and Liberman [22-25] (and most derivations since) are based on the following S-matrix (scattering matrix) equations for the superposition of the final states f (including ionized atoms, excited states, and elastic and inelastic scattered states) in a transition from the initial state i:

(eq 9)

(eq 10)

The scattering amplitudes T_fi in general are complex [26,27]. Most investigations have been restricted to coherent, forward scattering, and where changes in photon polarization do not occur.

All general theories make the isolated atom approximation and the IPA (independent particle approximation). Any variation between computations based on these theories are due to other limitations, not to the use of isolated atom or IPA. Experimental work relating to solids with very different near-edge structure from isolated atoms may be unable to be compared directly to these theoretical results. This can be used to investigate the redistribution of electron density in the formation of bonding in the solids, and can lead to improved XAFS calibration (see section 8.15). In some cases, this gives significant variation between one experiment and another. The comparison of different theoretical and computational schemes within these assumptions is unaffected by these solid state effects; and the conclusions below are largely independent of these concerns.

These approximations are usually combined with the electric dipole approximation to yield final computable results. In this sense all computations have made the same broad approximations. As seen below, most limitations in Chantler [16], Scofield [28], and Saloman and Hubbell [29] can be attributed to convergence problems rather than to higher-order corrections.