11. Limitations of the Current Tabulation

There are also regions in some elements where alternate tabulations are clearly superior to Refs. [17-20,24] and the current formulation. Delays of onset of photoabsorption for 20 eV (3%) above the edge, and peak amplification by factors of two for another 20-40 eV for monatomic gases (e.g.) also occur, not predicted within hydrogen-like calculations and those assuming plane wave continuum states [43]. Tabulation based on the local density approximation (LDA) or on experimental-theoretical syntheses [15,16,21,30] have potential advantages particularly in low energy regimes such as Fig. 6 where collective valence effects or dipole resonances lead to significant departures from the independent particle approximation. The omission of some outer orbitals, mentioned earlier, accentuates this problem in some medium-Z elements in this same energy range (i.e., up to 0.1 keV or so). In such low energy regimes, seminal and early experimental work and local transforms can still play an important role in revealing additional structure or correcting model-based assumptions [44].

Often the "best" tabulation in these cases may be fairly inaccurate, due to intrinsic limitations or additional contributions from exciton resonances or band structure. In the example of Fig. 6, early syntheses [7] differ substantially from later equivalents [15,16] and Refs. [17-20,24] differ markedly from that of the current calculations. Above or below the range of Fig. 6, predictions of the real and imaginary components of atomic form factors from Ref. [16] and herein often agree fairly closely. Discrepancies of an order of magnitude appear in the plotted range. The predictions of [16] for f₁ and f₂ (that is, the estimated total form factors), though probably the best available elsewhere for most of this region, are limited by the input data, theory and synthesis to be accurate to not much better than a factor of two. This limitation of general tabulations is also true in the near vicinity of edges affected by valence or molecular structure.

Conversely, syntheses [7,15,16] tend to diverge from experiment and theory where numerous edges occur or where data do not give detailed variation between and at edges. Divergence can be due to smoothing of edge structure, weighting of experimental data, or Z-interpolation. These are regions of particular usefulness of the current tabulation.

High-energy and high-momentum transfer limitations follow in part from dominance of nuclear resonance and pair-production channels. Each of these attenuation and scattering processes are ideally independent [38], or add together with a well-defined phase (as with the nuclear Thomson term in section 2) [12]. General neglect of other contributions does not intrinsically invalidate computations of atomic form factors, but it seriously limits their usefulness and applicability, and the precision to which measurements of these quantities may be made at these high-energy, high-momentum transfer regimes. Hubbell et al. [40] include corrections for radiative and double-Compton contributions to incoherent cross-sections, reaching 1% at 100 MeV energies. Nuclear-field pair production κ_n arises at 2m_ec² = 1.022 MeV and becomes dominant around 10 MeV. Electron-field pair production ("triplet production") begins at 2.044 MeV and contributes above this energy at the 1% level for high Z elements but up to 10% for fluorine and 50% for H (or 1/(1+Z)). Nuclear photoabsorption consists of one (or a few) peaks (giant resonances) between 10-24 MeV of width 3-9 MeV, contributing up to 10% of the total cross-section in this region. Elastic processes include high energy Delbruck and dipole resonance scattering in addition to Rayleigh and nuclear Thomson contributions mentioned above.

Below these high-energy limits but above 10-300 keV (dependent on Z), incoherent scattering forms the dominant attenuation process, and may be given to 1-2% by integration of the Klein-Nishina formula. Interference between this channel and photoabsorption can affect total cross-sections and form factors at the 5% level.

In the region 10-40 keV elastic scattering may exceed photoabsorption cross-sections (especially at Bragg peaks). This is of prime importance for estimates of total attenuation cross-sections µ_TOT or for attempts to compare f″ to experimental cross-sections, and is complicated by the coherence of scattering between adjacent atoms, leading to TDS (thermal diffuse scattering) and Rayleigh single-atom estimates in addition to accurate derivations from the form factor f₁ and diffraction formulae. Of course, a prime purpose of the current discussion has been to generate accurate form factors which can then be used to determine total attenuation (i.e., photoabsorption plus diffraction) for a given orientation of a sample to high precision. They can then be compared to experimental values.

Simple available and tabulated estimates of orientation-averaged scattering can err by orders of magnitude in special arrangements, although they are usually more accurate than this. The simple forms commonly used for such scattering assume that each atom is isolated, so integration neglects crystal orientations leading to Bragg-Laue peaks or hollows. This Rayleigh scattering may be replaced by a TDS cross-section if the crystal orientation is arranged to avoid Bragg-Laue peaks [45].