There are also regions in some elements where alternate tabulations are clearly superior to Refs. [1720,24] 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 2040 eV for monatomic gases (e.g.) also occur, not predicted within hydrogenlike calculations and those assuming plane wave continuum states [43]. Tabulation based on the local density approximation (LDA) or on experimentaltheoretical 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 mediumZ 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 modelbased 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. [1720,24] 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 f_{1} and f_{2} (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 Zinterpolation. These are regions of particular usefulness of the current tabulation.
Highenergy and highmomentum transfer limitations follow in part from dominance of nuclear resonance and pairproduction channels. Each of these attenuation and scattering processes are ideally independent [38], or add together with a welldefined 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 highenergy, highmomentum transfer regimes. Hubbell et al. [40] include corrections for radiative and doubleCompton contributions to incoherent crosssections, reaching 1% at 100 MeV energies. Nuclearfield pair production κ_{n} arises at 2m_{e}c^{2} = 1.022 MeV and becomes dominant around 10 MeV. Electronfield 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 1024 MeV of width 39 MeV, contributing up to 10% of the total crosssection 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 highenergy limits but above 10300 keV (dependent on Z), incoherent scattering forms the dominant attenuation process, and may be given to 12% by integration of the KleinNishina formula. Interference between this channel and photoabsorption can affect total crosssections and form factors at the 5% level.
In the region 1040 keV elastic scattering may exceed photoabsorption crosssections (especially at Bragg peaks). This is of prime importance for estimates of total attenuation crosssections µ_{TOT} or for attempts to compare f″ to experimental crosssections, and is complicated by the coherence of scattering between adjacent atoms, leading to TDS (thermal diffuse scattering) and Rayleigh singleatom estimates in addition to accurate derivations from the form factor f_{1} 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 orientationaveraged 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 BraggLaue peaks or hollows. This Rayleigh scattering may be replaced by a TDS crosssection if the crystal orientation is arranged to avoid BraggLaue peaks [45].
