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8. How to use these tables

These tables should be combined with the tables of C95 unless the full range of interest is covered in the tables here. Then the tables provide f1 and f2 form factors and [µ/ρ]PE for all elements up to Z = 92 from 0.001 - 0.01 keV to 1000 keV.

In isolation, these tables provide form factors, attenuation and scattering cross-sections for Z = 30-36 from E = 0.9 keV to E = 6.58 keV; for Z = 60-74, from E = 0.1 keV to E = 3.98 keV; and for Z = 75-89, from E = 0.5 keV to E = 8.54 keV. These regions relate directly to the regions of interest in the text, and are the regions where significant improvement has been made. Additionally we provide in Table 6 a coarse grid for Z = 30-36, Z = 60-89 from 0.1 keV to 10 keV following the "Grodstein grid" energies used in earlier tabulations and by other researchers.

Values for f1, f2 or [µ/ρ]PE should be extracted from the tables for the given element(s) and energies required. Linear interpolation of f1 should be adequate, while linear log-log interpolation of f2 or [µ/ρ]PE should be adequate on this scale, if required.

The energy range covered exceeds that for normal x-ray diffraction and crystallography studies but allows limitations and specialized experiments to be investigated with reference to updated and corrected theory. Discussion of solid target effects, correlation, nuclear resonances and uncertainties should be noted carefully in applications below 1 keV or above 100 keV.

The tabulation provides a sufficiently fine grid with accurate atomic edge structure to allow such experiments as DAFS (diffraction anomalous fine structure) to investigate fine structure and spatial distribution of atoms and electrons within materials [58]. Multilayer diffraction experiments may be pursued at lower energies in an analogous manner.

Table 6 presents results for the Grodstein grid energies in this region from 0.1 keV to 10 keV, particularly for comparison to other or earlier tables without interpolation. Although the interpolation process is very straightforward, it has been found that this brief summary is often useful for non-synchrotron applications.

8.1 Computation of form factors for forward scattering

Equation 4 should be used to obtain f′ using the negative value of frel as included at the top of each table for each element. For comparison to old data or computations, the value corresponding to ref. [6], denoted H82, may be used (following Cromer and Liberman [22-25] but omitting the Jensen [59] energy-dependent correction). More recent work has suggested not only that the Jensen term should be omitted but also that the appropriate relativistic correction is 3/5ths of the Cromer-Liberman value [60] [4] . This latter value is denoted 3/5CL at the top of each table for each element. Likewise, the nuclear Thompson term (also negative with respect to the atomic phase) is provided at the top of each table for each element.

For comparisons to other results in the forward scattering limit where the momentum transfer q = 0, the value of f0 = Z may be used and the real and imaginary components of f are then fully defined. As an example, the forward scattering limit for copper at 10.32 keV, in electrons per atom, is f = Re(f) + i Im(f) = 28.07(28) - 0.0876 - 0.000 726 + i 3.05(3). Clearly the uncertainty in the computation of f1 dominates over the relativistic and nuclear Thomson corrections in most cases.

8.2 Computation of form factors for high energies and large momentum transfers

For large scattering angles it is necessary to use a more appropriate value of f0 than f0 = Z, as may be gained from refs. [12-14], [29], or [61]. This is generally true for Bragg diffraction calculations. For example, metallic copper with a lattice spacing of 2d = 3.6150 Å will yield a momentum transfer q = 4π/2d = 3.476 Å-1 or x = 0.2766 Å-1. (The maximum momentum transfer for a back-reflected beam at this energy would be q = 4π/λ = 10.4598 Å-1 or x = 0.8324 Å-1.] We note that for q = 0 the tabulated values are exact, but uncertainties quoted in tabulations of f0 refer to 1 % to 5 % of the total, which would predict 0.29 e/atom uncertainty for q = 0. Nonetheless, we use 1 % here and add the uncertainties in quadrature. Then use of ref. [61] (for neutral copper atoms) gives f0 = 20.713 e/atom for the Bragg reflection, or f = Re(f) + i Im(f) = 28.07(28) - 0.0876 - 0.000 726 + [20.713(207) - 29] + i 3.05(3) = (19.69 0.35) + i (3.05 0.03) e/atom.

8.3 Computation of structure factors

Then the composition and arrangement of the material may be used as indicated in the introduction to provide structure factors (eq 1), refractive indices (eq 2), and Fresnel coefficients (eq 54 of ref. [33], for example), together with scattered, diffracted or transmitted intensities. More complex formula may be found in the relevant literature, allowing for thermal diffuse scattering, orientation effects, and the zeroth order reflection in particular.

8.4 Crystallography (diffraction)

For a general diffraction profile calculation, there is usually a need to consider at least two waves: the incident wave and the corresponding attenuation of this wave (represented by the zeroth order diffraction, the Fresnel equations for the interface, or equivalently the q = 0 forward scattering component) and the nearest Bragg-diffracted wave. There is often the need to consider multiple-beam diffraction, and in general the solution to a particular problem may require a dynamical theory of diffraction applied simultaneously to each of these waves. As a brief summary of some possible relevant formulae and applications, we refer to refs. [5], [62-64] (curved crystal diffraction), refs. [6-8], [65] (single layer or multilayer reflectivities and Fresnel equations), refs. [66,67] (flat perfect crystals), and ref. [68] (general discussion of many related issues). This is not intended as a complete list, but as a useful guide.

8.5 Electron density studies

As a simple extension of (eq 1), we note the field of difference density mapping uses the following equation for the exploration of bonding patterns:

eq 11 (eq 11)

8.6 Computation of sum rules

Sum rules have been discussed and investigated, particularly at relatively low energies. Good recent examples are given by Berkowitz [46], [69], Barkyoumb and Smith [70,71], and others [43], [72]. Such studies serve to highlight relativistic corrections to form factors and to confirm self-consistent tabulations. The relevant formulae are given by the high-energy limit of the Kramers-Kronig relation, and by other energy moments involving the form factors [73].

8.7 Computation of scattering cross-sections

The structure factors may be used to compute differential or integrated coherent and incoherent scattering cross-sections directly, rather than using the integrated sum given in the tables, which assumes Rayleigh scattering for the coherent component. Standard formulae for the Thomson scattering of unpolarized incident radiation, the intensity of Rayleigh (elastic, coherent) scattering, and the incoherent (inelastic) scattering are

eq 12 (eq 12)
and

eq 13 (eq 13)

In these equations, fj is the form factor for an individual orbital, leading to the sum f for the atomic form factor. Corresponding integrated cross-sections, as presented in sum in C95, this work, and (for example) refs. [2] and [13] are given by

eq 14 (eq 14)

eq 15 (eq 15)

where the large bracketed factor represents the recoil process for a free electron as given by the Klein-Nishina formula [74] and the binding effects are included by the incoherent scattering function I(q,Z) or S(q,Z).

However, for N atoms in a unit cell of volume Vc, the coherent scattering in a Bragg reflection should be summed in phase to give Icoh,H=hkl = Ie FH=hkl2 for the structure factor F from (eq 1). Use of the structure factor F then leads to (coherent) Bragg-Laue diffraction, with mH the multiplicity of the hkl reflection and dH the spacing of the hkl planes in the crystal yielding

eq 16 (eq 16)

This is a much larger value than the Rayleigh computation, and assumes alignment of the Bragg planes near a Bragg condition. The corresponding thermal diffuse scattering approximation assumes the scattering crystal is explicitly misaligned from any Bragg conditions, and leads to a much lower cross-section

eq 17 (eq 17)
or
eq 18 (eq 18)

Corresponding formulae may be found in refs. [2], [13], [19] and [68] for differential cross-sections. Because these various formulae have significant energy and angular dependence, and vary dramatically from monatomic gas to aligned or misaligned solid, it is often advisable to compute the scattering cross-sections directly rather than to use a simple approximation. However, the full version of the incoherent cross-section cannot be computed from the data in C95 or this work, because we do not present the orbital wavefunctions needed to compute the interference term of S(q,Z). It is however possible to compute the coherent cross-sections in any approximation, and to compute the estimates of S(q,Z) omitting that last term. For most low or medium-energy purposes this is quite adequate, but we also present the sum of coherent and incoherent cross-sections under the assumption of Rayleigh scattering, in the tabulation.

8.8 X-Ray Attenuation [Medical imaging, transmission studies]

For filters or filter materials, the photoelectric attenuation coefficient is provided in order to compare to appropriate experiments or to allow for objects in a beam-line. The conversion to this from f2 in appropriate units is provided at the top of the table. Use of barns/atom is also common, and the conversion factor for this is also provided. Often this column is not measured, and only the total observed mass attenuation coefficient

eq 19 (eq 19)

is observed. These latter two coefficients are angle-dependent and may in part be determined from appropriate structure factors for a given crystal orientation as described above. However a column is provided for the sum of these two latter coefficients in an average-over-angles for an atomic scatterer [45], [75]. These references should be consulted for details concerning the approximation involved, although the column in the current tabulation is a new computation of the sum (following C95). The main assumption is that Bragg-Laue peaks and troughs are avoided, or that the material is randomly oriented and preferably mosaic. If this is not true, it may be necessary to compute the dynamically diffracted intensities from the structure factor rather than rely on the approximation. However, simply summing these two columns allows the comparison of theory to experimental attenuation data. For most regions of interest for medical imaging, this is an adequate approximation. The accuracy of the scattering coefficients (within the Rayleigh approximation) is of order 5 %.

8.9 [High-energy] Radiation Shielding

For high energies (the transition depends upon Z), the coherent and incoherent cross-sections dominate over the photoelectric cross-section. In t his region the scattering coefficients of refs. [2] and [13] are recommended as a possibly higher precision computation. At this point the experimental evidence on this point is inconclusive, but we do not claim any higher accuracy than 5 % for these scattering estimates. At high energies there may also be interference between the photoeffect and coherent cross-sections, in which case the current tabulation is important in identifying such effects but not in computing them.

At 1 MeV energies and above, (or at γ-ray resonances), nuclear physics dominates and we recommend inclusion of corrections by Hubbell et al. [12,13] for radiative and double-Compton contributions to incoherent cross-sections, reaching 1 % at 100 MeV energies, and those of nuclear-field pair production κn beginning at 2mec2 = 1.022 MeV and becoming dominant around 10 MeV and above. 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 hydrogen [or 1/(1 + Z)]. Nuclear photoabsorption consists of one (or a few) peaks (giant resonances) between 10 MeV to 24 MeV of width 3 MeV to 9 MeV, contributing up to 10 % of the total cross-section in this region. Elastic processes include high energy Delbrück and dipole resonance scattering in addition to Rayleigh and nuclear Thompson contributions mentioned above.

8.10 VUV reflectivities and multi-layer computations

In addition to the discussions in refs. [6-8], relating to multilayer theory, experimental investigations in the VUV region suffer from the limited precision of theory (and of this current work). Our best recommendation regarding the estimation of either the magnitude of the form factor for an element in this region, or for a structural feature in this region, is to compare the results of the current approach to that of ref. [33], and to treat the difference as an estimate of the theoretical uncertainty in the region. The major problems arise from valence shell correlations, and hence poor convergence of orbitals, and from correlated excitations, phonons and other solid state interactions. At the current time, we only present the results of C95 and this work as a guide in the region below 100 eV.

8.11 Individual orbital cross-section studies, and fluorescence yields

The column providing the photoabsorption coefficient for the K-shell only is included for two purposes. The first is that at high energies this is the dominant contribution to the total photoabsorption, and provides a guide for the local energy dependence of the cross-section. Secondly, it serves as an illustration of the isolation of individual orbital cross-sections, particularly for higher energies.

The isolated K-shell cross-section is also important for experimental diagnostics and corrections. In particular, fluorescence yields from atoms are negligible for almost all orbitals except the K-shell, when compared to Auger and Coster-Kronig transitions. However, the fluorescence yield fraction for the K-shell is large, so the dependence of the cross-section upon energy is equally important. The qualitative result in an experimental ion chamber is significant - the fluorescence x-ray may escape from the ion chamber without conversion to (detectable) ion pairs. A more detailed discussion of this is provided elsewhere [42].

8.12 Comparisons to literature

The plots provide comparison to the theoretical results of Scofield [28,29], [34] the experimental compilation of Saloman et al. [34], and the experimental synthesis of Henke et al. [32,33]. This is considered by the author to be the most useful and convenient comparison of current work in the literature. Scofield is often cited and the original stimulation for the preparation of this work was a comparison with that theory. The plots indicate limitations regarding restricted ranges and tabulation steps, show good agreement over much of the energy range for many elements, and indicate regions of divergence, difficulty or concern. Some of these concerns have been addressed directly in this paper, while others remain. A naïve statement of uncertainty in Henke or this work arises from the divergence between the two. This may relate to local structure, absolute values, or global structure. An alternate error estimate is provided in Table 2.

8.13 Chemical Shifts

The edge energies used follow ref. [76] are provided at the top of each table so that criticism (or experimental investigation) may indicate a shift of the local energy scale which may be appropriate in a specific material or experiment. This is not encouraged or recommended; nonetheless, it is provided as a statement of the assumptions and basis of the computation.

8.14 Electron form factors and scattering

Within the isolated atom approximation for spherically symmetric atoms, the electron atomic form factor is given by an analogue of (eq 5), with the electron density replaced by the periodic potential φ(r):

eq 20 (eq 20)

Poisson's equation relates the potential and charge distributions, and leads to the Mott-Bethe formula for fB(q,Z) in terms of the x-ray atomic form factor f(q,Z):

eq 21 (eq 21)

On the basis of these formulae, numerous studies can be and have been conducted, and we refer simply to two summaries for elastic and inelastic scattering [77,78].

8.15 X-Ray Anomalous Fine Structure (XAFS) and Diffraction Anomalous Fine Structure (XAFS)

X-Ray Anomalous Fine Structure (XAFS) studies typically use a scaled reference line for atomic structure, relative to which the bonding, nearest neighbor, and structural information is extracted. This reference line should be derived from atomic theory for an isolated atom. If this reference theory were accurate to better than 1 %, XAFS and DAFS would be consistent and provide unambiguous determination of local structure. Not all theories provide a self-consistent reference for atomic theory near edges, which is a pre-condition for the correct interpretation of fine structure measurements.
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