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
f_{1} and f_{2} 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 f_{1}, f_{2} or [µ/ρ]_{PE} should be extracted from the
tables for the given element(s) and energies required. Linear interpolation of
f_{1} should be adequate, while linear log-log interpolation of
f_{2} 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
f_{rel} 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
f_{0} = 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 f_{1} 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
f_{0} than f_{0} = 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 f_{0} 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 f_{0} = 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:
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
and
In these equations, f_{j} 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
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 V_{c}, the
coherent scattering in a Bragg reflection should be summed in phase to give
I_{coh,H=hkl} =
I_{e} F_{H=hkl}^{2}
for the structure factor F from (eq 1).
Use of the structure factor F then leads to (coherent) Bragg-Laue
diffraction, with m_{H} the multiplicity of the hkl
reflection and d_{H} the spacing of the hkl planes in the
crystal yielding
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
or
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 f_{2} 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
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 κ_{n} beginning at
2m_{e}c^{2} = 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):
Poisson's equation relates the potential and charge distributions, and leads to
the Mott-Bethe formula for f^{B}(q,Z) in terms of
the x-ray atomic form factor f(q,Z):
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.