(eq 1) |

Equation (1) can be rewritten as

(eq 2) |

from which *μ*/*ρ* can be
obtained from measured values of *I*_{o}, *I* and *x*.

Note that the mass thickness is defined as the mass per unit area, and is
obtained by multiplying the thickness *t* by the density *ρ*, i.e., *x* = *ρ**t*.

The various experimental arrangements and techniques from which
*μ*/*ρ* can be obtained,
particularly in the crystallographic photon energy/wavelength regime, have
recently been examined and assessed by Creagh and
Hubbell (1987, 1990) as part of the International Union of Crystallography
(IUCr) X-Ray Attenuation Project. This has led to new tables of
*μ*/*ρ* in the 1992
*International Tables for Crystallography*
(Creagh and Hubbell, 1992). The current status of
*μ*/*ρ* measurements has also been
reviewed recently by Gerward (1993), and an updated
bibliography of measured data is available in
Hubbell (1994).

Present tabulations of *μ*/*ρ*
rely heavily on theoretical values for the total cross section per atom,
*σ*_{tot}, which is related to
*μ*/*ρ* according to

(eq 3) |

In (eq 3),
*u* (= 1.660 540 2 × 10^{-24} g
Cohen and Taylor 1986) is the atomic mass unit
(1/12 of the mass of an atom of the nuclide ^{12}C), *A* is the
relative atomic mass of the target element, and
*σ*_{tot} is the total
cross section for an interaction by the photon, frequently given in units of
b/atom (barns/atom), where b = 10^{-24} cm^{2}.

The attenuation coefficient, photon interaction cross sections and related quantities are functions of the photon energy. Explicit indication of this functional dependence has been omitted to improve readability.

The total cross section can be written as the sum over contributions from the principal photon interactions,

(eq 4) |

where *σ*_{pe} is the atomic
photoeffect cross section, *σ*_{coh} and *σ*_{incoh} are the coherent (Rayleigh) and the incoherent
(Compton) scattering cross sections, respectively, *σ*_{pair} and *σ*_{trip} are the cross sections for electron-positron
production in the fields of the nucleus and of the atomic electrons,
respectively, and *σ*_{ph.n.} is
the photonuclear cross section.

Photonuclear absorption of the photon by the atomic nucleus results most
usually in the ejection of one or more neutrons and/or protons. This interaction
can contribute as much as 5 % to 10 % to the total photon interaction
cross section in a fairly narrow energy region usually occurring somewhere
between 5 MeV and 40 MeV, depending on where the giant resonance of
the target nuclide falls (see, e.g., Hayward, 1970;
Fuller and Hayward, 1976; and
Dietrich and Berman, 1988; also the illustrative
tables in Hubbell, 1969, 1982). The effects of this
interaction can be observed in measurements of the total attenuation coefficient
(see, e.g., Gimm and Hubbell, 1978). However, this
cross section has not been included in previous tabulations because of the
difficulties due to (a) the irregular dependence of both the magnitude and
resonance-shape of the cross section as a function of both *Z* and
*A*; (b) the gaps in the available information, much of which is for
separated isotopes or targets otherwise differing from natural isotopic
mixtures; and (c) the lack of theoretical models for
*σ*_{ph.n.} comparable to those
available for calculations of the other cross sections of interest. The
practice of omitting the contribution of the photonuclear cross section in
tables of the mass attenuation coefficient has been continued in this work,
along with the neglect of other less-probable photon-atom interactions, such as
nuclear-resonance scattering and Delbrück scattering.

Our results for the elements are given in Table 3
for elements *Z* = 1 to 92 and photon energies 1 keV to
20 MeV, and have been calculated according to

(eq 5) |

Values for the relative atomic mass *A* of the target elements were taken from
Martin (1988) and can be extracted from the values of
*Z/A* given in Table 1; values for the
individual contributing cross sections are those found in the current NIST
database (see Berger and Hubbell, 1987), as
outlined below.

*Atomic photoeffect*. For photon energies from 1 keV to 1.5 MeV,
values of the photoelectric cross section, *σ*_{pe}, are those calculated by
Scofield (1973), based on his solution of the Dirac
equation for the orbital electrons moving in a static Hartree-Slater central
potential. No renormalization was performed using those factors given by
Scofield for the elements with *Z* = 2 to 54 to convert to
values expected from a relativistic Hartree-Fock model. This represents a break
with the practice by Hubbell (1977, 1982) and
Hubbell *et al.* (1980) in which this
renormalization had been done.

Absorption-edge fine structure can be experimentally observed using
continuum-energy photon sources and high-resolution detectors (see, e.g.,
Faessler, 1955;
Deslattes, 1969;
Hubbell *et al.*, 1974;
Lytle *et al.*, 1984; and
Del Grande, 1986, 1990), and the observed
variations are subject to chemical, phase and other environmental effects, such
as temperature (Lytle, 1963). As has been done in
the past, this fine structure is ignored in the present compilation. The cross
sections in the vicinity of absorption edges are instead assumed to have simple
sawtooth shapes. Values at the edge have been obtained by extrapolation of the
near-edge subshell cross sections of Scofield (1973)
to the threshold edge energies given by Bearden and
Burr (1967), according to the same procedure used by
Berger and Hubbell (1987) to prepare the NIST
database. The interpolation procedures used for the present tables are slightly
different from those used by Berger and Hubbell; here we use a cubic Hermite
interpolant for the individual subshell cross sections rather than a cubic
spline for the total photoeffect cross section, which results in occasional
small differences in the vicinity of M- and N-shell edges of high-*Z*
elements.

Scofield's (1973) photoeffect calculations were limited to photon energies of 1.5 MeV and below. His data were extended to higher energies (where the photoelectric cross section is quite small) by connecting them to the high-energy asymptotic values of Pratt (1960) through use of a semi-empirical formula (Hubbell, 1969).

*Coherent and incoherent scattering.* Values for the coherent (Rayleigh)
scattering cross section, *σ*_{coh}, are taken from Hubbell and
Øverbø (1979). These were calculated by numerical integration
of the Thomson (1906) formula weighted by
*F*^{2}(*q*,*Z*), where *F*(*q*,*Z*) is
the relativistic Hartree-Fock atomic form factor and *q* is the momentum
transfer. The compilation of *F*(*q*,*Z*) by Hubbell and
Øverbø was based on piecing together, over the
different ranges of *q* and *Z,* values given by
Pirenne (1946) for *Z* = 1, and
those of Doyle and Turner (1968),
Cromer and Waber (1974) and
Øverbø (1977, 1978) for the other
elements.

Values for the incoherent (Compton) scattering cross section,
*σ*_{incoh}, are from
Hubbell *et al.* (1975), obtained from
numerical integration of the Klein-Nishina (1929)
differential formula weighted by the incoherent scattering function
*S*(*q*,*Z*). For their compilation of
*S*(*q*,*Z*), Hubbell *et al.*
(1975) pieced together results given by Pirenne
(1946) (*Z* = 1), Brown (1970a,
1970b, 1971, 1972, 1974) (Z = 2 to 6, with configuration interaction)
and by Cromer and Mann (1967) and
Cromer (1969) (*Z* = 7 to 100, from
a non-relativistic Hartree-Fock model). Radiative and double-Compton
corrections from Mork (1971) were applied to the
integrated values for *σ*_{incoh}.

*Electron-positron pair and triplet production cross sections.* Cross
sections for the production of electron-positron pairs (e^{-},
e^{+}) in the field of the atomic nucleus, *σ*_{pair}, and for the production of triplets
(2e^{-}, e^{+}) in the field of the atomic electrons,
*σ*_{trip}, are taken from the
compilation of Hubbell *et al.* (1980). Their
synthesis combined the use of formulas from Bethe-Heitler theory with various
other theoretical models to take into account screening, Coulomb, and radiative
corrections. Different combinations were used in the near-threshold,
intermediate and high-energy regions to obtain the best possible agreement with
experimental cross sections (Gimm and Hubbell, 1978).

*Mixtures and compounds.* Values of the mass attenuation coefficient,
*μ*/*ρ*, for the 48 mixtures
and compounds (assumed homogeneous) are given in
Table 4, and were obtained according to simple
additivity:

(eq 6) |

where *w*_{i} is the fraction by weight of the i^{th}
atomic constituent, and the (*μ*/*ρ*)_{i} values are from Table 3.
The assumed fractions by weight are given in Table 2.
To obtain (*μ*/*ρ*)_{i}
values at all the absorption edges of all constituent elements, interpolation
has been performed separately for the cross sections indicated in (eq 5),
including the photoeffect cross sections for the individual atomic subshells.

Abstract **|**
Introduction **|
Mass Atten. Coef. |**
Mass Energy-Absorp. Coef. **|**
Summary **|**
References