2. Form Factors and Standard Definitions

(eq 3)

with λ in, e.g., Ångstroms; the "anomalous" scattering factor f′ (depending on x-ray energy E and the atomic number Z); and the small nuclear Thomson term f_NT [10,11]. f′ can also be expressed in terms of a small relativistic correction term f_rel, Z and the function f₁ often used to characterize these form factors:

(eq 4)

(eq 5)

The angular factor f₀ is identical to the values f(q) or F(x,Z) given in Hubbell et al. [12], Hubbell and Øverbø [13], and Schaupp et al. [14] and use q instead of x, with x = q/4π.

(eq 6)

The imaginary component Im(f) = f″ is directly related to the atomic photoabsorption cross-section given as τ_PE or σ_PE in different references:

(eq 7)

Fundamental constants and conversion factors are given in ref. [15]. Conventionally, the total interaction cross-section σ_tot is represented as a sum over the individual photon interaction cross-sections:

(eq 8)

These cross-sections would conventionally be given in barns/atom. This would be directly related to the linear attenuation coefficient (µ) in cm^-1, and the mass attenuation coefficient in cm²/g. The mass attenuation coefficient is conventionally given by the symbol [µ/ρ] = σ/uA, where σ is the cross-section in barns/atom, u is the atomic mass unit, and A is the relative atomic mass of the target element. Coefficients for converting between these units are given by many authors [16]. (See the table header in section 13 and the element header information accessed through the online database.)

This paper develops the approach covered in Chantler [16] and makes extensive reference to this earlier work, which will therefore be denoted in what follows as C95. Table 1 summarizes the type of use to which this tabulation (and that of C95) may be put. It summarizes the typical equation to use (with reference to column headings in the current tabulation) and gives the author's current personal recommendation of a useful or appropriate reference for additional information or coefficients as might be needed.

Table 1. Summary of particular uses of these tables (this work and Chantler [16])

Form factors for forwarding scattering	section 8.1	Direct use or interpolation, with (eq 4)
Form factors for significant momentum transfers	section 8.2	(eq 4) and (eq 5) or refs [13,14], [29] or [61]
Calculation of structure factors	section 8.3	As section 8.2 but also (eq 1)
Refractive indices	section 8.3	As section 8.2 but also (eq 2)
Crystallography (diffraction)	section 8.4	As section 8.3 but also see text for references
Multilayer reflectivity, transmission	section 8.4	As section 8.2 and section 8.3 but also see text for references
Electron density studies	section 8.5	As section 8.3 but also (eq 11)
Sum rules	section 8.6	As section 8.2 but also see text for references
Computation of scattering processes	section 8.7	As section 8.2 but also (eq 12), (eq 14), (eq 1), (eq 16), (eq 17), (eq 18), and (limited) (eq 13) and (eq 15) (see text)
Photoelectric cross-sections, linear absorption coefficient, or mass absorption coefficients	section 8.8	Direct use or interpolation, with conversion as given in table headers as needed
X-ray Attenuation [Medical imaging, transmission studies]	section 8.8	Direct use of total mass attenuation coefficient for Raleigh scattering
X-ray Attenuation studies with alternate scattering estimates	section 8.8	Direct use of mass absorption coefficient, with (eq 19) and possibly refs. [2], [13], [29]
X-ray Attenuation of crystalline samples	section 8.8	Direct use of mass absorption coefficient, with (eq 19) and section 8.7
Angle-dependent scattering processes	section 8.8	Not applicable in general - see text
High-energy attenuation, above 100 keV	section 8.9	Direct use of mass absorption coefficient, with (eq 19), (eq 8), and possibly refs [2], [12,13], and [29]
High-energy (γ-ray) attenuation, above 1 MeV	section 8.9	See refs [12,13]
VUV studies	section 8.10	Directly, but with caveats and see also ref. [33]
K-shell studies & fluorescence yields	section 8.11	Directly, but see text
Electron scattering	section 8.14	(eq 21), and see text