1. Introduction and Importance of Form Factors
In the x-ray energy range covered here, the primary
interactions of photons with atoms are photoabsorption and
coherent (elastic) scattering. Inelastic (Compton) scattering
becomes dominant for all elements as the higher
-ray
energies are approached. For light elements, this transfer of
dominance occurs at much lower energies (e.g., for hydrogen the
inelastic component dominates above 3-5 keV). Additional nuclear
scattering and absorption occur at MeV energies, including pair
production and Delbruck scattering from the nuclear field; and
nuclear resonant processes (such as nuclear Thomson scattering)
[1]. For XUV photons at the lower end of
the tabulated energy range, lattice phonon absorption, delocalized
plasmon excitation, excitons and dipole resonances may appear
[2]. Although these remain qualitatively
identifiable as photon interactions with bound electrons, they are
not associated with atomic orbitals or isolated atoms.
In the intermediate energy range, typically from 0.01-0.1 keV
through to 80-800 keV, the interaction of the incident photon
with the electrons, i.e., with the bound atomic orbitals, without
production of secondary x-rays of reduced energy, is the dominant
process. The photon is then either scattered without altering the
internal energy of the atom, or it is absorbed. This absorption is
usually into a single atomic orbital, with a consequent ejection of
a photoelectron and production of a singly-ionized species.
Both photoabsorption and (Rayleigh) scattering are described
by the structure factor F of the material in condensed or gas
phase. In particular, diffracted intensity or coherent scattering
is a complicated function of F, but for weak reflections is linear
or quadratic in F. Equally, transmission through a bulk material is
a complex function of F but local attenuation is a relatively
simple function of the imaginary component of F
[3-5]. This is
well known in the crystallographic community and is used
extensively in the multilayer community at lower energies
[6-8].
The structure factor for a given reflection (denoted hkl from the
Miller indices) is a sum over the atoms in the appropriate lattice
(for a crystal) of the atomic form factors or the x-ray scattering
factors fj of the jth atom:
where thermal diffuse scattering is neglected, Mj
is the thermal parameter for the given temperature, reflection and
atom, and the location of the atom in the unit cell is given by
(xj, yj, zj).
For an isolated atom or a single elemental lattice, a scaled atomic
form factor may therefore be substituted for the structure factor.
At grazing angles of incidence with solids, photons interact
with the surface, and the photoabsorption and reflection processes
may be given by Fresnel equations (while still dominated by
electron orbital interaction and governed by the structure factor
and form factors) [9].
If the atoms in a condensed system may be
considered to scatter as dipoles (i.e., for low energies or small
scattering angles) then the interaction of x-rays with matter may
be described using optical constants such as the complex index of
refraction nr or the complex dielectric constant
(E), which
are related to the form factors by
where nj is the atom number density and
r0 is the classical electron radius.
