Page 293 - Biomedical Engineering and Design Handbook Volume 2, Applications

THE PRINCIPLES OF X-RAY COMPUTED TOMOGRAPHY 271

fractional exchange of energy and the redirection of the path of propagation. The relative proportion
of either process to the primary beam attenuation varies with x-ray energy. Another scattering
process is Thompson scattering, which occurs at low energy and imparts very little energy to the
absorber. This involves the interaction of the photon wave and an electron, with the electron-emitting
electromagnetic radiation. Encounters of this type cause phase shifts in the incident photon wave
and are the basis of phase contrast imaging. However, for attenuation contrast imaging, the effect of
phase shift is assumed to be negligible.
For the photoelectric effect, a photon with energy hn encounters an atom, is completely annihi-
0
lated, and in the process ejects a bound electron from the atom with kinetic energy E . The vacant
k
atomic bound state is repopulated from an adjacent level with the emission of a fluorescent x-ray
photon. For Compton scattering, a photon with energy hn encounters a free electron, is redirected
0
though an angle q with changed energy hn ′, and the free electron recoils though an angle f with
received energy E . Both collision processes continuously remove photons from the x-ray beam as it
k
progresses through a material body.
The Photoelectric Effect. The K, L, and M shell electrons of the colliding atom are the principle
participants in the photoelectric interaction. When the incident photon possesses an energy hn 0
greater than the binding, or ionization energy I, the shell electron will be ejected with a kinetic
energy E according to the conservation of energy relation:
k
hn = E + I (10.5)
0 k
The value of the absorption cross section increases with increasing binding energy I, so that a K-shell
interaction is roughly 5 times more probable than an L-shell interaction. The event will most readily
occur if the photon energy is comparable to the binding energy, in a manner somewhat like a reso-
nance process. This behavior is borne out in practice where the photoelectric cross section s pe is
seen to vary with the inverse cube of the photon energy hn , according to
0
Z 4
σ pe = k (10.6)
ν h ( 0 ) 3
where the variation with atomic weight Z is a function of the penetration of the incident photon into
the Coulomb field presented by the electron shell and k is a constant for that particular shell.
The momentum of the incident photons is mostly imparted to the atomic nucleus with the direc-
tion f of the emitted photoelectrons predominantly toward larger angles (Fig. 10.3). The angle is
determined by the direction of the electric field associated with the incident radiation. If it is linearly
polarized, the photoelectron has a large component of velocity parallel to the electric field. However,
at higher energies the photoelectron acquires a forward component of velocity from the incident radi-
ation, as required from conservation of momentum.
The Compton Effect. The Compton effect refers to the scattering of incident photons by the inter-
action with free electrons. Here, an incident photon of energy hn is scattered into the angle q, with
0

FIGURE 10.3 Photoelectric absorption with energy transfer
to ionization and photoelectron motion.