Biomedical Engineering Reference
In-Depth Information
All of the details that we have worked out thus far for Compton scattering fol-
low kinematically from the energy and momentum conservation requirements
expressed by Eqs. (8.9)-(8.11). We have said nothing about how the photon and
electron interact or about the
probability
that the photon will be scattered in the
direction
θ
. The quantum-mechanical theory of Compton scattering, based on the
specific photon-electron interaction, gives for the angular distribution of scattered
photons the Klein-Nishina formula
2
m
2
c
4
ν
2
ν
ν
m
2
sr
-1
.
k
0
e
4
+
ν
ν
d
e
σ
d
-
sin
2
=
θ
(8.28)
ν
Here
d
e
σ
, called the
differential scattering cross section
, is the probability per unit
solid angle in steradians (sr) that a photon, passing normally through a layer of
material containing one electronm
-2
, will be scattered into a solid angle
d
at
angle
θ
. The integral of the differential cross section over all solid angles,
d
=
2
π
sin
θ
d
θ
, is called the
Compton collision cross section.
It gives the probability
e
σ
that the photon will have a Compton interaction per electronm
-2
:
/d
d
e
σ
d
sin
θ
d
θ
m
2
.
σ
= 2
π
(8.29)
e
The Compton cross section,
e
σ
, can be thought of as the cross-sectional area, like
that of a target, presented to a photon for interaction by one electron m
-2
.Itisthus
rigorously the stated interaction
probability.
However, the area
e
σ
, which depends
on the energy of the photon, is not the physical size of the electron.
The dependence of the differential cross section (8.28) on
θ
can be written ex-
plicitly with the help of Eq. (8.12) (Problem 24). Equation (8.28) can then be used
with the kinematic equations we derived to calculate various quantities of inter-
est. For example, the energy spectrum of Compton recoil electrons produced by
1-MeV photons is shown in Fig. 8.5. The relative number of recoil electrons de-
creases from
T
0.796
MeV,
where the spectrum has its maximum value (called the
Compton edge
in gamma-ray
spectroscopy). The most probable collisions are those that transfer relatively large
amounts of energy. The Compton electron energy spectra are similar in shape for
photons of other energies.
Of special importance for dosimetry is the average recoil energy,
T
avg
,ofComp-
ton electrons. For photons of a given energy
h
ν
, one can write the differential
Klein-Nishina energy-transfer cross section (per electronm
-2
)
=
0
until it begins to rise rapidly as
T
approaches
T
max
=
d
e
σ
tr
d
T
h
ν
d
e
σ
d
=
.
(8.30)
The average recoil energy is then given by
σ
tr
e
e
T
avg
=
h
ν
,
(8.31)
σ
