Biomedical Engineering Reference
In-Depth Information
Unlike collisional energy losses, no single analytic formula exists for calculating
the radiative stopping power
(-d
E
/d
x
)
rad
. Instead, numerical procedures are used
to obtain values, such as those in Table 6.1. Details of the analysis show that energy
loss by radiation behaves quite differently from that by ionization and excitation.
The efficiency of bremsstrahlung in elements of different atomic number
Z
varies
nearly as
Z
2
. Thus, for beta particles of a given energy, bremsstrahlung losses are
considerably greater in high-
Z
materials, such as lead, than in low-
Z
materials,
such as water. As seen from Eq. (6.1), the collisional energy-loss rate in an ele-
ment is proportional to
n
and hence to
Z
. In addition, the radiative energy-loss rate
increases nearly linearly with beta-particle energy, whereas the collisional rate in-
creases only logarithmically. At high energies, therefore, bremsstrahlung becomes
the predominant mechanism of energy loss for beta particles, as can be seen from
Table 6.1.
The following approximate formula gives the ratio of radiative and collisional
stopping powers for an electron of total energy
E
, expressed in MeV, in an element
of atomic number
Z
:
(-d
E
/d
x
)
rad
(-d
E
/d
x
)
col
ZE
800
.
=
(6.12)
This formula shows that in lead (
Z
= 82
), for example, the two rates of energy loss
are approximately equal at a total energy given by
82
E
800
=
1.
(6.13)
Thus
E
=
=
9.8
MeV, and the electron's kinetic energy is
T
=
E
-
mc
2
9.3
MeV.
In oxygen (
Z
= 8
), the two rates are equal when
E
=
100
MeV =
T
, an order-of-
magnitude higher energy than in lead. The radiative stopping power
(-d
E
/d
x
)
rad
for electrons is shown by the dashed curve in Fig. 5.6.
At very high energies the dominance of radiative over collisional energy losses
gives rise to electron-photon cascade showers. Since the bremsstrahlung photon
spectrum is approximately flat out to its maximum (equal to the electron's kinetic
energy), high-energy beta particles emit high-energy photons. These, in turn, pro-
duce Compton electrons and electron-positron pairs, which then produce addi-
tional bremsstrahlung photons, and so on. These repeated interactions result in
an electron-photon cascade shower, which can be initiated by either a high-energy
beta particle or a photon.
6.4
Radiation Yield
We have discussed the relative rates of energy loss by collision and by radiation.
Radiation yield is defined as the average fraction of its energy that a beta particle
radiates as bremsstrahlung in slowing down completely. Radiation yields are given
in Table 6.1 for electrons of various energies in water. At 100 MeV, for example,
