Biomedical Engineering Reference
In-Depth Information
Solution
Setting
T
82 in Eq. (6.14) gives for the fraction of the beta-particle
energy converted into photons
Y
=
=
2.28 and
Z
=
0.10. The total beta-particle energy released per
10
8
MeV s
-1
. Multiplica-
tion by
Y
gives, for the rate of energy emission by bremsstrahlung, ∼3.48 × 10
7
MeV.
The energy fluence rate at a distance of 1 m is therefore
10
8
s
-1
) (0.94 MeV)
second from the source is (3.7
×
=
3.48
×
10
7
MeV s
-1
)/(4
∼
(3.48
×
π
×
100
2
cm
2
)
277 MeV cm
-2
s
-1
. For assessing the radiation hazard, we assume that
the photons have an energy of 2.28 MeV. Therefore, the photon fluence rate at this
distance is
=
121 photons cm
-2
s
-1
. For comparison, we note that use of an
∼
227/2.28
=
13) shield to stop the beta particles would give
Y
=
aluminum (
Z
=
0.017, reducing
the bremsstrahlung by a factor of 5.9.
6.5
Range
The range of a beta particle is defined like that of heavy particles by Eq. (5.39) in
which the total stopping power (-d
E
/d
x
)
tot
is used. Unlike a heavy particle, how-
ever, its range is only a poor indicator of the depth to which a given electron is
likely to go into a target. Nevertheless, we shall employ electron csda ranges and
the assumption of straight-ahead travel in order to make at least rough estimates
in working problems. One must always bear in mind that this procedure over-
estimates electron penetration in matter. A more quantitative relationship between
the pathlength and the maximum penetration depth is considered in Section 6.7.
Table 6.1 gives electron ranges in water down to 10 eV. As with heavy charged
particles, the ranges expressed in g cm
-2
are approximately the same in different
materials of similar atomic composition. Electron ranges in H
2
O, muscle, bone,
Pb, and air are included in Figs. 5.7 and 5.8. For the same reasons as with heavy
charged particles (discussed at the end of Sect. 5.6), the collisional mass stopping
power for beta particles is smaller in high-
Z
materials, such as lead, than in water.
In Fig. 5.7, this fact accounts for the greater range of electrons in Pb compared with
H
2
O at energies below about 20 MeV. At higher energies, the radiative energy-loss
rate in Pb more than compensates for the difference in the collisional rate, and the
electron range in Pb is less than in H
2
O.
The following empirical equations for electrons in low-
Z
materials relate the
range
R
in g cm
-2
to the kinetic energy
T
in MeV:
For 0.01
2.5 MeV,
≤
T
≤
0.412
T
1.27-0.0954
ln
T
R
=
(6.15)
or
6.63 - 3.24(3.29 -
ln
R
)
1/2
;
ln
T
=
(6.16)
for
T
>2.5
MeV,
R
= 0.530
T
- 0.106,
(6.17)
or
