Biomedical Engineering Reference
In-Depth Information
giving
F
-
(
β
) = -0.110. From Eq. (6.6),
× 1.96
√
1.96 + 2) - 0.110 = 14.0.
G
-
(
β
) =
ln
(3.61 × 10
5
(6.9)
Finally, applying Eq. (6.5), we find
-
d
E
d
x
-
col
=
10
-31
10
29
5.08
×
×
3.34
×
[14.0 - 4.31]
0.886
1.86 MeV cm
-1
.
=
(6.10)
It is of interest to compare this result with that for a 1-MeV positron. The quanti-
ties
β
2
and
τ
are the same. Calculation gives
F
+
(
β
) = -0.312, which is a little larger
in magnitude than
F
-
(
). In place of (6.9) and (6.10) one finds
G
+
(
β
β
)
=
13.8 and
(-d
E
/d
x
)
col
= 1.82 MeV cm
-1
.The
β
+
collisional stopping power is practically equal
to that for
β
-
at 1 MeV in water.
The collisional, radiative, and total mass stopping powers of water as well as
the radiation yield and range for electrons are given in Table 6.1. The total stopping
power for
β
-
or
β
+
particles is the sum of the collisional and radiative contributions:
-
d
E
d
x
±
tot
=
-
d
E
d
x
±
+
-
d
E
d
x
±
(6.11)
,
col
rad
with a similar relation holding for the mass stopping powers. Radiative stopping
power, radiation yield, and range are treated in the next three sections. Table 6.1
can also be used for positrons with energies above about 10 keV.
The calculated mass stopping power of liquid water for electrons at low energies
is shown in Fig. 6.1. (Measurement of this important quantity does not appear to
be technically feasible.) The radiative stopping power is negligible at these ener-
gies, and so no subscript is needed to distinguish between the total and collisional
stopping powers. The threshold energy
(
Q
min
)
required for excitation to the lowest
lying electronic quantum state is estimated to be 7.4 eV. The curve in Fig. 6.1 joins
smoothly onto the electron stopping power at 10
-2
MeV in Fig. 5.6. The electron-
transport computer code, NOREC, was used to calculate the stopping power in
Fig. 6.1.
1)
The relative importance of ionization, excitation, and elastic scattering at ener-
gies up to 1 MeV can be seen from the plot of the respective attenuation coeffi-
cients,
µ
, in Fig. 6.2, also from the NOREC code. The ordinate gives the values of
µ
in units of reciprocal micrometers. Recall from the discussion at the end of Sec-
tion 5.4 that
µ
represents the probability of interaction per unit distance traveled,
which is also the inverse of the mean free path. Elastic scattering is the dominant
process at the lowest energies. Slow electrons undergo almost a random diffusion,
changing direction through frequent elastic collisions without energy loss. Even-
tually, the occasional competing inelastic excitation and ionization collisions bring
1
See Semenenko, V. A., Turner, J. E., and
Borak, T. B. in Section 6.8.
