Biomedical Engineering Reference
In-Depth Information
If
µ
is expressed in cm
-1
and
Q
in MeV, then Eq. (5.10) gives the stopping power
in MeV cm
-1
. The quantities
µ
and
Q
avg
, and hence
-d
E
/d
x
, depend upon the type
of particle, its energy, and the medium traversed.
Example
The macroscopic cross section for a 1-MeV proton in water is 410
m
-1
, and the
average energy lost in an electronic collision is 72 eV. What is the stopping power in
MeV cm
-1
and in J m
-1
?
µ
Solution
With
m
-1
and
Q
avg
=
µ
=
410
µ
72 eV, Eq. (5.10) gives
-
d
E
10
4
eV
m
-1
.
d
x
=
µ
Q
avg
=
410
×
72
=
2.95
×
µ
10
-6
MeV and 1
10
-4
cm, we obtain -d
E
/d
x
=
295 MeV cm
-1
.
Since 1 eV
=
µ
m
=
295 MeV cm
-1
10
-13
JMeV
-1
Converting units further, we have -d
E
/d
x
=
×
1.60
×
×
100 cmm
-1
10
-9
Jm
-1
.
=
4.72
×
In this example, note that the average energy loss of 72 eV for a 1-MeV pro-
ton is considerably larger than the range of the most probable energy losses in
Fig. 5.3. Table 5.1 shows that the maximum energy loss is 2200 eV, which lies be-
yond the horizontal scale in the figure by a factor of 22. The energy-loss distribution
W
(
Q
)
for heavy charged particles is thus very skewed for large losses out to
Q
max
.
Although
W
(
Q
)
is small when
Q
is large, the stopping power reflects an energy-
loss-weighted average,
QW
(
Q
)
, and hence substantial contributions from the tail of
the energy-loss distribution. In traveling short distances in matter, when the total
number of collisions is relatively small, the average energy lost by a charged parti-
cle (as implied by the stopping power) and the most probable energy lost can differ
substantially. This phenomenon of energy straggling is discussed in Section 7.5.
To understand what “short distances” mean in the present context, we recall that
the macroscopic cross section
µ
is the probability per unit distance of travel that an
electronic collision takes place. Its role in charged-particle penetration is analogous
to that of the decay constant
λ
, which is the probability of disintegration per unit
time in radioactive decay. Equation (4.22) showed that the reciprocal of the decay
constant is equal to the mean life. In the same way, the reciprocal of
µ
is the mean
distance of travel, or mean free path, of a charged particle between collisions. In
the last example, the mean free path of the 1-MeV proton is 1/
m
-1
)
µ
=
1/(410
µ
=
0.0024
µ
m = 24 Å. Atomic diameters are of the order of 1 Å to 2 Å.
5.5
Semiclassical Calculation of Stopping Power
Quantum mechanically, stopping power is the mean, or expectation, value of the
linear rate of energy loss. In 1913 Bohr derived an explicit formula giving the stop-
