Biomedical Engineering Reference
In-Depth Information
One sees that the (nonrelativistic) result (5.18) is identical with the Bethe formula
when
β
1
, and we write
V
=
βc
and
hf
=
I
/2
. Whereas the energy
hf
in the semi-
classical theory has only a rather vague meaning, the mean excitation energy
I
is
explicitly defined in the quantum theory in terms of the properties of the target
atoms. This quantity is discussed in the next section.
Figure 5.6 shows the stopping power of liquid water in MeV cm
-1
for a number
of charged particles as a function of their energy. The logarithmic term in Eq. (5.23)
leads to an increase in stopping power at very high energies (as
β
→
1
), just dis-
cernable for muons in the figure. At low energies, the factor in front of the bracket
in (5.23) increases as
β
→
0
. However, the logarithm term then decreases, causing
a peak (called the Bragg peak) to occur. The linear rate of energy loss is a maximum
there.
The mass stopping power of a material is obtained by dividing the stopping
power by the density
ρ
. Common units for mass stopping power, -d
E
/
d
x
, are
MeV cm
2
g
-1
. The mass stopping power is a useful quantity because it expresses
the rate of energy loss of the charged particle per g cm
-2
of the medium traversed.
In a gas, for example. -d
E
/d
x
depends on pressure, but -d
E
/
ρ
d
x
does not, be-
cause dividing by the density exactly compensates for the pressure. In addition,
the mass stopping power does not differ greatly for materials with similar atomic
composition. For example, for 10-MeV protons the mass stopping power of H
2
Ois
45.9 MeV cm
2
g
-1
and that of anthracene (C
14
H
10
) is 44.2 MeV cm
2
g
-1
. The curves
in Fig. 5.6 for water can be scaled by density and used for tissue, plastics, hydrocar-
bons, and other materials that consist primarily of light elements. For Pb (
Z
ρ
82
),
on the other hand,
-d
E
/
ρ
d
x
= 17.5
MeV cm
2
g
-1
for 10-MeV protons. Generally,
heavy atoms are less efficient on a g cm
-2
basis for slowing down heavy charged
particles, because many of their electrons are too tightly bound in the inner shells
to participate effectively in the absorption of energy.
=
5.7
Mean Excitation Energies
Mean excitation energies
I
for a number of elements have been calculated from the
quantum-mechanical definition obtained in the derivation of Eq. (5.23). They can
also be measured in experiments in which all of the quantities in Eq. (5.23) except
I
are known. The following approximate empirical formulas can be used to estimate
the
I
value in eV for an element with atomic number
Z
:
(5.24)
19.0 eV,
Z
=
1 (hydrogen)
I
=
(5.25)
11.2 + 11.7
Z
eV, 2
≤
Z
≤
13
52.8 + 8.71
Z
eV,
Z
> 13.
(5.26)
Since only the logarithm of
I
enters the stopping-power formula, values obtained
by using these formulas are accurate enough for most applications. The value of
I
for an element depends only to a slight extent on the chemical compound in which
