Biomedical Engineering Reference
In-Depth Information
Fig. 5.5
Annular cylinder of length d
x
centered about path of heavy charged particle. See text.
For the minimum impact parameter, the analysis implies that the particles' posi-
tions remain separated by a distance
b
min
at least as large as their de Broglie wave-
lengths during the collision. This condition is more restrictive for the less massive
electron than for the heavy particle. In the rest frame of the latter, the electron
has a de Broglie wavelength
λ =
h
/
mV
, since it moves approximately with speed
V
relative to the heavy particle. Accordingly, we choose
h
mV
.
b
min
∼
(5.17)
Combining the relations (5.15), (5.16), and (5.17) gives the semiclassical formula
for stopping power,
4
π
k
0
z
2
e
4
n
mV
2
ln
mV
2
hf
-
d
E
(5.18)
d
x
=
.
We see that only the charge
ze
and velocity
V
of the heavy charged particle enter
the expression for stopping power. This fact is consistent with the universality of
charged-particle energy-loss spectra in sudden collisions, mentioned in Section 5.3.
For the medium, only the electron density
n
(appearing merely as a multiplicative
factor) and the orbital frequency
f
appear in (5.18). The quantity
hf
in the denom-
inator of the logarithmic term is to be interpreted as an average energy associated
with the electronic quantum states of the medium. This energy is well defined
in the quantum-mechanical derivation. The essential correctness of much of the
physics in Bohr's derivation was vindicated by the later quantum stopping-power
formula, to which we turn in the next section.
Example
Calculate the maximum and minimum impact parameters for electronic collisions
for an 8-MeV proton. To estimate the orbital frequency
f
, assume that it is about the
same as that of the electron in the ground state of the He
+
ion.
