Biomedical Engineering Reference
In-Depth Information
To carry out the integration, we let
t
= 0
represent the time at which the heavy
charged particle crosses the
Y
-axis in Fig. 5.4. Since
cos
θ
=
b
/
r
and the integral is
symmetric in time, we write
∞
2
∞
0
2
b
∞
0
cos
θ
r
2
b
r
3
d
t
d
t
(
b
2
+
V
2
t
2
)
3/2
d
t
=
=
-
∞
2
b
b
2
(
b
2
+
V
2
t
2
)
1/2
∞
t
2
Vb
.
=
0
=
(5.12)
Combining this result with (5.11) gives, for the momentum transferred to the elec-
tron in the collision,
2)
2
k
0
ze
2
Vb
p
=
.
(5.13)
The energy transferred is
p
2
2
m
=
2
k
0
z
2
e
4
mV
2
b
2
Q
=
.
(5.14)
In traversing a distance
d
x
in a medium having a uniform density of
n
electrons
per unit volume, the heavy particle encounters
2
πnb
d
b
d
x
electrons at impact pa-
rameters between
b
and
b
+d
b
, as indicated in Fig. 5.5. The energy lost to these
electrons per unit distance traveled is therefore
2
πnQb
d
b
. The total linear rate of
energy loss is found by integration over all possible energy loses. Using Eq. (5.14),
we find that
-
d
E
n
Q
max
Q
min
b
max
4
π
k
0
z
2
e
4
n
mV
2
4
π
k
0
z
2
e
4
n
mV
2
d
b
b
=
ln
b
max
d
x
=
2
π
Qb
d
b
=
b
min
.
(5.15)
b
min
Here the energy limits of integration have been replaced by maximum and min-
imum values of the impact parameter. It remains to evaluate these quantities ex-
plicitly.
The maximum value of the impact parameter can be estimated from the physical
principle that a quantum transition is likely only when the passage of the charged
particle is rapid compared with the period of motion of the atomic electron. We de-
note the latter time by
1/
f
, where
f
is the orbital frequency. The duration of the col-
lision is of the order of
b
/
V
. Thus, the important impact parameters are restricted
to values approximately given by
b
V
<
1
V
f
(5.16)
or
b
max
∼
.
f
2
If one assumes that a constant force
F
k
0
ze
2
/
b
2
(equal to that at the distance of
closest approach) acts on the electron for a
time
t
∼
b
/
V
, then it follows that the
momentum transferred is
p
∼
k
0
ze
2
/
Vb
.
This simple estimate differs by a factor of 2
from (5.13), which is exact within the
conditions specified.
=
Ft
∼
