Biomedical Engineering Reference
In-Depth Information
ping power for heavy charged particles. Because quantum mechanics had not yet
been discovered, Bohr relied on intuition and insight to obtain the proper semiclas-
sical representation of atomic collisions. He calculated the energy loss of a heavy
particle in a collision with an electron at a given distance of passing and then aver-
aged over all possible distances and energy losses. The nonrelativistic formula that
Bohr obtained gave the correct physical features of stopping power as borne out by
experiment and by the later quantummechanical theory of Bethe. We present here
a derivation along the lines of Bohr.
In Fig. 5.4 we consider a heavy particle (charge
ze
and velocity
V
) that travels
swiftly past an electron (charge
-
e
and mass
m
) in a straight line at a distance
b
,
called the
impact parameter
. We assume that the electron is initially free and at rest
at the origin of the
XY
coordinate system shown. We assume, further, that the col-
lision is sudden: it takes place rapidly and is over before the electron moves appre-
ciably. Perpendicular components
F
x
and
F
y
of the Coulomb force
F
k
0
ze
2
/
r
2
that
the particle exerts on the electron at a given instant are shown in Fig. 5.4. With the
approximation that the electron remains stationary, the component
F
x
transfers no
net momentum to it over the duration of the collision. (This component acts sym-
metrically, first toward the left and then toward the right, its net effect being zero).
The charged particle transfers momentum to the electron through the action of the
other, perpendicular, force component
F
y
. The total momentum imparted to the
electron in the collision is
=
∞
∞
k
0
ze
2
∞
-
θ
cos
(5.11)
p
=
F
y
d
t
=
F
cos
θ
d
t
=
d
t
.
r
2
-
∞
-
∞
∞
Fig. 5.4
Representation of the sudden collision of a heavy
charged particle with an electron, located at the origin of
XY
coordinate axes shown. See text.
