Biomedical Engineering Reference
In-Depth Information
over that normally expected has been linked to the alpha-particle dose from inhaled
radon daughters.
Because of the statistical nature of energy losses by atomic collisions, all particles
of a given type and initial energy do not travel exactly the same distance before com-
ing to rest in a medium. This phenomenon, called range straggling, is discussed
in Section 7.6. The quantity defined by Eq. (5.39) provides the range in what is
called the
continuous-slowing-down approximation
,or
csda
. It ignores fluctuations of
energy loss in collisions and assumes that a charged particle loses energy contin-
uously along its path at the linear rate given by the instantaneous stopping power.
Unless otherwise indicated, we shall use the term “range” to mean the csda range.
For all practical purposes, the csda range at a fixed initial energy is the same as the
average pathlength that a charged particle travels in coming to rest. Heavy charged
particles of a given type with the same initial energy travel almost straight ahead in
a medium to about the same depth, distributed narrowly about the csda range. As
we shall see in the next chapter, electrons travel tortuous paths, with the result that
there is no simple relationship between their range and the depth to which a given
electron will penetrate.
4)
Electron transport is discussed more fully in the next two
chapters.
5.11
Slowing-Down Time
We can use the stopping-power formula to calculate the mean rate at which a heavy
charged particle slows down. The time rate of energy loss,
-d
E
/d
t
, can be expressed
in terms of the stopping power by using the chain rule of differentiation:
-d
E
/d
t
=
-(d
E
/d
x
)/(d
t
/d
x
) =
V
(-d
E
/d
x
)
, where
V
= d
x
/d
t
is the velocity of the particle. For a
proton with kinetic energy
T
= 0.5
MeV in water, for example, the rate of energy
loss is
-d
E
/d
t
=
10
11
MeV s
-1
.
A rough estimate can be made of the time it takes a heavy charged particle to
stop in matter, if one assumes that the slowing-down rate is constant. For a particle
with kinetic energy
T
, this time is approximately,
4.19
×
T
-d
E
/d
t
=
T
V
(-d
E
/d
x
)
.
(5.46)
τ
∼
For a 0.5-MeV proton in water,
τ
∼ (0.5 MeV)/(4.19 × 10
11
MeV s
-1
) = 1.2 × 10
-12
s.
Slowing-down rates and estimated stopping times for protons of other energies are
given in Table 5.4. Because, as seen from Fig. 5.6, the stopping power increases as
a proton slows down, actual stopping times are shorter than the estimates.
4
“Range” is sometimes used in the literature to
mean the depth of penetration for electrons.
