Biomedical Engineering Reference
In-Depth Information
2
At 1 MeV, for example, we find in Table 5.2 that
β
= 0.00213
and
F
(
β
) = 7.69
;
therefore Eq. (5.37) gives
-
d
E
0.170
0.00213
(7.69 - 4.31)
270 MeVcm
-1
.
d
x
=
=
(5.38)
This is numerically equal to the value of the stopping power plotted in Fig. 5.6 for
water. The curves in the figure were obtained by such calculations.
5.10
Range
The range of a charged particle is the distance it travels before coming to rest. The
reciprocal of the stopping power gives the distance traveled per unit energy loss.
Therefore, the range
R
(
T
)
of a particle of kinetic energy
T
is the integral of this
quantity down to zero energy:
-
d
E
d
x
-1
T
R
(
T
)
=
d
E
.
(5.39)
0
Table 5.3 gives the mass stopping power and range of protons in water. The latter
is expressed in g cm
-2
; that is, the range in cm multiplied by the density of water
(
ρ
=
1 gcm
3
). Like mass stopping power, the range in g cm
-2
applies to all materials
of similar atomic composition.
Although the integral in (5.39) cannot be evaluated in closed form, the explicit
functional form of (5.33) enables one to scale the proton ranges in Table 5.3 to
obtain the ranges of other heavy charged particles in water. Inspection of Eqs. (5.33)
and (5.39) shows that the range of a heavy particle is given by an equation of the
form
z
2
T
0
d
E
G
(
1
R
(
T
)
=
β
)
,
(5.40)
β
)
depends only on the velocity
β
.
in which
z
is the
charge
and the function
G
(
Since
E
=
Mc
2
/
1-
β
2
, where
M
is the particle's rest mass, the variable of inte-
gration in (5.40) can be expressed as
d
E
=
Mg
(
β
, where
g
is another function
of velocity alone. It follows that Eq. (5.40) has the form
β
)d
z
2
β
0
g
(
β
)
G
(
M
M
z
2
f
(
β
=
R
(
β
)
=
β
)
d
β
),
(5.41)
where the function
f
(
)
depends only on the initial velocity of the heavy charged
particle. The structure of Eq. (5.41) enables one to scale ranges for different parti-
cles in the following manner. Since
f
(
β
)
is the same for two heavy charged particles
at the same initial speed
β
, the ratio of their ranges is simply
β
z
2
M
1
z
1
M
2
,
R
1
(
)
R
2
(
β
)
=
β
(5.42)
