Biomedical Engineering Reference
In-Depth Information
Tab l e 5 . 4
Calculated Slowing-Down Rates, -d
E
/d
t
,and
Estimated Stopping Times
τ
for Protons in Water
Slowing-Down Rate
Estimated Stopping
Proton Energy T
-d
E
/d
t
Time
τ
(MeV s
-1
)
(MeV)
(s)
10
11
10
-12
0.5
4.19
×
1.2
×
10
11
10
-12
1.0
3.74
×
2.7
×
10
11
10
-11
10.0
2.00
×
5.0
×
10
10
10
-9
100.0
9.35
×
1.1
×
5.81 × 10
10
1.7 × 10
-8
1000.0
5.12
Limitations of Bethe's Stopping-Power Formula
The stopping-power formula (5.23) is valid at high energies as long as the inequality
γ
1
, mentioned before Eq. (5.7), holds (e.g., up to
∼
10
6
MeV for protons).
Other physical factors, not included in Bethe's theory, come into play at higher
energies. These include forces on the atomic electrons due to the particle's spin and
magnetic moment as well as its internal electric and magnetic structures (particle
form factors). Bethe's formula is also based on the assumption that the particle
moves much faster than atomic electrons. At low energies the formula (5.23) fails
because the term
ln
2
mc
2
m
/
M
2
/
I
eventually becomes negative, giving a negative value
β
for the stopping power.
In the low-energy region, also, a positively charged particle captures and loses
electrons as it moves, thus reducing its net charge and stopping power. Electron
capture becomes important when the speed
V
of the heavy particle is comparable
to or less than the speed that an electron needs in order to orbit about the particle
as a nucleus. Based on Eq. (2.9) of the discussion of Bohr's theory, the orbital speed
of an electron in the ground state about a nucleus of charge
ze
is
k
0
ze
2
/
h
.Thus,as
¯
a condition for electron capture and loss one has
k
0
ze
2
/
hV
1
. For electron capture
¯
by protons (
z
1), we see from Eq. (2.9) that
V
10
6
ms
-1
, corresponding to
=
=
2.2
×
a kinetic energy of ∼25 keV.
The dependence of the Bethe formula on
z
2
, the square of the charge of the
heavy particle, implies that pairs of particles with the same mass and energy but
opposite charge, such as pions,
π
±
, and muons,
µ
±
, have the same stopping power
and range. Departures from this prediction have been measured and theoretically
explained by the inclusion of
z
3
and higher powers of the charge in the stopping-
power formula. Bethe's formula is obtained by calculating the stopping power in
the first Born approximation in quantum mechanics. Successive Born approxima-
tions yield terms proportional to the higher powers
z
3
,
z
4
, and so on, of the incident
particle's charge.
