Biomedical Engineering Reference
In-Depth Information
15.2
Dielectric formalism for the electronic energy loss
of swift projectiles
A swift projectile travelling through a solid interacts with the target electrons and
nuclei, which reduces gradually its energy, and affects its direction of motion as well
as its charge state.
For projectile energies in the range of a few keV to several MeV the energy
loss due to electronic processes is dominant, whereas the energy loss resulting from
nuclear collisions is negligible [
23
].
In this section we provide the fundamentals of the dielectric formalism to
evaluate the relevant magnitudes for the energy loss distribution of swift projectiles
in condensed matter. These magnitudes are the stopping power S and the energy loss
straggling
2
, which are related to the mean value and the variance of the energy
loss distribution, respectively. More detailed information on the foundations of the
dielectric formalism can be found in [
12
-
14
,
24
].
When a projectile moves inside a target it can vary its charge state by exchanging
(capturing or losing) electrons with the target, reaching an equilibrium charge state
after a few femtoseconds. As the energy loss depends on the charge state of the
projectile, we write the stopping power, S , and the energy loss straggling,
2
,as
a weighted sum over the corresponding magnitudes (S
Q
and
2
Q
, respectively) for
the different charge states Q of the projectile:
X
Z
1
X
Z
1
2
Q
2
Q
:
S
D
Q
S
Q
;
D
(15.1)
D
D
Q
0
Q
0
In the above expressions
Q
represents the probability to find the projectile (with
atomic number Z
1
) in a given charge state Q. When dynamic equilibrium is
attained,
Q
is equivalent to the projectile charge-state fraction, which depends
on the target material, the projectile nature and energy. The energy dependence of
Q
for protons .Z
1
1/ in liquid water is obtained by the parameterization to
experimental data given by [
25
].
Based on the first Born approximation, the dielectric formalism provides the
following expressions [
12
] for the stopping power, S
Q
, and the energy loss
straggling,
2
Q
, of a material for a projectile with mass M , kinetic energy E,and
charge state Q:
D
d!!Im
;
Z
1
2
Q
.k/
Z
k
p
2E=M
Me
2
E
dk
k
1
".k; !/
S
Q
.E/
D
(15.2)
0
0
Z
1
2
Q
.k/
Z
k
p
2E=M
d!!
2
Im
:
e
2
E
M
„
dk
k
1
".k; !/
2
Q
.E/
D
(15.3)
0
0
