Physics and Engineering of Radiation Detection

118 Chapter 2. Interaction of Radiation with Matter

With Energy Straggling

(Single Particle)

Without Energy Straggling

(Particle Beam)

Residual Energy

Stopping Power

Figure 2.4.6: Bragg curves for

a single particle (dotted line)

and a particle beam (solid line).

A beam of particles, which is

originally mono-energetic, as-

sumes a distribution as it travels

through matter due to energy

straggling.

electrons later in the chapter. The equations we will present there will be applicable
to heavy charged particles passing through thin absorbers as well.
It is interesting to note that the Bethe-Block formulas 2.4.14 and 2.4.15 can not
be used to describe the behavior of a single particle. Because of energy straggling,
which is a stochastic process, these formulas actually represent the average stopping
power.
Since this process is probabilistic in nature and the distribution is skewed specially
for thin absorbers, therefore it is natural to look for the most probable energy loss
as well. This parameter has higher relevance to detector calibrations than the mean
energy loss. The reason is that the tail of the distribution generally gets buried in
the background and it becomes difficult to define it.
A consequence of the energy straggling is shown in Fig.2.4.6. The shape of
the Bragg curve slightly changes specially at the end of the particle track. Two
differences are clearly visible: the Bragg peak is more profoundly rounded and the
curve has a small tail at the end.

2.4.E RangeandRangeStraggling

It is very tempting to try to compute the average distance a particle beam will
travel (also called range) in a medium by integrating the stopping power over the
full energy spectrum of the incident particles, such as,

R(T)=

∫T

0

[

−

dE

dx

]− 1

dE. (2.4.20)

However due to multiple Coulomb scattering , the trajectory of a charged parti-
cle in a medium is not a straight line. Rather the particle moves in small straight
line segments. This implies that the range of a beam of particles would show sta-
tistical fluctuation around a mean value. In analogy with the energy straggling
phenomenon, this fluctuation is termed asrange straggling. It should be noted
that there is a fundamental difference between loss of energy and range and their
corresponding statistical fluctuations: the energy loss and energy straggling are dif-
ferential quantities while range and range straggling are integral quantities. It is