Physics and Engineering of Radiation Detection

4.5. Sources of Error in Liquid Filled Ionizing Detectors 239

Example:

Define the mean electron lifetimeτin terms of capture rate of electrons.

Solution:

We just saw that the electron concentration at any timetcan be computed

from equation 4.5.8, provided the impurity concentration does not change

with time. Let us Substitutingt=τfrom equation 4.5.9 into this equation to

determine the electron concentration after passage of one lifetime. This gives

Ce = Ce 0 e−^1

⇒

Ce

Ce 0

≈ 0. 37

This implies that after timeτ the surviving electron population is about

37% of the initial population. In other wordsτ is the time taken by the

impurity molecules to capture about 63% of electrons.

The reaction rate constantkis an important and widely used parameter to assess
the effect of impurities in detectors. It tells us how quickly orviolentlyan impurity
captures the free electrons in the liquid, and therefore gives us a measure to deter-
mine which impurities can be tolerated in a certain environment. Another advantage
of usingkis that it can be used to determine the lifetime of electrons.
The reaction rate constant can be determined from the following relation

k≈

∫∞

0

σ(E)f(E)dE, (4.5.10)

whereσ(E) is the energy dependent attachment cross section for electrons in the
liquid andf(E) is the distribution of electrons in the liquid. For most practical
purposes, we can safely assume that the electrons in a liquid can be described by
Maxwellian distribution even in the presence of high electric field. This is in contrast
to the case of gases where, as we saw in the previous chapter, the electrons do not
retain their Maxwellian distribution in the presence of high electric field. As a
reminder, the Maxwellian distribution is given by

f(E)=

2

√

πkBT

(

E

kBT

) 1 / 2

e−E/kBT, (4.5.11)

wherekBis the familiar Boltzmann’s constant andTis the absolute temperature.
The distribution assumes that the particles are in thermal equilibrium with the
surrounding. The average energy of the electrons in such a case is given by

=

3

2

kBT. (4.5.12)

This energy can be increased by the application of electric field across the electrodes,
something that is always done in liquid ionization and proportional chambers. As
stated earlier, this increase in the average energy does not significantly affect the
distribution of electrons and therefore the Maxwellian distribution can still be used
to determine the reaction rate constant.