238 Chapter 4. Liquid Filled Detectors
Substitutingλc=xin equations 4.5.4 and 4.5.5 gives us the definition of the capture
mean free path as
Pc =1−e−^1 ≈ 0. 63
Ps = e−^1 ≈ 0. 37 ,that is, it represents the distance for which the capture and survival probabilities of
an electron is about 63% and 37% respectively. In other words, if a population of
electrons travels a distanceλcthen only about 37% of the electrons will survive the
capture by molecules.
Up until now we have used the spatial variation of the electron flux due to the
capture process. We can also describe the change in electron concentration with
respect to time. For this, we argue that the rate of electron capture should be pro-
portional to not only the impurity concentration but also the electron concentration
in the liquid. In fact this has been observed by various experimenters. We can
therefore write the rate equation as
dCe
dt=−kCeCimp, (4.5.7)whereCstands for concentration (generally described innumber of particles per
mole of the liquid) with subscriptseandimprepresenting electrons and impurity
molecules respectively. kis the constant of proportionality that depends on the
type of impurity. This so calledreaction rate constantcan generally be found in
literature in units of per mole per second (i.e.M−^1 s−^1 ). The negative sign in the
above expression describes the decrease in electron population with time.
If we now assume that the concentration of impurity in the liquid does not change
with time then equation 4.5.7 can be solved to give
Ce=Ce 0 e−kCimpt, (4.5.8)where we have used the initial conditionCe=Ce 0 att= 0. It should be noted
that the constancy ofCimpwith time is not a strictly valid assumption specially for
trapping impurities since the loss of electrons also means loss of impurity molecules.
The reason is that most molecular ions thus formed loose their ability to capture
more electrons. However, since in liquid filled detectors that use high purity liquids
the capture rate is generally quite low, we can safely ignore the time dependence
of impurity concentration. For the reversible attachment impurity molecules this
assumption holds up to a good approximation. Such impurities release the captured
electron after a small time delay, which at most results in the longer transit time of
electrons and increases the slope of the rising edge of the output pulse by a small
amount. The signal height, however, is not affected.
Using equation 4.5.8 we can define themean electron lifetimeτas
τ=1
kCimp. (4.5.9)
τrepresents the time it takes an electron population to decrease by about 63% (see
Example below).