Physics and Engineering of Radiation Detection

3.4. Ionization Chambers 173

In the short circuit condition, this total energy should be equal to the total of kinetic
and potential energy inside the chamber volume, that is

Utotal = Uch+Tp+Tn

⇒

1

2

CV 02 =

1

2

CVnp^2 +N 0 eEvnt+N 0 eEvpt. (3.4.2)

The above equation can be rearranged to give an expression for the effective potential
Vef f=V 0 −Vnpas follows.

(V 0 −Vnp)(V 0 +Vnp)=

2 N 0 V 0 e

Cd

(vp+vn)t

⇒Vef f 

N 0 e

Cd

(vp+vn)t (3.4.3)

Here we have used the approximation

V 0 +Vnp≈ 2 V 0.

Since the electrons move much faster than ions (vn>> vp), therefore initial pulse
shape is almost exclusively due to electron motion. If we assume that the charge pairs
are produced at a distancexfrom the anode (see Fig.3.4.3), then the electrons will
taketn=x/vnto reach the anode. This will inhibit a sharp increase in pulse height
with the maximum value attained when all the electrons have been collected by the
anode. The ions, owing to their heavier mass, will keep on moving slowly toward
cathode until timetp=(d−x)/vp, increasing the pulse height further, however at a
much lower rate. The maximum voltage is reached when all the charges have been
collected. Based on these arguments, we can rewrite the expression for the output
pulse time profile for three distinct time periods as follows.

Vef f

N 0 e

Cd(vp+vn)t :0≤t≤tn

N 0 e

Cd(vp+x)t : tn≤t≤tp

N 0 e

C : t≥tp

(3.4.4)

The above equation is graphically depicted in Fig.3.4.4. However an actual pulse
measured through some associated electronic circuitry differs from the curve shown
here due to the following reasons.

An actual voltage readout circuit has a finite time constant.


The charge pairs are not produced in highly localized areas.

In the above derivation leading to the pulse profile of Fig.3.4.4 we have not
considered the effect of the inherenttime constantof the detector and associated
electronics on the pulse shape. Time constant is simply the product of resistance and
capacitance of the circuit (τ=RC). Every detector has some intrinsic capacitance
as well as the cable capacitance. These capacitances together with the installed
capacitor (if any) and load resistance of the output make up the effective time
constant of the circuit. The difference between this time constant and the charge