Physics and Engineering of Radiation Detection

204 Chapter 3. Gas Filled Detectors

HereV 0 is the applied electric potential andE+is the electric field intensity due
to the sheath of positive charges. Using these conditions in equation 3.7.23, the
integration constants can be found to be

C 1 = − 2

V 0 E+

d^2

and (3.7.26)

C 2 =

1

2

[

V 0

d

+E+

] 2

. (3.7.27)

Hence equation 3.7.23 becomes

E=

[

− 4

V 0 E+

d^2

x+

(

V 0

d

+E+

) 2 ]^1 /^2

. (3.7.28)

The dependence of the space charge induced electric field strengthE+on the effective
electric field intensityEfor a parallel plate chamber ofd=2cmat 2.5cm,3.5cm,
and 4.99cmfrom the cathode has been plotted in Fig.3.7.2. The applied potential
is 500V. It is apparent that even an space charge induced electric field equal to 10%
of the applied electric field can decrease the effective field intensity to unacceptable
levels. Of course the value ofE+depends on the number of positive charges in the
space charge sheath and the effect will therefore depend on the number of charge
pairs being created by the incident radiation. Now since we know that the number of
charge pairs created depends on the energy deposited by the radiation therefore we
can intuitively conclude that the effect of space charge will become more and more
prominent as the energy and/or the intensity of the incident radiation increases.
Since, due to very slow mobility of the positive ions, it is almost impossible to
completely eliminate this effect, therefore the question that should be answered is
that how much space charge can be tolerated. This depends on the application and
the type of detector. In ionization chambers, where we have a plateau over a large
range of applied potentials (and therefore effective electric field intensities), a small
space charge effect is of not much significance. On the other hand, in a proportional
chamber one must be careful in keeping the space charge to a minimum since it
could lead to a decrease in the electric field intensity to lower than the avalanche
threshold.
Since the effect of space charge induced electric field intensity is not negligible
therefore we will now try to derive a relation for a simple case of monoenergetic
photons incident uniformly over the whole detector. From theuniformity of the flux
we mean that the incident flux can be factorized into two partsφ 0 (y, z, t)andφ(x, t)
such that only the second factor varies as the photons get absorbed in the detector.
Let us now suppose that we want to find the number of photons absorbed in an
element of thicknessδxat a distancex(cf. Fig.3.7.1). Ifx′is a distance in this
element, then according to the exponential law of photon absorption in matter, the
number of photons being absorbed per unit time inx′fromxis given by

φabs = φ 0 (y, z, t)φ(x, t)

= φ 0 (y, z, t)e−μρx

(

1 −e−μρx

′)

. (3.7.29)

Hereμandρare the mass absorption coefficient and density of the filling gas.
The first exponential term in the above equation represents the number of photons