Physics and Engineering of Radiation Detection

3.7. Sources of Error in Gaseous Detectors 203

+

Incident

Radiation

x

Anode Cathode

δx

d

Figure 3.7.1: Schematic showing the drift of pos-

itive charge cloud towards cathode in a parallel

plate chamber.

be written as

·E =

ρ+



(3.7.17)

J+ = ρ+v+ (3.7.18)

·J+ = 0 (3.7.19)

v+ = μ+E. (3.7.20)

HereJ+represents the current density of the charge flow, v+is the drift velocity of
the charges, andμ+is their mobility. These equations can be shown to yield

·

[

E

(

·E

)]

=0. (3.7.21)

For our case of parallel plate geometry this equation can be written in one dimension
as
d
dx

[

E

(

dE

dx

)]

=0. (3.7.22)

This is justified if we assume that the distance between the electrodes is far less than
the length and width of the electrodes. The correct way to say this is that we have
assumed the electrodes to be infinitely long and wide, thus eliminating any need to
consider the non-uniformity of electric field intensity at the ends of the chamber. In
real detectors the electrodes are not that big but with proper designing (for example
by using end rings) on can ensure that the edge effects are minimal. The solution
to the above equation is
E=[2(C 1 x+C 2 )]^1 /^2 , (3.7.23)

whereC 1 andC 2 are constants of integration. To determine these constants we use
the following initial conditions.

E = −

V 0

d

−E+ atx= 0 (3.7.24)

E =

V 0

d

−E+ atx=d (3.7.25)