Physics and Engineering of Radiation Detection

206 Chapter 3. Gas Filled Detectors

e is the electronic charge,

Eγ is the photon energy,

W is the energy needed to create an electron-ion pair,

v+ is the drift velocity of the ions,

h is the height of the space charge sheath, and

b is the breadth of the space charge sheath.

After integration the above equation yields

Q=

eEγN 0

Wv+

e−μρx

[

δx−

e−μρx

μρ

(

1 −e−μρδx

)]

, (3.7.32)

where

N 0 =

∫h

y=0

∫b

z=0

φ 0 (y, z, t)dydz.

To simplify this relation we make a very valid assumption that the mean free path
of the photons is much larger than the elemental lengthδx,thatis

λγ=

1

μρ

>> δx.

In this case the above relation for the space charge becomes

Q=

eEγN 0

Wv+

e−μρxδx

(

1 −e−μρx

)

. (3.7.33)

Now that we know the amount of charge in the sheath of space charge, we can
compute the electric field intensity due to the whole space charge by using the
Gauss’s law. This law states that the net electric field intensity from a closed surface
is drivable from the amount of chargeQenclosed by that surface, that is
∮
E·nds=Q


, (3.7.34)

whereis the permeability of the medium. From the beginning we have assumed
that the variation in the number of charges is only inxdirection. We can now extend
this assumption to conclude that the components of the electric field intensity iny
andzdirections can be taken to be constant. Hence application of the Gauss’s law
on our case withQgiven by equation 3.7.33 yields

Eδx

[

2

∫h

y=0

∫b

z=0

dydz

]

=

eEγN 0

Wv+

e−μρxδx

(

1 −e−μρx

)

. (3.7.35)

The factor 2 on the left hand side of this equation has been introduced to account
for the fact that we have to integrate over both the left and the right surfaces out of