Physics and Engineering of Radiation Detection

232 Chapter 4. Liquid Filled Detectors

Now, the electrons that survive local recombination, encounter impurity molecules
as they move toward the anode. This could result in the their parasitic capture by
the impurities. Ifμcis the capture coefficient of the liquid, then after moving a
distancerthe number of electrons that survive the capture are given by

N 2 =N 1 e−μcr. (4.3.3)

This, however, is an oversimplification of the actual situation since in reality the
capture coefficient is itself a function of the electron energy, which in turn is a
function of the electric field intensity. The above equation should then be replaced
by

N 2 =N 1 exp

(

−

∫r

a

μc(r)dr

)

, (4.3.4)

whereais the anode wire radius. Note that we have written the capture coefficient
as a function of position since it depends on the electric field intensity, which is a
function of position. It has been found that the capture coefficient varies approxi-
mately inversely with the electric field and can be written in a general form as (6)

μc=A+

B

E(r)

, (4.3.5)

where the constantsAandBdepend on the characteristics of the liquid and are
determined experimentally.
The expression for the electric field intensity in the above equations depends
on the geometry of the chamber. For parallel plate geometry the field is uniform
throughout the active volume except at the edges. But such a geometry is not
suitable for operation in proportional region (see also chapter on gas filled detec-
tors). The reason is that the high field intensity needed to initiate the avalanche
in a parallel plate chamber requires application of extremely high potentials at the
electrodes. In liquid filled detectors the situation is even more demanding due to
higher probabilities of electron recombination and capture as compared to gases.
Therefore to ensure avalanche multiplication one should resort to cylindrical geom-
etry. For a cylindrical chamber having radiusband anode wire radiusa, the electric
field intensity is given by

E(r)=

V

rln(b/a)

, (4.3.6)

whereV is the applied potential. Hence for a cylindrical proportional counter the
capture coefficient can be written as

μc=A+

B

V

rln

(

b

a

)

. (4.3.7)

Substitution of this expression into equation 4.3.4 yields

N 2 = N 1 exp

[

−

∫r

a

{

A+

B

V

rln

(

b

a

)}

dr

]

= N 1 exp

[

−(r−a)

{

A+

(r+a)B

2 V

ln

(

b

a

)}]

≈ N 1 exp

[

−r

{

A+

rB

2 V

ln

(

b

a

)}]

, (4.3.8)