Physics and Engineering of Radiation Detection

176 Chapter 3. Gas Filled Detectors

r 0

−+

Cathode

a

Anode

Figure 3.4.7: Production of an electron ion

pair in a cylindrical ionization chamber at a

distance ofr 0 from the center of the anode

wire.

this can also be written as

U=

1

2

lClV 02. (3.4.9)

Differentiating both sides of this equation gives

dU=lClV 0 dV. (3.4.10)

Equations 3.4.8 and 3.4.10 can be equated to give

dV=−

Q

2 πl

1

r

dr. (3.4.11)

Let us now suppose that a charge pairs are produced at a radial distancer 0 (see
Fig.3.4.7). The electrons and ions thus produced move in opposite directions under
the influence of the electric potential. The change in potential due to the movement
of electrons having total charge−Qcan be computed by integrating the above
equation froma+r 0 toa. Hence we have

V− =

Q

2 πl

∫a

a+r 0

1

r

dr

= −

Q

2 πl

ln

[

a+r 0

a

]

. (3.4.12)

Similarly, since the ions having the total chargeQmove froma+r 0 tob, therefore
the change in potential caused by them can be calculated as

V+ =

Q

2 πl

∫b

a+r 0

1

r

dr

= −

Q

2 πl

ln

[

b

a+r 0

]

. (3.4.13)