Physics and Engineering of Radiation Detection

168 Chapter 3. Gas Filled Detectors

calledPaschen minimum, which is the voltage below which the breakdown is not
possible.
An expression for the Paschen minimum can be derived by differentiatingVbreak
with respect toPdand equating the result to zero (see example below).

(Pd)min=

e

A

ln

[

1+

1

γ

]

(3.3.13)

Here (Pd)minis the value at whichVbreakis minimum andeis the natural logarithm
number.

Pd (Torr cm)

1 10 102

V (volts)

100

200

300

400

500

600

700

Paschen Minimum

Figure 3.3.3: Paschen curve for he-

lium enclosed between two parallel

plate electrodes separated by distance

d. The gas pressure isPand the sec-

ond Townsend coefficient has been ar-

bitrarily chosen to be 0.1.

Example:

Derive equation 3.3.13 for Paschen minimum.

Solution:

We will start with equation 3.3.12 for the breakdown voltage.

Vbreak=

BPd

ln [AP d/ln (1 + 1/γ)]

To simplify the mathematical manipulations with respect toPdwe first write

this equation as

Vbreak =

BPd

ln [A/ln (1 + 1/γ)] + ln(Pd)

.

orVbreak =

Bu

ln(v)+ln(u)

.