Physics and Engineering of Radiation Detection

6.5. Photodetectors 387

gainμcan be written as

μ = δn

=

[

AVdyα

]n

= An

[

V

n+1

]αn

=

An

(n+1)αn

Vαn. (6.5.16)

HereV is the supply voltage applied between the anode and the cathode. This
voltage gets distributed to individual anodes through the voltage divider network
(cf. Fig.6.5.1). Here we have assumed that all the resistors across the dynodes have
the same value (hence we could useVdy=V/(n+ 1) in the above derivation). It
is apparent from this equation that the overall gain is proportional to the voltage
raised to the powerαn,thatis
μ∝Vαn,

For a tube having 10 dynodes,αntranslates into a number between 7 and 8. The
dependence of gain on voltage for a PMT having 10 dynodes as characterized by the
above relation is shown in Fig.6.5.16. Typical photomultiplier tubes are operated
between 1500V and 2000V with gains on the order of 10^5.

V (volts)

500 1000 1500 2000 2500

μ

10

102

103

104

105

106

Figure 6.5.16: Variation of gain with ap-

plied voltage for a PMT having 10 dyn-

odes.

Certainly the equation 6.5.16 is valid only for the case when all the dynodes have
equal gains. In case, individual dynodes have different gains, something that is not
uncommon in commercial PMTs, the overall gain should be computed from

μ=

∏n

i=1

δi, (6.5.17)