Physics and Engineering of Radiation Detection

5.1. Semiconductor Detectors 259

where

F=

x

i

(

wi

i

− 1

)

is called theFano factor. It is interesting to note here that even though we assumed
that the individual processes of ionization and excitations were Gaussian in nature,
but the spread in the output signal can be described by the Poisson process only if
we multiply it by another factor. The reason, of course, is that these processes are
not uncorrelated as required by a strictly Gaussian process. The Fano factor was
first introduced to explain the anomaly between the observed and expected variance
in the signal (14). The simple calculations we performed above do not produce
very accurate results and they were only meant to introduce the idea of the need
for introducing the Fano factor. For detailed calculations the interested reader is
referred to (2) or (43). The value of the Fano factor lies between 0 and 1: 0 for no
fluctuations and 1 for perfect Poisson process. It has been found that for germanium
and siliconF=0.1 gives satisfactory results.

Example:

Determine the relative statistical fluctuations in the number of charge pairs

produced in silicon if 2.5MeVof energy is deposited by the incident radiation.

Solution:

For silicon we havewi=3. 62 eV /charge-pair andF =0.1. The absolute

statistical fluctuations can be computed from substituting these values and

the deposited energy in equation 5.1.15. Hence we get

σi =

√

Fns=

√

F

Edep

wi

=

[

(0.1)

2. 5 × 106

3. 62

] 1 / 2

= 262.8 charge-pairs

IfN=Edep/wirepresents the mean number of charge pairs produced, the

corresponding relative fluctuations are

σi

N

=

σwi

Edep

× 100

=

(262.8)(3.62)

2. 5 × 106

× 100

≈ 0 .04%.

The amount of relative statistical fluctuations as computed above is a measure

of thephysicallimit of the system resolution. Of course in a semiconductor

detector there are a number of other sources of error that contribute to the

measurement error too and therefore the actual uncertainty is much larger.