Physics and Engineering of Radiation Detection

9.11. Counting Statistics 567

If we consider only the counts in the region of interestNi, then the error in mea-
surement will be given by

δNi=

1

√

Ni

. (9.11.5)

Of course, the value ofδNiwill depend on our choice of the area of interest. As
the area approachesNt, the error inNiapproaches the error inNt. This becomes
obvious if we divide equation 9.11.5 by equation 9.11.4.

δNi

δNt

=

√

Nt

Ni

⇒δNi =

√

Nt

Ni

δNt (9.11.6)

It was mentioned earlier that most peaks encountered in spectroscopic measure-
ments are Gaussian-like and generallyFWTMis used to define the region of interest.
For a Gaussian peak theFWTMis given by^2

FWTM≈ 4. 29 σt. (9.11.8)

Dividing both sides byNtgives

σt

Nt

≡δNt=

FWTM

4. 29 Nt

. (9.11.9)

We now substitute this expression in equation 9.11.6 and obtain

δNi =

√

Nt

Ni

FWTM

4. 29 Nt

=

FWTM

4. 29

√

NiNt

, (9.11.10)

which can be written in a more convenient form as

δNi

√

Nt=

FWTM

4. 29

√

Ni

. (9.11.11)

If the quantity on the left hand side of the above equation approaches unity, the
error inNiwill approach the least possible spread inNi,thatisσt=

√

Nt.This
implies that for best measurement precision we should have

FWTM

4. 29

√

Ni

=1

⇒FWTM =4. 29

√

Ni. (9.11.12)

(^2) This is obtained by using the definition of Gaussian distribution as follows:
−x
2
2 σ^2
=^1
10
⇒x = σ
p
2 ln(10)
⇒FWTM =2x=σ 2
p
2 ln(10)≈ 4. 29 σ
(9.11.7)
The reader is encouraged to verify that, for a Gaussian peak,FWHM≈ 2. 35 σ.