Physics and Engineering of Radiation Detection

568 Chapter 9. Essential Statistics for Data Analysis

Since it is not a requirement thatFWTMbe used to define the region of interest,
we can write the above equation in a general form as

w=

√

Ni, (9.11.13)

wherewis the width of the peak at the bottom of the region of interest and the
factordepends on the definition of the region of interest. We now want to write
equation 9.11.6 in terms ofFWHM, which is a commonly measured quantity in
spectroscopy. For that, we first note that

FWHM≈ 2. 35 σt. (9.11.14)

Dividing both sides byNtgives

σt

Nt

≡δNt=

FWHM

2. 35 Nt

. (9.11.15)

Substituting this expression in equation 9.11.6 gives

δNi =

√

Nt

Ni

FWHM

2. 35 Nt

=

FWHM

2. 35

√

NiNt

(9.11.16)

⇒δNi

√

Nt =

FWHM

2. 35

√

Ni

. (9.11.17)

As in the case ofFWTM, for the best precision in the measurement ofNi,theright
hand side of the above equation should approach unity, that is

FWHM

2. 35

√

Ni

=1

⇒FWHM =2. 35

√

Ni. (9.11.18)

Dividing equation 9.11.13 by equation 9.11.22 gives

w

FWHM

=



2. 35

⇒w =



2. 35

FWHM. (9.11.19)

If we define the region of interest using full width at tenth maximum, then=4. 29
and the above equation becomes

w≈ 1. 82 FWHM. (9.11.20)

In general, it is recommended that the region of interest is chosen such that the
widthwis approximately twice theFWHM.
Up until now we have not taken the background into consideration, which of
course is not a very realistic situation. In spectroscopic measurements, one generally
finds peaks embedded on a background. Fig.9.11.2 shows such a peak. It is apparent
that the background adds uncertainties to the measurements, which are not only
due to channel-by-channel variations but also statistical fluctuations in background