Tsoulfanidis N. Measurement and detection of radiation
Подождите немного. Документ загружается.
288
MEASUREMENT AND DETECTION OF RADIATION
obtained over time t,, the true net counting rate (Eq. 2.113) is
where
r
is the counter dead time. Usually the objective of the measurement is
to obtain the source strength
S
using a detection system of known
R,
F,
and
E.
Combining Eqs. 8.21 and 8.22, the source strength becomes
The error in the value of
S
is due to errors in the values of
R,
F,
E,
and the
statistical error of
r.
In many cases encountered in practice, the predominant
error is that of
r.
Then one obtains, from Eq. 8.23,
That is, the percent error of
S
is equal to the percent error of the true net
counting rate
r.
Example
8.4
The geometric setup shown in Fig. 8.22 was used for the
measurement of the strength of the radioactive source. The following data were
obtained:
G
=
6000
B
=
400
r
=
100 ps
E
=
0.60
f
0.005
t,
=
10 min t,
=
10 min
F
=
1
+
0.001
What is the strength
S
and its standard error?
Answer
The true net counting rate
r
is
Source
-
0.1
0
rn
I
Figure
8.22
Geometry
assumed
in
Example
8.4.
RELATIVE
AND
ABSOLUTE
MEASUREMENTS
289
The standard error of
r
is
(Eq.
2.114)
Therefore
The solid angle is
Using
Eq.
8.23,
r
s=-=
561
=
96,392 part./min
ln
FE
(0.0097)(0.60)
The standard error
of
S
is (Sec. 2.15.1)
PROBLEMS
8.1
Show that if
R,/d
<
1
and
R,/d
<
1,
the solid angle between two parallel disks with radii
R,
and
R,
a distance
d
apart is given to a good approximation by
where
J,
=
R,/d
and
o
=
R,/d.
8.2
Show that an approximate expression for the solid angle between two nonparallel disks is
where
0
is the angle between the planes of the two disks, and
lJJ
and
w
are defined as in Prob.
8.1.
290
MEASUREMENT
AND
DETECTION OF RADIATION
83
Show that the solid angle between a disk source and a detector with a rectangular aperture is
given, approximately, by
Eq.
8.13 under the conditions given in Sec. 8.2.6.
8.4
A 1-mCi point isotropic gamma source is located 0.10 m away from a 60" spherical shell of a
NaI detector, as shown in the figure below. Assuming that all the pulses at the output of the
photomultiplier tube are counted, what is the counting rate of the scaler? The gamma energy is 1.25
MeV.
8.5
Calculate the counting rate for the case shown in the figure below. The source has the shape of
a ring and emits
lo6
part./s isotopically. The background is zero. The detector efficiency is 80
percent, and
F
=
1.
8.6
Calculate the self-absorption factor for a
14c
source that has a thickness of 10 pg/cm2
kg/m2);
Em,,
=
156 keV.
8.7
An
attempt was made to measure the backscattering factor by placing foils of continuously
increasing thickness behind the source and observing the change in the counting rate. The foils were
of the same material as the source backing. The results of the measurements are given in the table
below. Calculate the saturation backscattering factor and the source backscattering factor.
Thickness
behind source
(rnm)
Counting
rate
(counts/rnin)
--
0.1
(Source
backing
only)
3015
0.15 3155
0.2 3365
0.25 3400
0.3 3420
0.35 3430
0.4 3430
8.8
What is the counting rate in a detector with a rectangular aperture measuring
1
mm
x
40
mm, if
a 1-mCi gamma-ray point isotropic source is 0.10 m away? The efficiency of the detector for these
gammas is 65 percent.
8.9
A radioactive source emits electrons isotropically at the rate of
lo4
electrons/s. A plastic
scintillator having the shape of a cylindrical disk with a 25-mm-radius is located 120 mm away from
the source. The efficiency of the detector for these electrons is 95 percent. The backscattering factor
is 1.02, and the source self-absorption factor is 0.98. Dead time of the counting system is 5 ps. How
long should one count, under these conditions, to obtain the strength of the source with a standard
error of
5
percent? Background is negligible. The only error involved is that due to counting
statistics.
RELATIVE
AND
ABSOLUTE MEASUREMENTS
291
8.10
How would the result of Prob.
8.9
change if the backscattering factor was known with an error
of
1
percent, the efficiency with an error of
f
0.5
percent, and the source self-absorption factor
with an error of
+
1
percent?
8.11
Calculate the strength of a point isotropic radioactive source if it is given that the gross
counting rate is
200
counts/min, the background counting rate is
25
counts/min, the counter
efficiency is
0.90,
the source detector distance is
0.15
m, and the detector aperture has a radius of
20
mm
(F
=
1).
What is the standard error of the results if the error of the gross counting rate is
known with an accuracy of
*5
percent and the background with
t3
percent? Dead time is
1
ps.
8.12
A point isotropic source is located at the center of a hemispherical
297
counter. The efficiency
of this detector for the particles emitted by the source is
85
percent. The saturation backscattering
factor is
1.5.
The background is
25
+
1
counts/min. What is the strength of the source if
3000
counts are recorded in
1
min? What is the standard error of this measurement?
REFERENCES
1.
Masket, A. V.,
Reu.
Sci.
Instrum.
28:191 (1957).
2.
Ruffle,
M.
P.,
Nucl.
Instrum. Meth.
52:354 (1967).
3.
Ruby,
L.,
and Rechen,
J.
B.,
Nucl.
Instrum. Meth.,
58:345 (1968).
4.
Gotoh,
H.,
and Yagi, H.,
Nucl.
Instrum. Meth.
96:485 (1971).
5.
Gardner, R. P., and Verghese, K.,
Nucl.
Instrum. Meth.
93:163 (1971).
6.
Verghese,
K.,
Gardner,
R.
P. and Felder,
R.
M.,
Nucl.
Instrum. Meth.
101:391 (1972).
7.
Williams,
I.
R.,
Nucl.
Instrum. Meth.
44:160 (1966).
8.
Green, M. V., Aamodt,
R.
L.,
and Johnston, G. S.,
Nucl.
Instrum. Meth.
117:409 (1974).
9.
Wielopolski,
L.,
Nucl.
Instrum. Meth.
143:577 (1977).
10.
Beam, G. B., Wielopolski,
L.,
Gardner, R.
P.,
and Verghese,
K,
Nucl.
Instrum. Meth.
154501
(1978).
11.
Tabata,
T.,
Ito, R., and Okabe, S.,
Nucl.
Instrum. Meth.
94509 (1971).
12.
Kuzminikh, V.
A.,
and Vorobiev, S.
A.,
Nucl.
Instrum. Meth.
129561 (1975).
13.
Deruytter,
A.
J.,
Nucl.
Instrum. Meth.
15164 (1962).
14.
Walker,
D.
H.,
Int.
J.
Appl.
Rad. Isot.
16:183 (1965).
15.
Hutchinson,
J.
M.
R.,
Nass, C. R., Walker, D.
H.,
and Mann,
W.
B.,
Int.
J.
Appl.
Rad. hot.
19:517 (1968).
16.
Waibel, E.,
Nucl.
Instrum. Meth.
74:236 (1969).
17.
Waibel,
E.,
Nucl.
Instrum. Meth.
86:29 (1970).
18.
Waibel,
E.,
Nucl.
Instrum. Meth.
131:133 (1975).
19.
Bell, P.
R.,
in
K.
Siegbahn (ed.),
Beta and Gamma-Ray Spectrometry,
Interscience Publishers,
New York,
1955,
Chap.
5.
20.
Heath,
R.
L.,
"Scintillation Spectrometry-Gamma-Ray Spectrum Catalogue,"
IDO-16880-1,
1964.
21.
Nakamura, T.,
Nucl.
Instrum. Meth.
86:163 (1970).
22.
Ruby,
L.,
Nucl.
Instrum. Meth.
337:531 (1994).
CHAPTER
NINE
INTRODUCTION TO SPECTROSCOPY
9.1
INTRODUCTION
Spectroscopy is the aspect of radiation measurements that deals with measuring
the energy distribution of particles emitted by a radioactive source or produced
by a nuclear reaction.
This introduction to spectroscopy is complemented by Chap.
11,
which
discusses methods of analysis of spectroscopic data, and Chaps.
12-14,
which
present details on spectroscopy of photons, charged particles, and neutrons. This
chapter discusses the following broad subjects:
1.
Definition of differential and integral spectra
2.
Energy resolution of the detector
3.
The function of a multichannel analyzer
(MCA)
9.2
DEFINITION OF ENERGY SPECTRA
A
particle energy spectrum is a function giving the distribution of particles in
terms of their energy. There are two kinds of energy spectra, differential and
integral.
The
differential energy spectrum,
the most commonly studied distribution, is
also known as an energy spectrum. It is a function
n(E)
with the following
294
MEASUREMENT
AND
DETECTION OF RADIATION
meaning:
n(E) dE
=
number of particles with energies between
E
and
E
+
dE
or
n( E)
=
number of particles with energy E per unit energy interval
The quantity n(E) dE is represented by the cross-hatched area of Fig. 9.1.
The integral
energy spectrum is a function N(E), where N(E) is the number
of particles with energy greater than or equal to
E.
The quantity N(E) is
represented by the hatched area of Fig. 9.1.
The
integral energy spectrum N(E)
and the differential energy spectrum n(E) are related by
The two examples that follow illustrate the relationship between a differential
spectrum and an integral spectrum.
Example
9.1
Consider a monoenergetic source emitting particles with en-
ergy
E,.
The differential energy spectrum n(E) is shown in Fig. 9.2. Since there
are no particles with energy different from E,, the value of n(E) is equal to
zero for any energy other than E
=
E,.
The corresponding integral spectrum N(E) is shown in Fig. 9.3. It indicates
that there are no particles with E
>
E,. Furthermore, the value of N(E) is
constant for
E
I
E,, since all the particles have energy
E,
and only those
particles exist. In other words,
N(E,)
=
number of particles with energy greater than or equal to E,
=
N(E,)
=
number of particles with energies greater than or equal to E, (Fig. 9.3)
Figure
9.1
A
differential energy spectrum. The quantity
n(E)
dE
is equal to the number
of
particles
between
E
and
E
f
dE
(cross-hatched area).
INTRODUCITON TO SPECTROSCOPY
295
1
E
Figure
9.2
A
monoenergetic spec-
€0
trum.
All
particles have energy
E,,.
Example
9.2
Consider the energy spectrum shown in Fig. 9.4. According to
this spectrum, there are 10 particles per MeV at 11, 12, and 13 MeV. The total
number of particles is 30. The integral spectrum is shown in Fig.
9.5.
Its values
at different energies are
N(14)
=
0
no particles above
E
=
14 MeV
N(13)
=
10 10 particles at
E
=
13
MeV and above
N(12)
=
20
20 particles at
E
=
12 MeV and above
N(11)
=
30 30 particles at
E
=
11 MeV and above
N(10)
=
30
30
particles at
E
=
10 MeV and above
N(0)
=
30
30 particles above
E
=
0
The determination of energy spectra is based on the measurement of
pulse-height spectra, as shown in the following sections. Therefore, the defini-
tions of differential and integral spectra given in this section in terms of energy
could be expressed equivalently in terms of pulse height. The relationship
between particle energy and pulse height is discussed in
Sec.
9.5.
9.3
MEASUREMENT
OF
AN
INTEGRAL SPECTRUM WITH A
SINGLE-CHANNEL ANALYZER
Measurement of an integral spectrum means to count all particles that have
energy greater than or equal to a certain energy
E
or, equivalently, to record all
particles that produce pulse height greater than or equal to a certain pulse
height
V.
A device is needed that can sort out pulses according to height. Such
a
device is a
single-channel analyzer
(SCA) operating as a discriminator (integral
mode). If the discriminator is set at
V,
volts, all pulses with height less than
Vo
will be rejected, while all pulses with heights above
Vo
will be recorded.
Therefore, a single discriminator can measure an integral energy spectrum. The
measurement proceeds as follows.
296
MEASUREMENT AND DETECTION OF RADIATION
1
I
I
I
Figure
9.3
The integral spectrum of
a
0
EI
E
0
monoenergetic source.
Consider the differential pulse spectrum shown in Fig.
9.6
for which all
pulses have exactly the same height Vo. To record this spectrum, one starts with
the discriminator threshold set very high (higher than VJ and then lowers the
threshold by a certain amount AV (or
AE) in successive steps. Table 9.1 shows
the results of this measurement, where
N(V) is the number of pulses higher
than or equal to V.
A
plot of these results is shown in Fig. 9.7.
9.4
MEASUREMENT
OF
A DIFFERENTIAL SPECTRUM WITH A
SINGLE-CHANNEL ANALYZER (SCA)
Measurement of a differential energy spectrum amounts to the determination of
the number of particles within a certain energy interval A
E
for several values of
energy; or, equivalently, it amounts to the determination of the number of
pulses within a certain interval AV, for several pulse heights. A SCA operating
in the differential mode is the device that is used for such a measurement.
If the lower threshold of the SCA is set at
Vl (or El) and the window has a
width AV (or AE), then only pulses with height between Vl and Vl
+
AV are
recorded. All pulses outside this range are rejected. To measure the pulse
spectrum of Fig. 9.6, one starts by setting the lower threshold at Vl, where
Vl
>
Vo, with a certain window AV (e.g., AV
=
0.1
V)
and then keeps lowering
the lower threshold of the SCA. Table
9.2
shows the results of the measurement,
where n(V) AV is the number of pulses with height between V and V
+
AV.
Figure
9.8
shows these results. It is assumed that the width is AV
=
I/:
-
V;,
,,
INTRODUCTION TO
SPECTROSCOPY
297
I
I
0
v
A
1111
I
E,
MeV
9
10 11 12 13
14
Figure
9.5
The integral spectrum corresponding to that of Fig.
9.4.
vo
Figure
9.6
A
differential pulse spectrum consisting of pulses with the same height
V,.
Table
9.1
Measurement
of
Integral Spectrum
Discriminator
Threshold
"3
C',
<
v,
"5
v
6
v;
<
Vo