Tsoulfanidis N. Measurement and detection of radiation
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68
MEASUREMENT
AND
DETECTION
OF
RADIATION
2.19 METHODS OF
ERROR
REDUCTION
In every radiation measurement it is extremely important to perform it in such a
way that the result is determined with the minimum possible error. In general,
the first task of the investigator is to improve the counting apparatus by
reducing the background as much as possible. Actually, the important quantity is
the ratio b/g or b/r and not the absolute value of the background. Assuming
that all possible improvements of background have been achieved, there is a
procedure that, if followed, will result in a smaller error. Two such procedures
will be discussed below. In addition, a method will be presented for the
calculation of the counting time necessary to measure a counting rate with a
desired degree of accuracy.
2.19.1 The Background Is Constant and There Is No Time Limit for
Its Measurement
In this case, the background is measured for a long period of time to minimize
the error introduced by it, i.e.,
t,
is so long that
Example
2.23
Suppose one obtains the following data:
G
=
400
t,
=
5 min
B
=
100
t,
=
2.5 min
Then
If the background is constant, this result can be improved by counting back-
ground for a long period of time, e.g., 250 min. In that case, the result is
100
B
=
-
X
250
=
10,000 counts
t,
=
250 min
2.5
400 10,000
r=---
5 250
=
40 counts/min
STATISTICAL
ERRORS
OF
RADIATION
COUNTING
69
2.19.2
There Is a Fixed Time T Available for Counting Both Background
and Gross Count
In this case, the question is, What is the optimum time to be used for gross and
background counting? Optimum time results in minimum statistical error for
the net counting rate. The optimum time is determined as follows.
An
estimate of the counting rates at the time of the measurement is
obtained with a short count (not the final one). Assume that one obtained the
approximate counting rates
Then, from Eq. 2.96 and also using
G
=
gt,,
B
=
bt,,
The best times t, and
t,
are those that minimize
a,
or (a,)2 subject to the
constraint
t,
+
t,
=
T
=
constant (2.99)
Considering
a,2
as a function of
t,
and t,, the minimum will be found by
differentiating (a;I2 and setting the differential equal to zero:
Differentiating the constraint, Eq. 2.99, one finds
dt,
=
-dt,
Substituting this value of dt, into d(a;)' gives
Therefore, if there is a fixed time
T
for the measurement, the optimum counting
times are determined from the two equations
2.19.3
Calculation of the Counting Time Necessary to Measure
a
Counting Rate with a Predetermined Statistical Error
Assume that the net counting rate of a radioactive sample should be measured
with an accuracy of
a
percent, i.e.,
a;/r
=
a
percent. Also assume that
a
counting system is provided with a background counting rate
b
and standard
70
MEASUREMENT
AND
DETECTION
OF
RADIATION
error
a,.
Both
b
and
a,
have been reduced as much as possible for this system
and have been determined earlier. The task is to determine the counting time
t,
necessary to result in
a
percent standard error for the net counting rate. The
time
t,
is calculated as follows.
The net counting rate and its standard error are
Therefore
Equation 2.101 solved for
t,
gives
It is assumed that an approximate gross counting rate is known.
Example
2.24
How long should a sample be counted to obtain the net
counting rate with an accuracy of
1
percent? It is given that the background for
the counting system is 100
f
2
counts/min.
Answer
The first step is to obtain an approximate gross counting rate.
Assume that the sample gave 800 counts in 2 min. Then
g
=
800/2
=
400
counts/ min and, using Eq. 2.102,
400
t,
=
=
80 min
(400
-
100)~(0.01)~
-
22
Indeed, if one counts for 80 min, the error of
r
is going to be
2.19.4
Relative Importance of Error Components
In every measurement, the observer tries to reduce the experimental error as
much as possible. If the quantity of interest depends on many variables, each
with its own error, the effort to reduce the error should be directed toward the
variable with the largest contribution to the final error.
STATISTICAL
ERRORS
OF
RADIATION
COUNTING
71
Consider the quantity
Q
=
x
+
y
-
z
and assume
x
=
3,
y
=
2, and
z
=
1.
Also assume that the corresponding standard errors are
The standard error of
Q
is
From the relative magnitude of the errors, one can see that if it is necessary to
reduce the error further, the effort should be directed toward reduction of
a;
first,
u,
second, and
uz
third. In fact, there is no need to reduce
u-
further
before
a;
and
a,
reach the same magnitude as
a,.
2.20
MINIMUM
DETECTABLE ACTIVITY
The minimum detectable activity (MDA) is the smallest net count that can be
reported with a certain degree of confidence that represents a true activity from
a sample and is not a statistical variation of the background. The term MDA is
not universally acceptable. In the general case, in measurements not necessarily
involving radioactivity, other terms such as lowest detection limit have been
used. Here, the notation and applications will be presented with the measure-
ment of a radioactive sample in mind.
Obviously, MDA is related to low count rates. In such cases of low count
rates, the person who performs the experiment faces two possible errors.
WE
I
error:
To state that the true activity is greater than zero when, in fact, it
is zero. If this is a suspected contaminated item, the person doing the
measurement will report that the item is indeed contaminated when, in fact,
it is not. This error is called
false positive.
TYPE
11
error:
To state that the true activity is zero when, in fact, it is not.
Using the previous example, the person doing the measurement reports that
the item is clean when, in fact, it is contaminated. This error is called
false
negative.
The outcomes of radiation measurements follow Poisson statistics, which
become, essentially, Gaussian when the average is greater than about 20 (see
Sec. 2.10.2). For this reason, the rest of this discussion will assume that the
results of individual measurements follow a normal distribution and the confi-
dence limits set will be interpreted with that distribution in mind. Following the
72
MEASUREMENT
AND DETECllON
OF
RADIATION
notation used earlier,
B
=
background with standard deviation uB
G
=
gross signal with standard deviation
a,
n
=
G
-
B
=
net signal with standard deviation
an
=
(a:
+
a;)
1/2
When the net signal is zero (and has a standard deviation un
=
uo), a
critical detection limit
(CDL)
is defined in terms of uo with the following
meaning:
1.
A
signal lower than
CDL
is not worth reporting.
2. The decision that there is nothing to report has a confidence limit of
1
-
a,
where
a
is a certain fraction of the normalized Gaussian distribution (Fig.
2.12). Take as an example
a
=
0.05. Then
CDL
=
k,uo (2.103)
with k,
=
1.645 (see Table 2.2). If
n
<
CDL,
one decides that the sample is
not contaminated, and this decision has a 95% confidence limit.
The
MDA
should obviously be greater than the
CDL.
Keeping in mind that
the possible
MDA
values also follow a normal distribution, a fraction
P
is
0
CDL
0
CDL
MDA
Figure
2.12
The meaning of the critical detection limit
(CDL)
and minimum detectable activity
(MDA)
in terms of the confidence limits defined by
a
and
P.
STATISTICAL
ERRORS
OF
RADIATION
COUNTING
73
established, meaning that a signal equal to MDA is reported as a correct/true
signal with a confidence limit
1
-
P.
The value of MDA is given by
MDA
=
CDL
+
kpuD (2.104)
where
a,
is the standard deviation of MDA (Fig. 2.12). Again, if
/3
=
0.05,
k,
=
1.645.
In most cases, in practice,
a
=
fl
=
0.05; then the CDL and MDA are
defined with a 95% confidence limit. For radioactivity specifically, remember
that
u2
=
uz
+
ui
=
(u:
+
ui)
+
a;
(2.105)
and for
n
=
0, un
=
6
=
0 and u
=
uo
=
dm
=
AuB. Then, if the
95% confidence limit is applied
(a
=
P
=
0.05), CDL
=
1.645~~
=
2.326~~.
The value of MDA turns out to be (see Prob. 2.24)
MDA
=
k2
+
2CDL
=
2.71
+
4.653~~ (2.106)
Example
2.25
Consider the data of a single measurement to be
G
=
465
counts/min,
B
=
400/min. Assume that from previous measurements in that
counting system it has been determined that uB
=
lO/min. The assumption is
made that the background is constant. What does one report in this case?
Answer
The net count rate is
n
=
465
-
400
=
65 counts/min.
The
mini-
mum detectable activity is, from Eq. 2.106, MDA
=
2.71
+
4.653
X
10
=
49.2.
Since MDA
<
65, one reports, with a 95% confidence limit, that this sample is
radioactive.
In most cases, the second term of
Eq.
2.106 is much larger than the first,
and the MDA is taken as
MDA
=
4.653~~ (2.107)
In using
Eq.
2.106 or 2.107, the user should keep in mind the underlying
assumption of the 95% confidence limit. The numerical factors will change if
one chooses a different confidence limit.
2.21
COUNTER DEAD-TIME CORRECTION AND
MEASUREMENT OF DEAD TIME
Dead time, or resolving time, of a counting system is defined as the minimum
time that can elapse between the arrival of two successive particles at the
detector and the recording of two distinct pulses. The components of dead time
consist of the time it takes for the formation of the pulse in the detector itself
and for the processing of the detector signal through the
preamplifier-amplifi-
er-discriminator-scaler (or
preamplifier-amplifier-MCA).
With modern electron-
ics, the longest component of dead time is that of the detector, and for this
74
MEASUREMENT
AND
DETECTION OF RADIATION
reason, the term "dead time" means the dead time of the detector. The
dead-time component of the preamplifier-amplifier-discriminator-scaler can be
ignored with any type of detector.
Because of counter dead time, the possibility exists that some particles will
not be recorded since the counter will not produce pulses for them. Pulses will
not be produced because the counter will be "occupied" with the formation of
the signal generated by particles arriving earlier. The counting loss of particles is
particularly important in the case of high counting rates. Obviously, the ob-
served counting rate should be corrected for the loss of counts due to counter
dead time. The rest of this section presents the method for correction as well as
a method for the measurement of the dead time.
Suppose
7
is the dead time of the system and g the observed counting rate.
The fraction of time during which the system is insensitive is g7. If n is the true
counting rate, the number of counts lost is
n(g7). Therefore
and
Equation 2.108 corrects the observed gross counting rate g for the loss of
counts due to the dead time of the counter.
Example
2.26
Suppose
7
=
200 ps and
g
=
30,000 counts/min. What frac-
tion of counts is lost because of dead time? What is the true counting rate?
Answer
The true counting rate is
Therefore, dead time is responsible for loss of
555
-
500 55
-
555 555
-
10% of the counts
Notice that the product g~
=
0.10, i.e., the product of the dead time and the
gross counting rate, is a good indicator of the fraction of counts lost because of
dead time.
The dead time is measured with the "two-source" method as follows. Let
n,,n,, n,, be the true gross counting rates from the first source only, from the
second source only, and from both sources, respectively, and let n, be the true
background rate. Let the corresponding observed counting rates be
g,, g,,
g,,,
b.
STATISTICAL
ERRORS
OF
RADIATION
COUNTING
75
The following equation holds:
True net True net True net
(counting rate)
,+
(counting rate)
,=
(counting rate
or
n,
+
n,
=
n,,
+
n,
Using Eq. 2.108,
It will be assumed now that br
4
1, in which case,
(If b~ is not much less than 1, the instruments should be thoroughly checked for
possible malfunction before proceeding with the measurement.)
The dead time
T
can be determined from Eq. 2.109 after g,, g,, g,,, and b
are measured. This is achieved by counting radioactive source
1,
then sources
1
and 2 together, then only source 2, and finally the background after removing
both sources. Equation 2.109 can be rearranged to give:
Equation 2.110 is a second-degree algebraic equation that can be solved for
T.
It
was derived without any approximations.
If the background is negligible, Eq. 2.110 takes the form
glg2gl2~~
-
2g1g27
+
gl
+
g2
-
g12
=
0
(2.111)
Solving for
T,
When dead-time correction is necessary, the net counting rate, called "true
net counting rate," is given by
It is assumed that the true background rate has been determined earlier with
the standard error
ub.
The standard error of
r,
a,,
is calculated from Eq. 2.113
using Eq. 2.84.
If
the only sources of error are the gross count
G
and the
76
MEASUREMENT AND DETECTION OF RADIATION
background, the standard error of
r
is
If
there is an error due to dead-time determination, a third term consisting
of that error will appear under the radical of
Eq.
2.114.
PROBLEMS
2.1
What is the probability when throwing a die three times of getting a four in any of the throws?
2.2
What is the probability when drawing one card from each of three decks of cards that all three
cards will be diamonds?
2.3
A
box contains 2000 computer cards. If five faulty cards are expected to be found in the box,
what is the probability of findiag two faulty cards
in
a sample of 250?
2.4
Calculate the average and the standard deviation of the probability density function f(x)
=
l/(b
-
a)
when
a
I
x
I
b. (This pdf is used for the calculation to round off errors.)
2.5
The energy distribution of thermal (slow) neutrons in a light-wave reactor follows very closely
the Maxwell-Boltzmam distribution:
N(E) dE
=
~&e-~/~~ d~
where N(E) dE
=
number of neutrons with kinetic energy between E and
E
+
dE
k
=
Boltzmann constant
=
1.380662
X
10-23~/0
K
T
=
temperature,
K
A
=
constant
Show that
(a) The mode of this distribution is
E
=
ikT.
(b) The mean is
E
=
+k~.
2.6
If the average for a large number of counting measurements is 15, what is the probability that a
single measurement will produce the result 20?
2.7
For the binomial distribution, prove
N
(a)
pi:),
=
1 (b)
3
=pN (c) u2
=
m(l -p)
n=O
2.8
For the Poisson distribution, prove
2.9
For the normal distribution, show
rn
a
P
=
1 (b)
i
=
m
(c) the variance is
o2
2.10
If n,, n,,
.
.
.
,
n, are mutually uncorrelated random variables with a common variance
u2,
show that
N-1
(ni
-
Ti)
=
-
u
N
STATISTICAL ERRORS OF RADIATION COUNTING
77
2.11
Show that in a series of
N
measurements, the result
R
that minimizes the quantity
is
R
=
E,
where
7i
is given by Eq. 2.31.
2.12
Prove
Eq.
2.62 using tables of the error function.
2.13
As
part of a quality control experiment, the lengths
of
10 nuclear fuel rods have been measured
with the following results in meters:
What is the average length? What is the standard deviation of this series of measurements?
2.14
At a uranium pellet fabrication plant the average pellet density is 17
X
lo3
kg/m3 with a
standard deviation equal to
lo3
kg/m3. What is the probability that a given pellet has a density less
than 14
x
lo3
kg/m3?
2.15
A radioactive sample was counted once and gave 500 counts in 1 min. The corresponding
number for the background is 480 counts. Is the sample radioactive or not? What should one report
based on this measurement alone?
2.16
A
radioactive sample gave 750 counts in 5 min. When the sample was removed, the scaler
recorded 1000 counts in 10 min. What is the net counting rate and its standard percent error?
2.17
Calculate the average net counting rate and its standard error from the data given below:
2.18
A counting experiment has to be performed in
5
min. The approximate gross and background
counting rates are 200 counts/min and 50 counts/min, respectively.
(a) Determine the optimum gross and background counting times.
(b) Based on the times obtained in (a), what is the standard percent error of the net counting
rate?
2.19
The strength of a radioactive source was measured with a
2
percent standard error by taking a
gross count for time
t
min and a background for time
2t
min. Calculate the time
t
if it is given that
the background is 300 counts/min and the gross count 45,000 counts/min.
2.20
The strength of radioactive source is to be measured with a counter that has a background of
120
f
8
counts/min. The approximate gross counting rate is 360 counts/min. How long should one
count if the net counting rate is to be measured with an error of 2 percent?
2.21
The buckling
B2
of a cylindrical reactor is given by
where
R
=
reactor radius
H
=
reactor height
If the radius changes by
2
percent and the height by
8
percent, by what percent will
B2
change? Take
R
=
1
m,
H
=
2 m.