Biomedical Engineering Reference
In-Depth Information
Although we have been discussing the number of disintegrations of a radio-
nuclide, the results can also be applied to the number of counts registered from
disintegrations in an experiment. Unless the efficiency
of a counter is 100%, the
number of counts will be less than the number of disintegrations. With
N
atoms
initially present, the probability that a given atom will disintegrate in time
t
and be
registered as a count is, in place of Eq. (11.2),
p
*
(1-e
-
λ
t
).
=
prob. of a count
=
p
=
(11.19)
The probability that the given atom will not give a count, either by not decaying or
by decaying but not being registered, is
q
*
=
prob. of no count
= 1-
p
*
= 1-
+
e
-
λ
t
.
(11.20)
The formalism developed thus far for the number of disintegrations can be applied
to the number of counts by using
p
*
and
q
*
in place of
p
and
q
. The binomial distrib-
ution function
P
n
then applies to the number of counts, rather than disintegrations,
obtained in time
t
.If
=
1, then Eqs. (11.19) and (11.20) are identical with (11.2)
and (11.1), respectively. Also, one can divide the number of disintegrations by the
time
t
and then apply the above formalism to obtain the average disintegration and
count
rates
over the observation time.
Example
An experimenter repeatedly prepares a large number of samples identical to that in
the example from the last section:
N
10 atoms of
42
Kattime
t
0. He does not
know how much activity is initially present, but he wants to estimate it by determin-
ing the mean number of disintegrations that occur in a given time. To this end, each
new sample is placed in a counter, having an efficiency
=
=
=
32%, and observed for
3 h, the same time period as before.
(a) What is the probability that exactly 3 counts will be observed?
(b) What is the expected number of counts in 3 h?
(c) What is the expected count rate, averaged over the 3 h?
(d) What is the expected disintegration rate, averaged over the 3 h?
(e) What is the standard deviation of the count rate over the 3 h?
(f ) What is the standard deviation of the disintegration rate over the 3 h?
(g) If
= 100%, the count rate would be equal to the disintegration rate. What
would then be the expected value and standard deviation of the disintegration rate?
Solution
(a) In the previous example, Eq. (11.8) gave a probability
P
3
=
0.136 for the occur-
rence of exactly 3 disintegrations. This, of course, remains true here. However, with
=
3 h is smaller,
because obtaining 3 counts will generally require more man 3 atoms to decay, with a
correspondingly lower probability. We let
n
*
represent the number of disintegrations
detected by the counter. The probability for exactly 3 counts, that is, for
n
*
0.32, the probability of observing exactly 3 counts in the time
t
=
3, is
given by Eq. (11.8) with
p
and
q
replaced by
p
*
and
q
*
. From Eqs. (11.19) and (11.4),
=
