Biomedical Engineering Reference
In-Depth Information
giving
r
g
r
b
.
t
g
t
b
=
(11.57)
The ratio of the optimum counting times is thus equal to the square root of the
ratio of the respective count rates.
Counting Short-Lived Samples
As we have seen, radioactive decay is a Bernoulli process. The distribution in the
number of disintegrations in a given time for identical samples of a pure radionu-
clide is thus described by the binomial distribution. When
p
1,the
Poisson and normal distributions give excellent approximations to the binomial.
However, for a rapidly decaying radionuclide, or whenever the time of observation
is not short compared with the half-life,
p
will not be small. The formalism pre-
sented thus far in this section cannot be applied for counting such samples. As an
illustration of dealing statistically with a short-lived radionuclide, the section con-
cludes with an analysis of a counting experiment in which the activity dies away
completely and background is zero.
We show how the binomial distribution leads directly to the formulas we have
been using when
λ
t
1
and then what the distribution implies when
λ
t
is large.
The expected number of atoms that disintegrate during time
t
in a sample of size
N
can be written, with the help of Eqs. (11.16) and (11.2),
1 and
N
µ
=
N
(1-e
-
λ
t
).
(11.58)
=
λ
t
,andso
µ
=
1
,
e
-
λ
t
For a long-lived sample (or short counting time),
λ
t
1-
N
λ
t
. The expected disintegration rate is
µ
/
t
=
λ
N
, as we had in an earlier chapter
[Eq. (4.2)]. With the help of Eqs. (11.18), (11.1), and (11.2), we see that the standard
deviation of the number of disintegrations is
N
(1-e
-
λ
t
)e
-
λ
t
σ
=
=
µ
e
-
λ
t
.
(11.59)
1
, we obtain
σ
=
√
µ
. This very important property of the binomial
distribution is exactly true for the Poisson distribution. Thus, a single observation
from a distribution that is expected to be binomial gives estimates of both the mean
and the standard deviation of the distribution when
λ
Again, for
λ
t
1.Ifthenumberofcounts
obtained is reasonably large, then it can be used for estimating
σ
.
Equations (11.58) and (11.59) also hold for long times (
λ
t
1), for which the
Poisson description of radioactive decay is not accurate. If we make an observation
over a time so much longer than the half-life that the nuclide has decayed away
(
λ
t
0
. The interpre-
tation of this result is straightforward. The expected number of disintegrations is
equal to the original number of atoms
N
in the sample and the standard deviation
of this number is zero. We have observed every disintegration and know exactly
t
→∞
), then Eqs. (11.58) and (11.59) imply that
µ
=
N
and
σ
=
