Biomedical Engineering Reference
In-Depth Information
One of the most common terms encountered in any discussion of radioactive materials
is
T
1/2
. This is because all radioactive substances follow the same general decay
pattern. After a certain interval of time, half of the original number of nuclei in a sample
will have decayed, then in a second time interval equal to the first, half of those nuclei
remaining will have decayed, and so on.
The half-life is the time required for half of a given number of radioactive nuclei to
decay. The half-life and decay constant are obviously related, since the larger the value of
the decay constant (
half-life,
), the faster the process of decay and consequently the shorter the
half-life. In any event, the decay constant is an unchanging value throughout the decay pro-
cess. Each radionuclide exhibits a distinctive disintegration process because of its inherent
properties—that is, its decay constant (
l
) and half-life (T
1/2
). All nuclides decay in the same
manner but not at the same rate, since this is a parameter determined by the unique nature
of the particular radioactive element in question.
Considering a number of atoms (
l
N
) of a specific type of radionuclide present at a time
t,
the transformation rate can be defined by -
(the minus sign denotes the decay/
decrease), which would be proportional to the number of atoms, or
dN/dt
dN
=
dt
¼ l
N
ð
15
:
1
Þ
where
is the transformation constant.
Taking the initial number of atoms as
l
N
0
(
N
¼
N
0
when
t
¼
t
0
), we get by integration
N
¼
N
0
e
l
t
ð
15
:
2
Þ
where
N
is the number of radioactive nuclei present at time
t
,
N
0
is the number present at
time
t
¼
0, and
e
¼
2.718
is the base of the natural logarithm (see Figure 15.3).
...
The unit of activity is the
curie
(Ci), defined as
10
10
decays
1Ci
3
:
7
=
s
This number of decay events per second was selected as the original activity unit because
it is the approximate activity of 1 g of radium. The SI unit of activity is the
becquerel
(Bq):
1Bq
¼
1 decay
=
s
10
10
Bq. The most commonly used units of activity are the millicurie
(10
3
Ci) and the microcurie (10
6
Ci).
The half-life, or
Therefore, 1 Ci
¼
3.7
T
1/2
(the time corresponding to transformation of 50 percent of the
nuclides, when
N
¼
N
0
/2), therefore may be obtained by solving for
l
T
1/2
in Eq. (15.2).
Therefore,
N
¼
N
0
e
l
t
and
T
1=2
¼
ln
2
=l ¼
0
:
693
=l
