Biomedical Engineering Reference
In-Depth Information
that of the parent. This condition is called secular equilibrium. The total activity
is
2
A
1
. In terms of the numbers of atoms,
N
1
and
N
2
, of the parent and daughter,
secular equilibrium can be also expressed by writing
λ
1
N
1
=
λ
2
N
2
.
(4.38)
Achainof
n
short-lived radionuclides can all be in secular equilibrium with a long-
lived parent. Then the activity of each member of the chain is equal to that of the
parent and the total activity is
n
+1times the activity of the original parent.
General Case
When there is no restriction on the relative magnitudes of
T
1
and
T
2
,wewritein
place of Eq. (4.31)
d
N
2
d
t
=
λ
1
N
1
-
λ
2
N
2
.
(4.39)
With the initial condition
N
20
=
0, the solution to this equation is
N
2
=
λ
1
N
10
λ
2
-
λ
1
(e
-λ
1
t
-e
-λ
2
t
),
(4.40)
as can be verified by direct substitution into (4.39). This general formula yields
Eq. (4.38) when
λ
2
λ
1
and
A
20
=
0
, and hence also describes secular equilibrium.
Transient Equilibrium (
T
1
T
2
)
Another practical situation arises when
N
20
= 0
and the half-life of the parent is
greater than that of the daughter, but not greatly so. According to Eq. (4.40),
N
2
and hence the activity
A
2
=
λ
2
N
2
of the daughter initially build up steadily. With
the continued passage of time,
e
-λ
2
t
eventually becomes negligible with respect
to
e
-λ
1
t
,since
λ
2
>
λ
1
. Then Eq. (4.40) implies, after multiplication of both sides
by
λ
2
,that
λ
2
N
2
=
λ
2
λ
1
N
10
e
-λ
1
t
λ
2
-
.
(4.41)
λ
1
Since
A
1
=
λ
1
N
1
=
λ
1
N
10
e
-λ
1
t
is the activity of the parent as a function of time, this
relation says that
λ
2
A
1
λ
2
-
A
2
=
λ
1
.
(4.42)
Thus, after initially increasing, the daughter activity
A
2
goes through a maximum
and then decreases at the same rate as the parent activity. Under this condition,
illustrated in Fig. 4.5, transient equilibrium is said to exist. The total activity also
