28
2 Radioactivity
λ A × N A − λ B × N B =
d N B
dt
= 0
(2.14)
Therefore, whatever be the relative half-lives of the parent or daughter, their
activities are equal at a time t m , that is, when the activity of the daughter reaches a
maximum value.
A detailed examination of Eq. (2.12) suggests that there can be three variations of
this equation, which are of importance. Two of them are for cases where the parent
is longer lived than the daughter (i.e., either λ B > λ A or λ B λ A ), and the third
case is when the daughter product is longer lived than its radioactive parent (i.e.,
λ A > λ B ). These cases we shall now deal with separately.
When λ B λ A
Under this condition, Eq. (2.12) can be rearranged by ignoring λ A into the following
form:
N B × λ B = λ A × N A0 (1 − e
−λ B t
)
(2.15)
One interesting feature comes out from this equation, i.e., if the mixture (i.e.,
atoms A and B) is allowed to grow together for a period 6–7 times the half-life of
the daughter B, after making sure that at t = 0 atom B was absent completely, the
exponential term becomes almost zero (i.e., e
−λ B t ) and the final equation takes a
form:
N B × λ B = λ A × N A0
(2.16)
Thus for λ B λ A , we see that the daughter B soon decays with the half-life
of the parent A and their activities are equal. The equilibrium represented by the
above equation is called secular equilibrium. Examples of secular equilibrium are
provided by the radioactive decay of Radium-226 (t 1/2 = 1600 years) to Radon-222
(t 1/2 = 3.82 days), Strontium-90 (t 1/2 = 29.12 years) to Yttrium-90 (t 1/2 = 64.0
hours), or Cesium-137 (t 1/2 = 30y) decaying to Barium-137 (t 1/2 = 2.6 min). All
these members of the former family emit α-particles, and the latter emit negative
β-particles. The growth of Barium-137 activity from freshly separated Cesium-137
is illustrated in Fig. 2.8.
When λ B > λ A
The second case is when λ B > λ A but not so great that λ A could be neglected.
However, for larger time “t” λ B could be neglected as compared to λ A . Under this
condition,
Equation (2.12) can be rewritten as
N B =
λ A
λ B − λ A
N A0 (e
−λ A t
)
(2.17)
Since
2 Radioactivity
λ A × N A − λ B × N B =
d N B
dt
= 0
(2.14)
Therefore, whatever be the relative half-lives of the parent or daughter, their
activities are equal at a time t m , that is, when the activity of the daughter reaches a
maximum value.
A detailed examination of Eq. (2.12) suggests that there can be three variations of
this equation, which are of importance. Two of them are for cases where the parent
is longer lived than the daughter (i.e., either λ B > λ A or λ B λ A ), and the third
case is when the daughter product is longer lived than its radioactive parent (i.e.,
λ A > λ B ). These cases we shall now deal with separately.
When λ B λ A
Under this condition, Eq. (2.12) can be rearranged by ignoring λ A into the following
form:
N B × λ B = λ A × N A0 (1 − e
−λ B t
)
(2.15)
One interesting feature comes out from this equation, i.e., if the mixture (i.e.,
atoms A and B) is allowed to grow together for a period 6–7 times the half-life of
the daughter B, after making sure that at t = 0 atom B was absent completely, the
exponential term becomes almost zero (i.e., e
−λ B t ) and the final equation takes a
form:
N B × λ B = λ A × N A0
(2.16)
Thus for λ B λ A , we see that the daughter B soon decays with the half-life
of the parent A and their activities are equal. The equilibrium represented by the
above equation is called secular equilibrium. Examples of secular equilibrium are
provided by the radioactive decay of Radium-226 (t 1/2 = 1600 years) to Radon-222
(t 1/2 = 3.82 days), Strontium-90 (t 1/2 = 29.12 years) to Yttrium-90 (t 1/2 = 64.0
hours), or Cesium-137 (t 1/2 = 30y) decaying to Barium-137 (t 1/2 = 2.6 min). All
these members of the former family emit α-particles, and the latter emit negative
β-particles. The growth of Barium-137 activity from freshly separated Cesium-137
is illustrated in Fig. 2.8.
When λ B > λ A
The second case is when λ B > λ A but not so great that λ A could be neglected.
However, for larger time “t” λ B could be neglected as compared to λ A . Under this
condition,
Equation (2.12) can be rewritten as
N B =
λ A
λ B − λ A
N A0 (e
−λ A t
)
(2.17)
Since