22 Chapter 1. Properties and Sources of Radiation
1.3.E Decay Equilibrium
Depending on the difference between the decay constants of parent and daughter
nuclides it is possible that after some time their activities reach a state of equilib-
rium. Essentially there are three different scenarios leading to different long term
states of the radioactive material. These are termed assecular equilibrium,tran-
sient equilibrium,andno equilibriumstates. For the discussion in this section we
will assume a radioactive material that has a parent and a daughter only. However
the assertions will also be valid for materials that go through a number of decays.
E.1 Secular Equilibrium
If the activity of parent becomes equal to that of the daughter, the two nuclides are
said to be in secular equilibrium. This happens if the half life of parent is much
greater than that of the daughter, i.e.
T 1 P/ 2 T 1 D/ 2 or λdPλdDLet us see if we can derive the condition of equal activity from equation 1.3.28. It
is apparent that ifλdP>λdDthen ast→∞
e−λdDte−λdPt,and hence we can neglect the second term on right side of equation 1.3.28. The
daughter activity in this case is given approximately by
AD
λdD
λdD−λdP
A 0 Pe−λdPt (1.3.32)=
λdD
λdD−λdPAP
Or
AP
AD
1 −
λdP
λdD. (1.3.33)
AsλdD λdP we can neglect second term on right hand side of this equation.
Hence
APAD
This shows that if the half life of parent is far greater than that of the daughter then
the material eventually reaches a state of secular equilibrium in which the activities
of parent and daughter are almost equal. The behavior of such a material with
respect to time is depicted in Fig.1.3.6.
An example of a material that reaches secular equilibrium is^23793 Np. Neptunium-
237 decays into protactinium-233 throughα-decay with a half life of about 2. 14 × 106
years. Protactinium-233 undergoesβ-decay with a half life of about 27 days.
Example:
How long would it take for protactanium-233 to reach secular equilibrium
with its parent neptunium-237?