18 Chapter 1. Properties and Sources of Radiation
1.3.D RadioactiveChain
We saw earlier that when a radionuclide decays, it may change into another element
or another isotope. This newdaughterradionuclide may as well be unstable and
radioactive. The decay mode and half life of the daughter may also be different
from theparent. Let us see how our radioactive decay equations can be modified for
such a situation.
I will start with a sample composed of a parent and a daughter radionuclide.
There will be two processes happening at the same time: production of daughter
(or decay of parent) and decay of daughter. The net rate of decay of the daughter
will then be the difference of these two rates, that is
dND
dt=λdPNP−λdDND, (1.3.24)where subscriptsPandDrepresent parent and daughter respectively.
UsingNP=N 0 Pe−λdPt, this equation can be written as
dND
dt+λdDND−λdPN 0 Pe−λdPt=0. (1.3.25)Solution of this first order linear differential equation is
ND=λdP
λdD−λdPN 0 P
(
e−λdPt−e−λdDt)
+N 0 De−λdDt. (1.3.26)HereN 0 PandN 0 Dare the initial number of parent and daughter nuclides respec-
tively. In terms of activityA(=λN), the above solution can be written as
AD=
λdD
λdD−λdPA 0 P
(
e−λdPt−e−λdDt)
+A 0 De−λdDt. (1.3.27)Equations 1.3.26 and 1.3.27 have decay as well as growth components, as one would
expect. It is apparent from this equation that the way a particular material decays
depends on the half lives (or decay constants) of both the parent and the daughter
nuclides. Let us now use equation 1.3.27 to see how the activity of a freshly pre-
pared radioactive sample would change with time. In such a material, the initial
concentration and activity of daughter nuclide will be zeroN 0 D=0,A 0 D=0. This
condition reduces equation 1.3.26 to
AD=
λdD
λdD−λdPA 0 P
(
e−λdPt−e−λdDt)
. (1.3.28)
The first term in parenthesis on the right side of this equation signifies the buildup
of daughter due to decay of parent while the second term represents the decay of
daughter. This implies that the activity of the daughter increases with time and,
after reaching a maximum, ultimately decreases (see figure 1.3.5). This point of
maximum daughter activitytmaxD can be easily determined by requiring
dAD
dt=0.
Applying this condition to equation 1.3.28 gives
tmaxD =ln(λdD/λdP)
λdD−λdP