Page 96 - Radiochemistry and nuclear chemistry

Unstable Nuclei and Radioactive Decay 85

~'1 ~'2 ~3
X 1 -* X 2 -'* X 3 --* X 4 ... (4.52)

The net rate of formation of the daughter atoms X 2 is the difference between the rate of
formation of the daughter and her rate of decay, i.e.

dN2/dt = N1 hi - N2 ~'2 (4.53)

where N 1 and N 2 are the number of parent and of daughter atoms, and )k I and )k2, the decay
constants of the parent and daughter, respectively. The solution of this equation is

N2 - " [Xl /()k2 - - Xl)] ~1 ( e-hIt - - e-k~) + ~2 e-h2t (4.54)

where/~1 and N~2 are the amounts of parent and daughter respectively at time t = 0. The
first term in this equation tells us how the number of daughter nuclei vary with time as a
consequence of the formation and subsequent decay of the daughter nuclei, while the second
term accounts for the decay of those daughter nuclei that were present at t = 0.
Let us illustrate this relationship by an example among the naturally occurring radioactive
decay series. In an old uranium mineral all the products in the decay chain can be detected
(see Fig. 1.2). Suppose now that we use a chemical separation to isolate two samples, one
containing only uranium and one containing only thorium (relation (1.3) and Fig. 1.1). At
the time of separation, which we designate as t = 0, there are/~1 atoms of 238U and/~2
atoms of 234Th. In the thorium fraction, which is free from uranium, /~1 = 0 and,
therefore, the thorium atoms decay according to the last term in (4.54). This sample gives
a simple exponential decay curve with a half-life of 24.1 d, as shown by the precipitate
curve in Figure 1.1. The uranium fraction at t = 0 is completely free of thorium; i.e./~2
= 0. However, after some time it is possible to detect the presence of 234Th. The change
in the number of 234Th with time follows the first term of (4.54); in fact, in Figure 1.1 the
measurements detect only the 234Th nuclide (B-emitting), since the detection system used
is not sensitive to the c~'s from 238U (ffa = 0). The time of observation is much smaller
than the half-life of the 238U decay, so there will be no observable change in the number
of atoms of uranium during the time of observation, i.e. N l = /~1" Further, since tv~ for
238U ~. tv2 for 234Th, i.e. X 1 ,~ X 2, we can simplify (4.54) to

N 2 = (~kl/~k2 ) NI (1 - e- xd) (4.55)

According to this equation, the number of 234Th atoms, N 2, increases with time with the
half-life of 234Th. In other words, after a period of 24.1 d there is 50 % of the maximum
value of 234Th, after 48.2 d there is 75% of the final maximum value, etc. This is
illustrated by the change in the uranium fraction activity in Figure 1.1. Further, from the
relationship (4.55) we can see that the maximum value of thorium (t = oo) is given by

N 2 ~k 2 = N 1 )k I (4.56)
These equations, based on ~'1 ~ )k2, show that the amount of daughter atoms becomes
constant after some time. At that time the rate of decay of the daughter becomes equal to