Introduction to Health Physics: Fourth Edition - Ruang Baca FMIPA UB

118 CHAPTER 4
and the first member of each series is a fission fragment. One of the most widely
known fission products, for example, 90 Sr, is the middle member of a five-member
series of beta emitters that starts with 90 Kr and finally terminates with stable 90 Zr
according to the following sequence:
90
36 Kr
Secular Equilibrium
33 s 90 2.74 min
−→ 37Rb −→ 90
28.8 yr
38Sr −→ 90
39
Y 64.2 hr
−→ 90
40 Zr.
The quantitative relationship among the various members of the series is of great
significance and must be considered in dealing with any of the group’s members.
Intuitively, it can be seen that any amount of 90 Kr will, in a time period of 10–15
minutes, have been transformed to such a degree that, for practical purposes, the
90 Kr may be assumed to have been completely transformed. Rubidium-90, the 90 Kr
daughter, because of its 2.74-minute half-life, will suffer the same fate after about
an hour. Essentially, all the 90 Kr is, as a result, converted into 90 Sr within about an
hour after its formation. The buildup of 90 Sr is therefore very rapid. The half-life
of 90 Sr is 28.8 years and its transformation, therefore, is very slow. The 90 Y daughter
of 90 Sr, with a half-life of 64.2 hours, transforms rapidly to stable 90 Zr. If pure 90 Sr
is prepared initially, its radioactive transformation will result in an accumulation of
90 Y. Because the 90 Y transforms very much faster than 90 Sr, a point is soon reached
at which the instantaneous amount of 90 Sr that transforms is equal to that of 90 Y.
Under these conditions, the 90 Y is said to be in secular equilibrium. The quantitative
relationship between radionuclides in secular equilibrium may be derived in the
following manner:
A λA
−→ B
λB
−→ C
where the half-life of isotope A is very much greater than that of isotope B. The
decay constant of A, λA, is therefore much smaller than λB, the decay constant for
B. C is stable and is not transformed. Because of the very long half-life of A relative
to B, the rate of formation of B may be considered to be constant and equal to K .
Under these conditions, the net rate of change of isotope B with respect to time, if
NB is the number of atoms of B in existence at any time t after an initial number, is
given by
rate of change = rate of formation − rate of transformation,
that is,
dNB
dt = K − λB NB. (4.32)
We will integrate Eq. (4.32), and solve for NB:
NB
NB 0
dNB
K − λB NB
=
t
0
dt. (4.33)