1. SERIAL TRANSFORMATION

2. In addition to the four chains of radioactive isotopes described above, a number of other groups of sequentially transforming isotopes are important to the health physicist and radiobiologist. Most of these series are associated with nuclear fission, and the first member of each series is a fission fragment. One of the most widely known fission products, for example, Sr-90, is the middle member of a five-member series of beta emitters that starts with Kr-90 and finally terminates with stable Zr-90 according to the following sequence:

4. Secular Equilibrium
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 Kr-90 will, in a time period of 10–15 minutes, have been transformed to such a degree that, for practical purposes, the Kr-90 may be assumed to have been completely transformed. Rubidium( Rb)-90, the Kr-90 daughter, because of its 2.74-minute half-life, will suffer the same fate after about an hour. Essentially, all the Kr-90 is, as a result, converted into Sr-90 within about an hour after its formation. The buildup of Sr-90 is therefore very rapid. The half-life of Sr-90 is 28.8 years and its transformation, therefore, is very slow. The Y-90 daughter of Sr-90, with a half-life of 64.2 hours, transforms rapidly to stable Zr-90.

5. If pure Sr-90 is prepared initially, its radioactive transformation will result in an accumulation of Y-90. Because the Y-90 transforms very much faster than Sr-90, a point is soon reached at which the instantaneous amount of Sr-90 that transforms is equal to that of Y-90. Under these conditions, the Y-90 is said to be in secular equilibrium. The quantitative relationship between radionuclides in secular equilibrium may be derived in the following manner:

6. 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:

7. and then solve for NB, we get:
rate of change = rate of formation − rate of transformation
(1)
Integrate Eq (1),
and then solve for NB, we get:

8. If we start with pure A, that is, if NB0 = 0, then Eq. (2) reduces to:
(3)
The rate of formation of B from A is equal to the rate of transformation of A. Therefore, K is simply equal to λANA. An alternative way of expressing Eq. (3), therefore, is
(4)

9. If the activity of the parent is given in becquerels or millicuries, then the activity of the daughter must also be in units of becquerels or millicuries, and Eq. (4) may be written as:
(5)
where QA and QB are the respective activities in becquerels or millicuries of the parent and daughter.

10. Equation (5), shows a buildup of daughter from zero to a maximum activity, which is equal to that of the parent from which it was derived. This buildup of daughter activity may be shown graphically by plotting Eq. (5).
A generally useful curve showing the buildup of daughter activity under conditions of secular equilibrium may be obtained if t is plotted in units of daughter half-life, as shown in the following Figure. As time increases, e−λt decreases and QB approaches QA. For practical purposes, equilibrium may be considered to be established after about seven half-lives of the daughter.

11. Secular equilibrium: Buildup of a very short lived daughter from a very long lived parent. The activity of the parent remains constant.

12. In addition, approaches zero faster as t increases. Therefore Eq
In addition, approaches zero faster as t increases. Therefore Eq. (5), can be modified to a good approximation:
(6)
Eq. (6), indicates, that, with increasing time, the rate of decay of the unstable daughter (NBλB) becomes equal to that of its long-lived parent (NAλA). This condition is known as “Secular Equilibrium”

13. Transient Equilibrium
In the case of secular equilibrium discussed above, the quantity of the parent remains substantially constant during the period that it is being observed. Since it is required for secular equilibrium that the half-life of the parent be very much longer than that of the daughter, it follows that secular equilibrium is a special case of a more general situation in which the half-life of the parent may be of any conceivable magnitude, but greater than that of the daughter. For this general case, where the parent activity is not relatively constant,

14. growth of species B at that time.
the time rate of change of the number of atoms of species B is given by the differential equation
(7)
In this equation, λANA is the rate of transformation of species A and is exactly equal to the rate of formation of species B, the rate of transformation of isotope B is λBNB,and the difference between these two rates at any time is the instantaneous rate of
growth of species B at that time.

15. We know that:
Equation (7) may be rewritten, after substituting the expression above for NA and transposing λBNB, as
(8)
Equation (8) is a first-order linear differential equation of the form:
(9)

16. and may be integrable by multiplying both sides of the equation by
and the solution to Eq. (9) is
(10)
Since NB, λB, and λANA0 e−λAt from Eq. (8) are represented in Eq. (10) by y , P, and Q, respectively, the solution of Eq. (8) is

17. or, if the two exponentials are combined, we have
And the solution is:
(11)

18. The constant C may be evaluated by applying the boundary conditions:
NB = 0 when t = 0
If the value for C, is substituted into Eq. (11), the solution for NB is found to be:
(12)

19. For the case in which the half-life of the parent is very much greater than that of the daughter, that is, when λA λB, Eq. (12) approaches the condition of secular equilibrium, which is the limiting case described by Eq. (6). Two other general cases should be considered—the case where the parent half-life is slightly greater than that of the daughter (λA < λB), and the case in which the parent half-life is less than that of the daughter (λB < λA). In the former case, where the half-life of the daughter is slightly smaller than that of the parent, the daughter activity (if the parent is initially pure and free of any daughter activity) starts from zero, rises to a maximum, and then seems to decay with the same half-life as that of the parent. When this occurs, the daughter is undergoing transformation at the same rate as it is being produced, and the two radionuclides are said to be in a state of transient equilibrium.

20. The quantitative relationships prevailing during transient equilibrium may be inferred from Eq. (12). If both sides of the equation are multiplied by λB, we have an explicit expression for the activity of the daughter:
(13)
Since λB is greater than λA, then, after a sufficiently long period of time, e−λB t will become much smaller than e−λAt . Under this condition, Eq. (13) may be rewritten as

21. or, in terms of activity units (becquerels, curies, etc.), As:
(14)
the mathematical expression for transient equilibrium, Eq. (14) may be rewritten as
or, in terms of activity units (becquerels, curies, etc.),
As:
(15)