Equations of Radioactive Decay and Growth

Equations of Radioactive Decay and Growth

EXPONENTIAL DECAY

Half Life. You have seen (Meloni) that a given radioactive species decays according to an exponential law: or , where N and A represent the number of atoms and the measured activity, respectively, at time t, and N0 and A0 the corresponding quantities when t = 0, and λ is the characteristic decay constant for the species. The half life is the time interval required for N or A to fall from any particular value to one half that value. The half life is conveniently determined from a plot of log A versus t when the necessary data are available, and is related to the decay constant:

Average Life. We may determine the average life expectancy of the atoms of a radioactive species. This average life is found from the sum of the times of existence of all the atoms divided by the initial number. If we consider N to be a very large number, we may approximate this sum by an equivalent integral, finding for the average life

We see that the average life is greater than the half life by the factor 1/0.693; the difference arises because of the weight given in the averaging process to the fraction of atoms that by chance survive for a long time.

It may be seen that during the time 1/ λ an activity will be reduced to just 1/e of its initial value.

Mixtures of Independently Decaying Activities. If two radioactive species, denoted by subscripts 1 and 2, are mixed together, the observed total activity is the sum of the two separate activities: A = A1 + A2 = c1 λ1N1+ c2 λ2N2. The detection coefficients c1 and c2 are by no means necessarily the same and often are very different in magnitude. In general, A1 ® A2 ® A3 ® ……® An for mixtures of n species.

For a mixture of several independent activities the result of plotting log A versus t is always a curve concave upward (convex toward the origin). This curvature results because the shorter-lived components become relatively less significant as time passes. In fact, after sufficient time the longest-lived activity will entirely predominate, and its half life may be read from this late portion of the decay curve. Now, if this last portion, which is a straight line, is extrapolated back to t = 0 and the extrapolated line subtracted from the original curve, the residual curve represents the decay of all components except the longest-lived. This curve may be treated again in the same way, and in principle any complex decay curve may be analyzed into its components. In actual practice experimental uncertainties in the observed data may be expected to make it difficult to handle systems of more than three components, and even two-component curves may not be satisfactorily resolved if the two half lives differ by less than about a factor of two. The curve shown in figure 1 is for two components with half lives differing by a factor of 10.

Time (h)

Figure 1- Analysis of composite decay curve: (a) composite decay curve; (b) longer-lived component (= 8.0 h); (c) shorter-lived component ( = 0.8 h).

The resolution of a decay curve consisting of two components of known but not very different half lives is greatly facilitated by the following approach. The total activity at time t is

By multiplying both sides by we obtain

Since A1 and A2 are known and A has been measured as a function of t, we can construct a plot of versus ; this will be a straight line with intercept and slope .

Least-squares analysis is a more objective method for the resolution of complex decay curves than the graphical analysis described. Computer programs for this analysis have been developed (J. B. Cumming, "CLSQ, The Brookhaven Decay-Curve Analysis Program," in Application of Computers to Nuclear and Radiochemistry (G. D. O'Kelly, Ed.), NAS-NRC, Washington, 1963, p. 25.) that give values of A° and its standard deviation for each of the components. Some of the programs can also be used to search for the "best values" of the decay constants.

Calculate the weight in grams w of 1 mCi of 14C from its half-life of 5720 years.

Growth of radioactive products

General Equation. We considered briefly a special case in which a radioactive daughter substance was formed in the decay of the parent. Let us take up the general case for the decay of a radioactive species, denoted by subscript 1, to produce another radioactive species, denoted by subscript 2.

The behavior of N1 is just as has been derived; that is,

and

where we use the symbol to represent the value of N1 at t = 0.

Now the second species is formed at the rate at which the first decays, , and itself decays at the rate . Thus

By multiplying both sides by :

what to be rewritten:

Integrating:

for t=0, N2 = :

(2)

The solution of this linear differential equation of the first order may be obtained by standard methods and gives

where is the value of N2 at t = 0. Notice that the first group of terms shows the growth of daughter from the parent and the decay of these daughter atoms; the last term gives the contribution at any time from the daughter atoms present initially.

Transient Equilibrium.

In applying (2) to considerations of radioactive (parent and daughter) pairs, we can distinguish two general cases, depending on which of the two substances has the longer half life.

If the parent is longer-lived than the daughter (λ1λ2), a state of so-called radioactive equilibrium is reached; that is, after a certain time the ratio of the numbers of atoms and, consequently, the ratio of the disintegration rates of parent and daughter become constant.

This can be readily seen from (2); after t becomes sufficiently large, is negligible compared with , and also becomes negligible; then

and, since

(3)

The relation of the two measured activities is found from , to be

(4)

In the special case of equal detection coefficients (c1=c2) the ratio of the two activities, , may have any value between 0 and 1, depending on the ratio of λ1 to λ2 that is, in equilibrium the daughter activity will be greater than the parent activity by the factor λ2/( λ2 – λ1).

In equilibrium both activities decay with the parent's half life.

As a consequence of the condition of transient equilibrium (λ2λ1), the sum of the parent and daughter disintegration rates in an initially pure parent fraction goes through a maximum before transient equilibrium is achieved.

This situation is illustrated in figure 2.

Figure 2 - Transient equilibrium: (a) total activity of an initially pure parent fraction; (b) activity due to parent (= 8.0 h); (c) decay of freshly isolated daughter fraction (= 0.80 h); (d) daughter activity growing in freshly purified parent fraction; (e) total daughter activity in parent-plus-daughter fractions

The more general condition for the total measured activity (A1+A2) of an initially pure parent fraction to exhibit a maximum is found to be c2/c1 > λ1/λ2. This condition holds regardless of the relative magnitudes of λ1, and λ2. The condition will give a maximum in the total measured activity that occurs at a negative time.

Secular Equilibrium. A limiting case of radioactive equilibrium in which and in which the parent activity does not decrease measurably during many daughter half lives is known as secular equilibrium.

Derive the equation as a useful approximation of (3):

or

In the same way (4) reduces to

and the measured activities are equal if c1 =c2.

Figure 2 presents an example of transient equilibrium with (actually with λ1/λ2 = 0.1); the curves represent variations with time of the parent activity and the activity of a freshly isolated daughter fraction, the growth of daughter activity in a freshly purified parent fraction, and other relations; in preparing the figure we have taken c1=c2.

Figure 3 is a similar plot for secular equilibrium; it is apparent that as λ1, becomes smaller compared to λ2 the curves for transient equilibrium shift to approach more and more closely the limiting case shown in figure 3.

Figure 3 - Secular equilibrium: (a) total activity of an initially pure parent fraction; (b) activity due to parent (); this is also the total daughter activity in parent-plus-daughter fractions; (c) decay of freshly isolated daughter fraction (); (d) daughter activity growing in freshly purified parent fraction.

The Case of No Equilibrium. If the parent is shorter-lived than the daughter (λ1λ2), it is evident that no equilibrium is attained at any time. If the parent is made initially free of the daughter, then as the parent decays the amount of daughter will rise, pass through a maximum, and eventually decay with the characteristic half life of the daughter. This is illustrated in figure 4; for this plot we have taken λ1/λ2= 10, and c1=c2. In the figure the final exponential decay of the daughter is extrapolated back to t=0.

Figure 4 - The case of no equilibrium: (a) total activity; (b) activity due to parent (); (c) extrapolation of final decay curve to time zero; (d) daughter activity in initially pure parent.

This method of analysis is useful if , for then this intercept measures the activity the atoms give rise to N2 atoms so early that may be set equal to the extrapolated value of N2 at t = 0. The ratio of the initial activity to this extrapolated activity gives the ratios of the half lives if the relation between c1 and c2 is known:

If λ2 is not negligible compared to λ1, it can be shown that the ratio λ1/λ2 in this equation should be replaced by and the expression involving the half lives changed accordingly.

Both the transient-equilibrium and the no-equilibrium cases are sometimes analyzed in terms of the time tm for the daughter to reach its maximum activity when growing in a freshly separated parent fraction.

This time we find from the general equation (2) by differentiating,

and setting when t = tm:

or

At this time the daughter decay rate is just equal to the rate of formation , [this is obvious from (1)]; in figures 2 and 4, in which we assumed c1=c2, we have the parent activity A1 intersecting the daughter growth curve d at the time tm. (The time tm is infinite for secular equilibrium.)

Many Successive Decays. If we consider a chain of three or more radioactive products, it is clear that the equations already derived for N1 and N2 as functions of time are valid, and N3 may be found by solving the new differential equation:

(5)

This is entirely analogous to the equation for , but the solution calls for more labor, since N2 is a much more complicated function than Nì. The next solution for N4 is still more tedious. H. Bateman (H. Bateman. "Solution of a System of Differential Equations Occurring in the Theory of Radio-active Transformations," Proc. Cambridge Phil. Soc. IS, 423 (1910) has given the solution for a chain of n members with the special assumption that at t = 0 the parent substance alone is present, that is, that . This solution is

where

......

If we do require a solution to the more general case with , we may construct it by adding to the Bateman solution for Nn, in an n-membered chain a Bateman solution for Nn in an (n-1)-membered chain with substance 2 as the parent, and, therefore, at t = 0, and a Bateman solution for Nn in an (n-2)-membered chain, and so on.

Branching Decay. The case of branching decay when a nuclide can decay by more than one mode is illustrated by

B
A
/ (
C

The two partial decay constants and must be considered when the general relations in either branch are studied because, for example, the substance B is formed at the rate

but A is consumed at the rate

The nuclide A has only one half life

where At = AB + AC + • • •. By definition the half life is related to the total rate of disappearance of a substance, regardless of the mechanism by which it disappears.

If the Bateman solution is to be applied to a decay chain containing branching decays, the ’s in the numerators of the equations defining C1, C2, and so on, should be replaced by the partial decay constants; that is, Ai in the numerators should be replaced by , where is the decay constant for the transformation of the ith chain member to the (i+1)th member. If a decay chain branches, and subsequently the two branches are rejoined as in the natural radioactive series, the two branches are treated by this method as separate chains; the production of a common member beyond the branch point is the sum of the numbers of atoms formed by the two paths.

EQUATIONS OF TRANSFORMATION DURING NUCLEAR REACTIONS

Stable Targets. When a target is irradiated by particles that induce nuclear reactions, a steady state can be reached in which radioactive products disintegrate at just the rate at which they are formed; the situation is analogous to that of secular equilibrium. If the irradiation is terminated before the steady state is achieved, then the disintegration rate of a particular active nuclide is less than its rate of formation R. The differential equation that governs the number of product atoms N present at time t during the irradiation is