Equations of Radioactive Decay and Growth

Equations of Radioactive Decay and GrowthEXPONENTIAL DECAYHalf 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 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/ 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 1N1+ c2 2N2. 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 productsGeneral 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 (1