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: N N0e t or A A0e t , 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 t1/2 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: ln2 0.693 t1 / 2 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 1 1 t 1 t 1 tdN NO tNdt te dt e 0 t NO 0 0 0 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 = A 1 + 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 (t1/2 = 8.0 h); (c) shorter-lived component (t1/2 = 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 A A10 e 1t A2 e 2t 0 By multiplying both sides by e 1t we obtain A A10 A2 e 1 2 t 0 Since A1 and A2 are known and A has been measured as a function of t, we can construct a plot of Ae 1t versus e 1 2 ; this will be a straight line with intercept A10 and slope A2 . t 0 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. ln2 0,693 3,83x 1012 s 1 t1 / 2 5720x 365x 24x 3600 dN N dt 1mCi 3,70x 107 dps A 3.7x 107 N 2,24x 10 4 g 3,83x 1012 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, dN1 1N1 dt and N1 N10 e 1t where we use the sy mbol N10 to represent the value of N 1 at t = 0. Now the second species is formed at the rate at which the first decays, 1N1 , and itself decays at the rate 2N2 . Thus dN2 1N1 2N2 dt dN2 1N10 e 1t 2N2 dt dN2 2N2 1N10 e 1t dt By multiplying both sides by e 2t : dN2 t e 2t 2N2e 2t 1N10 e 2 1 t dt what to be rewritten: d dt N2e2t 1N10e2 1 t Integrating: 1 N2e 2t N10 e 2 1 t C 2 1 1 N2 N10 e 1t Ce 2t 2 1 for t=0, N2 = N2 : 0 1 C N10 N2 0 2 1 1 N2 N10 e 1t e 2t N2 e 2t (2) 0 2 1 dN2 The solution of 1N1 2N2 0 this linear differential equation of the first order dt may be obtained by standard methods and gives 1 N2 N10 e 1t e 2t N2 e 2t 0 2 1 where N2 is the value of N2 at t = 0. Notice that the first group of terms shows the 0 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 (λ1<λ2), 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, e 2t is negligible compared with e 1t , and N2 e 2t also becomes negligible; then 0 1 N2 N10 e 1t 2 1 and, since N1 N10 e 1t The relation of the two measured activities is N1 1 found from A1 c11N1 , A2 c2 2N2 to be N2 2 A1 c1 2 1 (4) A2 c22 In the special case of equal detection coefficients (c 1 =c2) the ratio of the two A activities, 1 1 1 , may have any value between 0 and 1, depending on the ratio of A2 2 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 (λ2>λ1), 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 (t1/2 = 8.0 h); (c) decay of freshly isolated daughter fraction (t1/2 = 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 c 2/c1 > λ1/λ2. This condition holds 1 c2 1 regardless of the relative magnitudes of λ1, and λ2. The 2 condition 2 c1 2 will give a maximum in the total measured activity that occurs at a negative time. Secular E quilibriu m. A limiting case of radioactive equilibrium in which 1 2 and in which the parent activity does not decrease measur ably during many daughter half lives is known as secular equilibrium. Derive the equation as a useful approximation of (3): N1 1 or 1N1 2N2 N2 2 In the same way (4) reduces to A1 c 1 A2 c2 and the measured activities are equal if c1 =c2. Figure 2 presents an example of transient equilibrium with 1 2 (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 (t1/2 ); this is also the total daughter activity in parent-plus- daughter fractions; (c) decay of freshly isolated daughter fraction (t1 / 2 0.8h ); (d) daughter activity growing in freshly purified parent fraction. The Case of No Eq uilibriu m. If the parent is shorter-lived than the daughter ( λ1 >λ2 ), 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 l ife 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 (t1 / 2 0.8h ); (c) extrapolation of final decay curve to time zero; (d) daughter activity in initially pure parent. This method of analysis is useful if 1 2 , for then this intercept measures the activity c22N the N1 atoms give rise to N 2 atoms so early that N10 may be set equal to 1 0 0 the extrapolated value of N 2 at t = 0. The ratio of the initial activity c1 1N10 to this extrapolated activity gives the ratios of the half lives if the relation between c1 and c2 is known: t1 c11N10 c1 1 2 2 x c22N10 c2 2 t1 2 1 If λ 2 is not negligible compared to λ 1 , it can be shown that the ratio λ 1 /λ2 in this equation should be replaced by 1 2 and the expression involving the half lives changed 2 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 i n a freshly separated parent fraction. This time we find from the general equation (2) by differentiating, dN2 2 1 12 N10e 1t N10e 2t dt 2 1 2 1 dN2 and setting 0 when t = tm: dt 2 1 e 2 1 tm or tm ln 2 1 2 1 1 At this time the daughter decay rate 2N2 is just equal to the rate of formation 1N1 , [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: dN2 This is entirely analogous to the equation for dN3 N N , but the 2 2 3 3 (5) dt dt solution calls for more labor, since N 2 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 N20 N30 ........ Nn0 0 . This solution is Ni C1e 1t C2e 2t .... Cn e n t where 12 ...n 1 C1 N10 2 1 3 1 ..... n 1 12 ...n 1 C2 N10 1 2 3 2 ..... n 2 ............ 12 ...n 1 Cn N0 1 n 2 n ..... n 1 n 1 If we do require a solution to the more general case with N20 , N30 ,........, N30 0 , 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, N2 N20 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 B A C C The two partial decay constants B and C must be considered when the general relations in either branch are studied because, for example, the substance B is formed at the rate dNB BNB dt but A is consumed at the rate dNA B C NA dt The nuclide A has only one half life 0.693 t1 2 t 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 i* , where i* is the decay constant for the transformation of the i th 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 dN R N dt the solution to which is N R 1 e t For very large irradiation times ( T > > 1 / ) the disintegration rate N approaches the saturation value R. The factor ( 1 e t ) is often called the saturation factor. If the disintegration rate of a particular radioactive product at the end of a steady bombardment of known duration is divided by this saturation factor, the rate at which the product was formed during the bombardment is obtained. Occasionally a product is formed during irradiation both directly by nuclear reaction and by the decay of an active parent that is produced by another reaction [e.g., the product of a (p,pn) reaction, if unstable, may decay by positron emission or EC into the product of the (p, 2p) reaction on the same target]. Under these circumstances the number of atoms of the product of interest present at a time t, after the end of a bombardment of duration tb has three sources: 1. Those formed directly in nuclear reactions. 2. Those formed by the decay of the parent during bombardment. 3. Those formed by the decay of the parent during the interval tb (which may, for example, be the time between the end of bombardment and the chemical separation of daughter from parent). If R 1 and R 2 are the rates of the nuclear reactions that directly form the parent and daughter products, respectively, then the number of daughter atoms (characterized by subscript 2) arising from each of the three sources is R2 N2 ' 2 1 e 2tB e 2tS R R1 N2 1 1 e 2tB '' e 1tS e 2ts e 2tS 2 1 2 R2 1 e 2tB e 1tS e 2tS N ''' 2 1 2 Experimentally it is, of course, only the totality of the daughter atoms N ' 2 N2 N2 that is observed, but from a knowledge of the times tB and ts the '' ''' decay constants 1 , and 2 , and of the rate of formation R 1 of the parent (which can be determined in a separate experiment), it is possible to calculate R2. Radioactive Targets in a High-Flux Reactor. When nuclear reactions are induced in a radioactive nuclide, the rate of disappearance of the substance is no longer governed by the law of radioactive transformation alone but by a modified law that takes into account the disappearance by transmutation reactions also. Under most practical bombardment conditions the rate of transformation of radioactive species by nuclear reactions is negligible compared to the rate of radioactive decay. However, in the case of long-lived nuclides, and with the large neutron fluxes available in nuclear reactors, transformations by both mechanisms sometimes have to be considered. We state the modified transformation equations for the case of a neutron flux; they are equally applicable for any other bombarding particle. The treatment given here follows that developed by W. Rubinson . Consider N atoms of a single radioactive species of decay constant (in reciprocal seconds) and total neutron reaction cross section σ (in square centimeters) in a constant neutron flux nv (neutrons cm-2s-1). The rate of radioactive transformation is N, the rate of transformation by neutron reactions is nvσN, and the total rate of disappearance is dN nv N N (10) dt where may be considered as a modified decay constant. Equation 10 has the same form as the standard differential equation of radioactive decay and is integrated to give N N0e t (11) If we consider a parent-daughter pair, the parent disappears by both transmutation dN and decay: 1 1 nv1 N1 1N1 ; but the daughter grows by decay of the mother dt dN2 only and disappears by both processes: 1N1 2N2 , or, in more general notation, dt dNi1 iNi i1Ni1 dt Actually we may want to consider chains in which the transformation from one member to the next may occur by nuclear reaction as well as by radioactive decay. Then A, must be replaced by a modified decay constant i* i* nvi* , where the asterisks serve as a reminder that, if either the decay or reaction of the parent does not always lead to the next chain member, then i* must be the partial decay constant and i* must be the partial reaction cross section leading from the i th member to the (i+1)th member of the chain. With this notation the general solution is written, as in the Bateman equations, for N20 N30 ........ Nn0 0 : Ni C1e 1t C2e 2t .... Cn e n t (12) where 1 2 ...n 1 C1 N10 2 1 3 1 ..... n 1 1 2 ...n 1 C2 N10 and so on. 1 2 3 2 ..... n 2 As an illustration, we compu The numerical values to be substituted are reactions when 1 g 197Au is e t = 1.08xl05 s, of reactions is nv=1.x1014 cm-2 s-1 197 Au 99b n , 197 = 9.9x1023 cm2 198 = 2.5 x1020 cm 2 We use (12) for three-mem 1x6.02x1023 N0 197 3.05x1021 197 e 197 197 nv197 9.9x10 9 s 1 * Using these values, we get N199 197 198 199 * * * 198 198 nv198 3.0x10 6 2.5x10 6 5.5x10 6 s 1 e 0.00107 e 0.594 198 197 * N199 7.85x107 5.5x10 x2.55x10 5.5x10 x2.95x10 6 2.55x 6 6 6 198 nv198 2.5x10 s * 6 1 and N199 7.55x107 7.12x1010 3.40x1010 1.01x1011 3.2x1017 199 199 2.55x10 6 s 1 199 The disintegration rate of Au at the end of the irradiation is 199N199 0.82x1012 s 1 . For 198 comparison we compute the disintegration rate of Au in the sample [again from (12) for a two-membered chain]: e 197 t e 198t 0.999 0.522 198N198 198nv197N 0 197 9.06x10 x 7 7.36x1012 s1 198 198 197 198 5.5x10 6 Thus about 10 percent of the radioactive disintegrations in the sample occur in 199Au.
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