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1 Chapter 3 - Radioactive Decay Kinetics The number of nuclei in a radioactive sample that disintegrate during a given time interval decreases exponentially with time. Because the nucleus is insulated by the surrounding cloud of electrons, this rate is essentially independent of pressure, temperature, the mass action law, or any other rate- limiting factors that commonly effect chemical and physical changes. 1 As a result, this decay rate serves as a very useful means of identifying a given nuclide. Since radioactive decay represents the transformation of an unstable radioactive nuclide into a more stable nuclide, which may also be radioactive, it is an irreversible event for each nuclide. The unstable nuclei in a radioactive sample do not all decay simultaneously. Instead the decay of a given nucleus is an entirely random event. Consequently, studies of radioactive decay events require the use of statistical methods. With these methods, one may observe a large number of radioactive nuclei and predict with fair assurance that, after a given length of time, a definite fraction of them will have disintegrated but not which ones or when. 3.1 Basic Decay Equations 1 In the case of electron capture and internal conversion, the chemical environment of the electrons involved may affect the decay rate. For L-electron capture in 7 Be (t ½ = 53.3d), the ratio of is Similarly, a fully stripped radioactive ion cannot undergo either EC or IC decay, a feature of interest in astrophysics. 2 In order to make this statement completely correct, we should say that as we double the number of nuclei present, we double the rate of particle emission. This rate is equal to the number of particles emitted per unit time, provided that the time interval is small.

2 Radioactive decay is what chemists refer to as a first- order reaction; that is, the rate of radioactive decay is proportional to the number of each type of radioactive nuclei present in a given sample. So if we double the number of a given type of radioactive nuclei in a sample, we double the number of particles emitted by the sample per unit time. 2 This relation may be expressed as follows: Note that the foregoing statement is only a proportion. By introducing the decay constant, it is possible to convert this expression into an equation, as follows: (3-1) The decay constant, λ, represents the average probability per nucleus of decay occurring per unit time. Therefore we are taking the probability of decay per nucleus, λ, and multiplying it by the number of nuclei present so as to get the rate of particle emission. The units of rate are (disintegration of nuclei/time) making the units of the decay constant (1/time), i.e., probability/time of decay. To convert the preceding word equations to mathematical statements using symbols, let N represent the number of radioactive nuclei present at time t. Then, using differential calculus, the preceding word equations may be written as

3 -3- (3-2) Note that N is constantly reducing in magnitude as a function of time. Rearrangement of Equation (3-2) to separate the variables gives (3-3) If we say that at time t = 0 we have N0 radioactive nuclei present, then integration of Equation (3-3) gives the radioactive decay law (3-4) This equation gives us the number of radioactive nuclei present at time t. However, in many experiments, we want to know the counting rate that we will get in a detector as a function of time. In other words, we want to know the activity of our samples. Still, it is easy to show that the counting rate in one s radiation detector, C, is equal to the rate of disintegration of the radioactive nuclei present in a sample, A, multiplied by a constant related to the efficiency of the radiation measuring system. Thus (3-5) where ε is the efficiency. Substituting into Equation (3-4), we get C = C 0 e λt (3-6) where C is the counting rate at some time t due to a radioactive sample that gave counting rate C0 at time t = 0. Equations (3-4) and (3-6) are the basic equations governing the number of nuclei present in a radioactive sample and the number of counts observed in one s detector as a function of time. Equation (3-6) is shown graphically as Figure 3-1. As

4 -4- seen in Figure 3-1, this exponential curve flattens out and asymptotically approaches zero. If the same plot is made on a semi logarithmic scale (Figure 3-2), the decay curve is a straight line, with a slope equal to the value of - (λ/2.303). The half- life (t½) is another representation of the decay constant. The half- life of a radionuclide is the time required for its activity to decrease by one- half. Thus after one half- life, 50% of the initial activity remains. After two half- lives, only 25% of the initial activity remains. After three half- lives, only 12.5% is yet present and so forth. Figure 3-3 shows this relation graphically. The half- life for a given nuclide can be derived from Equation (3-6) when the value of the decay constant is known. In accordance with the definition of the term half- life, when A/A0 = ½, then t = t½. Substituting these values into Equation (3-6) gives (3-7) Hence (3-8) Note that the value of the expression for t½ has the units of 1/λ or dimensions of (time). The half- lives for different nuclides range from less than 10-6 sec to yr. The half- life has been measured for all the commonly used radionuclides. When an unknown radioactive nuclide is encountered, a determination of its half- life is normally one of the first steps in its identification. This determination can be done by preparing a semi log plot of a series of activity observations made over a period of time. A short- lived nuclide may

5 -5- be observed as it decays through a complete half- life and the time interval observed directly (Figure 3-4) Example Given the data plotted below for the decay of a single radionuclide, determine the decay constant and the half- life of the nuclide. Solution The data is plotted above. The slope (- λ) is given as - λ = - (6.06-0)/(220min - 0) λ = min - 1 t1/2 = ln 2/λ = 0.693/ = 25.2 min. What nuclide might this be?

6 -6- It is difficult to measure the half- life of a very long- lived radionuclide. Here variation in disintegration rate may not be noticeable within a reasonable length of time. In this case, the decay constant must be calculated from the absolute decay rate according to Equation (3-2). The absolute number of atoms of the radioisotope present (N) in a given sample can be calculated according to (3-9) The total mass of the radioisotope in the given sample can be determined once the isotopic composition of the sample is ascertained by such means as mass spectrometry. When the decay constant is known, the half- life can then be readily calculated. A table for the half- lives of a number of the known nuclei can be found in the Appendices. Although the half- life of a given radionuclide is a defined value, the actual moment of disintegration for a particular atom can be anywhere from the very beginning of the nuclide s life to infinity. The average or mean life of a population of nuclei can, however, be calculated. The mean life τ is naturally related to the decay constant and is, in fact, simply the reciprocal of the decay constant: (3-10) or the mean life can be expressed in terms of the half- life: (3-11) One can understand the preceding relationship by recalling that the decay constant, λ, was defined as the average probability of decay per unit time, so the 1/λ is the average time between decays. The concept of average life allows us to calculate of the total number of

7 -7- particles emitted during a defined decay period. This number is essential in determining total radiation dose delivered by a radioisotope sample, as in medical research and therapy. During the time equal to one mean life, τ, the activity falls to 1/e of its original value. For a sample of N0 nuclei with lifetimes ti, we can write for the mean life τ (3-12) The average or mean life is also of fundamental physical significance because it is the time to be substituted in the mathematical statement of the Heisenberg uncertainty principle, i.e., In this expression relating the uncertainty in energy of a system, ΔE, to its lifetime Δt, τ Δt. The quantity ΔE is called the width, Γ. The natural unit of radioactivity is disintegrations/time, such as disintegration per second (dps) or disintegrations per minute (dpm), etc. The SI unit of radioactivity is the Becquerel (Bq) where 1 Becquerel (Bq) 1 disintegration/sec. Counting rates in a detection system are usually given in counts per second (cps), counts per minute (cpm), etc., and differ from the disintegration rates by a factor representing the detector efficiency, ε. Thus

8 -8- (dpm) ε = (cpm) An older unit of radioactivity that still finds some use is the curie (Ci). It is defined as 1 curie (Ci) = 3.7x10 10 Bq = 3.7x10 10 dis/s The curie is a huge unit of radioactivity representing a very large amount and is approximately equal to the activity of one gram of radium. The inventories of radioactivity in a nuclear reactor upon shutdown are typically 10 9 Ci while radiation sources used in tracer experiments have activities of μci and the environmental levels of radioactivity are nci or pci. Note also that because radionuclides, in general, have different half- lives, the number of nuclei per curie will differ from one species to another. For example, let us calculate how many nuclei are in 1 MBq (~27 μci) of tritium( 3 H) (t½ = yr.). We know that But Thus The same calculation carried out for 14 C(t½) = 5730 yr) would give 2.60x10 17 nuclei/mbq. It is also interesting to calculate the mass associated with 1 MBq of tritium. We have

10 -10- Extrapolation of this long- time portion of the curve back to zero time gives the decay curve for component A and the activity of component A at t=0. The curve for component B is obtained by subtracting out, point by point, the activity values of component A from the total activity curve. If the half- lives of the two components in such samples are not sufficiently different to allow graphic resolution, a differential detection method may be applicable. If the radiation characteristics of the nuclides in the mixture are suitably distinct, i.e., emission of different particles or γ- rays, it may be possible to measure the activity of one component without interference from the radiation emitted by the other component. A case in point would be where one nuclide was a pure β- emitter, while the other emitted both β- and γ- rays. In the case where the half- lives of the components are known but are not sufficiently different to allow graphical resolution of the decay curve, computer techniques that utilize least squares fitting to resolve such a case are also available Example: Given the following decay data, determine the half- lives and initial activities of the radionuclides (B and C) present: t (h) A (cpm)

11 From the graph, we see: t1/2(b) = 8.0 h t1/2(c) = 0.8 h A0 (B) = 80 cpm A0(C) = 190 cpm 3.3 Radioactive Decay Equilibrium When a radionuclide decays, it does not disappear, but is transformed into a new nuclear species of lower energy and often differing Z, A, J, π, etc. The equations of radioactive decay discussed so far have focused on the decrease of the parent radionuclides but have ignored the formation (and possible decay) of daughter, granddaughter, etc., species. It is the formation and decay of these children that is the focus of this section.

12 -12- Let us begin by considering the case when a radionuclide 1 decays with decay constant λ1, forming a daughter nucleus 2 which in turn decays with decay constant λ2. Schematically we have 1 2 We can write terms for the production and depletion of 2, i.e., rate of change of 2 nuclei present at time t = rate of production of 2 - rate of decay of 2 (3-13) where N1 and N2 are the numbers of (1) and (2) present at time t. Rearranging and collecting similar terms (3-14) Remembering that (3-15) we have This is a first order linear differential equation and can be solved using the method of (3-16) integrating factors which we show below. Multiplying both sides by, we have (3-17) The left hand side is now a perfect differential (3-18) Integrating from t=0 to t=t, we have

13 -13- (3-19) λ 2 t N 2 e N 0 2 = λ 1 λ 2 λ 1 0 N 1 ( ( λ 2 λ 1)t e 1) (3-20) Multiplying by and rearranging gives (3-21) where is the number of species (2) present at t=0. The first term in Equation (3-21) represents the growth of the daughter due to the decay of the parent while the second term represents the decay of any daughter nuclei that were present initially. Remembering that A2 = λ2n2, we can write an expression for the activity of 2 as (3-22) These two equations, (3-21) and (3-22) are the general expressions for the number of daughter nuclei and the daughter activity as a function of time, respectively. The general behavior of the activity of parent and daughter species, as predicted by Equation (3-22), is shown in Figure 3-6. As one expects qualitatively for, the initial activity of the daughter is zero, rises to a maximum, and if one waits long enough, eventually decays. Thus there must be a time when the daughter activity is the maximum. We can calculate this by noting the condition for a maximum in the activity of (2) is

14 -14- (3-23) Taking the derivative of Equation (3-21) and simplifying, (3-24) Solving for t (3-25) All of this development may seem like something that would be best handled by a computer program or just represents a chance to practice one s skill with differential equations. But that is not true. It is important to understand the mathematical foundation of this development to gain insight into practical situations that such insight offers. Let us consider some cases that illustrate this point. Consider the special case where λ1 = λ2. Plugging into Equations (3-21) or (3-22), or a computer program based upon them leads to a division by zero. Does nature therefore forbid λ1 from equaling λ2 in a chain of decays? Nonsense! One simply understands that one must redo the derivation (Equations (3-13) through (3-21)) of Equations (3-21) and (3-22) for this special case (see homework). Let us now consider a number of other special cases of Equations (3-21) and (3-22) that are of practical importance. Suppose the daughter nucleus is stable (λ 2 = 0). Then we have (3-26) (3-27)

15 -15- (3-28) These relations are shown in Figure 3-7. They represent the typical decay of many radionuclides prepared by neutron capture reactions, the type of reaction that commonly occurs in a nuclear reactor. In Figure 3-8, we show the activity relationships for parent and daughter (as predicted by Equation (3-22)) for various choices of the relative values of the half- lives of the parent and daughter nuclides. In the first of these cases, we have t½ (parent) < t½ (daughter), i.e., the parent is shorter lived than the daughter. This is called the no equilibrium case because the daughter buildup (due to the decay of the parent) is faster than its loss due to decay. Essentially all of the parent nuclides are converted to daughter nuclides and the subsequent activity is due to the decay of the daughters only. Thus the name no equilibrium is used. Practical examples of this decay type are 131 Te 131 I, 210 Bi 210 Po, 92 Sr 92 Y. This situation typically occurs when one is very far from stability and the nuclei decay by β decay towards stability. A second special case of Equations (3-21) and (3-22) is called transient equilibrium (Figures 3-8b and 3-9a). In this case, the parent is significantly (~10x) longer- lived than the daughter and thus controls the decay chain. Thus λ2 > λ1 (3-29) In Equation (3-21), as t, (3-30) and we have

16 -16- (3-31) Substituting (3-32) we have (3-33) At long times, the ratio of daughter to parent activity becomes constant, and both species disappear with the effective half- life of the parent. The classic examples of this decay equilibrium are the decay of 140 Ba (t½ = 12.8 d) to 140 La (t½ = 40 hr) or the equilibrium between 222 Rn (t½ = 3.8 d) and its short- lived decay products. A third special case of Equations (3-21) and (3-22) is called secular equilibrium (Figure 3-8 (c+d), 3-9b). In this case, the parent is very much longer lived (~10 4 x) than the daughter or the parent is constantly being replenished through some other process. During the time of observation, there is no significant change in the number of parent nuclei present, although several half- lives of the daughter may occur. In the previous case of transient equilibrium, we had (3-34) Since we now also have λ1 < < < λ 2 (3-35) we can simplify even more to give (3-36)

17 -17- λ1n1 = λ2n2 (3-37) A1 = A2 In short, the activity of the parent and daughter are the same and the total activity of the sample remains effectively constant during the period of observation. The naturally occurring heavy element decay chains (see below) where 238 U 206 Pb, 235 U 207 Pb, 232 Th 208 Pb and the extinct heavy element decay series 237 Np 209 Bi are examples of secular equilibrium because of the long half- lives of the parents. Perhaps the most important cases of secular equilibrium are the production of radionuclides by a nuclear reaction in an accelerator, a reactor, a star or the upper atmosphere. In this case, we have Nuclear Reaction (2) (3-38) which produces the radionuclide 2 with rate R. If the reaction is simply the decay of a long- lived nuclide, then R=λ1N1 0 and N2 0 =0. Substitution into 3-21 gives the expression (3-39) If the reaction is slower than the decay or λ1 < < λ2 (3-40) It is most appropriate to say (since λ1 0) (3-41) or in terms of the activities (3-42)

18 -18- Equation (3-42) is known as the activation equation and is shown in Figure Initially the growth of the product radionuclide activity is nearly linear (due to the behavior of for small values of λt) but eventually the product activity becomes saturated or constant, decaying as fast as it is produced. At an irradiation time of one half- life, half the maximum activity is formed; after 2 half- lives, 3/4 of the maximum activity is formed, etc. This situation gives rise to the rough rule that irradiations that extend for periods that are greater than twice t½ of the desired radionuclide are usually not worthwhile. Equation (3-21) may be generalized to a chain of decaying nuclei of arbitrary length in using the Bateman equations (Bateman, 1910). If we assume that at t=0, none of the daughter nuclei are present,, we get (1) (2) (3),..., (n) where (3-43) These equations describe the activities produced in new fuel in a nuclear reactor. No fission or activation products are present when the fuel is loaded and they grow in as the reactions take place.

19 Example Consider the decay of a 1 µci sample of pure 222 Rn (t1/2 = 3.82 days). Use the Bateman equations to estimate the activity of its daughters ( 218 Po, 214 Pb, 214 Bi and 214 Po) after a decay time of 4 hours. The decay sequence is t1/2 3.82d 3.1m 26.8m 19.9m 164µsec activity A B C D E λ(10-4 s) x10 7 (Actually B/A = ) The reader should verify that for C, D and E, the only significant term is the term multiplying as it was for B. Thus for D/A, we have

20 -20- The reader should, as an exercise, compute the quantities of C and E present. 3.4 Branching Decay Some nuclides decay by more than one mode. Some nuclei may decay by either β + decay or electron capture; others by α- decay or spontaneous fission; still others by γ- ray emission or internal conversion, etc. In these cases, we can characterize each competing mode of decay by a separate decay constant λi for each type of decay where the total decay constant, λ, is given by the sum (3-44) Corresponding to each partial decay constant λi, there is a partial half- life where (3-45) and the total half- life, t1/2, is the sum of the reciprocals (3-46) The fraction of decays proceeding by the i th mode is given by the obvious expression (3-47) By analogy, the energy uncertainty associated with a given state, ΔE, through the Heisenberg uncertainty principle can be obtained from the lifetime contributed by each

21 -21- decay mode. If we introduce the definition ΔE Γ, the level width, then we can express Γ in terms of the partial widths for each decay mode Γi such that (3-48) where (3-49) where τi is the partial mean life associated with each decay mode. This approach is especially useful in treating the decay of states formed in nuclear reactions in which a variety of competing processes such as α emission, p emission, n emission, etc, may occur as the nucleus de- excites. In such cases, we can express the total width as Γ = Γ α + Γp + Γn (3-50) Example: Consider the nucleus 64 Cu (t1/2 = h). 64 Cu is known to decay by electron capture (61%) and β - decay (39%). What are the partial half- lives for EC and β - decay? What is the partial width for EC decay? Solution: λ = ln 2/12.700h = 5.46 x 10-2 h - 1 λ = λec + λβ = λec + (39/61)λEC λec = x 10-2 h - 1 t1/2 EC = (ln 2)/λEC = 20.8 h λβ = (39/61)λEC = x 10-2 h - 1

22 -22- t1/2 β = (ln 2)/λβ = 32.6 h τ EC = t1/2 EC /ln 2 = 30.0 h = s ΓEC = /τ EC = x MeV.s/ s = 6.1 x MeV All naturally occurring radioactive nuclei have extremely small partial widths. Did you notice that 64 Cu can decay into 64 Zn and 64 Ni? This is unusual but can occur for certain odd- odd nuclei (see Chapter 2) Natural Radioactivity There are approximately 70 naturally occurring radionuclides on earth. Most of them are heavy element radioactivities present in the natural decay chains, but there are several important light element activities, such as 3 H, 14 C, 40 K, etc. These radioactive species are ubiquitous, occurring in plants, animals, the air we breathe, the water we drink, the soil, etc. For example, in the 70 kg reference man, one finds ~ 4400 Bq of 40 K and ~ 3600 Bq of 14 C, i.e., about 8000 dis/s due to these two radionuclides alone. In a typical US diet, one ingests ~1 pci/day of 238 U, 226 Ra, and 210 Po. The air we breathe contains ~ 0.15 pci/l of 222 Rn, the water we drink contains >10 pci/l of 3 H while the earth s crust contains ~10 ppm and ~4 ppm of the radioelements Th and U. One should not forget that the interior heat budget of the planet Earth is dominated by the contributions from the radioactive decay of uranium, thorium, and potassium. The naturally occurring radionuclides can be classified as: (a) primordial - i.e., nuclides that have survived since the time the elements were formed (b) cosmogenic - i.e., shorter lived nuclides formed continuously by the interaction of cosmic rays with matter

23 -23- and (c) anthropogenic - i.e., a wide variety of nuclides introduced into the environment by the activities of man, such as nuclear weapons tests, the operation (or misoperation) of nuclear power plants, etc. The primordial radionuclides have half- lives greater than 10 9 years or are the decay products of these nuclei. This class includes 40 K (t1/2 = x 10 9 y), 87 Rb (t1/2 = 47.5 x 10 9 y), 238 U (t1/2 = x 10 9 y), 235 U (t1/2 = x 10 9 y) and 232 Th (t1/2 = x 10 9 y) as its most important members. (Additional members of this group are 115 In, 123 Te, 138 La, 144 Nd, 147 Sm, 148 Sm, 176 Lu, 174 Hf, 187 Re, and 190 Pt.) 40 K is a β - - emitting nuclide that is the predominant radioactive component of normal foods and human tissue. Due to the 1460 kev γ- ray that accompanies the β - decay, it is also an important source of background radiation detected by γ- ray spectrometers. The natural concentration in the body contributes about 17 mrem/year to the whole body dose. The specific activity of 40 K is approximately 855 pci/g potassium. Despite the high specific activity of 87 Rb of ~2400 pci/g, the low abundance of rubidium in nature makes its contribution to the overall radioactivity of the environment small. There are three naturally occurring decay series. They are the uranium (A = 4n + 2) series, in which 238 U decays through 14 intermediate nuclei to form the stable nucleus 206 Pb, the actinium or 235 U (A = 4n + 3) series in which 235 U decays through 11 intermediate nuclei to form stable 207 Pb and the thorium (A = 4n) series in which 232 Th decays through a series of 10 intermediates to stable 208 Pb (Figure 3-11). Because the half- lives of the parent nuclei are so long relative to the other members of each series, all members of each decay series are in secular equilibrium, i.e., the activities of each member of the chain are equal at equilibrium if the sample has not been chemically fractionated. Thus, the activity associated with 238 U in secular equilibrium with its

25 -25- importantly, 10 Be, 22 Na, 32 P, 33 P, 35 S and 39 Cl are also produced. These nuclei move into the troposphere through normal exchange processes and are brought by the earth s surface by rainwater. Equilibrium is established between the production rate in the primary cosmic ray interaction and the partition of the radionuclides amongst the various terrestrial compartments (atmosphere, surface waters, biosphere, etc.) leading to an approximately constant specific activity of each nuclide in a particular compartment. When an organism dies after being in equilibrium with the biosphere, the specific activity of the nuclide in that sample will decrease since it is no longer in equilibrium. This behavior allows these nuclides to act as tracers for terrestrial processes and for dating. 14 C (t1/2 = 5730 y) is formed continuously in the upper atmosphere by cosmic rays that produce neutrons giving the reaction n (slow) + 14 N 14 C + p or, in a shorthand notation, 14 N(n, p) 14 C. 14 C is a soft β - - emitter (Emax ~ 158 kev). This radiocarbon ( 14 C) reacts with oxygen and eventually exchanges with the stable carbon (mostly 12 C) in living things. If the cosmic ray flux is constant, and the terrestrial processes affecting 14 C incorporation into living things are constant, and there are no significant changes in the stable carbon content of the atmosphere, then a constant level of 14 C in all living things is found [corresponding to ~1 atom of 14 C for every atoms of 12 C or about 227 Bq/kg C]. When an organism dies, it ceases to exchange its carbon atoms with the pool of radiocarbon and its radiocarbon content decreases in accord with Equation (3-6). Measurement of the specific activity of an old object allows one to calculate the age of the object (see below).

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