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1 Lecture 12: Radioactivity Questions –How and why do nuclei decay? –How do we use nuclear decay to tell time? –What is the evidence for presence of now extinct radionuclides in the early solar system? –How much do you really need to know about secular equilibrium and the U-series? Tools –First-order ordinary differential equations

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2 Modes of decay A nucleus will be radioactive if by decaying it can lower the overall mass, leading to larger (negative) nuclear binding energy –Yet another manifestation of the 2nd Law of thermodynamics Nuclei can spontaneously transform to lower mass nuclei by one of five processes – -decay –positron emission –electron capture –spontaneous fission Each process transforms a radioactive parent nucleus into one or more daughter nuclei.

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3 -decay Emission of an -particle or 4 He nucleus (2 neutrons, 2 protons) The parent decreases its mass number by 4, atomic number by 2. Example: 238 U -> 234 Th + 4 He Mass-energy budget: 238 U amu 234 Th – He– mass defect amu = 6.86x J/decay = 1.74x10 12 J/kg 238 U = 7.3 kilotons/kg This is the preferred decay mode of nuclei heavier than 209 Bi with a proton/neutron ratio along the valley of stability

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4 -decay Emission of an electron (and an antineutrino) during conversion of a neutron into a proton The mass number does not change, the atomic number increases by 1. Example: 87 Rb -> 87 Sr + e – + Mass-energy budget: 87 Rb amu 87 Sr – mass defect amu = 4.5x J/decay = 3.0x10 11 J/kg 87 Rb = 1.3 kilotons/kg This is the preferred decay mode of nuclei with excess neutrons compared to the valley of stability

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5 -decay and electron capture Emission of a positron (and a neutrino) or capture of an inner- shell electron during conversion of a proton into a neutron The mass number does not change, the atomic number decreases by 1. Examples: 40 K -> 40 Ar + e V+ e – -> 50 Ti + + In positron emission, most energy is liberated by remote matter-antimatter annihilation. In electron capture, a gamma ray carries off the excess energy. These are the preferred decay modes of nuclei with excess protons compared to the valley of stability

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6 Spontaneous Fission Certain very heavy nuclei, particular those with even mass numbers (e.g., 238 U and 244 Pu) can spontaneously fission. Odd-mass heavy nuclei typically only fission in response to neutron capture (e.g., 235 U, 239 Pu) There is no fixed daughter product but rather a statistical distribution of fission products with two peaks (most fissions are asymmetric). Because of the curvature of the valley of stability, most fission daughters have excess neutrons and tend to be radioactive ( -decays). You can see why some of the isotopes people worry about in nuclear fallout are 91 Sr and 137 Cs. Recoil of daughter products leave fission tracks of damage in crystals about 10 m long, which only heal above ~300°C and are therefore useful for low-temperature thermochronometry.

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7 Fundamental law of radioactive decay Each nucleus has a fixed probability of decaying per unit time. Nothing affects this probability (e.g., temperature, pressure, bonding environment, etc.) [exception: very high pressure promotes electron capture slightly] This is equivalent to saying that averaged over a large enough number of atoms the number of decays per unit time is proportional to the number of atoms present. Therefore in a closed system: (Equation 3.1) – N = number of parent nuclei at time t – = decay constant = probability of decay per unit time (units: s –1 ) To get time history of number of parent nuclei, integrate 3.1: (3.2) – N o = initial number of parent nuclei at time t = 0.

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8 Definitions For an exponential process*, the mean life of a parent nuclide is given by the number present divided by the removal rate (recall this later when we talk about residence time): (3.3) The half life t 1/2 of a nucleus is the time after which half the parent remains: –This is also the e-folding time of the decay: The activity is decays per unit time, denoted by parentheses: (3.4) * For a linear process, this is off by a factor of 2

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9 Decay of parent Activity ln( N)–ln( N o ) Some dating schemes only consider measurement of parent nuclei because initial abundance is somehow known. 14 C- 14 N: cosmic rays create a roughly constant atmospheric 14 C inventory, so that living matter has a roughly constant 14 C/C ratio while it exchanges CO 2 with the environment through photosynthesis or diet. After death this 14 C decays with half life 5730 years. Hence even through the daughter 14 N is not retained or measured, age is calculated using:

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10 Radiocarbon dating in practice

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11 Radiocarbon dating in practice

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12 Evolution of daughter isotopes Consider the daughter isotope D resulting from decays of parent isotope N. There may be some D in the system at time zero, so we distinguish initial D o and radiogenic D*. (3.5) Under most circumstances, N o is unknown, so substitute Each decay of one parent yields one daughter (an extension is needed for branching decays and spontaneous fission…), so in a closed system

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13 Evolution of daughter isotopes Parent and daughter isotopes are frequently measured with mass spectrometers, which only measure ratios accurately, so we choose a third stable, nonradiogenic nuclide S such that in a closed system S(t) = S o : (3.6) Concentration ratios *

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14 Evolution of daughter isotopes When the initial concentration of daughter isotope can be taken as zero, a date can be obtained using a single measurement of (D/S) t and (N/S) t on the same sample. Example: 40 K- 40 Ar dating –Ar diffusivity is very high, so it is lost by minerals above some blocking temperature (~350 °C for biotite). We assume 40 Ar o = 0 and measure time since sample cooled through its blocking temperature. –If 36 Ar is used as the stable denominator isotope, an alternative to assuming 40 Ar o = 0 is to assume initial Ar of atmospheric composition. – 40 K/ 36 Ar ratios are hard to measure well, so 40 Ar- 39 Ar method is more accurate. The sample is irradiated with neutrons along with a neutron fluence standard of known age, converting 39 K into 39 Ar. 39 K/ 40 K is constant in nature, so one gets the 40 K content of the sample by step- heating and measuring 39 Ar/ 40 Ar ratios, which can be done very precisely. – 40 K has a branching decay; it can either electron capture to yield 40 Ar or -decay to 40 Ca. The relevant decay constant is therefore ( ec / 40 ) Another example is U,Th- 4 He thermochronometry, which dates the passage of apatite through the blocking temperature for 4 He retention, ~80°C (!). This is useful for dating the uplift of mountain ranges.

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15 K-Ar dating vs. Ar-Ar dating Here is an example of the relative precision of K-Ar and Ar-Ar methods. The top point below is an Ar-Ar measurement, the others are K-Ar.

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16 Isochron method Most often the initial concentration of neither parent nor daughter is known, and more than one measurement is required to extract a meaningful date and also solve for the initial (D/S) ratio. Ideally we need multiple samples of equal age with equal initial ratio (D/S) o but different ratios (N/S). In this case equation 3.6 defines a line on an isochron plot: D/SD/S y = intercept + x * slope

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17 Isochron method The best way to guarantee that all samples have the same initial (D/S) ratio is to use different isotopes of the same element as D and S so that at high temperature diffusion will equalize this ratio throughout a system. The best way to guarantee that all samples have the same age is to use different minerals from the same rock, which chemically fractionate N from D when they crystallize. The whole rock can also form a data point. Example 1: 87 Rb- 87 Sr –The parent is 87 Rb, half-life = 48.8 Ga –The daughter is 87 Sr, which forms only 7% of natural Sr. –The stable, nonradiogenic reference isotope is 86 Sr.

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18 Example 1: Rb-Sr systematics Rb is an alkali metal, very incompatible during melting, with geochemical affinity similar to K. Sr is an alkaline earth, moderately incompatible during melting, with geochemical affinity similar to Ca. Age of the Chondritic meteorites from Rb-Sr isochron: A compilation of analyses of many mineral phases from many chondrites define a high precision isochron with an age of 4.56 Ga and an initial 87 Sr/ 86 Sr of implies solar nebula in chondrite formation region was well-mixed for Sr isotope ratio and all chondrites formed in a short time. Igneous processes like melting and crystallization therefore readily separate Rb from Sr and generate a wide separation of parent-daughter ratios ideal for quality isochron measurements.

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19 Example 2: Sm-Nd systematics Parent isotope is 147 Sm, alpha decay half-life 106 Ga. Daughter isotope is 143 Nd, 12% of natural Nd. Stable nonradiogenic reference isotope is 144 Nd. Nd isotopes are useful not only for dating but as tracers of large- scale geochemical differentiation. For these purposes, Nd isotope ratios are given in the more convenient form Nd : where CHUR is the chondritic uniform reservoir, the evolution of a reservoir with bulk earth or bulk solar system Sm/Nd ratio and initial 143 Nd/ 144 Nd. (3.7)

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20 Example 2: Sm-Nd systematics Both Nd and Sm are Rare-Earth elements (REE or lanthanides), a coherent geochemical sequence of ions of equal charge (+3), smoothly decreasing ionic radius from La to Lu, and hence smooth variations in partition coefficients. In most minerals, Nd is more incompatible than Sm (opposite of Rb-Sr system, where daughter Sr is more compatible than parent Rb). Hence after a partial melting event, the rock crystallized from the extracted melt phase has a lower Sm/Nd ratio than the source whereas the residual solids have a higher Sm/Nd ratio than the source. Sample CI chondrite Normalizing concentration of each element to CI chondrite serves two purposes…it makes primitive (aka chondritic) compositions a flat line and it takes out the sawtooth pattern from the odd-even effect in the solar abundances.

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21 Example 2: Sm-Nd systematics Since the rock crystallized from the extracted melt phase has a lower Sm/Nd ratio than the source, it evolves with time to a less radiogenic isotope ratio. Since the residual solids have a higher Sm/Nd ratio than the source they evolve with time to a more radiogenic isotope ratio. Initial Nd isotope ratios are reported by extrapolating back to the measured or inferred age of the sample and comparing to CHUR at that time. Thus, Nd (t)=0 in an igneous rock implies that the source was chondritic (or primitive) at the time of melting. Typical continental crust has Nd (t)=-15 (requires remelting enriched source!) Typical oceanic crust has Nd (t)=+10 (requires remelting depleted source!). This is evidence that the upper mantle (from which oceanic crust recently came) is depleted, and that the complementary enriched reservoir is the continents. The mean age of depletion of the upper mantle is ~2.5 Ga. One-stage Nd evolution

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22 Example 3: Extinct nuclides Since the parent is extinct, we cannot use equation 3.6 to measure an isochron We can show that certain nuclei with half-lives between ~1 and 100 Ma were present in the early solar system even though they are extinct now. Chronometry based on these short-lived systems gives superior time resolution for studies of early solar system processes. Example: 26 Al- 26 Mg half-life of 26 Al is 0.7 Ma. It is present in supernova debris. Instead, to interpret measured (D/S) ratios we need another, stable isotope S2 of the same element as short-lived parent N, so that we can expect (N/S2) o was constant. This gives a new equation for a line (a fossil isochron):

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23 Example 3: Extinct nuclides Wasserburg used stable 27 Al as the second, stable isotope of Al to prove that 26 Al was present when the Ca,Al- rich inclusions in chondrites formed. He demonstrated a correlation between 26 Mg/ 24 Mg and Al/Mg among coexisting mineral phases. The correlation proves the presence of live 26 Al when the inclusion formed, and the slope is the initial 26 Al/Al ratio, ~5 x in the oldest objects. Given estimates of 26 Al production in supernovae, this places a maximum of a few million years between nucleosynthesis and condensation of solids in the solar system! Example: 26 Al- 26 Mg half-life of 26 Al is 0.7 Ma. It is present in supernova debris.

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24 Joys of the U,Th-Pb system 238 U decays to 206 Pb through an elaborate chain of 8 -decays and 6 -decays, each with its own decay constant. To understand U-Pb (or Th-Pb) geochronology, we need to understand decay chains.

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25 Decay chain systematics Consider a model system of three isotopes: Parent N 1 decays to N 2. Intermediate daughter N 2 decays to N 3. Terminal daughter N 3 is stable. Evolution of this system is governed by coupled equations: Solution for N 1 is already known (eqn. 3.2), so we have:

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26 Decay chain systematics The general solution for n isotopes in a chain was obtained by Bateman (1910); for our 3 isotope case: (3.8a) (3.8b) The behavior of this system depends on 1 / 2. Solutions fall into two classes. For 1 / 2 >1, all concentrations and ratios are transient:

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27 Decay chain systematics Consider the case 1 / s -1 ) In this case 2 – 1 ~ 2, so 3.8a simplifies to: (3.9) Since 2 > 1, the e – 2 t terms decay fastest, and after about 5 mean-lives of N 2, we have This is the condition of secular equilibrium: the activities of the parent and of every intermediate daughter are equal. The concentration ratios are fixed to the ratios of decay constants. (3.10)

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28 Decay chain systematics For 1 / 2 <<1, the system evolves to a state called secular equilibrium in which the ratio of parent to intermediate daughter is fixed: It takes about 5 mean-lives of N 2 to reach secular equilibrium. After this point the initial amount of N 2 is the system no longer matters. Note the N 3 does not participate in secular equilibrium, it just accumulates.

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29 Applications of U-series disequilibria Violations of secular equilibrium are extremely useful for studying phenomena on timescales comparable to the intermediate half-lives, e.g.: – 230 Th, t 1/2 = years – 226 Ra, t 1/2 = 1600 years – 210 Pb, t 1/2 = 21 years Some systems incorporate lots of daughter and essentially no parent when they form. The daughter is unsupported and acts like the parent of an ordinary short-lived radiodecay scheme. Example: measuring accumulation rates in pelagic sediments, where Th adsorbs on particles but U remains in solution. Some systems incorporate lots of parent and essentially no daughter. Surprisingly, the daughter grows in on the time scale of its own decay, not that of the parent. Example: corals readily incorporate U and exclude Th during CaCO 3 growth. In this case N 2 o = 0, e – 1 t ~1, and

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30 Applications of U-series disequilibria During partial melting, the partition coefficients of parents and daughters may differ, producing a secular disequilibrium in melt and residue. For the timescales of mantle melting and melt extraction to the crust, the relevant isotopes are 230 Th (75 ka), 231 Pa (33 ka), and 226 Ra (1.6 ka) During melting in the mantle at pressure 2.5 GPa, the mineral garnet preferentially retains U over Th, leading to excess ( 230 Th) in the melt. The melt would return to secular equilibrium within ~350 ka, so the presence of excess ( 230 Th) in erupted basalts proves both the role of garnet in the source region and fast transport of melt to the crust.

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31 U,Th-Pb geochronology On timescales long enough that all intermediate nuclei reach secular equilibrium, U and Th systems can be treated as simple one-step decays to Pb. 238 U, t 1/2 =4.5 Ga 235 U, t 1/2 =0.7 Ga 232 Th, t 1/2 =14 Ga

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32 U,Th-Pb geochronology Conveniently, 235 U/ 238 U is globally constant (except for an ancient natural fission reactor in Gabon, and perhaps near Oak Ridge, TN) at 1/138. One does not have to measure U at all for this method. Since 207 Pb- 206 Pb age depends only on Pb isotope ratios, not Pb or U concentration, it is not affected by recent alteration whether Pb-loss or U-loss. Only addition of contaminant Pb or aging after alteration will affect the measured age (still need to correct for common Pb). Each of these chronometers can be used independently. If they agree, the sample is said to be concordant. However, Pb is mobile in many environments, and samples often yield discordant ages from the 238 U- 206 Pb, 235 U- 207 Pb, and 232 Th- 208 Pb chronometers. Discordance due to recent Pb loss, such as during weathering, is resolved by coupling the two U-Pb systems to obtain a 207 Pb- 206 Pb date

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33 U,Th-Pb geochronology Any concordant group of samples plots on an isochron line in ( 207 Pb/ 204 Pb)*-( 206 Pb/ 204 Pb)* space; the age is calculable from its slope. Initial Pb isotope ratios can be neglected for many materials with very high U/Pb ratios (e.g., old zircons), or measured on a coexisting mineral with very low U/Pb ratio (e.g., feldspar, troilite). In 1955 C.C. Patterson measured initial Pb in essentially U-free troilite (FeS) grains in the Canyon Diablo meteorite and thereby determined the initial Pb isotope composition of the solar system. It follows from measurements of terrestrial Pb samples that the Pb-Pb age of the earth is 4.56 Ga, and that the earth has evolved with a =( 238 U/ 204 Pb) ratio of about 9 (chondrite value = 0.7)

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34 U,Th-Pb geochronology If Pb was lost long enough in the past for continued decay of U to have any significant effect on Pb isotopes, the 207 Pb- 206 Pb may be impossible to interpret correctly. In this case, we turn to the concordia diagram (G. Wetherill). Consider the family of all concordant compositions: These equations parameterize a curve in ( 206 Pb/ 238 U)*–( 207 Pb/ 235 U)* space, the concordia.

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35 U,Th-Pb geochronology Imagine that a suite of samples underwent a single short-lived episode of Pb-loss at some time. This event did not fractionate 206 Pb from 207 Pb, so it moved the samples along a chord towards the origin in the concordia plot: If these now discordant samples age as closed systems, they remain on a line, whose intercepts with the concordia evolve along the concordia with time

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36 U,Th-Pb geochronology Example: the oldest zircons on Earth (actually, the oldest anything on Earth), from the Jack Hills conglomerate in Australia… Peck et al. GCA 65:4215, 2001

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