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**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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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 a-decay b-decay positron emission electron capture spontaneous fission Each process transforms a radioactive parent nucleus into one or more daughter nuclei.

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a-decay Emission of an a-particle or 4He nucleus (2 neutrons, 2 protons) The parent decreases its mass number by 4, atomic number by 2. Example: 238U -> 234Th + 4He Mass-energy budget: 238U amu 234Th – 4He – mass defect amu = 6.86x10-13 J/decay = 1.74x1012 J/kg 238U = 7.3 kilotons/kg This is the preferred decay mode of nuclei heavier than 209Bi with a proton/neutron ratio along the valley of stability

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b-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: 87Rb -> 87Sr + e– + n Mass-energy budget: 87Rb amu 87Sr – mass defect amu = 4.5x10-14 J/decay = 3.0x1011 J/kg 87Rb = 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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**b+-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: 40K -> 40Ar + e+ + n 50V+ e– -> 50Ti + n + g 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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Spontaneous Fission Certain very heavy nuclei, particular those with even mass numbers (e.g., 238U and 244Pu) can spontaneously fission. Odd-mass heavy nuclei typically only fission in response to neutron capture (e.g., 235U, 239Pu) 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 (b-decays). You can see why some of the isotopes people worry about in nuclear fallout are 91Sr and 137Cs. Recoil of daughter products leave fission tracks of damage in crystals about 10 mm long, which only heal above ~300°C and are therefore useful for low-temperature thermochronometry.

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**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 l = 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) No = initial number of parent nuclei at time t = 0.

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Definitions For an exponential process*, the mean life t of a parent nuclide is given by the number present divided by the removal rate (recall this later when we talk about residence time): * For a linear process, this is off by a factor of 2 This is also the “e-folding” time of the decay: The half life t1/2 of a nucleus is the time after which half the parent remains: (3.3) The activity is decays per unit time, denoted by parentheses: (3.4)

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Decay of parent ln(lN)–ln(lNo) Activity Some dating schemes only consider measurement of parent nuclei because initial abundance is somehow known. 14C-14N: cosmic rays create a roughly constant atmospheric 14C inventory, so that living matter has a roughly constant 14C/C ratio while it exchanges CO2 with the environment through photosynthesis or diet. After death this 14C decays with half life 5730 years. Hence even through the daughter 14N is not retained or measured, age is calculated using:

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**Radiocarbon dating in practice**

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**Radiocarbon dating in practice**

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**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 Do and radiogenic D*. Each decay of one parent yields one daughter (an extension is needed for branching decays and spontaneous fission…), so in a closed system Under most circumstances, No is unknown, so substitute (3.5)

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**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) = So: (3.6) * Concentration ratios

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**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: 40K-40Ar dating Ar diffusivity is very high, so it is lost by minerals above some blocking temperature (~350 °C for biotite). We assume 40Aro = 0 and measure time since sample cooled through its blocking temperature. If 36Ar is used as the stable denominator isotope, an alternative to assuming 40Aro = 0 is to assume initial Ar of atmospheric composition. 40K/36Ar ratios are hard to measure well, so 40Ar-39Ar method is more accurate. The sample is irradiated with neutrons along with a neutron fluence standard of known age, converting 39K into 39Ar. 39K/40K is constant in nature, so one gets the 40K content of the sample by step-heating and measuring 39Ar/40Ar ratios, which can be done very precisely. 40K has a branching decay; it can either electron capture to yield 40Ar or b-decay to 40Ca. The relevant decay constant is therefore (lec/l40) Another example is U,Th-4He thermochronometry, which dates the passage of apatite through the blocking temperature for 4He retention, ~80°C (!). This is useful for dating the uplift of mountain ranges.

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**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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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/S y = intercept + x * slope

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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: 87Rb-87Sr The parent is 87Rb, half-life = 48.8 Ga The daughter is 87Sr, which forms only 7% of natural Sr. The stable, nonradiogenic reference isotope is 86Sr.

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**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. 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. 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 87Sr/86Sr of 0.698 implies solar nebula in chondrite formation region was well-mixed for Sr isotope ratio and all chondrites formed in a short time.

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**Example 2: Sm-Nd systematics**

Parent isotope is 147Sm, alpha decay half-life 106 Ga. Daughter isotope is 143Nd, 12% of natural Nd. Stable nonradiogenic reference isotope is 144Nd. 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 eNd: (3.7) where CHUR is the chondritic uniform reservoir, the evolution of a reservoir with bulk earth or bulk solar system Sm/Nd ratio and initial 143Nd/144Nd.

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**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. CI chondrite Sample • 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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**Example 2: Sm-Nd systematics**

One-stage Nd evolution 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, eNd(t)=0 in an igneous rock implies that the source was chondritic (or primitive) at the time of melting. Typical continental crust has eNd(t)=-15 (requires remelting enriched source!) Typical oceanic crust has eNd(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.

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**Example 3: Extinct nuclides**

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: 26Al-26Mg half-life of 26Al is 0.7 Ma. It is present in supernova debris. Since the parent is extinct, we cannot use equation 3.6 to measure an isochron 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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**Example 3: Extinct nuclides**

Example: 26Al-26Mg half-life of 26Al is 0.7 Ma. It is present in supernova debris. Wasserburg used stable 27Al as the second, stable isotope of Al to prove that 26Al was present when the Ca,Al-rich inclusions in chondrites formed. He demonstrated a correlation between 26Mg/24Mg and Al/Mg among coexisting mineral phases. The correlation proves the presence of live 26Al when the inclusion formed, and the slope is the initial 26Al/Al ratio, ~5 x 10-5 in the oldest objects. Given estimates of 26Al production in supernovae, this places a maximum of a few million years between nucleosynthesis and condensation of solids in the solar system!

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**Joys of the U,Th-Pb system**

238U decays to 206Pb through an elaborate chain of 8 a-decays and 6 b-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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**Decay chain systematics**

Consider a model system of three isotopes: Parent N1 decays to N2. Intermediate daughter N2 decays to N3. Terminal daughter N3 is stable. Evolution of this system is governed by coupled equations: Solution for N1 is already known (eqn. 3.2), so we have:

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**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 l1/l2. Solutions fall into two classes. For l1/l2>1, all concentrations and ratios are transient:

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**Decay chain systematics**

Consider the case l1/l2 << 1, which applies to all intermediates in the U and Th decay chains (parent l are all < s-1; intermediates l are all >10-12 s-1) In this case l2–l1 ~ l2, so 3.8a simplifies to: (3.9) Since l2 > l1, the e–l2t terms decay fastest, and after about 5 mean-lives of N2, we have (3.10) 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.

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**Decay chain systematics**

For l1/l2<<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 N2 to reach secular equilibrium. After this point the initial amount of N2 is the system no longer matters. Note the N3 does not participate in secular equilibrium, it just accumulates.

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**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.: 230Th, t1/2 = years 226Ra, t1/2 = 1600 years 210Pb, t1/2 = 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 CaCO3 growth. In this case N2o = 0, e–l1t~1, and

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**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 230Th (75 ka), 231Pa (33 ka), and 226Ra (1.6 ka) During melting in the mantle at pressure ≥2.5 GPa, the mineral garnet preferentially retains U over Th, leading to excess (230Th) in the melt. The melt would return to secular equilibrium within ~350 ka, so the presence of excess (230Th) 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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**U,Th-Pb geochronology 238U, t1/2=4.5 Ga 235U, t1/2=0.7 Ga**

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. 238U, t1/2=4.5 Ga 235U, t1/2=0.7 Ga 232Th, t1/2=14 Ga

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U,Th-Pb geochronology 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 238U-206Pb, 235U-207Pb, and 232Th-208Pb chronometers. Discordance due to recent Pb loss, such as during weathering, is resolved by coupling the two U-Pb systems to obtain a 207Pb-206Pb date Conveniently, 235U/238U 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 207Pb-206Pb 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).

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U,Th-Pb geochronology Any concordant group of samples plots on an isochron line in (207Pb/204Pb)*-(206Pb/204Pb)* 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 m=(238U/204Pb) ratio of about 9 (chondrite value = 0.7)

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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 207Pb-206Pb 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 (206Pb/238U)*–(207Pb/235U)* space, the concordia.

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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 206Pb from 207Pb, 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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**U,Th-Pb geochronology Peck et al. GCA 65:4215, 2001**

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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