Lecture 12: Radioactivity

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

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

3
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 -gt 234Th
4He Mass-energy budget 238U 238.0508 amu 234Th
234.0436 4He 4.00260 mass defect 0.0046
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
4
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 -gt 87Sr
e n Mass-energy budget 87Rb 86.909186
amu 87Sr 86.908882 mass defect 0.0003
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
5
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 -gt 40Ar
e n 50V e -gt 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
6
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 300C 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 s1)
• 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

• 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)

• 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
11
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, 80C
  (!). 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.

Normalizing concentration of each element to CI
chondrite serves two purposesit 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-15 (requires
  remelting enriched source!)
• Typical oceanic crust has eNd10 (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

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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/l2gt1, all
concentrations and ratios are transient
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Decay chain systematics
For l1/l2ltlt1, 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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Decay chain systematics

• Consider further the case l1/l2 ltlt 1, which
  applies to all intermediates in the U and Th
  decay chains (parent l are all lt 10-16 s-1
  intermediates l are all gt10-12 s-1)
• In this case l2l1 l2, so 3.8a simplifies to

(3.9)

• Since l2 gt l1, the el2t 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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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 75000 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, el1t1, 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

• 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, t1/24.5 Ga
235U, t1/20.7 Ga
232Th, t1/214 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

• Example the oldest zircons on Earth (actually,
  the oldest anything on Earth), from the Jack
  Hills conglomerate in Australia

Peck et al. GCA 654215, 2001