Basic notions of radiometric geochronology

Thanks to discoveries by Niels Bohr, Ernest Rutherford, Arnold
Sommerfeld, Joseph Thomson and James Chadwick, we know that rocks
and minerals are made of atoms, atoms are made of a nucleus and an
electron cloud, and the nucleus is made of nucleons of which there are two
kinds: protons and neutrons. The total number of nucleons in the atomic
nucleus is called the mass number (A). The number of protons (which
equals the number of electrons in a neutral atom) is called the atomic
number (Z). The chemical properties of a nuclide solely depend on the
atomic number, which therefore forms the basis of the Periodic Table
of Elements. The number of neutrons in the atomic nucleus may
take on a range of values for any given element, corresponding to
different isotopes of said element. For example, _{ 8}^{16}O is an isotope of
oxygen with 16 nucleons of which 8 are protons (and N = A-Z =
16-8 = 8 are neutrons). Adding one extra neutron to the nucleus
produces a second oxygen isotope, _{ 8}^{17}O, with identical chemical
properties as _{ 8}^{16}O, but slightly different physical properties (e.g. boiling
temperature). Adding another neutron produces _{ 8}^{18}O which, with 8
protons and 10 neutrons, is more than 10% heavier than _{ 8}^{16}O.
Due to this mass difference, the _{ 8}^{18}O/_{ 8}^{16}O ratio undergoes mass
fractionation by several natural processes, forming the basis of _{ 8}^{18}O/_{ 8}^{16}O
palaeothermometry (see the second half of this course). When we try to add
yet another neutron to the atomic nucleus of oxygen, the nucleus becomes
unstable and undergoes radioactive decay. Therefore, no _{ 8}^{19}O exists in
nature.

As mentioned before, the Periodic Table of Elements (aka ‘Mendeleev’s
Table’) arranges the elements according to the atomic number and the
configuration of the electron cloud. The equally important Chart
of Nuclides uses both the number of protons and neutrons as row
and column indices. At low masses, the stable nuclides are found
close to the 1 ÷ 1 line (N ≈ Z), with the radionuclides found at
higher and lower ratios. At higher atomic numbers, the stable nuclides
are found at higher mass numbers, reflecting the fact that more
neutrons are required to keep the protons together. For example,
_{ 82}^{208}Pb, which is the heaviest stable nuclide, has 44 more neutrons than
protons. The unstable nuclides (or radionuclides), such as _{ 82}^{209}Pb or
_{ 8}^{19}O may survive for time periods of femtoseconds to billions of
years depending on the degree of instability, which generally scales
with the ‘distance’ from the curve of stable nuclides. Radionuclides
eventually disintegrate to a stable form by means of a number of different
mechanisms:

- α-decay

The atomic nucleus (e.g.,_{ 92}^{238}U,_{ 92}^{235}U,_{ 90}^{232}Th,_{ 62}^{147}Sm) loses an α particle, i.e. the equivalent of a_{2}^{4}He nucleus. When these nuclei acquire electrons, they turn into Helium atoms, forming the basis of the U-Th-He chronometer, which is further discussed in Section 7.1. The recoil energy of the decay is divided between the α particle and the parent nucleus, which eventually relaxes into its ground state by emitting γ-radiation, i.e. photons with a wavelength of 10^{-12}m or less. In addition to the aforementioned U-Th-He method, α-decay is central to the^{147}Sm-^{143}Nd (Section 4.2),^{235}U-^{207}Pb,^{238}U-^{206}Pb and^{232}Th-^{208}Pb methods (Section 5). - β-decay

Comprises negatron (β^{-}) and positron (β^{+}) emission, in which either an electron or a positron is emitted from the nucleus, causing a transition of (N,Z) → (N-1,Z+1) for β^{-}decay and (N,Z) → (N+1,Z-1) for β^{+}decay. For example, the oxygen isotope_{ 8}^{19}O discussed in Section 2.1 decays to_{ 9}^{19}F by β^{-}emission. In contrast with α particles, which are characterized by discrete energy levels, β particles are characterised by a continuous energy spectrum. The difference between the maximum kinetic energy and the actual kinetic energy of any given emitted electron or positron is carried by a neutrino (for β^{+}decay) or an anti-neutrino (for β^{-}decay). Just like α decay, β decay is also accompanied by γ-radiation, arising from two sources: (a) relaxation into the ground state of the excited parent nucleus and (b) spontaneous annihilation of the unstable positron in β^{+}decay. β^{-}decay is important for the^{40}K-^{40}Ca,^{87}Rb-^{87}Sr (Section 4.2) and^{14}C-^{14}N (Section 4.1) clocks. It also occurs as part of the^{235}U-^{207}Pb,^{238}U-^{206}Pb and^{232}Th-^{208}Pb decay series (Sections 5 and 9). β^{+}decay is found in the^{40}K-^{40}Ar system (Section 6). - electron capture

This is a special form of decay in which an ‘extra-nuclear’ electron (generally from the K-shell) is captured by the nucleus. This causes a transformation of (N,Z) → (N+1,Z-1), similar to positron emission, with which it often co-exists. The vacant electron position in the K-shell is filled with an electron from a higher shell, releasing X-rays (~ 10^{-10}m wavelength), which is the diagnostic signal of electron capture. This mechanism occurs in the^{40}K-^{40}Ar decay scheme (Section 6). - nuclear fission

Extremely large nuclei may disintegrate into two daughter nuclei of unequal size, releasing large amounts of energy (~ 200 MeV). The two daughter nuclei move in opposite directions from the parent location, damaging the crystal lattice of the host mineral in their wake. The two daughter nuclides are generally radioactive themselves, giving rise to β-radiation before coming to rest as stable isotopes._{ 92}^{238}U is the only naturally occurring nuclide that undergoes this type of radioactive decay in measurable quantities, and even then it only occurs once for every ~2×10^{6}α-decay events. Nevertheless, the fission mechanism forms the basis of an important geochronological method, in which the damage zones or ‘fission tracks’ are counted (Section 7.2). Nuclear fission can also be artificially induced, by neutron irradiation of^{235}U, e.g.:(2.1) Note that every neutron on the left hand side of this formula generates three neutrons on the right hand side. The latter may react with further

^{235}U nuclei and generate a chain reaction. This forms the basis of nuclear reactors, the atom bomb and the ‘external detector’ method (Section 7.2).

A characteristic property of radioactive decay is its absolute independence of external physical and chemical effects. In other words, it is not affected by changes in pressure, temperature, or the molecular bonds connecting a radioactive nuclide to neighbouring atoms. This means that the rate at which a radioactive parent decays to a radiogenic daughter per unit time, i.e. dP∕dt only depends on P, the number of parent atoms present. The decay constant λ expresses the likelihood that a radioactive disintegration takes place in any given time (i.e., λ has units of atoms per atoms per year). This can be expressed mathematically with the following differential equation:

| (2.2) |

Integrating this equation over time yields:

| (2.3) |

where P_{∘} is the number of parent atoms present at time t = 0. Since this
number is generally unknown (one exception is ^{14}C, see Section 4.1),
Equation 2.3 generally cannot be used in this form. We can, however,
measure the present number of parent and daughter nuclides in the sample.
Rewriting Equation 2.3:

| (2.4) |

and bearing in mind that P_{∘} = P + D, we obtain:

| (2.5) |

This equation forms the foundation of most geochronological methods. It can be rewritten explicitly as a function of time:

| (2.6) |

The degree of instability of a radioactive nuclide can be expressed
by λ or by the half life t_{1∕2}, which is the time required for half of
the parent nuclides to decay. This follows directly from Equation
2.3:

| (2.7) |

As a rule of thumb, the detection limit of a radiometric geochronometer
is reached after about 10 half lives. Thus, ^{14}C goes back ~50,000 years,
^{10}Be 10 million and ^{40}K 10 billion years.

Sometimes the radiogenic daughter (D_{1}) of a radioactive parent is
radioactive as well, decaying to a daughter of its own (D_{2}), which may be
radioactive again etc., until a stable daughter (D_{*}) is reached. Considering
the simplest case of one intermediate daughter:

| (2.8) |

The increase (or decrease) of the number of atoms per unit time for each of the nuclides is given by:

for P | : dP∕dt = -λ_{P}P | (2.9) |

for D_{1} | : dD_{1}∕dt = λ_{P}P - λ_{1}D_{1} | (2.10) |

for D_{*} | : dD_{*}∕dt = λ_{1}D_{1} | (2.11) |

The number of parent atoms P can be written as a function of t:

| (2.12) |

Plugging Equation 2.12 into 2.10 yields

| (2.13) |

Solving this differential equation yields the evolution of D_{1} with time.
Assuming that D_{1} = 0 at t = 0:

| (2.14) |

If λ_{P} ≪ λ_{1} (by a factor of 10 or greater), then e^{-λ1t} becomes
vanishingly small relative to e^{-λPt} after a sufficiently long time so that
Equation 2.14 can be simplified:

| (2.15) |

or, alternatively:

| (2.16) |

This means that the ratio of D_{1} and P remains constant through time.
If λ_{P} ≪ λ_{1}, then λ_{1} - λ_{P} ≈ λ_{1}, from which it follows that:

| (2.17) |

Rearranging:

| (2.18) |

or, equivalently:

| (2.19) |

This is the secular equilibrium in which the number of atoms of both
radioactive members is proportional to their respective half lives. In the
geochronological isotope systems ^{235}U/^{207}Pb, ^{238}U/^{206}Pb and ^{232}Th/^{207}Pb,
the lead isotopes are the end points of a long decay series comprised of
several α and β^{-} disintegrations, in which the decay constants of the parent
nuclide is orders of magnitude shorter than the other nuclides in the
chain. For a decay series like that, Equation 2.18 can be generalised
to:

| (2.20) |

This means that the entire series is in equilibrium, so that all members
occur in mutually constant proportions. The number of atoms of the stable
end member D_{*} is given by:

| (2.21) |

Using Equation 2.20, this becomes:

| (2.22) |

or

| (2.23) |

Since each of the ratios λ_{P}∕λ_{1}, λ_{P}∕λ_{2}, etc. are vanishingly small, we can
simplify Equation 2.23 as:

| (2.24) |

This means that the accumulation of the final Pb isotope of the aforementioned three decay series is only a function of the decay of the parent isotope. All intermediate decay steps are therefore inconsequential. In rare cases, however, the isotopic equilibrium is disturbed by a dissolution or recrystallisation event, say. The intermediate parent/daughter pairs can then be used to date phenomena occurring over much shorter time scales than those probed by the U-Pb method (Section 9).