Chapter 2
Basic notions of radiometric geochronology

2.1 Isotopes and radioactivity

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,  ₈¹⁶O 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,  ₈¹⁷O, with identical chemical properties as  ₈¹⁶O, but slightly different physical properties (e.g. boiling temperature). Adding another neutron produces  ₈¹⁸O which, with 8 protons and 10 neutrons, is more than 10% heavier than  ₈¹⁶O. Due to this mass difference, the  ₈¹⁸O/ ₈¹⁶O ratio undergoes mass fractionation by several natural processes, forming the basis of  ₈¹⁸O/ ₈¹⁶O 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  ₈¹⁹O exists in nature.

2.2 Radioactivity

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,  ₈₂²⁰⁸Pb, which is the heaviest stable nuclide, has 44 more neutrons than protons. The unstable nuclides (or radionuclides), such as  ₈₂²⁰⁹Pb or  ₈¹⁹O 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:

1. α-decay
   The atomic nucleus (e.g.,  ₉₂²³⁸U,  ₉₂²³⁵U,  ₉₀²³²Th,  ₆₂¹⁴⁷Sm) loses an α particle, i.e. the equivalent of a ₂⁴He 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^-12m or less. In addition to the aforementioned U-Th-He method, α-decay is central to the ¹⁴⁷Sm-¹⁴³Nd (Section 4.2), ²³⁵U-²⁰⁷Pb, ²³⁸U-²⁰⁶Pb and ²³²Th-²⁰⁸Pb methods (Section 5).
2. 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  ₈¹⁹O discussed in Section 2.1 decays to  ₉¹⁹F 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 ⁴⁰K-⁴⁰Ca, ⁸⁷Rb-⁸⁷Sr (Section 4.2) and ¹⁴C-¹⁴N (Section 4.1) clocks. It also occurs as part of the ²³⁵U-²⁰⁷Pb, ²³⁸U-²⁰⁶Pb and ²³²Th-²⁰⁸Pb decay series (Sections 5 and 9). β⁺ decay is found in the ⁴⁰K-⁴⁰Ar system (Section 6).
3. 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^-10m wavelength), which is the diagnostic signal of electron capture. This mechanism occurs in the ⁴⁰K-⁴⁰Ar decay scheme (Section 6).
4. 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.  ₉₂²³⁸U 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 ~2×10⁶ 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 ²³⁵U, 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 ²³⁵U nuclei and generate a chain reaction. This forms the basis of nuclear reactors, the atom bomb and the ‘external detector’ method (Section 7.2).

2.3 The age equation

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 ¹⁴C, 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, ¹⁴C goes back ~50,000 years, ¹⁰Be 10 million and ⁴⁰K 10 billion years.

2.4 Decay series

Sometimes the radiogenic daughter (D₁) of a radioactive parent is radioactive as well, decaying to a daughter of its own (D₂), 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:

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 D₁ with time. Assuming that D₁ = 0 at t = 0:

(2.14)

If λ_P ≪ λ₁ (by a factor of 10 or greater), then e^-λ₁t becomes vanishingly small relative to e^-λ_Pt 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 D₁ and P remains constant through time. If λ_P ≪ λ₁, then λ₁ - λ_P ≈ λ₁, 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 ²³⁵U/²⁰⁷Pb, ²³⁸U/²⁰⁶Pb and ²³²Th/²⁰⁷Pb, 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∕λ₁, λ_P∕λ₂, 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).