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.
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:
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).
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:
Integrating this equation over time yields:
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:
and bearing in mind that P∘ = P + D, we obtain:
This equation forms the foundation of most geochronological methods. It can be rewritten explicitly as a function of time:
The degree of instability of a radioactive nuclide can be expressed by λ or by the half life t1∕2, which is the time required for half of the parent nuclides to decay. This follows directly from Equation 2.3:
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.
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:
The increase (or decrease) of the number of atoms per unit time for each of the nuclides is given by:
|for P||: dP∕dt = -λPP||(2.9)|
|for D1||: dD1∕dt = λPP - λ1D1||(2.10)|
|for D*||: dD*∕dt = λ1D1||(2.11)|
The number of parent atoms P can be written as a function of t:
Plugging Equation 2.12 into 2.10 yields
Solving this differential equation yields the evolution of D1 with time. Assuming that D1 = 0 at t = 0:
If λP ≪ λ1 (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:
This means that the ratio of D1 and P remains constant through time. If λP ≪ λ1, then λ1 - λP ≈ λ1, from which it follows that:
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:
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:
Using Equation 2.20, this becomes:
Since each of the ratios λP∕λ1, λP∕λ2, etc. are vanishingly small, we can simplify Equation 2.23 as:
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).