Principles of radioactive dating

Francis Albarède

The energy binding protons and neutrons in the nucleus is about a factor 106 larger than the energy of chemical reactions: radioactivity is therefore independent of the chemical bonding of atoms(in minerals or in solutions) as it is of the temperature and pressure inside the planets and even the Sun. In addition, radioactivity is a memoryless process (atoms do not age) and is therefore a nuclear event whose probability of occurrenceper unit of time, which wenote is independent of time. This probability, termed the decay constant, is specific to each radioactive nuclide. Radioactive decay, like incoming calls at a telephone exchange, is a prime example of a Poisson process, in which the number of events is proportional to the time over which the observation is made. In the absence of any other loss or gain, the proportion of parent atoms (or radioactive nuclides) disappearing per unit of time t is constant:

(1)

For a number of parent atoms P = P0 at time t = 0, this equation integrates as:

(2)

To determine the age of a system from the measurement of the number of parent atoms at the present time, we must know P0 and therefore,in this form, this equation is not a chronometer (a notable exception is the method 14C). For each parent atom, a daughter atom (or radiogenic nuclide) is created, usually of a single element, whose number can be noted D. In a closed system and for a stable daughter nuclide D, the number of parent and daughter atoms is constant, therefore:

(3)

The term is a measure of the accumulation of the radiogenic nuclide during timet and therefore:

(4)

Even if D and P are measured, this equation is no more a timing device than equation (2) unless we know the number of daughter atoms D0 at time t=0. A first class of chronometersarises when the condition D0D applies as for the U-Pb dating of zircons, for example:

and the K-Ar dating method. This age dates the isolation of the host mineral.

For a second class of chronometers, the condition D0D does not apply and we replace it by the principle of isotopic homogenization. We need not go into the existence of isotopic fractionation (both natural and analytical), a phenomenon which is of little importance in heavy elements and which, for the needs of geochronology, is simply eliminated by internal normalizationagainst some arbitrary reference ratio of stable isotopes, e.g., 86Sr/88Sr = 0.1194 for Sr. When carbonatesprecipitate out from seawater, the 87Sr/86Sr ratio is exactly the same in calcite as in the seawater from which it precipitates; as the mantle melts, 143Nd/144Nd is the same in the molten liquid as in the residue. Let us therefore divide Equation(3) by the number D’of atoms of a stable isotope of the same element as the radioactive nuclide represented by D. As the system is closed,D’ remains constantand therefore:

(5)

Example:

This is the standard ‘isochron’ equation.D/D’ represents the ratio of the radiogenic nuclide to its stable isotope (e.g., 87Sr/86Sr) and P/D’is the ‘parent/daughter’ ratio, in most cases proportional to an elemental ratio (e.g., 87Rb/86Sr). If two samples 1 and 2 formed at the same time from an isotopically homogeneous medium (ocean, magma), they share the same (D/D’)0 and, taking the 87Rb-87Sr system as an example, the time is obtained from:

This age dates the time at which the two samples last shared a same 87Sr/86Sr ratio. This method is commonly used for parent-daughter systems with a long half-life, typically 143Nd-144Nd, 176Lu-176Hf,187Re-187Os. A particular application combines the two chronometers 238U-206Pb and 235U-207Pb, in which the two parent nuclides on the one hand, and the two daughter nuclides on the other hand: as the U isotopes are not explicitly considered, this method is known as the Pb-Pb method.

A third class of chronometers relevant to astrobiology is that of extinct radioactivities. These have a short half-life (and therefore a large ). For large valuesof t, P becomes negligible and therefore the closed system condition reads:

(6)

Let uswrite this equation for a sample (spl) and divide it by D’:

(7)

in which we emphasize that the sample was born in isotopic equilibrium with the solar nebula (SN)(P’ is a stable isotope of the parent nuclide P). This equation is equivalent to:

(8)

The isotopic properties of the solar nebula are those of chondrites. For the example of the 26Al-26Mg chronometer, equation (7) reads:

This is the equation of the extinct radioactivity isochron (the conventional isochron cannot be used when all the parent nuclide has decayed away). When 26Al/24Mg is plotted against the 27Al/24Mg (note that the two nuclides of the latter ratio are stable), samples formed at the same moment in isotopic equilibrium with the solar nebula form a straight line with a slope indicative of the 26Al/27Al ratio of the gas at the time the system formed. The 26Al/27Al ratio of the solar nebula at the time the sample formed is obtained from:

If the 26Al/27Al ratio of the solar nebula at the reference time t=0 is assumed, the result can be converted into an age. This age dates the time at which the dated sample shared the same 26Al/27Al ratio as the solar nebula. If no history of the 26Al/27Al ratio is assumed for the solar nebula, dividing this equation for one sample by the same equation for a second sample gives the age difference between the two samples.This method is employed for a number of‘extinct’ short-lived nuclides, such as 41K-41Ca, 60Fe-60Ni, 53Mn-53Cr, 146Sm-142Nd, etc.