1.4       The law of radioactive decay

 

The rate of decay of a radioactive parent nuclide to a stable daughter product is proportional to the number of atoms, n, present at any time t (Rutherford and Soddy, 1902):

 

              dn

          !  ))  =  8 n                                                                 [1.1]

              dt

 

where 8 is the constant of proportionality, which is characteristic of the radionuclide in question and is called the decay constant (expressed in units of reciprocal time). The decay constant states the probability that a given atom of the radionuclide will decay within a stated time. The term dn/dt is the rate of change of the number of parent atoms, and is negative because this rate decreases with time. Rearranging equation [1.1], we obtain:

 

              dn

              ))   =   ! 8 n                                                            [1.2]

              dt

 

This expression is integrated from t = 0 to t, given that the number of atoms present at time t = 0 is n0.

 

        n /                                      t/

          |     dn                              |

          |     ))    =   !  8             |   dt                                       [1.3]

          |      n                               |

          / n0                                   /t=0

Hence:                    n

                        ln  ))    =    !  8 t                                           [1.4]

                             n0

 

which can also be written as:

 

            n  =  n0 e!8t                                                                  [1.5]

 

A useful way of referring to the rate of decay of a radionuclide is the ‘half-life’, t1/2, which is the time required for half of the parent atoms to decay. Substituting n=n0/2 and t = t1/2 into equation [1.5], and taking the natural log of both sides, we obtain:

 

                            ln 2               0.693

             t1/2  =     )))   =        ))))                                      [1.6]

                             8                   8

 

The number of radiogenic daughter atoms formed, D*, is equal to the number of parent atoms consumed:

 

               D* = n0 ! n                                                               [1.7]

 

but n0 = n e8t (from equation [1.5]); so substituting for n0 in equation [1.7] yields:

 

               D* =  n e8t ! n                                                          [1.8]

 

i.e:           D* = n (e8t ! 1)                                                         [1.9]

 

If the number of daughter atoms at time t = 0 is D0 , then the total number of daughter atoms after time t is given as:

 

               D = D0  +  n (e8t ! 1)                                                [1.10]

 

This equation is the fundamental basis of most geochronological dating tools.

 

            In the uranium series decay chains, the daughter products of radioactive decay (other than three Pb isotopes) are themselves radioactive. Hence the rate of decay of such a daughter product is given by the difference between its production rate from the parent and its own decay rate:

 

           dn2/dt  =  n1 81  !  n2 82                                               [1.11]

 

where n1 and 81 are the abundance and decay constant of the parent, and n2 and 82 correspond to the daughter.

 

            But equation [1.5] can be substituted for n1 in equation [1.11] to yield:

 

          dn2/dt  =  n1,initial e!81 t 81  !  n2 82                                 [1.12]

 

This equation is integrated for a chosen set of initial conditions, the simplest of which sets n2=0 at t=0. Then:

 

                            81

     n2 82  =         )))))   n1,initial (e!81 t ! e!82 t)                      [1.13]

                          82 ! 81

 

This type of solution was first demonstrated by Bateman (1910) and is named after him. Recently, Catchen (1984) examined more general initial conditions for these equations, leading to more complex solutions.

 

 

1.4.1    Uniformitarianism

 

When using radioactive decay to measure the ages of rocks we must apply the classic principle of uniformitarianism (Hutton, 1788), by assuming that the decay constant of the parent radionuclide has not changed during the history of the Earth. It is therefore important to outline some evidence that this assumption is justified.

 

            The decay constant of a radionuclide depends on nuclear constants, such as a (= elementary charge2/Plank’s constant/velocity of light). Shlyakhter (1976) argued that the neutron capture cross-section of a nuclide is very sensitively dependent on nuclear constants. Because neutron absorbers (such as 143Nd and 145Nd) in the 1.8 Byr-old Oklo natural reactor give rise to the expected abundance increases in the product isotopes (Fig. 1.13) this constrains nuclear constants to have remained more or less invariant over the last 2 Byr.

 

            The possibility that physical conditions (e.g. pressure and temperature) could affect radionuclide decay constants must also be examined. Because radioactive decay is a property of the nucleus, which is shielded from outside influence by orbital electrons, it is very unlikely that physical conditions influence " or $ decay, but electron capture decay could be affected. Hensley et al. (1973) demonstrated that the electron capture decay of 7Be to 7Li is increased by 0.59% when BeO is subjected to 270 " 10 kbars pressure in a diamond anvil. This raises the question of whether the electron capture decay of 40K to 40Ar could be pressure dependant, affecting K)Ar dates. In fact this is very unlikely, because at high pressure-temperature conditions at depth in the Earth, K)Ar systems will be chemically open and unable to yield dates at all, while at crustal depths the pressure dependence of 8 will be negligible compared with experimental errors.

 

            There are a few ways in which the invariance of decay constants has been experimentally verified. Uranium series dates have been calibrated against coral growth bands back to one thousand years (section 12.4.2), the radiocarbon method has been calibrated against tree rings (dendrochronology) back to 10 kyr (section 14.1.4), and K-Ar ages have been calibrated against sea floor spreading rates over periods of several Myr (section 10.4). In addition, age agreements between systems with very different decay constants also provides supporting evidence.

 

            There have been many attempts to test the consistency of different decay constants by comparing ages for geologically ‘well behaved’ systems using different dating methods (e.g. Begemann et al., 2001). As well as verifying the uniformitarian assumption, such geological decay constant comparisons allow the values for poorly known decay schemes to be optimised by comparison with better-known decay schemes. At present, the uranium decay constants are the most well established, and therefore the basis for most comparisons. However, a promising new calibration is the ‘astronomical timescale’, based on the tuning of glacial cycles to the Earth’s orbital motions (section 10.4).

 

            The best approach to the improvement of decay constants is new and better measurement. Where this is not possible, a useful alternative is acceptance by the geological community of a ‘recommended value’. The most successful application of  this procedure was the IUGS Subcommission on Geochronology (Steiger and Jager, 1977). The recommendations served as a useful standard for 20 years but are now in need of revision. Therefore, Table 1.1 presents these values where applicable, together with a compilation of the best decay constant determinations obtained since that time, by various methods. Full references and further details are given in the appropriate chapters.

 

Table 1.1          Summary of decay constants and half lives of long-lived nuclides

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Nuclide            S & J 1977                  ‘Best value’                  Half life Reference

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

40K (40Ar)         5.81   E-11                                                          11.93   Byr Beckinsale and Gale (1969)

40K (40Ca)        4.962 E-10                                                            1.397 Byr

40K (total)         5.543 E-10                                                            1.25   Byr

 

87Sr                  1.42   E-11                  1.402   E-11                    49.44 Byr     Minster et al. (1982)

147Sm                                                   6.54     E-12                  106.0   Byr     Lugmair and Marti (1978)

176Lu                                                    1.867   E-11                    37.1   Byr     Scherer et al. (2001)

186Re                                                    1.666   E-11                    41.6   Byr     Smoliar et al. (1996)

190Pt                                                     1.477   E-12                  469.3   Byr     Brandon et al. (1999)

 

230Th                                                    9.1577 E-6                      75.69 kyr     Cheng et al. (2000)

232Th                                                    4.9475 E-11                    14.01 Byr     Jaffey et al. (1971)

231Pa                                                    2.116   E-5                      32.76 kyr     Robert et al. (1969)

234U                                                      2.826   E-6                    245.25 kyr     Cheng et al. (2000)

235U                  9.8485   E-10              unchanged                      703.8   Myr    Jaffey et al. (1971)

238U                  1.55125 E-10              unchanged                    4468.0   Myr    Jaffey et al. (1971)

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

 

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