Geochronology - The Solid Earth - page 238

all the radioactive heating in the Earth (see Chapter 7). All the radiations produced
(except the neutrinos 6 that accompany every decay) contribute heat as they are
absorbed within about 30 cm of their origin. The neutrinos interact so little with
matter that they penetrate the whole Earth and escape.
Radioactive decay schemes are not always simple. A series of disintegrations
may take place before a stable daughter nuclide is formed. Of those isotopes
that are commonly used for geological dating, 87 Rb and 147 Sm undergo a simple
decay to stable daughters (

decay, respectively); some of the others
undergo a sequence of decays (Table 6.2).

decay and

Decay series
The long series of decays undergone by the isotopes of uranium and thorium can
be described by a series of equations like Eq. (6.2). Suppose that the radioactive
isotope decays in series to X 1 ,X 2 ,...,X n and the isotope X n is the stable daughter.
Let the number of atoms of each isotope present at time t be P , P 1 , P 2 ,..., P n − 1 ,
D and let the decay constants be λ n − 1 .

The number of atoms of each isotope in the series increases due to the decay
of its parent and is reduced as the result of its own radioactive decay. Equation
(6.2) can be applied to each disintegration in the series:
d P
d t =− λ P
d P 1
d t

λ , λ 1 , λ 2 ,...,

=− λ 1 P 1 + λ P

d P 2
d t

=− λ 2 P 2 + λ 1 P 1
.

(6.13)

d P n − 1
d t

=− λ n − 1 P n − 1 + λ n − 2 P n − 2

d D
d t = λ n − 1 P n − 1

The general solution to these equations is more complicated than Eq. (6.5).
The solution for, say, the i th isotope in the chain involves exponentials for all
the decays from the parent to isotope i inclusive. (In physics, these equations are
known as the Bateman relations .)
For the decay schemes of uranium and thorium, the half-life of the first decay
is many orders of magnitude greater than those of the subsequent disintegrations

6

A neutrino is a particle produced during nuclear reactions, which has no charge and little interaction
with matter. A remarkable illustration of the neutrino's penetrating ability is that the burst of
neutrinos detected on 23 February 1987, which came from the supernova explosion in the Greater
Magellanic Cloud high in the southern sky, was detected in the northern hemisphere, in Japan and
the U.S.A., after they had passed through the Earth.