Geochronology and radiogenic tracers - Geochemistry: An Introduction - page 77

and a little mathematics give the expression:

R 0
λ + v/ l e − z / l

N ( z ) = (4.21)

It can be seen, then, that if we know the rate of production R 0 and the attenuation distance
l , the abundance N ( z =0)of 10 Be at the surface provides a direct estimate of the rate v of

erosion.

4.1.3 The thorium-230 excess method

This is a somewhat different case, as this nuclide, with a radioactive half-life of 75 000
years, is not created by radiation but by the decay of a parent nuclide whose half-life is
long enough for its rate of production to be considered constant on time scales of less than
one million years. It is one of the examples of clocks based on a chain of radioactive decay
( Fig. 4.5 ) in which nuclides decay from one to another by β − radioactive processes.

The vast majority of intermediate nuclides have half-lives too short to act as useful clocks.
There are only four natural radioactive chains. These chains, which have a long-lived,
heavy nuclide as their parent, are those of 232 Th, 235 U, 238 U, and 237 Np. The first three
end with three isotopes of lead, 208 Pb, 207 Pb, 206 Pb, and we will see that their relative
abundances in modern lead are utilized as clocks. The fourth chain is that of neptunium-
237, an extinct radiogenic nuclide ending with the single isotope bismuth-209. It is not
detectable in natural products.
Uranium-238 decays to 234 Th, then to 234 U and, finally, to 230 Th, which is itself radioac-
tive. The two intermediate nuclides, 234 Th and 234 U, are so short-lived that we can ignore
them here. The change in the number N 230 Th of 230 Th nuclides can therefore be written as
the difference between production by the parent 238 U and radioactive decay:
d N 230 Th
d t

α or

=− λ 230 Th + λ 238 U (4.22)

The complete set of equations and their solutions were found by Bateman (1910) . After
a few hundreds of thousand years, the number of 230 Th nuclides reaches steady state
and the two terms of the right-hand side cancel out. This state, where all activity levels
are equal for all the nuclides in the chain, is known as secular equilibrium. The activity
238 U remains essentially constant during the time it takes to establish secular equilib-
rium (roughly 1/ λ 230 Th , or several 100, 000 years) and variation d 238 U /d t with time is

therefore zero. Allowing for this property and multiplying (4.22) by λ 230Th , we obtain:

d 230 Th − 238 U
d t

=−λ 230 Th 230 Th

238 U

(4.23)
in which the square brackets represent activity. The difference 230 Th − 238 U is known
as 230 Th excess (the term excess is taken by reference to the amount present at secular
equilibrium), and is written 230 Th ex . Equation (4.23) integrates as:
230 Th

−

230 Th

e − λ 230 Th t

ex =

(4.24)

ex,0