Geoscience Reference
In-Depth Information
This equation expresses the number of daughter atoms
D
in terms of the number of
parent atoms
P
, both measured at time
t
.Taking the natural logarithm of Eq. (6.11)
allows the time
t
that has elapsed to be expressed in terms of the numbers of the
parent
P
and daughter
D
atoms present:
log
e
1
+
1
λ
D
P
t
=
(6.12)
This is therefore a clock based on the gradual reduction in the number of parent
atoms and build-up in the number of daughter atoms. The equations use the
probability that any atom will survive a time
t
. The actual proportion of parent
atoms that survive in any particular case is subject to statistical fluctuations, but
these are usually very small because
P
0
and
P
are very large numbers. In practice,
measurements of the ratio
D
P
would be made with a mass spectrometer.
5
The
half-life of the radioactive isotope used to estimate the age of a rock should ideally
be comparable to the age of the sample because, if they are very different, the
measurement of
D
/
P
in Eq. (6.12)may become impracticable.
Age determination is rarely as simple as applying Eq. (6.12). In most cases
there will have been an initial concentration of the daughter in the sample, and thus
the assumption that all the measured daughter atoms are a product of the parent
is not necessarily valid. Also, all systems are not
closed
,ashas been assumed
here; in other words, over time there may have been some exchange of the parent
and/or daughter atoms with surrounding material (in other words, the parent and
daughter atoms have not been kept in a 'closed' box). It is important to remember
that the parent atoms keep on decaying to their daughters regardless of what is
happening to the rock. The numbers of parent and daughter atoms can be used
to act as a simple clock only if the rock system is closed so that none of them
escape. In other words, the stop watch starts ticking only when the system closes
(the box shuts). If, when more than one dating method is used to estimate
t
, the
resulting ages are within analytical error, then these potential problems can be
neglected; in this case the various dates are
concordant
.Inthe case of
discordant
ages, the possibility of a non-closed or
open
system or the initial presence of the
daughter must be considered and, when possible, appropriate corrections made.
(See 'Other important factors' later in this section.)
The value for
t
that is estimated does not necessarily establish the time that
has passed since the formation of a rock; it may represent the time since the rock
crystallized or since cooling following a metamorphic event that heated the rock
such that chemical changes took place (see Section 7.8.5 for discussion).
Table 6.2 shows the decay products, half-lives and decay constants for those
radioactive isotopes most often used in dating geological samples. Also given are
the
heat-generation rates
for the four isotopes that are responsible for effectively
/
5
A mass spectrometer is an instrument used to establish the ratio of numbers of isotopes with
different masses. It depends for its operation on the fact that moving ions with different masses can
be separated by magnetic fields.
