Simple parent-daughter pairs

There are two stable isotopes of carbon: ^{12}C and ^{13}C, and one naturally
occurring radionuclide: ^{14}C. The half life of ^{14}C is only 5,730 years, which is
orders of magnitude shorter than the age of the Earth. Therefore, no
primordial radiocarbon remains and all ^{14}C is cosmogenic (see Section 8 for
related methods). The main production mechanism is through secondary
cosmic ray neutron reactions with ^{14}N in the stratosphere: _{7}^{14}N (n,p) _{6}^{14}C.
Any newly formed ^{14}C rapidly mixes with the rest of the atmosphere
creating a spatially uniform carbon composition, which is incorporated into
plants and the animals that eat them. Prior to the industrial revolution, a
gram of fresh organic carbon underwent 13.56 (β^{-}) decays per minute.
When a plant dies, it ceases to exchange carbon with the atmosphere
and the ^{14}C concentration decays with time according to Equation
2.2:

| (4.1) |

where λ_{14} = 0.120968 ka^{-1}. Thus, the radiocarbon concentration is
directly proportional to the radioactivity, which can be measured by
β-counting. This can then be used to calculate the radiocarbon age by
rearranging Equation 2.3:

| (4.2) |

where (d^{14}C∕dt)_{∘} is the original level of β activity. This method was
developed by Willard Libby in 1949, for which he was awarded the Nobel
Prize in 1960. As mentioned before, (d^{14}C∕dt)_{∘} was 13.56 prior to the
industrial revolution, when thousands of tonnes of ‘old’ carbon were
injected into the atmosphere, resulting in a gradual lowering of the
radiocarbon concentration until 1950, when nuclear testing produced an
opposite effect, leading to a doubling of the atmospheric ^{14}C activity in
1963. Since the banning of atmospheric nuclear testing, radiocarbon
concentrations have steadily dropped until today, where they have almost
fallen back to their pre-industrial levels. But even prior to these
anthropogenic effects, ^{14}C concentrations underwent relatively large
fluctuations as a result of secular variations of the Earth’s magnetic field
and, to a lesser extent, Solar activity. These variations in (d^{14}C∕dt)_{∘} can be
corrected by comparison with a precisely calibrated production rate curve,
which was constructed by measuring the ^{14}C activity of tree rings
(dendrochronology).

Since the 1980’s, β-counting has been largely replaced by accelerator
mass spectrometry (AMS, see Section 3.1), in which the ^{14}C concentration
is measured directly relative to a stable isotope such as ^{13}C. Although this
has not significantly pushed back the age range of the radiocarbon method,
it has nevertheless revolutionised the technique by reducing the sample size
requirements by orders of magnitude. It is now possible to analyse
individual seeds or tiny fragments of precious objects such as the Turin
Shroud, which was dated at AD1260-1390.

Trace amounts of Rb and Sr are found in most minerals as substitutions for
major elements with similar chemical properties. Rb is an alkali metal
that forms single valent positive ions with an ionic radius of 1.48 Å,
which is similar to K^{+} (1.33 Å). Rb is therefore frequently found
in K-bearing minerals such as micas, K-feldspar and certain clay
minerals. Strongly evolved alkalic rocks such as syenites, trachites
and rhyolites often contain high Rb concentrations. Rb contains
two isotopes of constant abundance: ^{85}Rb (72.1854%) and ^{87}Rb
(27.8346%). Sr is an alkaline earth metal that forms bivalent positive ions
with a radius of 1.13 Å, similar to Ca^{2+} (ionic radius 0.99 Å). It
therefore substitutes Ca^{2+} in many minerals such as plagioclase, apatite,
gypsum and calcite in sites with 8 neighbours, but not in pyroxene
where Ca^{2+} has a coordination number of 6. Native Sr^{2+} can also
substitute K^{+} in feldspars (where radiogenic Sr is expected to be
found), but this substitution is limited and requires the simultaneous
replacement of Si^{4+} by Al^{3+} in order to preserve electric neutrality. Sr
therefore predominantly occurs in Ca-rich undifferentiated rocks
such as basalts. Sr contains four isotopes (^{84}Sr, ^{86}Sr, ^{87}Sr and ^{88}Sr)
with variable abundance due to the variable amount of radiogenic
^{87}Sr. However, the non-radiogenic ^{84}Sr/^{86}Sr and ^{86}Sr/^{88}Sr-ratios
are constant with values of 0.056584 and 0.1194, respectively. The
Rb-Sr chronometer is based on the radioactive decay of ^{87}Rb to
^{87}Sr:

| (4.3) |

Where ν indicates an antineutrino. The number of radiogenic ^{87}Sr atoms
produced by this reaction after a time t is given by:

| (4.4) |

where ^{87}Rb is the actual number of ^{87}Rb atoms per unit weight and
λ_{87} is the decay constant 1.42 × 10^{-11} a^{-1} (t_{1∕2} = 4.88×10^{10}a). In
addition to this radiogenic ^{87}Sr, most samples will also contain some
‘ordinary’ Sr. The total number of ^{87}Sr atoms measured is therefore given
by:

| (4.5) |

with ^{87}Sr_{∘} the initial ^{87}Sr present at the time of isotopic closure.
Combining Equations 4.5 and 4.3, we obtain:

| (4.6) |

Dividing this by the non-radiogenic ^{86}Sr yields

| (4.7) |

The method can be applied to single minerals or to whole rocks. Given
the very long half life, the optimal time scale ranges from the formation of
the solar system to the late Palaeozoic (300-400 Ma). To measure a Rb/Sr
age, the weight percentage of Rb is measured by means of X-ray
fluorescence, ICP-OES or similar techniques, and the ^{87}Sr/^{86}Sr ratio is
determined by mass spectrometry (isotope dilution). The ^{87}Rb/^{86}Sr-ratio is
then calculated as:

| (4.8) |

Where Ab(⋅) signifies ‘abundance’ and A(⋅) ‘atomic weight’.

Equation 4.8 can be used in one of two ways. A first method is to use an
assumed value for (^{87}Sr∕^{86}Sr)_{∘}, based on the geological context of
the sample. This method is only reliable for samples with a high
Rb/Sr ratio (e.g., biotite) because in that case, a wrong value for
(^{87}Sr∕^{86}Sr)_{∘} has only a minor effect on the age. A second and much better
method is to analyse several minerals of the same sample and plot
them on a (^{87}Rb∕^{86}Sr) vs. (^{87}Sr∕^{86}Sr) diagram (Figure 4.1). Due to
Equation 4.7, this should form a linear array (the so-called isochron)
with slope (e^{λ87t} - 1) and intercept (^{87}Sr∕^{86}Sr)_{∘}. Both parameters
can be determined by linear regression, allowing us to quantify the
‘goodness of fit’ of the data and obviating the need to assume any initial
Sr-ratios.

The elements Neodymium (Z=60) and Samarium (Z=62) are so-called ‘rare
earth elements’. All elements of this family have similar chemical properties.
They nearly all form 3+ ions of roughly equal albeit slightly decreasing size
with atomic number. The ionic radius of Nd and Sm is 1.08 and 1.04
Å, respectively. As the name suggests, rare earth elements rarely
form the major constituents of minerals. One notable exception is
monazite, which is a rare earth phosphate. In most cases, the rare
earth elements are found in trace amounts of up to 0.1% in apatite
[Ca_{5}(PO_{4})_{3}(OH,Cl,F)] and zircon [ZrSiO_{4}]. Both Sm and Nd are
slightly enriched in feldspar, biotite and apatite and thus tend to be
found in higher concentrations in differentiated (alkalic) magmatic
rocks.

Because their chemical properties are so similar, geological processes are
rarely capable of fractionating the Sm and Nd concentrations. Therefore,
the Sm/Nd ratio in most rocks generally falls in a narrow range of
0.1 to 0.5 (the Sm/Nd ratio of the solar system being 0.31). One
exception is garnet, in which Sm/Nd ratios > 1 have been found. Partial
melting of mafic minerals such as pyroxene and olivine produces
lower Sm/Nd ratios in the fluid phase than the solid residue. The
Sm/Nd ratio of magmatic rocks therefore decreases with increasing
differentiation.

Natural Sm contains seven naturally occurring isotopes, three of
which are radioactive (^{147}Sm, ^{148}Sm and ^{149}Sm). Only ^{147}Sm has a
sufficiently short half life to be useful for geochronology. Nd also
contains seven isotopes, of which only one is radioactive (^{144}Nd)
but with a very long half life. ^{143}Nd is the radiogenic daughter of
^{147}Sm and is formed by α-decay. This forms the basis of the Sm-Nd
chronometer. Analogous to the Rb-Sr method (Equation 4.4), we can
write:

| (4.9) |

Hence:

| (4.10) |

With λ_{147} = 6.54×10^{-12}a^{-1} (t_{1∕2} = 1.06×10^{11}a). Since most samples
contain some initial Nd, the preferred way to calculate Sm/Nd ages is by
analysing several minerals in a rock and create an isochron, similar to the
Rb/Sr method (Section 4.3):

| (4.11) |

All measurements are done by mass spectrometry using isotope dilution.
Because of the identical atomic masses of ^{147}Sm and ^{147}Nd, it is
necessary to perform a chemical separation between Sm and Nd prior to
analysis.

The Sm/Nd method is generally applied to basic and ultrabasic igneous rocks (basalt, peridotite, komatiite) of Precambrian to Palaeozoic age. The method thus complements the Rb/Sr method, which is preferentially applied to acidic rock types. The Sm/Nd method can also be applied to high grade metamorphic rocks (granulites, eclogites) as well as meteorites (shergottites, nahklites). Since the rare earths are significantly less mobile than Rb and Sr, the Sm/Nd is more reliable in rocks that have been disturbed by weathering or metamorphism.