The K-Ar system

Potassium has three naturally occurring isotopes: ^{39}K, ^{40}K and ^{41}K. ^{40}K
is radioactive and undergoes branched decay to ^{40}Ca (by electron
emission λ_{β-} = 4.962 × 10^{-10}yr^{-1}) and ^{40}Ar (by electron capture
λ_{e} = 0.581 × 10^{-10}yr^{-1}) with a combined half life of 1.248 billion years.
The positron emission mechanism mentioned in Chapter 2.2 has an
extremely long half life and can therefore safely be neglected. In addition to
^{40}Ar, argon has two more stable isotopes: ^{36}Ar and ^{38}Ar. Argon makes up
~1% of the terrestrial atmosphere, with a fixed isotopic composition of
^{40}Ar/^{36}Ar = 298.5 and ^{38}Ar/^{36}Ar = 0.187. The argon contained in
K-bearing minerals is made up of a mixture of radiogenic (^{40}Ar^{*}) and
non-radiogenic gas (^{40}Ar_{∘}):

| (6.1) |

with λ the total decay constant of ^{40}K (λ = λ_{e} + λ_{β-} =
5.543 × 10^{-10}yr^{-1}).

The ^{40}K →^{40}Ar^{*} decay scheme forms the basis of the K-Ar geochronometer,
with the following age equation:

| (6.2) |

Taking into account the ‘contaminated’ (aka ‘excess’ or ‘inherited’)
argon component ^{40}Ar_{∘} and analysing several cogenetic rocks or minerals
with different K (and therefore ^{40}Ar^{*}) contents, an isochron equation can
be formed by division through ^{36}Ar:

| (6.3) |

which can be solved for t. Alternatively, we can simply assume
that all the inherited argon has an atmospheric origin, so that
(^{40}Ar∕^{36}Ar)_{∘} = 298.5.

From an analytical perspective, K-Ar dating is a two step process. Because
K (an alkali metal) and Ar (a noble gas) cannot be measured on the
same analytical equipment, they must be analysed separately on two
different aliquots of the same sample. This limitation is overcome by
the ^{40}Ar/^{39}Ar technique, which is a clever variation of the K-Ar
method. The idea is to subject the sample to neutron irradiation and
convert a small fraction of the ^{39}K to synthetic ^{39}Ar, which has a
half life of 269 years. The age equation can then be rewritten as
follows:

| (6.4) |

where ‘x’ stands for ‘sample’ and J is a constant of proportionality
which encapsulates the efficiency of the ^{39}K (n,p) ^{39}Ar reaction and into
which the factor λ∕λ_{e} is folded as well. The J-value can be determined by
analysing a standard of known age t_{s} which was co-irradiated with the
sample:

| (6.5) |

In which the subscript ‘s’ stands for ‘standard’. The great advantage of
equation 6.4 over 6.2 is that all measurements can be completed on the
same aliquot and using a single instrument, namely a noble gas mass
spectrometer, which can analyse extremely small (down to μg-sized)
samples.

The ^{40}Ar/^{39}Ar-method also allows the analyst to investigate the extent
of argon loss by means of stepwise heating experiments. This is done
by degassing the sample under ultra-high vacuum conditions in a
resistance furnace. At low temperatures, the weakly bound Ar is released,
whereas the strongly bound Ar is released from the crystal lattice at
high temperatures until the sample eventually melts. Plotting the
apparent ages against the cumulative fraction of ^{39}Ar released yields
an ^{40}Ar/^{39}Ar age spectrum (Figure 6.1). If a rock or mineral has
remained closed since its formation, the ^{40}Ar/^{39}Ar-ratio should remain
constant over the course of the different heating steps, forming an ‘age
plateau’. More complex (e.g. rising) release spectra, on the other
hand, are diagnostic of complex thermal histories featuring partial
argon loss. ‘saddle’ shaped release spectra are indicative of ‘excess’
argon. The composition of the inherited argon gas can be determined
using a variant of the isochron method, assuming that all ^{36}Ar is
inherited:

| (6.6) |

If the Ar contamination is constant throughout the entire sample, then the -measurements will be arranged along a linear trend whose slope is a function of and, hence, the age.

The K-Ar and ^{40}Ar/^{39}Ar-methods are some of the most widely used
geochronometers and important tools in the calibration of the geologic time
scale. The method is applicable to rocks and minerals > 10^{6}yr. Obviously,
younger materials require more careful treatment of the inherited argon
components.

- Magmatic rocks: formation ages can only be obtained for rapidly cooled volcanic rocks, using either mineral separates (sanidine, biotite, hornblende) or whole rocks. Pyroclastics and obsidian may yield reliable ages only if they are unaltered and contain little non-radiogenic argon. Plutonic rocks typically cool much slower than volcanic rocks and generally yield cooling ages rather than formation ages.
- Sedimentary rocks: K-Ar dating of authigenic mineral phases has often been attempted but remains difficult. Glauconite has been used successfully in some cases. Dating detrital minerals such as white mica (muscovite, phengite) in fluvial sediments is frequently used to study the metamorphic history of the hinterland.
- Metamorphic rocks: pelitic metamorphic rocks tend to be rich
in K-bearing micas and amphiboles, which can easily be dated
with the K-Ar and
^{40}Ar/^{39}Ar methods, but require careful interpretation. In high grade metamorphic terranes, the apparent ages can either reflect the metamorphic crystallisation history or the postmetamorphic cooling history. Low grade metamorphic terranes, on the other hand, carry a risk of containing inherited argon components from previous evolutionary stages.