K–Ca dating

Potassium has three naturally occurring isotopes: stable ³⁹K, ⁴¹K and radioactive ⁴⁰K. ⁴⁰K exhibits dual decay: through β-decay (E = 1.33 MeV), 89% of ⁴⁰K decays to ⁴⁰Calcium, and the rest decays to ⁴⁰Ar via electron capture (E = 1.46 MeV).[1] While ⁴⁰K comprises only 0.001167% of total potassium mass, ⁴⁰Ca makes up 96.9821% of total calcium mass; thus, ⁴⁰K decay leads to significantly greater ⁴⁰Ca enrichment than any other isotope.[5] The decay constant for the decay to ⁴⁰Ca is denoted as λ_β and equals 4.962 × 10⁻¹⁰ yr⁻¹; the decay constant to ⁴⁰Ar is denoted as λ_EC and equals 5.81 × 10⁻¹¹ yr⁻¹.

The general equation for the decay time of a radioactive nucleus that decays to a single product is:

${\displaystyle t=-\lambda \ln \left[{\frac {N}{N_{0}}}\right]=-{\frac {\ln(2)}{t_{\frac {1}{2}}}}\ln \left[{\frac {N}{N_{0}}}\right]}$

Where λ is the decay constant, t_1/2 is the half-life, N₀ is the initial concentration of the parent isotope, and N is the final concentration of the parent isotope.

Similarly, the equation for the decay time of a radioactive nucleus that decays to more than one product is:

${\displaystyle t=-{\frac {1}{\lambda _{t}}}\ln \left[{\frac {\lambda _{t}}{\lambda _{a}}}{\frac {N}{N_{0}}}+1\right]}$

Where a is the daughter product of interest, λ_a is the decay constant for daughter product a, and λ_t is the sum of decay constants for daughter products a and b.

This approach is taken in potassium-calcium dating where argon and calcium are both products of decay and can be expressed as:

${\displaystyle t=-{\frac {1}{\lambda _{t}}}\ln \left[{\frac {\lambda _{t}}{\lambda _{\beta }}}{\frac {\ce {Ca^{\ast }}}{\ce {K0}}}+1\right]}$

Where Ca^* is the measured amount of radiogenic ⁴⁰Ca in terms of parent isotope ⁴⁰K, and K₀ is the initial concentration of ⁴⁰K.

Age determination using Potassium-Calcium dating is best done using the isochron technique.[4] The isochron constructed for Pike’s Peak in Colorado and the K/Ca age for the granites in the area were found to be 1041 ± 32 Ma. Rb-Sr dating of the same batholith gave results of 1008 ± 13 Ma,[4] supporting the practicality of this method of dating. For comparison, the isochron method uses non-radiogenic ⁴²Ca to develop an isochron.

The following equation is used in the construction of the isochron plot:

${\displaystyle t={\frac {1}{\lambda _{t}}}\ln \left({\frac {{\ce {Ca}}-{\ce {Ca0}}}{{\ce {K}}\xi }}+1\right)}$

• t is time elapsed
• ξ is the branching ratio (= λ_β / λ _total) = 0.8952
• Ca₀ is the initial ⁴⁰Ca/⁴²Ca isotope ratio
• Ca is the ⁴⁰Ca/⁴²Ca isotope ratio
• K is the ⁴⁰K/⁴²Ca isotope ratio

• K–Ar dating
• Rb-Sr dating