An isotope of potassium $^{40}_{19}K$ is used to date rocks - AQA - A-Level Physics - Question 1 - 2017 - Paper 1

The fundamental interaction responsible for this decay is the weak interaction. This interaction is crucial for processes like electron capture, which involves a proton in the nucleus converting into a neutron with the emission of a beta-minus particle (electron) and an antineutrino.

The energy of the emitted gamma photon can be calculated using the relationship:

$E = hf$

where (E = 1.46 \text{ MeV} = 1.46 \times 10^6 \times 1.6 \times 10^{-19} \text{ J}$\approx 2.33 \times 10^{-13} \text{ J}$. The wavelength (\lambda) can be calculated using (\lambda = \frac{hc}{E}), where (h = 6.63 \times 10^{-34} \text{ J s} and \(c = 3.0 \times 10^8 \text{ m/s}.

Plugging in the values, we get:

$\lambda = \frac{(6.63 \times 10^{-34})(3.0 \times 10^8)}{2.33 \times 10^{-13}} \approx 8.52 \times 10^{-12} \text{ m}$.

To confirm the decay process occurring, one could detect the emitted beta particles (electrons) and antineutrinos in electron capture. Additionally, detecting gamma radiation in the case of gamma emission allows for differentiation between the decay pathways, including measurements of energy levels to distinguish between $^{40}K$ and $^{40}Ca$.