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 the decay is the weak interaction (also known as weak nuclear force). This interaction is responsible for processes involving leptons and hadrons, including electron capture, where a proton in the nucleus is transformed into a neutron while emitting a positron and an antineutrino.

To calculate the wavelength of the photon, we use the energy-wavelength relation given by Planck's equation:
$E = \frac{hc}{\lambda}$
Where:

• $E$ is the energy (1.46 MeV),
• $h$ is the Planck constant ($6.626 \times 10^{-34} \, Js$),
• $c$ is the speed of light ($3.00 \times 10^8 \, m/s$), and
• $\lambda$ is the wavelength.
  First, convert the energy from MeV to Joules:
  $E = 1.46 \times 1.602 \times 10^{-13} \, J = 2.335 \times 10^{-13} \, J$
  Now use the rearranged equation to find $\lambda$:
  $\lambda = \frac{hc}{E}$
  Substituting the values:
  $\lambda = \frac{(6.626 \times 10^{-34} \, Js)(3.00 \times 10^8 \, m/s)}{2.335 \times 10^{-13} \, J}$
  Calculating yields:
  $\lambda \approx 8.52 \times 10^{-13} \text{ m}$
  This is the wavelength of the emitted photon.

The emissions from the decaying potassium nucleus can be analyzed using detection of emitted particles. For the electron capture process, one would expect to detect an X-ray or a characteristic gamma photon, which corresponds to the transition of an electron from a higher energy level to fill the vacancy left by the captured electron. In contrast, if the decay process involves beta decay, one would detect beta particles (electrons) being emitted. Therefore, by measuring the type of emissions, we can determine which decay process is occurring.