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 of $^{40}K$ into argon is the weak interaction (or weak nuclear force). This interaction allows for decay processes involving leptons and quarks and is essential for processes like beta decay, where changes in the composition of the nucleus occur.

To find the wavelength, we can use the energy-wavelength relation. Given that the energy of the gamma photon is 1.46 MeV, we first convert this energy into joules:

$E = 1.46 \text{ MeV} = 1.46 \times 1.602 \times 10^{-13} \text{ J} = 2.33 \times 10^{-13} \text{ J}$

Next, using the relation:

$E = \frac{hc}{\lambda}$

we can rearrange to find the wavelength:

$\lambda = \frac{hc}{E}$

Substituting in the values of $h$ (Planck's constant, $6.626 \times 10^{-34} \text{ J s}$) and $c$ (the speed of light, $3.00 \times 10^{8} \text{ m/s}$), we have:

$\lambda = \frac{(6.626 \times 10^{-34} \text{ J s})(3.00 \times 10^{8} \text{ m/s})}{2.33 \times 10^{-13} \text{ J}}$

Calculating this gives:

$\lambda \approx 8.52 \times 10^{-13} \text{ m}$

The emissions from a decaying potassium nucleus can be analyzed using a detector like a Geiger counter or a scintillation detector. Each decay process emits specific particles or energy signatures. For electron capture, the emission would primarily be low-energy photons, while beta decay would produce beta particles (electrons) along with higher-energy photons. By measuring the type and energy of the emitted particles, one can determine which decay process is occurring.