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

The process is governed by the weak interaction, which is responsible for electron capture. In this interaction, a proton in the potassium nucleus is transformed into a neutron, releasing an electron (beta particle) and an electron antineutrino.

To find the wavelength, we can use the energy-frequency relationship:

$E = h f$

where $E$ is the energy emitted (1.46 MeV), $h$ is Planck's constant ($6.63 \times 10^{-34} J s$), and $f$ is frequency. We first convert the energy to joules:

$1.46 MeV = 1.46 \times 1.6 \times 10^{-13} J = 2.336 \times 10^{-13} J$

Now, we can find the wavelength using the relationship:

$\lambda = \frac{c}{f}$

Substituting for frequency:

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

where $c$ is the speed of light ($3 \times 10^8 m/s$).

Substituting in the values:

$\lambda = \frac{(6.63 \times 10^{-34} J s)(3 \times 10^8 m/s)}{2.336 \times 10^{-13} J} \approx 8.52 \times 10^{-12} m$

The emissions from a decaying potassium nucleus can be analyzed using spectroscopy. The distinct energy levels associated with the emitted photons (from both the electron capture and the secondary decay) will provide a unique signature. Observing the presence of specific gamma emissions can confirm the occurrence of electron capture or beta decay process. If emissions consist of beta particles, it suggests a beta decay process, whereas if photon emissions are prominent in the spectrum, it would indicate electron capture.