Nuclear and Quantum Physics - Radioactive Decay

Radioactive decay is a random and spontaneous process. The probability of decay per unit time is called the decay constant $\lambda$.

The activity $A$ of a sample is the rate of decay, defined as $A = -\frac{dN}{dt} = \lambda N$, measured in Becquerels ($1 \text{ Bq} = 1 \text{ decay per second}$).

The number of undecayed nuclei $N$ decreases exponentially over time according to the law $N = N_0 e^{-\lambda t}$.

Half-life $T_{1/2}$ is the time taken for the number of undecayed nuclei (or the activity) to reduce to half of its initial value. It is related to the decay constant by $T_{1/2} = \frac{\ln 2}{\lambda}$.

In $\alpha$ decay, a nucleus emits a $^4_2\text{He}$ nucleus. In $\beta^-$ decay, a neutron transforms into a proton, emitting an electron ($e^-$) and an anti-neutrino ($\bar{\nu}_e$). In $\beta^+$ decay, a proton transforms into a neutron, emitting a positron ($e^+$) and a neutrino ($\nu_e$).

Gamma emission ($\gamma$) occurs when a nucleus transitions from an excited state to a lower energy state, emitting high-energy photons without changing the atomic or mass number.