Nuclei - Radioactivity

Radioactivity is a spontaneous process by which an unstable nucleus emits radiations like $\alpha$, $\beta$, or $\gamma$ rays to achieve stability.

The Law of Radioactive Decay states that the number of nuclei disintegrating per unit time is directly proportional to the total number of nuclei present at that instant: $\frac{dN}{dt} = -\lambda N$.

Decay Constant ($\lambda$) is the probability of decay of a nucleus per unit time. It is characteristic of the radioactive substance and independent of physical conditions.

Half-life ($T_{1/2}$) is the time interval during which the number of radioactive nuclei reduces to half of its initial value.

Mean life ($\tau$) is the average lifetime of all the nuclei in a radioactive sample, given by $\tau = \frac{1}{\lambda}$.

Activity ($R$) of a sample is the rate of disintegration. Its SI unit is the Becquerel ($1 \text{ Bq} = 1 \text{ disintegration per second}$). Another common unit is the Curie ($1 \text{ Ci} = 3.7 \times 10^{10} \text{ Bq}$).

$\alpha$-decay: The nucleus loses 2 protons and 2 neutrons, represented as $_Z^A X \rightarrow _{Z-2}^{A-4} Y + _2^4 He$.

$\beta^-$-decay: A neutron converts into a proton, emitting an electron and an antineutrino: $_Z^A X \rightarrow _{Z+1}^{A} Y + e^- + \bar{\nu}$.

$\gamma$-decay: The nucleus transitions from an excited state to a lower energy state by emitting a high-energy photon, with no change in $A$ or $Z$.