Decay of Radioactive Parent into Stable/Unstable Daughter

Decay Process Overview

The simplest form of radioactive decay involves a radioactive parent nucleus P decaying with decay constant λ_P into a stable or unstable daughter nucleus D:

P → D (λ_P)

The rate of depletion of the number of radioactive parent nuclei N_P(t) is equal to the activity A_P(t) at time t, which is defined as the product λN(t) from Eq. (1.7):

dN_P(t)/dt = -λ_P N_P(t) = -A_P(t)

This leads to the following general integral form of the differential equation for N_P(t):

N_P(t) = N_P(0) e^-λ_Pt

Where N_P(0) is the initial number of radioactive nuclei at time t = 0.

Activity of Parent Nuclei

The activity A_P(t) of the parent nuclei at time t can be expressed as:

A_P(t) = λ_P N_P(t) = λ_P N_P(0) e^-λ_Pt

Where A_P(0) = λ_P N_P(0) is the initial activity of the radioactive substance.

Half-Life and Mean Life

The half-life T_1/2P of a radioactive substance P is defined as the time during which the number of radioactive nuclei decays to half of the initial value:

N_P(T_1/2P) = 1/2 N_P(0)

It is related to the decay constant λ_P through the following equation:

T_1/2P = (ln 2) / λ_P = 0.693 / λ_P

The mean life τ_P of a radioactive parent P is defined as the time required for the number of radioactive atoms or activity to fall to 1/e (approximately 36.8%) of the initial value. The relationship between the mean life and the decay constant is:

τ_P = 1 / λ_P

The mean life is related to the half-life by:

τ_P = 1.44 T_1/2P

Decay Curve Example

A typical radioactive decay is shown in the figure below, with the plot of activity A_P(t) of the parent substance against time. The decay follows the equation:

A_P(t) = A_P(0) e^-λ_Pt

Note: The activity decreases exponentially with time, and after one half-life, it falls to half of its initial value.