Radioactive Decay

There are about 2500 known nuclides, but only less than 300 of them are stable, the rest are unstable. Those unstable nuclides decay to form other nuclides by emitting particles and electromagnetic radiation. This emission of particles and electromagnetic radiation is a process called radioactivity.

If a given sample contains N radioactive nuclei, the rate at which nuclei will decay is proportional to N. This is expressed as

\(-{dN \over dt} = \lambda N\),     (1)

where \(\lambda\) is the decay constant or disintegration constant and has specific value for every radionuclide. Its unit is inverse second, \(s^{-1}\).

To find N, first, we rearranged equation (1),

\({dN \over N} = -\lambda dt\),     (2)

integrating (2), we have

\(\int \limits_{N_0}^N {dN \over N} = -\lambda \int \limits_{t_0}^t\),

or

\(\text{ln}\; N – \text{ln}\; N_0 = -\lambda (t – t_0)\),     (3)

where \(N_0\) is the number of radioactive nuclei in the sample at time \(t_0\).

Rearranging (3), and letting \(t_0=0\), (3) will become

\(\text{ln}\; {N \over N_0} = -\lambda t\).     (4)

Taking the exponential of both sides of equation (4),

\({N \over N_0} = e^{-\lambda t}\)

rearranging the above equation, we now have

\(N=N_0 e^{-\lambda t}\).     (5)

Equation (5) is the radioactive decay in which \(N_0\) is the number of radioactive nuclei in the sample at \(t=0\) and N is the remaining number of radioactive nuclei at any succeeding time t.

To find the decay rate R, we differentiate equation (5),

\(R=-{dN \over dt}=\lambda N_0 e^{-\lambda t}\)

or

\(R=R_0 e^{-\lambda t}\),     (6)

where \(R_0\) is the decay rate at \(t=0\) and R is the rate at any following time t.

From equation (6), equation (1) can be rewritten as

\(R=\lambda N\),     (7)

where R and N that have not yet undergone decay must be evaluated at the same time.

The activity of a sample is the total decay rate R of a sample of one or more radionuclides. The SI unit for activity is becquerel (Bq), named after Henri Becquerel who discovered radioactivity.

1 Bq = 1 decay per second

Another unit for activity is the curie (Ci) in which

1 Ci = \(3.7 \times 10^{10}\) Bq

Some radioactive detectors measure only proportional of the true activity of a radioactive sample. This proportional activity cannot be described in becquerel units but just in counts per unit time. The two common time measures are the mean life and the half-life. The mean life, \(\tau\) or average life is the time when R and N are reduced to \(e^{-1}\) of their initial values. Half-life, \(T_{1/2}\) is the time in which R and N are reduced to one-half of their initial values.

To get the equation for \(T_{1/2}\), first, we let \(R=\frac 12 R_0\) and \(t=T_{1/2}\) in equation (6),

\(\frac 12 R_0 = R_0e^{-\lambda T_{1/2}}\),     (7)

then take the natural logarithm of both sides of (7), and solve for \(T_{1/2}\)

\(T_{1/2}={\text{ln}\; 2 \over \lambda}\).     (8)

Letting \(R=e^{-1} R_0\) and substituting \(\tau\) to t in equation (6), we get the equation for the mean life as

\(\tau={1 \over \lambda}\).     (9)

Summarizing equations (7), (8), and (9), we have

\(T_{1/2}={\text{ln}\; 2 \over \lambda}=\tau \;\text{ln}\; 2\).     (10)