Law of Radioactive Decay

What is being described

• Materials which undergo alpha, beta or gamma decays are called radioactive materials.
• Radioactivity is statistical: given a single radioactive atom, it can never be predicted how long it will take to decay. Given $N_0$ atoms at $t=0$, all that can be said is that the number will decrease with time as some nuclei, and it cannot be said which, decay.

Statement of the law

• The number of nuclei undergoing decay per unit time is proportional to the number of unchanged (undecayed) nuclei present at that instant.

Derivation of $N = N_0e^{-\lambda t}$

• Let $N(t)$ be the number of parent nuclei left at time $t$, and $dN$ the number that decay in the short interval between $t$ and $t+dt$. The larger $N(t)$ is, and the longer the interval $dt$ is, the more nuclei decay. Hence

$$dN \propto -N(t)\,dt \qquad\Rightarrow\qquad dN = -\lambda N(t)\,dt$$

• where $\lambda$ is the constant of proportionality, called the decay constant or disintegration constant. The negative sign indicates that the change in the number of parent nuclei is negative, i.e. $N(t)$ decreases with time.
• Separating the variables and integrating, with $N_0$ the number of parent atoms at $t=0$:

$$\int_{N_0}^{N(t)}\dfrac{dN}{N(t)} = \int_0^t -\lambda\,dt$$

$$\therefore \log_e\dfrac{N(t)}{N_0} = -\lambda t$$

$$\therefore N(t) = N_0e^{-\lambda t}$$

• This is the decay law of radioactivity. Graph in words: an exponential curve starting at $N_0$ on the vertical axis, falling steeply at first and then more and more slowly, approaching but never reaching the time axis.

Activity

• The rate of decay, i.e. the number of decays per unit time, is called the activity $A(t)$:

$$A(t) = -\dfrac{dN}{dt} = \lambda N(t) = \lambda N_0e^{-\lambda t} = A_0e^{-\lambda t}$$

• where $A_0 = \lambda N_0$ is the activity at $t=0$. Activity therefore falls off with exactly the same exponential as the number of nuclei.
• Units: the SI unit is the becquerel (Bq), one becquerel being one decay per second. Another unit is the curie (Ci), where $1\ \text{Ci} = 3.7\times10^{10}$ decays per second, i.e. $1\ \text{Ci} = 3.7\times10^{10}\ \text{Bq}$.
• To get the activity of a sample of given mass, first find the number of nuclei in it: $N = \dfrac{m}{M_{\text{mol}}}\times N_A$, with $m$ in gram and $M_{\text{mol}}$ the molar mass in gram per mol, then use $A = \lambda N$.

Half-life

• The time taken for the number of parent nuclei of a particular species to reduce to half its value is called the half-life $T_{1/2}$ of that species.
• Putting $N(t) = \dfrac{N_0}{2}$ at $t = T_{1/2}$ in the decay law:

$$\dfrac{N_0}{2} = N_0e^{-\lambda T_{1/2}} \qquad\Rightarrow\qquad e^{\lambda T_{1/2}} = 2$$

$$\therefore \lambda T_{1/2} = \log_e 2 = 0.693 \qquad\Rightarrow\qquad T_{1/2} = \dfrac{0.693}{\lambda}$$

• The number does not go to zero after two half-lives. In the first $T_{1/2}$ it falls from $N_0$ to $\dfrac{N_0}{2}$; in the next $T_{1/2}$ it falls to $\dfrac{N_0}{4}$, and so on. In each interval equal to a half-life the number is reduced by a factor of one half.
• Convenient working form: $N = N_0\left(\dfrac{1}{2}\right)^{t/T_{1/2}}$.

Average (mean) life

• Different nuclei of a species decay at different times, so they have different lifetimes; the average lifetime is defined by weighting each lifetime $t$ by the number of nuclei that decay at that time.
• The number decaying between $t$ and $t+dt$ is $\lambda N_0e^{-\lambda t}dt$, and their lifetime is $t$, so

$$\tau = \dfrac{1}{N_0}\int_0^\infty t\,\lambda N_0e^{-\lambda t}\,dt = \lambda\int_0^\infty t\,e^{-\lambda t}\,dt$$

• Carrying out the integration,

$$\tau = \dfrac{1}{\lambda}$$

• Relation with half-life, using $T_{1/2} = \dfrac{0.693}{\lambda}$:

$$T_{1/2} = 0.693\,\tau = \tau\log_e 2$$

• so the average life is always longer than the half-life, $\tau = 1.44\,T_{1/2}$.

Radioactive equilibrium

• If the daughter nucleus produced in a decay is itself unstable, let $N_p$ and $N_d$ be the numbers of parent and daughter nuclei at time $t$ and $\lambda_p$, $\lambda_d$ their decay constants.
• Daughters produced in time $dt$: $\lambda_pN_p\,dt$. Daughters decaying in the same interval: $\lambda_dN_d\,dt$.
• For $N_d$ to remain constant these must be equal, so the required condition is

$$\lambda_pN_p = \lambda_dN_d$$

Working with two readings

• If the activity is $A_1$ at time $t_1$ and $A_2$ at time $t_2$, dividing $A_1 = A_0e^{-\lambda t_1}$ by $A_2 = A_0e^{-\lambda t_2}$ gives

$$\dfrac{A_1}{A_2} = e^{\lambda(t_2-t_1)} \qquad\Rightarrow\qquad \lambda = \dfrac{1}{t_2-t_1}\log_e\!\left(\dfrac{A_1}{A_2}\right)$$

• Dating a specimen uses the same relation with the original activity per gram $A_0$ and the present activity per gram $A$: $t = \dfrac{1}{\lambda}\log_e\!\left(\dfrac{A_0}{A}\right)$.

Other exponential decays for comparison

• The amplitude of a damped simple pendulum, $A = A_0e^{-bt}$, where $b$ is the damping factor.
• The cooling of an object in an open surrounding, $T = T_0e^{-kt}$.
• The discharging of a capacitor through a pure resistor, $Q = Q_0e^{-t/RC}$, where $RC$ is the time constant. Charging a capacitor is exponential too, but is called exponential growth.

Standard results collected

• $N(t) = N_0e^{-\lambda t}$; $A(t) = A_0e^{-\lambda t}$; $A = \lambda N$.
• $T_{1/2} = \dfrac{0.693}{\lambda}$; $\tau = \dfrac{1}{\lambda}$; $T_{1/2} = 0.693\,\tau$.
• $1\ \text{Bq} = 1$ decay per second; $1\ \text{Ci} = 3.7\times10^{10}\ \text{Bq}$.
• $N = \dfrac{m}{M_{\text{mol}}}N_A$ with $N_A = 6.022\times10^{23}\ \text{mol}^{-1}$.