Radioactive decay (A-level only)

AQA·Physics·3.8.1.3

Random radioactive decay

An unstable nucleus may decay, but it is impossible to predict exactly when an individual nucleus will decay. For a given nuclide, each undecayed nucleus has the same constant probability of decay per unit time. This probability does not increase as the nucleus gets older.

The decay constant, $$\lambda$$, describes this probability. For a sufficiently short time interval $$\Delta t$$, the probability that a surviving nucleus decays during the interval is approximately $$\lambda\Delta t$$. The units of $$\lambda$$ are inverse time, such as $$\mathrm{s}^{-1}$$.

Decay of a population of nuclei

Although individual decay times are unpredictable, the expected behaviour of a large population is predictable. If $$N$$ nuclei remain undecayed, each has the same chance of decay over a short interval. A larger $$N$$ therefore means more decays are expected in that interval.

The expected rate of change of $$N$$ is proportional to $$N$$. Over a sufficiently short interval:

ΔNΔt≈−λN\frac{\Delta N}{\Delta t} \approx -\lambda N

The minus sign shows that the number of undecayed nuclei decreases. In continuous form, the relationship for the expected population is:

dNdt=−λN\frac{dN}{dt} = -\lambda N

As nuclei decay, fewer remain available to decay, so the expected rate of decrease becomes smaller. If $$N_0$$ nuclei are initially present, the expected number remaining after time $$t$$ is:

N=N0e−λtN = N_0\mathrm{e}^{-\lambda t}

This is an exponential decay law. It describes the expected population behaviour, not the exact timing of individual decays. Actual numbers fluctuate randomly around the expectation, particularly when the population is small.

Simulating random decay

A simulation can model decay by giving each surviving nucleus the same probability of decay in each equal time interval. For short intervals, this probability can be taken as $$\lambda\Delta t$$. The simulation randomly selects which nuclei decay, removes them, and repeats the process with the survivors. Each survivor has the same chance in the next interval, regardless of how long it has already survived.

Finding the initial number of nuclei

If the mass $$m$$ is the mass of the nuclide in question, its number of nuclei can be found from its molar mass $$M$$:

N=mMNAN = \frac{m}{M}N_{\mathrm A}

Here $$m/M$$ is the amount of the nuclide in moles, and $$N_{\mathrm A}$$ is the Avogadro constant. Use consistent mass units for $$m$$ and $$M$$. If the sample contains other nuclides, use only the mass of the nuclide being considered.

Radioactive activity

A radioactive sample contains unstable nuclei that decay over time. Its activity, $$A$$, is the number of nuclear decays per unit time. Activity is a rate of decay, not the number of undecayed nuclei in the sample.

The SI unit of activity is the becquerel ($$\mathrm{Bq}$$). An activity of $$1,\mathrm{Bq}$$ corresponds to one decay per second.

Activity and undecayed nuclei

For a population with $$N$$ undecayed nuclei and decay constant $$\lambda$$, the activity is

A=λN=−dNdtA=\lambda N=-\frac{dN}{dt}

The decay constant $$\lambda$$ describes the probability per unit time that an undecayed nucleus will decay. Its unit is $$\mathrm{s^{-1}}$$. For a given decay constant, a larger population of undecayed nuclei has a greater activity.

As nuclei decay, $$N$$ decreases, so $$dN/dt$$ is negative. The minus sign in $$-dN/dt$$ makes activity a positive decay rate.

Exponential decrease in activity

The number of undecayed nuclei falls exponentially. Because activity is proportional to $$N$$, activity also falls exponentially:

A=A0e−λtA=A_0\mathrm{e}^{-\lambda t}

Here $$A_0$$ is the initial activity and $$t$$ is the time elapsed. As time passes, fewer nuclei remain available to decay, so the activity decreases.

Radioactive half-life

A radioactive population contains nuclei that have not yet decayed. As decay occurs, the number of these undecayed nuclei, $$N$$, falls. The population’s activity, $$A$$, is its rate of decay. Activity also falls because fewer undecayed nuclei remain.

The half-life, $$T_{\frac12}$$, is the time taken for $$N$$ or $$A$$ to fall to half its initial value. For example, after one half-life, half the initial number of nuclei remains; after two half-lives, one quarter remains; and after three, one eighth remains.

Under the exponential decay model, each successive halving takes the same length of time. The half-life therefore describes a fixed fractional decrease, not a fixed number of decays.

Half-life and decay constant

The decay constant, $$\lambda$$, describes how rapidly a nuclide decays. It is related to the half-life by

T12=ln⁡2λT_{\frac12}=\frac{\ln 2}{\lambda}

A larger decay constant means a shorter half-life. If $$\lambda$$ is measured in inverse seconds, the half-life calculated from this equation is in seconds.

Finding half-life from graphs

On a decay curve of $$N$$ or $$A$$ against time, choose a value and read the time at which it occurs. Then read the time at which the value has fallen to half of it. The time interval between the two readings is the half-life. The first reading need not be at the start of the graph: under the exponential model, any successive halving takes the same time.

A graph of the natural logarithm $$\ln N$$ or $$\ln A$$ against time is a straight line. Its gradient $$m$$ is related to the decay constant by

m=−λm=-\lambda

The gradient is negative because $$N$$ and $$A$$ decrease with time. Find $$\lambda$$ from the gradient, then use $$T_{\frac12}=\ln 2/\lambda$$.

If the graph uses common logarithms instead, plotting $$\log_{10}N$$ or $$\log_{10}A$$ against time, the gradient is

m=−λln⁡10m=-\frac{\lambda}{\ln 10}

For this graph, calculate $$\lambda=-m\ln 10$$ before using the half-life equation.

Radioactive dating

Radioactive dating estimates how much time has passed by comparing the amount of a radioactive nuclide remaining in a material with the amount present at the time being dated. If the nuclide’s decay rate is known, the fraction remaining indicates the elapsed time.

The comparison can use either the number of radioactive nuclei or their activity, the number of decays per unit time. For a particular nuclide, activity is proportional to the number of radioactive nuclei, so both decrease by the same fraction over the same time.

Radioactive dating and exponential decay

The remaining quantity is related to its initial value by the exponential decay law:

Q=Q0e−λtQ = Q_0 e^{-\lambda t}

Here, $$Q$$ is the number of radioactive nuclei remaining or the current activity, and $$Q_0$$ is the corresponding initial number or activity. The elapsed time is $$t$$, and $$\lambda$$ is the decay constant of the nuclide. Time and the decay constant must use consistent units.

If the current and initial quantities are known, rearranging the decay law gives:

t=1λln⁡(Q0Q)t = \frac{1}{\lambda}\ln\left(\frac{Q_0}{Q}\right)

The decay constant is related to the half-life, the time taken for the number of radioactive nuclei or the activity to halve, by $$t_{1/2} = \ln 2/\lambda$$. For example, if one-quarter of the initial quantity remains, two half-lives have elapsed: the quantity has halved twice.

Conditions for a reliable date

A measured quantity alone does not establish an elapsed time: its initial value must also be established or inferred. The estimate therefore depends on assumptions about the material’s initial composition.

The estimate also assumes that radioactive material has not been gained or lost in a way that changes the measured quantity independently of decay. If material has entered or left the sample, the change cannot be attributed to decay alone, so the calculated time may be inaccurate.

The nuclide’s half-life must be suitable for the timescale being investigated. If the half-life is much longer than the elapsed time, too little decay may have occurred to measure the change reliably. If it is much shorter, too little radioactive material may remain to measure reliably.

Radioactive waste storage

Radioactive waste must be stored so that people and the environment are protected for as long as its radiation presents an unacceptable hazard. Storage must contain the radioactive material and limit exposure to its radiation.

Activity and storage time

Radioactive nuclei decay over time. Activity is the number of nuclear decays per second, so the activity of a radioactive sample decreases as its nuclei decay.

A nuclide’s half-life is the time taken for its activity to fall to half its starting value. After one half-life, half the starting activity remains; after two half-lives, one quarter remains. A shorter-lived nuclide’s activity falls more rapidly, while a long-lived nuclide can remain radioactive for a long time. Half-life therefore helps indicate how activity changes over a storage period.

What determines the storage needed?

Half-life alone does not determine how hazardous waste is or how it should be stored. The assessment also depends on:

  • Activity: how many decays occur each second.
  • Radiation emitted: different types of radiation present different exposure hazards.
  • Potential for escape or contact: radioactive material may cause exposure if it escapes from storage or is encountered.

Storage must therefore provide suitable containment and limit exposure for an appropriate period. A long half-life does not, by itself, establish that waste is more hazardous; the other factors must also be considered.