doPhysics · Modern Physics
Radioactivity
Why some nuclei are unstable, and the three ways they change: alpha, beta and gamma decay, with the energy each releases. The random decay of one nucleus and the exact law of many, half-life and mean life, decays in parallel and in chains, and the clocks of carbon-14 and uranium that date charcoal and rocks.
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Why this chapter
Why this chapter exists
In 1896 a photographic plate in a closed drawer darkened next to a uranium salt. Nothing had shone on it. The salt was giving out rays of its own, and the nuclei inside it were changing into other nuclei. This chapter asks why a nucleus does that, and what comes out when it does. It then finds a strange mixture of chance and certainty. Nobody can say when one nucleus will decay, yet a sample of many decays by an exact law. That law sets the dose of a hospital's tracer, and it dates the charcoal of a Harappan hearth and the oldest rocks of the Earth.
You will learn
By the end of this chapter you will
- Explain why a nucleus can be unstable, and name the decay that cures each of the four causes.
- Write the equations of alpha, beta-minus, beta-plus and gamma decay and of electron capture, and find the daughter from the conservation of A and Z.
- Find the Q value of a decay from atomic masses, and share it between an alpha particle and the recoiling nucleus.
- Explain the continuous beta spectrum and why it needed the neutrino.
- Count the alpha and beta decays in a decay series.
- State the decay law, derive \(N = N_0e^{-\lambda t}\), and find the activity of a sample in becquerel.
- Use the half-life and the mean life, and find the fraction left after any time.
- Handle two decay routes at once, steady production with decay, and a parent feeding a daughter, including secular equilibrium.
- Date a once-living sample by carbon-14 and a rock by uranium-238 and lead-206, and say what each method assumes.
Before you begin
Before you begin
From the chapter on Matter Waves and Special Relativity, you should know that \(E = mc^2\). So \(1\,\text{u}\) of mass is \(931.5\,\text{MeV}\) of energy. The chapter on Momentum and Its Conservation showed that a body at rest that bursts in two gives the pieces equal and opposite momenta. From the chapter on the Bohr atom, you should know that an atom falling between energy levels gives out one photon. The nucleus and its binding energy are the subject of the chapter just before this one. The few facts needed here are recalled where they are used. You should be able to differentiate and integrate \(e^{x}\) and \(1/x\).
Lessons
1
Why Nuclei Decay: Alpha Decay
A nucleus is held together by a strong but short-ranged force against the repulsion of its protons. Four kinds of imbalance make a nucleus unstable, and each has its own decay. A heavy nucleus sheds an alpha particle, releasing an energy Q fixed by the lost mass, and momentum decides how that energy is shared.
Four causes of instabilityAlpha decaySharing the energy of an alpha decay
2
Beta and Gamma Decay
A neutron turns into a proton, or a proton into a neutron, and an electron or a positron is created with a neutrino. The electrons come out with every energy up to Q, which is how the neutrino was found. An excited nucleus gives out gamma photons, and a chain of decays leads from uranium to lead.
Beta minus beta plus and electron captureThe continuous beta spectrum and the neutrinoGamma decayCounting decays in a series
3
The Decay Law
Nobody can say when one nucleus will decay, yet a large sample decays by an exact law. Each nucleus has a fixed chance per second, the decay constant; from it the number left falls exponentially, and so does the activity, counted in becquerel.
Decay is random for one nucleusThe decay constantExponential decayActivity
4
Half-Life and Mean Life
Every half-life, a sample halves, whatever its size, so \(T_{1/2} = 0.693/\lambda\). The average lifetime of a nucleus is the mean life \(\tau = 1/\lambda = 1.44\,T_{1/2}\). After any time \(t\) the fraction left is \((1/2)\) to the power \(t/T_{1/2}\), which answers most questions about decay in one line.
Half lifeMean lifeFraction left after some half lives
5
Two Paths and Chains
A nucleus with two ways to decay uses both, and the decay constants add. Nuclei made at a steady rate while they decay build up to a fixed number. A parent that feeds a radioactive daughter drives the daughter up to a peak, and a long-lived parent holds it in secular equilibrium.
Parallel decayProduction and decay togetherSuccessive decay and equilibrium
6
Radioactive Dating
Living things hold a steady share of carbon-14, and at death its clock starts. The fall in the activity dates charcoal, grain and bone up to about fifty thousand years. Uranium-238, turning into lead-206 with a half-life of 4.47 billion years, dates the rocks of the Earth.
Carbon 14 in living thingsDating with carbon 14Uranium lead dating of rocks
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- activity
- The number of decays per second in a sample, \(A = \lambda N\). It falls with time in the same exponential way as \(N\).
- alpha particle
- A helium-4 nucleus, two protons and two neutrons, given out by a heavy nucleus in alpha decay.
- becquerel
- The SI unit of activity, one decay per second, written Bq.
- beta-minus decay
- A decay in which a neutron in the nucleus turns into a proton, and an electron and an antineutrino are given out. \(A\) stays the same and \(Z\) rises by one.
- beta-plus decay
- A decay in which a proton in the nucleus turns into a neutron, and a positron and a neutrino are given out. \(A\) stays the same and \(Z\) falls by one.
- branching ratio
- The fraction of the nuclei that decay by one route when two or more routes are open, \(\lambda_i/\lambda\).
- curie
- An older unit of activity, \(3.7\times10^{10}\,\text{Bq}\), about the activity of one gram of radium-226.
- daughter nucleus
- The nucleus left after a decay. It may itself be radioactive.
- decay constant
- The chance per unit time that one nucleus decays, written \(\lambda\). It is fixed for each nuclide, and a sample obeys \(dN/dt = -\lambda N\).
- decay series
- A chain of decays from a heavy parent, through radioactive daughters, to a stable end, such as uranium-238 to lead-206.
- electron capture
- A decay in which the nucleus takes in one of the atom's own inner electrons. A proton turns into a neutron, and a neutrino is given out. \(Z\) falls by one.
- gamma decay
- The emission of a high-energy photon by a nucleus that falls from an excited state to a lower one. \(A\) and \(Z\) do not change.
- half-life
- The time in which half the nuclei of a sample decay, \(T_{1/2} = \ln 2/\lambda = 0.693/\lambda\). It is the same whatever the size of the sample.
- mean life
- The average lifetime of a nucleus of a sample, \(\tau = 1/\lambda = 1.44\,T_{1/2}\). After one mean life \(1/e\), about 37 %, of the nuclei are left.
- neutrino
- A neutral particle of almost no mass, given out in beta decay. It shares the energy of the decay with the electron or positron, and it passes through matter almost untouched. Beta-minus decay gives out its antiparticle, the antineutrino.
- nuclide
- A nucleus of one particular kind, fixed by its proton number \(Z\) and its neutron number \(N\), and written \({}^{A}_{Z}\text{X}\) with \(A = Z + N\).
- parent nucleus
- The nucleus that decays.
- positron
- The antiparticle of the electron: the same mass, and a charge of \(+e\).
- Q value
- The energy released in a decay or a reaction, the loss of rest mass times \(c^2\): \(Q = (m_{\text{before}} - m_{\text{after}})c^2\). For a decay it is shared as kinetic energy among the products.
- radioactive dating
- Finding the age of a sample from how far a radioactive nuclide in it has decayed. Carbon-14 dates once-living matter; uranium-238 dates rocks.
- radioactivity
- The spontaneous change of an unstable nucleus into another, with the emission of an alpha particle, an electron or positron, or a gamma photon. It happens inside the nucleus, so heat, pressure and chemical combination do not affect it.
- secular equilibrium
- The state of a chain in which a long-lived parent feeds a short-lived daughter, and the daughter decays as fast as it is made. The two activities are then equal, and \(N_B/N_A = T_B/T_A\).
- stability belt
- The narrow band of the chart of neutron number against proton number in which the stable nuclei lie. It starts along \(N = Z\), bends above it to about \(N/Z = 1.5\) at lead, and ends at \(Z = 83\).