Radioactivity and Decay Modes

A radioactive nucleus decays to a lower-energy state of a different nuclide by emitting radiation. Rutherford's classification survives: alpha rays are ${}^4\mathrm{He}$ nuclei, beta rays are electrons or positrons, and gamma rays are short-wavelength photons, ranked by penetrating power ($\gamma >\beta >\alpha$) and inversely by ionizing power. The decay of any single nucleus is a random event, and that randomness fixes the time dependence of the whole sample.

The decay law

The number of nuclei decaying in $\mathbf{d}t$ is proportional to $\mathbf{d}t$ and to the number present:

where $\lambda$, the decay constant, is the probability per unit time that a given nucleus decays. Integrating,

Both the population and the decay rate $R$ fall exponentially with the same constant; the rate is what a detector measures.

After each half-life the sample and its rate halve; after each mean life they fall to $1/e$.

Activity is measured in becquerel ($1\mathrm{Bq}=1decay/s$) or the older curie ($1\mathrm{Ci}=3.7\times 10^{10}\mathrm{Bq}$, the decay rate of one gram of radium).

When a decay product is itself radioactive, a sequential decay or chain results. If a long-lived parent feeds a short-lived daughter, the daughter reaches secular equilibrium, decaying as fast as it is produced, and the whole chain shares the parent's slow rate.

Conservation laws

Every decay conserves the same quantities as any other physical process: relativistic energy, electric charge, linear and angular momentum, nucleon number, and lepton number.¹ The last two are the bookkeeping that distinguishes the modes and, historically, forced the neutrino into existence.

Alpha decay

Heavy nuclei ($Z>83$) are unstable to emission of an alpha particle because the parent mass exceeds the sum of daughter and alpha masses. The energy released is the $Q$ value,

shared as kinetic energy of the daughter and the alpha. Emission requires the alpha to tunnel through the Coulomb barrier: inside the nucleus its energy $E_{\alpha }$ lies below the barrier top, and it escapes only by quantum tunneling.

Because a narrower or lower barrier is tunneled far more easily, the half-life is exquisitely sensitive to $E_{\alpha }$: a small change in energy swings the half-life over many orders of magnitude. This is the empirical Geiger-Nuttall rule,

with $A,B$ constants. The wave-mechanical derivation, treating $\lambda$ as the product of a barrier transmission coefficient $T$ and the frequency $v/2R$ at which the alpha strikes the wall, reproduces it and even provides an independent route to the nuclear radius.²

An alpha step lowers both $N$ and $Z$ by $2$ and $A$ by $4$, so all members of a chain share $A\mathrm{mod}4$. There are four alpha-decay series ($A=4n$, $4n+1$, $4n+2$, $4n+3$); three occur in nature, but the $4n+1$ series is absent because its longest-lived member, ${}^{237}\mathrm{Np}$ ($t_{1/2}=2\times 10^6\mathrm{y}$), decayed away long ago.

The chart of nuclides and a decay chain

An alpha decay moves down and to the left on the $(Z,N)$ chart; it leaves the daughter on the neutron-rich side of the stability line, which then usually ${\beta }^{-}$-decays back up. The thorium ($4n$) series threads this way from ${}^{232}\mathrm{Th}$ to the stable ${}^{208}\mathrm{Pb}$.

Beta decay

Three processes change $Z$ and $N$ by one while leaving $A$ fixed, converting a nucleus into its isobaric neighbor.

• ${\beta }^{-}$ decay: a neutron becomes a proton, $n\to p+e^{-}+{\bar{\nu }}_e$. On the free neutron ($t_{1/2}\approx 10.8\min$) the energy release is $0.78\mathrm{MeV}$, the neutron-proton-electron rest-energy difference.
• ${\beta }^{+}$ decay: a proton becomes a neutron, $p\to n+e^{+}+{\nu }_e$. Forbidden for a free proton but allowed inside a nucleus.
• Electron capture (EC): a proton captures an atomic electron (usually a $1s$ electron), $p+e^{-}\to n+{\nu }_e$, competing with ${\beta }^{+}$ when the mass difference is under $2m_e{c}^2$.

Neutron-rich nuclei favor ${\beta }^{-}$; proton-rich nuclei favor ${\beta }^{+}$ or EC. In atomic-mass terms the $Q$ values are

so ${\beta }^{+}$ decay requires at least $2m_e{c}^2=1.022\mathrm{MeV}$ of mass difference.

The decisive feature is the electron's continuous energy spectrum. If only the daughter and the electron shared $Q$, the electron energy would be fixed; instead it varies from zero to a maximum $E_{\max }\approx Q$. Energy, momentum, and angular momentum all appeared to fail.

To rescue the conservation laws, Pauli (1930) proposed an unseen third particle, neutral, nearly massless, carrying the balance. Fermi (1933) built the successful quantum theory of beta decay around it and named it the neutrino; Cowan and Reines detected it in 1956. Two facts follow.

• The decay of the free neutron reads $n\to p+e^{-}+{\bar{\nu }}_e$, and a typical nuclear example is ${}^{198}\mathrm{Au}\to {}^{198}\mathrm{Hg}+e^{-}+{\bar{\nu }}_e$, where lepton conservation demands an antineutrino accompany the electron.
• Electrons and neutrinos feel neither the strong nor (for the neutral neutron) the electromagnetic force, so beta decay needs a new interaction. Its long lifetimes compared to the nuclear timescale ($\sim 10^{-23}\mathrm{s}$) show it is weaker than the strong force: the weak interaction, short-ranged like the strong force but far feebler. This is the province of the weak interaction.

Because the mass formula is quadratic in $Z$ at fixed $A$, a cut across the energy valley at constant $A$ is a parabola (one for odd $A$, two — even-even below odd-odd — for even $A$). Beta decays walk down the parabola toward the stable isobar at the bottom.

Gamma decay

An excited nucleus drops to a lower state of the same nuclide by emitting a photon, the nuclear analog of atomic light emission. Nuclear level spacings are of order $\mathrm{MeV}$ (versus $\mathrm{eV}$ in atoms), so gamma wavelengths are about $10^{-3}\mathrm{nm}$:

Gamma emission usually follows alpha or beta decay, which typically leaves the daughter excited. Conservation of momentum gives the nucleus a small recoil energy $E_r=(hf{)}^2/2M{c}^2\ll hf$, so the photon energy is very nearly the level difference. Selection rules govern the rates: a large spin change is strongly suppressed, which is why the isomeric first excited state of ${}^{93}\mathrm{Nb}$ (spin $\frac{1}{2}$, ground state $\frac{9}{2}$) has a $13.6$-year half-life.

Two competing channels round out gamma de-excitation.

• Internal conversion: the excitation energy is transferred directly to an inner ($K$ or $L$) electron, ejected with kinetic energy equal to the transition energy minus its binding energy — a one-step process, not a photon reabsorbed.
• Isomers: excited states whose selection-rule-forbidden decays give them anomalously long lives, from hours to years.

The recoilless emission of gamma rays from nuclei bound in a crystal lattice, the Mössbauer effect, removes even the small recoil energy and yields frequency precision fine enough to measure the gravitational redshift.