--- title: Radioactivity and Decay Modes module: Radioactive Decay moduleNumber: 4 lessonNumber: 1 order: 401 summary: > Unstable nuclei decay at a rate proportional to how many remain, giving the exponential law N(t) = N0 e^(-lambda t) with half-life t = 0.693/lambda. We work through the three common modes: alpha decay as Coulomb-barrier tunneling with the Geiger-Nuttall rule, beta decay whose continuous spectrum demands the neutrino, and gamma de-excitation, and follow a decay chain across the chart of nuclides. topics: [Radioactive Decay] sources: - book: Tipler & Llewellyn ref: "Ch. 11 — Nuclear Physics; §11-3 Radioactivity, Sequential Decays" - book: Tipler & Llewellyn ref: "§11-4 Alpha, Beta, and Gamma Decay; Energetics of Alpha Decay" draft: false --- 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 $\d t$ is proportional to $\d t$ and to the number present: $$ \d N = -\lambda N\,\d t, $$ where $\lambda$, the **decay constant**, is the probability per unit time that a given nucleus decays. Integrating, $$ N(t) = N_0\,e^{-\lambda t}, \qquad R(t) = -\frac{\d N}{\d t} = \lambda N_0\,e^{-\lambda t} = R_0\,e^{-\lambda t}. $$ Both the population and the decay rate $R$ fall exponentially with the same constant; the rate is what a detector measures. > **Definition (Half-life and mean life).** The **half-life** $t_{1/2}$ is the > time for $N$ to fall to half its value, and the **mean life** $\tau$ is the > average lifetime of a nucleus: > $$ > \tau = \int_0^\infty t\,\lambda e^{-\lambda t}\,\d t = \frac{1}{\lambda}, > \qquad > t_{1/2} = \frac{\ln 2}{\lambda} = 0.693\,\tau. > $$ After each half-life the sample and its rate halve; after each mean life they fall to $1/e$. $$ % caption: Radioactive decay halves the population every t-half; the mean life % tau = 1/lambda is where the curve reaches 1/e of its start. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (7.6,0) node[right, black!70] {$t$}; \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {$N(t)$}; % exponential curve N0=4 (units), decay so that at t=2 half \draw[acc, very thick] (0,4.0) .. controls (0.8,2.9) and (1.4,2.3) .. (2.0,2.0) .. controls (3.0,1.5) and (3.6,1.15) .. (4.0,1.0) .. controls (5.2,0.7) and (5.8,0.55) .. (6.6,0.42); % N0 \node[anchor=east, black!70] at (0,4.0) {$N_0$}; % half markers \draw[black, dashed] (0,2.0) -- (2.0,2.0) -- (2.0,0); \node[anchor=east, black, font=\scriptsize] at (0,2.0) {$\tfrac{1}{2}N_0$}; \node[anchor=north, black!70] at (2.0,0) {$t_{\frac{1}{2}}$}; \draw[black, dashed] (0,1.0) -- (4.0,1.0) -- (4.0,0); \node[anchor=east, black, font=\scriptsize] at (0,1.0) {$\tfrac{1}{4}N_0$}; \node[anchor=north, black!70] at (4.0,0) {$2\,t_{\frac{1}{2}}$}; % mean life 1/e level \draw[black, dotted] (0,1.47) -- (2.9,1.47) -- (2.9,0); \node[anchor=east, black, font=\scriptsize] at (0,1.47) {0.37 $N_0$}; \node[anchor=north, black, font=\scriptsize] at (2.9,0) {mean life}; \end{tikzpicture} $$ Activity is measured in **becquerel** ($1\,\mathrm{Bq} = 1\,\mathrm{decay/s}$) or the older **curie** ($1\,\mathrm{Ci} = 3.7 \times 10^{10}\,\mathrm{Bq}$, the decay rate of one gram of radium). > **Example (Counting rate).** A source with $t_{1/2} = 1\,\mathrm{min}$ reads > $2000\,\mathrm{counts/s}$ at $t = 0$. Then $\tau = t_{1/2}/\ln 2 = 1.44\,\mathrm{min}$ > and $\lambda = 1/\tau = 1.16 \times 10^{-2}\,\mathrm{s^{-1}}$. Since one half-life > halves the rate, at $t = n$ minutes $R = (1/2)^n \cdot 2000$: $1000$, $500$, and > $250\,\mathrm{counts/s}$ at $1$, $2$, $3\,\mathrm{min}$, and $\approx 2\,\mathrm{counts/s}$ > at $10\,\mathrm{min}$. 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.[^tl-decay] 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, $$ Q_\alpha = \bigl[\,M_P - (M_D + M_\alpha)\,\bigr]c^2 > 0, $$ 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](/nuclear-physics/alpha-decay/alpha-decay-gamow-theory). $$ % caption: An alpha particle of energy E-alpha is trapped in the nuclear well % behind the Coulomb barrier V(r); it escapes only by tunneling through the % shaded classically forbidden region. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (7.6,0) node[right, black!70] {$r$}; \draw[->, black] (0,-1.6) -- (0,3.6) node[above, black!70] {$V(r)$}; % nuclear well: deep square well for r, acc] (3.6,1.1) -- (4.6,1.1); \end{tikzpicture} $$ 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**, $$ \log t_{1/2} = A\,E_\alpha^{-1/2} + B, $$ 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.[^tl-alpha] $$ % caption: Geiger-Nuttall rule: the alpha-decay half-life falls by twenty orders % of magnitude as the emitted energy rises from 4 to 9 MeV. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.6,0) node[right, black!70] {$E$ (MeV)}; \draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {$\log t_{\frac{1}{2}}$}; \foreach \x/\lab in {0.6/4,2.0/6,3.4/8,4.4/9} \node[anchor=north, black, font=\scriptsize] at (\x,-0.05) {\lab}; % steeply falling line (semilog) \draw[acc, very thick] (0.5,4.3) -- (4.7,0.3); % data-like dots \foreach \p in {(0.7,4.05),(1.6,3.1),(2.4,2.35),(3.2,1.5),(4.0,0.75)} \fill[black] \p circle (1.6pt); \node[black, font=\scriptsize, anchor=west] at (2.6,3.4) {lower $E$ lives longer}; \end{tikzpicture} $$ An alpha step lowers both $N$ and $Z$ by $2$ and $A$ by $4$, so all members of a chain share $A \bmod 4$. 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 $^{208}\mathrm{Pb}$. $$ % caption: A decay chain on the Z-N chart: alpha steps move down-left by (2,2), % beta-minus steps move down-right by (Z up 1, N down 1), zig-zagging toward the % stable end point. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.4,0) node[right, black!70] {$Z$}; \draw[->, black] (0,0) -- (0,5.0) node[above, black!70] {$N$}; % nodes: start top-right, alpha (down-left), beta (down-right) \fill[black] (5.2,4.6) circle (2pt); \node[black, anchor=west, font=\scriptsize] at (5.3,4.6) {parent}; \fill[black] (4.6,4.0) circle (2pt); % alpha \fill[black] (5.0,3.6) circle (2pt); % beta- \fill[black] (4.4,3.0) circle (2pt); % alpha \fill[black] (4.8,2.6) circle (2pt); % beta- \fill[black] (4.2,2.0) circle (2pt); % alpha \fill[black] (3.9,1.3) circle (2pt); % stable end \node[black, anchor=west, font=\scriptsize] at (4.0,1.15) {stable}; % arrows \draw[->, black] (5.2,4.6) -- (4.66,4.06); \draw[->, black] (4.6,4.0) -- (4.94,3.66); \draw[->, black] (5.0,3.6) -- (4.46,3.06); \draw[->, black] (4.4,3.0) -- (4.74,2.66); \draw[->, black] (4.8,2.6) -- (4.26,2.06); \draw[->, black] (4.2,2.0) -- (3.96,1.36); % legend \draw[->, black] (0.6,4.4) -- (1.2,4.4); \node[anchor=west, black, font=\scriptsize] at (1.25,4.4) {alpha: down-left}; \draw[->, black] (0.6,3.9) -- (1.2,3.9); \node[anchor=west, black, font=\scriptsize] at (1.25,3.9) {beta: down-right}; \end{tikzpicture} $$ ## 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 ($t_{1/2} \approx 10.8\,\mathrm{min}$) 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 $$ Q_{\beta^-} = (M_P - M_D)c^2, \qquad Q_{\beta^+} = (M_P - M_D - 2m_e)c^2, \qquad Q_{\mathrm{EC}} = (M_P - M_D)c^2, $$ 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. $$ % caption: Beta electrons come out with every energy up to E-max, not the single % value a two-body decay would give; the missing energy and momentum are carried % by the neutrino. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.6,0) node[right, black!70] {electron energy}; \draw[->, black] (0,0) -- (0,4.0) node[above, black!70] {$N(E)$}; % continuous hump rising then to zero at Emax \draw[acc, very thick] (0.1,0.15) .. controls (0.9,2.6) and (1.9,3.4) .. (3.0,3.0) .. controls (4.1,2.6) and (4.9,0.8) .. (5.4,0.05); % Emax marker \draw[black, dashed] (5.4,0) -- (5.4,0.6); \node[anchor=north, black!70] at (5.4,0) {$E_{\max}$}; % mean energy marker \node[acc, font=\scriptsize, anchor=south] at (2.4,3.15) {continuous}; \end{tikzpicture} $$ 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 **anti**neutrino 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](/nuclear-physics/beta-decay/weak-interaction-parity-violation). 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 $10^{-3}\,\mathrm{nm}$: $$ hf = E_{\text{high}} - E_{\text{low}}. $$ Gamma emission usually follows alpha or beta decay, which typically leaves the daughter excited. Conservation of momentum gives the nucleus a small recoil energy $E_r = (hf)^2/2Mc^2 \ll hf$, 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 $\tfrac{1}{2}$, ground state $\tfrac{9}{2}$) 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. [^tl-decay]: Tipler & Llewellyn, §11-3. [^tl-alpha]: Tipler & Llewellyn, §11-4.