--- title: Nuclear Reactions, Fission, and Fusion module: Nuclear Reactions moduleNumber: 8 lessonNumber: 1 order: 801 summary: > A nuclear reaction X(x, y)Y is governed by its Q value and its cross section, the effective target area for a given process. Splitting the curve of binding energy near iron in either direction releases energy: fission of heavy nuclei by neutron capture and a chain reaction, and fusion of light nuclei that powers the Sun and needs Lawson's density-confinement criterion to be practical. topics: [Nuclear Reactions] sources: - book: Tipler & Llewellyn ref: "Ch. 11 — Nuclear Physics; §11-7 Nuclear Reactions" - book: Tipler & Llewellyn ref: "§11-8 Fission and Fusion; Nuclear Power" draft: false --- Firing a particle at a nucleus can scatter it, excite the nucleus, or transmute it into a new nuclide with a new particle emitted. The general reaction of a projectile $x$ on a target $X$ producing $Y$ and $y$ is written $$ x + X \to Y + y + Q, \qquad\text{or}\qquad X(x, y)Y. $$ Two numbers control it: the $Q$ value, which says whether energy is released, and the cross section, which says how likely the reaction is. ## Energy conservation and the Q value > **Definition (Q value).** The energy released in $X(x,y)Y$ is > $$ > Q = (m_x + m_X - m_y - m_Y)\,c^2. > $$ > $Q > 0$ is **exothermic** (mass converts to kinetic energy); $Q < 0$ is > **endothermic** and needs an energy input to proceed. The same reaction run forward and backward has opposite $Q$. For example, $$ n + {}^1\mathrm{H} \to {}^2\mathrm{H} + \gamma + 2.22\,\mathrm{MeV} \quad(\text{exothermic}), $$ $$ \gamma + {}^2\mathrm{H} \to {}^1\mathrm{H} + n - 2.22\,\mathrm{MeV} \quad(\text{endothermic}). $$ An endothermic reaction needs a threshold energy. In the center-of-mass frame the threshold is just $|Q|$, but a lab target at rest must recoil, carrying kinetic energy that momentum conservation forbids from vanishing. For a projectile of mass $m$ on a target of mass $M$, the lab threshold is $$ E_{\text{th}} = \frac{m + M}{M}\,|Q|. $$ > **Example (Q value of $p + {}^7\mathrm{Li} \to {}^4\mathrm{He} + {}^4\mathrm{He}$).** > Initial mass $1.007825 + 7.016003 = 8.023828\,\mathrm{u}$; final mass > $2(4.002602) = 8.005204\,\mathrm{u}$. The deficit $\Delta m = 0.018624\,\mathrm{u}$ > gives > $$ > Q = \Delta m\,c^2 = (0.018624\,\mathrm{u})(931.5\,\mathrm{MeV/u}) = 17.35\,\mathrm{MeV}, > $$ > exothermic. Using atomic masses lets the electron masses cancel. ## Cross section > **Definition (Cross section).** If $I$ is the incident intensity (particles per > unit area per unit time) and $R$ the number of reactions per nucleus per unit > time, the cross section is > $$ > \sigma = \frac{R}{I}. > $$ > It has units of area; the natural unit is the **barn**, > $1\,\mathrm{barn} = 10^{-28}\,\mathrm{m^2}$, of order the square of a nuclear > radius. Each possible outcome — elastic scattering $X(x,x)X$, inelastic $X(x,x')X^\ast$, capture, or a transmutation — has its own **partial cross section**, and the total is their sum. A given nucleus presents different effective sizes to different projectiles and energies because the reaction probability depends on the target's internal energy levels. The energy dependence is sharpest through the **compound nucleus**. Bohr (1936) described low-energy reactions as two stages: the projectile is absorbed and its energy shared among all nucleons, forming an excited compound nucleus that lives far longer ($\sim 10^{-16}\,\mathrm{s}$) than the $\sim 10^{-21}\,\mathrm{s}$ crossing time, then decays independently of how it formed. A peak in $\sigma(E)$ marks a **resonance**: an energy at which the projectile matches an allowed level of the compound nucleus. The width $\Gamma$ of a resonance fixes the level's lifetime through $\Gamma\tau \approx \hbar$. $$ % caption: The reaction cross section versus energy: a smooth background pierced % by resonance peaks where the projectile energy matches a level of the compound % nucleus; the peak width sets the level lifetime. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (7.4,0) node[right, black!70] {energy}; \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {cross section}; % smooth background rising slightly \draw[black, dashed] (0.3,1.0) .. controls (3.0,0.8) and (5.0,0.7) .. (7.0,0.65); % resonance peaks on top \draw[acc, very thick] (0.3,1.0) .. controls (1.0,1.0) and (1.3,3.6) .. (1.7,3.6) .. controls (2.1,3.6) and (2.3,1.0) .. (2.8,0.85) .. controls (3.3,0.8) and (3.6,2.6) .. (4.0,2.6) .. controls (4.4,2.6) and (4.6,0.8) .. (5.2,0.75) .. controls (5.8,0.72) and (6.2,1.6) .. (6.5,1.6) .. controls (6.7,1.6) and (6.85,0.7) .. (7.0,0.65); \node[acc, anchor=south, font=\scriptsize] at (1.7,3.6) {resonance}; \node[black, anchor=west, font=\scriptsize] at (4.6,0.95) {background}; \end{tikzpicture} $$ Neutrons are special projectiles: uncharged, they feel no Coulomb barrier and can be captured at any energy. A neutron scattered many times slows to thermal energy $kT \approx 0.025\,\mathrm{eV}$ (a **thermal neutron**). At low energy the capture cross section rises as $1/v$ — slower neutrons spend more time near the nucleus — punctuated by resonances that can reach thousands of barns. This $1/v$ law and the huge resonances underlie both neutron activation analysis and reactor control. ## The energy in mass The [binding-energy-per-nucleon curve](/nuclear-physics/nuclear-properties/nuclear-constituents-nuclide-chart) peaks near $^{56}\mathrm{Fe}$. Plotting the rest-mass excess per nucleon (its negative) shows that both very light ($A < 20$) and very heavy ($A \approx 200$) nuclei sit higher than mid-mass nuclei. Moving toward the middle from either end lowers the total mass and releases the difference as energy. - **Fission** splits a heavy nucleus into two mid-mass fragments, releasing about $1\,\mathrm{MeV}$ per nucleon, so roughly $200\,\mathrm{MeV}$ per event. - **Fusion** joins two light nuclei; the $^2\mathrm{H} + {}^3\mathrm{H}$ reaction releases $17.6\,\mathrm{MeV}$, about $3.5\,\mathrm{MeV}$ per nucleon — less per event but several times more per unit mass. $$ % caption: Rest-mass excess per nucleon versus A is the inverted binding-energy % curve: fusing light nuclei or splitting heavy ones both move downhill toward % the mid-mass minimum near iron and release energy. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0.4) -- (7.8,0.4) node[right, black!70] {$A$}; \draw[->, black] (0,0.4) -- (0,4.4) node[above, black!70] {mass excess per nucleon}; % inverted curve: high at left, dips near Fe, rises slowly at right \draw[acc, very thick] (0.2,4.0) .. controls (0.7,1.6) and (1.6,0.9) .. (2.4,0.85) .. controls (4.0,0.75) and (5.6,1.2) .. (7.4,1.7); \fill[acc] (2.4,0.85) circle (1.6pt); \node[acc, anchor=north, font=\scriptsize] at (2.6,0.8) {$^{56}$Fe}; \node[black, font=\scriptsize, anchor=west] at (0.3,3.5) {$^2$H}; \draw[->, black] (0.9,2.6) -- (1.9,1.2) node[midway, sloped, below, font=\scriptsize] {fusion}; \draw[->, black] (6.6,1.45) -- (4.4,0.9) node[midway, sloped, above, font=\scriptsize] {splitting}; \end{tikzpicture} $$ ## Fission Hahn and Strassmann discovered in 1938 that neutron bombardment of uranium produces mid-mass elements. A representative reaction is $$ {}^{235}\mathrm{U} + n \to {}^{92}\mathrm{Kr} + {}^{142}\mathrm{Ba} + 2n + 179\,\mathrm{MeV}. $$ When $^{235}\mathrm{U}$ captures a thermal neutron, the compound nucleus $^{236}\mathrm{U}$ is excited by $6.5\,\mathrm{MeV}$, above its critical fission energy of $6.2\,\mathrm{MeV}$; it splits about $85\%$ of the time. The liquid-drop picture explains this as an oscillation: surface tension resists deformation while Coulomb repulsion drives it, and once the drop stretches past a critical shape the Coulomb barrier is overcome and it splits. $$ % caption: Liquid-drop fission: neutron capture excites the drop, which % oscillates, stretches past the critical shape, and splits into two fragments % plus a few prompt neutrons. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % stage 1: sphere + incoming neutron \draw[thick] (0.9,0) circle (0.55); \node[black, font=\scriptsize] at (0.9,-1.0) {capture}; \draw[->, black] (-0.2,0.6) -- (0.35,0.25); \node[black, font=\scriptsize, anchor=east] at (-0.15,0.65) {$n$}; % stage 2: excited sphere \draw[thick] (3.0,0) circle (0.6); \node[black, font=\scriptsize] at (3.0,-1.0) {excited}; % stage 3: ellipsoid (deformed) \draw[thick] (5.2,0) ellipse (0.85 and 0.45); \node[black, font=\scriptsize] at (5.2,-1.0) {deformed}; % stage 4: pinched (two lobes) \draw[thick] (7.3,0) ellipse (0.4 and 0.42); \draw[thick] (8.3,0) ellipse (0.4 and 0.42); \node[black, font=\scriptsize] at (7.8,-1.0) {split}; % emitted neutrons \draw[->, black] (8.7,0.35) -- (9.3,0.6); \draw[->, black] (8.7,-0.35) -- (9.3,-0.6); % flow arrows \draw[->, black] (1.6,0) -- (2.3,0); \draw[->, black] (3.7,0) -- (4.25,0); \draw[->, black] (6.15,0) -- (6.8,0); \end{tikzpicture} $$ Not all heavy nuclei are **fissile**. Capturing a neutron on $^{238}\mathrm{U}$ excites $^{239}\mathrm{U}$ by only $5.2\,\mathrm{MeV}$, below its $5.9\,\mathrm{MeV}$ critical energy, so it de-excites by radiation rather than fissioning. The fission fragments are neutron-rich (they lie far left of the stability line), so each event emits an average of about $2.4$ prompt neutrons and the fragments then beta-decay toward stability. The emitted neutrons make a **chain reaction** possible: each fission's neutrons can trigger further fissions. Fermi's group achieved the first self-sustaining chain reaction in 1942. $$ % caption: A branching chain reaction: each fission releases neutrons that induce % further fissions, so the reaction rate grows if more than one neutron per event % goes on to fission. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % generation 0 \fill[black] (0,2) circle (2.4pt); % generation 1 (two) \fill[black] (2.2,3.0) circle (2.4pt); \fill[black] (2.2,1.0) circle (2.4pt); \draw[->, black] (0.2,2.1) -- (2.0,2.95); \draw[->, black] (0.2,1.9) -- (2.0,1.05); % generation 2 (four) \fill[black] (4.6,3.7) circle (2.4pt); \fill[black] (4.6,2.5) circle (2.4pt); \fill[black] (4.6,1.5) circle (2.4pt); \fill[black] (4.6,0.3) circle (2.4pt); \draw[->, black] (2.4,3.1) -- (4.4,3.65); \draw[->, black] (2.4,2.9) -- (4.4,2.55); \draw[->, black] (2.4,1.1) -- (4.4,1.55); \draw[->, black] (2.4,0.9) -- (4.4,0.35); % labels \node[black, font=\scriptsize, anchor=north] at (0,1.7) {1}; \node[black, font=\scriptsize, anchor=north] at (2.2,0.7) {2}; \node[black, font=\scriptsize, anchor=west] at (4.8,2.0) {4, 8, \ldots}; \end{tikzpicture} $$ > **Example (Energy from $^{235}\mathrm{U}$).** One gram of $^{235}\mathrm{U}$ > holds $N = (6.02\times10^{23})/235 = 2.56\times10^{21}$ nuclei. At > $200\,\mathrm{MeV}$ per fission the energy is > $$ > (2.56\times10^{21})(200\,\mathrm{MeV})(1.6\times10^{-13}\,\mathrm{J/MeV}) \approx 8.2\times10^{10}\,\mathrm{J} \approx 2.3\times10^{4}\,\mathrm{kW{\cdot}h}, > $$ > about the electricity a household uses in over a year, from one gram of fuel. A reactor sustains the chain at a controlled rate: a **moderator** (water, graphite) slows neutrons to thermal energy where the $^{235}\mathrm{U}$ cross section is large, and neutron-absorbing **control rods** (cadmium, boron) tune the multiplication so that on average exactly one neutron per fission goes on to the next. ## Fusion Fusion joins light nuclei. The workhorse laboratory reaction is $$ {}^2\mathrm{H} + {}^3\mathrm{H} \to {}^4\mathrm{He} + n + 17.6\,\mathrm{MeV}. $$ The obstacle is the Coulomb barrier: the nuclei must approach within about $10^{-14}\,\mathrm{m}$ for the nuclear force to grab, requiring kinetic energies of order $1\,\mathrm{MeV}$. An accelerated beam loses far more energy to scattering than it recovers, so the fuel must instead be **heated** until random thermal collisions (aided by tunneling and by the high-energy tail of the distribution) drive fusion. A temperature of $kT \approx 10\,\mathrm{keV}$, about $10^8\,\mathrm{K}$, suffices — the fuel is then a **plasma** of bare ions and electrons. Achieving net energy requires both high density and long confinement. The heating energy scales with the ion density $n$, while the fusion output scales with $n^2$ times the confinement time $\tau$, giving **Lawson's criterion**: $$ n\tau \gtrsim 10^{20}\ \mathrm{s{\cdot}particles/m^3}. $$ Two schemes pursue it: **magnetic confinement**, holding the plasma in a toroidal field (the tokamak, e.g. ITER), and **inertial confinement**, compressing a frozen deuterium-tritium pellet with laser or particle beams so briefly that its own inertia confines it. $$ % caption: The fusion cross section is negligible until the ions carry enough % energy to tunnel the Coulomb barrier, then rises steeply; thermal fusion relies % on the high-energy tail of the plasma distribution. \begin{tikzpicture}[>=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (7.4,0) node[right, black!70] {ion energy}; \draw[->, black] (0,0) -- (0,4.2) node[above, black!70] {fusion rate}; % near-zero then steep rise \draw[acc, very thick] (0.3,0.05) .. controls (2.0,0.05) and (2.8,0.15) .. (3.4,0.5) .. controls (4.2,1.1) and (4.9,2.6) .. (5.6,3.6) .. controls (6.0,3.9) and (6.4,3.95) .. (6.9,3.6); % barrier threshold marker \draw[black, dashed] (3.4,0) -- (3.4,0.5); \node[anchor=north, black, font=\scriptsize] at (3.4,-0.03) {barrier}; \node[acc, anchor=west, font=\scriptsize] at (4.9,2.3) {tunneling}; \end{tikzpicture} $$ Fusion powers the stars. In the Sun's $1.5\times10^7\,\mathrm{K}$ core the **proton-proton cycle** burns hydrogen to helium: $$ {}^1\mathrm{H} + {}^1\mathrm{H} \to {}^2\mathrm{H} + e^+ + \nu_e + 0.42\,\mathrm{MeV}, $$ $$ {}^2\mathrm{H} + {}^1\mathrm{H} \to {}^3\mathrm{He} + \gamma + 5.49\,\mathrm{MeV}, $$ $$ {}^3\mathrm{He} + {}^3\mathrm{He} \to {}^4\mathrm{He} + 2\,{}^1\mathrm{H} + 12.86\,\mathrm{MeV}. $$ The first step is extraordinarily slow — only protons in the far tail of the distribution react, and it proceeds through the weak interaction — which is why the Sun burns steadily over billions of years. The neutrinos escape the core directly, our only direct probe of the solar interior; the deficit in their measured flux (the **solar-neutrino problem**) was resolved by neutrino oscillation, which converts electron neutrinos to other flavors en route. Stellar fusion and the life of stars are developed in the astrophysics subject. [^tl-reactions]: **Tipler & Llewellyn**, _Modern Physics_, §11-7 — Nuclear Reactions: the $Q$ value, laboratory threshold, cross section and the barn, and the compound-nucleus resonance picture. [^tl-fission]: **Tipler & Llewellyn**, _Modern Physics_, §11-8 — Fission and Fusion: liquid-drop fission of $^{236}\mathrm{U}$, the chain reaction, Lawson's criterion, and the proton-proton cycle in the Sun.