Nuclear Reactions, Fission, and Fusion

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, or 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

The same reaction run forward and backward has opposite $Q$ . For example,

n +^{1} H \to^{2} H + γ + 2.22 MeV (exothermic),

γ +^{2} H \to^{1} H + n - 2.22 MeV (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_{th} = \frac{m + M}{M} ∣ Q ∣.

Cross section

Each possible outcome — elastic scattering $X (x, x) X$ , inelastic $X (x, x^{'}) X^{*}$ , 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 ( $\sim 1 0^{- 16} s$ ) than the $\sim 1 0^{- 21} s$ crossing time, then decays independently of how it formed. A peak in $σ (E)$ marks a resonance: an energy at which the projectile matches an allowed level of the compound nucleus. The width $Γ$ of a resonance fixes the level's lifetime through $Γ τ \approx ℏ$ .

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.

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 $k T \approx 0.025 eV$ (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 peaks near $^{56} Fe$ . 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 $1 MeV$ per nucleon, so roughly $200 MeV$ per event.
Fusion joins two light nuclei; the $^{2} H +^{3} H$ reaction releases $17.6 MeV$ , about $3.5 MeV$ per nucleon — less per event but several times more per unit mass.

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.

Fission

Hahn and Strassmann discovered in 1938 that neutron bombardment of uranium produces mid-mass elements. A representative reaction is

^{235} U + n \to^{92} Kr +^{142} Ba + 2 n + 179 MeV .

When $^{235} U$ captures a thermal neutron, the compound nucleus $^{236} U$ is excited by $6.5 MeV$ , above its critical fission energy of $6.2 MeV$ ; 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.

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.

Not all heavy nuclei are fissile. Capturing a neutron on $^{238} U$ excites $^{239} U$ by only $5.2 MeV$ , below its $5.9 MeV$ 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.

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.

A reactor sustains the chain at a controlled rate: a moderator (water, graphite) slows neutrons to thermal energy where the $^{235} U$ 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} H +^{3} H \to^{4} He + n + 17.6 MeV .

The obstacle is the Coulomb barrier: the nuclei must approach within about $1 0^{- 14} m$ for the nuclear force to grab, requiring kinetic energies of order $1 MeV$ . 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 $k T \approx 10 keV$ , about $1 0^{8} K$ , 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 $τ$ , giving Lawson's criterion:

n τ ≳ 1 0^{20} 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.

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.

Fusion powers the stars. In the Sun's $1.5 \times 1 0^{7} K$ core the proton-proton cycle burns hydrogen to helium:

^{1} H +^{1} H \to^{2} H + e^{+} + ν_{e} + 0.42 MeV,

^{2} H +^{1} H \to^{3} He + γ + 5.49 MeV,

^{3} He +^{3} He \to^{4} He + 2^{1} H + 12.86 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.