13.3 Nuclear Energy: Fission and Fusion

In December 1938 the chemists Otto Hahn and Fritz Strassmann in Berlin bombarded uranium with neutrons and found, to their astonishment, barium among the products — an element with roughly half the mass of uranium. Lise Meitner and Otto Frisch worked out over the Christmas holidays what had happened: the uranium nucleus had split in two, releasing about $200$ MeV, and they called the process fission. Within four years the first nuclear reactor was running in Chicago. Today about one-tenth of the world's electricity comes from fission reactors, in countries from France (about two-thirds of its electricity) to India, whose reactors are being developed to run eventually on thorium.

The opposite process, fusion — joining light nuclei — is what makes the Sun and every other star shine. On Earth it has been achieved in an uncontrolled way in thermonuclear weapons, and in experiments such as the JET tokamak in the United Kingdom, the National Ignition Facility in the USA and the international ITER project under construction in France, scientists are trying to tame it. In Section 13.2 we saw why both processes release energy: they move nucleons up the binding-energy-per-nucleon curve. In this section we calculate how much energy, see how a chain reaction of fissions can be started and controlled, and follow the fusion reactions in the Sun.

Nuclear Reactions and Their $Q$-value

In a nuclear reaction, nuclei (and particles such as neutrons, protons and alpha particles) collide or break apart and new nuclei are formed. We write a reaction like a chemical equation, for example $${}^{2}_{1}\text{H}+{}^{3}_{1}\text{H}\to{}^{4}_{2}\text{He}+{}^{1}_{0}n.$$ Two quantities must balance on both sides: the total charge (the sum of the lower numbers, $1+1=2+0$) and the total number of nucleons (the upper numbers, $2+3=4+1$). The total mass does not balance exactly — and the difference is the energy released.

Nuclear Fission

The intermediate nucleus ${}^{236}\text{U}^{*}$ is formed in a highly excited state; it wobbles like a liquid drop, stretches, and the electric repulsion between its two halves drives them apart. Many different pairs of fragments are possible — barium and krypton, xenon and strontium, and dozens of others — for example $${}^{235}_{92}\text{U}+{}^{1}_{0}n\to{}^{141}_{56}\text{Ba}+{}^{92}_{36}\text{Kr}+3\,{}^{1}_{0}n,$$ $${}^{235}_{92}\text{U}+{}^{1}_{0}n\to{}^{140}_{54}\text{Xe}+{}^{94}_{38}\text{Sr}+2\,{}^{1}_{0}n.$$ Most of the energy appears as kinetic energy of the two fragments (about $170$ MeV), which fly apart and heat the surrounding material; the rest comes out as prompt neutrons and gamma rays and, later, from the decay of the radioactive fragments. Because heavy nuclei have more neutrons per proton than medium ones, the fragments are neutron-rich and spare neutrons are released — which is what makes a chain reaction possible.

Chain Reactions and Nuclear Reactors

Each fission is started by one neutron and releases on average about $2.5$. If at least one of these goes on to cause another fission, the process keeps itself going: a chain reaction. Enrico Fermi's team built the first controlled one, a pile of graphite and uranium under a sports stadium in Chicago, on 2 December 1942.

Not every neutron produces a fission. Some escape through the surface of the fuel; others are captured by uranium-238 or other materials without causing fission. The fraction lost through the surface depends on the ratio of surface area to volume, which falls as the lump grows. So for a given shape there is a critical size (and corresponding critical mass) below which $k\lt 1$ however the fuel is arranged.

A second difficulty is that the neutrons from fission are fast (about $2$ MeV), while uranium-235 is far more likely to absorb slow neutrons (about $0.025$ eV, "thermal" neutrons). A nuclear reactor is designed to keep $k$ exactly at $1$ using slowed-down neutrons. Its main parts are:

Fuel — uranium enriched to about $3$–$5\%$ uranium-235 (natural uranium is only $0.7\%$ uranium-235), in sealed rods. Moderator — ordinary water, heavy water ($\text{D}_2\text{O}$) or graphite, which slows the neutrons by elastic collisions. A neutron loses most energy per collision with a nucleus of similar mass (as in a head-on collision of equal balls), which is why light nuclei such as hydrogen, deuterium and carbon are used. Control rods — made of cadmium or boron, which absorb neutrons strongly; pushing them in lowers $k$, pulling them out raises it. Coolant — water, heavy water, carbon dioxide or liquid sodium, which carries the heat to a steam generator that drives a turbine. Shielding — thick concrete and steel to absorb neutrons and gamma rays.

Why can $k$ be held at exactly $1$ by mechanical rods? About $0.7\%$ of fission neutrons are emitted not at once but seconds later, by the decay of some fragments. These delayed neutrons stretch the effective generation time to about $0.1$ s, slow enough for the control system to respond.

Nuclear Fusion and the Energy of the Sun

The Coulomb barrier. Two nuclei can fuse only if they come within the range of the nuclear force, a few femtometres. Both are positively charged, so they repel each other, and they must approach with enough kinetic energy to climb this electric "hill". In a gas, the energy comes from temperature: the average kinetic energy of a particle is $\tfrac32k_BT$, where $k_B=1.38\times10^{-23}$ J/K is the Boltzmann constant (Class 11 Physics, Chapter 12 — Kinetic Theory). Fusion therefore needs temperatures of millions of kelvin — hence "thermonuclear". It happens in practice at lower temperatures than the simple estimate below suggests, because some particles move much faster than average and because quantum mechanics lets nuclei "tunnel" through the top of the barrier.

Fusion in the Sun. In the Sun's core, at about $1.5\times10^7$ K and a density over $100$ times that of water, hydrogen is turned into helium mainly by the proton–proton chain: $$\begin{aligned}{}^{1}_{1}\text{H}+{}^{1}_{1}\text{H}&\to{}^{2}_{1}\text{H}+e^{+}+\nu\quad(\text{twice}),\\ {}^{2}_{1}\text{H}+{}^{1}_{1}\text{H}&\to{}^{3}_{2}\text{He}+\gamma\quad(\text{twice}),\\ {}^{3}_{2}\text{He}+{}^{3}_{2}\text{He}&\to{}^{4}_{2}\text{He}+2\,{}^{1}_{1}\text{H}.\end{aligned}$$ Here $e^{+}$ is a positron (which soon annihilates with an electron) and $\nu$ a neutrino, an almost massless neutral particle that escapes the Sun freely. The net effect is $$4\,{}^{1}_{1}\text{H}+2e^{-}\to{}^{4}_{2}\text{He}+2\nu+26.7\ \text{MeV}.$$ The very first step is extremely slow (a given proton waits, on average, billions of years), which is why the Sun burns steadily for ten billion years instead of exploding. Heavier stars also burn helium into carbon and oxygen, and so on up to the iron peak.

Fusion on Earth. The easiest reaction to start is deuterium–tritium (Example 1). Deuterium is plentiful — about one hydrogen atom in $6400$ in seawater is deuterium — and tritium can be made inside the reactor from lithium. The difficulty is to hold a plasma at $10^8$ K long enough, and dense enough, for more energy to come out than goes in. Tokamaks such as JET and ITER hold the plasma in a doughnut-shaped magnetic field (charged particles spiral along field lines — Section 4.1); inertial confinement machines crush a tiny fuel pellet with powerful lasers. In December 2022 the National Ignition Facility for the first time released more fusion energy than the laser energy delivered to the pellet — a milestone, though still far from a power station.

Exercises