Nuclear Fusion and Stellar Energy

Nuclear fusion is the joining of two light nuclei into a heavier nucleus with the release of energy because the binding energy per nucleon increases; it powers the Sun and other stars, where four hydrogen nuclei fuse into one helium-4 nucleus, releasing about 26.7 MeV, and it requires temperatures of millions of kelvins to overcome the electrical repulsion between the nuclei. In brief: the Sun converts about 4.3 million tonnes of mass into energy every second (L = 3.83 × 10²⁶ W), fusing about 6 × 10¹¹ kg of hydrogen per second. Quantum tunneling lets protons fuse at the solar core temperature of 1.5 × 10⁷ K. On Earth, the aim is to fuse deuterium and tritium (D-T, 17.6 MeV) at about 10⁸ K in a magnetic confinement device or by inertial confinement.

Fusion is the process behind the energy of stars and hydrogen bombs, and the goal of research for clean power. This guide explains why fusion releases energy and why high temperatures are needed, describes the proton-proton chain and the later stages of stellar fusion, discusses fusion reactors, compares fission and fusion in a table, works through calculations of energy and mass loss, and corrects common mistakes.

Why does fusion release energy?

On the curve of binding energy per nucleon, light nuclei are on the rising part: hydrogen and deuterium are loosely bound, helium-4 is very tightly bound (7.07 MeV per nucleon). When light nuclei fuse into a nucleus closer to the iron peak, the product has a higher binding energy per nucleon and a smaller mass than the reactants, and the mass difference appears as the kinetic energy of the products (E = Δm c²). Per unit mass of fuel, fusion releases several times more energy than fission: deuterium-tritium fusion about four times as much as uranium-235 fission, and the hydrogen-to-helium conversion in stars, which releases 0.7 percent of the mass as energy, about eight times as much (fission of uranium releases about 0.09 percent). Fusion of nuclei heavier than iron-56 would absorb energy, so fusion in stars ends with the production of iron.

What conditions are needed?

Nuclei are positively charged and repel each other with a Coulomb barrier that, for two protons at a distance of about 1 fm (where the strong force takes over), is about 1 MeV, equivalent to a temperature of ~10¹⁰ K. In the Sun, the core temperature is only about 1.5 × 10⁷ K (the average kinetic energy is about 2 keV), but fusion occurs because of (1) the fast tail of the Maxwell-Boltzmann distribution and (2) quantum tunneling through the barrier. Even so, the proton-proton reaction has a very low probability: an individual proton in the core waits about 10 billion years on average before fusing, which is why the Sun burns slowly and will last for about 10 billion years. The conditions for a fusion reactor are summarized by the Lawson criterion: a sufficient combination of temperature (about 10⁸ K for D-T), density and confinement time (the triple product nTτ must exceed about 3 × 10²¹ keV·s/m³).

How do stars produce energy?

• Proton-proton chain (Sun, and stars of similar or lower mass): (1) ¹H + ¹H → ²H + e⁺ + ν_e; (2) ²H + ¹H → ³He + γ; (3) ³He + ³He → ⁴He + 2 ¹H. Net: 4 ¹H → ⁴He + 2e⁺ + 2ν_e + energy of 26.7 MeV (including the annihilation of the positrons with electrons, but excluding the energy carried away by the neutrinos, about 0.5 MeV in the main branch).
• CNO cycle in more massive stars (hotter than about 1.7 × 10⁷ K): carbon, nitrogen and oxygen act as catalysts for the fusion of hydrogen into helium.
• Later stages: after the hydrogen in the core is used, helium fuses into carbon and oxygen (triple-alpha process at about 10⁸ K), and in massive stars carbon, neon, oxygen and silicon burn in successive shells, ending with an iron core that collapses in a supernova. Elements heavier than iron are formed in supernovae and neutron-star collisions. The energy is transported outward by radiation and convection, taking about 10⁵ years for the energy to travel from the core to the surface, and then 8 minutes to reach Earth.

What about fusion on Earth?

The easiest reaction is deuterium-tritium: ²H + ³H → ⁴He (3.5 MeV) + n (14.1 MeV), total 17.6 MeV. Deuterium is abundant in seawater (1 in 6,700 hydrogen atoms), and tritium would be bred from lithium by neutrons in the blanket of the reactor. At temperatures of 10⁸ K, matter is a plasma, which cannot touch walls; two main approaches: magnetic confinement (tokamak and stellarator: strong magnetic fields hold the plasma in a toroidal shape; the ITER tokamak in France is designed for 500 MW of fusion power from 50 MW of heating) and inertial confinement (lasers compress and heat a small fuel pellet: in December 2022 the National Ignition Facility in the US achieved fusion ignition in which the fusion energy output (3.15 MJ) exceeded the laser energy delivered to the target (2.05 MJ), though much more energy was needed to run the lasers). Fusion would offer abundant fuel, no greenhouse gas emissions, no chain-reaction risk and less long-lived waste than fission, but a commercial plant has not yet been achieved.

Fission and fusion compared

Worked example: energy of the Sun and fusion reactions

1. Mass defect of the hydrogen-to-helium conversion. 4 × 1.007825 u (four hydrogen atoms) − 4.002603 u (helium-4 atom) = 0.028697 u. Energy = 0.028697 × 931.5 = 26.7 MeV. As a fraction of the mass: 0.028697 ÷ 4.0313 = 0.71 percent.
2. Mass lost by the Sun. L = 3.83 × 10²⁶ W. By E = mc², the mass converted per second is m = L ÷ c² = 3.83 × 10²⁶ ÷ (3.00 × 10⁸)² = 4.3 × 10⁹ kg/s (4.3 million tonnes).
3. Hydrogen burned per second: 4.3 × 10⁹ ÷ 0.0071 = 6.0 × 10¹¹ kg/s. The Sun has about 2 × 10³⁰ kg and about 10 percent will be used in the core, giving a lifetime of about 10¹⁰ years, consistent with the estimated age of 4.6 billion years of a total of about 10 billion years on the main sequence.
4. Deuterium-tritium reaction. The energy released is 17.6 MeV per reaction. The number of reactions in 1.0 kg of a D-T mixture (average mass 2.5 u per nucleus; each reaction uses one D and one T, 5.03 u): N = 1.0 kg ÷ (5.03 × 1.66 × 10⁻²⁷ kg) = 1.20 × 10²⁶. Energy = 1.20 × 10²⁶ × 17.6 × 1.602 × 10⁻¹³ J = 3.4 × 10¹⁴ J, about four times the energy from 1 kg of uranium-235 in fission.
5. Temperature equivalent. A plasma temperature of 10⁸ K corresponds to a mean thermal energy of (3/2)kT = 1.5 × 8.617 × 10⁻⁵ eV/K × 10⁸ K = 12.9 keV.
6. Neutrino flux at the Earth. Two neutrinos are produced per helium nucleus formed: the rate in the Sun is (L ÷ 26.7 MeV) × 2 ≈ 1.8 × 10³⁸ per second, and at the Earth’s distance (1.5 × 10¹¹ m), the flux is about 6 × 10¹⁴ per m² per second, passing through our bodies without interacting.

How was fusion understood?

Arthur Eddington proposed in 1920 that stars shine by converting hydrogen into helium. George Gamow showed that quantum tunneling allows nuclei to fuse at stellar temperatures (1928), and Hans Bethe worked out the CNO cycle and the proton-proton chain in 1938 to 1939 (Nobel Prize 1967). The first thermonuclear (hydrogen) bomb was tested in 1952. Tokamaks were developed in the Soviet Union in the 1950s by Igor Tamm and Andrei Sakharov; the JET tokamak in the UK produced 59 MJ of fusion energy in a 5-second pulse in 2021 to 2023. ITER is under construction, and several private companies are developing compact designs.

Which mistakes are common?

Mistake 1: Thinking that fusion is easy at room temperature

The Coulomb repulsion makes it very difficult. Claims of “cold fusion” have not been confirmed.

Mistake 2: Saying that the Sun burns like a fire

The Sun’s energy is nuclear fusion in the core, not chemical combustion.

Mistake 3: Confusing which direction releases energy for heavy nuclei

Fusion releases energy for nuclei lighter than iron; fission releases energy for heavy nuclei. Fusion of heavy elements absorbs energy.

Mistake 4: Believing that the Sun’s core temperature is high enough for classical fusion

It is far lower than the temperature at which classical collisions can overcome the barrier. Quantum tunneling is essential.