Nuclear Fusion

Look up on a clear day. The Sun is about 150 million kilometres away, which is roughly 8 light-minutes. Every photon hitting the cricket pitch at the Wankhede in Mumbai right now left the Sun eight and a half minutes ago. If you could follow any one of those photons backwards through its last few million years — photons in the Sun's dense interior bounce off charged particles so relentlessly that it takes something like 10^5 years for a photon of gamma-ray energy to diffuse from the core to the surface — you would find it in the centre of the Sun, in a region about 20% of the solar radius across, where four hydrogen nuclei were squeezing themselves into one helium nucleus. The 26.7 MeV released by that fusion event is what the photon carried to your face today.

Every second, the Sun's core converts roughly 6.2 \times 10^{11} kilograms of hydrogen into 6.16 \times 10^{11} kilograms of helium. The missing 4 \times 10^9 kilograms per second — four million tonnes — has become energy, via E = mc^2. That is the Sun's luminosity: 3.85 \times 10^{26} watts. It has been doing this for 4.6 billion years and has another 5 billion or so left before the hydrogen in the core runs out.

Everything about that description is physics. This article derives the pieces — why fusion releases so much energy, why the Coulomb barrier is the obstacle, why the Sun fuses at a temperature that should be classically impossible, and what the machines at Gandhinagar and Cadarache are trying to do to bring the same process to a power grid near you.

Where the energy comes from — the binding-energy curve, again

Plot the binding energy per nucleon B/A against mass number A. The curve rises very steeply on the light side — from B/A = 0 for a free proton (there is nothing to bind), past B/A = 1.11 MeV for deuterium, 2.83 MeV for tritium, a sharp spike to 7.07 MeV for ^4He, through a smooth rise to the peak at ^{56}Fe (8.79 MeV per nucleon) — then falls slowly on the heavy side all the way to uranium (7.6 MeV).

Binding energy per nucleon is "how much glue is holding each nucleon in place." A higher B/A means a more tightly bound, more stable nucleus. Any nuclear process that moves the constituents up the curve — to a higher B/A — releases energy; any process that moves them down consumes energy. Fission is "heavy \to middle," moving up the curve from the right. Fusion is "light \to middle," moving up the curve from the left.

The fusion side of the curve is the steeper side. Climbing from hydrogen to helium is worth about 6 MeV per nucleon; climbing from uranium down toward middle masses is worth only 0.9 MeV per nucleon. This is the fundamental statement that fusion should, gram for gram, release roughly seven times more energy than fission. Everything else is engineering.

Energy of D–T fusion from the mass defect

Compute the energy release of the central terrestrial fusion reaction, deuterium + tritium:

The atomic masses (in unified atomic mass units, u):

• m(^2\text{D}) = 2.014102 u
• m(^3\text{T}) = 3.016049 u
• m(^4\text{He}) = 4.002602 u
• m_n = 1.008665 u

Compute the mass defect:

Why: atomic masses cancel the electron count (D and T each have one electron, ^4He has two, and that pairing works out on both sides). So you can use atomic masses directly here without tracking electrons, which is the conventional shortcut for fusion-reaction energy calculations.

Convert to MeV using 1\ \text{u}\,c^2 = 931.494\ \text{MeV}:

Why: this is the energy released per D–T fusion event. Most of it (14.1 MeV) goes to the neutron; the remaining 3.5 MeV goes to the alpha particle. The mass of the products is smaller than the mass of the reactants, and the shortfall has left the reaction as kinetic energy of the \alpha and the n. This is the Einstein bookkeeping.

Per nucleon: 17.59\ \text{MeV}/5 = 3.52\ \text{MeV} per nucleon. Compare with uranium fission, which releases 200/236 \approx 0.85\ \text{MeV} per nucleon. D–T fusion gives roughly four times more energy per nucleon than uranium fission — and for pure hydrogen fusion (as in the Sun) it is even higher.

Why fusion needs extreme temperatures — the Coulomb barrier

Fusion looks easy on paper. You just put two hydrogen nuclei close enough that the short-range nuclear force takes over and glues them together. The problem is "close enough." Two nuclei are both positively charged; they repel each other electrostatically. To get them within the nuclear-force range (about r_0 \approx 2\ \text{fm} = 2 \times 10^{-15}\ \text{m}), you have to push them against a mountain of Coulomb energy.

Deriving the barrier height

The Coulomb potential energy of two point charges Z_1 e and Z_2 e at separation r is

Why: the standard Coulomb potential-energy formula, applied to the two nuclei. Positive energy (repulsive) since both charges have the same sign.

The nuclei stop being point-like and the nuclear (strong) force takes over at r \approx r_0 \approx R_1 + R_2, the sum of the two nuclear radii. For proton–proton, both radii are about 1.0\ \text{fm}, so r_0 \approx 2\ \text{fm}.

Plug in numbers for proton–proton (Z_1 = Z_2 = 1):

Why: this is the height of the Coulomb barrier in SI units. Plug in the two fundamental charges and the relevant distance, do the arithmetic.

Convert to electron-volts using 1\,\text{eV} = 1.6 \times 10^{-19}\ \text{J}:

Why: convert to the energy unit of nuclear physics. About 0.7 MeV is the height of the Coulomb barrier for proton–proton — a number worth memorising.

For D–T fusion, both charges are 1 (both are hydrogen isotopes) and r_0 \approx 3.5\ \text{fm}, giving a slightly lower barrier of about 0.4\ \text{MeV}. For heavier nuclei the barrier grows as Z_1 Z_2/r_0 \propto Z_1 Z_2/(A_1^{1/3} + A_2^{1/3}), so fusion between two carbon nuclei (as in advanced stellar burning) sees a barrier of about 8\ \text{MeV} — ten times higher.

Classical temperature estimate

If every particle needs kinetic energy E \sim U_\text{C}(r_0) to climb the barrier, and the average kinetic energy in a thermal plasma is \tfrac{3}{2} k_B T, then the required temperature is

Why: set the average thermal kinetic energy equal to the barrier height, solve for temperature. This is a direct consequence of the Maxwell–Boltzmann distribution's mean.

That is five billion kelvin. The Sun's core is only 1.5 \times 10^7 kelvin — four hundred times cooler than this estimate. If fusion required each individual nucleus pair to climb the barrier classically, the Sun would not shine, and neither would any other star. Something else is going on.

The Sun cheats — quantum tunnelling through the barrier

A proton approaching another proton does not behave like a ball rolling toward a hill. It behaves like a quantum wavefunction, spread out over a region about one de Broglie wavelength across. When that wavefunction overlaps the far side of the barrier, there is a finite probability the proton will "tunnel through" and end up on the other side without ever having had the classical energy to climb.

This is the same tunnelling that alpha decay exploits to let alpha particles escape heavy nuclei. The probability of tunnelling through a Coulomb barrier at centre-of-mass energy E is the Gamow factor:

where \mu is the reduced mass of the two nuclei. The constant b encapsulates the barrier. Its important feature is that it is large — so the tunnelling probability is very small. For proton–proton at the Sun's core temperature, P \sim 10^{-9} per collision.

Fusion rate has two competing factors. Higher E gives a bigger tunnelling probability (the Gamow factor \exp(-b/\sqrt{E}) grows sharply with E). But fewer particles sit at higher E (the Maxwell–Boltzmann distribution is \propto \exp(-E/k_B T), falling with E). Their product,

peaks at a specific energy E_\text{G} called the Gamow peak — the energy at which most stellar fusion actually happens. For the Sun's core (k_B T \approx 1.3\ \text{keV}), E_\text{G} \approx 5-6 keV, still two orders of magnitude below the 720 keV classical barrier. The Sun fuses almost entirely on the high-energy tail of its thermal distribution, tunnelling through a barrier whose top it cannot classically reach.

The full calculation is in the going-deeper section. Here is the qualitative picture: the reaction rate depends on temperature through a steep exponential in T^{-1/3}. At T = 1.5 \times 10^7 K the Sun's core makes just enough fusions per second to power the Sun. Raise the temperature by a factor of two and the fusion rate would go up by a factor of about 2 \times 10^4. Halve the temperature and fusion would essentially stop. This extreme sensitivity is what makes stellar cores self-regulating — any local increase in temperature accelerates fusion and pushes the gas outward (cooling it), any local decrease chokes fusion and lets gravity recompress the core.

Explore — reaction rate versus temperature

Drag the slider. The take-away: terrestrial fusion reactors aim for the steep rise, not the peak. The plasma density and confinement time at 10 keV are already as hard as any engineering challenge on the planet; pushing to 60 keV for a few extra units of reactivity is not worth it.

The proton–proton chain — how the Sun makes helium

The Sun fuses four protons into one helium nucleus. It does not do this in one step — four-body collisions are too rare. Instead, it takes a sequence of two-body steps called the proton–proton chain (pp-I):

Step 1. Two protons fuse into a deuteron (with the emission of a positron and a neutrino):

One of the protons has to beta-plus-decay into a neutron during the collision (a proton becomes a neutron + positron + neutrino). This is a weak interaction process, which is why it is rare. The mean lifetime of a proton in the Sun's core before it undergoes this fusion is about 10^{10} years. This is the bottleneck that controls the Sun's entire luminosity.

Why: a single fast process in a chain does not set the rate — the slowest step does. The pp fusion of two protons is the slowest step by an enormous margin, which is why the Sun burns steadily over billions of years rather than in a flash.

Step 2. The deuteron captures another proton to form ^3He:

This is a strong-interaction capture, so it is fast — a deuteron in the Sun's core lives only about one second before being captured.

Step 3. Two ^3He nuclei fuse into a ^4He plus two protons:

Net reaction (Step 1 happens twice, Step 2 happens twice, Step 3 happens once):

The positrons annihilate with ambient electrons, adding 4 m_e c^2 \approx 2.04\ \text{MeV} of extra gamma rays. The neutrinos escape the Sun almost immediately carrying about 0.59\ \text{MeV} between them. The remaining 26.14\ \text{MeV} is deposited as thermal energy in the core — this is what heats the Sun and eventually leaks out as the light hitting your face.

Worked examples

Controlled fusion — magnetic and inertial confinement

A fusion reactor has three numbers to maximise simultaneously: plasma density n, confinement time \tau_E, and temperature T. Their product, the Lawson triple product nT\tau_E, must exceed about 3 \times 10^{21}\ \text{keV}\cdot\text{s}/\text{m}^3 for a D–T reactor to produce more power than it consumes. This is called the ignition criterion, and no reactor on Earth has yet achieved it in a sustained way (though the ignition milestone at the National Ignition Facility in December 2022 crossed it for a single shot, and ITER is projected to do so in the 2030s).

Two strategies compete:

Magnetic confinement. A plasma at 10^8\ \text{K} will instantly vaporise any solid container. Instead, use a magnetic bottle — charged particles (nuclei and electrons) spiral along magnetic field lines, so a magnetic field that closes on itself can trap them. The tokamak is a doughnut (torus) shaped vessel with a helical magnetic field, and it is the leading candidate reactor design. The plasma is kept in the middle of the doughnut, never touching the walls.

India operates two tokamaks at the Institute for Plasma Research, Gandhinagar:

• Aditya-U (upgraded from the original 1989 Aditya): a compact tokamak with major radius 0.75 m, used for plasma-physics experiments, diagnostics development, and student training.
• SST-1 (Steady State Tokamak-1): a superconducting tokamak designed for long-pulse (1000-second) plasma confinement — the longer pulses that commercial reactors will need.

And India is a 9.1%-stakes member of ITER at Cadarache, France, supplying (among other things) the cryostat — the world's largest stainless-steel vacuum vessel, built by Larsen & Toubro at their Hazira facility in Gujarat.

Inertial confinement. A different strategy: compress a millimetre-scale pellet of frozen D–T with 192 ultra-high-power lasers (as at NIF in the United States), so briefly and intensely that the fuel fuses before its own inertia lets it fly apart. You don't need to contain the plasma; you just need it to fuse in the nanosecond before it disassembles itself. India has a small inertial-fusion programme at BARC using lasers at the Raja Ramanna Centre for Advanced Technology in Indore, focused on inertial-confinement physics studies.

Common confusions

• "Fusion and fission are opposites, so fusion must consume the energy that fission releases." No — both release energy, for different nuclei. Fission works only for heavy nuclei (past the peak), fusion only for light nuclei (before the peak). Every nuclear reaction that climbs the B/A curve releases energy; every one that descends consumes it. Both fusion and fission climb — from opposite sides.

• "The Sun fuses hydrogen to helium, so its core must contain mostly helium by now." Only about 50% of the Sun's core hydrogen has been fused to helium over its 4.6 billion year lifetime; the Sun's whole composition is still about 70% hydrogen by mass (because the core is a small fraction of the total volume). When the core does run out of hydrogen, in roughly 5 billion years, the Sun will leave the main sequence, the core will contract, and the outer envelope will expand into a red giant — enough to engulf Mercury and Venus.

• "Fusion doesn't produce radioactive waste." Mostly true — the main fusion product, helium-4, is a stable noble gas. But the 14\ \text{MeV} neutrons from D–T fusion activate the reactor walls, producing short-lived radioactive isotopes of iron, cobalt, and other structural-metal elements. The waste problem is much smaller and much shorter-lived than fission (tens to hundreds of years rather than tens of thousands), but it is not zero. Aneutronic fuel cycles (D–^3He, p–^{11}B) avoid this but have even higher temperature requirements.

• "Cold fusion works." No. The 1989 Fleischmann–Pons claim to have observed room-temperature fusion in a palladium electrode was never reproduced, and there is no theoretical mechanism that would allow it. Fusion requires either high temperatures (thermonuclear) or extraordinary measures to compensate (muon-catalysed fusion, inertial-electrostatic confinement) — all of which are either too costly or too slow to produce net power.

• "A tokamak is just a big magnet; why is it so hard?" The plasma is a conductive fluid. It is unstable in dozens of ways — kinks, sausages, tearing modes, edge-localised modes — and each instability has to be understood, predicted, and suppressed. The engineering problems are formidable but the physics problems are harder. ITER is as much a plasma-physics experiment as it is a reactor.

• "Hydrogen bombs are fusion bombs." They are, in a sense — the primary stage is a fission bomb that compresses a secondary fusion fuel to ignition. A pure fusion bomb (without a fission trigger) has never been built, because the only way to reach fusion temperatures rapidly enough is with a fission explosion. Thermonuclear weapons thus carry radioactive fallout from their fission trigger, not from the fusion itself.

Where this leads next

• Mass Defect and Binding Energy — the B/A curve that drives both fusion and fission energetics.
• Nuclear Fission — the other route to nuclear energy, working from the opposite side of the same curve.
• Radioactivity — Types of Decay — the same quantum tunnelling that lets protons fuse is what lets alpha particles escape.
• Radioactive Decay Law and Half-Life — the statistical mathematics of nuclear processes, applicable to every decay rate appearing above.
• The Nucleus — Composition and Size — what the protons and neutrons actually look like before they start fusing.