Fusion reactions: from the Sun to deuterium and tritium

The proton–proton chain

1. Two protons fuse into a deuteron, $^{2}_{1}\text{H}$. One proton turns into a neutron, so a positron $\text{e}^+$ and a neutrino $\nu$ are emitted.
2. The deuteron captures another proton and becomes helium-3; a gamma-ray photon $\gamma$ carries off energy.
3. Two helium-3 nuclei, each from steps 1 and 2, fuse into helium-4 and give back two protons.

In total, four protons become one helium-4 nucleus, and $26.7\ \text{MeV}$ is released; about $2\ \%$ of it escapes with the neutrinos. The first step is extremely slow, because turning a proton into a neutron needs the weak interaction. An average proton in the Sun's core waits about $9$ billion years before it fuses. The Sun shines only because it holds an enormous number of protons.

Key idea

The temperature of the Sun's core is about $15.7$ million kelvin. There, the average kinetic energy of a nucleus is only $\frac{3}{2}kT \approx 2\ \text{keV}$, more than $200$ times too little. By classical physics, practically no nucleus would ever get over the barrier, and the Sun could not shine.

The answer is quantum tunnelling. A nucleus behaves partly like a wave, and it has a small probability of appearing on the other side of the barrier without having the energy to climb over it. George Gamow found this effect in 1928 for alpha decay, and in 1929 Robert Atkinson and Fritz Houtermans used it to calculate fusion in stars. Tunnelling is most likely for the fastest nuclei and for barriers that are low and thin, so fusion needs high temperatures, and it is easiest for the hydrogen isotopes, which have the smallest charge. Because of tunnelling, deuterium and tritium already fuse readily at collision energies of about $0.1\ \text{MeV}$, well below the $0.5\ \text{MeV}$ top of the barrier, and in a reactor plasma (about $15\ \text{keV}$) the fastest nuclei do most of the fusing.

Sharing the energy of D–T fusion

The easiest reaction for a reactor is the fusion of deuterium and tritium:

$${}^{2}_{1}\text{H} + {}^{3}_{1}\text{H} \to {}^{4}_{2}\text{He} + {}^{1}_{0}\text{n} + 17.6\ \text{MeV}$$

The reaction is balanced: $2 + 3 = 4 + 1$ nucleons and $1 + 1 = 2 + 0$ elementary charges. The energy appears as kinetic energy of the products. If the fuel nuclei are almost at rest, the total momentum is about zero, so the two products fly apart with equal and opposite momenta, $p_\alpha = p_\text{n} = p$. With $E_\text{k} = p^2/(2m)$, the lighter particle gets more energy:

$$\begin{aligned} \frac{E_\text{n}}{E_\alpha} &= \frac{m_\alpha}{m_\text{n}} \approx 4 \\ E_\text{n} &= \frac{m_\alpha}{m_\alpha + m_\text{n}} \cdot Q \end{aligned}$$

So the neutron carries $14.1\ \text{MeV}$ and the helium nucleus, the alpha particle, $3.5\ \text{MeV}$.

Worked example

Fusion in the laboratory

A power plant cannot simply copy the Sun:

The Sun's reaction is far too slow for a machine, and its gravity holds a dense core for billions of years. A reactor must use the fastest reaction, D–T, in a thin plasma that it can hold only for seconds or less. To make enough nuclei tunnel in that short time, the plasma must be about ten times hotter than the Sun's core, which ITER plans at about $150$ million kelvin. "Hotter than the Sun" refers to the temperature only; the Sun's core has a far higher density and pressure.

The neutron has no charge, so magnetic fields cannot hold it: it leaves the plasma with $80\ \%$ of the energy and heats the wall around it, where its energy can be collected and where it can breed new tritium from lithium. The alpha particle is charged and stays in the plasma, keeping it hot.

Common misconception

"The Sun fuses deuterium and tritium, like a fusion reactor." The Sun fuses ordinary hydrogen through the slow proton–proton chain. D–T fusion is chosen for reactors because it is the fastest reaction at reachable temperatures; tritium hardly exists in nature and must be made.

"A reactor plasma is hotter than the Sun, so it is more extreme in every way." Only the temperature is higher. The Sun's core, $150\ \text{g/cm}^3$, is more than a hundred billion times denser than a reactor plasma, which is about a million times thinner than air, and its pressure is far higher.

"Nuclei fuse when they have enough energy to get over the Coulomb barrier." At the temperatures of the Sun and of reactors, almost all fusions happen by tunnelling, below the top of the barrier.

Summary

• The Sun fuses four protons into helium-4 through the proton–proton chain and releases $26.7\ \text{MeV}$; its first step is very slow, because it needs the weak interaction.
• Positive nuclei repel: the Coulomb barrier is about $0.5\ \text{MeV}$, far more than the thermal energy: a few keV in the Sun, about $10\text{–}20\ \text{keV}$ in a reactor. Fusion happens by quantum tunnelling, mostly for the fastest nuclei.
• Oliphant, Harteck and Rutherford made the first fusion reactions in a laboratory in 1934 (D–D), discovering tritium and helium-3.
• D–T fusion releases $17.6\ \text{MeV}$; by momentum conservation the neutron gets $14.1\ \text{MeV}$ and the alpha particle $3.5\ \text{MeV}$.
• A reactor uses the fastest reaction, D–T, in a thin plasma held by magnets, so it needs about ten times the temperature of the Sun's core.