Fusion and the Sun

The Sun does not burn

A coal fire burns for a few hours. If the Sun were a ball of coal, it would burn out in a few thousand years. But the Sun has been shining for about $4.6$ billion years. What gives it so much energy? In 1938 Hans Bethe showed that the Sun gets its energy from nuclear fusion, the joining of light nuclei into heavier ones.

The curve of binding energy per nucleon tells us that joining light nuclei moves up the curve, and releases energy. Two deuterons make helium-4 and give $23.85$ MeV. This is about $6$ MeV for each nucleon, which is seven times more than in fission.

Why fusion is hard to start

All nuclei are positive, so two of them repel one another. For them to fuse, they must come within about $3$ fm, where the nuclear force, which has a short range, takes over. The electric potential energy of two nuclei with charges $Z_1e$ and $Z_2e$ at a distance $d$ is

Take two deuterons ($Z = 1$). Their radii are $1.2\ \text{fm} \times 2^{1/3} = 1.51$ fm each, so they touch at $d = 3.02$ fm. Then

This is the height of the barrier. The nuclei need about this much kinetic energy. The average energy of a particle in a gas is $\tfrac{3}{2}kT$, so we would need $T = \dfrac{2U}{3k} = \dfrac{2 \times 0.48 \times 10^{6}\ \text{eV}}{3 \times 8.617 \times 10^{-5}\ \text{eV/K}} = 3.7 \times 10^{9}$ K.

The core of the Sun is at $1.5 \times 10^{7}$ K, which is a hundred times cooler. The average energy of a proton there is only $\tfrac{3}{2}kT = 1.9$ keV, about $250$ times lower than the barrier. How does fusion then happen? There are two reasons.

• Some nuclei are much faster than the average. In a gas the energies have a wide spread, and a tiny part of the nuclei have many times the mean energy.
• Quantum tunnelling. A nucleus can pass through the barrier even if it has less energy than the top. The chance is small but not zero. The Sun has a huge number of nuclei, so enough get through.

So a fusion plant on Earth needs a temperature of about $10^{8}$ K. At this temperature atoms lose their electrons, and the gas of nuclei and electrons is a plasma.

The Sun: the proton-proton chain

In the core of the Sun, four protons turn into one helium-4 nucleus in three steps.

1. ${}^{1}_{1}\text{H} + {}^{1}_{1}\text{H} \to {}^{2}_{1}\text{H} + e^+ + \nu$ ($0.42$ MeV)
2. ${}^{2}_{1}\text{H} + {}^{1}_{1}\text{H} \to {}^{3}_{2}\text{He} + \gamma$ ($5.49$ MeV)
3. ${}^{3}_{2}\text{He} + {}^{3}_{2}\text{He} \to {}^{4}_{2}\text{He} + 2\,{}^{1}_{1}\text{H}$ ($12.86$ MeV)

Steps $1$ and $2$ must happen twice to give the two helium-3 nuclei for step $3$. The first step is very slow, since a proton must change into a neutron while the two are together. This is why the Sun burns so slowly and lasts so long. Adding the steps, with the two positrons annihilating two electrons ($2 \times 1.022$ MeV), the net result is

Check with masses: $4 \times 1.007825\ \text{u} - 4.002603\ \text{u} = 0.028697$ u, and $0.028697\ \text{u} \times 931.494\ \text{MeV/u} = 26.73$ MeV.

How much does the Sun burn? The Sun radiates $3.8 \times 10^{26}$ W. Each completed chain gives $26.7 \times 1.602 \times 10^{-13}\ \text{J} = 4.28 \times 10^{-12}$ J. So the chains happen $\dfrac{3.8 \times 10^{26}\ \text{W}}{4.28 \times 10^{-12}\ \text{J}} = 8.9 \times 10^{37}$ times each second. Each uses $4$ protons of $1.67 \times 10^{-27}$ kg, so the Sun turns $6 \times 10^{11}$ kg of hydrogen into helium each second. That is $600$ million tonnes. It still has a huge supply of hydrogen.

In heavier stars another cycle, the carbon-nitrogen-oxygen cycle, in which carbon-12 acts as a catalyst, has the same net result. Stars cannot get energy from the fusion of nuclei past iron, since iron is at the top of the curve. Those nuclei are made in stars that end in explosions.

Fusion on Earth

The easiest reaction uses two forms of hydrogen:

The barrier is lower than for helium, at $0.44$ MeV. Deuterium is in sea water, about one atom in $6400$ of hydrogen. Tritium is radioactive and rare, but it can be made inside the plant. Lithium-6 takes a neutron and gives tritium: ${}^{1}_{0}n + {}^{6}_{3}\text{Li} \to {}^{3}_{1}\text{H} + {}^{4}_{2}\text{He}$.

To hold a plasma at $10^{8}$ K, no wall can touch it. There are two ways. In magnetic confinement, as in a tokamak, magnetic fields in a ring hold the charged particles. In inertial confinement, laser beams heat and squeeze a tiny pellet so quickly that it fuses before it flies apart. To get more energy than we put in, the plasma must be hot enough, dense enough and held for long enough. This is the Lawson criterion. Experiments have reached it only for very short times, and a working power plant has not yet been built.

The energy for each unit of mass is also larger. The D-T reaction gives $17.59$ MeV from $5.03$ u of fuel, which is about $3.5$ MeV per u. For fission it is about $0.85$ MeV per u. The fuel is easy to find, and a fusion plant makes no chain reaction that can run away. Its waste is much less radioactive and lasts a shorter time than the waste from fission.

Start with $Z_1 = Z_2 = 1$ and $d = 3$ fm. What is the ratio of the barrier to the thermal energy at $15$ million kelvin? Now raise $Z_1$ to $2$. What does that do to the barrier, and why do heavier stars need to be hotter to burn helium?

Practise with new numbers

Try it in Python

The program finds the electric barrier for four pairs of nuclei, and the temperature at which the average energy equals the barrier.

Common mistakes

• Saying fusion starts at the temperature where $\tfrac{3}{2}kT$ equals the barrier. Real fusion happens at a much lower temperature, because of tunnelling and the fast nuclei.
• Saying that fusion in the Sun needs a very high density only. It needs a high temperature and a high density together.
• Mixing up fission and fusion energy. Fission gives more energy for one event, about $200$ MeV against $17.59$ MeV, but fusion gives more for each unit of mass.
• Saying that the Sun burns hydrogen as a fire does. It is fusion, not a chemical burning.
• Adding charges for the barrier. The potential energy depends on the product $Z_1Z_2$.
• Forgetting that iron is the end of the line. Fusion beyond iron absorbs energy.

Remember these

• Fusion joins light nuclei and releases energy. Deuterium and tritium give $17.59$ MeV.
• The barrier is $U = \dfrac{1.44\,Z_1Z_2}{d}$ in MeV, with $d$ in fm. For two deuterons it is about $0.48$ MeV.
• The mean energy of a particle in a gas is $\tfrac{3}{2}kT$. The core of the Sun, at $1.5 \times 10^{7}$ K, has only about $1.9$ keV.
• Fusion in the Sun is possible because of tunnelling and the fast nuclei in a gas.
• The Sun's chain: four protons make a helium-4 nucleus, two positrons and two neutrinos, and $26.7$ MeV.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Why do two nuclei need a very high temperature to fuse?

2. In the Sun, how many protons and how many neutrinos are involved when one helium-4 nucleus is made?

3. Why can no material container hold a plasma at $10^{8}$ K?

4. Which gives more energy for each kilogram of fuel, fusion of deuterium and tritium, or fission of uranium-235?

References

• NCERT Physics Part II (Class 12, rationalised edition), Chapter 13 Nuclei, on nuclear fusion and energy generation in stars.
• OpenStax University Physics Volume 3, Chapter 10 Nuclear Physics, on nuclear fusion.
• AME2020 atomic mass evaluation (Wang and others, Chinese Physics C, 2021), for the atomic masses.