What Is Fusion

Fusion is a reaction in which light atomic nuclei combine to form a heavier nucleus. In this process, a tiny amount of mass is lost and converted into energy, releasing an enormous amount of it. This page explains why fusion releases energy, why it is so hard to achieve, and how fusion in the Sun differs from fusion on Earth, building up the ideas step by step from the high school level to the PhD level.

Start with Intuition (High School)

At the center of every atom is a nucleus that carries positive electric charge. A nucleus is made of protons and neutrons packed together. Fusion is the reaction of smashing these tiny nuclei together and making them stick.

Here something curious happens. When two light nuclei stick together to form a single heavier nucleus, the resulting nucleus weighs slightly less than the sum of the two original masses. This “missing mass” is what turns into energy and comes out. That is the true source of the energy released in fusion.

The idea that “mass turns into energy” is exactly what Einstein’s famous equation $E = mc^2$ expresses. Here $c$ is the speed of light, a very large number. So even a tiny change in mass becomes a tremendous amount of energy.

But why is it so hard to make nuclei stick together? Every nucleus carries positive charge. Just as like poles of a magnet repel each other, positive charges also repel each other strongly. As you try to bring nuclei closer, this repulsive force grows stronger and stronger, as if an invisible wall is blocking the way. This wall is called the Coulomb barrier.

To get over this wall, you have to smash the nuclei together at an enormous speed. Moving fast means being at a high temperature. The center of the Sun is about 15 million degrees, and the devices aiming for fusion on Earth need roughly 100 million degrees, an unimaginably high temperature. The essence of the difficulty of achieving fusion lies in how to create and maintain this “ultra-high temperature for clearing the wall.”

Understand the Physics (Undergraduate)

The reason fusion releases energy can be explained through binding energy. Binding energy is the energy needed to break a nucleus apart into its separate protons and neutrons. Put the other way around, it is also the energy released when separate nucleons come together to form a nucleus.

The mass of a nucleus is smaller than the sum of the masses of the nucleons that make it up. This difference is called the mass defect. The mass defect $\Delta m$ and the binding energy $E_b$ are connected by the following relation.

$$E_b = \Delta m \, c^2$$

This equation reads as “the lost mass $\Delta m$ multiplied by the square of the speed of light $c^2$ equals the energy released.” If you plot the binding energy per nucleon against atomic number, you get a curve that peaks near iron ($^{56}$Fe). Lighter nuclei have smaller binding energy per nucleon, so fusing them to move closer to iron lets you extract energy equal to the difference. This is fusion; conversely, splitting heavy nuclei to obtain energy is fission.

Let us look concretely at the reaction of deuterium (D) and tritium (T), considered the easiest to achieve.

$$\text{D} + \text{T} \rightarrow {}^{4}\text{He} + n + 17.6 \text{ MeV}$$

In this reaction, a helium-4 nucleus (an alpha particle) and a neutron $n$ are produced, and 17.6 MeV of energy is released. This energy is distributed between the two particles according to conservation of momentum, with the neutron carrying away about 14.1 MeV and the alpha particle about 3.5 MeV.

Next, let us estimate why such a high temperature is needed from the height of the Coulomb barrier. The Coulomb potential energy when two nuclei with charges $Z_1 e$ and $Z_2 e$ approach to a distance $r$ can be written as follows.

$$U(r) = \frac{1}{4\pi\varepsilon_0} \frac{Z_1 Z_2 e^2}{r}$$

Here $\varepsilon_0$ is the vacuum permittivity and $e$ is the elementary charge. If you use this equation to bring the nuclei to the distance at which they touch (roughly a few femtometers), the height of the barrier reaches several hundred keV. Yet the thermal energy corresponding to 100 million degrees is only about 10 keV as $k_B T$. In other words, the energy of a typical particle is far below the barrier. Fusion still occurs because of the following effect from quantum mechanics.

Deepen the Theory (Graduate)

In classical mechanics, a particle with energy below the barrier can never cross it. But in quantum mechanics, a particle can “slip through” the barrier with a certain probability. This is quantum tunneling. Even at temperatures where the average energy is below the barrier, fusion proceeds with a finite probability thanks to tunneling.

The tunneling probability is approximated by the Gamow factor. The probability that a particle with relative kinetic energy $E$ penetrates the barrier is roughly proportional to the following.

$$P \sim \exp\!\left(-\sqrt{\frac{E_G}{E}}\right)$$

Here $E_G$ is a quantity called the Gamow energy, determined by the charges of the nuclei and the reduced mass. This equation shows that “the higher the energy $E$, the smaller the content of the exponent, and the more rapidly the penetration probability increases.” The cross section $\sigma(E)$, which expresses how readily a reaction occurs, is described by combining this tunneling probability with the effect that slower particles are more likely to collide (a factor of $1/E$).

In a real plasma, the particles have speeds following a Maxwell distribution. How readily a reaction occurs is expressed by the reaction rate coefficient $\langle \sigma v \rangle$, which is the product of the cross section $\sigma(E)$ and the relative velocity $v$ averaged over the velocity distribution.

$$\langle \sigma v \rangle = \int_0^\infty \sigma(E)\, v\, f(E)\, dE$$

In this integral, the Maxwell distribution, which falls off sharply on the high-energy tail, is multiplied by the tunneling probability, which falls off sharply on the low-energy side, and an intermediate energy band where the product of the two is maximized dominates the reaction. This band is called the Gamow peak. The $\langle \sigma v \rangle$ of the D-T reaction reaches a practical magnitude in the range of roughly 100 million to 200 million degrees, and this is the theoretical basis for the target temperature.

Quantifying the conditions for a reaction to occur further, the fusion triple product $n \tau_E T$, which combines the density $n$, the energy confinement time $\tau_E$, and the temperature $T$, must exceed a certain value. This is the idea behind the Lawson criterion, which is treated in detail on a separate page.

Research Frontier (PhD)

Research aimed at making fusion an energy source is broadly divided into two approaches: magnetic confinement, which confines the plasma with magnetic fields, and inertial confinement, which momentarily compresses fuel pellets. The former is represented by the tokamak and the stellarator, and the latter by laser fusion.

In inertial confinement, in December 2022, the National Ignition Facility (NIF) in the United States achieved ignition, extracting more fusion energy than the laser energy put in. This is known as the first demonstration of scientific target gain. However, the path to a practical reactor, including the electrical efficiency of the entire laser system, is still studied as a major challenge.

The following keywords appear frequently among today’s major research topics: how to suppress turbulent transport in the plasma; tritium breeding, in which the fuel tritium is produced within the reactor itself; plasma-facing materials that can withstand the fast 14 MeV neutrons; and the physics of self-heating (burning plasma), in which the plasma is sustained by self-heating from alpha particles. These are studied worldwide as unsolved problems that determine whether a practical reactor can be realized.

When reading papers, terms such as the reaction rate $\langle \sigma v \rangle$ obtained by integrating the cross section, the beta value that expresses the dimensionless normalized pressure, the energy confinement time $\tau_E$ that expresses confinement performance, and the fusion gain $Q$ as a performance metric appear repeatedly. Grasping their meanings lets you dive into specialized literature.

Check Your Understanding

Q1. Fusion releases energy because of what happens when nuclei stick together?

A The electrons around the nuclei are accelerated B The mass of the nucleus after the fusion reaction becomes smaller than before, and the mass defect turns into energy C The nuclei split apart and fragments scatter D New electric charge is created in the reaction

Q2. Why is an ultra-high temperature needed to make fusion happen?

A Without high temperature, the nuclei do not separate from the electrons B The speed to overcome the repulsion between positively charged nuclei (the Coulomb barrier) is required C High temperature increases the mass of the nuclei D At low temperature the neutrons freeze and cannot move

Q3. Why does fusion proceed even though the energy of a typical particle is below the Coulomb barrier?

A Because of quantum tunneling, particles can slip through with a finite probability even at energies below the barrier B The barrier naturally lowers over time C The electrons neutralize the barrier D Gravity presses the nuclei together

Q4. How is the 17.6 MeV released in the D-T reaction distributed?

A Exactly half each to the neutron and the alpha particle B All released as gamma rays C The neutron carries away about 14.1 MeV and the alpha particle about 3.5 MeV D The alpha particle carries away about 14.1 MeV and the neutron about 3.5 MeV

Q5. Which of the following is a correct main difference between fusion in the Sun and the fusion aimed for on Earth?

A The Sun uses the D-T reaction, and Earth uses the pp chain B The Sun confines fuel for a long time by gravity and proceeds at about 15 million degrees, while Earth uses the high-reaction-rate D-T reaction at about 100 million degrees or more C Both the Sun and Earth use the same temperature and reaction D Earth can achieve fusion at a lower temperature than the Sun

Related Topics

• If you want to learn more about the types of fusion reactions and their energy balance, see Fusion Reactions.
• To learn quantitatively about the conditions for sustaining fusion, the Lawson Criterion is helpful.
• For how fusion research has developed, see History of Fusion Research.