Nuclear fusion is a process where two or more atomic nuclei join to form a larger nucleus. The difference in mass between the starting materials and the final product results in the release or absorption of energy. This mass difference happens because the energy that holds the nucleus together, called nuclear binding energy, changes before and after the fusion process. Nuclear fusion is the process that powers all active stars through many different methods.

Fusion requires very high levels of temperature, density, and confinement time. These extreme conditions are found only in the centers of stars, in advanced nuclear weapons, and are being tested in experiments for fusion power.

A fusion process that creates atomic nuclei lighter than nickel-62 usually releases energy because the nuclear binding energy curve shows a positive trend for lighter nuclei. The easiest nuclei to fuse are the lightest ones, such as deuterium, tritium, and helium-3. The opposite process, nuclear fission, releases the most energy when very heavy nuclei, like those in the actinide group, are split.

Fusion has many uses, including fusion power, thermonuclear weapons, boosted fission weapons, neutron sources, and the production of superheavy elements.

History

American chemist William Draper Harkins first suggested the idea of nuclear fusion in 1915. In 1919, Francis William Aston created the mass spectrometer, which showed that four hydrogen atoms are heavier than one helium atom. This helped Arthur Eddington correctly predict in 1920 that fusing hydrogen into helium could be the main way stars produce energy.

Quantum tunneling, a process related to how electrons behave, was discovered by Friedrich Hund in 1927. In 1928, George Gamow used this idea to explain nuclear processes, first with alpha decay and later with fusion. In 1929, Robert Atkinson and Fritz Houtermans made the first estimates of how fast fusion happens in stars.

In 1938, Hans Bethe and Charles Critchfield described the proton–proton chain, a process that powers stars like the Sun. In 1939, Bethe also explained the CNO cycle, which is common in larger stars.

During the 1920s, Patrick Blackett performed experiments on artificial nuclear changes at the Cavendish Laboratory. There, John Cockcroft and Ernest Walton built a machine inspired by Gamow’s work. In April 1932, they published results showing a reaction where beryllium-8, a very short-lived atom, was formed. This experiment is considered the first artificial fusion reaction.

In 1933, Ernest Lawrence and others at the University of California Radiation Laboratory accidentally created the first deuterium–deuterium fusion reactions using a cyclotron. They mistakenly thought the reaction was a type of breakdown, but it was later corrected. In 1934, Mark Oliphant, Paul Harteck, and Ernest Rutherford intentionally studied deuterium fusion and discovered tritium and helium-3. This experiment is widely seen as the first successful fusion demonstration.

In 1938, Arthur Ruhlig at the University of Michigan observed deuterium–tritium fusion and identified its 14 MeV neutrons, which are now known as the most efficient fusion reaction.

Research on fusion for military use started in the 1940s as part of the Manhattan Project. In 1941, Enrico Fermi and Edward Teller discussed how a fission bomb might create conditions for fusion. In 1942, Emil Konopinski shared Ruhlig’s work on deuterium–tritium fusion with the project. Scientists used early computers like ENIAC to study fusion reactions starting in 1945.

The first artificial thermonuclear fusion happened in 1951 during the US Greenhouse George test, using deuterium–tritium gas. This test produced 225 kilotons of energy, the largest yield at that time. The first true two-stage thermonuclear weapon, Ivy Mike, tested in 1952, used liquid deuterium and produced over 10 megatons of energy. The Teller–Ulam design made this possible.

The Soviet Union began its hydrogen bomb program earlier and tested RDS-6s in 1953, the first air-deliverable fusion bomb. It produced 400 kilotons of energy but was limited by its single-stage design. In 1955, the Soviet Union tested RDS-37, a two-stage bomb using a version of the Teller–Ulam design, producing 1.5 megatons of energy.

Modern fusion devices use solid lithium deuteride enriched with lithium-6. This helps create tritium for the deuterium–tritium reaction, which produces a lot of energy.

While fusion bombs were studied for energy, scientists focused on controlled fusion for peaceful use. Research on fusion reactors began in the 1930s. In 1958, the Scylla I device at Los Alamos National Laboratory produced the first laboratory fusion.

The first major controlled fusion experiments used Tokamaks, which are doughnut-shaped reactors. In the 1990s, experiments at PPPL’s TFTR produced 1.6 gigajoules of fusion energy. Later, JET achieved 16 megawatts of fusion power. In 2024, JET produced 69 megajoules of energy using small amounts of deuterium and tritium.

The US National Ignition Facility, which uses lasers to compress fusion fuel, aimed to achieve a fusion energy gain factor greater than one. In 2022, scientists reached a major milestone by producing more energy from fusion than was used to trigger the reaction.

Before this, controlled fusion could not produce enough energy to be self-sustaining. Two main methods are being studied: magnetic confinement (using toroidal reactors) and inertial confinement (using lasers). The ITER project, expected to begin full fusion experiments in 2039, aims to create a reactor that produces ten times more energy than it uses.

In 2021, private companies working on fusion energy received $2.6 billion in funding. Companies like Commonwealth Fusion Systems, Helion Energy, and TAE Technologies are developing commercial fusion technologies.

A recent breakthrough occurred in France’s WEST reactor, where scientists maintained a plasma at 90 million degrees Celsius for six minutes. This reactor is similar to the upcoming ITER reactor and uses the tokamak design.

Process

The release of energy during the fusion of light elements happens because of two forces that work against each other: the nuclear force, which holds protons and neutrons tightly together in the nucleus, and the Coulomb force, which pushes positively charged protons apart. Lighter nuclei, which are smaller than iron and nickel, are small enough and have few protons to allow the nuclear force to overcome the Coulomb force. This is because the nucleus is small enough that all particles feel the short-range nuclear force more strongly than the long-range Coulomb repulsion. When lighter nuclei combine through fusion, energy is released because the nuclear force creates a net attraction between particles. However, larger nuclei do not release energy because the nuclear force cannot act across their larger size.

Fusion powers stars and creates most elements lighter than cobalt through a process called nucleosynthesis. The Sun is a main-sequence star that produces energy by fusing hydrogen nuclei into helium. In its core, the Sun fuses 620 million metric tons of hydrogen into 616 million metric tons of helium every second. Fusion releases energy and mass, as seen in the fusion of two hydrogen nuclei into helium, where 0.645% of the mass is converted into kinetic energy or other forms of energy, such as light.

It requires a lot of energy to make nuclei fuse, even for hydrogen, the lightest element. When nuclei move fast enough, they overcome the electrostatic repulsion and get close enough for the nuclear force to pull them together. Once close, the strong force increases rapidly, allowing the nuclei to merge. This process releases energy and is called exothermic.

Most nuclear reactions release far more energy than chemical reactions because the energy that holds a nucleus together is much greater than the energy that holds electrons to a nucleus. For example, adding an electron to a hydrogen nucleus releases 13.6 eV of energy, which is less than one-millionth of the 17.6 MeV released in a deuterium-tritium fusion reaction. Fusion reactions have much higher energy density than nuclear fission, even though individual fission reactions release more energy than individual fusion reactions. Fusion reactions are also millions of times more energetic than chemical reactions. According to the mass-energy equivalence principle, fusion converts about 0.7% of the mass of the reactants into energy. This efficiency is only exceeded in extreme cases, such as when matter falls into neutron stars or black holes, which can reach up to 40% efficiency, or in antimatter annihilation, which reaches 100% efficiency. For example, converting one gram of matter completely would release 9 × 10¹⁴ joules of energy.

In astrophysics

Fusion creates most elements lighter than iron in space. This includes processes like Big Bang nucleosynthesis and stellar nucleosynthesis. Other processes, such as the s-process and r-process in neutron mergers and supernova nucleosynthesis, help form elements heavier than iron.

A key fusion process is stellar nucleosynthesis, which powers stars, including the Sun. In the 20th century, scientists learned that energy from nuclear fusion explains why stars stay hot and bright for long periods. Fusion in a star combines hydrogen and helium into new elements, releasing energy. The specific reactions depend on the star’s mass, which affects the temperature and pressure in its core.

Around 1920, Arthur Eddington suggested that nuclear fusion in stars produces their energy. At the time, scientists did not know how stars generated energy. Eddington correctly guessed that hydrogen fuses into helium, releasing energy as shown by Einstein’s equation E = mc². This was significant because fusion and the fact that stars are mostly hydrogen were not yet understood. Eddington’s paper argued:

• The idea that stars shrink to create energy would cause them to spin faster, but observations of Cepheid variable stars showed this did not happen.
• The only known energy source was converting matter into energy, as Einstein had shown.
• Francis Aston found that helium atoms are slightly lighter than four hydrogen atoms, suggesting energy could be released if they combined.
• If stars had 5% fusible hydrogen, it could explain their energy. Today, most stars are about 70% to 75% hydrogen.
• Scientists later speculated that stars might create heavier elements by fusing lighter ones, but more accurate measurements were needed.

These ideas were confirmed in later years.

The main energy source for the Sun and similar stars is fusing hydrogen into helium (the proton-proton chain), which happens at 14 million kelvin in the Sun’s core. This process creates helium, releases energy, and produces two positrons and two neutrinos. In heavier stars, the CNO cycle is more important. As stars use up hydrogen, they begin fusing heavier elements. In massive stars, silicon burning creates iron and nickel.

Fusion becomes less likely for elements heavier than nickel because it requires more energy. These elements form through non-fusion processes like the s-process, r-process, and others in giant stars, supernovae, or neutron star mergers.

Brown dwarfs fuse deuterium and, in rare cases, lithium.

Carbon–oxygen white dwarfs that gain mass from other stars or mergers approach the Chandrasekhar limit of 1.44 solar masses. When this happens, carbon burning starts, causing a Type Ia supernova explosion within seconds.

Helium white dwarfs may merge without exploding, but this can start helium burning in a special type of helium star.

Some neutron stars gain hydrogen and helium from companion stars. When helium builds up, a thermonuclear reaction occurs on the star’s surface in about one second.

Like stars, extreme conditions near black holes can allow fusion. Calculations show the most energetic reactions happen around lower-mass black holes (under 10 solar masses). Beyond five Schwarzschild radii, helium-3 and carbon burning dominate. Closer to lower-mass black holes, nitrogen, oxygen, neon, and magnesium may fuse. In extreme cases, silicon burning can occur.

Between 10 seconds and 20 minutes after the Big Bang, the universe cooled enough for protons and neutrons to combine into deuterium, starting a chain that created helium-4 and small amounts of lithium, beryllium, and boron.

Evidence shows early gas clouds in the universe collapsed under gravity, forming the first stars about 13.6 billion years ago.

Requirements

A large amount of energy is needed to overcome the repelling force between positively charged particles before fusion can happen. When two nuclei are far apart, they push each other away because of the repelling force between their protons. If nuclei get close enough, they can pass through the repelling force through a process called quantum tunneling.

The nuclear force pulls nucleons toward each other, but mainly affects nearby nucleons because the force doesn't reach far. Nucleons inside a nucleus have more neighbors than those on the outside. Smaller nuclei have more surface area compared to their size, so the energy that holds them together increases as the nucleus grows, but it eventually reaches a limit. Because nucleons are quantum objects, it's not possible to tell them apart, so quantum mechanics is needed for accurate calculations.

The electrostatic force decreases with distance, so a proton added to a nucleus feels repulsion from all other protons. The energy from this force increases as the nucleus becomes larger. The combined effect of the repelling electrostatic force and the attractive nuclear force means the energy that holds nuclei together increases up to iron and nickel, then decreases for heavier nuclei. Very heavy nuclei are unstable. The most tightly bound nuclei are nickel, iron, iron, and nickel. Even though nickel is more stable, iron is more common in stars because it is harder to create through certain processes.

An exception is helium-4, which has a higher binding energy than lithium. This is because protons and neutrons cannot occupy the same energy state due to a rule called the Pauli exclusion principle. Helium-4 has two protons and two neutrons, all in the lowest energy state, making it extremely stable. It is often treated as a single particle in physics, called an alpha particle.

When two nuclei approach each other, all their protons repel each other. Fusion only happens when the nuclei get close enough for the strong nuclear force to overcome the repelling force. This is called overcoming the Coulomb barrier. The energy needed to do this can be lower than the barrier itself because of quantum tunneling.

The Coulomb barrier is smallest for hydrogen isotopes because their nuclei have only one positive charge. Fusion of deuterium and tritium requires about 0.1 MeV of energy. The result is an unstable helium nucleus that quickly releases a neutron with 14.1 MeV of energy. The remaining helium nucleus has 3.5 MeV of energy, so the total energy released is 17.6 MeV, much more than the energy needed to start the reaction.

The reaction cross section measures the chance of fusion based on the speed of nuclei. The reactivity is the average of this chance multiplied by speed. The fusion rate depends on the number of nuclei and their speeds. If a nucleus reacts with itself, the calculation changes slightly.

Reactivity increases from nearly zero at room temperature to meaningful levels at temperatures of 10–100 keV/kB. At these temperatures, fusion reactants exist as a plasma. The importance of reactivity at a given temperature is determined by the Lawson criterion, which is a major challenge to overcome on Earth. This explains why fusion research has taken many years to advance.

Artificial fusion

Thermonuclear fusion is a process where atomic nuclei combine, or "fuse," at very high temperatures. These temperatures cause the matter to become a plasma, a state where particles move quickly. If the plasma is kept in a controlled area, fusion reactions can happen because the particles collide with enough energy. There are two types of thermonuclear fusion: uncontrolled, where the energy is released suddenly, like in hydrogen bombs or stars; and controlled, where the energy can be captured and used for power.

Temperature measures the average energy of particles. When material is heated, the particles gain energy. Once the temperature is high enough, as shown by the Lawson criterion, the energy from collisions can overcome a barrier called the Coulomb barrier, allowing nuclei to fuse.

For example, in a deuterium–tritium fusion reaction, the energy needed to overcome the Coulomb barrier is 0.1 MeV. Converting this energy to temperature shows that the barrier is overcome at temperatures above 1.2 billion kelvin.

Two factors help lower the actual temperature needed for fusion. First, temperature is an average, so some particles have more energy than others. The ones with the most energy are more likely to fuse. Second, a process called quantum tunnelling allows particles to pass through the barrier even if they don’t have enough energy. Because of these effects, fusion can occur at lower temperatures, though more slowly.

Thermonuclear fusion is being studied as a way to create fusion power. If successful, it could greatly reduce the world’s carbon emissions.

Accelerator-based light-ion fusion uses particle accelerators to give ions enough energy to fuse. This process is simple and can be done with a vacuum tube, electrodes, and a high-voltage transformer. Fusion can be observed with as little as 10,000 volts. Ions can be directed at a target (beam–target fusion) or sent toward each other (beam–beam fusion). However, most ions lose energy through radiation instead of fusing. Small devices called sealed-tube neutron generators use this method to produce neutrons, which are used in the petroleum industry to find oil reserves.

Some experiments have tried to reuse ions that didn’t collide. In the 1970s, a system called Migma used a ring to capture ions and return them to the reaction area. However, scaling this up for power was difficult. In the 1990s, a new method using a field-reversed configuration (FRC) was proposed and is still studied today. Companies like TAE Technologies and Helion Energy are exploring similar approaches. These methods use high ion energies and may use alternative fuels like p-B, which are harder to use with other techniques.

Fusing heavy nuclei with accelerated ion beams is how scientists create new elements. In the 1930s, scientists used deuteron beams to discover elements like technetium, neptunium, and plutonium. Later, heavy ion beams helped find superheavy elements like darmstadtium and oganesson.

Muon-catalyzed fusion happens at normal temperatures. Scientists studied it in the 1980s, but it hasn’t produced net energy because creating muons is expensive and they decay quickly.

Other fusion methods include:
– Antimatter-initialized fusion: Uses tiny amounts of antimatter to trigger fusion. It’s studied for space propulsion but is not practical for power.
– Pyroelectric fusion: Uses heat and electric fields to accelerate deuterium into a target, creating neutrons. It’s not used for power because it uses more energy than it produces.
– Hybrid fusion-fission power: Combines fusion and fission to generate energy. It was studied in the past and is being revisited.
– Project PACER: Explored using underground explosions of hydrogen bombs to generate energy, but it would require many bombs, making it costly.
– Bubble fusion (sonofusion): Proposed using sound waves to create fusion, but experiments failed to replicate results.

Confinement in thermonuclear fusion

The main challenge in achieving thermonuclear fusion is how to hold the hot plasma in place. Because the plasma is extremely hot, it cannot touch any solid material, so it must be placed in a vacuum. High temperatures also create high pressure, causing the plasma to expand. A force must be applied to prevent this expansion. This force can be gravity, as in stars; magnetic forces, as in magnetic confinement fusion reactors; or inertia, where the fusion reaction happens quickly before the plasma has time to expand.

Gravity is one force that can hold fusion fuel long enough to meet the Lawson criterion. However, the amount of mass needed is so large that gravitational confinement only occurs in stars. The smallest stars that can sustain fusion are red dwarfs, while brown dwarfs can fuse deuterium and lithium if they have enough mass. In larger stars, after hydrogen is used up in their cores, helium begins to fuse into carbon. In the most massive stars (at least 8–11 times the mass of the Sun), fusion continues until lighter elements are turned into iron. Iron has one of the highest binding energies, so creating heavier elements usually requires energy. These heavier elements are not formed during normal star life but are produced in supernova explosions. Some lighter stars also create these elements over time by absorbing energy and neutrons from fusion processes inside the star.

Elements heavier than iron can release energy if split back into smaller elements, like iron, through nuclear fission. This process releases energy that was stored during earlier fusion reactions in stars.

Electrically charged particles, such as fuel ions, follow magnetic field lines. This means fusion fuel can be trapped using strong magnetic fields. Different magnetic setups are used, including the donut-shaped designs of tokamaks and stellarators, and open-ended mirror systems.

A third method uses a quick burst of energy to compress a fusion fuel pellet, causing it to implode and heat rapidly. If the fuel is dense and hot enough, fusion can occur before the plasma spreads out. To achieve this, the fuel must be compressed explosively. Inertial confinement is used in hydrogen bombs, where x-rays from a fission bomb drive the process. In controlled fusion experiments, lasers, ion beams, or electron beams are used. Conventional explosives have also been tested to compress fuel. The UTIAS facility used a method involving a stoichiometric mixture of deuterium and oxygen or a miniature Voitenko compressor to create fusion conditions.

Some devices use electrostatic fields to trap ions, like the fusor. This device has a cathode inside an anode wire cage. Positive ions are pulled toward the cage and heated by the electric field. If they miss the cage, they may collide and fuse. However, most ions hit the cathode, causing high energy losses. Fusion rates in fusors are low due to other energy losses, like light radiation. New designs, such as plasma oscillating devices, Penning traps, and polywells, aim to improve efficiency. These technologies are still under development, and many questions remain.

Since 1999, some amateurs have built homemade fusion reactors using fusors. Other inertial electrostatic confinement devices include the Polywell, MIX POPS, and Marble concepts.

Important reactions

In the cores of stars, fusion reactions occur very slowly because of the high temperatures and densities. For example, at the center of the Sun (temperature ≈ 15 million Kelvin and density ≈ 160 grams per cubic centimeter), the energy released is only 276 microwatts per cubic centimeter—about a quarter of the heat produced by a resting human body. Replicating these conditions in a laboratory for nuclear fusion energy is not practical. Fusion reaction rates depend on both temperature and density. Most fusion methods use low densities, so they require much higher temperatures than found in stars. The fusion rate depends on temperature (exp(−E/kT)), which means that temperatures in artificial reactors must be 10–100 times higher than in stars: T ≈ (0.1–1.0) × 10^8 K.

In artificial fusion, the fuel is not limited to protons, and higher temperatures can be used. This allows scientists to choose reactions with larger cross-sections. Neutrons are produced during fusion, which can damage reactor structures but also help extract energy and create tritium. Reactions that do not produce neutrons are called aneutronic.

For fusion to be a useful energy source, reactions must meet several criteria. Few reactions satisfy these conditions. The following reactions have the largest cross-sections:

In reactions with two products, energy is divided between them based on their masses. In reactions with three products, energy distribution varies. For reactions that can form multiple products, branching ratios are given.

Some reactions can be ruled out. The D–Li reaction is not better than p–5B because it is just as hard to start but produces more neutrons from side reactions. A p–3Li reaction has a very low cross-section, except at extremely high temperatures, where another reaction becomes significant. A p–4Be reaction is also difficult to start, and 4Be can split into alpha particles and a neutron.

Neutron reactions are important for producing tritium in fusion bombs and reactors:

The second equation was not known when the U.S. tested the Castle Bravo bomb in 1954. Designers understood lithium’s role in tritium production but did not realize that lithium fission would greatly increase the bomb’s power. Lithium has a small neutron cross-section at low energies but a larger one above 5 MeV. The test produced 15 megatons of energy—150% more than predicted—and caused unexpected radiation exposure.

To evaluate reactions, scientists consider reactants, products, energy released, and nuclear cross-sections. Fusion devices have a maximum plasma pressure they can sustain. The most efficient fusion output occurs at a temperature where ⟨σv⟩/T is maximized. This is also the temperature where the triple product nTτ for ignition is minimized (Lawson criterion). A plasma is "ignited" if fusion reactions generate enough heat to maintain temperature without external heating. Optimal temperatures and ⟨σv⟩/T values for some reactions are shown in a table.

Many reactions form chains. For example, a reactor using 1T and 2He produces 1D, which can later react with 2He if conditions are right. Combining reactions (8) and (9) could work if 2He from reaction (8) reacts with 3Li before thermalizing. However, detailed analysis shows this idea is not effective, highlighting a case where the usual assumption of a Maxwellian plasma is not valid.

Any of the above reactions could be used for fusion power. In addition to temperature and cross-sections, scientists consider total fusion energy (E_fus), energy from charged particles (E_ch), and the atomic number (Z) of non-hydrogenic reactants.

The 1D–1D reaction has challenges. It requires averaging over two branches (2i) and (2ii). Deciding how to handle 1T and 2He products is also difficult. 1T burns quickly in a deuterium plasma and is hard to extract, while the 1D–2He reaction works best at higher temperatures. Assuming 1T burns but not 2He, the total reaction becomes the sum of (2i), (2ii), and (1):

For calculating reactor power (where the D–D step determines the reaction rate), the energy per D–D fusion is E_fus = (4.03 MeV + 17.6 MeV) × 50% + (3.27 MeV) × 50% = 12.5 MeV. The energy in charged particles is E_ch = (4.03 MeV + 3.5 MeV) × 50% + (0.82 MeV) × 50% = 4.2 MeV. The energy per deuteron consumed is 5.0 MeV (about 225 million MJ per kilogram of deuterium).

The 1D–1D reaction has only one reactant, which affects the reaction rate calculation.

Parameters for four important reactions are listed in a table. The last column shows the neutronicity, the fraction of fusion energy released as neutrons. This is important for problems like radiation damage and safety. For the first two reactions, neutronicity is (E_fus − E_ch)/E_fus. For the last two, values are estimates based on side reactions.

Reactants should be mixed in optimal proportions. This occurs when each reactant ion and its electrons account for half the pressure. Assuming total pressure is fixed, the particle density of non-hydrogenic ions is smaller than hydrogenic ions by a factor of 2/(Z + 1). This reduces the reaction rate. However, the 1D–1D reaction has only one reactant, so its rate is twice as high as when fuel is split between two hydrogenic species, making it more efficient.

Non-hydrogenic fuels have a "penalty" of 2/(Z + 1) because they require more electrons

Mathematical description of cross section

In a classical model, nuclei can be seen as hard spheres that push each other away due to the Coulomb force but join together when the spheres are close enough to touch. Estimating the radius of an atomic nucleus as about one femtometer, the energy required for two hydrogen atoms to fuse is:

This suggests that in the sun's core, where the temperature is about 1.4 keV and follows a Boltzmann distribution, the chance of hydrogen reaching the energy needed for fusion is 10, meaning fusion would not occur. However, fusion in the sun does happen because of quantum mechanics.

The likelihood of fusion increases significantly compared to the classical model, due to the spreading of the effective radius as the de Broglie wavelength and quantum tunneling through the potential barrier. To calculate the rate of fusion reactions, the most important value is the cross section, which describes the probability of particles fusing by providing a characteristic area of interaction. An estimate of the fusion cross-sectional area is often divided into three parts:

where σ geometry is the geometric cross section, T is the barrier transparency, and R represents the reaction characteristics of the specific interaction.

σ geometry is approximately the square of the de Broglie wavelength, σ geometry ≈ λ² = (ℏ / (m_r v))² ∝ 1/ε, where m_r is the reduced mass of the system and ε is the center of mass energy of the system.

T can be approximated by the Gamow transparency, which is expressed as T ≈ e^(-√(ε_G / ε)), where ε_G = (π α Z₁Z₂)² × 2m_r c² is the Gamow factor, derived from estimating the quantum tunneling probability through the potential barrier.

R includes all nuclear physics details of the specific reaction and varies greatly depending on the interaction type. However, for most reactions, changes in R(ε) are small compared to changes in the Gamow factor, so R is often approximated by a function called the astrophysical S-factor, S(ε), which changes little with energy. Combining these factors, one approximation for the fusion cross section as a function of energy is:

More detailed forms of the cross-section can be calculated using nuclear physics-based models and R-matrix theory.

The Naval Research Lab's plasma physics formulary provides the total cross section in barns as a function of the energy (in keV) of the incident particle toward a target ion at rest, using the formula:

Bosch-Hale also reports R-matrix calculated cross sections that match observational data using Padé rational approximating coefficients. With energy in units of keV and cross sections in millibarns, the formula is:

where σ Bosch-Hale(ε) = S Bosch-Hale(ε) / (ε exp(√(ε_G / ε))).

In fusion systems in thermal equilibrium, particles follow a Maxwell–Boltzmann distribution, meaning they have a range of energies centered around the plasma temperature. The sun, magnetically confined plasmas, and inertial confinement fusion systems are well modeled as being in thermal equilibrium. In these cases, the important value is the fusion cross-section averaged across the Maxwell–Boltzmann distribution. The Naval Research Lab's plasma physics formulary lists Maxwell-averaged fusion cross sections and reactivities in cm³/s.

For energies T ≤ 25 keV, the data can be represented by:

with T in units of keV.