Nuclear fusion is a
reaction in which two or more
atomic nuclei combine to form a larger nuclei, nuclei/
neutron
The neutron is a subatomic particle, symbol or , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
by-products. The difference in mass between the reactants and products is manifested as either the release or
absorption of
energy
Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
. This difference in mass arises as a result of the difference in
nuclear binding energy between the atomic nuclei before and after the fusion reaction. Nuclear fusion is the process that powers all active
stars
A star is a luminous spheroid of plasma held together by self-gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night; their immense distances from Earth make them appear as fixed points of ...
, via many
reaction pathways.
Fusion processes require an extremely large
triple product
In geometry and algebra, the triple product is a product of three 3- dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the ve ...
of temperature, density, and confinement time. These conditions occur only in
stellar cores, advanced
nuclear weapons
A nuclear weapon is an explosive device that derives its destructive force from nuclear reactions, either nuclear fission, fission (fission or atomic bomb) or a combination of fission and nuclear fusion, fusion reactions (thermonuclear weap ...
, and are approached in
fusion power experiments.
A nuclear fusion process that produces atomic nuclei lighter than
nickel-62 is generally
exothermic, due to the positive gradient of the
nuclear binding energy curve. The most fusible nuclei are among the lightest, especially
deuterium,
tritium, and
helium-3. The opposite process,
nuclear fission
Nuclear fission is a reaction in which the nucleus of an atom splits into two or more smaller nuclei. The fission process often produces gamma photons, and releases a very large amount of energy even by the energetic standards of radioactiv ...
, is most energetic for very heavy nuclei, especially the
actinides.
Applications of fusion include
fusion power,
thermonuclear weapons,
boosted fission weapons,
neutron sources, and
superheavy element
Superheavy elements, also known as transactinide elements, transactinides, or super-heavy elements, or superheavies for short, are the chemical elements with atomic number greater than 104. The superheavy elements are those beyond the actinides in ...
production.
History
Theory

American chemist
William Draper Harkins was the first to propose the concept of nuclear fusion in 1915.
[
] Francis William Aston's 1919 invention of the
mass spectrometer allowed the discovery that four hydrogen atoms are heavier than one helium atom. Thus in 1920,
Arthur Eddington correctly predicted fusion of hydrogen into helium could be the primary source of stellar energy.
Quantum tunneling was discovered by
Friedrich Hund in 1927, with relation to electron levels. In 1928,
George Gamow was the first to apply tunneling to the nucleus, first to
alpha decay
Alpha decay or α-decay is a type of radioactive decay in which an atomic nucleus emits an alpha particle (helium nucleus). The parent nucleus transforms or "decays" into a daughter product, with a mass number that is reduced by four and an a ...
, then to fusion as an inverse process. From this, in 1929,
Robert Atkinson and
Fritz Houtermans made the first estimates for stellar fusion rates.
In 1938,
Hans Bethe worked with
Charles Critchfield to enumerate the
proton–proton chain that dominates Sun-type stars. In 1939, Bethe published the discovery of the
CNO cycle common to higher-mass stars.
Early experiments

During the 1920s,
Patrick Blackett made the first conclusive experiments in artificial
nuclear transmutation at the
Cavendish Laboratory. There,
John Cockcroft and
Ernest Walton built
their generator on the inspiration of Gamow's paper. In April 1932, they published experiments on the reaction:
: +
p → → 2
where the intermediary nuclide was later confirmed to be the extremely short-lived
beryllium-8.
This has a claim to the first artificial fusion reaction.
In papers from July and November 1933,
Ernest Lawrence et. al. at the
University of California Radiation Laboratory, in some of the earliest
cyclotron
A cyclotron is a type of particle accelerator invented by Ernest Lawrence in 1929–1930 at the University of California, Berkeley, and patented in 1932. Lawrence, Ernest O. ''Method and apparatus for the acceleration of ions'', filed: Januar ...
experiments, accidentally produced the first
deuterium–deuterium fusion reactions:
: + → + p
: + → +
The Radiation Lab, only detecting the resulting energized protons and neutrons,
misinterpreted the source as an exothermic disintegration of the deuterons, now known to be impossible.
In May 1934,
Mark Oliphant,
Paul Harteck, and
Ernest Rutherford at the Cavendish Laboratory,
published an intentional deuterium fusion experiment, and made the discovery of both
tritium and
helium-3. This is widely considered the first experimental demonstration of fusion.
In 1938, Arthur Ruhlig at the
University of Michigan
The University of Michigan (U-M, U of M, or Michigan) is a public university, public research university in Ann Arbor, Michigan, United States. Founded in 1817, it is the oldest institution of higher education in the state. The University of Mi ...
made the first observation of
deuterium–tritium (DT) fusion and its characteristic 14 MeV neutrons, now known as the most favourable reaction:
: + → +
Weaponization
Research into
fusion for military purposes began in the early 1940s as part of the
Manhattan Project. In 1941, Enrico Fermi and Edward Teller had a conversation about the possibility of a fission bomb creating conditions for thermonuclear fusion. In 1942,
Emil Konopinski brought Ruhlig's work on the deuterium–tritium reaction to the projects attention.
J. Robert Oppenheimer initially commissioned physicists at Chicago and Cornell to use the Harvard University cyclotron to secretly investigate its cross-section, and that of the lithium reaction (see below). Measurements were obtained at Purdue, Chicago, and Los Alamos from 1942 to 1946. Theoretical assumptions about DT fusion gave it a similar cross-section to DD. However, in 1946
Egon Bretscher discovered a
resonance
Resonance is a phenomenon that occurs when an object or system is subjected to an external force or vibration whose frequency matches a resonant frequency (or resonance frequency) of the system, defined as a frequency that generates a maximu ...
enhancement giving the DT reaction a cross-section ~100 times larger.
From 1945, John von Neumann, Teller, and other Los Alamos scientists used
ENIAC
ENIAC (; Electronic Numerical Integrator and Computer) was the first Computer programming, programmable, Electronics, electronic, general-purpose digital computer, completed in 1945. Other computers had some of these features, but ENIAC was ...
, one of the first electronic computers, to simulate thermonuclear weapon detonations.
The first artificial thermonuclear fusion reaction occurred during the 1951 US
Greenhouse George nuclear test, using a small amount of
deuterium–tritium gas. This produced the largest yield to date, at 225 kt, 15 times that of
Little Boy. The first "true"
thermonuclear weapon
A thermonuclear weapon, fusion weapon or hydrogen bomb (H-bomb) is a second-generation nuclear weapon design. Its greater sophistication affords it vastly greater destructive power than first-generation nuclear bombs, a more compact size, a lowe ...
detonation i.e. a two-stage device, was the 1952
Ivy Mike test of a
liquid
Liquid is a state of matter with a definite volume but no fixed shape. Liquids adapt to the shape of their container and are nearly incompressible, maintaining their volume even under pressure. The density of a liquid is usually close to th ...
deuterium-fusing device, yielding over 10 Mt. The key to this jump was the full utilization of the fission blast by the
Teller–Ulam design.
The Soviet Union had begun their focus on a hydrogen bomb program earlier, and in 1953 carried out the
RDS-6s test. This had international impacts as the first air-deliverable bomb using fusion, but yielded 400 kt and was limited by its single-stage design. The first Soviet two-stage test was
RDS-37 in 1955 yielding 1.5 Mt, using an independently reached version of the Teller–Ulam design.
Modern devices benefit from the usage of solid
lithium deuteride with an enrichment of lithium-6. This is due to the
Jetter cycle involving the exothermic reaction:
: + → +
During thermonuclear detonations, this provides tritium for the highly energetic DT reaction, and benefits from its neutron production, creating a closed neutron cycle.
Fusion energy
While fusion bomb detonations were
loosely considered for energy production, the possibility of controlled and sustained reactions remained the scientific focus for peaceful fusion power. Research into developing controlled fusion inside
fusion reactors has been ongoing since the 1930s, with
Los Alamos National Laboratory's Scylla I device producing the first laboratory thermonuclear fusion in 1958, but the technology is still in its developmental phase.
The first experiments producing large amounts of controlled fusion power were the experiments with mixes of deuterium and tritium in
Tokamaks.
Experiments in the
TFTR at the
PPPL in
Princeton University
Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...
Princeton NJ, USA during 1993–1996 produced created 1.6 GJ fusion energy.
The peak fusion power was 10.3 MW from reactions per second, and peak fusion energy created in one discharge was 7.6 MJ.
Subsequent experiments in the
JET in 1997 achieved a peak fusion power of 16 MW ().
The central ''Q'', defined as the local fusion power produced to the local applied heating power, is computed to be 1.3.
A JET experiment in 2024 produced 69 MJ of fusion power, consuming 0.2 mgm of D and T.
The US
National Ignition Facility, which uses laser-driven
inertial confinement fusion, was designed with a goal of achieving a
fusion energy gain factor (Q) of larger than one; the first large-scale laser target experiments were performed in June 2009 and ignition experiments began in early 2011.
On 13 December 2022, the
United States Department of Energy
The United States Department of Energy (DOE) is an executive department of the U.S. federal government that oversees U.S. national energy policy and energy production, the research and development of nuclear power, the military's nuclear w ...
announced that on 5 December 2022, they had successfully accomplished break-even fusion, "delivering 2.05 megajoules (MJ) of energy to the target, resulting in 3.15 MJ of fusion energy output." The rate of supplying power to the experimental test cell is hundreds of times larger than the power delivered to the target.
Prior to this breakthrough, controlled fusion reactions had been unable to produce break-even (self-sustaining) controlled fusion. The two most advanced approaches for it are
magnetic confinement (toroid designs) and inertial confinement (laser designs). Workable designs for a toroidal reactor that theoretically will deliver ten times more fusion energy than the amount needed to heat plasma to the required temperatures are in development (see
ITER). The ITER facility is expected to finish its construction phase in 2025. It will start commissioning the reactor that same year and initiate plasma experiments in 2025, but is not expected to begin full deuterium–tritium fusion until 2035.
Private companies pursuing the commercialization of nuclear fusion received $2.6 billion in private funding in 2021 alone, going to many notable startups including but not limited to
Commonwealth Fusion Systems,
Helion Energy Inc.,
General Fusion,
TAE Technologies Inc. and
Zap Energy Inc.
One of the most recent breakthroughs to date in maintaining a sustained fusion reaction occurred in France's WEST fusion reactor. It maintained a 90 million degree plasma for a record time of six minutes. This is a tokamak style reactor which is the same style as the upcoming ITER reactor.
Process

The release of energy with the fusion of light elements is due to the interplay of two opposing forces: the
nuclear force, a manifestation of the
strong interaction
In nuclear physics and particle physics, the strong interaction, also called the strong force or strong nuclear force, is one of the four known fundamental interaction, fundamental interactions. It confines Quark, quarks into proton, protons, n ...
, which holds protons and neutrons tightly together in the
atomic nucleus
The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford at the Department_of_Physics_and_Astronomy,_University_of_Manchester , University of Manchester ...
; and the
Coulomb force, which causes positively
charged proton
A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
s in the nucleus to repel each other. Lighter nuclei (nuclei smaller than iron and nickel) are sufficiently small and proton-poor to allow the nuclear force to overcome the Coulomb force. This is because the nucleus is sufficiently small that all nucleons feel the short-range attractive force at least as strongly as they feel the infinite-range Coulomb repulsion. Building up nuclei from lighter nuclei by fusion releases the extra energy from the net attraction of particles.
For larger nuclei, however, no energy is released, because the nuclear force is short-range and cannot act across larger nuclei.
Fusion powers
star
A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s and produces most elements lighter than cobalt in a process called
nucleosynthesis
Nucleosynthesis is the process that creates new atomic nuclei from pre-existing nucleons (protons and neutrons) and nuclei. According to current theories, the first nuclei were formed a few minutes after the Big Bang, through nuclear reactions in ...
. The Sun is a main-sequence star, and, as such, generates its energy by nuclear fusion of hydrogen nuclei into helium. In its core, the Sun fuses 620 million metric tons of hydrogen and makes 616 million metric tons of helium each second. The fusion of lighter elements in stars releases energy and the mass that always accompanies it. For example, in the fusion of two hydrogen nuclei to form helium, 0.645% of the mass is carried away in the form of kinetic energy of an
alpha particle or other forms of energy, such as electromagnetic radiation.
It takes considerable energy to force nuclei to fuse, even those of the lightest element,
hydrogen
Hydrogen is a chemical element; it has chemical symbol, symbol H and atomic number 1. It is the lightest and abundance of the chemical elements, most abundant chemical element in the universe, constituting about 75% of all baryon, normal matter ...
. When accelerated to high enough speeds, nuclei can overcome this electrostatic repulsion and be brought close enough such that the attractive
nuclear force is greater than the repulsive Coulomb force. The
strong force grows rapidly once the nuclei are close enough, and the fusing nucleons can essentially "fall" into each other and the result is fusion; this is an
exothermic process.
Energy released in most
nuclear reactions is much larger than in
chemical reaction
A chemical reaction is a process that leads to the chemistry, chemical transformation of one set of chemical substances to another. When chemical reactions occur, the atoms are rearranged and the reaction is accompanied by an Gibbs free energy, ...
s, because the
binding energy
In physics and chemistry, binding energy is the smallest amount of energy required to remove a particle from a system of particles or to disassemble a system of particles into individual parts. In the former meaning the term is predominantly use ...
that holds a nucleus together is greater than the energy that holds
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s to a nucleus. For example, the
ionization energy
In physics and chemistry, ionization energy (IE) is the minimum energy required to remove the most loosely bound electron of an isolated gaseous atom, Ion, positive ion, or molecule. The first ionization energy is quantitatively expressed as
: ...
gained by adding an electron to a hydrogen nucleus is —less than one-millionth of the released in the
deuterium–
tritium (D–T) reaction shown in the adjacent diagram. Fusion reactions have an
energy density many times greater than
nuclear fission
Nuclear fission is a reaction in which the nucleus of an atom splits into two or more smaller nuclei. The fission process often produces gamma photons, and releases a very large amount of energy even by the energetic standards of radioactiv ...
; the reactions produce far greater energy per unit of mass even though ''individual'' fission reactions are generally much more energetic than ''individual'' fusion ones, which are themselves millions of times more energetic than chemical reactions. Via the
mass–energy equivalence, fusion yields a 0.7% efficiency of reactant mass into energy. This can be only be exceeded by the extreme cases of the
accretion process involving neutron stars or black holes, approaching 40% efficiency, and
antimatter
In modern physics, antimatter is defined as matter composed of the antiparticles (or "partners") of the corresponding subatomic particle, particles in "ordinary" matter, and can be thought of as matter with reversed charge and parity, or go ...
annihilation at 100% efficiency. (The complete conversion of one gram of matter would expel of energy.)
In astrophysics
Fusion is responsible for the astrophysical production of the majority of elements lighter than iron. This includes most types of
Big Bang nucleosynthesis and
stellar nucleosynthesis
In astrophysics, stellar nucleosynthesis is the creation of chemical elements by nuclear fusion reactions within stars. Stellar nucleosynthesis has occurred since the original creation of hydrogen, helium and lithium during the Big Bang. As a ...
. Non-fusion processes that contribute include the
s-process and
r-process
In nuclear astrophysics, the rapid neutron-capture process, also known as the ''r''-process, is a set of nuclear reactions that is responsible for nucleosynthesis, the creation of approximately half of the Atomic nucleus, atomic nuclei Heavy meta ...
in neutron merger and
supernova nucleosynthesis, responsible for elements heavier than iron.
Stars

An important fusion process is the
stellar nucleosynthesis
In astrophysics, stellar nucleosynthesis is the creation of chemical elements by nuclear fusion reactions within stars. Stellar nucleosynthesis has occurred since the original creation of hydrogen, helium and lithium during the Big Bang. As a ...
that powers
star
A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s, including the Sun. In the 20th century, it was recognized that the energy released from nuclear fusion reactions accounts for the longevity of stellar heat and light. The fusion of nuclei in a star, starting from its initial hydrogen and helium abundance, provides that energy and synthesizes new nuclei. Different reaction chains are involved, depending on the mass of the star (and therefore the pressure and temperature in its core).
Around 1920,
Arthur Eddington anticipated the discovery and mechanism of nuclear fusion processes in stars, in his paper ''The Internal Constitution of the Stars''.
At that time, the source of stellar energy was unknown; Eddington correctly speculated that the source was fusion of hydrogen into helium, liberating enormous energy according to
Einstein's equation . This was a particularly remarkable development since at that time fusion and thermonuclear energy had not yet been discovered, nor even that stars are largely composed of
hydrogen
Hydrogen is a chemical element; it has chemical symbol, symbol H and atomic number 1. It is the lightest and abundance of the chemical elements, most abundant chemical element in the universe, constituting about 75% of all baryon, normal matter ...
(see
metallicity). Eddington's paper reasoned that:
# The leading theory of stellar energy, the contraction hypothesis, should cause the rotation of a star to visibly speed up due to
conservation of angular momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
. But observations of
Cepheid variable stars showed this was not happening.
# The only other known plausible source of energy was conversion of matter to energy; Einstein had shown some years earlier that a small amount of matter was equivalent to a large amount of energy.
#
Francis Aston had also recently shown that the mass of a helium atom was about 0.8% less than the mass of the four hydrogen atoms which would, combined, form a helium atom (according to the then-prevailing theory of atomic structure which held atomic weight to be the distinguishing property between elements; work by
Henry Moseley and
Antonius van den Broek would later show that nucleic charge was the distinguishing property and that a helium nucleus, therefore, consisted of two hydrogen nuclei plus additional mass). This suggested that if such a combination could happen, it would release considerable energy as a byproduct.
# If a star contained just 5% of fusible hydrogen, it would suffice to explain how stars got their energy. (It is now known that most 'ordinary' stars are usually made of around 70% to 75% hydrogen)
# Further elements might also be fused, and other scientists had speculated that stars were the "crucible" in which light elements combined to create heavy elements, but without more accurate measurements of their
atomic masses nothing more could be said at the time.
All of these speculations were proven correct in the following decades.
The primary source of solar energy, and that of similar size stars, is the fusion of hydrogen to form helium (the
proton–proton chain reaction), which occurs at a solar-core temperature of 14 million kelvin. The net result is the fusion of four
protons
A proton is a stable subatomic particle, symbol , H+, or 1H+ with a positive electric charge of +1 ''e'' ( elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an electron (the pro ...
into one
alpha particle, with the release of two
positrons and two
neutrinos (which changes two of the protons into neutrons), and energy. In heavier stars, the
CNO cycle and other processes are more important. As a star uses up a substantial fraction of its hydrogen, it begins to fuse heavier elements. In massive cores,
silicon-burning is the final fusion cycle, leading to a build-up of iron and nickel nuclei.
Nuclear binding energy makes the production of elements heavier than nickel via fusion energetically unfavorable. These elements are produced in non-fusion processes: the
s-process,
r-process
In nuclear astrophysics, the rapid neutron-capture process, also known as the ''r''-process, is a set of nuclear reactions that is responsible for nucleosynthesis, the creation of approximately half of the Atomic nucleus, atomic nuclei Heavy meta ...
, and the variety of processes that can produce
p-nuclei. Such processes occur in giant star shells, or
supernovae
A supernova (: supernovae or supernovas) is a powerful and luminous explosion of a star. A supernova occurs during the last evolutionary stages of a massive star, or when a white dwarf is triggered into runaway nuclear fusion. The original ob ...
, or
neutron star mergers.
Brown dwarfs
Brown dwarfs fuse deuterium and in very high mass cases also fuse lithium.
White dwarfs
Carbon–oxygen
white dwarfs, which accrete matter either from an active stellar companion or white dwarf merger, approach the
Chandrasekhar limit of 1.44 solar masses. Immediately prior,
carbon burning fusion begins, destroying the Earth-sized dwarf within one second, in a
Type Ia supernova
A Type Ia supernova (read: "type one-A") is a type of supernova that occurs in binary systems (two stars orbiting one another) in which one of the stars is a white dwarf. The other star can be anything from a giant star to an even smaller white ...
.
Much more rarely, helium white dwarfs may merge, which does not cause an explosion but begins
helium burning in an extreme type of
helium star.
Neutron stars
Some neutron stars accrete hydrogen and helium from an active stellar companion. Periodically, the helium accretion reaches a critical level, and a thermonuclear burn wave propagates across the surface, on the timescale of one second.
Black hole accretion disks
Similar to stellar fusion, extreme conditions within
black hole
A black hole is a massive, compact astronomical object so dense that its gravity prevents anything from escaping, even light. Albert Einstein's theory of general relativity predicts that a sufficiently compact mass will form a black hole. Th ...
accretion disks can allow fusion reactions. Calculations show the most energetic reactions occur around lower
stellar mass black holes, below 10 solar masses, compared to those above 100. Beyond five
Schwarzschild radii,
carbon-burning and fusion of helium-3 dominates the reactions. Within this distance, around lower mass black holes, fusion of nitrogen,
oxygen
Oxygen is a chemical element; it has chemical symbol, symbol O and atomic number 8. It is a member of the chalcogen group (periodic table), group in the periodic table, a highly reactivity (chemistry), reactive nonmetal (chemistry), non ...
,
neon
Neon is a chemical element; it has symbol Ne and atomic number 10. It is the second noble gas in the periodic table. Neon is a colorless, odorless, inert monatomic gas under standard conditions, with approximately two-thirds the density of ...
, and magnesium can occur. In the extreme limit, the
silicon-burning process can begin with the fusion of silicon and selenium nuclei.
Big Bang
From the period approximately 10 seconds to 20 minutes after the
Big Bang, the universe cooled from over 100 keV to 1 keV. This allowed the combination of protons and neutrons in deuterium nuclei, and beginning a rapid fusion chain into tritium and helium-3 and ending in predominantly helium-4, with a minimal fraction of lithium, beryllium, and boron nuclei.
Requirements

A substantial energy barrier of electrostatic forces must be overcome before fusion can occur. At large distances, two naked nuclei repel one another because of the repulsive
electrostatic force between their
positively charged protons. If two nuclei can be brought close enough together, however, the electrostatic repulsion can be overcome by the quantum effect in which nuclei can
tunnel through coulomb forces.
When a
nucleon such as a
proton
A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
or
neutron
The neutron is a subatomic particle, symbol or , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
is added to a nucleus, the nuclear force attracts it to all the other nucleons of the nucleus (if the atom is small enough), but primarily to its immediate neighbors due to the short range of the force. The nucleons in the interior of a nucleus have more neighboring nucleons than those on the surface. Since smaller nuclei have a larger surface-area-to-volume ratio, the binding energy per nucleon due to the
nuclear force generally increases with the size of the nucleus but approaches a limiting value corresponding to that of a nucleus with a diameter of about four nucleons. It is important to keep in mind that nucleons are
quantum objects. So, for example, since two neutrons in a nucleus are identical to each other, the goal of distinguishing one from the other, such as which one is in the interior and which is on the surface, is in fact meaningless, and the inclusion of quantum mechanics is therefore necessary for proper calculations.
The electrostatic force, on the other hand, is an
inverse-square force, so a proton added to a nucleus will feel an electrostatic repulsion from ''all'' the other protons in the nucleus. The electrostatic energy per nucleon due to the electrostatic force thus increases without limit as nuclei atomic number grows.

The net result of the opposing electrostatic and strong nuclear forces is that the binding energy per nucleon generally increases with increasing size, up to the elements
iron
Iron is a chemical element; it has symbol Fe () and atomic number 26. It is a metal that belongs to the first transition series and group 8 of the periodic table. It is, by mass, the most common element on Earth, forming much of Earth's o ...
and
nickel, and then decreases for heavier nuclei. Eventually, the
binding energy
In physics and chemistry, binding energy is the smallest amount of energy required to remove a particle from a system of particles or to disassemble a system of particles into individual parts. In the former meaning the term is predominantly use ...
becomes negative and very heavy nuclei (all with more than 208 nucleons, corresponding to a diameter of about 6 nucleons) are not stable. The four most tightly bound nuclei, in decreasing order of
binding energy
In physics and chemistry, binding energy is the smallest amount of energy required to remove a particle from a system of particles or to disassemble a system of particles into individual parts. In the former meaning the term is predominantly use ...
per nucleon, are , , , and . Even though the
nickel isotope, , is more stable, the
iron isotope is an
order of magnitude more common. This is due to the fact that there is no easy way for stars to create through the
alpha process.
An exception to this general trend is the
helium-4
Helium-4 () is a stable isotope of the element helium. It is by far the more abundant of the two naturally occurring isotopes of helium, making up about 99.99986% of the helium on Earth. Its nucleus is identical to an alpha particle, and consi ...
nucleus, whose binding energy is higher than that of
lithium
Lithium (from , , ) is a chemical element; it has chemical symbol, symbol Li and atomic number 3. It is a soft, silvery-white alkali metal. Under standard temperature and pressure, standard conditions, it is the least dense metal and the ...
, the next heavier element. This is because protons and neutrons are
fermion
In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s, which according to the
Pauli exclusion principle cannot exist in the same nucleus in exactly the same state. Each proton or neutron's energy state in a nucleus can accommodate both a spin up particle and a spin down particle. Helium-4 has an anomalously large binding energy because its nucleus consists of two protons and two neutrons (it is a
doubly magic nucleus), so all four of its nucleons can be in the ground state. Any additional nucleons would have to go into higher energy states. Indeed, the helium-4 nucleus is so tightly bound that it is commonly treated as a single quantum mechanical particle in nuclear physics, namely, the
alpha particle.
The situation is similar if two nuclei are brought together. As they approach each other, all the protons in one nucleus repel all the protons in the other. Not until the two nuclei actually come close enough for long enough so the strong attractive
nuclear force can take over and overcome the repulsive electrostatic force. This can also be described as the nuclei overcoming the so-called
Coulomb barrier. The kinetic energy to achieve this can be lower than the barrier itself because of quantum tunneling.
The
Coulomb barrier is smallest for isotopes of hydrogen, as their nuclei contain only a single positive charge. A
diproton is not stable, so neutrons must also be involved, ideally in such a way that a helium nucleus, with its extremely tight binding, is one of the products.
Using
deuterium–tritium fuel, the resulting energy barrier is about 0.1 MeV. In comparison, the energy needed to remove an
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
from
hydrogen
Hydrogen is a chemical element; it has chemical symbol, symbol H and atomic number 1. It is the lightest and abundance of the chemical elements, most abundant chemical element in the universe, constituting about 75% of all baryon, normal matter ...
is 13.6 eV. The (intermediate) result of the fusion is an unstable
5He nucleus, which immediately ejects a neutron with 14.1 MeV. The recoil energy of the remaining
4He nucleus is 3.5 MeV, so the total energy liberated is 17.6 MeV. This is many times more than what was needed to overcome the energy barrier.
The reaction
cross section (σ) is a measure of the probability of a fusion reaction as a function of the relative velocity of the two reactant nuclei. If the reactants have a distribution of velocities, e.g. a thermal distribution, then it is useful to perform an average over the distributions of the product of cross-section and velocity. This average is called the 'reactivity', denoted . The reaction rate (fusions per volume per time) is times the product of the reactant number densities:
:
If a species of nuclei is reacting with a nucleus like itself, such as the DD reaction, then the product
must be replaced by
.
increases from virtually zero at room temperatures up to meaningful magnitudes at temperatures of . At these temperatures, well above typical
ionization energies (13.6 eV in the hydrogen case), the fusion reactants exist in a
plasma state.
The significance of
as a function of temperature in a device with a particular energy
confinement time is found by considering the
Lawson criterion. This is an extremely challenging barrier to overcome on Earth, which explains why fusion research has taken many years to reach the current advanced technical state.
Artificial fusion
Thermonuclear fusion
Thermonuclear fusion is the process of atomic nuclei combining or "fusing" using high temperatures to drive them close enough together for this to become possible. Such temperatures cause the matter to become a
plasma and, if confined, fusion reactions may occur due to collisions with extreme thermal kinetic energies of the particles. There are two forms of thermonuclear fusion: ''uncontrolled'', in which the resulting energy is released in an uncontrolled manner, as it is in
thermonuclear weapon
A thermonuclear weapon, fusion weapon or hydrogen bomb (H-bomb) is a second-generation nuclear weapon design. Its greater sophistication affords it vastly greater destructive power than first-generation nuclear bombs, a more compact size, a lowe ...
s ("hydrogen bombs") and in most
star
A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s; and ''controlled'', where the fusion reactions take place in an environment allowing some or all of the energy released to be harnessed.
Temperature is a measure of the average
kinetic energy
In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion.
In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
of particles, so by heating the material it will gain energy. After reaching sufficient temperature, given by the
Lawson criterion, the energy of accidental collisions within the
plasma is high enough to overcome the
Coulomb barrier and the particles may fuse together.
In a
deuterium–tritium fusion reaction, for example, the energy necessary to overcome the
Coulomb barrier is 0.1
MeV. Converting between energy and temperature shows that the 0.1 MeV barrier would be overcome at a temperature
in excess of 1.2 billion kelvin.
There are two effects that are needed to lower the actual temperature. One is the fact that
temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
is the ''average'' kinetic energy, implying that some nuclei at this temperature would actually have much higher energy than 0.1 MeV, while others would be much lower. It is the nuclei in the high-energy tail of the
velocity distribution that account for most of the fusion reactions. The other effect is
quantum tunnelling. The nuclei do not actually have to have enough energy to overcome the Coulomb barrier completely. If they have nearly enough energy, they can tunnel through the remaining barrier. For these reasons fuel at lower temperatures will still undergo fusion events, at a lower rate.
''Thermonuclear'' fusion is one of the methods being researched in the attempts to produce
fusion power. If thermonuclear fusion becomes favorable to use, it would significantly reduce the world's
carbon footprint.
Beam–beam or beam–target fusion
Accelerator-based light-ion fusion is a technique using
particle accelerator
A particle accelerator is a machine that uses electromagnetic fields to propel electric charge, charged particles to very high speeds and energies to contain them in well-defined particle beam, beams. Small accelerators are used for fundamental ...
s to achieve particle kinetic energies sufficient to induce light-ion fusion reactions.
Accelerating light ions is relatively easy, and can be done in an efficient manner—requiring only a vacuum tube, a pair of electrodes, and a high-voltage transformer; fusion can be observed with as little as 10 kV between the electrodes. The system can be arranged to accelerate ions into a static fuel-infused target, known as ''beam–target'' fusion, or by accelerating two streams of ions towards each other, ''beam–beam'' fusion. The key problem with accelerator-based fusion (and with cold targets in general) is that fusion cross sections are many orders of magnitude lower than Coulomb interaction cross-sections. Therefore, the vast majority of ions expend their energy emitting
bremsstrahlung radiation and the ionization of atoms of the target. Devices referred to as sealed-tube
neutron generators are particularly relevant to this discussion. These small devices are miniature particle accelerators filled with deuterium and tritium gas in an arrangement that allows ions of those nuclei to be accelerated against hydride targets, also containing deuterium and tritium, where fusion takes place, releasing a flux of neutrons. Hundreds of neutron generators are produced annually for use in the petroleum industry where they are used in measurement equipment for locating and mapping oil reserves.
A number of attempts to recirculate the ions that "miss" collisions have been made over the years. One of the better-known attempts in the 1970s was
Migma, which used a unique particle
storage ring to capture ions into circular orbits and return them to the reaction area. Theoretical calculations made during funding reviews pointed out that the system would have significant difficulty scaling up to contain enough fusion fuel to be relevant as a power source. In the 1990s, a new arrangement using a
field-reversed configuration (FRC) as the storage system was proposed by
Norman Rostoker and continues to be studied by
TAE Technologies . A closely related approach is to merge two FRC's rotating in opposite directions, which is being actively studied by
Helion Energy. Because these approaches all have ion energies well beyond the
Coulomb barrier, they often suggest the use of alternative fuel cycles like p-
11B that are too difficult to attempt using conventional approaches.
Element synthesis
Fusion of very heavy target nuclei with accelerated ion beams is the primary method of element synthesis. In early 1930s nuclear experiments, deuteron beams were used, to discover the first synthetic elements, such as
technetium
Technetium is a chemical element; it has Symbol (chemistry), symbol Tc and atomic number 43. It is the lightest element whose isotopes are all radioactive. Technetium and promethium are the only radioactive elements whose neighbours in the sense ...
,
neptunium, and
plutonium:
Fusion of very heavy target nuclei with heavy ion beams has been used to discover
superheavy elements:
Muon-catalyzed fusion
Muon-catalyzed fusion is a fusion process that occurs at ordinary temperatures. It was studied in detail by
Steven Jones in the early 1980s. Net energy production from this reaction has been unsuccessful because of the high energy required to create
muons, their short 2.2 μs
half-life Half-life is a mathematical and scientific description of exponential or gradual decay.
Half-life, half life or halflife may also refer to:
Film
* Half-Life (film), ''Half-Life'' (film), a 2008 independent film by Jennifer Phang
* ''Half Life: ...
, and the high chance that a muon will bind to the new
alpha particle and thus stop catalyzing fusion.
Other principles

Some other confinement principles have been investigated.
*
Antimatter-initialized fusion uses small amounts of
antimatter
In modern physics, antimatter is defined as matter composed of the antiparticles (or "partners") of the corresponding subatomic particle, particles in "ordinary" matter, and can be thought of as matter with reversed charge and parity, or go ...
to trigger a tiny fusion explosion. This has been studied primarily in the context of making
nuclear pulse propulsion, and
pure fusion bombs feasible. This is not near becoming a practical power source, due to the cost of manufacturing antimatter alone.
*
Pyroelectric fusion was reported in April 2005 by a team at
UCLA
The University of California, Los Angeles (UCLA) is a public land-grant research university in Los Angeles, California, United States. Its academic roots were established in 1881 as a normal school then known as the southern branch of the C ...
. The scientists used a
pyroelectric crystal heated from , combined with a
tungsten needle to produce an
electric field
An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
of about 25 gigavolts per meter to ionize and accelerate
deuterium nuclei into an
erbium deuteride target. At the estimated energy levels, the D–D fusion reaction may occur, producing
helium-3 and a 2.45 MeV
neutron
The neutron is a subatomic particle, symbol or , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
. Although it makes a useful neutron generator, the apparatus is not intended for power generation since it requires far more energy than it produces. D–T fusion reactions have been observed with a tritiated erbium target.
*
Nuclear fusion–fission hybrid (hybrid nuclear power) is a proposed means of generating
power by use of a combination of nuclear fusion and
fission processes. The concept dates to the 1950s, and was briefly advocated by
Hans Bethe during the 1970s, but largely remained unexplored until a revival of interest in 2009, due to the delays in the realization of pure fusion.
*
Project PACER, carried out at
Los Alamos National Laboratory (LANL) in the mid-1970s, explored the possibility of a fusion power system that would involve exploding small
hydrogen bomb
A thermonuclear weapon, fusion weapon or hydrogen bomb (H-bomb) is a second-generation nuclear weapon design. Its greater sophistication affords it vastly greater destructive power than first-generation nuclear bombs, a more compact size, a lo ...
s (fusion bombs) inside an underground cavity. As an energy source, the system is the only fusion power system that could be demonstrated to work using existing technology. However, it would also require a large, continuous supply of nuclear bombs, making the economics of such a system rather questionable.
*
Bubble fusion also called sonofusion was a proposed mechanism for achieving fusion via
sonic cavitation which rose to prominence in the early 2000s. Subsequent attempts at replication failed and the principal investigator,
Rusi Taleyarkhan, was judged guilty of
research misconduct in 2008.
Confinement in thermonuclear fusion
The key problem in achieving thermonuclear fusion is how to confine the hot plasma. Due to the high temperature, the plasma cannot be in direct contact with any solid material, so it has to be located in a
vacuum. Also, high temperatures imply high pressures. The plasma tends to expand immediately and some force is necessary to act against it. This force can take one of three forms: gravitation in stars, magnetic forces in magnetic confinement fusion reactors, or
inertia
Inertia is the natural tendency of objects in motion to stay in motion and objects at rest to stay at rest, unless a force causes the velocity to change. It is one of the fundamental principles in classical physics, and described by Isaac Newto ...
l as the fusion reaction may occur before the plasma starts to expand, so the plasma's inertia is keeping the material together.
Gravitational confinement
One force capable of confining the fuel well enough to satisfy the
Lawson criterion is
gravity
In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
. The mass needed, however, is so great that gravitational confinement is only found in
star
A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...
s—the least massive stars capable of sustained fusion are
red dwarf
A red dwarf is the smallest kind of star on the main sequence. Red dwarfs are by far the most common type of fusing star in the Milky Way, at least in the neighborhood of the Sun. However, due to their low luminosity, individual red dwarfs are ...
s, while
brown dwarfs are able to fuse
deuterium and
lithium
Lithium (from , , ) is a chemical element; it has chemical symbol, symbol Li and atomic number 3. It is a soft, silvery-white alkali metal. Under standard temperature and pressure, standard conditions, it is the least dense metal and the ...
if they are of sufficient mass. In stars
heavy enough, after the supply of hydrogen is exhausted in their cores, their cores (or a shell around the core) start fusing
helium to carbon. In the most massive stars (at least 8–11
solar masses), the process is continued until some of their energy is produced by
fusing lighter elements to iron. As iron has one of the highest
binding energies, reactions producing heavier elements are generally
endothermic. Therefore, significant amounts of heavier elements are not formed during stable periods of massive star evolution, but are formed in
supernova explosions.
Some lighter stars also form these elements in the outer parts of the stars over long periods of time, by absorbing energy from fusion in the inside of the star, by absorbing neutrons that are emitted from the fusion process.
All of the elements heavier than iron have some potential energy to release, in theory. At the extremely heavy end of element production, these heavier elements can
produce energy in the process of being split again back toward the size of iron, in the process of
nuclear fission
Nuclear fission is a reaction in which the nucleus of an atom splits into two or more smaller nuclei. The fission process often produces gamma photons, and releases a very large amount of energy even by the energetic standards of radioactiv ...
. Nuclear fission thus releases energy that has been stored, sometimes billions of years before, during stellar
nucleosynthesis
Nucleosynthesis is the process that creates new atomic nuclei from pre-existing nucleons (protons and neutrons) and nuclei. According to current theories, the first nuclei were formed a few minutes after the Big Bang, through nuclear reactions in ...
.
Magnetic confinement
Electrically charged particles (such as fuel ions) will follow
magnetic field
A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
lines (see
Guiding centre). The fusion fuel can therefore be trapped using a strong magnetic field. A variety of magnetic configurations exist, including the toroidal geometries of
tokamak
A tokamak (; ) is a device which uses a powerful magnetic field generated by external magnets to confine plasma (physics), plasma in the shape of an axially symmetrical torus. The tokamak is one of several types of magnetic confinement fusi ...
s and
stellarators and open-ended mirror confinement systems.
Inertial confinement
A third confinement principle is to apply a rapid pulse of energy to a large part of the surface of a pellet of fusion fuel, causing it to simultaneously "implode" and heat to very high pressure and temperature. If the fuel is dense enough and hot enough, the fusion reaction rate will be high enough to burn a significant fraction of the fuel before it has dissipated. To achieve these extreme conditions, the initially cold fuel must be explosively compressed. Inertial confinement is used in the
hydrogen bomb
A thermonuclear weapon, fusion weapon or hydrogen bomb (H-bomb) is a second-generation nuclear weapon design. Its greater sophistication affords it vastly greater destructive power than first-generation nuclear bombs, a more compact size, a lo ...
, where the driver is
x-rays
An X-ray (also known in many languages as Röntgen radiation) is a form of high-energy electromagnetic radiation with a wavelength shorter than those of ultraviolet rays and longer than those of gamma rays. Roughly, X-rays have a wavelength ran ...
created by a fission bomb. Inertial confinement is also attempted in "controlled" nuclear fusion, where the driver is a
laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...
,
ion, or
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
beam, or a
Z-pinch. Another method is to use conventional high
explosive material to compress a fuel to fusion conditions. The UTIAS explosive-driven-implosion facility was used to produce stable, centred and focused hemispherical implosions to generate
neutron
The neutron is a subatomic particle, symbol or , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
s from D–D reactions. The simplest and most direct method proved to be in a predetonated stoichiometric mixture of
deuterium–
oxygen
Oxygen is a chemical element; it has chemical symbol, symbol O and atomic number 8. It is a member of the chalcogen group (periodic table), group in the periodic table, a highly reactivity (chemistry), reactive nonmetal (chemistry), non ...
. The other successful method was using a miniature
Voitenko compressor, where a plane diaphragm was driven by the implosion wave into a secondary small spherical cavity that contained pure
deuterium gas at one atmosphere.
Electrostatic confinement
There are also
electrostatic confinement fusion devices. These devices confine
ions using electrostatic fields. The best known is the
fusor. This device has a cathode inside an anode wire cage. Positive ions fly towards the negative inner cage, and are heated by the electric field in the process. If they miss the inner cage they can collide and fuse. Ions typically hit the cathode, however, creating prohibitory high
conduction losses. Also, fusion rates in
fusors are very low due to competing physical effects, such as energy loss in the form of light radiation. Designs have been proposed to avoid the problems associated with the cage, by generating the field using a non-neutral cloud. These include a plasma oscillating device, a
Penning trap and the
polywell. The technology is relatively immature, however, and many scientific and engineering questions remain.
The most well known Inertial electrostatic confinement approach is the
fusor. Starting in 1999, a number of amateurs have been able to do amateur fusion using these homemade devices. Other IEC devices include: the
Polywell, MIX POPS and Marble concepts.
Important reactions
Stellar reaction chains
At the temperatures and densities in stellar cores, the rates of fusion reactions are notoriously slow. For example, at solar core temperature (''T'' ≈ 15 MK) and density (160 g/cm
3), the energy release rate is only 276 μW/cm
3—about a quarter of the volumetric rate at which a resting human body generates heat. Thus, reproduction of stellar core conditions in a lab for nuclear fusion power production is completely impractical. Because nuclear reaction rates depend on density as well as temperature, and most fusion schemes operate at relatively low densities, those methods are strongly dependent on higher temperatures. The fusion rate as a function of temperature (exp(−''E''/''kT'')), leads to the need to achieve temperatures in terrestrial reactors 10–100 times higher than in stellar interiors: ''T'' ≈ .
Criteria and candidates for terrestrial reactions
In artificial fusion, the primary fuel is not constrained to be protons and higher temperatures can be used, so reactions with larger cross-sections are chosen. Another concern is the production of neutrons, which activate the reactor structure radiologically, but also have the advantages of allowing volumetric extraction of the fusion energy and
tritium breeding. Reactions that release no neutrons are referred to as
''aneutronic''.
To be a useful energy source, a fusion reaction must satisfy several criteria. It must:
; Be
exothermic : This limits the reactants to the low ''Z'' (number of protons) side of the
curve of binding energy. It also makes helium the most common product because of its extraordinarily tight binding, although and also show up.
; Involve low atomic number (''Z'') nuclei : This is because the electrostatic repulsion that must be overcome before the nuclei are close enough to fuse (
Coulomb barrier) is directly related to the number of protons it contains – its atomic number.
; Have two reactants : At anything less than stellar densities, three-body collisions are too improbable. In inertial confinement, both stellar densities and temperatures are exceeded to compensate for the shortcomings of the third parameter of the Lawson criterion, ICF's very short confinement time.
; Have two or more products : This allows simultaneous conservation of energy and momentum without relying on the electromagnetic force.
; Conserve both protons and neutrons : The cross sections for the weak interaction are too small.
Few reactions meet these criteria. The following are those with the largest cross sections:
:
For reactions with two products, the energy is divided between them in inverse proportion to their masses, as shown. In most reactions with three products, the distribution of energy varies. For reactions that can result in more than one set of products, the branching ratios are given.
Some reaction candidates can be eliminated at once. The D–
6Li reaction has no advantage compared to
p+– because it is roughly as difficult to burn but produces substantially more neutrons through – side reactions. There is also a
p+– reaction, but the cross section is far too low, except possibly when ''T''
''i'' > 1 MeV, but at such high temperatures an endothermic, direct neutron-producing reaction also becomes very significant. Finally there is also a
p+– reaction, which is not only difficult to burn, but can be easily induced to split into two alpha particles and a neutron.
In addition to the fusion reactions, the following reactions with neutrons are important in order to "breed" tritium in "dry" fusion bombs and some proposed fusion reactors:
:
The latter of the two equations was unknown when the U.S. conducted the
Castle Bravo fusion bomb test in 1954. Being just the second fusion bomb ever tested (and the first to use lithium), the designers of the Castle Bravo "Shrimp" had understood the usefulness of
6Li in tritium production, but had failed to recognize that
7Li fission would greatly increase the yield of the bomb. While
7Li has a small neutron cross-section for low neutron energies, it has a higher cross section above 5 MeV.
[Subsection 4.7.4c]
. Kayelaby.npl.co.uk. Retrieved 19 December 2012. The 15 Mt yield was 150% greater than the predicted 6 Mt and caused unexpected exposure to fallout.
To evaluate the usefulness of these reactions, in addition to the reactants, the products, and the energy released, one needs to know something about the
nuclear cross section. Any given fusion device has a maximum plasma pressure it can sustain, and an economical device would always operate near this maximum. Given this pressure, the largest fusion output is obtained when the temperature is chosen so that is a maximum. This is also the temperature at which the value of the triple product required for
ignition is a minimum, since that required value is inversely proportional to (see
Lawson criterion). (A plasma is "ignited" if the fusion reactions produce enough power to maintain the temperature without external heating.) This optimum temperature and the value of at that temperature is given for a few of these reactions in the following table.
Note that many of the reactions form chains. For instance, a reactor fueled with and creates some , which is then possible to use in the – reaction if the energies are "right". An elegant idea is to combine the reactions (8) and (9). The from reaction (8) can react with in reaction (9) before completely thermalizing. This produces an energetic proton, which in turn undergoes reaction (8) before thermalizing. Detailed analysis shows that this idea would not work well, but it is a good example of a case where the usual assumption of a
Maxwellian plasma is not appropriate.
Abundance of the nuclear fusion fuels
Neutronicity, confinement requirement, and power density
Any of the reactions above can in principle be the basis of
fusion power production. In addition to the temperature and cross section discussed above, we must consider the total energy of the fusion products ''E''
fus, the energy of the charged fusion products ''E''
ch, and the atomic number ''Z'' of the non-hydrogenic reactant.
Specification of the – reaction entails some difficulties, though. To begin with, one must average over the two branches (2i) and (2ii). More difficult is to decide how to treat the and products. burns so well in a deuterium plasma that it is almost impossible to extract from the plasma. The – reaction is optimized at a much higher temperature, so the burnup at the optimum – temperature may be low. Therefore, it seems reasonable to assume the but not the gets burned up and adds its energy to the net reaction, which means the total reaction would be the sum of (2i), (2ii), and (1):
:5 → + 2
n0 + +
p+, ''E''
fus = 4.03 + 17.6 + 3.27 = 24.9 MeV, ''E''
ch = 4.03 + 3.5 + 0.82 = 8.35 MeV.
For calculating the power of a reactor (in which the reaction rate is determined by the D–D step), we count the – fusion energy ''per D–D reaction'' as ''E''
fus = (4.03 MeV + 17.6 MeV) × 50% + (3.27 MeV) × 50% = 12.5 MeV and the energy in charged particles as ''E''
ch = (4.03 MeV + 3.5 MeV) × 50% + (0.82 MeV) × 50% = 4.2 MeV. (Note: if the tritium ion reacts with a deuteron while it still has a large kinetic energy, then the kinetic energy of the helium-4 produced may be quite different from 3.5 MeV, so this calculation of energy in charged particles is only an approximation of the average.) The amount of energy per deuteron consumed is 2/5 of this, or 5.0 MeV (a
specific energy of about 225 million
MJ per kilogram of deuterium).
Another unique aspect of the – reaction is that there is only one reactant, which must be taken into account when calculating the reaction rate.
With this choice, we tabulate parameters for four of the most important reactions
The last column is the
neutronicity of the reaction, the fraction of the fusion energy released as neutrons. This is an important indicator of the magnitude of the problems associated with neutrons like radiation damage, biological shielding, remote handling, and safety. For the first two reactions it is calculated as . For the last two reactions, where this calculation would give zero, the values quoted are rough estimates based on side reactions that produce neutrons in a plasma in thermal equilibrium.
Of course, the reactants should also be mixed in the optimal proportions. This is the case when each reactant ion plus its associated electrons accounts for half the pressure. Assuming that the total pressure is fixed, this means that particle density of the non-hydrogenic ion is smaller than that of the hydrogenic ion by a factor . Therefore, the rate for these reactions is reduced by the same factor, on top of any differences in the values of . On the other hand, because the – reaction has only one reactant, its rate is twice as high as when the fuel is divided between two different hydrogenic species, thus creating a more efficient reaction.
Thus there is a "penalty" of for non-hydrogenic fuels arising from the fact that they require more electrons, which take up pressure without participating in the fusion reaction. (It is usually a good assumption that the electron temperature will be nearly equal to the ion temperature. Some authors, however, discuss the possibility that the electrons could be maintained substantially colder than the ions. In such a case, known as a "hot ion mode", the "penalty" would not apply.) There is at the same time a "bonus" of a factor 2 for – because each ion can react with any of the other ions, not just a fraction of them.
We can now compare these reactions in the following table.
The maximum value of is taken from a previous table. The "penalty/bonus" factor is that related to a non-hydrogenic reactant or a single-species reaction. The values in the column "inverse reactivity" are found by dividing by the product of the second and third columns. It indicates the factor by which the other reactions occur more slowly than the – reaction under comparable conditions. The column "
Lawson criterion" weights these results with ''E''
ch and gives an indication of how much more difficult it is to achieve ignition with these reactions, relative to the difficulty for the – reaction. The next-to-last column is labeled "power density" and weights the practical reactivity by ''E''
fus. The final column indicates how much lower the fusion power density of the other reactions is compared to the – reaction and can be considered a measure of the economic potential.
Bremsstrahlung losses in quasineutral, isotropic plasmas
The ions undergoing fusion in many systems will essentially never occur alone but will be mixed with
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s that in aggregate neutralize the ions' bulk
electrical charge and form a
plasma. The electrons will generally have a temperature comparable to or greater than that of the ions, so they will collide with the ions and emit
x-ray
An X-ray (also known in many languages as Röntgen radiation) is a form of high-energy electromagnetic radiation with a wavelength shorter than those of ultraviolet rays and longer than those of gamma rays. Roughly, X-rays have a wavelength ran ...
radiation of 10–30 keV energy, a process known as
Bremsstrahlung.
The huge size of the Sun and stars means that the x-rays produced in this process will not escape and will deposit their energy back into the plasma. They are said to be
opaque to x-rays. But any terrestrial fusion reactor will be
optically thin for x-rays of this energy range. X-rays are difficult to reflect but they are effectively absorbed (and converted into heat) in less than mm thickness of stainless steel (which is part of a reactor's shield). This means the bremsstrahlung process is carrying energy out of the plasma, cooling it.
The ratio of fusion power produced to x-ray radiation lost to walls is an important figure of merit. This ratio is generally maximized at a much higher temperature than that which maximizes the power density (see the previous subsection). The following table shows estimates of the optimum temperature and the power ratio at that temperature for several reactions:
The actual ratios of fusion to Bremsstrahlung power will likely be significantly lower for several reasons. For one, the calculation assumes that the energy of the fusion products is transmitted completely to the fuel ions, which then lose energy to the electrons by collisions, which in turn lose energy by Bremsstrahlung. However, because the fusion products move much faster than the fuel ions, they will give up a significant fraction of their energy directly to the electrons. Secondly, the ions in the plasma are assumed to be purely fuel ions. In practice, there will be a significant proportion of impurity ions, which will then lower the ratio. In particular, the fusion products themselves ''must'' remain in the plasma until they have given up their energy, and ''will'' remain for some time after that in any proposed confinement scheme. Finally, all channels of energy loss other than Bremsstrahlung have been neglected. The last two factors are related. On theoretical and experimental grounds, particle and energy confinement seem to be closely related. In a confinement scheme that does a good job of retaining energy, fusion products will build up. If the fusion products are efficiently ejected, then energy confinement will be poor, too.
The temperatures maximizing the fusion power compared to the Bremsstrahlung are in every case higher than the temperature that maximizes the power density and minimizes the required value of the
fusion triple product. This will not change the optimum operating point for – very much because the Bremsstrahlung fraction is low, but it will push the other fuels into regimes where the power density relative to – is even lower and the required confinement even more difficult to achieve. For – and –, Bremsstrahlung losses will be a serious, possibly prohibitive problem. For –,
p+– and
p+– the Bremsstrahlung losses appear to make a fusion reactor using these fuels with a quasineutral, isotropic plasma impossible. Some ways out of this dilemma have been considered but rejected. This limitation does not apply to
non-neutral and anisotropic plasmas; however, these have their own challenges to contend with.
Mathematical description of cross section
Fusion under classical physics
In a classical picture, nuclei can be understood as hard spheres that repel each other through the Coulomb force but fuse once the two spheres come close enough for contact. Estimating the radius of an atomic nuclei as about one femtometer, the energy needed for fusion of two hydrogen is:
:
This would imply that for the core of the sun, which has a
Boltzmann distribution with a temperature of around 1.4 keV, the probability hydrogen would reach the threshold is , that is, fusion would never occur. However, fusion in the sun does occur due to quantum mechanics.
Parameterization of cross section
The probability that fusion occurs is greatly increased compared to the classical picture, thanks to the smearing of the effective radius as the
de Broglie wavelength as well as
quantum tunneling through the potential barrier. To determine the rate of fusion reactions, the value of most interest is the
cross section, which describes the probability that particles will fuse by giving a characteristic area of interaction. An estimation of the fusion cross-sectional area is often broken into three pieces:
:
where
is the geometric cross section, is the barrier transparency and is the reaction characteristics of the reaction.
is of the order of the square of the de Broglie wavelength
where
is the reduced mass of the system and
is the center of mass energy of the system.
can be approximated by the Gamow transparency, which has the form:
where
is the
Gamow factor and comes from estimating the quantum tunneling probability through the potential barrier.
contains all the nuclear physics of the specific reaction and takes very different values depending on the nature of the interaction. However, for most reactions, the variation of
is small compared to the variation from the Gamow factor and so is approximated by a function called the astrophysical
S-factor,
, which is weakly varying in energy. Putting these dependencies together, one approximation for the fusion cross section as a function of energy takes the form:
:
More detailed forms of the cross-section can be derived through nuclear physics-based models and
R-matrix theory.
Formulas of fusion cross sections
The Naval Research Lab's plasma physics formulary gives the total cross section in
barns as a function of the energy (in keV) of the incident particle towards a target ion at rest fit by the formula:
:
with the following coefficient values:
Bosch-Hale
also reports a R-matrix calculated cross sections fitting observation data with
Padé rational approximating coefficients. With energy in units of keV and cross sections in units of millibarn, the factor has the form:
:
, with the coefficient values:
where
Maxwell-averaged nuclear cross sections
In fusion systems that are in thermal equilibrium, the particles are in a
Maxwell–Boltzmann distribution, meaning the particles have a range of energies centered around the plasma temperature. The sun, magnetically confined plasmas and inertial confinement fusion systems are well modeled to be in thermal equilibrium. In these cases, the value of interest is the fusion cross-section averaged across the Maxwell–Boltzmann distribution. The Naval Research Lab's plasma physics formulary tabulates Maxwell averaged fusion cross sections reactivities in
.
For energies
the data can be represented by:
:
:
with in units of keV.
See also
References
Further reading
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External links
NuclearFiles.org– A repository of documents related to nuclear power.
Annotated bibliography for nuclear fusion from the Alsos Digital Library for Nuclear IssuesNRL Fusion Formulary
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Physical phenomena
Energy conversion
Neutron sources
Nuclear chemistry
Nuclear physics