Nuclear 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 weapons ("hydrogen bombs") and in most stars; and controlled, where the fusion reactions take place in an environment allowing some or all of the energy released to be harnessed for constructive purposes.

Temperature is a measure of the average kinetic energy 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 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.

Accelerator-based light-ion fusion is a technique using particle accelerators to achieve particle kinetic energies sufficient to induce light-ion fusion reactions.[23]

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-reverse configuration (FRC) as the storage system was proposed by Norman Rostoker and continues to be studied by TAE Technologies as of 2021. A closely related approach is to merge two FRC's rotating in opposite directions,[24] 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.[25]

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, and the high chance that a muon will bind to the new alpha particle and thus stop catalyzing fusion.[26]

The Tokamak à configuration variable, research fusion reactor, at the École Polytechnique Fédérale de Lausanne (Switzerland).

Some other confinement principles have been investigated.

• Antimatter-initialized fusion uses small amounts of antimatter 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 scientists used a pyroelectric crystal heated from −34 to 7 °C (−29 to 45 °F), combined with a tungsten needle to produce an electric field of about 25 gigavolts per meter to ionize and accelerate deuterium nuclei into an erbium deuteride target. At the estimated energy levels,[27] the D–D fusion reaction may occur, producing helium-3 and a 2.45 MeV neutron. Although it makes a useful neutron generator, the apparatus is not intended for power generation since it requires far more energy than it produces.[28][29][30][31] D–T fusion reactions have been observed with a tritiated erbium target.[32]
• 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.[33]
• 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 bombs (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.[34]

One force capable of confining the fuel well enough to satisfy the Lawson criterion is gravity. The mass needed, however, is so great that gravitational confinement is only found in stars—the least massive stars capable of sustained fusion are red dwarfs, while brown dwarfs are able to fuse deuterium and lithium 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 thus releases energy that has been stored, sometimes billions of years before, during stellar nucleosynthesis.

Electrically charged particles (such as fuel ions) will follow magnetic field 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 tokamaks and stellarators and open-ended mirror confinement systems.

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, where the driver is x-rays created by a fission bomb. Inertial confinement is also attempted in "controlled" nuclear fusion, where the driver is a laser, ion, or electron beam, or a Z-pinch. Another method is to use conventional high explosive material to compress a fuel to fusion conditions.[35][36] The UTIAS explosive-driven-implosion facility was used to produce stable, centred and focused hemispherical implosions[37] to generate neutrons from D-D reactions. The simplest and most direct method proved to be in a predetonated stoichiometric mixture of deuterium-oxygen. The other successful method was using a miniature Voitenko compressor,[38] where a plane diaphragm was driven by the implosion wave into a secondary small spherical cavity that contained pure deuterium gas at one atmosphere.[39]

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.[40] 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,[41] a Penning trap and the polywell.[42] 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.[43][44][45][46] Other IEC devices include: the Polywell, MIX POPS[47] and Marble concepts.[48]

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/cm³), the energy release rate is only 276 μW/cm³—about a quarter of the volumetric rate at which a resting human body generates heat.[49] 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 ≈ (0.1–1.0)×10⁹ K.

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 ⁴
He
the most common product because of its extraordinarily tight binding, although ³
He
and ³
H
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 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:[50] [51]

(1)

2 1D

+

3 1T

→

4 2He

(

3.52 MeV

)

+

n0

(

14.06 MeV

)

(2i)

2 1D

+

2 1D

→

3 1T

(

1.01 MeV

)

+

p+

(

3.02 MeV

)

50%

(2ii)

→

3 2He

(

0.82 MeV

)

+

n0

(

2.45 MeV

)

50%

(3)

2 1D

+

3 2He

→

4 2He

(

3.6 MeV

)

+

p+

(

14.7 MeV

)

(4)

3 1T

+

3 1T

→

4 2He

+

2 n0

+

11.3 MeV

(5)

3 2He

+

3 2He

→

4 2He

+

2 p+

+

12.9 MeV

(6i)

3 2He

+

3 1T

→

4 2He

+

p+

+

n0

+

12.1 MeV

57%

(6ii)

→

4 2He

(

4.8 MeV

)

+

2 1D

(

9.5 MeV

)

43%

(7i)

2 1D

+

6 3Li

→

2 4 2He

+

22.4 MeV

(7ii)

→

3 2He

+

4 2He

+

n0

+

2.56 MeV

(7iii)

→

7 3Li

+

p+

+

5.0 MeV

(7iv)

→

7 4Be

+

n0

+

3.4 MeV

(8)

p+

+

6 3Li

→

4 2He

(

1.7 MeV

)

+

3 2He

(

2.3 MeV

)

(9)

3 2He

+

6 3Li

→

2 4 2He

+

p+

+

16.9 MeV

(10)

p+

+

11 5B

→

3 4 2He

+

8.7 MeV

Nucleosynthesis
Stellar nucleosynthesis Big Bang nucleosynthesis Supernova nucleosynthesis Cosmic ray spallation
Related topics
Astrophysics Nuclear fusion R-process S-process Nuclear fission

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⁺–¹¹
₅B
because it is roughly as difficult to burn but produces substantially more neutrons through ²
₁D
–²
₁D
side reactions. There is also a p⁺–⁷
₃Li
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⁺–⁹
₄Be
reaction, which is not only difficult to burn, but ⁹
₄Be
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:

n0	+	6 3Li	→	3 1T	+	4 2He + 4.784 MeV
n0	+	7 3Li	→	3 1T	+	4 2He + n0 − 2.467 MeV

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.[52] 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 ⟨σv⟩/T² is a maximum. This is also the temperature at which the value of the triple product nTτ required for ignition is a minimum, since that required value is inversely proportional to ⟨σv⟩/T² (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 ⟨σv⟩/T² 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 ³
₁T
and ³
₂He
creates some ²
₁D
, which is then possible to use in the ²
₁D
–³
₂He
reaction if the energies are "right". An elegant idea is to combine the reactions (8) and (9). The ³
₂He
from reaction (8) can react with ⁶
₃Li
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.

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 ²
₁D
–²
₁D
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 ³
₁T
and ³
₂He
products. ³
₁T
burns so well in a deuterium plasma that it is almost impossible to extract from the plasma. The ²
₁D
–³
₂He
reaction is optimized at a much higher temperature, so the burnup at the optimum ²
₁D
–²
₁D
temperature may be low. Therefore, it seems reasonable to assume the ³
₁T
but not the ³
₂He
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
₁D
→ ⁴
₂He
+ 2 n⁰ + ³
₂He
+ 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 ²
₁D
–²
₁D
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,[53] 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 ²
₁D
–²
₁D
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 (E_fus − E_ch)/E_fus. 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 2/(Z + 1). Therefore, the rate for these reactions is reduced by the same factor, on top of any differences in the values of ⟨σv⟩/T². On the other hand, because the ²
₁D
–²
₁D
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 2/(Z + 1) 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 ²
₁D
–²
₁D
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 ⟨σv⟩/T² 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 1.24×10⁻²⁴ by the product of the second and third columns. It indicates the factor by which the other reactions occur more slowly than the ²
₁D
–³
₁T
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 ²
₁D
–³
₁T
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 ²
₁D
–³
₁T
reaction and can be considered a measure of the economic potential.

The ions undergoing fusion in many systems will essentially never occur alone but will be mixed with electrons 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 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 ²
₁D
–³
₁T
very much because the Bremsstrahlung fraction is low, but it will push the other fuels into regimes where the power density relative to ²
₁D
–³
₁T
is even lower and the required confinement even more difficult to achieve. For ²
₁D
–²
₁D
and ²
₁D
–³
₂He
, Bremsstrahlung losses will be a serious, possibly prohibitive problem. For ³
₂He
–³
₂He
, p⁺–⁶
₃Li
and p⁺–¹¹
₅B
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.[54][55] This limitation does not apply to non-neutral and anisotropic plasmas; however, these have their own challenges to contend with.

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:

${\displaystyle E_{\ce {thresh}}={\frac {1}{4\pi \epsilon _{0}}}{\frac {Z_{1}Z_{2}}{r}}{\ce {->[{\text{2 protons}}]}}{\frac {1}{4\pi \epsilon _{0}}}{\frac {e^{2}}{1\ {\ce {fm}}}}\approx 1.4\ {\ce {MeV}}}$

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 ${\displaystyle 10^{-290}}$, that is, fusion would never occur. However, fusion in the sun does occur due to quantum mechanics.

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:

${\displaystyle \sigma \approx \sigma _{\text{geometry}}\times T\times R,}$

where ${\displaystyle \sigma _{\text{geometry}}}$ is the geometric cross section, T is the barrier transparency and R is the reaction characteristics of the reaction.

${\displaystyle \sigma _{\text{geometry}}}$ is of the order of the square of the de Broglie wavelength ${\displaystyle \sigma _{\text{geometry}}\approx \lambda ^{2}={\bigg (}{\frac {\hbar }{m_{r}v}}{\bigg )}^{2}\propto {\frac {1}{\epsilon }}}$ where ${\displaystyle m_{r}}$ is the reduced mass of the system and ${\displaystyle \epsilon }$ is the center of mass energy of the system.

T can be approximated by the Gamow transparency, which has the form: ${\displaystyle T\approx e^{-{\sqrt {\epsilon _{G}/\epsilon }}}}$ where ${\displaystyle \epsilon _{G}=(\pi \alpha Z_{1}Z_{2})^{2}\times 2m_{r}c^{2}}$ is the Gamow factor and comes from estimating the quantum tunneling probability through the potential barrier.

R 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 ${\displaystyle R(\epsilon )}$ is small compared to the variation from the Gamow factor and so is approximated by a function called the astrophysical S-factor, ${\displaystyle S(\epsilon )}$, 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:

${\displaystyle \sigma (\epsilon )\approx {\frac {S(\epsilon )}{\epsilon }}e^{-{\sqrt {\epsilon _{G}/\epsilon }}}}$

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

The Naval Research Lab's plasma physics formulary[56] 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:

${\displaystyle \sigma ^{\text{NRL}}(\epsilon )={\frac {A_{5}+{\big (}(A_{4}-A_{3}\epsilon )^{2}+1{\big )}^{-1}A_{2}}{\epsilon (e^{A_{1}\epsilon ^{-1/2}}-1)}}}$ with the following coefficient values:

Bosch-Hale[57] 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:

${\displaystyle S^{\text{Bosch-Hale}}(\epsilon )={\frac {A_{1}+\epsilon {\bigg (}A_{2}+\epsilon {\big (}A_{3}+\epsilon (A_{4}+\epsilon A_{5}){\big )}{\bigg )}}{1+\epsilon {\bigg (}B_{1}+\epsilon {\big (}B_{2}+\epsilon (B_{3}+\epsilon B_{4}){\big )}{\bigg )}}}}$, with the coefficient values:

Bosch-Hale coefficients for the fusion cross section

	DT(1)	DD(2ii)	DHe3(3)	THe4
${\displaystyle \epsilon _{G}}$	31.3970	68.7508	31.3970	34.3827
A1	5.5576×104	5.7501×106	5.3701×104	6.927×104
A2	2.1054×102	2.5226×103	3.3027×102	7.454×108
A3	−3.2638×10−2	4.5566×101	−1.2706×10−1	2.050×106
A4	1.4987×10−6	0	2.9327×10−5	5.2002×104
A5	1.8181×10−10	0	−2.5151×10−9	0
B1	0	−3.1995×10−3	0	6.38×101
B2	0	−8.5530×10−6	0	−9.95×10−1
B3	0	5.9014×10−8	0	6.981×10−5
B4	0	0	0	1.728×10−4
Applicable Energy Range [keV]	0.5–5000	0.3–900	0.5–4900	0.5–550
${\displaystyle (\Delta S)_{\text{max}}\%}$	2.0	2.2	2.5	1.9

where ${\displaystyle \sigma ^{\text{Bosch-Hale}}(\epsilon )={\frac {S^{\text{Bosch-Hale}}(\epsilon )}{\epsilon \exp(\epsilon _{G}/{\sqrt {\epsilon }})}}}$

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 ${\displaystyle \mathrm {cm^{3}/s} }$.

For energies ${\displaystyle T\leq 25{\text{ keV}}}$ the data can be represented by:

${\displaystyle ({\overline {\sigma v}})_{DD}=2.33\times 10^{-14}\cdot T^{-2/3}\cdot e^{-18.76\ T^{-1/3}}\mathrm {~{cm}^{3}/s} }$

${\displaystyle ({\overline {\sigma v}})_{DT}=3.68\times 10^{-12}\cdot T^{-2/3}\cdot e^{-19.94\ T^{-1/3}}\mathrm {~{cm}^{3}/s} }$

with T in units of keV.