# Nuclear fusion

File:D-t-fusion.png
The deuterium-tritium (D-T) fusion reaction is considered the most promising for producing fusion power.

In physics, nuclear fusion is the process by which two nuclei join together to form a heavier nucleus. It is accompanied by the release or absorption of energy depending on the masses of the nuclei involved. Iron and nickel nuclei have the largest binding energies of all nuclei and so are the most stable. The fusion of two nuclei to produce a nucleus lighter than iron or nickel generally gives off energy while the fusion of nuclei heavier than them absorbs energy.

Nuclear fusion of light elements releases the energy that causes stars to shine and hydrogen bombs to explode. Nuclear fusion of heavy elements occurs in the extreme conditions of supernova explosions. Nuclear fusion in stars and supernovae is the primary process by which new natural elements are created.

This article deals with the fusion reaction itself. See the article on fusion power for information on controlling the fusion reaction to produce useful power.

It takes considerable energy to force nuclei to fuse, even those of the least massive element, hydrogen. But the fusion of lighter nuclei, which creates a heavier nucleus and a free neutron, will generally release more energy than it took to force them together — an exothermic process that can produce self-sustaining reactions.

The energy released in most nuclear reactions is much larger than that for chemical reactions, because the binding energy that holds a nucleus together is far greater than the energy that holds electrons to a nucleus. For example, the ionization energy gained by adding an electron to hydrogen is 13.6 electron volts -- less than one-millionth of the 17 MeV released in the D-T (deuterium-tritium) reaction shown to the right.

## Requirements for fusion

A substantial energy barrier must be overcome for fusion to occur. Nuclei repel one another because of the electrostatic force between their positively charged protons. If two nuclei can be brought close enough together, however, the electrostatic force is overwhelmed by the more powerful strong nuclear force which only operates over short distances.

When a nucleon (proton or neutron) is added to a nucleus, the strong force attracts it to other nucleons, 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-to-volume ratio, the binding energy per nucleon due to the strong force generally increases with the size of the nucleus but approaches a limiting value corresponding to that of a fully surrounded nucleon.

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 get larger.

The net result of these opposing forces is that the binding energy per nucleon generally increases with increasing size, up to the elements iron and nickel, and then decreases for heavier nuclei. Eventually, the binding energy becomes negative and very heavy nuclei are not stable. The four most tightly bound nuclei, in decreasing order of binding energy, are 62Ni, 58Fe, 56Fe, and 60Ni [1]. Even though the nickel isotope 62Ni is more stable, the iron isotope 56Fe is an order of magnitude more common. This is due to a greater disintegration rate for 62Ni in the interior of stars due to photon absorption.

A notable exception to this general trend is the helium nucleus whose binding energy is higher than lithium's which is the next heavier. The Pauli exclusion principle provides an explanation for this exceptional behavior: it says that because protons and neutrons are fermions, they cannot exist in exactly the same state. Each proton or neutron energy state in a nucleus can accommodate both a spin up particle and a spin down particle. Helium has an anomolously large binding energy because its nucleus consists of two protons and two neutrons: so all four of its nucleons can be in the ground state. Any additional nucleons have to go into higher energy states.

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 in contact can the strong nuclear force take over. Consequently, even when the final energy state is lower, there is a large energy barrier that must first be overcome. In chemistry, one would speak of the activation energy. In nuclear physics it is called the Coulomb barrier.

The Coulomb barrier is smallest for isotopes of hydrogen - they contain only a single positive charge in the nucleus. A bi-proton 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 D-T fuel, the resulting energy barrier is about 0.1 MeV. In comparison, the energy needed to remove an electron from hydrogen is 13 eV, about 7,500 times less energy. Once the fusion reaction is complete, the new nucleus drops to a lower-energy configuration and gives up additional energy by ejecting a neutron with 17.59 MeV, considerably more than what was needed to fuse them in the first place. This means that the D-T fusion reaction is very highly exothermic, making it a powerful energy source.

If the energy to initiate the reaction comes from accelerating one of the nuclei, the process is called beam-target fusion; if both nuclei are accelerated, it is beam-beam fusion. If the nuclei are part of a plasma near thermal equilibrium, one speaks of thermonuclear fusion. Temperature is a measure of the average kinetic energy of particles, so by heating the nuclei they will gain energy and eventually have enough to overcome this 0.1 MeV barrier. Converting the units between eV and kelvins shows that the barrier would be overcome at a temperature in excess of 1 GK, obviously a very high temperature.

There are two effects that lower the actual temperature needed. 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 tunneling. 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 this reason fuel at lower temperatures will still undergo fusion events, at a lower rate.

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 with thermonuclear fusion, then it is useful to perform an average of over the distributions of the product of cross section and velocity. The reaction rate (fusions per volume per time) is <σv> times the product of the reactant number densities:

$\displaystyle f = n_1 n_2 \langle \sigma v \rangle$

If a species of nuclei is reacting with itself, such as the DD reaction, then the product $\displaystyle n_1n_2$ must be replaced by $\displaystyle (1/2)n^2$ .

$\displaystyle \langle \sigma v \rangle$ increases from virtually zero at room temperatures up to meaningful magnitudes at temperatures of 10 - 100 keV. At these temperatures, well above typical ionization energies (13 eV in the hydrogen case), the fusion reactants exist in a plasma state.

The significance of <σv> as a function of temperature in a device with a particular energy confinement time is found by considering the Lawson criterion.

### Methods of fuel confinement

The fusion reaction can sustain itself if enough of the energy produced goes into keeping the fuel hot.

Gravitational confinement 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. Even if the more reactive fuel deuterium were used, a mass about the size of the Moon would be needed.

Magnetic confinement Since plasmas are very good electrical conductors, magnetic fields can also confine fusion fuel. A variety of magnetic configurations can be used, the most basic distinction being between mirror confinement and toroidal confinement, especially tokamaks and stellarators.

Inertial confinement A third confinement principle is to apply a rapid pulse of energy to 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.

Some other confinement principles have been investigated, such as muon-catalyzed fusion, the Farnsworth-Hirsch fusor (inertial electrostatic confinement), and bubble fusion.

## Important fusion reactions

File:FusionintheSun.png
Nuclear Fusion in the Sun.

### Astrophysical reaction chains

The most important fusion process in nature is that which powers the stars. The net result is the fusion of four protons into one alpha particle, with the release of two positrons, two neutrinos, and energy, but several individual reactions are involved, depending on the mass of the star. For stars the size of the sun or smaller, the proton-proton chain dominates. In heavier stars, the CNO cycle is more important. See stellar nucleosynthesis.

### Criteria and candidates for terrestrial reactions

In man-made fusion, the primary fuel is not constrained to be protons and higher temperatures can be used, so reactions with larger cross-sections are chosen. This implies a lower Lawson criterion, and therefore less startup effort. 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.

In order to be useful as a source of energy, a fusion reaction must satisfy several criteria. It must:

• ... be exothermic. This one is obvious, but it limits the reactants to the low Z side of the curve of binding energy. It also makes helium He-4 the most common product because of its extraordinarily tight binding, although He3 and T also show up.
• ... involve low Z nuclei. This is because the electrostatic repulsion must be overcome before the nuclei are close enough to fuse.
• ... have two reactants. At anything less than stellar densities, three body collisions are too improbable.
• ... have two or more products. This allows simultaneous conservation of energy and momentum without relying on the (weak!) electromagnetic force.
• ... and conserve both protons and neutrons. The cross sections for the weak interaction are too small.

Few reactions meet these criteria. The most interesting are the following:

 (1) D + T → 4He (3.5 MeV) + n (14.1 MeV) (2) D + D → T (1.01 MeV) + p (3.02 MeV) (50%) (3) → 3He (0.82 MeV) + n (2.45 MeV) (50%) (4) D + 3He → 4He (3.6 MeV) + p (14.7 MeV) (5) T + T → 4He + 2 n + 11.3 MeV (6) 3He + 3He → 4He + 2 p + 12.9 MeV (7) 3He + T → 4He + p + n + 12.1 MeV (51%) (8) → 4He (4.8 MeV) + D (9.5 MeV) (43%) (9) → 4He (0.5 MeV) + n (1.9 MeV) + p (11.9 MeV) (6%) (10) D + 6Li → 2 4He + 22.4 MeV (11) p + 6Li → 4He (1.7 MeV) + 3He (2.3 MeV) (12) 3He + 6Li → 2 4He + p + 16.9 MeV (13) p + 11B → 3 4He + 8.7 MeV

p (proton), D (deuterium), and T (tritium) are shorthand notation for the first three isotopes of hydrogen. 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.[2] The D-6Li reaction has no advantage compared to p-11B because it is roughly as difficult to burn but produces substantially more neutrons. There is also a p-7Li reaction, but the cross section is far too low except possible for Ti > 1 MeV, but at such high temperatures an endothermic, direct neutron-producing reaction also becomes very significant. Finally there is also a p-9Be reaction, which is not only difficult to burn, but 9Be can be easily induced to split into two alphas 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:

n + 6Li → T + 4He
n + 7Li → T + 4He + n

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 cross section. Any given fusion device will have a maximum plasma pressure that it can sustain, and an economical device will 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. This optimum temperature and the value of <σv>/T² at that temperature is given for a few of these reactions in the following table.

fuel T [keV] <σv>/T² [m³/s/keV²]
D-T 13.6 1.24×10-24
D-D 15 1.28×10-26
D-3He 58 2.24×10-26
p-6Li 66 1.46×10-27
p-11B 123 3.01×10-27

Note that many of the reactions form chains. For instance, a reactor fueled with T and 3He will create some D, which is then possible to use in the D + 3He reaction if the energies are "right". An elegant idea is to combine the reactions (11) and (12). The 3He from reaction (11) can react with 6Li in reaction (12) before completely thermalizing. This produces an energetic proton which in turn undergoes reaction (11) before thermalizing. A detailed analysis shows that this idea will not really work well, but it is a good example of a case where the usual assumption of a Maxwellian plasma is not appropriate.

### 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 Efus, the energy of the charged fusion products Ech, and the atomic number Z of the non-hydrogenic reactant.

Specification of the D-D reaction entails some difficulties, though. To begin with, one must average over the two branches (2) and (3). More difficult is to decide how to treat the T and 3He products. T burns so well in a deuterium plasma that you probably can't get it out even if you want to. The D-3He reaction is optimized at a much higher temperature, so the burnup at the optimum D-D temperature may be low, so it seems reasonable to assume the T but not the 3He gets burned up and adds its energy to the net reaction. Thus we will count the DD fusion energy as Efus = (4.03+17.6+3.27)/2 = 12.5 MeV and the energy in charged particles as Ech = (4.03+3.5+0.82)/2 = 4.2 MeV.

Another unique aspect of the D-D 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.

fuel Z Efus [MeV] Ech [MeV] neutronicity
D-T 1 17.6 3.5 0.80
D-D 1 12.5 4.2 0.66
D-3He 2 18.3 18.3 ~0.05
p-11B 5 8.7 8.7 ~0.001

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 (Efus-Ech)/Efus. 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 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 D-D reaction has only one reactant, the rate is twice as high as if the fuel were divided between two hydrogenic species.

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. There is at the same time a "bonus" of a factor 2 for D-D due to the fact that 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.

fuel <σv>/T² penalty/bonus reactivity Lawson criterion power density
D-T 1.24e-24 1 1 1 1
D-D 1.28e-26 2 48 30 68
D-3He 2.24e-26 2/3 83 16 80
p-11B 3.01e-27 1/3 1240 500 2500

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 "reactivity" are found by dividing (1.24e-24) by the product of the second and third columns. It indicates the factor by which the other reactions occur more slowly than the D-T reaction under comparable conditions. The column "Lawson criterion" weights these results with Ech and gives an indication of how much more difficult it is to achieve ignition with these reactions, relative to the difficulty for the D-T reaction. The last column is labeled "power density" and weights the practical reactivity with Efus. It indicates how much lower the fusion power density of the other reactions is compared to the D-T reaction and can be considered a measure of the economic potential.

### Bremsstrahlung losses

The ions undergoing fusion will essentially never occur alone but will be mixed with electrons that neutralize the ions' 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 Bremsstrahlung. The Sun and stars are opaque to Bremsstrahlung, but essentially any terrestrial fusion reactor will be optically thin at relevant wavelengths. Bremsstrahlung is also difficult to reflect and difficult to convert directly to electricity, so the ratio of fusion power produced to Bremsstrahlung radiation lost 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 the rough optimum temperature and the power ratio at that temperature for several reactions.[3]

fuel Ti (keV) Pfusion/PBremsstrahlung
D-T 50 140
D-D 500 2.9
D-3He 100 5.3
3He-3He 1000 0.72
p-6Li 800 0.21
p-11B 300 0.57

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 plasma is assumed to be composed purely of fuel ions. In practice, there will be a significant proportion of impurity ions, which will lower the ratio. In particular, the fusion products themselves must remain in the plasma until they have given up their energy, and will remain 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 D-T very much because the Bremsstrahlung fraction is low, but it will push the other fuels into regimes where the power density relative to D-T is even lower and the required confinement even more difficult to achieve. For D-D and D-3He, Bremsstrahlung losses will be a serious, possibly prohibitive problem. For 3He-3He, p-6Li and p-11B the Bremsstrahlung losses appear to make a fusion reactor using these fuels impossible. Some ways out of this dilemma are considered — and rejected — in Fundamental limitations on plasma fusion systems not in thermodynamic equilibrium.