# Nuclear fusion

In physics, nuclear fusion (a thermonuclear reaction) is a process in which two nuclei join, forming a larger nucleus and releasing energy. Nuclear fusion is the energy source which causes stars to shine, and hydrogen bombs to explode.

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 even 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 glues 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 reaction shown below.

## Requirements for fusion

A substantial energy barrier opposes the fusion reaction. The positive electrical charges of the nuclei repel each other via the electromagnetic force, attempting to break any nuclei apart. Opposing this is the slightly more powerful strong nuclear force, which tries to hold them together. It is the tension between these two powerful forces that makes nuclear reactions so powerful.

The strong force only operates over short distances, unlike the electromagnetic force. As the nucleus grows by adding additional protons and neutrons, it eventually reaches a size (at the element iron) where the strong force becomes overwhelmed and the nucleus spontaneously starts to "fall apart". This is what causes radioactivity in heavier elements.

In order to fuse nuclei, the electromagnetic repulsive force between protons must be overcome, and the nuclei brought close enough together for the strong force to start to act again. The combination of these two determines the threshold energy required for a fusion reaction. Since the repulsive force is generated solely by the electrically charged protons, this Coulomb barrier is a minimum for hydrogen, which contains only one proton. The strong force is generated by both protons and neutrons, so the threshold energy is likewise minimized by adding neutrons. Thus it is lowest for heavy isotopes of hydrogen, deuterium (D) and tritium (T), which have only one proton keeping them apart, but several neutrons pulling them together.

In the D-T fuel, the resulting energy barrier is about 0.1 MeV. In comparison, the energy needed to remove an electron from hydrogen is 13eV, about 75000 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.

### Fusion cross section

The "easiest" way to provide the energy needed to overcome the Coulomb barrier is to heat the nuclei up. Temperature is a measure of the average kinetic energy in a volume, 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. Additional effects, such as quantum tunneling, lower the energy barrier slightly.

However it is important to remember 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. For this reason fuel at lower temperatures will still undergo fusion events, at a lower rate.

The reaction cross section combines the effects of the Coulomb barrier and thermal velocity distribution of the nuclei into an "effective area" for fusion collisions. The cross section forms an equation

$f = n \sigma \nu$

where n is the density of nuclei, σ is the cross section, ν is the thermal velocity, and f is the frequency of fusion producing collisions.

Increasing any of these three quantities will increase the fusion-causing collision frequency, and thus the overall rate of fusion. Interestingly, the cross section is itself also a function of thermal energy in the nuclei. Cross section 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 (13eV in the hydrogen case), the fusion reactants exist in a plasma state.

### The fusion triple product

For any given amount of fuel in a particular state, the rate of fusion in the fuel, f, is constant. Thus the measure of the actual net energy being released is a function of f (and in turn, the temperature), the number of particles in a particular area (its density), and the amount of time they remain together (the confinement time). This can be quantified by what is commonly called the fusion triple product, nTτ or where p=nT.

Releasing useful energy from a fuel can thus take place at a low value of f. For instance, the conditions inside the sun are actually quite "poor", and the nuclei only undergo fusion once in every 1029 seconds. However, the fact that the sun contains 1059 nuclei means that the net reaction rate is actually quite high, and since the sun is around for billions of years, eventually the fuel is used up and the total energy released is huge.

### Break-even and ignition temperature

At some point the energy being generated in the reactions will be equal to the energy that had to be added to the system to get it to that temperature in the first place. This is the so-called break-even point, also referred to as Q=1. Q is used to denote the ratio of energy going in to coming out, with numbers larger than one meaning that net energy is being released from the system.

Plasmas tend to be leaky in terms of energy; the neutrons that carry away the energy often simply leave the reaction. However, their energy is so high that if even a small fraction of them are "captured" in the plasma, they will heat the plasma back up, allowing other nuclei to undergo fusion. Thus at some temperature where Q >> 1, the reaction becomes a self-sustaining chain reaction, a point known as the critical ignition temperature. A plasma, once ignited, will burn all of the fuel as long as the conditions remain the same. Ignition is considered vital for a practical fusion system.

For D-T fusion, the temperature where this reaction becomes self-sustaining is about 45 MK, still a very high temperature, but about 1/10th the energy of the Coulomb barrier itself.

### Lawson's criterion

The above discussion assumes that the reaction occurs in a steady state, as it does in the Sun for instance. Containing a reaction at these sorts of energies is difficult in an Earth-bound lab, however, and the reaction will tend to "blow up" as the temperature increases long before the fuel is spent. Fusion fuel is expensive, so in order to make use of it a reaction should attempt to use up as much as possible.

In 1957 J. D. Lawson explored the transient cases, and demonstrated that practical systems needed to have particular values of the triple product. For D-T fusion, nTτ needs to be greater than 1014 s / cm³. For fusion power applications, this number has proven very difficult to achieve. Some have attempted it by creating huge densities for small times, others with lower densities for longer times, but to date no system has maintained the Lawson criterion while operating at ignition temperature.

## Fusion reactions

(D is a shorthand notation for 2H, deuterium, and T is short for 3H, tritium.

Fusion powers the Sun and other stars, where the fuel is contained by the pull of its own gravity. In stars the size of the sun or smaller, the proton-proton chain predominates; in larger stars, the CNO cycle is the dominant reaction. Both of these cycles have considerably higher threshold temperatures than reactions being studied on Earth, and the latter reaction rates are therefore much lower.

For Earth-bound fusion reactors the primary concern is a low threshold energy. This implies a lower Lawson Criterion, and therefore less startup effort. Another concern is the production of neutrons, which are difficult to use and control. Reactions that release no neutrons are referred to as the aneutronic reactions and are of considerable interest, but those that release lower-energy neutrons are equally interesting.

Low threshold energy reactions:

D-T reaction (lowest threshold energy, ~50 keV)

D + T → 4He (3.5 MeV) + n (14.1 MeV)

D-D reaction

D + D → T (1.01 MeV) + p (3.02 MeV) (50%)
D + D → 3He (0.82 MeV) + n (2.45 MeV) (50%)

T-T reaction

T + T → 4He + 2 n (11.3 MeV)

Other interesting reactions, mostly aneutronic:

3He reactions

3He + 3He → 4He + 2 p
D + 3He → 4He (3.6 MeV) + p (14.7 MeV)
T + 3He → 4He (0.5 MeV) + n (1.9 MeV) + p (11.9 MeV) (51%)
T + 3He → 4He (4.8 MeV) + D (9.5 MeV) (43%)
T + 3He → 5He (2.4 MeV) + p (11.9 MeV) (6%)

6Li reactions

p + 6Li → 4He (1.7 MeV) + 3He (2.3 MeV)
D + 6Li → 2 4He (22.4 MeV)
3He + 6Li → 2 4He + p (16.9 MeV)

Tritium "breeder" reactions used in "dry" fusion bombs and some proposed fusion reactors:

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

11B reaction

p + 11B → 3 4He (8.7 MeV)

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". The two most studied aneutronic reactions are the T + 3He and D + 6Li, the latter forms the basis for thermonuclear bombs. However all of these, even the aneutronic ones, do not operate "cleanly" and a number of less interesting reactions will occur at the same time, some of those producing neutrons.

## Fusion in the sun

The Sun and other stars are powered by the fusion of hydrogen or helium.

Solar neutrinos are a different type of radiation emitted by the nuclear reactions in stars. Electrons and positrons (anti-electrons) are delocalised because the matter in stars is a plasma. These leptons may also be considered a form of solar radiation, but they do not travel far from the solar body. Fusion begins with the combination of four hydrogen-1 nuclei to create two hydrogen-2 nuclei. As a result, two positrons (positive electrons) and two neutrinos are released. These two hydrogen-2 nuclei, together with another two hydrogen-1 nuclei, form two helium-3 nuclei and release gamma radiation. These two unstable isotopes of helium fuse to form helium-4 and two particles of hydrogen-1.

The whole process can be summed up by saying that four protons undergo fusion to produce a helium nucleus and energy. The energy radiated away in the form of gamma radiation, as well as the positrons and neutrinos, is solar radiation. The hydrogen-1 nuclei are not radiation, by its strict definition, as they are usually used again as an input in the fusion chain-reaction.

Also note that energy is conserved, so by calculating the mass of the four protons, and the mass of the helium nucleus, and subtracting you can calculate the mass of energy (energy and mass are interchangeable) emitted in gamma and positron radiation. The equation E = mc2 can be used to convert between mass and energy in joules. Since these values are very small, it is useful to convert joules into electron volts. An eV (electron volt) is equal to 1.6 × 10-19 joules, but in most cases the energy released in reactions is measured in MeV or mega-electron volts, or larger quantities.

## Fuel confinement

Gravitational confinement All mass and energy in general creates a gravitational force. One way to hold the fuel together long enough to undergo fusion is to put enough of it in one place that the gravity created by the fuel is enough to hold it together, as in stars. Stars are self-regulating, the force holding the star out against its own gravity is the heat being generated by the fusion inside. Thus if the rate of fusion rises, the star expands and the rate slows. Some simple math can demonstrate that the mass of fuel needed to make a star using the D-D reaction is about the size of the Moon.

Inertial confinement The fuel can be explosively compressed with external photons or other particles. Of course with an explosive, this implies that the containment time will tend to be quite small. However if the compression is high enough this is of little concern, as the fuel will still undergo significant fusion.

This is the process used in the hydrogen bomb, where a huge explosion, provided by a nuclear fission bomb, compresses a small cylinder of fusion fuel. In these systems the x-rays generated by a fission device "boils" a plastic foam, so rapidly it explodes outward. This creates a shock wave that is focused onto a "trigger" cylinder containing a liquid D-T mix, initiating fusion.

Other forms of inertial confinement have been attempted for fusion power, including using large lasers focused on a small pellet of fuel, using ions of the fuel itself accelerated into a central region as in the Farnsworth-Hirsch Fusor, or using acoustic cavitation in theoretical bubble fusion.

Magnetic confinement A plasma consists of charged particles which can then be confined with appropriate magnetic fields. A variety of magnetic fields can be used to confine and insulate a fusion plasma. However, the confined plasma interacts with different confining magnetic fields in ways that affect the heating and confinement efficiency of the system. The nature of the fusion reactor will also be profoundly affected by the particular magnetic configuration. There are only two basic magnetic structures which have been shown to confine plasmas of fusion interest: the magnetic mirror and the magnetic torus. However, each of these magnetic confinement systems has several variants. These confinement systems differ in practice by emphasizing particular principles of fusion science to improve plasma confinement or to simplify the technical requirements for producing the magnetic fields. Historically, the tokamak, a toroidal confinement concept, embodied a set of principles which was comparatively easy to implement in the laboratory. As a result, most of the scientific progress has been made with this concept.

## Fusion as a power source

For many years, considerable theoretical and experimental effort has gone into tapping fusion, initially to generate electricity, isotopes, and possibly as a spacecraft propulsion rocket far more efficient than chemical or nuclear fission rockets. See fusion power for an extensive discussion.