Nuclear Fusion – The Theory

Nuclear Fusion – The Theory

By Barrie Lawson, UK

The Holy Grail of nuclear energy, nuclear fusion is the process by which the Sun generates its prodigious energy providing us with the warmth and light we receive. It is the process by which the nuclei two light atoms combine to form a single, bigger nucleus of a new atom releasing large amounts of energy as a consequence. In 1939 Hans Bethe explained that this process occurs in the stars all over the universe but up to now we have not been able to successfully duplicate this process on earth despite over 70 years of trying, but at last we are coming close.

The great attractions of nuclear fusion as an energy source are that the fuel, mostly isotopes of hydrogen, is plentiful and easy to obtain, and the elements produced as a result of the fusion are usually light and stable atoms rather than the heavy radioactive products which result from nuclear fission. Furthermore, the potential release of energy per unit mass of the fuel is much higher in the case of fusion than in fission since reactions allowing greater increases in binding energy are possible with fusion reactions. See the graph of binding energy above.

The Coulomb Barrier

Unfortunately, although bringing about fusion is theoretically possible, achieving it in practice is fraught with major difficulties.
Coulomb Barrier

For fusion between two positively charged nuclei to take place, they must get close enough to eachother to undergo a nuclear reaction. For this to occur the nuclei must overcome the energy barrier due to the "repulsive" electrostatic Coulomb force, known as the Coulomb barrier, between the nuclei, to force them close enough to each other to come within the influence of, and be captured by, the "attractive" Strong nuclear force which holds the nucleons in each nucleus together.

The magnitude of the Coulomb barrier corresponds to the work done or energy needed for the two nuclei overcome the Coulomb force to come together. This is equal to the potential energy between the particles and is given by the Coulomb force between two nuclei multiplied by the distance between them integrated over the distance over which the force is effective. Thus for fusion to occur, the nuclei must have at least enough kinetic energy to exceed the Coulomb repulsion and this is proportional to the mass of the nuclei and the square of their velocity. The magnitude of the this kinetic energy is also directly proportional to the temperature of the nuclei.

All fusion reactions require the fusing nuclei to have extremely high kinetic energies in order to overcome the Coulomb barrier and get close enough to each other to fuse. Their temperature must therefore be extremely high, 100 Million°C or more, so that the nuclei collide with each other at great speed.

A Note About Temperatures:
The temperature and energy of a moving gas or plasma particle are directly proportional to its velocity squared.

In nuclear physics, temperature is typically used to express the "average" energy of the particles in a fuel sample. Conversely, it is also common to express very high temperatures in terms of energy units (Joules or electronVolts).

The energy in Joules is given by multiplying the temperature in degrees Kelvin by Boltzmann’s constant of 1.38 X 10-23 Joules/degree Kelvin (J/K)


1 Joule (J) ≡ 6.24 x 1012 Million electronVolts (MeV)


1000 electronVolts (KeV) ≡1.16 X 107 °K

Fusion temperatures are so high that °Kelvin and °Celsius are almost the same and are often used interchangeably

As a point of reference, the temperature at the core of the Sun is around 1.5 X 107 °K. This does not mean that all the atomic particles are at that temperature. This temperature represents the average energy of the atomic particles in the core. In thermal equilibrium, the atoms have a range of velocities (energies) described by the Maxwell-Boltzmann distribution which represents the number of particles at each energy level in the sample.

Maxwell-Boltzmann Temperature Distribution
The energy distribution shows a very high percentage of particles with energy levels around the average level and relatively low percentage of very low and very high energy particles at the lower and upper tails of the distribution. In absolute terms however, because of the extremely high particle density at the centre of the Sun, there will be a very large number of particles in the longer, upper tail with energies or temperatures many times higher than the mean

The Coulomb barrier between two protons in free space is 3.43 MeV and this corresponds to a temperature of 4 × 1010 °K. This is over 1000 times the temperature of 1.3 KeV (1.5 X 107 °K) at the core of the Sun. How then can proton fusion occur on the Sun with such low particle energy levels? There are two explanations for this apparent anomaly.

  • The calculation of the Coulomb barrier determines the "average" energy per proton needed for fusion, however these energies are distributed according to the bell shaped Maxwell-Boltzmann distribution. Within its long tail there is a sufficiently large number of particles whose energy is much larger than the average and thus enough to initiate fusion.
  • It is not necessary for the incident protons to have sufficient energy to overcome the Coulomb barrier entirely. Due to the wave-particle duality and the statistical probabilities of the properties of very small particles, as explained by quantum mechanics, the protons can also pass through the barrier by a phenomenon known as quantum tunnelling, provided the barrier height is not too high above the kinetic energy of the incoming particle.
Fuel Density

For significant fusion to take place, the particle density of the fuel must also be very high to provide sufficient opportunities for collisions between the particles to occur.

In a star, the high temperatures are provided by the self sustaining nature of the nuclear reaction itself and the density of the fuel is maintained by the star’s massive gravity. On earth, attempts have been made to create these extreme conditions in a plasma of ionised gases but containment is a serious problem since no known materials for containing the fuels can withstand such high temperatures. An alternative is to capture the energy from a series of tiny, controlled thermonuclear explosions which has so far been more successful. See Nuclear Fusion Reactors

The story so far

Fusion Fuels
Fusing two light nuclei can liberate more than the fission of Uranium-235 or Plutonium-239. Ideal fuels are the lightest elements since they experience the lowest repulsive force, or Coulomb barrier, between their nuclei, so that the energy needed to bring about fusion is reduced. The most common fuels used in fusion attempts are Deuterium and Tritium, gaseous isotopes of hydrogen, in the so called D-T reaction.

  • Deuterium
    Deuterium (2H) also known as heavy Hydrogen is a naturally occurring, stable isotope of Hydrogen found in the earth’s oceans where it accounts for approximately 0.015% of the Hydrogen atoms, or 0.030% of the weight of Hydrogen. Deuterium is extracted from the water by a variety of separation methods which exploit the small differences in physical and chemical properties between Deuterium and Hydrogen.

    The Deuterium in one gallon of sea water has the energy content of 300 gallons of gasoline and the oceans contain enough energy to supply the world’s energy needs for thousands of years..

  • Tritium
    Tritium (3H) is also an isotope of Hydrogen. It is radioactive with a half-life of 12.3 years decaying at the rate of 5.5% per year by beta decay into Helium-3 with the release of an electron and of 18.6  keV of energy. Because of its short half-life, only tiny quantities are found naturally as the result of the reactions of cosmic rays with atmospheric gases. Supplies of Tritium are produced as a by-product of other nuclear reactions, notably those involving Lithium and much of it is reserved for nuclear weapons. It is consequently both rare and very expensive.

    Due to their low energy, the emitted electrons can not penetrate the skin and so Tritium is not considered an external radiation hazard though it could cause damage if inhaled or ingested. Tritium is however often used as a biological tracer in medical research because of its short half-life and low radiation.

The fusion of Hydrogen nuclei (protons) in a proton-proton reaction is not practical since it would require too much energy or too high a temperature to start. The fusion of Deuterium with Tritium needs a temperature of 100 million to 150 million degrees Centigrade. All other fusion reactions need even higher temperatures.

Fusion Energy Release

The D-T fusion reaction between Deuterium and Tritium is shown below:

Nuclear Fusion

2H1    +    3H1    ⇒    4He2    +   1 n0    +    @ 17.6 MeV

The diagram above shows that the fusion of one atom of Deuterium with one atom of Tritium to form one atom of Helium (also called an alpha particle) releases about 17.6 MeV of energy.

Note that this reaction results in the release of a troublesome surplus neutron which can cause problems in practical fusion reactors, such as the Tokamak. Since they carry no electric charge, neutrons are not constrained in the the plasma by the magnetic field and can migrate to the walls of the reactor where they can react with parts of the reactor construction materials which may consequently become radioactive.

This unavoidable side effect can however be turned to an advantage. By using a Lithium metal blanket to capture the surplus neutrons in the reactor, two useful fission reactions are possible, both of which result in the release of alpha particles (Helium nuclei) and, more importantly, the production of Tritium, the scarce, radioactive, isotope of Hydrogen, which is one of the basic fuels for the fusion reaction.
The first reaction is exothermic and uses the isotope Lithium-6 to capture slow neutrons producing Tritium while releasing an alpha particle and 4.8 MeV of energy as follows:

6Li3 + 1 n04He2 + 3T1+ @ 4.8 MeV

The second reaction is endothermic using the isotope Lithium-7 to capture fast neutrons, also producing Tritium and releasing an alpha particle while leaving a free neutron and absorbing 2.5 MeV of energy.

7Li3 + 1 n04He2 + 3T1+1 n0 – @ 2.5 MeV

Natural Lithium is relatively abundant in the earth’s crust and is typically composed of 92.6% of the isotope Lithium-7 and 6.4% of Lithium-6.

The energy released at the atomic level by the fusion of Deuterium and Tritium can be calculated from the binding energies of the parent and daughter atoms as shown in the following table:

The table shows that the combined binding energy of the Deuterium and Tritium atoms of 10.7 MeV increases to 28.3 MeV when the atoms fuse into Helium releasing energy of 17.6 MeV, equivalent to 2.8 X 10-12 Joules.

Note that 80% of the released energy is carried by the neutron with the Helium alpha particle accounting for only 20%

The energy released from practical amounts of fuel can be calculated as follows:

The atomic mass of the Deuterium nuclide is 2 amu = 3.34 X 10-27Kg
1 Kg of Deuterium therefore contains 1Kg/2 amu = 2.99 X 1026 atoms.

The atomic mass of the Tritium nuclide is 3 amu and hence 1.5 Kg of Tritium also contains 2.99 X 1026 atoms.

The energy released by fusion of 1 atom of Deuterium with 1 atom of Tritium is 17.6 Mev = 2.82 X 10-12 Joules.

The energy liberated by the fusion of 1 Kg of Deuterium with 1.5 Kg of Tritium is therefore 2.82 X 10-12 X 2.99 X 1026 = 8.43 X 1014 Joules = (8.43 X 1014) / (3.6 X 1012) GWHours = 234 GWHours.

This energy appears in the form of heat. If it was used to generate electricity in a conventional steam turbine power plant with an efficiency of 38%, it would provide 88,900 MWH of electricity which is near enough equivalent to one year’s operation with a constant output power of 10 MWatts.

Fission – Fusion Energy Comparison

Note that the 234 GWH (8.43 X 1014 Joules) released by the fusion of 2.5 Kg of the fuel in the D-T (40-60 proportion) reaction above is equivalent to 93.6 GWH (3.37 X 1014 Joules) per Kg. This is over four times the 22.5 GWHours (8.1 X 1013 Joules) of energy released by the fission of 1 Kg of Uranium – 235. The fusion reaction also uses safer fuels which are inexpensive, more plentiful, and easier to manage, leaving behind much more benign waste products resulting from the process.

To put these values into perspective, 1 Kg of petrol (gasoline) has an energy content of about 4.7 X 107 Joules or (4.7 X 107) / (3.6 X 106) kWH = 13 kWH, less than one millionth of the energy from either nuclear fission or fusion. See also Energy Content Comparison Table

Other Fusion Fuels

There are actually four possible fusion reactions which could take place in a reactor fueled by Deuterium only.

2H1    +      2H1    ⇒    3He2    +   1 n0     +    @ 3.3 MeV
2H1    +      2H1    ⇒    3H1      +    1 H1    +    @ 4.0 MeV
   2H1    +      3H1    ⇒    4He2    +    1 n0    +    @ 17.6 MeV
2H1    +     3He2   ⇒    4He2    +   1 H1    +    @ 18.3 MeV

The results of these four reactions can be summarised as follows:

62H1   ⇒    21H1   +   24He2   +   21 n0    +    @ 43.2 MeV

The first two of the above reactions using only Deuterium as the fuel, known as D-D reactions are equally probable. This fuel combination has the advantage that the Deuterium fuel does not have the mildly radioactive properties of Tritium, but the energy release is relatively low.

  • The first reaction produces the He3 isotope of Helium and an energetic neutron.
  • The second reaction produces Hydrogen and Tritium which is mildly radioactive.
  • The third and fourth reactions involve the Deuterium fuel reacting with the fusion products of the first two reactions producing a much higher energy release.
  • The third reaction is the well known D-T reaction between Deuterium and Tritium which produces a Helium atom (alpha particle) and a high energy neutron.
  • The fourth reaction between Deuterium and the Helium isotope He3 produces a Helium nucleus (alpha particle) and 2 energetic neutrons and is known as the D-He3 reaction.

Other promising fusion fuel candidates are the Hydrogen-1 (proton) with / Boron reaction, which releases Helium (alpha particles).

Solar Fusion Chain

Solar fusion is initiated by the fusion of two Hydrogen nuclei (protons) in the following reaction in which the two protons fuse to form a Deuterium atom as one proton is transformed into a neutron, with the release of a positron and a neutrino and 0.42 MeV of energy.

1H1    +     1H1    ⇒    2H1    +    e+    +    ν     @ 0.42 MeV

At the same time, the positron emitted by the beta decay of the proton is almost immediately annihilated with an electron, and their combined mass energy, as well as their kinetic energy, is carried off by two gamma ray photons.

e+    +    e    ⇒    2 γ   @1.02 MeV

Proton – Proton (p-p) Chain Reaction
After the initial reaction, solar fusion continues with more protons reacting with the Deuterium produced in the first reaction to form a light isotope of Helium releasing more energy in the following reaction and initiating a chain of reactions.

2H1    +     1H1    ⇒    3He2    +    ν     @ 5.49 MeV

In turn, the products of this second reaction fuse with even more particles to form ever more complex, heavier nuclei releasing more energy in further reactions such as the Deuterium D – D reactions above.

Power from Nuclear Fusion

Unfortunately immense amounts of energy are needed to create the conditions for self-sustaining fusion to take place and in practice there are serious technical problems to overcome in order to achieve a net energy gain. These are considered in the section on Nuclear Fusion Reactors

About the author, Barrie Lawson:

Barrie graduated from Birmingham University with a degree in Electrical and Electronic Engineering in 1964. Since then he as has worked at Director level in many branches of the electronics industry including military electronics, telecommunications, computers, automotive and consumer electronics. During the last 10 years he has been involved in the battery business, originally as Chairman of MPower Batteries, a custom battery pack making company in Scotland which he helped to found and later in China where he set up a similar business. He is currently Chairman of CHE EVC, another battery startup company pioneering some interesting new technologies. In his spare time he writes and maintains the Electropaedia web site, a comprehensive knowledge base about batteries and energy sources.