A nuclear fusion reaction is the interaction of two atomic nuclei to form a heavier nucleus. Fusion of light nuclei (atomic wt. Ai < 50) is generally exothermic, especially where helium-4 is formed. Examples are:
Here p, D and T stand for proton, deuteron and triton (isotopes of hydrogen); n for a neutron, e+ is a positron, ν a neutrino, and γ a photon; 1 MeV = 1.6 × 10−13J; [ ] is the normalized cross-section in (KeV) m2 × 10−28 [see Bahcall and Pinsonneault (1992)]. For fusion to occur, the interacting nuclei must penetrate or surmount the Coulomb barrier. The fusion reactions 1(a) to 1(c) occur in stars, providing 97% of the solar luminosity. Reaction 2(a) starts the C-N cycle whereby 12C and 14N catalyze the fusion of hydrogen to helium. Because of the higher Coulomb barrier, it predominates in higher temperature stars. Reactions 3(a)(b) are used in terrestrial fusion reactors. 3(b) provides the highest rates; the tritium is produced by neutron reactions in lithium:
Fusion Reactors are assemblies of nuclei undergoing fusion reactions. The stars are natural fusion reactors, held together by gravity, with their temperature and pressure sustained by fusion energy. Hydrogen bombs are transient fusion reactors, where the fusion fuel is compressed and heated by radiation from a nuclear fission explosion. Teller (1987) reports that they can be cylindrical devices, a foot or so in diameter and a few feet in length, and may produce an energy equivalent to a few hundred thousand tonnes of high explosive (1 tonne of H.E. is 4.2 × 109J).
Inertial Confinement Reactors are conceived as containable fusion explosions of up to about 3 × 108J. The fuel is compressed by radiation provided by converging pulsed energy from lasers or possibly particle accelerators. The most fusion energy, so far reported (1993), from a single event is about 50J, achieved at Lawrence Livermore National Laboratory with the pulsed laser, NOVA (up to 50kJ of 0.35 μm light in 10−9 s), focused on to a small sphere (D ~ 0.3 mm) containing D-T fuel. [See references in IAEA (1993) vol.3.]
Magnetic Fusion Reactors use magnetic fields to contain fusion fuel at pressure up to about 10 bar, heated by particle beams and, ultimately, by the 3.5 MeV helium nuclei produced by reaction 3(b). The principal system researched, the Tokamak (a Russian acronym), uses a large current flowing in the fuel, inside a metal torus, to provide the confining magnetic fields; externally-generated fields stabilize the configuration. Fusion power lasting 2s peaking at 1.7 MW has been obtained in the Joint European Torus (JET) at Culham, UK, using 3.1 MA current to confine dilute D-T fuel. When optimized, both JET and the USA's Tokamak Fusion Test Reactor are expected to produce fusion power up to the 10 MW level, using >10 MW power inputs from external sources.
Thermonuclear fusion reactions occur when the velocities of the colliding nuclei arise from high temperature of the fuel. The reaction rate R12 is given by
where n1 and n2 are the number densities of two reacting species, σR(u) is the reaction cross-section for a relative velocity u, and the average < > is taken over the Maxwellian distribution of u. Figure 1 shows < u σR(u) > as a function of temperature. The values provide a criteria for a net energy output from a self-sustained thermonuclear D-T fusion reactor, namely:
Here n = n1 + n2, T is the temperature of about 108 K, and τE is the energy confinement time (a measure of the thermal insulation of the fuel) defined as the thermal energy of the fuel divided by the power used to sustain it. Nearly 1028 m−3Ks has been reached in tokamaks. Solar fusion power is thermonuclear. Up to about half the fusion power in JET is thermonuclear, the rest arises from fusion reactions of suprathennal particles used for heating, see IAEA (1993).
Heat transfer in fusion reactors is by radiation, by thermal conduction and by convection. The atoms of the hot (>106 K) fuel are all ionized and form plasma, the highly-conducting, high-temperature state of matter, see Dendy (1993). Fully ionized plasma radiates energy through electron scattering on ions of charge Ze and number density ni at a rate PB
with photon energies up to a few kB T, where kB is the Boltzmann constant and Z the atomic number. In laboratory-scale plasma the radiation escapes. Stars are easily large enough to re-absorb the radiation, so that the radiant energy diffuses out at the rate
where a is the radiation density constant, c the speed of light, κ the mass absorption coefficient and ρ the mass density. Radiative diffusion dominates in the inner region of the sun. In the outer regions , the heat is transferred by convective cells.
The thermal conductivity of field-free plasma is due to the electrons and is approximately
where ln Λ is the so-called Coulomb logarithm, a slowly varying function of n, T and Z given in Wesson (1987), having a value of about 10. This conduction occurs in inertial confinement plasma (T ~ 107–108K, n ~ 1030–1031 m−3), and along lines of force in magnetic fields. The conductivity normal to a strong magnetic induction B is much reduced from (9), so that magnetic fields provide thermal insulation as well as confining plasma pressure. Heat and mass transfer due to electron and ion collisions across the confining magnetic fields of a tokamak are interdependent, are dependent on geometry and electric fields, and are predicted by neoclassical theory, see Wesson (1987). The thermal conductivity is due to the ions and is about
where ε ≡ r/rM is ratio of minor to major radius of the torus, and B is the induction arising from the confining electric current A semi-empirical expression due to Bohm for convective thermal and plasma diffusivity across the magnetic field is
The experimentally observed values of δ and D are generally intermediate between (10) and (11), and are in the range 1-100 m2/s in large (~3m bore) tokamaks, where, typically, n ~ 1020m−3, T ~ 108K B ~ 3T, and the currents are < 7MA.
Envisaged industrial fusion reactors have a central fusion reactive zone of plasma, surrounded by a lithium-containing blanket about 1 m thick, where the 14.1 MeV fusion neutrons transfer energy by collisions with atoms to a heat transfer fluid used to raise steam to power turbo-alternators. The tritium fuel is produced by reactions 4 in the blanket. The helium exhaust system has also to handle the heat convected out with the exhaust, and is a key factor in the designs.
Bahcall, J. N. and Pinsonneault, M. H. (1992) Standard Solar Models and solar neutrinos. Rev. Mod. Phys., 64, 885.
Teller, E. (1987) Fusion Devices Explosive, Encyclopedia of Physical Science and Technology, 5, 723, Academic Press.
IAEA (1993), Plasma Physics and Controlled Nuclear Fusion Research, Vols. 1 & 3 (IAEA, Vienna).
Wesson, J. (1987) Tokamaks, Clarendon Press, Oxford.
Dendy, R. (1993) Introduction to Plasma Physics, CUP.