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4.7.4    Nuclear fusion

Fusion reactions

The first generation of controlled thermonuclear (fusion) reactors (CTR) producing electrical energy from nuclear fusion reactions between light ions will almost certainly exploit D–T reactions occurring in a hot plasma that is confined magnetically, although a successful inertial confinement reactor cannot be ruled out. The D–T reaction is specified because of its high cross-section at low ion kinetic energies and its large energy release. However, even in a plasma containing equimolar quantities of deuterium and tritium there will occur fusion reactions between like as well as unlike ion species and between the energetic charged particle reaction products and the fuel (and impurity) ions. In low power test reactors, the use of radio-frequency wave heating may lead to acceleration of light ions to MeV energies so that nuclear reactions between these accelerated ions and fuel (and impurity) ions may occur; the resulting reaction rates may be significant in comparison with D–D (but not D–T) fusion rates. The important fusion reactions are listed below. Particle energies (in MeV) are quoted for the two-particle exothermic reactions, calculated relativistically for zero energy reactants; the total energy release is given for the other reactions.

Fusion reactions of CTR interest

Fusion reaction cross-sections

Plasma reactivity calculations require reaction cross-sections for energies well below those at which direct measurements are practicable, so it is necessary to extrapolate downwards using the theoretical formula

where the cross-section is expressed in centre-of-mass units, E = mv², m = m₁m₂/(m₁ + m₂) and v is the relative velocity of the interacting particles which have masses m₁ and m₂ and charges Z₁ and Z₂, respectively. The constants e, and c have their usual meaning. S(E) = A exp(−βE) and the parameters A, β and R are given in Table 1. Note that laboratory energies may be used if the substitution E = (m/m₁)E_lab is made.

Table 1. Low-energy cross-section parametrization

Fusion reaction rates

The reaction rate, r, between two species of ion with densities n_i and n_j in a plasma is given by r = n_in_j(1 + δ_ij)^{− 1} σv, where σv is the appropriate average of the fusion cross-section σ over the relative velocities v, and δ_ij is the Kronecker delta function. The relative ion energy distribution in a plasma is customarily taken to be of Maxwellian form,

where E is the relative energy and θ =kT, so that

Provided the cross-sections have the simple form σ(E) = (A/E) exp(−βE − R/), applicable at low energies, the integration can be performed approximately by saddle-point integration to give

in m³ s^{− 1}, where A is in barns, θ in keV and m is the relative mass in a.m.u. Using the values of A, R and β from Table 1, σv values for temperatures well below the Coulomb barrier may be calculated. It is customary to neglect terms involving β.

For calculations of plasma reactivities at high energies, an accurate representation of σ is required, necessitating a numerical integration of the σv product over energy. Thermal reactivity values for the primary fusion reactions are plotted in Fig. 2. Bosch and Hale (Nuclear Fusion, 1992, 22, 611) have provided convenient parametrizations for the thermal reactivities for the reactions D(d, n)3He, D(d, p)T, T(d, n)4He and 3He(d, p)4He. Harris, Fowler, Caughlan and Zimmermann (Ann. Rev. Astron. and Astrophys., 1983, 21, 165) have presented a comprehensive set of stellar thermonuclear reaction rate formulae applicable over a very wide range of temperatures.

Thermonuclear neutron energy spectra

In a Maxwellian plasma containing two sets of particles at a common temperature θ, the mean energy of the centre-of-mass of the colliding pairs is

and the relative mean kinetic energy is

where V is the centre-of-mass velocity and v is the relative velocity. The mean relative velocity of reacting particles is obtained by weighting with the reactivity σv and Brysk (Plasma Physics, 1973, 15, 611) has shown that there exists a convenient relationship

Use of the simple Gamov cross-section weighted by a Maxwellian energy distribution results in

      The mean energy and energy distribution. of neutrons emitted from D–D and D–T plasmas provide information on the plasma temperature. The average neutron energy is

f(En) =   1  exp −{En − En}2 σw 2σ2w

where

The distribution fwhm is 2σ_w. The width of the Gaussian distribution is a measurable quantity, thus providing a valuable diagnostic technique for determining the temperature, θ, of a Maxwellian plasma. At low ion temperatures, the relationship between F and θ is F = 82.6√θ for D–D neutrons and F = 177.2√θ for D–T neutrons, where F and θ are in keV. The multipliers are weakly temperature-dependent.

Neutron blanket reactions

A reactor based on the D–T reaction will consume appreciable quantities of tritium. In a pure fusion reactor system, this tritium will have to be derived from nuclear reactions between the 14 MeV fusion neutrons and lithium in a breeding blanket surrounding the reaction chamber. The breeding reactions are

and

Because of parasitic neutron capture and penetration losses, the number of tritium atoms produced per 14 MeV neutron may fall below unity. In this case a neutron multiplier such as beryllium will be needed:

      Reaction and scattering cross-sections needed for neutron transport and moderation considerations are discussed in sections 4.7.2 and 4.7.3.

Neutron activation reactions

Since the D–D and D–T plasmas are intense sources of neutrons it will be possible to employ activation techniques for plasma diagnostic and for reactor dosimetric purposes. Standard reactions which may be employed for this purpose are listed in Table 2, ordered by approximate threshold energies. Most of the cross-sections are taken from the DOSCROS84 Dosimetry File (Zijp, Nolthenius and Verhaag (1984) ECN Petten, The Netherlands, Report ECN-160). Related information on these activation reactions may be found in the Nuclear Data Guide for Reactor Neutron Metrology (Baard, Zijp and Nolthenius (1989) Kluwer Academic Publishers, ISBN 0-7923-0486-1).

The structural materials of which the vacuum vessel and breeding blanket can be constructed will become so highly radioactive that direct access to these inner regions of the fusion reactor will not be possible for maintenance purposes. Further, when the reactor reaches the end of its operational lifetime the problem of disposal of the highly radioactive structural materials will have to be considered. This activation problem can be minimized by appropriate choice of structural material. The most important radionuclides produced in each of the chemical elements most likely to be employed in a structural material are listed in Table 3. The Handbook of Fusion Activation Data (Parts 1 and 2) prepared by Forty, Forrest, Compton and Rayner (AEA FUS 180 (1992) and 232 (1993), AEA Technology, Fusion, Culham, Abingdon, UK) contains extensive information for elements from hydrogen to bismuth.

The induced radioactivity R remaining after time T due to a particular reaction channel is given to first order by

for an irradiation of duration brief compared with the decay time T and radionuclide half-life T_1/2. F is the fractional abundance of the target nuclide in the irradiated element, A is the nuclear number, is the energy-averaged neutron reaction cross-section in barns (10 ^{− 28} m²), is the neutron fluence (n · m⁻²) and the times are in years.

The above formula takes only first generation reaction products into account and ignores the burnup of target and daughter nuclides. For orientation purposes, can be taken to be the reaction cross-section averaged over the energy spectrum of neutrons emitted from a D–T plasma at 20 keV ion temperature as provided in Table 4 for the significant particle-emitting reactions. More detailed cross-section information can be obtained from The European Activation File EAF-3 with Neutron Activation and Transmutation Cross-sections, by Kopecky, van der Kamp, Gruppelaar and Nierop (1992) ECN Petten Report ECN-C-92-058.

The surface gamma dose-rate from a thick slab of material can be estimated from

where S_V is the rate of gamma radiation energy emission (in MeV kg^{− 1} s^{− 1}), μ_a is the energy absorption coefficient in air, μ_m is the mass absorption coefficient of the element and B is the photon buildup factor, all appropriate to the gamma radiation energy E_γ (MeV). The absorption coefficients may be obtained from Storm and Israel (1970) Nuclear Data Tables, A7, 565. Typically, B ≈ 2 and (μ_a/μ_m) ≈ E_γ/2, which leads to the simplified form

where f is the number of quanta E_γ emitted per decay and R is the specific activity. If several gamma emissions occur, then D must be summed over all of them.

Table 2. Fusion neutron dosimetry threshold reactions

* The cross-sections are given as group averages from 2.40 to 2.55 and from 14.0 to 14.1 MeV.

 
Table 3. Important contributing radionuclides, by half-life*

* Note the lack of important nuclides with 100 years < T_1/2 < 500 years.

Table 4. Reaction cross-section (in barns), averaged for 14 MeV fusion neutrons

O.N.Jarvis

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