Table of contents

The production processes and spatial distribution of fast ions resulting from tangential injection of a diffuse neutral beam into a tokamak are discussed. The spatial distribution of fast ions for various injection trajectories and absorption mean free paths are calculated and discussed in detail. Maximum beam absorption for a parabolic density profile is shown to occur for injection roughly halfway between the inner wall of the torus and the magnetic axis; however, since this maximum is near unity and only weakly dependent on the injection trajectory, this is not the most important possible optimization. Since the drift orbit surface area over which the fast ions are distributed is roughly proportional to the distance from the magnetic axis, the fast ion density is found to be strongly peaked at the magnetic axis for present experiments where the absorption mean free path λ is comparable to the plasma radius a. This geometric peaking effect is strong enough to overcome the exponential beam attenuation and cause the fast-ion density and consequent beam energy deposition to be peaked at the plasma centre as long as λ₀ ≳ a/4. Charge exchange of the fast ions with neutrals in the plasma can deplete the fast-ion population, particularly near the plasma edge. When charge exchange is an important loss mechanism, beam injection nearly tangent to the magnetic axis is found to maximize the beam effectiveness in heating.

The authors have examined the cross-section sensitivity of the tritium-breeding ratio in a fusion reactor blanket model which is representative of a large class of blanket configurations under current investigation. This blanket model exhibits the following tritium-breeding and neutronic characteristics: (1) liquid lithium of natural isotopic abundance is used for breeding; (2) the tritium-breeding ratio is in the "high" range, ∼ 1. 5; (3) graphite is employed for neutron moderation; and (4) the neutron leakage from the blanket is relatively low, ∼4% of the source neutrons. The sensitivity of the breeding ratio was evaluated by altering reference cross-section sets to reflect estimates of the uncertainties in the cross-section of the⁶Li(n, α)t, the ⁷Li(n, n'α)t, and the ⁹³Nb(n,2n) reactions. Our analysis leads to the following observations:

1. The status of the ⁶Li(n,α)t cross-section data appears adequate for accurate (of the order of ∼1%) tritium-breeding calculations. We emphasize that this observation applies to the class of blanket configurations considered here and that other classes must be examined to determine the generality of the observation.

2. Uncertainties in the cross-section and secondary-neutron energy distribution of the ⁷Li(n, n'α)t reaction attach uncertainties in excess of 5% to the tritium-breeding ratio. A re-evaluation of the relevant data in the energy range above ∼ 10 MeV might provide a smaller uncertainty, and we recommend that such a re-evaluation be undertaken. Additional measurements may be required to reduce the uncertainty in the tritium-breeding ratio to the order of 1 to 2%.

3. Uncertainties in the (n, 2n) cross-section of ⁹³Nb attach uncertainties of ∼ 2.4% to the tritium-breeding ratio. New experimental data on the ⁹³Nb(n, 2n) cross-section are available, and a recent evaluation of this data suggests that the variations adopted here may reflect too pessimistic an estimate of the uncertainty. This point will be pursued in future studies

4. When the cross-sections of the ⁶Li(n, α)t, the ⁷Li(n, n'α) t, and the ⁹³Nb(n, 2n) reactions are varied simultaneously, the calculated uncertainty in the tritium-breeding ratio is ∼ 7 to 8%. This uncertainty approaches 10% when uncertainties in the secondary-neutron energy distribution of the ⁷Li(n, n'α)t reaction are included.

The author formulates the conditions under which cyclotron oscillations randomize the motion of charged particles in an adiabatic trap. When the amplitude of the cyclotron wave is considerable, the randomization is due to the resonance of the longitudinal oscillations between the magnetic mirrors with transverse oscillations in the cyclotron wave field. If the amplitude is moderate, then the effect is due to phase shift at the resonance point. It is demonstrated that in the latter case the loss of particles from the trap may be insignificant, provided the oscillations have a short wave-length and the plasma is collisionless.

The effect of injection fuel energy on the plasma dynamics stability of a stationary fusion reactor has been investigated, using a mathematical model based on the energy density and particle density balance of four plasma constituents. For this purpose, numerical calculations of eigenvalues have been made after the linearization of a set of plasma equations. The result suggests that a neutral-beam injection gun of moderate energy favours, from the point of view of dynamics, the realization of a stable plasma rather than a pellet injection method of very low energy. Further calculations have been made to study the inner mechanism of thermal instability of a plasma. Finally, a feasible way of plasma stabilization by feedback control has been examined.

Effects of neutral beam injection upon the equilibrium of a toroidal plasma are considered. The distribution function of energetic ions produced by the beam in the plasma is calculated for injection both parallel and perpendicular to the magnetic field taking account of effects due to the toroidal geometry. The effect of trapped particles on the current induced in the plasma by such a beam is calculated, together with the associated cross-field diffusion. Loss mechanisms for the momentum deposited in the plasma by the neutral beam are considered. Ripples in the toroidal magnetic field strength are particularly efficient at destroying toroidal momentum and lead to flow velocities much less than the sound speed for typical injection parameters.

Results are presented from studies of microinstabilities in a model mirror plasma. The model has a Gaussian density distribution of ions and electrons along a varying magnetic field, and a radial density gradient perpendicular to this field. Particles are contained self-consistently by a fictitious confining potential modelling mirror confinement. The ion velocity-space distribution is the finite-plasma equivalent of a Guest-Dory loss-cone distribution, and the electron distribution allows for finite-temperature effects, including Landau damping. Resonant and non-resonant loss-cone instabilities are discussed. It is found that resonant instabilities are stabilized when the plasma length is less than a critical length which is dependent upon electron temperature, but modes corresponding to flute modes in homogeneous plasmas (Dory-Guest-Harris and drift-cone) are found to remain unstable at all plasma lengths.

Current transport theory is extrapolated to predict the performance characteristics of future tokamak power reactors operating in a quasi-steady-state mode. Criteria for the suppression of macroscopic instabilities are used to delimit a stable reactor regime. Microscopic instabilities are accounted for by enhancement of the transport coefficients. The effect of variations in reactor size and aspect ratio, toroidal current, injected ion source strength, and toroidal magnetic field strength are evaluated. Results are presented in terms of engineering parameters such as fractional fuel burn-up and power output.

Turbulent heating of electrons and ions by a collisionless shock wave propagating in a plasma across the magnetic field is investigated. Plasma heating in isothermal plasma occurs as a result of the development of hydrodynamic current instabilities. Heating of plasma subject to hydrodynamic current instabilities is caused by a stochastic mechanism. The temperature variation of the plasma particles at the shock front as well as an estimate of the heating time are obtained. The estimates agree well with the known experimental data.

Turbulent heating of ions of a non-isothermal plasma by the kinetic electron-sound instability is investigated. The heating of ions exposed to the electron-sound instability is caused by a quasi-linear heating mechanism. The ion temperature rises owing to the absorption of the unstable-oscillation energy by ions under the conditions of Cherenkov resonance. The heating time under the electron-sound instability is estimated.

An integral equation which generalizes the dispersion relation for electrostatic plasma modes with wave numbers in the presence of a large-scale magnetoacoustic (pump) oscillation is derived. This calculation is accurate to all orders in the parameter ⋅ where is the amplitude of the particle displacement induced by the pump wave. Expanding in the latter parameter, we obtain a second-order differential equation including three-wave processes, non-linear frequency shifts and nonlinear Landau damping. The electromagnetic wave may disintegrate into two Bernstein waves (three-wave interaction) if the appropriate resonance condition (between the real parts of the frequencies) can be satisfied approximately. However, a Bernstein wave can still be excited when the resonance condition is not fulfilled. In the latter case, instability arises through inverse non-linear Landau damping, and the non-linear dispersion relation can be approximated, to a high degree of accuracy, by a three-by-three matrix. In the former case, the dispersion relation reduces, to a good approximation, to the standard two-by-two matrix only if the linear decrements of both electrostatic waves are sufficiently small; otherwise, the three-wave process identifies with the non-linear Landau instability calculated in the appropriate limit (of matching between the real parts of the frequencies). The relative importance of resonant and off-resonant coupling of the pump wave to the plasma is discussed.

The vertical stability of an elongated tokamak plasma with infinite conductivity is numerically investigated under simple assumptions, in toroidal co-ordinates. The rectangular configuration is stable for an elongation ratio of less than 3 ∼ 4, depending upon the current profile. It is shown that the stabilization at high elongation ratio is related to the rotational-transform constraint.

The authors consider the influence of kinetic effects on wave propagation in an inhomogeneous plasma. It is shown that effects such as phase focusing of particles by an inhomogeneity, perturbation transfer by resonance electrons and echo phenomena can lead to a substantial increase in plasma transparency.

This Symposium on the engineering problems in fusion research attracted approximately 280 participants including representatives from the USSR, Japan, Italy, the United Kingdom, and the Federal Republic of Germany. Two simultaneous sessions were held during all four days and approximately 130 papers were presented. One session was devoted exclusively to fusion-reactor studies with about 40 papers while the other session included the engineering of experimental hardware for present and future fusion experiments. The program included a visit to the Princeton Plasma Physics Laboratory to view the experiments there. The proceedings of this Symposium will be published by the IEEE in April 1974.

The nuclear fusion R & D programs underwaytoday in various countries are generally directed toward the near-term objective of scientific feasibility by any of several concepts involving either magnetic or inertial confinement of the plasma. In the last few years, the level of support for fusion research has markedly increased, first, because of the significant technical advances made in the field, and, more recently, because of the high priorities given to such work as a result of the world energy crisis.

A translation of the abstracts of all the articles for this issue into French, Russian and Spanish.