Mirror Machine Research Articles

Radial distribution of plasma formed in simple mirror machines by field ionization of fast neutral atoms

The radial distribution of plasma which results from field ionization of excited atoms injected into simple mirror machines is calculated and compared with measurements made in the Phoenix machine. The excited atoms are assumed to have a level population distribution given by An−3 where n is the principal quantum number. The fraction of the atoms ionized per unit interval along their path is obtained by calculating the quantum mechanical barrier penetration probability at each point. Finite Larmor orbit diameter, precessional drift around the field axis and finite atom beam dimensions are taken into account in obtaining the radial distribution of plasma that results from this method of injection. The plasma distribution rises to a maximum on the axis in contrast to the distribution obtained by assuming that ionization occurs immediately a critical field is reached. Good agreement with experiment is obtained both for the absolute density and for radial distributions. The fraction ionized per unit interval at the axis is calculated for the Alice and Ogra II experiments without their stabilising fields; this fraction is approximately proportional to the square root of the radial field gradient coefficient. However, the fraction ionized per unit interval is also calculated for an artificial flat-topped field shape and the results show that even in the absence of field gradient, useful trapping can be obtained. Calculations are also made of the effect of pre-ionization on the radial extent of the plasma.

Charge Exchange Loss of High Temperature Plasma Confined by Mirror Magnetic Field

Relaxation of the distribution function of ions confined in a mirror field is in investigated theoretically taking both the charge exchange and scattering effects into account. The result is compared with experiment. If the particles are not thermalized sufficiently, the ion density decays exponentially due to the charge exchange, but only linearly due to the mirror loss. The life time, τ c x , is proportional to the ion velocity while that of the mirror loss, τ m , is proportional to it to the power three. The experiment is performed on the HX-0 device, a mirror machine for high temperature experiments. A decay time of 26 µs is obtained for high energy protons which are trapped in the mirror magnetic field at the background pressure of 10 -5 mm Hg.

Fusion energy balance in mirror machines

Motivated by the encouraging outlook for stable confinement of fusion plasmas by `minimum-B' magnetic mirrors, we have recalculated Q, the ratio of total fusion energy released to the injection energy required to replenish mirror losses by collisions. We solve numerically Fokker-Planck equations for ion and electron energy distributions with special attention toward ion cooling by electrons and the effect of the ambipolar potential on the velocity loss cone. We find that both these effects enhance mirror losses much more than was supposed previously. Consequently, for a D-T plasma our Q is a factor 4 or more smaller than previous estimates, and Q is maximum at a higher energy, 200-300 keV. For a mirror ratio R = 3, the maximum is Q = 1.5, and, for larger R, Q scales approximately as log10R. We find a negligible benefit from ion heating by collisions with trapped α-particles from fusion reactions.

Stochastic motion of particles in a non-adiabatic magnetic trap

The properties of a mirror machine with a spatially modulated central field are investigated. The modulation is simulated by a square oscillation and leads to non-conservation of magnetic moment. Equations are set up which describe, in various degrees of approximation, the equilibrium state of an ensernble of particles interacting only with the magnetic field, given a constant input of particles and taking mirror loss into account. The magnetic mirrors are approximated by reflection of particles without change of magnetic moment but with random Larmor phase. An integral equation is obtained involving transmission and reflection operators for the non-adiabatic central region. In successive approximations, an ergodic hypothesis of Larmor phase in the central region and a Fokker-Planck expansion lead, respectively, to a set of coupled integral equations and their Fokker-Planck counterpart, a set of diffusion equations. Results from the diffusion equations are compared with a numerical study of this magnetic trap reported recently by Dunnett et al.

Critical Length Determination for Convective Instabilities in Weakly Inhomogeneous Plasmas

A convergent perturbation expansion procedure for determining the reflection coefficient for unstable convective modes in a plasma with small changes in the density is derived. If the total variation of the local wave vector is small, the lowest-order reflection coefficient can be used to determine the critical length of the system. The technique is used to determine the critical length of mirror machines which are subject to the convective loss-cone instability. Even for small rapid or large gentle changes in the local wave vector, critical lengths less than ten e-folding distances can be realized.

Theory of the flute instability in mirror machines

The flute instability of a low β and neutral plasma contained in a mirror machine is investigated. The particle motion in a real mirror field is fully taken into account by using a Hamiltonian method for the solution of the Vlasov equation. An eigenvalue equation is obtained for the potential of the flute-like perturbation assuming an arbitrary radial dependence of plasma density and of the electron and ion energies.

Flute Instabilities at Ion Gyrofrequency

Flute modes at the ion gyrofrequency, propagating perpendicular to the magnetic field, are considered. A dispersion relation for inhomogeneous plasmas is obtained from which density thresholds for instabilities are derived. Several plasma distributions are considered. For a loss cone in velocity space, as in mirror machines, the density threshold is given by (ωpi/ωi) ≈ (R/rli)¾. In the case of a plasma shell with a radial density proportional to r2 at r = 0, (ωpi/ωi) ≈ (R/rli)½. Here, ωpi is the ion plasma frequency, ωi is the ion gyrofrequency, R is the plasma radius, and rli is the ion gyroradius.

Concerning the possibility of a self-sustaining thermonuclear reaction in a mirror machine

An investigation is made of the energy balance in thermonuclear reactors of various types employing magnetic mirror trapping. Only the energy loss due to the escape of particles through the mirrors is taken into account. It is very likely that a self-sustaining thermonuclear reaction cannot be produced in such devices.

Plasma stability in a mirror machine with stabilizing rods

Plasma stability in a mirror machine with stabilizing rods

Unstable Precession under the Influence of Drag Forces

Coherent precession of plasma particles in ▿B is generally subjected to drag forces associated with viscous or resistive energy dissipation in background plasma. The dissipated energy can be supplied by continual displacement of the processing entity along — ▿B. Precession in mirror machines therefore tends to be unstable against drag, while the converse is true for minimum-B geometries. A heuristic mechanical model is first described. The precession of the E-layer is next shown to be drag-unstable if one uses the ``super-particle model,'' which does not allow for a finite spread in particle precession frequencies. Similar results are obtained for ordinary mirror-machine plasmas in the two limits that are ordinarily considered stable by virtue of the high-density and low-density finite-Larmor-radius conditions.

A numerical study of particle behaviour in non-adiabatic magnetic traps

The behaviour of charged particles in a mirror machine with a spatially-modulated central field is examined by adopting a model in which the modulation has the form of a square wave. This enables the equations of motion to be solved algebraically and particle orbits may be computed for long periods in the machine. The non-conservation of magnetic moment allows particles injected through the mirror to be temporarily trapped, and by considering that mirror reflections randomize the Larmor phase of a particle, a statistical distribution of lifetimes is obtained corresponding to the injection of a beam having a finite spread in initial parameters.

On the origin of rotation in magnetically confined plasmas

Abstract The main part of this paper surveys the mechanisms that can lead to rotation of plasma in an electrodeless discharge (commonly called a theta pinch) and in a mirror machine. Eight proposed mechanisms are examined, of which two are found to be incorrect in some way. Another mechanism, in which the reaction is purely electromagnetic, applies only to the tenuous plasma of a mirror machine. The remaining five theories apply to the highly compressed plasma of a theta pinch. These all rely fundamentally on the large difference in mass between an ion and an electron, i.e. either on Hall currents or the correction for finite ion Larmor radius in the stress tensor. The mechanisms which incorporate the Hall effect cause an equal and opposite mechanical reaction either on the wall (Roberts and Taylor), or on external conductors (Haines), or on the plasma at each end of the discharge axis (Bostick). The mechanisms which use the collisionless stress tensor predict either radial division into oppositely rotati...

Theory of the flute instability in mirror machines

The flute instability of a low β and neutral plasma contained in a mirror machine is investigated. The particle motion in a real mirror field is fully taken into account by using a Hamiltonian method for the solution of the Vlasov equation. An eigenvalue equation is obtained for the potential of the flute-like perturbation assuming an arbitrary radial dependence of plasma density and of the electron and ion energies. Using a variational method, the eigenvalue equation has been discussed in detail for a plasma with no energy gradients. The stability condition has been calculated taking into account terms up to second order in Larmor radius. Finite Larmor radius stabilization is effective for the higher-order modes and cold electron plasmas. The lowest mode, k = 1, is practically not stabilized by finite Larmor radius effects. The discussion of the eigenvalue equation with finite energy gradients has been limited to the case of a hot electron plasma surrounded by a sheath of cold plasma. If the density of the cold plasma is an order of magnitude higher than the density of the hot plasma, the system is stable at all densities. The stabilization is due to the high dielectric constant of the cold plasma.

A numerical study of particle behaviour in non-adiabatic magnetic traps

The behaviour of charged particles in a mirror machine with a spatially-modulated central field is examined by adopting a model in which the modulation has the form of a square wave. This enables the equations of motion to be solved algebraically and particle orbits may be computed for long periods in the machine. The non-conservation of magnetic moment allows particles injected through the mirror to be temporarily trapped, and by considering that mirror reflections randomize the Larmor phase of a particle, a statistical distribution of lifetimes is obtained corresponding to the injection of a beam having a finite spread in initial parameters. Containment properties are derived for a variety of resonant and non-resonant modulation systems.

High-frequency heating of electrons in a mirror machine

High-frequency heating of electrons in a mirror machine

Remarks on the finite larmor radius stabilization theory for mirror machines

Several existing theoretical treatments of the low-frequency drift instability for low β and low-density plasmas in mirror machines are compared. We point out the importance of boundary conditions, and show that finite Larmor radius stabilization is important for the m = 1 mode. The equation for the electrostatic potential for high densities is solved for some ratios a/r0 of the Larmor radius to the plasma radius. It is found that in the Phoenix experiment, under the present operating conditions, a ratio of a/r0 ⩾ 0.5, corresponding to the field of ⩽5 kG is sufficient for stability.

Synchrotron Radiation Measurements from a Plasma in a Magnetic Mirror Machine

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Electron cyclotron resonance instability in a mirror machine

Electron cyclotron resonance instability in a mirror machine

Experimental and Theoretical Studies of Instabilities in a High-Energy Neutral Injection Mirror Machine

Recent observations with the Phoenix high-energy neutral injection experiment are described. At densities where the electron plasma frequency is greater than the ion cyclotron frequency, strong emission at the ion cyclotron frequency and ½ the ion cyclotron frequency is observed. This is interpreted on the basis of the theory of electrostatic instability developed by Harris and others. Experimentally, the instability results in strong scattering of ions out of the transverse direction but so far as can be observed, there is no actual loss of plasma. Above a density of 5 × 108 particles/cm3, accumulation of particles is limited by the development of strong low-frequency (∼100 kc/sec) oscillations which appear to be an m = 1 displacement of ions and electrons rotating at half the ion precession frequency. This mode is approximately independent of density. During periods of strong emission at the ion cyclotron frequency, however, another mode is observed, the frequency of which is approximately proportional to density. These observations suggest an m = 1 precessional drift instability. The existing theories concerning this low-frequency instability are fully discussed. We present a new calculation in which the cylindrical plasma shape and the associated boundary conditions are taken into account. Its predictions are in reasonable agreement with the experiment. The electric field inside the plasma is found to be nonuniform even for the m = 1 mode, so that finite Larmor radius effects are to be expected for this mode, contrary to common belief. The full equation for the electric potential valid for all densities and including the first order finite Larmor radius effect is presented. However, the finite Larmor radius effect is found to be small at densities around 108 particles/cm3, and at magnetic fields above 10 kG.

The passage of a plasmoid through an adiabatic mirror machine

The passage of a plasmoid through an adiabatic mirror machine