Magnetic Mirrors

(2.83) |

because . Thus, we obtain

(2.84) |

where

is the lowest order total particle energy. Not surprisingly, a charged particle neither gains nor loses energy as it moves around in non-time-varying electromagnetic fields. Because and are constants of the motion, we can rearrange Equation (2.85) to give

(2.86) |

Thus, charged particles can drift in either direction along magnetic field-lines in regions where . However, particles are excluded from regions where (because they cannot have imaginary parallel velocities). Evidently, charged particles must reverse direction at those points on magnetic field-lines where . Such points are termed

Let us now consider how we might construct a device to confine a
collisionless (in other words, very high temperature) plasma. Obviously, we cannot use conventional
solid walls, because they would melt. However, it is possible to confine a
hot plasma using a magnetic field (fortunately, magnetic field-lines cannot melt). This
technique is
called *magnetic confinement*.
The electric field in confined plasmas is
usually weak (that is,
), so that the
drift
is similar in magnitude to the magnetic and curvature drifts. In this
case, the bounce point condition,
, reduces to

Consider the magnetic field configuration illustrated in Figure 2.1. As indicated, this configuration is most easily produced by two Helmholtz coils. Incidentally, this type of magnetic confinement device is called a

It is clear, from the figure, that the magnetic field-strength on a magnetic field-line situated close to the axis of the device attains a local minimum at , increases symmetrically as increases until reaching a maximum value at about the locations of the two field-coils, and then decreases as is further increased. According to Equation (2.87), any particle that satisfies the inequality

is trapped on such a field-line. In fact, the particle undergoes periodic motion along the field-line between two symmetrically placed (in ) mirror points. The magnetic field-strength at the mirror points is

(2.89) |

On the midplane, and . (From now on, for ease of notation, we shall write .) Thus, the trapping condition, Equation (2.88), reduces to

Particles on the midplane that satisfy this inequality are trapped. On the other hand, particles that do not satisfy the inequality escape along magnetic field-lines. A magnetic mirror machine is incapable of trapping charged particles that are moving parallel, or nearly parallel, to the direction of the magnetic field. In fact, the previous inequality defines a

If plasma is placed inside a magnetic mirror machine then all
of the particles whose velocities lie in the loss cone promptly escape, but the
remaining particles are confined. Unfortunately, that is not
the end of the story. There is no such thing as an absolutely collisionless
plasma. Collisions take place at a low rate, even in very hot plasmas.
One important
effect of collisions is to cause diffusion of particles in velocity space (Hazeltine and Waelbroeck 2004).
Thus, collisions in a mirror machine continuously scatter trapped particles into
the loss cone, giving rise to a slow leakage of plasma out of the device.
Even worse, plasmas whose distribution functions deviate strongly from an isotropic
Maxwellian (for instance, a plasma confined
in a mirror machine) are prone to *velocity-space instabilities* (see Chapter 8) that tend to
relax the distribution function back to a Maxwellian. Such instabilities
can have a disastrous effect on plasma confinement in a mirror machine.