Magnetic Mirrors

Consider the important case in which the electromagnetic fields do not vary in time. It follows that ${\bf E}= -\nabla{\mit\Phi}$, where ${\mit\Phi}$ is the electrostatic potential. Equation (2.64) yields

$$\frac{dK}{dt} = -e\,{\bf U}\cdot\nabla{\mit\Phi} = - \frac{d(e\,{\mit\Phi})}{dt},$$ (2.83)

because $d/dt = \partial/\partial t+{\bf U}\cdot\nabla$. Thus, we obtain

$$\frac{d{\cal E}}{dt} = 0,$$ (2.84)

where

$${\cal E} = K + e\,{\mit\Phi} = \frac{m}{2}\,(U_\parallel^{2} + {\bf v}_E^{2}) +\mu\,B + e\,{\mit\Phi}$$ (2.85)

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 ${\cal E}$ and $\mu$ are constants of the motion, we can rearrange Equation (2.85) to give

$$U_\parallel = \pm \left[(2/m)\,({\cal E} -\mu\,B-e\,{\mit\Phi})-{\bf v}_E^{2}\right]^{1/2}.$$ (2.86)

Thus, charged particles can drift in either direction along magnetic field-lines in regions where ${\cal E} > \mu\,B +e\,{\mit\Phi} + m\,{\bf v}_E^{2}/2$. However, particles are excluded from regions where ${\cal E} < \mu\,B +e\,{\mit\Phi} + m\,{\bf v}_E^{2}/2$ (because they cannot have imaginary parallel velocities). Evidently, charged particles must reverse direction at those points on magnetic field-lines where ${\cal E} = \mu\,B +e\,{\mit\Phi} + m\,{\bf v}_E^{2}/2$. Such points are termed bounce points or mirror points.

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, $E\ll B\,v$), so that the ${\bf E}\times{\bf B}$ drift is similar in magnitude to the magnetic and curvature drifts. In this case, the bounce point condition, $U_\parallel = 0$, reduces to

$${\cal E} = \mu\,B.$$ (2.87)

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 magnetic mirror machine. The magnetic field configuration obviously possesses axial symmetry. Let $z$ be a coordinate that measures distance along the axis of symmetry. Suppose that $z=0$ corresponds to the midplane of the device (that is, halfway between the two field-coils).

Figure 2.1: Schematic cross-section of a magnetic mirror machine employing two Helmholtz coils.

It is clear, from the figure, that the magnetic field-strength $B(z)$ on a magnetic field-line situated close to the axis of the device attains a local minimum $B_{\rm min}$ at $z=0$, increases symmetrically as $\vert z\vert$ increases until reaching a maximum value $B_{\rm max}$ at about the locations of the two field-coils, and then decreases as $\vert z\vert$ is further increased. According to Equation (2.87), any particle that satisfies the inequality

$$\mu> \mu_{\rm trap} = \frac{{\cal E}}{B_{\rm max}}$$ (2.88)

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

$$B_{\rm mirror} = \frac{\mu_{\rm trap}}{\mu}\,B_{\rm max} < B_{\rm max}.$$ (2.89)

On the midplane, $\mu = m\, v_\perp^{2}/(2\, B_{\rm min})$ and ${\cal E} = m\,(v_\parallel^{2} + v_\perp^{2})/2$. (From now on, for ease of notation, we shall write ${\bf v} = v_\parallel\,{\bf b} + {\bf v}_\perp$.) Thus, the trapping condition, Equation (2.88), reduces to

$$\frac{\vert v_\parallel\vert}{\vert v_\perp\vert} < (B_{\rm max}/B_{\rm min} - 1)^{1/2}.$$ (2.90)

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 loss cone in velocity space. (See Figure 2.2.)

Figure 2.2: A loss cone in velocity space. Particles whose velocity vectors lie inside the cone are not reflected by the magnetic field.

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 7) 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.