Magnetic Mirrors

Consider the important case in which the electromagnetic fields do not vary in time. It immediately follows from Eq. (99) that

$$\frac{d{\cal E}}{dt} =0,$$ (114)

where

$${\cal E} = K + e\,\phi = \frac{m}{2}\,(U_\parallel^{~2} + {\bf v}_E^{~2}) +\mu\,B + e\,\phi$$ (115)

is the total particle energy, and $\phi$ is the electrostatic potential. Not surprisingly, a charged particle neither gains nor loses energy as it moves around in non-time-varying electromagnetic fields. Since both ${\cal E}$ and $\mu$ are constants of the motion, we can rearrange Eq. (115) to give

$$U_\parallel = \pm \sqrt{(2/m)[{\cal E} -\mu\,B-e\,\phi]-{\bf v}_E^{~2}}.$$ (116)

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

Let us now consider how we might construct a device to confine a collisionless (i.e., very hot) 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 fields do not melt!): this technique is called magnetic confinement. The electric field in confined plasmas is usually weak (i.e., $E\ll B\,v$), 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

$${\cal E} = \mu\,B.$$ (117)

Consider the magnetic field configuration shown in Fig. 1. This is most easily produced using 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 which measures distance along the axis of symmetry. Suppose that $z=0$ corresponds to the mid-plane of the device (i.e., halfway between the two field-coils).

Figure 1: Motion of a trapped particle in a mirror machine.

It is clear from Fig. 1 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 location of the two field-coils, and then decreases as $\vert z\vert$ is further increased. According to Eq. (117), any particle which satisfies the inequality

$$\mu> \mu_{\rm trap} = \frac{{\cal E}}{B_{\rm max}}$$ (118)

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

$$B_{\rm mirror} = \frac{\mu_{\rm trap}}{\mu}\,B_{\rm max} < B_{\rm max}.$$ (119)

Now, on the mid-plane $\mu = m\, v_\perp^{~2}/2\, B_{\rm min}$ and ${\cal E} = m\,(v_\parallel^{~2} + v_\perp^{~2})/2$. (n.b. From now on, we shall write ${\bf v} = v_\parallel\,{\bf b} + {\bf v}_\perp$, for ease of notation.) Thus, the trapping condition (118) reduces to

$$\frac{\vert v_\parallel\vert}{\vert v_\perp\vert} < (B_{\rm max}/B_{\rm min} - 1)^{1/2}.$$ (120)

Particles on the mid-plane which satisfy this inequality are trapped: particles which do not satisfy this inequality escape along magnetic field-lines. Clearly, a magnetic mirror machine is incapable of trapping charged particles which are moving parallel, or nearly parallel, to the direction of the magnetic field. In fact, the above inequality defines a loss cone in velocity space--see Fig. 2.

Figure 2: Loss cone in velocity space. The particles lying inside the cone are not reflected by the magnetic field.

It is clear that 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. Thus, in a mirror machine collisions 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 (e.g., a plasma confined in a mirror machine) are prone to velocity space instabilities, which tend to relax the distribution function back to a Maxwellian. Clearly, such instabilities are likely to have a disastrous effect on plasma confinement in a mirror machine. For these reasons, magnetic mirror machines are not particularly successful plasma confinement devices, and attempts to achieve nuclear fusion using this type of device have mostly been abandoned.