Magnetic Mirror

A magnetic mirror confines plasma between the two ends of a straight magnetic field, each end sealed off by a strong field. Its greatest feature is that it does not need to be bent into a doughnut shape like a tokamak: the structure is straight and simple. On this page we start from the intuition of why particles bounce back at the two ends, then move on to a rigorous understanding based on adiabatic invariants, the escape route known as the loss cone, ways of overcoming instabilities, and the recent reappraisal enabled by high-temperature superconducting coils.

Start with Intuition (High School)

Picture a charged particle (an ion or an electron) traveling through a straight magnetic field, spiraling as it winds around a field line. Where the field lines are sparse and the field is weak, the spiral is wide and gentle. But as the particle approaches an end where the field lines are bunched tightly together and the field grows stronger, the spiral becomes narrower and tighter.

The key point here is that the total energy the particle carries does not change. As it enters the stronger field and its gyration (the motion across the field lines) becomes more vigorous, its forward momentum (the motion along the field lines) is gradually lost. It is just like a ball rolling up a slope, losing speed until it stops and rolls back down. A particle that uses up its forward momentum is reflected there and returns the way it came. This turning point is likened to a mirror and called the “mirror point,” and because the strong fields at both ends play the role of mirrors, the scheme is named the “magnetic mirror.”

This structure, with a mirror at each end, is also called a “magnetic bottle.” The image is that the mouth of the bottle is narrowed by a strong field to trap the particles inside. However, this bottle has a weakness that can never be completely sealed. Particles whose forward momentum is too strong to begin with, that is, particles that barely spiral at all and drive straight ahead, slip through the mouth and escape before the mirror can reflect them. Later we will organize this “group of easily escaping particles” using the term “loss cone.” For now, hold onto the image that both ends are mirrors able to reflect many particles, but that they miss the particles that drive straight ahead with strong momentum.

Understand the Physics (Undergraduate)

The true nature of the bounce-back is a conserved quantity called an adiabatic invariant. When the magnetic field varies slowly in space (barely changing during one gyration of the particle), a quantity called the magnetic moment

$$\mu = \frac{m v_\perp^2}{2 B}$$

is kept nearly constant. Here $m$ is the particle mass, $v_\perp$ is the velocity component perpendicular to the field line, and $B$ is the field strength. That $\mu$ stays constant means that if $B$ increases, $v_\perp^2$ increases by the same proportion.

Meanwhile, since the magnetic field does no work, the total kinetic energy is conserved, so

$$\frac{1}{2} m v_\parallel^2 + \frac{1}{2} m v_\perp^2 = \text{constant}$$

holds, where $v_\parallel$ is the velocity component along the field line. As the particle moves toward the stronger field and $B$ increases, $v_\perp^2$ increases because $\mu$ is constant, and by energy conservation $v_\parallel^2$ decreases correspondingly. The point where $v_\parallel = 0$ is the mirror point, and there the particle is reflected. This is the mathematical explanation of the bounce-back.

Whether a particle is reflected depends on the direction of its velocity in the weak-field region (field $B_\text{min}$), that is, on its pitch angle $\theta$ ($\tan\theta = v_\perp / v_\parallel$). Defining the mirror ratio as

$$R_m = \frac{B_\text{max}}{B_\text{min}}$$

the condition for reflection can be written as

$$\sin^2\theta > \frac{1}{R_m}$$

In other words, only particles whose pitch angle is larger than a certain critical angle (those spiraling strongly enough) are confined, while particles with a pitch angle smaller than the critical angle are not reflected and escape from the ends. The cone-shaped region in velocity space occupied by these escaping particles is called the loss cone. The larger the mirror ratio $R_m$, the narrower the loss cone and the greater the fraction of confined particles, but since $R_m$ cannot be made infinitely large, a certain escape route always remains.

The problem is that even confined particles are not safe. In a plasma, Coulomb collisions gradually change the direction of a particle’s velocity, so a particle that was once outside the loss cone is eventually scattered into it and escapes. This end loss is the essential difficulty of the mirror scheme, and the confinement time is roughly set by the time for scattering by collisions. Raising the mirror ratio to narrow the loss cone improves matters only logarithmically in $R_m$, so achieving the confinement needed for fusion with a simple mirror alone is far from easy.

Deepen the Theory (Graduate)

The magnetic moment $\mu$ is, more precisely, the first adiabatic invariant, proportional to the action integral of the gyromotion $\oint p_\perp \, dl$. “Adiabatic” refers to the condition that the scale length of the spatial variation of the field is much larger than the Larmor radius of the particle, and that its temporal variation is much slower than the gyration period. Where this condition breaks down under strong gradients, the conservation of $\mu$ fails and particles jump across the boundary of the loss cone. The breakdown of $\mu$ conservation at the steep field gradients of the mirror ends is a correction that cannot be ignored when estimating end losses.

To discuss the physics of confinement, we must first pin down macroscopic stability. In a simple mirror, the field lines curve concavely from the high-pressure center, where the field is weaker, outward toward the weaker field. This configuration has the same structure as the Rayleigh-Taylor instability that arises when a light fluid supports a heavy fluid, and it causes the plasma surface to ripple, producing the interchange instability, also known as the flute instability. Stability theory requires that to suppress this, one should create a configuration in which the field grows stronger the farther the plasma bulges outward, that is, a minimum-B configuration in which the field is smallest at the center. The baseball coil and the yin-yang coil are windings that twist the field three-dimensionally to realize this minimum B.

The idea of reducing the end loss itself leads to the tandem mirror. Proposed independently in 1976 by Dimov and by Fowler and Logan, this scheme places smaller plug cells that confine more strongly at each end of a long central cell (the central cell). By accumulating ions in the plug cells to a higher density than the electrons, charge separation creates a hill of positive electrostatic potential, and this potential barrier electrostatically pushes back the ions of the central cell. The key point is to make ion confinement rely not only on the magnetic mirror but also on the electrostatic potential. Furthermore, in the concept of the thermal barrier, the electron density is lowered locally to create a dip in the potential, thermally decoupling the plug-region electrons from the central-cell electrons, so that a high potential barrier can be maintained with little power.

Here it is worth touching on the field-reversed configuration (FRC) as a relative in the same family of straight systems as the mirror. In an FRC, the current flowing in the plasma itself cancels the external field and reverses it inside, creating a compact torus made of closed field lines within a straight vessel. Unlike a mirror, which relies solely on external coils, it generates the closed field lines itself, so the structure of its end losses is different, and it can in principle allow ultra-high beta, with the beta value (the ratio of plasma pressure to magnetic pressure) close to 1. Mirrors and FRCs share the arena of straight open-ended systems while contrasting in whether confinement relies on an external field or on a self-organized closed field, and recent private fusion efforts are exploring such intermediate, hybrid ideas as well.

Research Frontier (PhD)

In the United States, the large device MFTF-B was completed in 1986 only to be canceled, before it was ever operated, because of budget cuts, and magnetic mirror research stagnated for a long time. Since the 2010s, however, a reappraisal (the mirror revival) has been advancing against the backdrop of several technical and academic developments.

A major factor driving this reappraisal is high-field coils using high-temperature superconductor (HTS). Rare-earth REBCO tape conductors are making it possible to generate strong fields, previously difficult, with compact coils, and the prospect of taking a large mirror ratio $R_m$ to narrow the loss cone has come into view. In this trend, WHAM (Wisconsin HTS Axisymmetric Mirror), centered at the University of Wisconsin, has been built and operated as a demonstration device for high-field, axisymmetric mirrors, and the private company Realta Fusion that spun off from it is aiming at power-generation applications. Axisymmetric configurations have simple coil structures and are engineering-friendly, but as they stand they are vulnerable to the interchange instability, so the theory and experiments of boundary conditions that control eddy currents and of stabilization by rotation and kinetic effects (kinetic stabilization) are being actively pursued.

The GDT (Gas Dynamic Trap) at the Budker Institute in Russia is an important experiment that achieved a plasma beta value on the order of 60% in an axisymmetric mirror, showing that a high-beta axisymmetric mirror can be stably established. The lineage of the GDT also connects to the design of next-generation devices and to studies of applications as a neutron source. In Japan, GAMMA 10/PDX at the University of Tsukuba has operated for many years as a representative tandem-mirror device and has been a hub for research on the physics of potential confinement and on the interaction of the plasma with walls and divertors (plasma-material interaction) at open ends.

Among the main current research themes, the following keywords appear frequently in the literature: potential formation in the expander and the outer expansion region, which bears on how to reduce end losses; suppression of the drift cyclotron loss cone (DCLC) instability, a microinstability inherent to loss-cone distributions; vortex confinement and sheared rotation, which provide macroscopic stabilization of axisymmetric configurations; and transport and beta limits in high-beta, high-field operation. It is also argued, as grounds supporting the reappraisal, that because a mirror has open ends it needs no divertor and can exhaust heat and particles straightforwardly, that it is compatible with steady-state operation and direct energy conversion, and that it has applications as a fusion neutron source and as the drive source for a fusion-fission hybrid. Nothing can be stated with certainty, but the present situation is that an old scheme is being reexamined in a new design space under the new toolset of HTS high-field coils.

Check Your Understanding

Q1. Focusing on the exchange of energy, which of the following correctly explains why a particle is reflected in the strong-field region?

A The magnetic field does work on the particle, robbing it of kinetic energy and bringing it to a complete stop B Because the magnetic moment is kept nearly constant, the perpendicular velocity increases on the strong-field side, so the velocity along the field line decreases correspondingly and the particle is reflected at the point where it reaches zero C On the strong-field side the particle's mass increases so that it can no longer move D The strong field reverses the particle's charge, reversing the direction of the force

Q2. How does confinement change when the mirror ratio Rm is increased?

A The loss cone widens and more particles escape B End losses vanish completely and the confinement time becomes infinite C The loss cone narrows and the fraction of confined particles increases, but the improvement is only logarithmic in Rm and an escape route always remains D It has no effect on confinement and only the magnetic field energy increases

Q3. Why is end loss essentially unavoidable in a simple mirror?

A Because Coulomb collisions gradually change the direction of a particle's velocity, so even confined particles are eventually scattered into the loss cone and escape B Because the magnetic field naturally weakens over time C Because the position of the mirror point is fixed outside the device D Because the strong-field coils inevitably absorb a fixed fraction of the particles

Q4. Which is the correct combination of the reason a simple mirror is prone to the interchange (flute) instability and the measure against it?

A It is because the field lines curve concavely on the outside, and the measure is to lower the mirror ratio further B It is because the plasma temperature is too high, and the measure is to cool the fuel C It is because the field lines curve concavely at the center, forming a structure in which a weak field supports high-pressure plasma, and the measure is to create a minimum-B configuration in which the field grows stronger toward the outside D It is because the mass difference between electrons and ions is large, and the measure is to remove the electrons

Q5. How does a tandem mirror suppress end losses?

A It places plug cells at each end of the central cell to accumulate ions, creating a hill of positive electrostatic potential whose potential barrier electrostatically pushes back the ions of the central cell B It cools the central cell to stop Coulomb collisions C It removes the mirror coils at both ends to eliminate the magnetic field D It rotates the entire plasma to confine it by centrifugal force alone

Related Topics

• For the basics of adiabatic invariants and spiral motion, see Motion of Charged Particles.
• Its place among the full range of field configurations such as mirrors, tokamaks, and stellarators is covered in Magnetic Configurations.
• For the trends of private companies pursuing WHAM and FRC, see Private Fusion Ventures.