The Magnetic Moment Adiabatic Invariant

The conserved ratio of perpendicular energy to field strength that governs magnetic mirroring.

The Invariant

The magnetic moment mu = m v_perp^2 / (2 B) is the first adiabatic invariant of charged-particle motion. It is conserved whenever the magnetic field changes slowly compared with the gyration period and gently over a gyroradius. Unlike energy or momentum, mu is not an exact constant of the motion but an adiabatic invariant, preserved to high accuracy under slow variation.

Magnetic Mirroring

Because mu is conserved and total energy is conserved, as a particle moves into stronger field its perpendicular energy m v_perp^2/2 = mu B must rise, which drains its parallel energy. If the field is strong enough, the parallel velocity reaches zero and the particle is reflected: this is the magnetic mirror. The reflection condition depends only on the pitch angle and the mirror ratio, the ratio of maximum to minimum field.

Loss Cone

Particles whose pitch angle at the low-field point is too small relative to the mirror ratio are not reflected and escape through the high-field throat. In velocity space these form the loss cone. A mirror-confined plasma continuously loses particles into the loss cone, and collisions refill it, setting a fundamental confinement constraint for open magnetic systems.

Relevance

The magnetic moment invariant is the operating principle of both toroidal trapped-particle physics and open mirror machines. The Kronos D-3He burner is a tandem-mirror generator that confines plasma between high-field plugs, with the throat field reaching 17 T and the plug field 26.49 T; mirror confinement and loss-cone control are central to its design. Tandem-mirror end plugs raise the confining potential to reduce end losses. These are design-stage figures for a machine in simulation.