The Magnetic Moment as an Adiabatic Invariant
The conserved quantity mu that makes magnetic mirrors work and simplifies kinetic theory.
An approximate conservation law
The magnetic moment of a gyrating particle is the ratio of its perpendicular kinetic energy to the field strength:
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mu = m v_perp^2 / (2B)When the field changes slowly compared to the gyration (the adiabatic condition), mu is conserved to high accuracy. It is called an adiabatic invariant because it is not an exact constant of motion but is preserved as long as conditions vary gradually.
Why it is invariant
The magnetic moment is the action associated with the periodic gyromotion. Classical mechanics guarantees that the action of a periodic motion is conserved under slow changes of the system, which is exactly the situation of a gyrating particle in a slowly varying field.
Consequences
- Conservation of mu produces the mirror force that reflects particles from strong-field regions
- It reduces velocity space from three dimensions to two (v_parallel and mu) in drift and gyrokinetics
- It underlies the second (bounce) and third (drift) adiabatic invariants used in orbit theory
When it breaks
The invariance fails where the field varies on the scale of a gyroradius, for example at a field null or a sharp gradient. There, mu is not conserved and particles can be scattered across the loss cone, a mechanism relevant to end losses and to certain instabilities.
Fusion relevance
Magnetic-moment conservation is the physical basis of mirror confinement, central to the Kronos burner, a D-3He tandem-mirror generator. It also simplifies the kinetic equations solved for tokamaks like the Hyperion breeder, where mu is a coordinate in gyrokinetic and drift-kinetic descriptions.