Magnetic Mirror Confinement Equations

The invariants and loss-cone conditions that govern how a converging magnetic field traps charged particles.

The mirror force

A charged particle spiraling into a region of stronger magnetic field feels a force pushing it back toward weaker field. This mirror force arises from conservation of the magnetic moment mu = m v_perp^2 / (2B), an adiabatic invariant. As B rises, v_perp must rise to keep mu constant, and since total energy is conserved, v_parallel falls until the particle reflects.

mu = m v_perp^2 / (2B) = constant
(1/2) m v^2 = constant

The mirror ratio and loss cone

A particle is trapped only if it reflects before reaching the strong-field throat. The condition depends on the mirror ratio R = B_max / B_min and the particle's pitch angle. Particles whose velocity vector lies within the loss cone escape:

sin^2(theta_loss) = B_min / B_max = 1/R

A larger mirror ratio gives a smaller loss cone and better confinement, but the loss cone never closes: mirrors leak, which is the central challenge of mirror confinement.

Tandem mirrors and ambipolar plugs

A tandem mirror places high-field plugs at each end of a long central cell. Electrostatic potentials built at the plugs (through ambipolar physics) confine central-cell ions electrostatically, plugging the loss cone that magnetic mirroring alone cannot close. This combines magnetic and electrostatic confinement.

How it is analyzed numerically

• Track guiding centers with mu conservation to map trapped and passing regions
• Solve the bounce-averaged Fokker-Planck equation for loss-cone scattering and end losses
• Balance ambipolar potentials against electron and ion end-loss rates

The Kronos burner is a D-3He tandem-mirror generator with a 26.49 T plug field and 17 T throat, using strong mirror ratios and ambipolar plugging. These equations set its confinement and end-loss physics; the machine is a design and simulation study.