Magnetic Mirror Bottle

What is modelled

A charged particle in a slowly changing magnetic field keeps its magnetic moment, which is its perpendicular kinetic energy divided by the field. When the particle moves into a stronger field its perpendicular energy must grow, so its parallel energy shrinks, and if the field is strong enough the particle stops and turns back. A particle with a small pitch angle never meets a field that strong and leaves through the end: that is the loss cone. In a parabolic field well the turning motion is a harmonic oscillation whose period does not depend on its amplitude.

Equations

• Field B(z) = B0 (1 + κ z2) with κ = 4 (R − 1) / L2.
• Invariant μ = m v⊥2 / 2B, so v∥ = v √[1 − sin2α B(z) / B0].
• Loss cone sin2αc = 1 / R, trapped share of an isotropic gas √(1 − 1/R).
• Turning point zt = √[(1/sin2α − 1) / κ], bounce period 2π / (v sinα √κ).

Checks run in the page

• Boris full-orbit bounce period against the harmonic result.
• Boris turning point against the field condition.
• Magnetic moment and kinetic energy along the full orbit.
• Isotropic sampling against the loss-cone share.

Honest simplifications

The field is an idealised parabolic well on the axis with its radial part from the divergence-free condition, the ion is a single hydrogen ion with no collisions, no electric fields and no instabilities, and the picture is a sketch of the helix, not the exact orbit. Real mirror machines lose particles by collisions and drifts as well.

• Press Just inside the loss cone and watch the ion escape.
• Press Strong mirror to shrink the loss cone.
• Press Hot ion and read the longer gyroradius and bounce period.