Magnetic mirror - 2D loss-cone view

Mirror force and the magnetic moment

In a field that grows along the axis a gyrating particle feels F_∥ = -μ dB/dz with magnetic moment μ = m v_⊥² / (2B). While the field changes slowly over one gyro-orbit (ρ << L) the moment is an adiabatic invariant, so v_⊥²/B stays constant and, with energy conservation, the particle turns round where B = B₀ / sin²θ₀, with θ₀ the pitch angle at the centre.

Loss cone

If the mirror throat has B_max = R_m B₀, a particle escapes when sin²θ₀ < 1/R_m. For an isotropic distribution the leaked fraction is 1 - √(1 - 1/R_m), so a mirror ratio of 3 leaks about 18 % and a ratio of 10 about 5 %. Collisions scatter the pitch angle and keep refilling the cone, so a real mirror leaks continuously.

The code integrates the full 3D Lorentz force with the Boris scheme in a parabolic on-axis profile B(z) = B₀[1 + (R_m - 1)(z/L)²], with B_r = -(r/2) dB/dz from div B = 0. Units are normalised (q B₀/m = 1, speed = 1).

Honest simplifications: test particles only, with no plasma self-field, electric potential, or radial drifts of a real device; a particle is counted as lost when it crosses |z| = L; real machines also lose particles through drifts and instabilities, and the 3D page shows the apparatus as an illustrative reconstruction.

• Set the tracked pitch angle just below and just above the cone edge (the Cone edge and Just trapped presets) and see one particle escape and the other reflect.
• Raise the mirror ratio and watch the live leaked fraction follow the theory curve down.
• Turn on collisions: the loss cone refills and the confined number decays over time.
• Enlarge the launch radius to see finite-gyroradius corrections to the invariant.