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The Loss Cone

Source lecture(s): PC368 Lec 8–9

Intuition

A magnetic mirror traps a particle by converting parallel motion into perpendicular motion. But the conversion has a budget: a particle only turns around if it has enough \(v_\perp\) to begin with. Particles moving too nearly along the field escape — and "too nearly along" is a cone in velocity space. Whether a mirror machine confines a plasma depends entirely on how fast collisions refill that cone.

The condition

Two conservation laws do all the work. Energy: \(v_\parallel^2 + v_\perp^2 = v^2\) (magnetic forces do no work). Magnetic moment:

\[\mu = \frac{mv_\perp^2}{2B} = \text{const}\]

At the turning point all the energy is perpendicular: \(v_\perp^2 = v^2\) where \(B = B_{\max}\). Tracking \(\mu\) back to the midplane where \(B = B_0\) and the pitch angle is \(\alpha\) (\(\sin\alpha = v_\perp/v\)):

\[\frac{v^2\sin^2\alpha}{B_0} = \frac{v^2}{B_{\max}} \quad\Longrightarrow\quad \sin^2\alpha_{\rm LC} = \frac{B_0}{B_{\max}} = \frac{1}{R_m}\]

Loss cone

$\(\sin\alpha_{\rm LC} = \frac{1}{\sqrt{R_m}},\qquad R_m \equiv \frac{B_{\max}}{B_0}\)$ Particles with pitch angle \(\alpha < \alpha_{\rm LC}\) at the midplane escape. Everything else bounces.

The devastating feature is what is absent: no \(m\), no \(q\), no \(v\). The loss cone is purely geometric. You cannot escape it by heating the plasma or choosing a different ion — only by raising the mirror ratio, and \(\sin\alpha_{\rm LC} = R_m^{-1/2}\) is a punishing scaling.

Worked example

A mirror machine has \(B_0 = 1\) T at the midplane and \(B_{\max} = 4\) T at the throats. What fraction of an isotropic plasma is lost immediately?

\(R_m = 4\), so \(\sin\alpha_{\rm LC} = 1/2\), \(\alpha_{\rm LC} = 30°\). The escaping solid angle is two cones of half-angle 30°:

\[\frac{\Omega_{\rm LC}}{4\pi} = 1 - \cos\alpha_{\rm LC} = 1 - \cos 30° = 0.134\]

13.4% of the plasma leaves at once — and collisions keep scattering more particles into the cone, so the loss continues. Reaching even 1% loss would need \(R_m \approx 2500\). This, and not engineering, is why mirror machines lost the race to tokamaks.

Where it matters anyway

  • The Van Allen belts. Earth's dipole is a natural mirror; trapped particles bounce between hemispheres in seconds. Particles scattered into the loss cone precipitate into the atmosphere near the poles — that precipitation is the aurora.
  • The South Atlantic Anomaly. The field is weaker there, so the mirror points sit lower; particles reach deeper into the atmosphere, and satellites take a measurable radiation hit.
  • Tokamak trapped particles. The \(1/R\) toroidal field is itself a mirror: particles on the outboard side can be trapped, tracing banana orbits in the poloidal plane. These drive neoclassical transport and the bootstrap current.

Common mistakes

  • Thinking the loss cone empties once and stops. Collisions continuously scatter particles into it. The confinement time is set by the scattering time into the cone, not by the transit time.
  • Trying to fix it with more energy. The condition has no \(v\) in it. A 100 keV ion and a 1 eV ion at the same pitch angle share the same fate.
  • Forgetting the loss cone is defined at a specified location. The pitch angle changes as a particle moves along the field, so "\(\alpha < 30°\)" only means anything once you say where.

Knowledge graph position

Prerequisites: Magnetic mirror, adiabatic invariants. Leads to: mirror confinement scaling, radiation-belt physics, neoclassical transport.

Quiz

Q1 (conceptual). Why is the loss-cone angle independent of particle mass, charge and energy?

Answer

Both conserved quantities — energy and \(\mu\) — scale with \(mv^2\), so those factors cancel when you take their ratio. What survives is purely the field-strength ratio \(B_0/B_{\max}\). The trap is geometric.

Q2 (computational). What mirror ratio confines all but 1% of an isotropic distribution?

Answer

Need \(1 - \cos\alpha_{\rm LC} = 0.01\), so \(\cos\alpha_{\rm LC} = 0.99\), \(\sin\alpha_{\rm LC} = 0.141\), and \(R_m = 1/\sin^2 = 50\). With \(B_0 = 1\) T that means 50 T at the throat — beyond steady-state magnet technology, which is the practical verdict on simple mirrors.

Q3 (MCQ). The aurora occurs because:

  • (a) the solar wind directly reaches the atmosphere at the poles
  • (b) trapped particles scattered into the loss cone precipitate along field lines into the atmosphere
  • (c) the magnetic field accelerates particles to relativistic energies
  • (d) reconnection heats the upper atmosphere
Answer

(b). The belts are a natural mirror; pitch-angle scattering drops particles into the loss cone, they escape along the converging field into the polar atmosphere, and collisional excitation of oxygen and nitrogen does the rest.