Mirror Ratio and the Loss Cone

How the ratio of peak to minimum field defines which particles are trapped and which escape a magnetic mirror.

Defining the mirror ratio

The mirror ratio is the ratio of the maximum magnetic field at the throat to the minimum field at the mid-plane:

R = B_max / B_min

It is the single most important parameter of a magnetic mirror. Combined with magnetic-moment conservation, it fixes the fraction of particles that can be confined.

The loss cone in velocity space

Whether a particle is trapped depends on its pitch angle, the angle between its velocity and the field. Reflection requires the pitch angle at the mid-plane to exceed a critical value:

sin^2(theta) > B_min / B_max = 1/R

Particles with pitch angles inside this cone (moving too nearly parallel to the field) are not reflected and stream out the ends. In velocity space this is a cone of loss, hence the term loss cone.

Consequences of an open loss cone

• The confined fraction is 1 - sqrt(1 - 1/R), approaching 1 only as R grows large
• The loss-cone distribution is non-Maxwellian and can drive microinstabilities
• Collisions scatter particles into the loss cone, setting an end-loss rate

Improving confinement

Because a magnetic mirror alone always leaks, real machines raise the effective confinement by increasing R, adding electrostatic potential barriers (tandem mirrors), or using multiple-mirror and sloshing-ion configurations. The loss-cone-driven instabilities are managed by shaping the velocity distribution and providing warm plasma.

Burner relevance

The Kronos burner, a D-3He tandem-mirror generator, uses a high plug field (26.49 T) to create a large mirror ratio and steep loss cone, then plugs the residual loss cone electrostatically. Loss-cone and mirror-ratio analysis directly informs its end-loss and confinement modeling as a design study.