Editing Magnetic mirror (section)

== Mathematical derivation ==
<!-- Deleted image removed: [[File:Gas Dynamic Trap Overhead.jpg|thumbnail|The Gas Dynamic Trap a modern magnetic mirror machine operated in Russia<ref>P.A. Bagryansky et. al., Physical Review Letters 114, 205001 (2015).</ref>]] -->

The mirror effect can be shown mathematically. Assume [[Adiabatic invariant#The first adiabatic invariant, μ|adiabatic invariance of the magnetic moment]], i.e. that the particle's magnetic moment and total energy do not change.<ref>F. Chen, Introduction to Plasma Physics and Controlled Fusion (Plenum, New York, 1984), Vol. 1, pp. 30–34. {{ISBN|978-0-306-41332-2}}</ref> Adiabatic invariance is lost when a particle occupies a null point or zone of no magnetic field.<ref>TG Northrop, "The Adiabatic Motion of Charged Particles" (Interscience, New York, 1963)</ref> The magnetic moment can be expressed as:
<math display="block">\mu=\frac{m v_{\perp}^2}{2 B}</math>

It is assumed that μ will remain constant while the particle moves into the denser magnetic field. Mathematically, for this to happen the velocity perpendicular to the magnetic field <math>v_{\perp}</math> must also rise. Meanwhile the total energy of the particle <math>\mathcal{E}</math> can be expressed as:
<math display="block">\mathcal{E} = q\phi + \tfrac{1}{2} m v_{\parallel}^2 + \tfrac{1}{2} m v_{\perp}^2</math>

In regions with no electric field, if the total energy remains constant then the velocity parallel to the magnetic field must drop. If it can go negative then there is a motion repelling the particle from the dense fields.{{citation needed|date=August 2019}}

=== Mirror ratios ===
Magnetic mirrors themselves have a '''mirror ratio''' this is expressed mathematically as:<ref>"Particle Loss Rates from Electrostatic Wells of Arbitrary Mirror Ratios." Physics of Fluids 28.1 (1985): 352-57. Web. 15.</ref>
<math display="block"> r_\text{mirror} = \frac{B_\text{max}}{B_\text{min}} </math>

At the same time, particles within the mirror have a [[Pitch angle (particle motion)|pitch angle]]. This is the angle between the particles' velocity vector and the magnetic field vector.<ref>Dolan, T. J. "Magnetic Electrostatic Plasma Confinement." Plasma Physics and Controlled Fusion 36 (1994): 1539-593. Print.</ref> Surprisingly, the particles with the small pitch angle can escape the mirror.<ref>G Gibson, Willard C Jordan, Eugene Lauer, Physical Review Letters, 5: 141 (1960)</ref> These particles are said to be in the '''loss cone'''. The reflected particles meet the following criteria:<ref>Principles of Plasma Physics, N Krall, 1973, Page 267</ref>
<math display="block">\frac{v_\perp}{v_\parallel} > \frac{1}{\sqrt{r_\text{mirror}}}</math>

Where <math>v_\perp</math> is the particle velocity perpendicular to the magnetic field and <math>v_\parallel</math> is particle velocity parallel to the magnetic field.

This result was surprising because it was expected that heavier and faster particles, or those with less electric charge, would be harder to reflect. It was also expected that a smaller magnetic field would reflect fewer particles. However, the [[gyroradius]] in those circumstances is also larger, so that the radial component of the magnetic field seen by the particle is also larger. It is true that the minimum volume and magnetic energy is larger for the case of fast particles and weak fields, but the mirror ratio required remains the same.