while the particle moves into the denser magnetic field. Mathematically, for this to happen the velocity perpendicular to the magnetic field must also rise. Meanwhile the total energy of the particle can be expressed as: 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. Mirror ratios Magnetic mirrors themselves have a mirror ratio this is expressed mathematically as: At the same time, particles within the mirror have a pitch angle. This is the angle between the particles' velocity vector and the magnetic field vector. Surprisingly, the particles with the small pitch angle can escape the mirror. These particles are said to be in the loss cone. The reflected particles meet the following criteria: Where is the particle velocity perpendicular to the magnetic field and is the particle speed. 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. Adiabatic invariance The properties of magnetic mirrors can be derived using the adiabatic invariance of magnetic flux under changes in magnetic field strength. As the field gets stronger, the velocity increases proportionally to the square root of B, and the kinetic energy is proportional to B. This can be thought of as an effective potential binding the particle. Magnetic bottles A magnetic bottle is two magnetic mirrors placed close together. For example, two parallel coils separated by a small distance, carrying the same current in the same direction will produce a magnetic bottle between them. Unlike the full mirror machine which typically had many large rings of current surrounding the middle of the magnetic field, the bottle typically has just two rings of current. Particles near either end of the bottle experience a magnetic force towards the center of the region; particles with appropriate speeds spiral repeatedly from one end of