Adiabatic invariant of a charged particle moving in a magnetic field with a constant gradient

Abstract

This paper presents the calculation of the adiabatic invariant for the motion of a charged particle in a two-dimensional magnetic field with a constant gradient. Magnetic field intensity is equal to zero along the neutral line for this field model. The mathematical expression for the invariant depends upon whether the particle crosses the neutral line. For trajectories that do not cross the neutral line, the adiabatic invariant reduces to the familiar expression for the magnetic moment, μ 0 = v 2 / B, for small values of the magnetic field gradient. The two expressions for the adiabatic invariant can be matched continuously across the change in the type of trajectory. When the magnetic field parameters smoothly change in time, the adiabatic invariant is conserved exponentially well as long as the type of the particle trajectory remains the same. If, however, the trajectory of a particle initially crosses the neutral line but after the magnetic field evolution stops crossing it (or vice versa), the adiabatic invariant is not conserved.