Breaking of the first adiabatic invariants of charged particles in time-dependent magnetic fields: Computer simulations and theory

The mechanics of the first adiabatic invariant {μ} of nonrelativistic charged particles in time-dependent magnetic inductions B(t) are studied by means of computer simulations and analytic theory. Linear-ramp magnetic-induction profiles B=B₀+({Δ}B/{Δ}t)t are utilized, as well as hyperbolic-tangent ramps and sine half-wave ramps. The change in {μ} that results from an induction change {Δ}B that occurs over a time {Δ}t is quantified for all values of {Δ}B and {Δ}t, as well as for all values of the particle position. It is found that the cases fall into two categories with very different {μ} behavior: cases in which the change in the magnetic induction occurs over a time {Δ}t that is exactly equal to an integer number of gyroperiods (textbook case) or cases in which the change in the induction occurs over a time {Δ}t that is not equal to an integer number of gyroperiods (more general case). In both categories {μ} is an adiabatic invariant, although the conservation of {μ} is much poorer in the latter category. It is pointed out that, in addition to the well-known constraints on {Δ}B and {Δ}t, there is a constraint on the particle's initial position in the magnetic field if the change in the adiabatic invariant is to be kept small.