Motions of charged particles in plasmas

The motion of a single charged particle in magnetic and electric fields B and E is described by the basic non-relativistic equation of motion in which radiation damping is neglected.

With emphasis on vector methods, data are obtained for the drift-free reference case of motion in a circular helix, and then the drift velocity perpendicular to B due to electric field perturbation of the circular case is used to obtain the familiar gravitational and curved magnetic line drifts. Application of the gravitational drift to the plasma physics Rayleigh-Taylor instability is examined critically in the light of information which can be obtained from the macroscopic equation of motion for a plasma in a magnetic field.

For a spatially perturbed magnetic field a guiding centre treatment using the standard orthogonal curvilinear co-ordinate system of differential geometry yields, after time averaging, the first-order Alfvén and curvature drifts, the latter being in agreement with the result obtained less rigorously from the electric drift velocity. The magnetic field geometry is described in terms of an extended set of Frenet-Serret partial derivatives which involve six scalar quantities, five of which are independent when a scalar magnetic potential exists. After time averaging there also exists a first order velocity component of unusual form parallel to B. For the usually encountered case of B irrotational, this component vanishes.

The three equations of motion for the components of the particle velocity are also obtained in terms of this curvilinear co-ordinate system, and when time averaged the equation of motion for the particle velocity component parallel to B enables the spatial adiabatic invariance of a spiralling particle's magnetic moment to be rapidly established.

The case of B dependent on the time, which can lead to radial compression of a plasma, is discussed from drift and non-drift view points, and the difference between Larmor's frequency and the gyrofrequency is emphasized. A simple microscopic treatment of the polarization drift is also included.

Finally the historical and mathematical significance of adiabatic invariance is discussed, with particular reference to the magnetic moment in plasma physics.