Charged Particle Orbits in Magnetised Plasma

LARMOR ORBITS IN HOMOGENEOUS MAGNETIC FIELD

Consider the motion of a charged particle of mass m and charge Ze (where -e is the electron charge) in a magnetic field B which is constant in space and time. The equation of motion for such a particle has the form

so the vector of particle acceleration lies in the plane perpendicular to B. The velocity of the particle parallel to В does not depend on the magnetic field and remains constant. Clearly, the parallel, K_h and perpendicular, K±, components of the kinetic energy of the particle are constants,

and

as a constant magnetic field cannot change the kinetic energy of particle, K=const.

For finding the velocity perpendicular to B, we introduce the Cartesian coordinate system with its axis z directed along B. so that the components of the velocity along .v and у satisfy in accordance with (2.1):

By introducing a complex value of

one can combine two equations (2.4) and (2.5) to obtain

The solution of (2.7) has the form

which describes a circle orbit of the particle in the plane perpendicular to В rotating with angular frequency

This frequency is called cyclotron frequency or gyro-frequency. For electrons, the value of this frequency is

and for ions with charge Ze and mass m = M = AM_H. it is

Here, Z and A are the charge and mass numbers of the ions, respectively, M_H is the hydrogen ion (proton) mass, and magnetic field is measured in Gauss of the CGS units.

The motion of a charged particle in a homogeneous constant magnetic field is a super-position of the cyclotron rotation (0_B across the magnetic field and free motion along the magnetic field with velocity v,|. The cyclotron frequency depends on the value of the magnetic field, while the radius p of the cyclotron orbit (also called Larmor radius) is determined by the balance between the centrifugal and the magnetic forces,

and depends on the value of the perpendicular velocity of the particle:

For thermal motion of particles in the plane perpendicular to В, the Larmor radii of such particles have a distribution similar to that of the thermal velocities, that is, Maxwell distribution.

By considering the momentum conservation law for cyclotron rotation, that is,

the value of v| / В is conserved, thus, implying a conservation of the magnetic moment of the Larmor orbit

The magnetic moment is in the direction of the unit vector normal to the area covered by the current loop due to the circular motion of a charged particle. In the present case, the direction of//is antiparallel to the direction of B.

so that a plasma is diamagnetic.

The conservation law (2.15) is valid for particle motion in a homogeneous stationary magnetic field. In a more general case of a weakly varying magnetic field in time and space, this conservation law is approximately valid. If the variation rate and inhomogeneity of the magnetic field in time and/or space satisfy

these slow variations of the magnetic field are called adiabatic. The magnetic moment is a nearly conserved quantity for such a system, and is called adiabatic invariant.

DRIFT MOTION OF LARMOR ORBITS IN THE PRESENCE OF EXTERNAL FORCES AND INHOMOGENEOUS MAGNETIC FIELD

If a not very dense plasma is affected by a strong external force F, one could neglect the interaction between the particles and consider plasma as a sum of independently charged particles moving along their own orbits determined by the externally applied forces. The only internal plasma field that could be never neglected is the electric polarisation field resulting from the charge separation and delivering the quasi-neutrality condition. For the motion of a charged particle in the presence of given external forces, one has

Equation (2.19) has a vector form and can be solved analytically in only some simple cases. In general cases, an approximate technique called drift approach is applied to describe the solutions of (2.19). The drift approach is valid for plasmas in the presence of external forces strong enough for neglecting the interaction between the particles of the plasma. Under these conditions, the motion of a charged particle could be decomposed into three contributions:

• 1. Fast cyclotron rotation in the plane perpendicular to В with gyro-frequency (2.9),
• 2. Drift motion of the Larmor circle across B,
• 3. Particle motion along В not affected by the Lorentz force.

The motion of the Larmor circle of the particle forms a so-called guiding centre trajectory kinetic or guiding centre approach to describing the particle orbits (Figure 2.1).

We represent the external force as a sum of the forces along the magnetic field and perpendicular to it:

so that (2.19) has the following projections in the Cartesian coordinate system:

FIGURE 2.1 Relationship between guiding centre trajectory and actual trajectory.

Here, Eq. (2.23) for the parallel particle motion describes acceleration/deceleration of the particle along В depending on the directivity of F_h and a free motion of the particle along В if F_n = 0. This equation is decoupled from the equations for the particle motion across В and corresponds to the above introduced contribution (3) of the particle motion.

Equations (2.21) and (2.22) describing the motion of the charged particle across В could be represented via the complex variable v* = v_x + iv_y as follows:

The solution of (2.24) is a sum of the general solution of the homogeneous equation and the particular solution of the inhomogeneous equation,

provided the force F_± is constant in both time and space. One can see that (2.25) consists of a fast cyclotron rotation and drift velocity which is determined by the force applied. In the vector form, the drift velocity is