Magnetic mirror and pitch angle of plasmas

Question

I've been learning about magnetic mirrors and how the pitch angle determines whether a particle will be trapped in it or not. I've also learnt about how, when a plasma is trapped, collisions between the electrons and ions in the plasma change their pitch angles, meaning that some are able to scatter out of the mirror.

But I'm confused because I thought that the pitch angle of a particle changed depending on the position of the particle in its gyro-orbit about the magnetic field. So does the pitch angle in the first two sentences mean the pitch angle of the particle at the point of going into the stronger field in the magnetic mirror, or does it mean something else?

I ask because this hasn't been specified anywhere that I've seen, and instead the information I've read makes it sound like pitch angle is a constant property of a specific particle's gyro-orbit (like in the first paragraph), so I want to make sure I'm not misunderstanding the pitch angle.

Answer 1

In many problems, such as magnetic mirrors, the change in magnetic field over the gyro-orbit of a particle is negligible. Therefore, on this timescale, the particle can be considered to follow a helical trajectory with constant pitch angle, as in the uniform magnetic field. However, for non-uniform fields, pitch angle does change along a particle's trajectory over longer timescales. It depends on the strength of the magnetic field at its location and it is not a constant of motion.

Making the above assumption and neglecting collisions, kinetic energy $KE$ and magnetic moment $\mu$ (first adiabatic invariant) are constants of motion for a charged particle in a magnetic field. These can be expressed as: $$ KE = \frac{m v_{\parallel}^2}{2} + \frac{m v_{\perp}^2}{2} $$ $$ \mu = \frac{m v_{\perp}^2}{2 B} $$ where $v_{\parallel}$ and $v_{\perp}$ are respectively the parallel and perpendicular velocity components and $B$ is the magnetic field strength.

As $B$ changes, $v_{\perp}$ must change for $\mu$ to be constant. Therefore $v_{\parallel}$ must also change for $KE$ to be constant. Thus the pitch angle changes. The mirror point occurs where $v_{\parallel} = 0$. This corresponds to the maximum value of $v_{\perp}$ the particle can have given its kinetic energy, and therefore the maximum value of $B$ that it can reach.