The Plasma Dielectric Tensor

The frequency-dependent response tensor that encodes how a magnetized plasma polarizes and conducts under an oscillating field.

Definition

For a wave of frequency omega the plasma responds to the electric field through a current, and this response is packaged into a dielectric tensor epsilon(omega, k). The wave equation becomes a matrix eigenvalue problem: n x (n x E) + epsilon . E = 0, where n = c k / omega is the refractive index vector. Nontrivial solutions require the determinant to vanish, which is the dispersion relation.

Cold-Plasma Form

In the cold-plasma limit with B along z, the tensor has the Stix form with diagonal element S, off-diagonal +/- iD in the plane perpendicular to B, and P along B. These are built from the plasma frequencies and cyclotron frequencies of each species: P = 1 - sum omega_ps^2/omega^2, and S and D contain omega/(omega +/- omega_cs). Setting the determinant to zero yields the ordinary, extraordinary, R, and L wave branches.

Hot-Plasma Extension

When thermal motion matters, the tensor is obtained from the linearized Vlasov equation and involves the plasma dispersion function and sums over cyclotron harmonics. This warm form captures cyclotron absorption, Landau damping, and finite-Larmor-radius corrections, which are essential for radio-frequency heating and current-drive calculations.

Application

Resonances (where a diagonal element diverges) and cutoffs (where the refractive index goes to zero) determine where launched waves are absorbed or reflected. Designing electron-cyclotron or ion-cyclotron heating for the Hyperion breeder, or evaluating wave accessibility to the high-field burner plug, both rest on the dielectric tensor. These are design computations for machines not yet built.