Cold-Plasma Dispersion Relation

It is convenient to define a vector

$${\bf n} = \frac{c}{\omega}\,{\bf k}$$ (5.40)

that points in the same direction as the wavevector, ${\bf k}$ , and whose magnitude, $n$ , is the refractive index (i.e., the ratio of the velocity of light in vacuum to the phase-velocity). Equation (5.9) can be rewritten

$${\bf M}\cdot{\bf E} =({\bf n}\cdot{\bf E})\,{\bf n} - n^2\,{\bf E}+ {\bf K}\cdot{\bf E} = {\bf0}.$$ (5.41)

Without loss of generality, we can assume that the equilibrium magnetic field is directed along the $z$ -axis, and that the wavevector, ${\bf k}$ , lies in the $x$ -$z$ plane. Let $\theta$ be the angle subtended between ${\bf k}$ and ${\bf B}_0$ . The eigenmode equation (5.41) can be written

$$\left(\!\begin{array}{ccc} S - n^2\,\cos^2\theta, & -{\rm i}\,D, ... ...right)\left(\! \begin{array}{c} E_x\\ E_y \\ E_z \end{array}\!\right) = {\bf0}.$$ (5.42)

The condition for a nontrivial solution is that the determinant of the square matrix be zero. With the help of the identity

$$S^2 - D^{\,2} \equiv R\,L,$$ (5.43)

we find that (Hazeltine and Waelbroeck 2004)

$${\cal M}(\omega,{\bf k}) \equiv A\,n^4- B\,n^2 + C = 0,$$ (5.44)

where

$$A = S\,\sin^2\theta + P\,\cos^2\theta,$$ (5.45)

$$B = R\,L\,\sin^2\theta + P\,S\,(1+\cos^2\theta),$$ (5.46)

$$C = P\,R\,L.$$ (5.47)

The dispersion relation (5.44) is evidently a quadratic in $n^2$ , with two roots. The solution can be written

$$n^2 = \frac{B\pm F}{2\,A},$$ (5.48)

where

$$F^{\,2} = (B^{\,2}-4\,A\,C) = (R\,L - P\,S)^2\,\sin^4\theta + 4\,P^{\,2} \,D^{\,2}\,\cos^2\theta.$$ (5.49)

Note that $F^{\,2}\geq 0$ . It follows that $n^2$ is always real, which implies that $n$ is either purely real, or purely imaginary. In other words, the cold-plasma dispersion relation describes waves that either propagate without evanescense, or decay without spatial oscillation. The two roots of opposite sign for $n$ , corresponding to a particular root for $n^2$ , simply describe waves of the same type propagating, or decaying, in opposite directions.

The dispersion relation (5.44) can also be written

$$\tan^2\theta = -\frac{P\,(n^2-R)\,(n^2-L)}{(S\,n^2 - R\,L)\,(n^2-P)}.$$ (5.50)

For the special case of wave propagation parallel to the magnetic field (i.e., $\theta=0$ ), the previous expression reduces to

$$P =0,$$ (5.51)

$$n^2 = R,$$ (5.52)

$$n^2 = L.$$ (5.53)

Likewise, for the special case of propagation perpendicular to the field (i.e., $\theta=\pi/2$ ), Equation (5.50) yields

$$n^2 = \frac{R\,L}{S},$$ (5.54)

$$n^2 = P.$$ (5.55)