The cold-plasma dispersion relation

It is convenient to define a vector

$${\bf n} = \frac{{\bf k}\,c}{\omega},$$ (464)

which points in the same direction as the wave-vector, ${\bf k}$, and whose magnitude $n$ is the refractive index (i.e., the ratio of the velocity of light in vacuum to the phase-velocity). Note that $n$ should not be confused with the particle density. Equation (434) can be rewritten

$${\bf n} \times({\bf n}\times {\bf E}) + {\bf K}\!\cdot\!{\bf E} = 0.$$ (465)

We may, without loss of generality, assume that the equilibrium magnetic field is directed along the $z$-axis, and that the wave-vector, ${\bf k}$, lies in the $xz$-plane. Let $\theta$ be the angle subtended between ${\bf k}$ and ${\bf B}_0$. The eigenmode equation (465) can be written

$$\left(\!\begin{array}{ccc} S - n^2\,\cos^2\theta & -{\rm i}\... ...egin{array}{c} E_x\\ E_y \\ E_z \end{array}\!\right) = {\bf0}.$$ (466)

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

$$S^2 - D^2 \equiv R\,L,$$ (467)

we find that

$${\cal M}(\omega,{\bf k}) \equiv A\,n^4- B\,n^2 + C = 0,$$ (468)

where

$$A = S\,\sin^2\theta + P\,\cos^2\theta,$$ (469)

$$B = R\,L\,\sin^2\theta + P\,S\,(1+\cos^2\theta),$$ (470)

$$C = P\,R\,L.$$ (471)

The dispersion relation (468) is evidently a quadratic in $n^2$, with two roots. The solution can be written

$$n^2 = \frac{B\pm F}{2\,A},$$ (472)

where

$$F^2 = (R\,L - P\,S)^2\,\sin^4\theta + 4\,P^2 \,D^2\,\cos^2\theta.$$ (473)

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 which either propagate without evanescense, or decay without spatial oscillation. Note that the two roots of opposite sign for $n$, corresponding to a root for $n^2$, simply describe waves of the same type propagating, or decaying, in opposite directions.

The dispersion relation (468) can also be written

$$\tan^2\theta = -\frac{P\,(n^2-R)\,(n^2-L)}{(S\,n^2 - R\,L)\,(n^2-P)}.$$ (474)

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

$$P = 0,$$ (475)

$$n^2 = R,$$ (476)

$$n^2 = L.$$ (477)

Likewise, for the special case of propagation perpendicular to the field (i.e., $\theta=\pi/2$), Eq. (474) yields

$$n^2 = \frac{R\,L}{S},$$ (478)

$$n^2 = P.$$ (479)