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Next: Polarization Up: Waves in Cold Plasmas Previous: Cold-Plasma Dielectric Permittivity

Cold-Plasma Dispersion Relation

It is convenient to define a vector
\begin{displaymath}
{\bf n} = \frac{{\bf k}\,c}{\omega},
\end{displaymath} (488)

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 (458) can be rewritten
\begin{displaymath}
{\bf M}\cdot{\bf E} =({\bf n}\cdot{\bf E})\,{\bf n} - n^2\, {\bf K}\!\cdot\!{\bf E} = {\bf0}.
\end{displaymath} (489)

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 (489) can be written

\begin{displaymath}
\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}.
\end{displaymath} (490)

The condition for a nontrivial solution is that the determinant of the square matrix be zero. With the help of the identity
\begin{displaymath}
S^2 - D^2 \equiv R\,L,
\end{displaymath} (491)

we find that
\begin{displaymath}
{\cal M}(\omega,{\bf k}) \equiv A\,n^4- B\,n^2 + C = 0,
\end{displaymath} (492)

where
$\displaystyle A$ $\textstyle =$ $\displaystyle S\,\sin^2\theta + P\,\cos^2\theta,$ (493)
$\displaystyle B$ $\textstyle =$ $\displaystyle R\,L\,\sin^2\theta + P\,S\,(1+\cos^2\theta),$ (494)
$\displaystyle C$ $\textstyle =$ $\displaystyle P\,R\,L.$ (495)

The dispersion relation (492) is evidently a quadratic in $n^2$, with two roots. The solution can be written

\begin{displaymath}
n^2 = \frac{B\pm F}{2\,A},
\end{displaymath} (496)

where
\begin{displaymath}
F^2 = (R\,L - P\,S)^2\,\sin^4\theta + 4\,P^2 \,D^2\,\cos^2\theta.
\end{displaymath} (497)

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. 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 (492) can also be written

\begin{displaymath}
\tan^2\theta = -\frac{P\,(n^2-R)\,(n^2-L)}{(S\,n^2 - R\,L)\,(n^2-P)}.
\end{displaymath} (498)

For the special case of wave propagation parallel to the magnetic field (i.e., $\theta=0$), the above expression reduces to
$\displaystyle P$ $\textstyle =$ $\displaystyle 0,$ (499)
$\displaystyle n^2$ $\textstyle =$ $\displaystyle R,$ (500)
$\displaystyle n^2$ $\textstyle =$ $\displaystyle L.$ (501)

Likewise, for the special case of propagation perpendicular to the field (i.e., $\theta=\pi/2$), Eq. (498) yields
$\displaystyle n^2$ $\textstyle =$ $\displaystyle \frac{R\,L}{S},$ (502)
$\displaystyle n^2$ $\textstyle =$ $\displaystyle P.$ (503)


next up previous
Next: Polarization Up: Waves in Cold Plasmas Previous: Cold-Plasma Dielectric Permittivity
Richard Fitzpatrick 2011-03-31