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\markright{{\sf Propagation in the Ionosphere and Magnetosphere}}
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\begin{document}

\section{7.6\quad Propagation in the Ionosphere and Magnetosphere.}

\subsection{7.6.1}

\quad \thinspace \textbf{a.} For an ionized plasma, the dielectric constant $%
\varepsilon =1-\omega _p^2/\omega ^2$ can be derived from the simple model
of $N$ free electrons of mass $m,$ charge $e.$ Each electron's position is
described by $-m\omega ^2\mathbf{x=}$ $e\mathbf{E,}$ the current is $\mathbf{%
J=}$ $Ne\mathbf{v=}-i\omega Ne\mathbf{x=}$ $i\left( Ne^2/m\omega \right) 
\mathbf{E,}$ so that $\sigma =iNe^2\mathbf{/}m\omega $ and $\varepsilon
=1+4\pi i\sigma /\omega $ given the desired result, with $\omega _p^2$ $%
=4\pi Ne^2/m.$

\smallskip\ 

\textbf{b}. The relation between $k$ and $\omega ,$ $k^2$ $=\varepsilon
\omega ^2/c^2$, gives

\begin{equation}
k^2=\frac{\omega ^2-\omega _p^2}{c^2}\,\,\,\,\,\,\,\,\text{or}%
\,\,\,\,\,\,\,\omega ^2=\,\,\omega _p^2+k^2c^2.  \label{1}
\end{equation}
Considered as a relation for $\hbar \omega $ as a function of $\hbar k,$ (%
\ref{1}) gives the energy-momentum relation for a new kind of
``quasiparticle'', the transverse plasmon.

\smallskip\ 

\textbf{c. }The plasmon dispersion curve in the figure shows once again (see
7.5.7 d) that propagation is not possible at frequencies below $\omega _p,$
which acts as a cut-off frequency. It is also interesting to note that
plasmons have the same energy-momentum relation as relativistic particles of
mass $\hbar \omega _p/c^2$. For $kc\ll $ $\omega _p$, they behave like
newtonian particles of this mass.

\newpage 

\subsection{7.6.2}

\quad \thinspace \textbf{a.} Wave propagation in the ionosphere is further
complicated by the presence of the slowly varying magnetic field of the
earth, which for the present purposes will be assumed to have the locally
constant value $\mathbf{B}_0$. Then the medium is anisotropic (see 7.5.1
c(i)), which implies that birefringence occurs at least in certain
directions of propagation.

\smallskip\ 

\textbf{b.} It is clear on physical grounds that the response will be
different if $\mathbf{E}$ is parallel to $\mathbf{B}_0$ than it is for $%
\mathbf{E}$ perpendicular to $\mathbf{B}_0.$ In fact, for $\mathbf{E}$ $//$ $%
\mathbf{B}_0$ it is as if there were no $\mathbf{B}_0$, while for $\mathbf{E}
$ $\perp \mathbf{B}_0$ the two circularly polarized components have
different response functions, but are different of each other. Thus the
three directions $\mathbf{\varepsilon }_{\pm }=\mathbf{\varepsilon }_1\pm i%
\mathbf{\varepsilon }_2$ and $\mathbf{\varepsilon }_3,$ with $\mathbf{%
\varepsilon }_1,\mathbf{\varepsilon }_2,\mathbf{\varepsilon }_3,$ mutually
orthogonal and $\mathbf{\varepsilon }_3$ $//\mathbf{\ B}_0,$ diagonalize the
dielectric tensor $\varepsilon _{pq}.$

\smallskip\ 

\textbf{c}. In the simple, independent electron model, we have

\begin{equation}
m\frac{d\mathbf{v}}{dt}+\frac ec\mathbf{B}_0\times \mathbf{v=eE.}  \label{2}
\end{equation}
If $\mathbf{E=E}_0(\mathbf{\varepsilon }_1\pm i\mathbf{\varepsilon }_2)\exp
(-i\omega t),$ we find $\mathbf{v=}-ie\mathbf{E}$ $/m(\omega \pm \omega _B),$
with $\omega _B=\left| e\right| \mathbf{B}_0/mc,$ the cyclotron frequency.
One can understand the frequency shift by changing to a frame rotating with
frequency $\pm \omega _B$. Setting $Ne\mathbf{v=}$ $\sigma \mathbf{E}$ and $%
\varepsilon =1+4\pi i\sigma /\omega ,$ one has

\begin{equation}
\mathbf{\varepsilon }_{\pm }=1-\omega _p^2/\omega (\omega \pm \omega _B).
\label{3}
\end{equation}

\newpage 

\subsection{7.6.3}

\quad \thinspace \textbf{a.} For $\mathbf{k}$ parallel to $\mathbf{B}_0$,
the relation between $k$ and $\omega $ for left-circularly polarized $\left(
+\right) $ waves is not too different from the case $\mathbf{B}_0=0;$ but
for right-circularly polarized $\left( -\right) $ waves, the relation

\begin{equation}
c^2k^2=\omega \left( \omega -\frac{\omega _p^2}{\omega -\omega _B}\right) 
\label{4}
\end{equation}
is qualitatively different for $\omega <\omega _B,$ where $k^2$ is positive.
For $\omega \ll \omega _B$ and $\ll \omega _p^2/\omega _B,$ we have $%
c^2k^2=\omega \omega _p^2/\omega _B.$ The energy-momentum relation is
quadratic, as for a non-relativistic particle in quantum mechanics. This
wave is called a whistler in radio reception and an helicon in solid state
plasma physics.

\smallskip\ 

\textbf{b. }Whistlers are generated by thunderstorms and travel down the
magnetic field lines to the antipodal point on the surface of the earth.

\smallskip\ 

\textbf{c.} For $\omega >\omega _B,\,\,\,\varepsilon _{\pm }$ are both
negative up to a frequency that is essentially $\omega _p$ modified by the
presence of $B_0.$ If the electron density $N$ varies slowly, $k$ will
change smoothly and remain positive as long as the local plasma frequency is
below $\omega .$ The physics is the same as for a quantum mechanical
particle moving in a slowly varying potential. If the density is such that
the local $\omega _p$ is above $\omega ,$ a turning point is reached and the
wave is reflected.

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