7.6 Propagation in the Ionosphere and Magnetosphere.

7.6.1

a. For an ionized plasma, the dielectric constant ε = 1 − ω_{p}^{2}/ω^{2} can be derived from the simple model of N free electrons of mass m, charge e. Each electron’s position is described by −mω^{2}x = eE, the current is J = Nev = − iωNex = i (Ne^{2}/mω) E, so that σ = iNe^{2}/mω and ε = 1 + 4πiσ/ω gives the desired result, with ω_{p}^{2} = 4πNe^{2}/m.

b. The relation between k and ω, k^{2} = εω^{2}/c^{2}, gives

k^{2} = \frac{ω^{2} − ω_{p}^{2}}{c^{2}} or ω^{2} = ω_{p}^{2} + k^{2}c^{2}.  (6.1)

Considered as a relation for h̄ω as a function of h̄k, (6.1) gives the energymomentum relation for a new kind of “quasiparticle”, the transverse plasmon.

c. The plasmon dispersion curve in the figure shows once again (see 7.5.7d) that propagation is not possible at frequencies below ω_{p}, which acts as a cut-off frequency. It is also interesting to note that plasmons have the same energymomentum relation as relativistic particles of mass h̄ω_{p}/c^{2}. For kc ≪ ω_{p}, they behave like newtonian particles of this mass. For kc ≫ ω_{p}, they behave like photons.

Plot of ω/ω_{p} vs. x = kc/ω_{p}. The dotted line is ω = kc.

7.6.2

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 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.

b. It is clear on physical grounds that the response will be different if E is parallel to B_{0} than it is for E perpendicular to B_{0}. In fact, for E // B_{0} it is as if there were no B_{0}, while for E ⊥ B_{0} the two circularly polarized components have different response functions, but are independent of each other. Thus the three directions ϵ_{±} = ϵ_{1} ± iϵ_{2} and ϵ_{3}, with ϵ_{1}, ϵ_{2}, ϵ_{3}, mutually orthogonal and ϵ_{3} // B_{0}, diagonalize the dielectric tensor ε_{pq}.

c. In the simple, independent electron model, we have

m\frac{dv}{dt} + \frac{e}{c}B_{0} × v =eE.  (6.2)

If E =E_{0}(ϵ_{1} ± iϵ_{2}) exp(−iωt), we find

v = −\frac{ieE}{m(ω ∓ ω_{B})} ,

v =−ieE /m(ω±ω_{B}), with ω_{B} = |e| B_{0}/mc, the cyclotron frequency. One can understand the frequency shift by changing to a frame rotating with frequency ±ω_{B}. Setting Nev = σE and ε = 1 + 4πiσ/ω, one has

ε_{±} = 1 − \frac{ω_{p}^{2}}{ω(ω ∓ ω_{B})} .  (6.3)

7.6.3

a. For k parallel to B_{0}, the relation between k and ω for left-circularly polarized (+) waves is not too different from the case B_{0} = 0; but for right-circularly polarized (−) waves, the relation

c^{2}k^{2} = ω (ω − \frac{ω_{p}^{2}}{ω − ω_{B}} )  (6.4)

is qualitatively different for ω < ω_{B}, where k^{2} is positive. For ω ≪ ω_{B} and ≪ ω_{p}^{2}/ω_{B}, we have c^{2}k^{2} = ωω_{p}^{2}/ω_{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.

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

c. For ω > ω_{B}, ε_{±} are both negative up to a frequency that is essentially ω_{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 ω. 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 ω_{p} is above ω, a turning point is reached and the wave is reflected.