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# Alfven waves consider a solid with an equal concentration n …

Alfven waves consider a solid with an equal concentration n of electrons of mass m, and holes of mass m . This situation may arise in a semimetal or in a compensated semiconductor. Place the solid in a uniform magnetic field B = B . Introduce the coordinate ζ = x + iy appropriate for circularly polarized motion, with ζ having time dependence e . Let w = eB/m and w = eB/m c. (a) In CGS units, show that ζ = eE /mℓw(w + wℓ); ζ = – eE /rn w(w – w ) are the displacements of the electrons and holes in the electric field E e = (E + iE ) e . (b) Show that the dielectric polarization P+ = ne(ζh – ζe) in the regime w << wℓ, wh may be written as P = nc (m + m )E /B , and the dielectric function ε(w) = ε + 4πP /E = ε + 4πc p/B , where ε , is the dielectric constant of the host lattice and p = n(m + m ) is the mass density of the carriers. If ε may be neglected, the dispersion relation w ε (w) = c K becomes, fur electromagnetic waves propagating in the z direction, w = (B /4πp)K . Such waves are known m Alfven waves; they propagate with the constant velocity B/ (4πp) . If B = 10kG; n = 10 cm ; m = 10 g, the velocity is ~10 cm s . Alfven waves have been observed in semi-metals and in electron-hole drops in germanium.

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