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Graphic illustration of the dispersion relation solutions; real roots exist at the intersections (circles) of the beam plasma wave (solid) and the lattice phonon branch (dashed) curves (a) , four real roots (b) and (c) , two real roots. The other two roots are complex (instability) if the encircled branches do not intersect. All frequencies normalized to .
Dispersion relation of the unstable, longitudinal (ES), beam plasmon/phonon polaron branch (a) (b) vs for various values of .
Graphic illustration of the roots, at the intersections of the beam plasma wave (solid) and the lattice phonon branch (dashed) curves, for the sixth order electromagnetic (transverse) mode dispersion relation. There are always four real solutions—the two remaining roots are always complex (instability).
Dispersion relation of the unstable, transverse (EM), beam plasmon/phonon polaron branch, (solid) and (dashed) vs for , .
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