Dispersion Relation for Relativistic Streams of Finite Radius Page: 5 of 14
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DISPERSION RELATION FOR RELATIVISTIC STREAMS
OF FINITE RADIUS
Lawrence Radiation Laboratory, University of California
December 21, 1964
We have obtained a new dispersion relation for longitudinal plasma os-
cillations of a finite plasma, which includes relativistic and retardation effects,
temperature, and streaming. A considerable unification of previous results1-6
has thereby been accomplished. The equations exhibit a discontinuity when
the phase velocity of the plasma wave equals the light velocity c.
The solution of these equations will be useful in the plasma oscillations
of thin beams and will decide whether the two-stream instability can produce
The model considered is that of a cylinder of radius a and infinite length
populated with charged particles and immersed in an infinitely strong magnetic
field, directed along the cylinder axis, so that only longitudinal motion occurs.
The magnetic field thus acts only as a constraint. Other discussions of longi-
tudinal oscillations dispense with this artifice and consider only motion
parallel to some vector kz, which in our case is along the cylinder axis.
Each charged particle stream is assumed to be uniform in density across
the cylinder and may be moving relativistically. The velocity distribution of
each stream is also uniform throughout the cylinder, but is assumed suffi-
ciently narrow so that the beam may be characterized by one value of -y (the
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Landau, R. W. Dispersion Relation for Relativistic Streams of Finite Radius, report, December 21, 1964; Livermore, California. (https://digital.library.unt.edu/ark:/67531/metadc1030342/m1/5/: accessed March 25, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.