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Influence of profile shape on the extraordinary‐mode stability properties of relativistic non‐neutral electron flow in a planar diode with applied magnetic field

### Abstract

The cold‐fluid extraordinary‐mode eigenvalue equation is solved numerically to determine the influence of equilibrium profile shape on the detailed stability properties of relativistic non‐neutral electron flow in a planar diode with cathode located at *x*=0 and anode at *x*=*d*. Stability properties are investigated for the class of equilibrium energy profiles γ_{ b }(*x*) specified by γ_{ b }(*x*)=λ cosh α_{1} *x*+(1−λ) {[1−α^{2} _{2}(*b* ^{2}−*x* ^{2})]^{1} ^{/} ^{2}/[1−α^{2} _{2} *b* ^{2}]^{1/2}} over the interval 0≤*x*≤*b*. Here α_{1} and α_{2} are constants (with α^{2} _{2} *b* ^{2}<1), *x*=*b* is the outer edge of the electron layer, and λ is a constant parameter in the range of 0≤λ≤1. The corresponding equilibrium profiles for *B* _{ z }(*x*), *n* _{ b }(*x*), and *E* _{ x }(*x*) are determined self‐consistently from the steady‐state (∂/∂*t*=0) cold‐fluid‐Maxwell equations. As the parameter λ is varied from unity to zero there is a large change in the equilibrium profile for *n* _{ b }(*x*)/γ_{ b }(*x*), ranging from *n* _{ b }(*x*)/γ_{ b }(*x*)=const over the interval 0≤*x*<*b* when λ=1, to monotonic decreasing profiles for *n* _{ b }(*x*)/γ_{ b }(*x*) when λ<1. The numerical analysis of the extraordinary‐mode eigenvalue equation shows that the detailed stability properties are very sensitive to the shape of the equilibrium profiles. As λ is reduced from unity, and the profile for *n* _{ b }(*x*)/γ_{ b }(*x*) becomes monotonic decreasing, it is found that the instability growth rate Im ω is reduced. Moreover, the more rapid the decrease in *n* _{ b }(*x*)/γ_{ b }(*x*) (i.e., the smaller the value of λ), the more the growth rate is reduced. Indeed, in some parameter regimes, the instability growth rate can be reduced to zero over the range of wavenumber *k* examined numerically.

© 1985 American Institute of Physics

Received 23 October 1984
Accepted 07 January 1985

/content/aip/journal/pof1/28/4/10.1063/1.865042

http://aip.metastore.ingenta.com/content/aip/journal/pof1/28/4/10.1063/1.865042

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/content/aip/journal/pof1/28/4/10.1063/1.865042

1985-04-01

2016-02-13

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