The diocotron (or slipping stream) instability of low density (ωp « ωc) electron beams in crossed fields is considered for a cylindrical geometry. For a simple density distribution, the normal modes of the electron beam correspond to a continuum of eigenvalues, plus two discrete eigenvalues. Work due to Case and Dikii appears to show that the continuous spectrum is not important in stability studies of this type. The condition for stability considering the discrete modes only is derived; under suitable geometrical and electrical conditions, it is shown that these modes can be stable. The analogy between the electromagnetic problem considered here and the problem of the stability of an ideal rotating fluid is discussed. It is shown that stability conditions derived for the latter problem depend on the possibility of axial perturbations; what this implies for the electron beam problem is briefly discussed.

Topics
Plasma instabilities, Vortex dynamics
1.
G. C.
MacFarlane
 and
H. G.
Hay
,
Proc. Roy. Soc. (London)
63B
,
409
(
1953
).
Google Scholar
 
2.
O. Buneman, in Crossed Field Microwave Devices, edited by E. Okress (Academic Press Inc., New York 1961), Chap. 5.
3.
R. W. Gould, Electron Tube and Microwave Laboratory, California Institute of Technology, Technical Report No. 3 (1955).
4.
B.
Epsztein
,
Compt. Rend.
240
,
408
(
1955
).
Google Scholar
 
5.
W. Knauer, Hughes Research Laboratory, Research Report 332 (1960).
6.
K. M.
Case
,
Phys. Fluids
3
,
143
(
1960
).
Google Scholar
Crossref
Search ADS
 
7.
K. M.
Case
,
Phys. Fluids
3
,
149
(
1960
).
Google Scholar
Crossref
Search ADS
 
8.
L. A.
Dikii
,
Dokl. Akad. Nauk SSSR
135
,
1068
(
1960
)
Google Scholar
 
[English transl.:
L. A.
Dikii
,
Soviet Phys.‐Doklady
5
,
1179
(
1960
)].
Google Scholar
 
9.
S.
Goldstein
,
Proc. Roy. Soc. (London)
A132
,
524
(
1931
).
Google Scholar
 
10.
The mode l = 0 has no nontrivial solution. This can be seen as follows: Eq. (13) shows that for this mode dφ/dr as well as φ is continuous at r = b,c. The eigenfunction φ = A+B In r is therefore valid in all three regions. If φ(a) = φ(d) = 0,A = B = 0.
11.
S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Clarendon Press, Oxford, 1961).
12.
C. C. Lin, The Theory of Hydrodynamic Stability (Cambridge University Press, London, 1955).
13.
Lord Rayleigh, Scientific Papers (Cambridge University Press, London, 1920), Vol. VI, p. 447.
14.
O.
Buneman
,
J. Electron.
3
,
1
(
1957
).
Google Scholar
 
15.
J.
Feinstein
 and
G. S.
Kino
,
Proc. I. R. E.
45
,
10
(
1957
).
Google Scholar
Crossref
Search ADS
 
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© 1965 The American Institute of Physics.
1965
The American Institute of Physics
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