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KINETIC DESCRIPTION OF COUPLED TRANSVERSE OSCILLATIONS IN AN INTENSE RELATIVISTIC ELECTRON BEAM-PLASMA SYSTEM
Uhm
Han S. and Ronald C.
JUNE,
Davidson
1979
PFC/JA-79-9
KINETIC DESCRIPTION OF COUPLED TRANSVERSE OSCILLATIONS IN AN INTENSE RELATIVISTIC ELECTRON BEAM-PLASMA SYSTEM Han S. Uhm Naval Surface Weapons Center
White Oak, Silver Spring, Maryland 20910 Ronald C. Davidson Plasma Fusion Center Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
The stability properties of an intense relativistic electron beam
propagating through a collisionless background plasma are investigated within the framework of the Vlasov-Maxwell equations.
Vi /y
It is assumed that
4), the system can be most unstable for parameter values other than (f , k 2c/wb)=(OO) [Figs. 3(b) and 3(c)]. The dependence of stability properties on conducting wall location is illustrated in Fig. 4 where Imw/wcb and Rew/wcb are plotted versus R /R
for k=l, f e=0 , k Abc/wcb=O, and parameters otherwise identical
to Fig. 3.
It is evident from Fig. 4 that the k=l mode can be completely
stabilized by increasing the value of R /Rc beyond some critical value. For example, in Fig. 4, the k=l mode is stable for R /Rc in the range
20 0.59 < R /R p
2, the influence of the conducting wall radius on stability behavior is found to be negligibly small. Moreover, for Z=1, the dependence of stability properties on other system parameters is summarized in Ref. 10. We conclude this section by summarizing the following points. First, for sufficiently low beam density satisfying wpb
0.5, the cbb2Il05,th
higher harmonic perturbations (2 > 2) are more unstable than the k=1 mode.
Moreover, generally speaking, the growth rates for k > 2 are
larger than for k=1. (fekzabc/wcb)=(Oo).
Second, for k < 3, the system is most unstable for Third, the Z=1 mode is completely
stabilized if the conducting wall is sufficiently close to the plasma boundary.
However, the influence of the conducting wall on
stability behavior for X > 2 is negligibly small.
21 V.
ION RESONANCE INSTABILITY
The ion resonance instability12 is one of the fundamental instabilities that can occur in a relativistic beam-plasma system with both ion and electron components.
Previous investigations of this instability for a
relativistic electron component
10
have been carried out only for the
In this section, we investigate ion resonance
fundamental mode (k=l).
stability properties for higher k values within the context of the kinetic dispersion relation (57).
To make the analysis tractable, we
assume that the plasma electron density is equal to zero,
(74)
i =0 .
e
In this case, the dispersion relation in Eq. (57) can be expressed as
(1gf-
)w
(-a
2)bR
rb(w)
-2 2 Zv
2
b2
2v
v1 2
(2 - A
R2
pb p
b W
'
gf-
2
= W
2v
2 b
.
2
i
(75)
As a comparison with previous analyses,10,12 we first consider the 2=l mode, neglecting the effects of the conducting wall (R /Rc-+0) and substituting Eq. (66) into Eq. (75).
We obtain
2 W(w-k 2 bf e(l1ib) )_. 2pb bC)7_cbo [(w-k A)b kz bc) [( *kzb
x
((-k0
C) 2l+W
z c i
1 f
(w,k01c)-1 (be
= e k
from Eq. (75).
z 1
2
Ybme
4
.
pb
1
b)2
pb'%a)
mi
(76)
2 abb
'
Equation (76) is identical to the result obtained by
Davidson and Uhm10 within the framework of a rigid beam model.
In
22
obtaining Eq. (76), use has been made of the definition of fractional charge neutralization f
Throughout this section, we assume
in Eq. (63).
that the ion motion is nonrelativistic (i.e.,
II
2 are the most
Moreover, the fundamental mode (Z=l) is completely
stabilized whenever the conducting wall is sufficiently close to the The ion resonance instability was investigated in
plasma boundary.
Sec. V, including the interaction of the beam electrons with the plasma ions.
From the numerical analysis of the dispersion relation, it is found
that the instability domain extends to large values of axial wavenumber for high harmonic numbers X.
Furthermore, the maximum growth rate of the
ion resonance instability increases slowly with increasing azimuthal harmonic number k.
26
ACKNOWLEDGEMENTS
This research was supported in part by the National Science Foundation and in part by the Office of Naval Research under the auspices of a Joint Program with the Naval Research Laboratory.
The research by one of
the authors (H.S.U.) was supported in part by the Independent Research Fund at the Naval Surface Weapons Center.
27
REFERENCES
1.
W. W. Destler, H. S. Uhm, H. Kim and M. P. Reiser, J. Appl. Phys. 50_, in press (1979).
2.
V. L. Granatstein, P. Sprangle, R. R. Parker, H. Herndon and S. P. Schlesinger, J. Appl. Phys. 46, 3800 (1975).
3.
E. J. Lauer, R. J. Briggs, T, J. Fessenden, R. E. Hester and E. P.. Lee, Phys. Fluids 21, 1344 (1978).
4.
C. A. Kapetanakos, Appl. Phys. Lett. 25, 481 (1974).
5.
M. N. Rosenbluth, Phys. Fluids 3, 932 (1960).
6.
D. A. Hammer and N. Rostoker, Phys. Fluids 13, 1931 (1970).
7.
E. P. Lee, Phys. Fluids 21, 1327 (1978).
8.
R. C. Davidson and H. S. Uhm, J. Appl. Phys. 50, 696 (1979).
9.
R. C. Davidson and H. S. Uhm, Fluids 22, in press (1979).
10.
R. C. Davidson and H. S. Uhm, "Coupled Dipole Oscillations in an Intense Relativistic Electron Beam," submitted for publication (1979).
11.
R. C. Davidson and B. Hui, Annals of Physics 94, 209 (1975).
12.
R. C. Davidson and H. S. Uhm, Phys. Fluids 21, 60 (1978).
13.
R. C. Davidson and H. S. Uhm, Phys. Fluids 20, 1938 (1977).
28
FIGURE CAPTIONS
Fig. 1
Equilibrium configuration and coordinate system.
Fig. 2
Stability domain [Eq. (69)] in the parameter space
W cbb) for yb=5, (fi e/b b- pb b'
f e=0,
R /R =0.5, kz
p
c
z bC/Wb=0, WcbO
and
an
(a) 2=1, (b) .=3, and (c) 2=5. Fig. 3
Stability boundaries [Eq. (69)] in the parameter space (f , kzb c/Wcb) for yb=5,
R /Rc=0.5,
2e
cbw'
b=0.5,
and
(a) k=1,2,3, (b) Z=4, and (c) X=5. Fig. 4
Plots of (a) normalized growth rate Imo/&cb and (b) normalized real oscillation frequency Rew/wb versus Rb/Rc [Eq.
(69)] for k=,
f e=0 , k zb c/W cb=0, and parameters otherwise identical to Fig. 3. Fig. 5
Stability domain [Eq. (78)] in the parameter space (f , kz bc/Wcb) for
Fig. 6
b=5,
w 2 b/W2 b1, R /RC=0.5, and (a) Z=1, (b) Z=2, and (c) Z=3.
Plots of (a) normalized growth rate Imw/ cb and (b) normalized real oscillation frequency Rew/wcb versus kz b c/Wcb for f =0.3, k=1,3, and 5, and parameters otherwise identical to Fig. 5.
29
N
L-
Q)
0
N < (1) 1 054
(9
z
C) D
Uhm
Han S. and Ronald C.
JUNE,
Davidson
1979
PFC/JA-79-9
KINETIC DESCRIPTION OF COUPLED TRANSVERSE OSCILLATIONS IN AN INTENSE RELATIVISTIC ELECTRON BEAM-PLASMA SYSTEM Han S. Uhm Naval Surface Weapons Center
White Oak, Silver Spring, Maryland 20910 Ronald C. Davidson Plasma Fusion Center Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
The stability properties of an intense relativistic electron beam
propagating through a collisionless background plasma are investigated within the framework of the Vlasov-Maxwell equations.
Vi /y
It is assumed that
4), the system can be most unstable for parameter values other than (f , k 2c/wb)=(OO) [Figs. 3(b) and 3(c)]. The dependence of stability properties on conducting wall location is illustrated in Fig. 4 where Imw/wcb and Rew/wcb are plotted versus R /R
for k=l, f e=0 , k Abc/wcb=O, and parameters otherwise identical
to Fig. 3.
It is evident from Fig. 4 that the k=l mode can be completely
stabilized by increasing the value of R /Rc beyond some critical value. For example, in Fig. 4, the k=l mode is stable for R /Rc in the range
20 0.59 < R /R p
2, the influence of the conducting wall radius on stability behavior is found to be negligibly small. Moreover, for Z=1, the dependence of stability properties on other system parameters is summarized in Ref. 10. We conclude this section by summarizing the following points. First, for sufficiently low beam density satisfying wpb
0.5, the cbb2Il05,th
higher harmonic perturbations (2 > 2) are more unstable than the k=1 mode.
Moreover, generally speaking, the growth rates for k > 2 are
larger than for k=1. (fekzabc/wcb)=(Oo).
Second, for k < 3, the system is most unstable for Third, the Z=1 mode is completely
stabilized if the conducting wall is sufficiently close to the plasma boundary.
However, the influence of the conducting wall on
stability behavior for X > 2 is negligibly small.
21 V.
ION RESONANCE INSTABILITY
The ion resonance instability12 is one of the fundamental instabilities that can occur in a relativistic beam-plasma system with both ion and electron components.
Previous investigations of this instability for a
relativistic electron component
10
have been carried out only for the
In this section, we investigate ion resonance
fundamental mode (k=l).
stability properties for higher k values within the context of the kinetic dispersion relation (57).
To make the analysis tractable, we
assume that the plasma electron density is equal to zero,
(74)
i =0 .
e
In this case, the dispersion relation in Eq. (57) can be expressed as
(1gf-
)w
(-a
2)bR
rb(w)
-2 2 Zv
2
b2
2v
v1 2
(2 - A
R2
pb p
b W
'
gf-
2
= W
2v
2 b
.
2
i
(75)
As a comparison with previous analyses,10,12 we first consider the 2=l mode, neglecting the effects of the conducting wall (R /Rc-+0) and substituting Eq. (66) into Eq. (75).
We obtain
2 W(w-k 2 bf e(l1ib) )_. 2pb bC)7_cbo [(w-k A)b kz bc) [( *kzb
x
((-k0
C) 2l+W
z c i
1 f
(w,k01c)-1 (be
= e k
from Eq. (75).
z 1
2
Ybme
4
.
pb
1
b)2
pb'%a)
mi
(76)
2 abb
'
Equation (76) is identical to the result obtained by
Davidson and Uhm10 within the framework of a rigid beam model.
In
22
obtaining Eq. (76), use has been made of the definition of fractional charge neutralization f
Throughout this section, we assume
in Eq. (63).
that the ion motion is nonrelativistic (i.e.,
II
2 are the most
Moreover, the fundamental mode (Z=l) is completely
stabilized whenever the conducting wall is sufficiently close to the The ion resonance instability was investigated in
plasma boundary.
Sec. V, including the interaction of the beam electrons with the plasma ions.
From the numerical analysis of the dispersion relation, it is found
that the instability domain extends to large values of axial wavenumber for high harmonic numbers X.
Furthermore, the maximum growth rate of the
ion resonance instability increases slowly with increasing azimuthal harmonic number k.
26
ACKNOWLEDGEMENTS
This research was supported in part by the National Science Foundation and in part by the Office of Naval Research under the auspices of a Joint Program with the Naval Research Laboratory.
The research by one of
the authors (H.S.U.) was supported in part by the Independent Research Fund at the Naval Surface Weapons Center.
27
REFERENCES
1.
W. W. Destler, H. S. Uhm, H. Kim and M. P. Reiser, J. Appl. Phys. 50_, in press (1979).
2.
V. L. Granatstein, P. Sprangle, R. R. Parker, H. Herndon and S. P. Schlesinger, J. Appl. Phys. 46, 3800 (1975).
3.
E. J. Lauer, R. J. Briggs, T, J. Fessenden, R. E. Hester and E. P.. Lee, Phys. Fluids 21, 1344 (1978).
4.
C. A. Kapetanakos, Appl. Phys. Lett. 25, 481 (1974).
5.
M. N. Rosenbluth, Phys. Fluids 3, 932 (1960).
6.
D. A. Hammer and N. Rostoker, Phys. Fluids 13, 1931 (1970).
7.
E. P. Lee, Phys. Fluids 21, 1327 (1978).
8.
R. C. Davidson and H. S. Uhm, J. Appl. Phys. 50, 696 (1979).
9.
R. C. Davidson and H. S. Uhm, Fluids 22, in press (1979).
10.
R. C. Davidson and H. S. Uhm, "Coupled Dipole Oscillations in an Intense Relativistic Electron Beam," submitted for publication (1979).
11.
R. C. Davidson and B. Hui, Annals of Physics 94, 209 (1975).
12.
R. C. Davidson and H. S. Uhm, Phys. Fluids 21, 60 (1978).
13.
R. C. Davidson and H. S. Uhm, Phys. Fluids 20, 1938 (1977).
28
FIGURE CAPTIONS
Fig. 1
Equilibrium configuration and coordinate system.
Fig. 2
Stability domain [Eq. (69)] in the parameter space
W cbb) for yb=5, (fi e/b b- pb b'
f e=0,
R /R =0.5, kz
p
c
z bC/Wb=0, WcbO
and
an
(a) 2=1, (b) .=3, and (c) 2=5. Fig. 3
Stability boundaries [Eq. (69)] in the parameter space (f , kzb c/Wcb) for yb=5,
R /Rc=0.5,
2e
cbw'
b=0.5,
and
(a) k=1,2,3, (b) Z=4, and (c) X=5. Fig. 4
Plots of (a) normalized growth rate Imo/&cb and (b) normalized real oscillation frequency Rew/wb versus Rb/Rc [Eq.
(69)] for k=,
f e=0 , k zb c/W cb=0, and parameters otherwise identical to Fig. 3. Fig. 5
Stability domain [Eq. (78)] in the parameter space (f , kz bc/Wcb) for
Fig. 6
b=5,
w 2 b/W2 b1, R /RC=0.5, and (a) Z=1, (b) Z=2, and (c) Z=3.
Plots of (a) normalized growth rate Imw/ cb and (b) normalized real oscillation frequency Rew/wcb versus kz b c/Wcb for f =0.3, k=1,3, and 5, and parameters otherwise identical to Fig. 5.
29
N
L-
Q)
0
N < (1) 1 054
(9
z
C) D
