Technical Physics, Vol. 46, No. 10, 2001, pp. 1331–1334. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 10, 2001, pp. 128–130. Original Russian Text Copyright © 2001 by Dubinov.
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Characteristic Features of the Development of Two-Stream Instability of Electron Beams in the Presence of Spatially Localized Perturbations A. E. Dubinov All-Russia Research Institute of Experimental Physics, Russian Federal Nuclear Center, Sarov, Nizhni Novgorod oblast, 607188 Russia e-mail:
[email protected] Received December 13, 2000
Abstract—A study is made of the evolution of a spatially localized perturbation in the form of a Gaussian packet during two-stream instability in a plasma. It is shown that, on the whole, the moving perturbation is decelerated and its shape is distorted; moreover, the higher the group velocity of the packet, the higher the deceleration rate. © 2001 MAIK “Nauka/Interperiodica”.
The multistream (in particular, two-stream) instabilities of parallel electron beams in a vacuum and in plasma are a striking example illustrating the instability of nonequilibrium electron distribution. In the simplest cases, the dispersion relation describing the linear stage of the multistream instability can be solved and analyzed exactly. This analysis is given in many textbooks and monographs on plasma physics (see, e.g., [1, 2] for two-stream instability and [3] for multistream instability). A fairly comprehensive review of the literature on two-stream instability can be found in [4]. However, multistream (in particular, two-stream) instabilities are of interest not only from a methodological point of view. Thus, Fedorchenko et al. [5] experimentally investigated the excitation of high-frequency oscillations during multistream instability of a system of parallel electron beams. Additionally, in microwave devices such as vircators, reflex klystrons, and Barkhausen–Kurtz oscillators with a decelerating field, two-stream instability can develop in colliding beams (for vircators, the possible onset of two-stream instability was pointed out in [6]). The linear stage of the two-stream instability is usually analyzed by linearizing the basic equations under the assumption that the initial perturbations of the system depend on the coordinate and time as ∝ exp[i(kz – ωt )]. In other words, the initial perturbation is assumed to be distributed uniformly over the interval from –∞ to +∞, which corresponds to a system of infinitely long beams and, generally speaking, to the infinitely high energy of the perturbation. In this context, it is of interest to study how twostream instability develops when the initial perturbation is localized in space. As a matter of fact, the idea of investigating the characteristic features of the development of various instabilities in the case of spatially
localized perturbations is not new (see, e.g., [7, 8]). However, from a methodological standpoint, it is somewhat difficult to analyze such perturbations by expanding them in harmonic waves [7], because the energy of a localized perturbation is finite (this is evidenced by the convergence of the corresponding integral), while each harmonic component of the perturbation has infinite energy. That is why, in my opinion, the simplest and most illustrative way of studying the instability of spatially localized perturbations is that which does not involve the Fourier analysis of the perturbations. As will be shown below, this approach provides a fairly simple analytical examination of two-stream instability. We start with the traditional equations [3] ∂n --------α + ∇ ⋅ n α v α = 0, ∂t
(1)
∂v e --------α + (v α ⋅ ∇)v α = ---- E, m ∂t
(2)
∇ ⋅ E = 4πe
∑ mn α
α
– N i ,
(3)
where nα and vα are the density and velocity of the αth electron beam, e and m are the charge and mass of an electron, and E is the electric field. Equation (1) is the continuity equation for an electron beam, Eq. (2) is the equation of motion, and Eq. (3) is Gauss’s law. For simplicity, we assume that parallel electron beams propagate against the background of immobile ions with density Ni, which serve merely to provide the charge and current neutralization of the unperturbed electron beams.
1063-7842/01/4610-1331$21.00 © 2001 MAIK “Nauka/Interperiodica”
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Solving Eqs. (7) and (8) with respect to vα and nα yields e E - , (10) v α = ---- -------------------------------------------------------------------------------------m 2χ 2 ( z – V g t ) ( V g – V α ) + i ( kV α – ω ) N α E [ – 2χ ( z – V g t ) + ik ] e - . (11) n α = – ---- --------------------------------------------------------------------------------------------2 m [ 2χ ( z – V t ) ( V – V ) + i ( kV – ω ) ] 2 g g α α 2
We insert these expressions for vα and nα into Eq. (9) to arrive at the desired dispersion relation
Fig. 1. Instantaneous shape of a perturbation in the form of a Gaussian packet.
We direct the z-axis along the propagation direction of the beams and assume that the system is uniform in transverse directions. In this case, Eqs. (1)–(3) constitute a set of one-dimensional equations for the densities and the projections of the velocity and electric field onto the z-axis. We represent the unknowns in the equations as 2 2 n α = N α + n˜ α exp [ i ( kz – ωt ) ] exp [ – χ ( z – V g t ) ], (4)
v α = V α + v˜ α exp [ i ( kz – ωt ) ] exp [ – χ ( z – V g t ) ], (5) 2
2
2 2 E = E˜ exp [ i ( kz – ωt ) ] exp [ – χ ( z – V g t ) ].
(6)
We can see that the perturbation is spatially localized and is described by a harmonically modulated Gaussian function with the parameters χ (which has units of inverse length and determines the spatial width of the perturbation) and Vg (which is the group velocity of the perturbation). In the theory of wavelets, the function describing the perturbation coincides (to within a normalizing factor, which is unimportant for our purposes here) with the Gabor function [8]; in [7], this function is referred to as a Gaussian packet. The plot of this function is shown in Fig. 1. In what follows, we assume that the perturbation amplitudes in representations (4)–(6) are small and omit the tilde from the perturbed quantities. We substitute representations (4)–(6) into the basic equations and, to the first order in the small perturbation amplitudes, obtain n α [ 2χ ( z – V g t )V g – iω ] + ( N α v α + n α V α ) 2
× [ – 2χ ( z – V g t ) + ik ] = 0, 2
(7)
e 2 × v α V α [ – 2χ ( z – V g t ) + ik ] = ---- E, m
∑ 4πen α
α
where ωpα = (4πe2Nα/m)1/2 is the Langmuir frequency of the αth beam. Now, we proceed to an analysis of the dispersion relation (12). First, note that, when deriving this dispersion relation, we failed to completely eliminate the dependence on the coordinate and time. However, this circumstance is usually associated with the expansion of functions in a series of the basic localized functions—wavelets [8, 9]—and makes it possible to analyze the processes in the {z, t} and {k, ω} spaces simultaneously. Note also that, for unlocalized perturbations (χ 0), the dispersion relation (12) passes over to the familiar dispersion relation for harmonic perturbations. In the case of two parallel beams with arbitrary unperturbed densities and velocities, Eq. (12) contains many parameters, so that its analysis, although straightforward, is very lengthy. Here, we examine only the particular case of two counterpropagating identical beams with the same values of ωpα = ωp and Vα = V. For such beams, the two-stream instability of a traditional harmonic perturbation is absolute in nature. We investigate the onset of the instability at different characteristics C = z – Vgt = const of the perturbation envelope. In this case, the dispersion relation has the form ωp 1 = – --------------------------------------------------------------------------------------2 2 [ 2χ ( z – V g t ) ( V g – V ) + i ( kV – ω ) ] 2
2
2
2
∑
2
ωp -. – ---------------------------------------------------------------------------------------2 2 [ 2χ ( z – V g t ) ( V g + V ) – i ( kV + ω ) ]
v α [ 2χ ( z – V g t )V g – iω ]
E [ – 2χ ( z – V g t ) + ik ] =
1 = –
ω pα --------------------------------------------------------------------------------------------- , (12) 2 2 [ 2χ ( z – V g t ) ( V g – V α ) + i ( kV α – ω ) ]
α.
(13)
(8)
(9)
We can readily see that, at the top of the envelope (C = 0), the instability develops in the same way as in the case of a harmonic perturbation. However, at the other parts of the envelope, the instability growth rates can differ substantially from those in the case of a harTECHNICAL PHYSICS
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CHARACTERISTIC FEATURES OF THE DEVELOPMENT
monic perturbation. Since Eq. (13) is a quartic equation in the variable ω, we are dealing with the four roots
Im(ω1,2,3,4)
(a)
ω 1, 2, 3, 4 = 2iCV g 2
2
2
2
2
(14)
2
3
2
2 2
2
2
4
2
2
2
2
Figure 2a refers to a nonpropagating (Vg = 0) perturbation with a certain value of k. At the top of the envelope, the second and third roots vanish, ω2, 3 = 0, while the first and fourth roots ω1, 4 undergo a jump equal to the doubled instability growth rate for a harmonic perturbation. The growth rates increase with distance from the top, because the evolving perturbation acquires the nature of a uniform harmonic perturbation, so that the effective width of the envelope of the evolving perturbation increases (this effect is known as the spreading of a Gaussian packet). For a perturbation with a low group velocity (0 < Vg < V), the dependence of Im(ω1, 2, 3, 4) on C is somewhat different from that in the previous case. From Fig. 2b, we can see that the leading edge of the perturbation grows slower than the trailing edge. As a result, first, the centroid of the perturbation is displaced toward the trailing edge, indicating the deceleration of the perturbation; and, second, the leading edge of the perturbation becomes flatter than the trailing edge. The leading edge of a perturbation with a sufficiently high group velocity (Vg > V) can even decay, while its trailing edge can grow at a very high rate (Fig. 2c). If the two beams under consideration are in an electrodynamic system of the carcinotron type, in which case we have Vg < –|V | < 0 < V (Fig. 2d) and the characteristics C are positive at the trailing edge of the perturbation, then the perturbation is also decelerated and its shape is deformed. Although the above features of the evolution of a spatially localized perturbation were revealed in the linear approximation, they cannot be established in terms of a harmonic perturbation. These features can be described by expanding a spatially localized perturbaVol. 46
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1
2
In expression (14) and below, the first root is identified by the plus signs in front of and under the square root, the second, by the minus and plus signs, the third, by the plus and minus signs, and the fourth, by the minus signs. The imaginary parts of the roots, Im(ω1, 2, 3, 4), calculated as functions of the position C on the envelope of the perturbation under the natural assumption k2 Ⰷ χ2 are illustrated graphically in Fig. 2.
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2
4
ω p + 4ω p k V – 16C χ ω p V + 16iCχ ω p kV . 4
C
0
3
where D=
4
1
± k V + ω p – 4χ C V + 4iCχ kV ± D , 2
1333
(b)
1 4
2
3 C
0
3
2
4
1 (c)
1
4
3
4
2 C
4 0 3 2 1 (d)
4
2
1 0 2
3
C 1
3 4 Fig. 2. Imaginary parts of the roots calculated for (a) Vg = 0, (b) 0 < Vg < V, (c) Vg > V, and (d) Vg < –|V| < 0 < V.
tion in harmonic waves, but this way is far more lengthy because it involves the calculation of integrals like the Duhamel’s convolution. ACKNOWLEDGMENTS I am grateful to A.A. Rukhadze for consultations on some problems associated with multistream instabilities.
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REFERENCES 1. N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics (Academic, New York, 1973; Mir, Moscow, 1975). 2. P. C. Clemmow and J. P. Dougherty, Electrodynamics of Particles and Plasmas (Addison-Wesley, Reading, 1990; Mir, Moscow, 1996). 3. T. H. Stix, The Theory of Plasma Waves (McGraw-Hill, New York, 1962; Atomizdat, Moscow, 1965). 4. R. J. Briggs, in Advances in Plasma Physics, Ed. by A. Simon and W. Thompson (Wiley, New York, 1971; Mir, Moscow, 1974), Vol. 4.
5. V. D. Fedorchenko, Yu. P. Mazalov, A. S. Bakaœ, and B. N. Rutkevich, Zh. Éksp. Teor. Fiz. 65, 2225 (1973) [Sov. Phys. JETP 38, 1111 (1973)]. 6. A. E. Dubnov, Radiotekh. Élektron. (Moscow) 45, 875 (2000). 7. A. B. Mikhailovskii, Theory of Plasma Instabilities (Atomizdat, Moscow, 1971; Consultants Bureau, New York, 1974), Vol. 1. 8. S. K. Zhdanov and B. A. Trubnikov, Quasi-Gaseous Unstable Media (Nauka, Moscow, 1991). 9. N. M. Astaf’eva, Usp. Fiz. Nauk 166, 1145 (1996) [Phys. Usp. 39, 1085 (1996)].
Translated by O. Khadin
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