Electron beam excitation of a potential well in a magnetized plasma waveguide

Physics LettersA 165 (1992) 63—68 North-Holland

PHYSICS LETTERS A

Electron beam excitation of a potential well in a magnetized plasma waveguide V.1. Maslov Institute of Physics and Technology, 310108 Kharkov, Ukraine Received 17 June 1991; revised manuscript received 13 February 1992; acceptedfor publication 20 February 1992 Communicated by M. Porkolab

The evolution and properties of a potential well being excited in a magnetized plasma-filled waveguide by a hot electron beam have been investigated. An equation of KdV-type describing the potential well has been obtained. It is shown that a potential shock is formed near the potential well, the front ofthe well is steeper than the back, the width ofthe well decreases with increasing amplitude. The well is determined by two modes with a linear dispersion relation: by the mode in a magnetized plasma-loaded waveguide and by the electron-acoustic mode.

I.

Introduction

The energy exchange between particles and nonhomogeneous oscillations as well as the nonlinear generation of higher harmonics by these oscillations, if their dispersion relation is nearly linear, can lead to the formation of a localized wave structure. In particular, the possible formation of a correlated perturbation the potential well has been investigated with an ion-acoustic instability development in refs. [1—4].Also a solitary perturbation can form in a magnetized plasma-loaded waveguide [5—10].Instead of treating this potential well emergence we describe its evolution and properties when it is excited by a “hot” electron beam. We obtain an equation ofKdV-type describing the potential well. We show that a potential shock is formed near the potential well, the front of the well is steeper than the back, the width of the well decreases with increasing amplitude. The well is determined by two modes with a linear dispersion relation: the mode in a magnetized plasma-loaded waveguide and the electron-acoustic mode. —

—

2. Properties of the potential well Let us consider the excitation of a potential well by a “hot” electron beam in a magnetized plasma-loaded waveguide. We consider the case of a strong external longitudinal magnetic field, H0—~oo.Then the electron dynamics is one-dimensional. The velocity distribution functions of the plasma and beam electrons, f=f~ +f~,, are described by the Vlasov equation, (1) ~9t

8z

môz8V

Their initial distribution functions are equal, 2/2V~h), f~= (nbo/ Vbth~J~) exp[— (V— Vb)2/2V~~h] f~= (fbi Vth ~ exp(— V The potential ~ is described by the Poisson equation, .

0375-960l/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

(2)

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Volume 165, number 1

A~= A

~~

PHYSICS LETTERS A

=4~e(n_no_nbo)=4xe(

J

dVf—n

0

_flbO)~

4 May 1992

(3)

where A is the transverse Laplacian; n0, ~bO are the initial plasma and beam electron densities, j/b is the beam velocity; ~th, Vb~hare the thermal velocities of the plasma and beam electrons. We choose the initial perturbation in the form of a potential well co_-~J0(k1r)co~(z, I) of small amplitude (depth) ~, of width E~.zsmaller than the system length L and moving with a velocity V~,satisfying the inequalities Vb~Vb~hY(c0o)1~2(c0o)

this case the changing velocity distribution of the resonant electrons moves from the well with relative velocity equal to ~r Then the nonhomogeneous potential distribution takes the form shown in fig. 1. We describe the well evolution using an expanding in the small parameter a = yAz/ Vtr. Below we neglect the nonadiabatic dynamics ofthe electrons with velocities I V— V 0~~ ~r in the field ofthe potential tails. The latter moves with a velocity ~r Taking into account nonadiabatic dynamics leads to additional terms of higher order as compared to the ones used. The electron trajectories are described by the equation of the characteristics of the Vlasov equation (1), In

Fig. I. Potential distribution of asymmetric well.

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~=u,

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4 May 1992

(4)

ã=(e/m)ço’(t,c~) ,

and are determined by the equation ~=~mu2+e~(t,~)=const

(5)

in the rest frame of the well. Here a dot means aiot, a prime means

aio~,~=z—

V

0t. Fig. 2 shows the phasespace. It is seen that the resonant region is wider in front of the well, ~> 0, by the value of the potential shock, than behind the well, ~< 0. Taking into account that the resonant electrons are reflectedfrom the well, we obtain from (1) the expression for the velocity distribution function of the beam electrons, 2—2ecoim)”2+ V 2, ~‘.0, fb(1, u) =fb((u 0.— Vb), u>A(ço)= [2e(ço00+ço)/m]~ u> —A(~), ~<0, ~,

=fb((u2—2ecoim)”2— V 0 + Vb),

u
~>0,

u<—A(~),

~

=ço0J0(k1 r),

~<0,

(6)

neglecting the nonadiabatic dynamics of the electrons in the field of the potential tails. In order to describe the slow evolution of the system we shall look for a solution of eqs. (1), (3) in the form (6) and for the potential distribution in the form of a well with slowly varying amplitude in time ~ ( I). Let us consider the evolution of the well under the assumption of small ecoi Tb, (Vb V0) / Vb~h, where Tb = m V~th.Integrating (6) over the velocities, we obtain the expression for the density perturbation of the beam electrons in first order of (Vb V0 ) / Vblh and second order of eç~/Tb, —

—

~ Tb

Vblh

+~ln(

~

)]},

(7) where sgn(~)=—l (+1) for~.<0(~>0). From eq. (1) we obtain the expression for the linear perturbation of the plasma electron distribution function, (8) Then from eq. (1) we obtain the nonlinear and time-dependent terms of the perturbation 2~=(e/m)2(Vo—V)~ x~ço2a.~(V öf~ 0—Vy’8f~/aV 3)=(V 2ç~ôf~/ôV. ~f~ 0— V)~~f”~=(e/m)(V0— V)

(9)

U

Fig. 2. Electron phase-space.

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From (8), (9) the expression for the density perturbation of the plasma electrons follows, ~n=6n

~

1+ön2,

6n’2=—2en0çb/mV~.

(10)

Substituting (7), (10) in eq. (3) and neglecting higher-order nonlinear terms, we derive the evolution equation V~)

~

____

VbthVm

~

)]}=o

(11)

ofKdV-type but with a nonquadratic nonlinearity. Here V~is the maximum phase velocity of a wave in a waveguide with no beam, Vm = Vblh ( flO/flbO)’ /2 is the maximum phase velocity of the electron-acoustic mode [111, w~=4mn~e~/m, /3=J7~° J~(k1r)r dr/J~° J~(k1r)r dr. Eq. (11) describes the space distribution of a nonhomogeneous potential, its deformation, the increase of~0(t)in time and also the motion of the potential well with velocity V0. The last term in eq. (11) is determined by the resonant electrons. Let us obtain from (11) the expressions for the properties of the potential well. For this we choose symmetric, Ø(~),and asymmetric, öØ(~),parts of the potential distribution q~(~) =Ø(~)+~Ø(~). ~Ø(~)describes the asymmetric potential, added in the presence of a beam to the symmetric well in the approximation Aç~/ coo. öØ(~)changes from —Atp at ~5< —Az to Aco at ~>Az. Assuming Aco<< coo we obtain the equations for 0 and öØfrom (11), 2+ V2 V~2)+Ø2fl(e/2m)(3w~/V~+d~2V~) (12) —

Ø”=Øw~(V ~

(13)

~

d~=~ From (12) we derive the equation describing the space distribution of the potential,

where

(14)

Ø’2=w~Ø2(V~2+V~2—V~2)+Ø3~

From (14) and O’I~,,_~=0 we obtain the expression for the well velocity, V 0=V00+AV(co0),

Voo=(v2~~)l/2~

AV(coo)=(~+-~t~).

(15)

The well velocity increases with increasing amplitude and is equal to the electron-acoustic velocity, J,,, at Vt>> Vm and to the maximum phase velocity of the wave in the magnetized plasma waveguide, V~,at Vm2’> Vc. From (14), (15) it may be concluded that the well is determined by two modes with a linear dispersion relation: the mode in a magnetized plasma-loaded waveguide and the electron-acoustic mode. In the quasi-stationary approximation from (13), (15) at ~> Az we obtain the rough expression for the potential shock near the well, m 66

~

(Vb—V0)/VbIh V~+V~/V~0)~

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In the case being considered the shock is larger by a large factor TbiecoO in comparison with the case when fronts of electrons reflected from the well reach the boundaries of the system [4]. The profile of the well is characterized by its width. We roughly determine the width as follows, Az= cool I co’~(co,,= ~coo)I. Taking into account the asymmetric part of the potential distribution we obtain in the quasistationary approximation from (3), (15), —

(ecoofl

A

L~Z

ri

Iri2

[~Wp/

\2j_ ‘I..!

ri

~—2

r~l,UbVbthJ

00)

V

—

1/2

~

.

(17)

/

bth

The well is asymmetric in the presenceof the beam: the front is steeper than the back. The well width decreases with increasing amplitude. To describe the well evolution in time we determine the well amplitude growth rate y= 8 ln coo/8t. For this, assuming q~= coo in (13), we obtain —

ri3 / coo — 2~2~ b 2(2/ OOj d—

)I/2

T/

Ti lim Vblh 0 o-.-c~/l+0/coo

b

+fl6Ø’(Ø= —co We let approach lim

ø--co

0

2V~)+6Ø”(Ø= —coo)). 0)x2co0(e/m)(w~/V~o +~d~ to in (14) to determine the limit in (18),

(18)

co312fl(e/m)”2(w2/V4+ ~d~2Vb~)”2

(19)

—~

I0’I

~Jl +0/coo

.

Substituting (19) in (18) and using (16), (17) we obtain V~(16~,/~i~co l66Ø’(O=—coo)

+o

20

—

coo= 3 (Az)2 0 (0——coo) 2,~2i~, ( Az) Changing 60’ (0= coo) —~Aco/Azwe obtain approximately —

.

(

)

—

16

8Ø’(Ø= —coo)

>>

6Ø”(Ø= —coo),

Vb—VO

2,/~Vbth

(ecoofl/m)1/2.~

~

(21)

The nonlinear growth rate increases with increasing well amplitude.

3. Conclusion Thus we have described the evolution and properties of a potential well being a nonlinear collective correlated perturbation. The well is excited by a hot electron beam in a magnetized plasma-filled waveguide. The investigation shows that the potential well is determined by two modes with a linear dispersion relation. The well is excited by resonant electrons which have a nonequilibrium distribution function and are reflected from it. The resonant electrons transfer energy to the well. Owing to the resonant electrons’ reflection from the well, whose velocity distribution is asymmetric relative to the well velocity, the electrons are short in the front of the well and they are surplus in the back. The restoration of the latter leads to the self-consistent formation 67

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of a potential shock near the well. The well is asymmetric: its front is steeper than the back. The well is accelerated, its width decreases and the growth rate of the amplitude increases with increasing amplitude.

References [1] H. Schamel, Phys. Rep. 140 (1986)163. [2] T. Sato and H. Okuda. Phys. Rev. Lett. 44 (1980) 740. [3] K. Nishihara et al., Riso Nat. Lab. R. 472 (1982) 4 L [4] VI. Maslov, Plasma Phys. 16 (1990) 759. [5] S.M. Krivoruchkoetal., JETP 67 (1974) 2092. [6]1.P. Lynov et al., Phys. Scr. 20 (1979) 328. [7] H. Ikezietal., Phys. Fluids 14(1971)1997. [8] B.N. Rutkevich et al.. Zh. Tekh. Fiz. 42 (1972) 493. [9] VI. Kurilko and A.P. Tolstolushskij, JTP 44 (1974) 1418. [10] V.K. Sayal and S.R. Sharma, Phys. Lett. A 149 (1990) 155. [II] V.K. Jam and N.T. Tsintsadze, IC/89/ ISO (1989).

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