Plasma Heating in a Beam-Plasma System

Transcription

1 Adv. Studies Theo. hys., Vol. 4,, no., lasma Heating in a Beam-lasma System B. M. Dakhel Geneal Requied Couses Depatment, Jeddah Community College King Adul Aziz Univesity, Jeddah, Saudi Aaia dakhel@yahoo.com Astact We study the eam-plasma heating due to the electon eam unde the effect of an extenal static magnetic field. We conside a longitudinal one-dimensional oscillations in a plasma, which is inhomogeneous and ounded in the diection of eam motion. Keywods: lasma Heating / Beam-lasma System INTRODUCTION Electon eam has many applications in aeas like mateial studies, compact tous fomation, geneation of x-ay and micowave, ion acceleation, etc, whee it is desiale to have enegy souce supplied ove a long duation of time. As a potential application, use of an electon eam to heat the plasma to a high tempeatue has attacted a lot of attention, oth theoetically [-3] and expeimentally [4-6] in the ecent past. The polem of electon eam inteaction with unmagnetized plasma was studied y many authos, whee this inteaction takes the fom of an amplification of waves y eam [7,8]. It is shown thatdue to the esonance ise of the wave field with plasma dielectic pemeaility educing to zeo, the powe asoed y the plasma is finite and independent of the value of the dissipation. In this case the eam not only amplifies waves in the plasma, ut also povides fo effective asoption of these waves y the plasma (plasma heating)[9-7].

2 6 B. M. Dakhel In the pesent pape we conside a -polaized wave, E E, E,), ( x y popagating in the plasma unde the effect of extenal static magnetic field diected along Z diection, H ext. (,, H ). We suppose also that the electon eam popagates along the diection of the magnetic field V (,, V ), to inteact with plasma. Both the plasma and eam ae cold. Ions ae sleeping. If the Z axis coincides with the diection of popagation of a eam with fequency ω fo the extenal souce whee the dependence of the electic field E on the spatial coodinate x and the time t can take the fom iω t x, t) e + C. C. () This equation implies that the oscillation is consideed to e unifom along the magnetic field. FUNDAMENTAL WAVES Fo simplicity, we conside the case of one-dimensional electostatic oscillations when the diection of eam popagation, plasma density gadient and wave electic field coincide with the x-axis. We also, conside that the phase velocity of waves is much less compaed to the electon eam velocity. The initial lineaized set of equations (the equation of motion and the continuity equation) desciing the oscillations in -D, fo a elativistic electon eam, which tavels along the magnetic field, ae: V e + ( V ) V E; V V + V, V V ez () m N + ( N V ) ; N n + n (3) whee; V is the component of eam velocity in the z-diection. The initial lineaized set of equations (the equation of motion and the continuity equation) desciing the oscillations in -D, fo inhomogeneous plasma electons in the oscillating electic field and a static magnetic field H ext. pependicula to the plasma density gadient ae given y: - V e + ( V ) V [ E + ( V H ext. )] (4) m C N + ( N V ) ; N n + n (5) In equations (-5), V, n ae the unpetued velocity and density of the eam while n, n ae the unpetued and petued density of the plasma, espectively. All othe tems have thei usual meaning.

3 lasma heating in a eam-plasma system 6 Fom ()-(5), we can deive the following expessions fo the petued densities: - en i( ω / V x i V x i V x i V x n e ) ( ω / ) ( ω / ) ( ω / ) ( ) e [ ( i / V ) e e x ) d x d x mv ] + ω (6) e n ( ) ~ [ n ] (7) m ωω x whee, ~ / eh ω ( ω ωc ), ωc is the electon cycloton fequency. mc Using oisson's equation de 4π e( n + n ), (8) dx in comination with (6) and (7) in (8), the following second ode diffeential equation, which descies the electic field due to eam-plasma inteaction, is otained: ( iω + V ) [ ( ] + ω (9) x ω x e n x ( ) 4π e ( ),(, ) whee, ( ~, ω ( e, ωω m ω F( i Intoducing the vaiale ( ) x, e equation (9) educes V ( d F( ) to + F( ) () d ( ) ω ω ( ) whee, ( ), ( ). ω ωω~ The solution of equation () unde the condition that the plasma density is zeo at, inceases with incease of, having then its maximum within the inteval < < l and then deceases to zeo when. Besides, λ << a, povided the following inequalities satisfied d <<, λ << a () d This condition enales us to find solution fo () using the method of geometical optics, when eam popagation in an inhomogeneous plasma is appoximately the same as in a homogeneous medium with vaiale dielectic pemittivity ( ). Then the solution of equation () in the thee egions of the plasma as / 4 ir ir F( ) [ ce + ce ] at < () / 4 ( R + ir3 ) R ir3 F( ) [ c e + c e ] at > l (3) / 4 R4 R [ ce + ce F( ) i 4 ] at < < l (4) Whee; l R d, R d, R3 d, R4 d l

4 6 B. M. Dakhel c and c ae integations constants. At, l, condition () is invalid to find solutions simila to that of ()- (4). In fact, it is inteesting to find the solution at l, whee the field E is exponentially lage. Using Taylo expansion fo the dielectic nea the point l, we get: Re ( ), / l (5) λ λ d Then the solution of equation () looks like: () () F( ) aη { c3 H ( aη ) + c4h ( aη )} (6) () () H ( ) and H ( ) ae the Hankel functions of the fist and second kinds espectively; c 3 and c 4 ae the integations constants. Accodingly, we can get the following solutions fo equation () : F ( ) ( aη) / c3 exp i aη π + c4 exp i aη π at a η >> (7) π 4 4 F ( ) i ( c 3 + c4 ) at a η << (8) π Using expession (5) in the expessions () and (3) with the solution (7) o (8), then we can otain c 3 and c 4 as c π 3 c3 exp i π + R / 4 (9) λ 4 c π 3 c4 exp i π R / 4 () λ 4 Then we can wite the final solution of equation () in the simple fom: const. i E e () It follows fom () that, just as in the case of electomagnetic wave popagation in the inhomogeneous plasma in the asence of a eam, the inteval whee (i.e., ω ω ) is that of the esonant ise of the electic field of the wave. It is also clea that the existence of an static magnetic field may lead to the limitation of the shap incease of E when ω ω (whee in this case > ). At <<, E and this elation is valid fo any elation of eam and plasma densities. CONCLUSIONS We investigated in this pape the polem of eam-plasma heating using electon eam unde the effect of an extenal static magnetic field. Both the eam and plasma ae cold. We deived and solved an equation which descies the electic field due to the density petuations in the plasma and the eam. In this study, we consideed that the vaiation of the dielectic constant is insignificant within the

5 lasma heating in a eam-plasma system 63 inteval that is of the ode of one wavelength, so that we wee ale to use the method of geometic optics to otain solution fo the wave equation. It is found that just as in the case of electomagnetic wave popagation in the inhomogeneous plasma in the asence of a eam, the inteval whee (i.e., at ω ω ) is that the esonance ise of the electic field E of the wave. It is also clea that the existence of an static magnetic field may lead to the limitation of the shap incease of E when ω ω (whee in this case > ). At <<, E and this elation is valid fo any elation of eam and plasma densities. ACKNOWLEDGMENT I would like to expess my sincee thanks to of. Kh. H. El-Shoagy fo his suggestion of this polem and also fo his help and guidance in the calculations of this pape. REFERENCES [] L.E. Thode, hys. Fluids, (977). [] G. C. A. M. Janssen, J. H. M. Bonnie, E.H. A. Ganneman, V. I. Kementsov, and H. J.Hopman, hys. Fluids, 7 (984) 76. [3] G.. Gupta, T. Vijayan, and V.K. Rohatgi, hys. Fluids, 3 (988) 66. [4] D. A. Hamme, K. A. Gee, and A.W. Ali, IEE Tans. lasma Sci, s-7 (986) 83. [5].H. De Hann, G. C. A. M. Janssen, H.J. Hopman and E.H.A.Ganneman, hysics Fluids, 5 (98) 59. [6] M. Fiedman, V. Selin, A. Doot, and A. Mondelli, IEEE Tans. lasma Sci., s- 4 (986). [7] W. H. Amein, V. V. Dologpolov, A. M. Hussein, and K.E. Zayed, (a) lasma hysics 7 (975) 497 ) hysica 79c (975) 68. [8] A.A. Ivanov, N.G. opkov, J. Wilhelm, and R.Winkle, Beit. lasma hys., 7 (984) 3. [9] N. G. Zaki and Kh. H. El-Shoagy, Egpt. J. hys., 9 (998) 3. [] W. M. Sahyouni and Kh. H. El-Shoagy, Aa J. Nucl. Sc. Appl., 3 (999)85.

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