Сollisionless damping of electron waves in non-maxwellian plasma 1

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1 Сollisionless damping of electron waes in non-mawellian plasma V. N. Soshnio Plasma Physics Dept., All-Russian Institute of Scientific and Technical Information of the Russian Academy of Sciences (VINITI, Usieitcha, 535 Moscow, Russia) Abstract In this paper we hae criticized the so-called andau damping theory. We hae analyzed solutions of the standard dispersion equations for longitudinal (electric) and transersal (electromagnetic and electron) waes in half-infinite slab of the uniform collisionless plasmas with non-mawellian and Mawellian-lie electron energy distribution functions. One considered the most typical cases of both the δ type distribution function (the stream of plasma with monochromatic electrons) and distribution functions, different from Mawellian ones as with a surplus as well as with a shortage in the Mawellian distribution function tail. It is shown that there are present for the considered cases both collisionless damping and also non-damping electron waes een in the case of non-mawellian distribution function. PACS numbers: 5.5 Dg; 5.35 Fp. Key words: plasma waes; andau damping. Introduction As it is well nown, dispersion equations of plasma waes include indefinitely (logarithmically) diergent integrals (IDI), what implies only one sense: it is caused by an information lac in original inetic and Mawell equations. Therefore for the choice of way of taing these integrals it ought to supply original physical conditions with some additional ones. Our analysis of the simplest one-dimensional problem for a half-infinite isotropic plasma slab has shown that such additional clear physical conditions for considered cases are both absence of the so-called inematical waes and absence of fast bacward waes [,]. These conditions are satisfied for IDI s taen in the principal alue sense following to the primary supposition by Vlaso [3] but are not satisfied for the contour sense of IDI s proposed by andau, the latter ones leading to the ubiquitously recognized now ect of collisionless andau damping for Mawellian plasma [4]. Neertheless collisionless damping is possible also for IDI s in the principal alue sense, but only for the electron elocity distribution function f different from Mawellian one [5]. In this connection it is interesting to inestigate the possibilities of collisionless wae damping or wae building up in a plasma with some more realistic non-mawellian distribution function on the base of before proposed by us method of characteristic (ectie) elocity alue [6]. This method simplifies drastically ealuation of IDI s in dispersion equations, though at the price of some uncertainty in the alue of ectie elocity in the end result. In the following we hae used standard dispersion equations It is English ersion of the paper published in Russian The Integrated Scientific Journal, 6, n.4, p.54. Krasnodarsaya str., 5--68, Moscow 9559, Russia. it3363@yande.ru.

2 f f G = + d d p = + = p + p, pp () where p = i, p = i are aplace transform parameters [6] (for the forward wae Re > ) for longitudinal waes, and ( ) p p f f dd G = p + dd = + + = z z pp z z z c c z p+ p c c z () for transersal waes with asymptotical solutions of the type ( it i) ep, where is gien frequency and in the general case comple alues are defined by dispersion equations, where is direction of wae propagation in the half-infinite plasma slab; is some ectie alue of elocity field Ez waes; in the farther described epressions. In the following E is either transersal electric for the case of transersal waes or longitudinal field E for the case of longitudinal is angmuir electron frequency; f is electron elocity distribution function normalized to unity. In the case of Mawellian distribution function approimate solutions of dispersion equations are c (3) c + c (high elocity electromagnetic waes at account for the smallness of the term denominator of epression ()); in + (4) c (low elocity transerse waes at account for the negligible smallness of the term c in equation ()); = (5) (longitudinal electron waes according to equation ()). In these epressions one supposes that equals approimately the alue within a factor ~(,5 ). It was shown before [5] that at the case of Mawellian-lie distribution function with some eceeding oer Mawellian distribution in the distribution tail there has place high collisionless damping of transerse electron waes of the low elocity electron mode. In the following we consider the most characteristic ariants of collisionless solutions (as constituent parts of the general solution for low-collision plasma) at non-mawellian distribution functions.

3 Effectie alue method In the preceding papers [],[],[5],[6],[7] we used an eident principle at taing integrals in dispersion equations: for some function Φ ( ), for eample Mawellian-lie distribution or any other function w( ) there can be found such a alue that Φ ( ) =Φ ( ), (6) where in general case. In the more general case with presence some additional parameters, e.g. ( ) Φ=Φ,, Eq.6 will transform to ( ) ( ) Φ, =Φ,, (7) where the aeraging integral is taen in the principal alue sense. Thus functions of the type ( ) Φ ( ) w( ) Φ ( ) w or (8) can be equalized zero near = at integrating in the principal alue sense. is ariant at different, in the first linear approimation one can suppose either Since ( ) ( ) ( ) + α (9) or ( ) ( ) + α, () where or are some gien alues at real alues ( ) ( ) or (6). The factor α is determined by the type of functions ( ) according to condition Φ and ( ) w ; for the case of Mawellian distribution it is assumed ery small or equals zero due to ery sharp lowering down w = f in eponential tail of ( ) ( ). At as a whole monotonically falling down function f Φ ( ) some surplus of ( ) tail of distribution function f ( ) leads to lowering, that is α < Φ ( ) > with the distribution tail below Mawellian distribution Φ in the, and ice ersa at lowering f it is equialent to α >. In the following we consider forward plasma waes with Re >. An important moment is here that parameter can be in general comple alue determined as solution of dispersion equation. At real alue it means that factorization coicient η at ( ) in Eq. must be the real linear combination of products of comple and them conjugated alues α and with Reη, Imη = (possible constrained as Re η > a, where 3

4 a is some constant alue), namely η = Re( + α). One can find an equation for determination comple α with calculation of (either ) by the direct numerical calculation of the principal alues of integrals in dispersion equation at two selected alues of comple. In the simplest interpretation one could find out een if qualitatiely the fact itself of possibility of damping or building up forward wae with comple, Re >, neglecting further imaginary part Im α. It is assumed that een such simplification gies ey to understanding features of transersal electron (low-elocity) wae in the case of Mawellian distribution function ep( ξ ) with some surplus oer the latter at ξ ~4, as it was discoered before [5] and in other analogous cases. The real and imaginary parts of (and correspondingly α ) are connected to a certain etent by the requirement of absence of bacward longitudinal waes or bacward transerse electromagnetic waes. To satisfy this requirements the root of dispersion equation of for bacward wae Re < (at > ) must be real alue, that is with iolation of the symmetry relatie to ± Im in forward and bacward waes [7]. Qualitatie interpretation of damping of transerse low-elocity electron waes in the case of Mawellian-lie distribution function Par eample, according to Eq.4 and Eq. we can obtain as one of ariants, () + α where ( α = ) = ( ), and ( ) can be taen ~( ) with α. Thus.5 - corresponding, () + α and α =. (3) α The root sign is selected corresponding to a limit at α. An addition of some positie surplus oer Mawellian distribution tail displaces to more great alues leading to smaller alues of α. At different α in Eq.3 we obtain the following table of : Table α /4 / i.866.5i i These results are in qualitatie agreement with the strong damping of the transerse low- ep ξ at the range elocity electron mode in [5] at surplus oer Mawellian distribution ( ) 4

5 ξ 3 4. Additions in the more remote tail of distribution function at ξ 5 lead to some lower α up to disappearing of damping. The damping is absent at α.4, that is at small surpluses or some gap in the Mawellian distribution tail with α >. It ought also to note that the range ξ 3 5 in the case of gas discharge strongly ionized plasma corresponds usually to energy ecitation of atomic and molecular leels up to energies of the order ev. Fast (electromagnetic) waes in Mawellian-lie plasmas According to the ectie elocity method dispersion equation of the fast elocity electromagnetic wae is (after corresponding replacing in Eq.3) again c, (4) c + + α c ( ) is determined as α =, and ought to be taen near the alue Thus, is in this case equal to independently of α and distribution function. ongitudinal waes in Mawellian-lie plasma Analogously to the foregoing, dispersion equation () transforms to. m + α T B + α =, (5), and approimately it was supposed here m z =, = + α, (6) T B ( α = ) =. (7) After elementary transformations one obtains solution α, (8) α α α where root sign is selected so that at α also. This solution is analogous to the solution (3) at the case of transerse low elocity electron wae with the possibility of strong damping at negatie alues α. 5

6 ongitudinal waes in plasma with δ -lie electron elocity distribution function For the more clearness of the latter transformations with δ function the latter can be presented as a trapezium with smoothed angles and the width approaching to zero and the height approaching to infinity at eeping constant the area S = (see Fig.). Integration by parts in epression () with ( ) δ ( ) f = (9) leads to f f f d d = = ( ). () = ( ) Thus dispersion equation at p = i, p = i G pp = = () ( ) has two solutions with forward waes ± =. () At < and > there are no solutions in the form of forward waes. Transerse waes in plasma with δ lie electron distribution function Since ( y) ( z) ( ) f = (3) δ δ δ, Fig.. Clear presentation of function f = δ ( ) as a trapezium with constant area S = ; height, approaching to infinity; width, approaching correspondingly to zero. The deriatie is also shown (not scaled). Integration by parts is possible because the function δ ( ) in all interc al ( ± b) equals zero, it approaches also to zero after crossing oer the point = c (the latter is an arbitrary constant) with its transition into region ± ( a+ b) at approaching to δ function limit. then δ ( ) z z dz = ; δ ( y) dy =, (4) z and dispersion equation is then written as c + =. (5) In the case of fast elocity (electromagnetic) wae with c dispersion equation can be written approimately as 6

7 c c + with solution for the forward wae = (6). (7) c c ( ) In the case of electron transerse wae, as before, neglecting in Eq. term c (that is unit in Eq.5), supposing = + τ, τ, =, (8) we obtain an equation for determining τ c + τ + ( τ ), (9) whence it follows τ. (3) c Thus, in transerse electron waes at δ lie distribution function there is no eponential damping/growing of amplitudes. ongitudinal waes in Mawellian plasma which moes with elocity In this case distribution function at stream elocity (that is Mawellian electron current in plasma medium) is m ( ) f ep T. (3) B At a more accurate consideration than epressed by relations () and in a more correspondence with the aboe-stated method of ectie alue, the analogue of epressions () will be u + = + u where + f u f u d du ( ) u ( ) ue ff, T B u ; u ; u η; u >, (33) m = (3) and f is now renormalized to unity in integration limits. 7

8 The solution of Eq.3 is now η u = ± η + ( η ) u η u η u, (34) where waes can eist only at Re <, that is η <. At = there is u = =± u u. (35) This is generalization of Eq.5, where one assumed waes can eist only at condition u u. It ought to note that propagating u >. u (36) One may assume that this relation is satisfied also at. Then in Eq.34 is real alue, that is the waes (if they eist) hae no damping/growing features. Transerse electron waes in Mawellian plasma which moes with elocity Analogously to preceding section, with the distribution function m m BT T ( ) z B f e e (37) with following renormalization of f, we can rewrite epressions () in the form (, ) ( ) ( ) ( ) ( ) f f + + d d = + + d + z z z z c c u + c c u + = ; u, (38a) c c ( ) u and then in equialent form ( ) u + ( ) u ( ) =. c (38b) c For the low-elocity branch, neglecting terms of the order dispersion equation in the form ~ u c, c, one obtains ( ) u = (39) with its triial solutions 8

9 = ; =. u + u (4) Here < u c, and slow-traeling waes eist at a condition Re < and also > since the branch of the slow-elocity waes is coupled with the branch of the fast waes for which wae number is gien by (3). Conclusion We hae analyzed dispersion equations of electron waes in half-infinite slab of isotropic homogeneous plasma for characteristic ariants of electron elocity distribution function: Mawellian distribution with small (without sign changing of deriate) positie or negatie additions in the distribution tail; monochromatic or Mawellian-lie beams moing as a whole with elocity. Eponential growing of traeling waes is obsered in no ariants, although this is not ecluded in some other ariants of distribution function. Eponential damping appears only in the case of relatiely small (without the change of ep ξ at limited region deriatie sign) positie addition in Mawellian distribution tail ( ) ξ 3 4 (see [5]) corresponding in the case of gas discharge plasma to electron energies from some ev up to ev. In all cases we considered only forward waes in half-infinite slab of homogeneous electron plasma, what is physically justified for longitudinal electron and fast transerse electromagnetic waes. As it was shown in [8], slow bacward waes which form at their reflection from the structure of fast waes can superpose oer slow forward electron waes, therefore in the case of half infinite plasma slab slow transerse waes contain a constituent of oscillatory standing waes. It appears howeer doubtless to be slow dependence of ectie alues, u on, so that ( ) ( ) + α. (4) But this leads to more high orders of algebraic dispersion equations, that is to possible presence of some additional wae modes n with a necessity of selection physical and unphysical roots. With a series of our preceding papers this wor is directed to the better understanding the nature of the real ect of collisionless damping of plasma waes together with some outlined ways of analytical and numerical soling corresponding them dispersion equations, including account for non-linear term and the case of low-collision plasma. References. Soshnio V.N. ogical contradictions of andau damping// Soshnio V.N. Notes on the ideology of plasma waes//engineering Physics, 4, n., p.48 (in Russian). See also physics/898 ( 3. Vlaso A.A. On ibration properties of electron gas//jetp (USSR), 938,.8, p.9 (in Russian); Uspehi Fiz. Nau, 967,.93, p.444 (reprint in Russian). 4. andau.d. On electron plasma oscillations//jetp (USSR), 946,.6, p.574 (in Russian); Uspehi Fiz. Nau, 967,.93, p.57 (reprint in Russian). 5. Soshnio V.N. Strong collisionless damping of the low-speed electromagnetic wae in a plasma with Mawellian-lie electron energy distribution function//engineering Physics, 5, n.3, p.39 (in Russian). 9

10 6. Soshnio V.N. Damping of electron waes in Mawellian non-linear low-collision ionelectron plasma//engineering Physics, 5, n. p.4 (in Russian). See also: Soshnio V.N. Damping of plasma-electron oscillations and waes in low-collision electron-ion plasmas// arxi.org/physics/ Soshnio V.N. On some general characteristics of waes in electron plasma//the Integrated Scientific Journal, 6, n.6, p.48 (in Russian). 8. Soshnio V.N. The relation of fast and slow electron waes at propagation of electromagnetic waes in Mawellian collisionless plasma//engineering Journal, 5, n.3, p.48 (in Russian). See also: Soshnio V.N. On some general features of propagation of the pulse signal in electron Mawellian plasma and its bac response//