Equilibrium of ensemble of ions, produced in plasma placed in crossed fields with positive radial electric field

The equilibrium of ensemble of ions produced in crossed fields with positive (directed outward) radial electric field is studied. The equilibrium ion distribution function and its moments – an ion density and hydrodynamic rotation frequency – are determined.

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Datum:	2017
1. Verfasser:	Yeliseyev, Yu.N.
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Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2017
Schriftenreihe:	Вопросы атомной науки и техники
Schlagworte:	Фундаментальная физика плазмы
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spelling	irk-123456789-1221202017-06-28T03:02:54Z Equilibrium of ensemble of ions, produced in plasma placed in crossed fields with positive radial electric field Yeliseyev, Yu.N. Фундаментальная физика плазмы The equilibrium of ensemble of ions produced in crossed fields with positive (directed outward) radial electric field is studied. The equilibrium ion distribution function and its moments – an ion density and hydrodynamic rotation frequency – are determined. Рассматривается равновесие ансамбля ионов, образовавшихся в скрещённых полях в положительном (направленном наружу) радиальном электрическом поле. Определена равновесная функция распределения ионов и её моменты – распределение плотности ионов и гидродинамическая частота вращения. Розглядається рівновага ансамблю іонів, що утворилися в схрещених полях у позитивному (спрямованому назовні) радіальному електричному полі. Визначена рівноважна функція розподілу іонів і її моменти – розподіл густини іонів і гідродинамічна частота обертання. 2017 Article Equilibrium of ensemble of ions, produced in plasma placed in crossed fields with positive radial electric field / Yu.N. Yeliseyev // Вопросы атомной науки и техники. — 2017. — № 1. — С. 72-75. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 05.20.Dd; 52.25.Dg; 52.55.-s http://dspace.nbuv.gov.ua/handle/123456789/122120 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Фундаментальная физика плазмы Фундаментальная физика плазмы
spellingShingle	Фундаментальная физика плазмы Фундаментальная физика плазмы Yeliseyev, Yu.N. Equilibrium of ensemble of ions, produced in plasma placed in crossed fields with positive radial electric field Вопросы атомной науки и техники
description	The equilibrium of ensemble of ions produced in crossed fields with positive (directed outward) radial electric field is studied. The equilibrium ion distribution function and its moments – an ion density and hydrodynamic rotation frequency – are determined.
format	Article
author	Yeliseyev, Yu.N.
author_facet	Yeliseyev, Yu.N.
author_sort	Yeliseyev, Yu.N.
title	Equilibrium of ensemble of ions, produced in plasma placed in crossed fields with positive radial electric field
title_short	Equilibrium of ensemble of ions, produced in plasma placed in crossed fields with positive radial electric field
title_full	Equilibrium of ensemble of ions, produced in plasma placed in crossed fields with positive radial electric field
title_fullStr	Equilibrium of ensemble of ions, produced in plasma placed in crossed fields with positive radial electric field
title_full_unstemmed	Equilibrium of ensemble of ions, produced in plasma placed in crossed fields with positive radial electric field
title_sort	equilibrium of ensemble of ions, produced in plasma placed in crossed fields with positive radial electric field
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2017
topic_facet	Фундаментальная физика плазмы
url	http://dspace.nbuv.gov.ua/handle/123456789/122120
citation_txt	Equilibrium of ensemble of ions, produced in plasma placed in crossed fields with positive radial electric field / Yu.N. Yeliseyev // Вопросы атомной науки и техники. — 2017. — № 1. — С. 72-75. — Бібліогр.: 9 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT yeliseyevyun equilibriumofensembleofionsproducedinplasmaplacedincrossedfieldswithpositiveradialelectricfield
first_indexed	2025-07-08T21:09:39Z
last_indexed	2025-07-08T21:09:39Z
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fulltext	EQUILIBRIUM OF ENSEMBLE OF IONS, PRODUCED IN PLASMA PLACED IN CROSSED FIELDS WITH POSITIVE RADIAL ELECTRIC FIELD Yu. N. Yeliseyev Institute of Plasma Physics, National Science Center “Kharkov Institute of Physics and Technology” Kharkov, Ukraine e-mail: eliseev2004@rambler.ru The equilibrium of ensemble of ions produced in crossed fields with positive (directed outward) radial electric field is studied. The equilibrium ion distribution function and its moments – an ion density and hydrodynamic rotation frequency – are determined. PACS: 05.20.Dd; 52.25.Dg; 52.55.-s 1. INTRODUCTION Collisionless plasma placed in crossed fields (ExB) is used in numerous technological devices. The ions produced due to ionization of atoms and molecules by electron impact, do often appear unmagnetized and the ensemble of such ions has to be described kinetically. In the absence of collisional relaxation the equilibrium ion distribution function (DF) should reflect the peculiari- ties of an initial state of ions. In paper [1] the kinetic equilibrium of ensemble of ions in plasma placed in crossed longitudinal magnetic field and negative (di- rected inward) radial electric field is described. In the present paper the equilibrium of ensemble of ions pro- duced in crossed fields with positive (directed outward) radial electric field is studied. The equilibrium ion DF and its moments – ion density and hydrodynamic rota- tion frequency – are determined. 2. THE GEOMETRY AND EQUILIBRIUM MODEL We consider the case when the basic plasma is pro- duced by an external source and completely fills a cy- lindrical waveguide of radius a bounded by a metal casing. The plasma is under action of crossed fields: homogeneous magnetic field with strength H (induc- tion B ) directed along the axis of a cylinder (axis z ) and positive (directed outward) radial electric field with intensity rE ( 0rE > ). The radial electric field of any polarity (and, proba- bly, of any radial dependence) can be created in plasma by means of the end electrodes, when applying to them corresponding potentials, which are transferred along the magnetic field by plasma electrons [2]. Just in such a way the positive radial electric field is created in the installation Archimedes Plasma Mass Filter [3]. If not specified, everywhere below the potential of positive electric field 0 ( )rΦ is a quadratic function of radius: ( )2 2 0 0( ) ( ) /r a r aΦ = Φ and ( )0 0aΦ < . We consider the radial motion of ions to be finite, and the electric field is not too strong: r crE E< . Here 2 4 0cr r i ci iE m r e≡ ω > is the critical electric field and ie , im , ci i ie B m cω = are charge, mass, and cyclotron frequency of ion. The vapours of mixture of substances are introduced (as atoms and/or molecules) into the basic plasma and are ionized by electron impact within the whole plasma volume, having the form of a cylinder with radius a . We want to describe the kinetic equilibrium of ensemble of these ions that move without collisions. Because of big mass difference between electrons and neutrals, one may consider that ions are produced in a state of rest. The kinetic energy they gain in any noticeable electric field does considerably exceed the initial ion kinetic energy which is close to thermal energy of neutrals. -2 -1 0 1 2 -2 -1 0 1 2 0ϕ 0t = x/r0 y/r0 Fig. 1. The trajectory of the ion produced in a state of rest at radius 0r and azimuth angle 0ϕ in crossed fields with positive radial electric field: / 0.7cr r rE E = , 0 / 3ϕ = π , ( ) ( )max 0 0 1.83ci ir r r= ω Ω ≈ . The dashed circles correspond to radii 0r and ( ) ( )max 0 0 ci ir r r= ω Ω The ion produced in a positive radial electric field at radius 0r moves along epicycloid (see Fig. 1) and reaches the maximum radius (an outer turning point on radius) [4] ( ) ( )max 0 0 ci ir r r= ω Ω . (1) Here ( ) 1 2 24i ci i r ie E m r constΩ ≡ ω − = (2) is the “modified” ion cyclotron frequency in crossed fields [5], ( )0rE r d r rdr const= − Φ = . With positive electric field ci iω > Ω . From (1) it follows that those ions produced at the periphery of plasma 0 ir b> , where ( )i i cib a≡ Ω ω , (3) hit the chamber wall and stop to exist [4]. Such ions are not considered in the steady-state model. The ions pro- duced closer to the axis, 0 ir b< , are confined; they move freely alone the trajectories, that do not cross the wall throughout many periods of radial oscillations. Only such ions we consider in the steady-state model. The quantity ib is naturally to name as the radius of confinement of i –kind ion [4]. The value of radius ib depends on an ion mass. In a mixture of ions of different masses radii ib are different. Namely, the heavier ions have the shorter ib . For heaviest ions, which move infinitely along ra- dius, there is no equilibrium state: they are thrown out onto the chamber wall from the whole plasma volume. 3. EQUILIBRIUM DISTRIBUTION FUNCTION OF ENSEMBLE OF IONS The equilibrium DF of ions produced in crossed fields with positive radial electric field can be found by the use of the procedure developed in [1] for negative electric field. In this section the radial dependence of the positive field, ( ) 0rE r > , is considered to be arbitrary, but such that ions are confined in the volume of basic plasma by magnetic field, i.e., there is an outer turning point at radius r a< , where 0rv = . According to as- sumed model of plasma, the total density of ion produc- tion ( )0iN r is considered non-zero only in the region 0 ir b< , and the ions produced here are only confined, but ( )0 0iN r ≡ in the region 0 ir b> . At the plasma periphery ( ir b> ) the expression for ion density distribution has the form ( ) min 0 0 ( ) 2( ) , , ib i i r rr r N rn r dr r T v M r⊥ = ε∫ . (4) In the inner area ir b< the density amounts: ( ) min 0 0 ( ) 2( ) , , r i i r rr r N rn r dr r T v M r⊥ = ε∫ . (5) In (4), (5) the value ( ) ( ) ( )max 0 0 0 2 , , r r r r rr drT T r v M r⊥ = = ε∫ is the period of radial oscillations of the ion, born at rest at radius 0r , ( ), ,rv M r⊥ε is the radial velocity of ion at radius r ; max 0( )r r in rT is the radius of its outer turning point; min ( )r r in (4), (5) is the radius of inner turning point (point of ion production) corresponding to the outer turning point r . In the electric field with the square-law radial dependence of potential these quantities are: ( ) ( )min i cir r r= Ω ω , ( ) ( )max 0 0 ci ir r r= ω Ω . The energy and the generalized angular momentum of ion, ( ) 2 0 00, / 2 0i i cie r M m r⊥ε = Φ < = ω > (6) are the integrals of motion. The relationships (6) are the consequence of the fact that ions were born at rest. When deriving (4), (5) it was supposed, following [6], that the probability Pdr for the ion produced at radius 0r to be located in a ring ( ),r r dr+ , is propor- tional to the ratio: time of intersection of the ring / rdr v to the period of radial oscillation rT . Keeping in mind that, moving along its trajectory, the particle crosses that ring twice, the result should be doubled: ( )2 , ,r rPdr dr T v M r⊥= ε   . Using the density distributions (4) and (5) let’s de- rive the equilibrium DF, which has to be the function of integrals of motion ( ), , zF F M v⊥= ε . From (6) it fol- lows that for the ions born at rest at an arbitrary radius 0r , there exists the connection between energy and momentum ( )( )1 2 0 2i i cie M m⊥ε = Φ ω . (7) In the phase space , M⊥ε the ions are present only along the line (7), and the rest part of the space is emp- ty. Taking into account (7), the equilibrium DF should have the form: ( ) ( )( )( ) ( ) 1 2 0 2i i ci zF f e M m v⊥ ⊥= ε δ ε − Φ ω δ , (8) where the function ( )f ⊥ε has to be found. This can be done by integrating function F (8) over the velocity space. The ion density ( )in r , found in such a way, we compare with (4), (5). Thus: (9) ( ) ( )2 2 3 2 0 0 2 , , r r i z i rv v Fn r Fd v d dM dv m r v M r +∞ ⊥ ⊥−∞≥ ≥ = = ε ε∫ ∫∫ ∫ . In (9) the transition from velocities to new variables – integrals of motion , M⊥ε – was made. The Jacobian of transition is equal ( ) ( ) 2, ,r i rM v v m r v⊥ ϕ∂ ε ∂ = [7]. The integration was made over that space area , M⊥ε , where 2 0rv ≥ . Integrating over zv in (9), changing the order of in- tegration over , M⊥ε , then integrating over ⊥ε , and passing from M to a variable 0r (in accordance with the second inequality (6)), we come to the expression ( ) ( )( ) ( )( )min 0 0 0 0 2 ( ) 0 0 0 2 . , 2 , r ici i i r r r i i ci f e r r dr n r m r v e r m r r Φω = Φ ω∫ (10) When comparing this expression with (4), the value of function f at plasma periphery, can be found: ( ) ( ) ( )( )0i r i ci if N T m Y e b⊥= ω ε − Φ , ir b> . (11) Thus, the equilibrium DF of ions produced in posi- tive radial electric field, in the area ir b> has the form: ( )( ) ( ) 1 2 0 0 2 ,i i i i i z r ci i ci N m MF Y e b e v T m⊥ ⊥      = ε − Φ δ ε − Φ δ   ω ω    (12) The presence in (12) of a Heaviside step function ( )( )0i iY e b⊥ε − Φ means that in an ensemble of ions only those ions that are produced at radii from the inter- val ( )min ( ), ir r b are present, and there are no ions born in the interval ( ),ib r , as it is supposed in the equilibrium model. Comparing (10) and (5), the expression for f inside the area where ions are produced, ir b< is: ( ) ( )i i r cif N m T= ω , ir b< . (13) The DF in this area has the form: ( ) 1 2 0 2i i i z r ci i ci N m MF e v T m⊥      = δ ε − Φ δ   ω ω    . (14) The absence of Heaviside step function in (13) and (14) means that all ions that are produced in the interval [ ]min ( ),r r r reach radius r and give the contribution to the DF. The ions produced outside of this interval, can- not reach radius r and cannot contribute to the DF. Expressions (12) and (14) were derived for arbitrary dependence of potential ( )0 rΦ on radius r ( 0rE > ), for arbitrary dependences ( )0iN r and ( )0rT r . In de- pendences ( )0iN r and ( )0rT r it is necessary to express 0r in terms of ⊥ε or M , using the first or second rela- tion (6). By the use of the second relation (6), it always can be made explicitly. As it’s seen from (12), (14), the equilibrium DF of ions, produced due to ionization of neutral particles (atoms, molecules) by electron impact in crossed fields, is a non-Maxwellian and anisotropic one. When the potential ( )0 rΦ has a quadratic depen- dence on radius (thus, dependence (7) becomes linear) and quantities iN and rT are constant quantities, the DF (14) becomes ( ) ( ) ( ) ( ) , .i i r ci e z iF N m T M v r b⊥= ω δ ε −ω δ < (15) This DF belongs to the type of "a rigid rotator» [5] with a rotation frequency e rcE Brω = − . In the region ir b> the DF (12) cannot have the form of "a rigid rotator» because apart from a connec- tion between energy ⊥ε and the momentum M , which is contained in δ - function, there is, at least, one more dependence on the energy ⊥ε , containing in the Heavi- side step function ( )( )0i iY e b⊥ε − Φ . The latter makes the DF (12) similar to a degenerate Fermi-Dirac one. 4. MOMENTS OF THE EQUILIBRIUM ION DISTRIBUTION FUNCTION Let's define the density distribution of ions ( ) 3 in r Fd v= ∫ and "hydrodynamic" frequency of rota- tion ( ) ( )( ) 31i ir n r r Fv d vϕ ω =   ∫ of ensemble of ions having the DF like (12), (14). The potential of electric field is assumed to be a square-law function of radius ( )2 2 0 0( ) ( ) /r a r aΦ = Φ at r a< . Thus rE r const= and 2r iT const= π Ω = . We consider also ( )0iN r const= at 0 ir b< and ( )0 0iN r = at 0 ir b> . By integration it can be found 2 2 2 2 2 ( ) , 21 1 arcsin 1 1 , 2 i i i i ci i i i ci ci i i n r N r b aN r r b = Ω ≤ ω =    Ω Ω − − −   ω π ω −Ω      ≥ (16) A "hydrodynamic" rotation frequency ( )i rω of en- semble of ions is: i ( )rω = (17) e i 1 22 22 2 2 2 2 2 2 2 e 2 2 2 2 2 i , 1 1 1 1 2 , 2 arcsin 1 1 2 . i i ci i ci i i ci i r b a a r r a r r b ω <      Ω Ω  − − −     ω −Ω ω −Ω       ω +   Ωπ − − −  ω −Ω       > where ( ) ( )2 24 1 0e r ci i cicE Brω = − = ω Ω ω − < . So, / 4e ciω < ω . 5. DISCUSSION The radial dependence of ion density in positive and zero radial electric fields is illustrated in Fig. 2. It is seen from the expression (16) and Fig. 2 that in positive electric field the density satisfies the inequality ( )i in r N< (lines 1, 2, 3). Its dependence does sharply differ in the inner ( ir b< ) and outer ( ir b> ) areas. In zero radial electric field the density ( )i in r N const= = within the whole area r a< (line 4). The radial dependence of rotation frequency of en- semble of ions ( )i rω (17) is presented on Fig. 3. The frequency of rotation is negative at all radii, i.e. the en- semble of ions rotates counterclockwise around z axes. Its dependence sharply differs in the inner area ir b< where it is constant and equal eω , and in the outer area ir b> where rotation frequency ( )i rω is strongly in- homogeneous. Equality ( )i erω = ω in the inner area is a consequence of the fact that the DF of ensemble of ions (15) is of «a rigid rotator» type [5] with the rotation frequency equals to eω . In radial electric fields of equal magnitude, but di- rected inversely, the rotation frequencies of ensemble of ions in the inner area ( ir b< ) are identical in size ( e e+ − ω ≈ ω ) but are inverse in direction. In the outer area ( ir b> ) the rotation frequency in a positive electric field exceeds the frequency with negative field: ( ) ( )i e ir r + − ω ≥ ω ≥ ω . The quantity ib is less than the corresponding quantity ( )min ( ) ci ir a a − = ω Ω in a negative electric field, i.e. min ( )ib r a< . 0 1 0,0 0,5 1,0 4 ( )i in r N 3 2 r/a 1 Fig. 2. The density distribution of the ions produced in a state of rest in crossed fields with positive (curves 1, 2, 3) and zero (straight line 4) radial electrical fields. The values of parameter i ciΩ ω for curves 1-4 are 0.3, 0.6, 0.8, 1i ciΩ ω = , respectively It is interesting that in the inner area ir b< the rota- tion frequency of ensemble of ions ( )i rω does not de- pend on the ion mass and the ensemble of ions rotates with the same frequency eω , as electrons. In the outer area ir b> in a positive field the ions rotate even faster than the ensemble of electrons, ( ( )i er + ω ≥ ω ). At the plasma edge ( r a→ ) we have ( ) 2i erω → ω . These peculiarities of rotation of ensemble of ions do not have a direct relevance to plasma stability, because the dis- persion equation includes characteristics of one-particle ion motion ( ( ) 2i ci i ±ω = −ω ±Ω [5]), instead of the whole ensemble. It should be noted, that the expressions for density distribution (16) and the rotation frequency (17) derived in positive radial electric field, have appreciable similar- ity with the corresponding expressions derived in the negative radial electric field [6, 8], however, there is no complete coincidence. Expression (16) and Fig. 2 give the flavour not only of the dynamic density distribution of ions in plasma, but also of the radial distribution of ions leaving the plasma. So, they provide an indication on radial distri- bution of light ions settled on the face collector (meant for collection of light ions) in the Archimedes Plasma Mass Filter device [9]. The expression (16) and curves 1-3 in Fig. 2 allow to understand the peculiarities ob- served in experiment [9] as for the distribution of amount of matter deposited on the face collector de- pending on radial electric field intensity. 0 1 -0,5 0,0 2 1 3 4 r/a ( )i ci rω ω Fig. 3. "Hydrodynamic" rotation frequency of ensemble of the ions produced in crossed fields, with positive (curves 1, 2, 3) and zero (a straight line 4) radial elec- tric fields. Curves 1, 2, 3 are calculated by formula (17). Values of parameter i ciΩ ω are the same, as in Fig. 2 REFERENCIES 1. V.G. Dem’yanov, Yu.N. Yeliseyev, Yu.A. Kiroch- kin, et al. // Fiz. Plazmy. 1988, v. 14, p. 840 [Sov. J. Plasma Phys. 1988, v.14, p. 494]. 2. A. Morozov and S. Lebedev. Plasma-Optics, Re- views of Plasma Physics, issue 8, M. Leontovich, Ed., New York: Consultants Bureau, 1975, p. 247 [Мoscow: Atomizdat, 1974, p.247–281]. 3. T. Ohkawa Plasma Mass Filter// US Patent #6,096,220 (August 1, 2000). 4. Yu. N. Yeliseyev//Problems of Atomic Science and Technology, Series ‘Plasma Phys.’.2016, № 6 (106). 5. R.C. Davidson. Theory of Nonneutral Plasmas. Ben- jamin, New York,1974.[Мoscow: Mir, 1978, 215 p.] 6. G.E. Sieger, R.C. Conolly, H.W. Lefevre //Phys. Fluids. 1984, v. 27, p. 291. 7. R.H. Levy, J.D. Daugherty, O. Buneman //Phys. Fluids. 1969, v. 12, p. 2616. 8. M.A. Vlasov, E.I. Dobrochotov and A.V. Zharinov // Nuclear Fusion. 1966, v. 6, p. 24. 9. J. Gilleland, C. Ahlfeld, C. Little //Waste Manage- ment Conference 2005 (WM05). Tucson, Arizona, USA, Febr. 27 – March 3, 2005. РАВНОВЕСИЕ АНСАМБЛЯ ИОНОВ, ОБРАЗОВАВШИХСЯ В ПЛАЗМЕ В СКРЕЩЕННЫХ ПОЛЯХ ПРИ ПОЛОЖИТЕЛЬНОМ РАДИАЛЬНОМ ЭЛЕКТРИЧЕСКОМ ПОЛЕ Ю.Н. Елисеев В работе рассматривается равновесие ансамбля ионов, образовавшихся в скрещенных полях в положитель- ном (направленном наружу) радиальное электрическое поле. Определена равновесная функция распределения ионов и ее моменты – распределение плотности ионов и гидродинамическая частота вращения. РІВНОВАГА АНСАМБЛЮ ІОНІВ, ЩО УТВОРИЛИСЯ В ПЛАЗМІ В СХРЕЩЕНИХ ПОЛЯХ ПРИ ПОЗИТИВНІМ РАДІАЛЬНІМ ЕЛЕКТРИЧНІМ ПОЛІ Ю.М. Єлісеєв У роботі розглядається рівновага ансамблю іонів, що утворилися в схрещених полях у позитивному (спрямованому назовні) радіальному електричному полі. Визначена рівноважна функція розподілу іонів і її моменти – розподіл густини іонів і гідродинамічна частота обертання. http://context.reverso.net/%D0%BF%D0%B5%D1%80%D0%B5%D0%B2%D0%BE%D0%B4/%D0%B0%D0%BD%D0%B3%D0%BB%D0%B8%D0%B9%D1%81%D0%BA%D0%B8%D0%B9-%D1%80%D1%83%D1%81%D1%81%D0%BA%D0%B8%D0%B9/an+exact+match Ю.Н. Елисеев Ю.М. Єлісеєв

Equilibrium of ensemble of ions, produced in plasma placed in crossed fields with positive radial electric field

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