CYBERNETCS AND PHYSCS, VOL 3, NO 4, 14, 161 165 ACCELERATON OF THE HEAVY ONS N THE FLOW OF THE ELECTRONS Alexander Chikhachev All-Russian Electrotechnical nstitute Krasnokazarennaya 1, oscow, 1115, Russia Abstract The rocess of ion flow extraction fro lasa is one of the ost iortant objects being interesting while creating effective electrojet engine Due to this fact a self-consistent roble of ion acceleration in electron cloud is considered in this research with additional electron flux taken into account The kinetic descrition in the case of collision absence is used Electron flow is described by unconventional solution of kinetic equation which deendent not only on the energy integral t is shown that cold ion can be accelerated to the energy exceeding the electron teerature, ie ion velocity can exceed ion-acoustic velocity Key words Kinetic equation, Poisson equation, ion flow, electron flow 1 ntroduction To study the real rocesses of heavy ion extraction fro lasa the lot of researches is carried out which roose the different extraction odels n work [Rieann, 1991] it is shown that lasa is left by ions with the seeds exceeding ion-acoustic velocity n ractice teerature of electrons T significantly ore than teerature of ions T i T >> T i ) the nuber of the accelerated ions is exonential sall hence the ion current and a traction of the electrojet engine are sall too We will note, however, the work [Kovalenko, Chernyshev, and Chikhachev, 11a] studying acceleration of a thin ionic bea n this work it is shown that the velocity of ions can surass ion-acoustic velocity when the bea radius is changed n work [Sternberg and Godyak, 7] the condition of the accelerated strea cold ion flux existence in the cloud of hot electrons is found n articular, in [Sternberg and Godyak, 7] it is shown that the transitional layer in lasa vacuu syste infinite Current of electrons is equal to zero Work [Kovalenko and Chikhachev, 13] studies conditions for the ion flux oving in the electron layer erendicularely to the direction of the electron flux n this work the axiu energy which ions can gain in a layer, is equal the electron teerature T, however current of ions is not exonential sall t should be noted here that exeriental works exist studying the various asects of the roble of the ower electrojet engine develoent, as exale the work [Erilov and Kovalenko, at al, 8] that studies the traction characteristics such engine here n the real work the consecutive kinetic descrition of ion-electronic syste in the resence of a nonzero strea of electrons is used Thus function of distribution of electrons deends not only on otion integrals Proble Definition We will describe articles by eans of collisionless kinetic equations for both electrons and ions considering the roble for the sake of silicity as onediensional For articles by eans of the kinetic equitation takes the for of: f x + edφ dx f = 1) where is the ass of electron, e is the charge, Φ is a otential, x is a coordinate, is aoentu, f is a article distribution function Equation 1) always has a solution ) f = ΨH) = Ψ eφx), where Ψ - arbitrary function This solution is characterized by article zero flux Γ e along axis x : Γ e = + dψh) = because of antisyetry of integrand At that density n e = + dψh) = + eφx) ΨH)dH H + eφ)
16 CYBERNETCS AND PHYSCS, VOL 3, NO 4, 14 f an exonential energy distribution is used Ψ = κ ex H ), the density has the for of: By the siilar way the ion flow can be described Just like at exression 4) for the ion density we will get: n e = κ π ex ) eφ, ) n i = κ i πi eφ ex i ) where κ is Boltsans constant To describe the non-zero electron flux we will use equation solution 1) that is not only function of the integral of the otion H Equation: f = σ ) C + eφ) ΨH) 3) Here σx) is the Heaviside function, C eφx) for any x t s easy to ake sure that equation 3) corresonds to equation 1) After differentiation in accordance with equation 1) we will get: δ C + eφ)) deφ dx 1 ), C + eφ) ie zero Exression 3) deterines non-zero electron flow: Γ e = C+eΦ) d ΨH) = dhψh) C )) C1 eφ 1 erf, i here is the ion ass, T i is the ion teerature Using asytotic decoosition for error function integral at T i we will get: n i = κ i i ex C ) 1 i i C 1 eφ ) Uon that Γ i = κ i i ex C 1 i, whence it follows: n i = Γ i Let us equate flow density C 1 eφ) as: Γ i = n i υ, n i is an initial ion density, υ is an initial ion velocity and exress in ters of C 1 = υ t should be indicated that here ion descrition corresonds absolutely to hydrodynaic descrition of cold ion flow So we get: n the case of exonential distribution Γ e = κ ex C ) So the electron density ay be exressed by the next forulae: ) π eφ n e = κ ex )) C + eφ 1 erf 4) x Here erfx) = π ex y )dy is an error function integral n 3) the role of the ultilier σ ) C + eφ) is substantial - on the one hand, the shae of article density changes, on the other hand article current is not equal zero Using of a such ultilier for the ais of the bea descrition gives us the ossibility to study the new tye of the bea equilibria Cobining the otion integrals and the ultilier in the for of ste-function one can significantly affect on the balance of the syste if the own agnetic field of the electron flux is taken into account see [Kovalenko and Chikhachev, 13]) Note, the kinetic equation solution with Heaviside function is used in aer [Lohder and Ulyanov, 13] when studying the henoena in the gas discharge Here υ s = equation: e ε n i = Γ i υ i = n i υ eφ = n i υ υ s υ u s, u = eφ Let us write down Poisson d Φ dx = e n e n i ) = ε n i exu)1 erf C T + u) n i υ υ s u s 5), π where n e = κ, ε is the vacuu constant After these diensionless arguents should be introduced t = x κε l, l = T en e, let us set C = ζ We will get: d u dt = exut)) 1 erf ) ζ + u Here ν i = n i n e υ υ s s ν i ut) 6)
CYBERNETCS AND PHYSCS, VOL 3, NO 4, 14 163 3 Results of Coutational Solution Let us set solution results of equation 6), where ζ = 4, ν i = 8, υ υ = 1 n the function of initial condition let us set u) = 4, u) = Deenance of non- s diensional otential ut)/ curve ) fro nondiensional co-ordinate is deictured in Fig1 This solution has a eriodical character t is also reresented qualities of functional deendence of ion density fro co-ordinate ) and electron density fro co-ordinate ) Fro the character of these deendencies we ay conclude: ions accelerate away fro oint t = 4379 where u = 16 ) to oint t = where u = 4 ) Electrons, on the contrary, decelerate in such otion Electrons can accelerate during the otion in the oosite direction: fro oint t = to oint t = 4379 Functional deendence of electron velocity fro otential is deictured in Fig, curve, curve on this icture reresented the deendence of ion velocity fro otential Directions of the velocities strictly oosite Kinetic energy at the oint t = is deterined by the value ζ f this energy equals to 416T t is fourfold as uch as ion teerature The ore is an absolute value ζ, the ore is the value of kinetic energy ions at the exit fro lasa area with the electrons described by distribution 3) This fact confirs the result of the work [5] in which is shown that in a layer of the electrons oving erendicularly to a strea of ions, ions can reach the value of energy which is not exceeding electronic teerature Flat ga is the electrode where t = under otential which is equal to 4T and the second electrode where t = 437 is under otential 16T Electron flow with energy ore then 4T has to fall within flat ga fro electrode with negative otential fro the siilar electrode ions with slow velocity enter in flat ga These ions accelerate to kinetic energy 4T eans of the hydrodynaic equations n the resence of a strea of electrons the ressure of electronic gas P is not equal to P = n e T The decision of the self consistent syste of the hydrodynaic equations is rovided in work [Kovalenko, Chernyshev, and Chikhachev, 13] 4 Three-Part Syste Three-art syste shall be understood as situation, eerging in the event when excet electron flow and ions the oen interval has electrons consisting a cloud with a zero average velocity These articles are described by eans of usual axwell distribution function f ex H/ ) The density of this articles is roortional to exu) Figure 35 3 5 15 1 5 u Deendences of the ions velocity on otential curve ), deendences average velocity flow of the electrons on otential curve ) and deendences average velocity of the electrons on otential when co-existent flow and cloud curve ) 18 16 14 1 1 8 6 4 1 According to entioned above let us add at right side of an equation 4) addend 1 exu) Then the equation for otential 6) turns into following exression: d u dt = exut)) 11 erf ) ζ + u 7) 8 6 4 4 6 8 1 t s ν i ut) Figure 1 Co-ordinate ion density-relationshi curve ), Coordinate electron density-relationshi curve ) and co-ordinate otential curve ) t should be noted that the real consideration significantly differs fro descrition of the syste by A inor addition leads to considerable changes in otential solution Let us set: ζ = 4, υ/ s = 1, ν i = 1 We also use initial conditions u) = 38, u) = Fig 3 shows that electron cloud resence with zero average velocity hardly influences the behavior attern of otential The sae figure deonstrates the change of electron density n such case an additional axiu of electronic density aears at
164 CYBERNETCS AND PHYSCS, VOL 3, NO 4, 14 the sae lace where the axiu of ion density aears The average electron velocity calculated with account of because of electron cloud resence with zero average velocity) over density detects a otential for uward otion in the sae direction that ion velocity lease see Fig 4) t is ossible to create a echanis where a siultaneous acceleration in the sae direction of electrons and ions takes lace Though in such case electron acceleration asses in average uon article deceleration constituting electron flow Sall difference of otential, however, aears only at a sall additive of the axvellian electrons Figure 3 8 6 4 4 6 8 1 5 4 3 1 Co-ordinate ion density-relationshi curve ) and coordiante electron density-relationshi curve ) Co-ordinate otential-relationshi curve ) in case of flow resence and constituting in electron ediu when density clouds 1 exu) strea is locked by the own charge The curve in Fig reresents deendence of the electron velocity on otential in this case deendence isn t onotonous velocity has an extreu, ie there is an area of values of otential where there is an acceleration both the ions and the electrons 5 Conclusions This research studied the behavior of ion and electronic collisionless syste with self-consistent electric field Electrons can be reresented as article flow and cloud flow with zero average velocity and relatively high teerature Cold ions can be accelerated to energy exceeding teerature of electrons, ie their velocity can exceed ion-acoustic velocity if there an electron flow with high directing velocity As axiu otential difference over a articular eriod of the tie is deterined by the value C = ζ, the ain roble is to create electron flow characterized with high directed velocity f ζ = 4 drift velocity shall constitute υ T e n the case of both electrons and clouds exist in the syste, there is an area of values of araeters at which there is a siultaneous acceleration of electrons and ositively loaded ions The technique of the descrition of the selfcoordinated syste of the real work can be useful to the solution of ore colex challenge studying of the electronic and ion bunch liited in the cross direction Probles of the real work were studied also in aers [Kovalenko, Chernyshev, and Chikhachev, 11b], [Chikhachev, 13] Figure 4 6 4 4 4 Coordinate ion density-relationshi curve ) and coordiante electron density-relationshi curve ) Coordinate otentialrelationshi curve ) in case of flow resence and constituting in electron ediu when density clouds exu) f we add to the right art 4)coosed exu) instead of 1 exu) one can find that the character of the decision will sharly change see Fig 4) The decision isn t eriodic now, there are ositive values that results in divergence of the right art and the ionic References Chikhachev, AS 13) on-electronic enseble in electric field Usehi rikladnoy Fiziki, vol 1, no, 189 193 Erilov, AN, Kovalenko, YuA, at al 8) Exeriental investigations of a odel breadboard of Hall thruster High Teerature, vol 46, no 4, 535 541 Kovalenko, YuA, Chernyshev, TV, and Chikhachev, AS 11) Acceleration of heavy ions in the quasyneutral regie Technical Physics, vol 56, no 5, 75 77 Kovalenko, YuA, Chernyshev, TV, and Chikhachev, AS 11) Acceleration of heavy ions at equality of streas of ions and electrons zvestiya Akad Nauk, ser Energetika, no 4, 4 8 Kovalenko, YuA, and Chikhachev, AS 13) Dynaics of an ion flow in an electron layer Technical Physics, vol58, no 5, 77 773 Kovalenko, YuA, Chernyshev, TV, and Chikhachev, AS 13) Acceleration of an ion-electron flow in a lane ga Technical Physics, vol 58, no 4, 68 611 Lohder, Ya, and Ulyanov, KN 13) Theory of the
CYBERNETCS AND PHYSCS, VOL 3, NO 4, 14 165 negative anode dro in low-ressure discharges High Teerature, vol 51, no 1, 7 16 Rieann, K-U 1991) The Boh criterion and sheath foration JPhysD: Al Phys vol 4, 493 519 Sternberg, N, and Godyak, V 7) The Boh lasa-sheas odel and the Boh criterion revisted EEE Transaction on Plasa Science, vol 35, no 5, 1341 1349