1.Introduction

The most interesting effects discovered in gas discharge are associated,we believe,with the generation of ions with energies considerably exceeding(by a factor of up to 100)the values equivalent to the discharge voltage.There are two notable cases where this effect was observed:the production of accelerated ion beams in the plasma of pulsed vacuum discharges and in stationary systems with a hollow cathode.A pioneering study into the generation patterns of accelerated ion beams in vacuum discharge plasmas(sparks)is due to research efforts made by Plyutto’s team[1–3].It should be emphasized that at the time of generation of the electron beam their experiments recorded energetic ions traveling along the direction of propagation of the electron beam.As reported in[1],the energy of a part of the heavy component(C+)reached 4eVd(Vd = 9kV is the amplitude of the accelerating voltage)and the energy of the lightest component(H+)reached 2eVd.In this case,the mean energies were estimated at 2.5keV.The authors considered as possible acceleration mechanisms the ambipolar acceleration by electrons during the expansion of the plasma as a whole and the acceleration in the boundary double layer with an electric field intensity of the order of 104–105Vm-1.

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Plyutto et al[2]reported on the predominant recording of light ions with maximum energies Wm,exceeding eVdby factors of 10 to 100.Korop and Plyutto[3]reported on the determination of commonalty of the processes in the case of vacuum breakdown and the generation of electron beams from plasma emitters.

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Luce diodes provided a basis for solving the problem of generating powerful ion beams[4,5]where protons and deuterons with energies of up to 15 MeV were obtained in the case of electron energy We = 4MeV.Later,results reported in[1–5]were summarized into the overall scheme of acceleration in systems with a virtual cathode[6,7].

The model of anomalous ion acceleration based on the concept of a nonstationary potential well was developed consistently in[8–10].It was shown that in the unstable mode of a vacuum discharge the electron emission current exceeds the Child–Langmuir limit,which is accompanied by the generation of a virtual cathode with amplitude of up to 80%of Vd[8].The depth of the stationary potential well is independent of the value of the external accelerating field and is 8We/3,where Weis the kinetic energy of injected electrons.The energy of singly charged ions was W ≈ 2.7We.

An alternative ‘non-macroscopic’approach to determining the ion acceleration mechanism has also been actively developed.Adler et al[11]were the first to suggest that electron–ion instability should be taken into consideration as the mechanism of impulsive acceleration of ions.Kondratenko and Kostenko[12]investigated more comprehensively the collective acceleration of ions as they are captured into the potential well of the Langmuir wave excited at the onset of a two-dimensional instability of an electron beam in plasma[12].

It is obvious that the ions in gas discharges also travel along a ‘natural direction’,i.e.along the external accelerating electric field.Pulsed vacuum discharges were reported in[13],where a pulsed electric field was applied to the plasma filled discharge gap.Ions accelerated toward the cathode were recorded.Intense beams of charged particles were generated at the time of formation(in the current-carrying plasma)of a‘disruption’,the region of spatial charge within the plasma in which the discharge potential difference was concentrated.The mean energies of singly,doubly and triply ionized carbon ions were equivalent to the discharge voltage eVd.The maximum energies exceeded eVdand were dependent on the multiplicity of ion charges,which was testimony to the potential character of particle acceleration in the layer.

In stationary plasma thrusters,and in generators of ion flows with hollow cathodes(e.g.[14,15]),the energies of xenon ions,as measured with a retarding potential analyzer(RPA),reached the highest probable values corresponding to the maximum of the distribution function Wm ≈ 1.5eVdwith maximum values of the order of 3.5eVd[16].An increase in discharge current led more conspicuously to an increase in the ion energy in contrast to the increase in ion energy with an increase in discharge voltage.Conceivably the most energetic ions with maximum energies W ～ 10eVdwere observed with an electrostatic energy analyzer by the authors of[17].

The authors of[18]were the first to point out the fact of interest to us here that the maximum of the distribution function of xenon ions shifted toward high energies exceeding eVdby factors of 2 to 3 times,with an increase in discharge current Id.The same result was obtained in[19],were with an increase in Id the ion spectrum broadened toward low as well as high energies.

The most familiar devices with an E × B discharge are plasma accelerators(thrusters)with a closed drift of electrons(e.g.[20]),in which ions with energies exceeding the‘discharge voltage’are also recorded[21,22].The distribution function of the working gas ions in such accelerators was measured with different analyzers:RPA ‘total’,and analyzers featuring a Wien velocity filter or using the method of laser induced fluorescence(LIF).

Results for singly charged xenon ions in a stationary plasma thruster(SPT),obtained by the LIF method(its importance for us should be emphasized here,i.e.identification of the ion with the known charge multiplicity),showed[23,24]that the main acceleration of particles occurs in the layer with an axial size of about 3mm,with the length of the acceleration channel totaling 15mm.A bi-modal spectrum of ions is generated,with beam velocity Ubeam ≈ 1.4Ubulk.An increase in the xenon flow rate leads to a slight increase in the amplitude of the potential jump(it tends to eVd)and to its steepening.For a lowpower SPT,when Vd = 250V,an almost Maxwellian distribution function for Xe+was recorded.The velocity of Xe+in the azimuthal direction increased with increasing magnetic field induction.A high-energy tail of Xe2+ions(as high as 163eV)is recorded in the axial direction along the SPT axis[25].

The ‘low-voltage’operation mode of SPT is described in[26]:Vd = 100–120 V.As the neutralizer,the SPT uses the plasma source with a hollow cathode located axially with respect to it,and the gas flow rate(CCF)through it influences the SPT discharge current.When CCF ≥ 0.24ACF(ACF being the gas flow through the anode),the SPT changes to low-current’mode,Id.low,whose value is about 10%smaller than the value of Id.highin ‘high current’mode.These two modes differ greatly in the energy spectra of xenon ions.In the high-current mode,the energy of the maximum of the Xe+distribution function begins to exceed the value of eVd(measurements made far into the plume).

As an example of the RPA-based measurement of the energy distribution function of xenon ions,we refer the reader to[27,28]where ions were recorded with energies larger than eVdin the tail of the distribution function.

In the case where the anodic current in SPT is a pulsating current(when Vd = const)and,as a consequence,the ionic current also pulsates,ion energy spectra can be measured using the RPA located at different distances LRPAfrom the SPT output(LRPA = 35–70cm)by resorting to the transit time technique.We were able to identify a more intense component on the ionic signal,Xe+,and a second, ‘faster’,component which can be generated by Xe2+ions as well as by ‘superfast’singly charged xenon ions.The fast component was clearly identified at LRPA = 70cm.When Vd = 220V,the difference between the mean energies of the two components is estimated at 15eV;here,no ions have energies exceeding eVd[29].

2.Formulation of the problem

This paper forms part of the research into the physical principles of plasma-optical mass separation and of the efforts aimed at the creation of the first model of a plasma-optical mass separator POMS-E-3[30,31].Among the objectives of the investigations is the generation of a multi-component ion beam.The device used for beam generation is a two-chamber plasma accelerator with an anodic layer,PAAL(with a cold cathode)[32].The second PAAL chamber also serves as the azimuthator,which is represented by the region with an applied radial(transverse to the ion velocity)magnetic field(up to 0.5T);when passing through it,the ions with different masses acquire different azimuthal velocities,i.e.they are separated by their masses.It was necessary to determine the capabilities of a PAAL operating in the range of relatively large　discharge　voltagesVd = 600–2000 V,and　radial magnetic fields in the anode region BrA = 10–2to 10–1T.In the process of investigating a two-chamber PAAL,we detected ions with bulk energy exceeding eVdby factors of 1.5 to 2;therefore,our goal was to determine the modes of operation of the PAAL in which such ions exist,and to find the parameters governing the energy of anomalously accelerated ions.

Figure 1.Schematic of the two-chamber PAAL:1,anode;2,cathode 1;3,cathode 2(azimuthator);4,insulator;5,screens;6,gas distributor.

3.Experimental results

The two-chamber PAAL used in this experiment is schematically shown in figure 1.The first cathode(cathode 1,position 2)of the PAAL,made of stainless steel(nonmagnetic)can be electrically isolated from the magnetic circuit.In this case,an electric potential VC1from a separate source is applied to it,or it can be under a floating potential V fl.Thus we have a two-stage PAAL.If cathode 1 is grounded,we have a one-stage PAAL.Cathode 2(azimuthator;position 3)always is under a zero potential;it forms part of the magnetic circuit and generates two magnetic poles.The supply voltage sources with the plasma accelerator in operation have a relative level of oscillations ΔV/Vd ≤ 2%,which is much smaller than the amount by which the energy of our recorded ‘super-accelerated’ions exceeded eVd.The frequency of pulsations of the discharge voltage did not exceed 200Hz.The discharge chamber in the anode–cathode region is con fined by metallic wall(shields,5),which were under the floating potential.The working discharge voltages for the one-as well as the two-stage modes were Vd = 900　and　1100V.The　anode–cathode　separation was 8mm,and the separation of cathode 1 and cathode 2(azimuthator)was 30mm.In the anode–cathode 1 gap,Br increased monotonically,and the value of the longitudinal(Bz)component did not exceed 13%of Br.The experiment was conducted for two values of magnetic induction on the anode:B1rA ≈ 3.12 × 10–2T;in this case,BrC1 ≈ 4.6 × 10–2T on cathode 1,and on cathode 2(azimuthator)B1rC2 ≈ 0.29T and B2rA ≈ 3.78 × 10–2T,B2rC1 ≈ 5.7 × 10–2T,B2rC2 ≈ 0.36T.The working gases were:helium(He),nitrogen(N2)and argon(Ar).The gas flow rate was set in the range 0.1–0.8mgs-1;the pressure at the azimuthator output did not exceed 2 × 10–4Torr.

Langmuir and emissive probes were used for diagnostics of plasma parameters.The overall volt–ampere characteristics(VAC)of the flat and cylindrical probes in the discharge region between the anode and cathode of the PAAL were difficult to measure because of electrical breakdowns to the probes at probe voltages Vp ≥ 80 V.Good results from measuring electron temperature Tewere obtained by processing the VAC near the floating potential where the VAC was approximated most accurately by an exponential function.

A three-grid retarding potential analyzer was used to measure the ion energy distribution functions.The external grid and the RPA body were grounded.Secondary electron emission from the collector was suppressed by applying to it a positive potential of 10 V.Ion energy spectra were measured immediately at the output of the first stage,‘spectra in PAAL’,in the second chamber when the PAAL was operated in the two-stage mode,and at the output of the cathode–azimuthator in the ‘near plume’.The RPA input accommodated the collimator(a tube of nonmagnetic metal 4.5mm in diameter and 45 mm long)which was introduced through the azimuthator to,for example,the PAAL output in order to fix the ‘point of collection of ions’.

3.1.Evolution of the energy spectra of ions

The effect of anomalous acceleration of ions was discovered experimentally[33].Examples of measurements at fixed Vdand Brare given in figure 2,where argon was used as the working gas,and in figure 3 for helium(the distance z is measured from the PAAL anode,where z = 0).Flow rates of the gases determining pressure P ≥ 12 × 10–5Torr were not set,because the PAAL operation mode was unstable in this case.

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With an increase in pressure higher than P ≈ 9 × 10–5 Torr(spectra 1–3 in figure 2(a),and spectrum 4 in figure 3),the energy distribution function of ions as a whole traveled mainly beyond the boundary W = eVd(the heavy vertical lines in figures 2 and 3).The most probable energies of argon ions( figure 2(a))were Wm ≈ 1125 eV,1362 eV and 1581eV,respectively.In the near plume( figure 2(b),z = 60mm,output from the cathode 2),the energy of the ions is less than in the gap between cathode 1 and cathode 2.The reason is probably that the ions are decelerated by the ambipolar electric field produced by the separation of ions and electrons at the magnetic barrier of cathode 2(of the azimuthator).

It follows from the data in figure 9 that with a decrease in magnetic field induction at fixed values of J,the value of the potential jump also decreases.In order for a potential jump of,for example,1.6Vdto be reached,the total ionic current density must exceed a critical density by a factor of about 3.

Figure 2.Two-chamber PAAL(Vd = 1100 V,VC1 = V fl).Evolution of energy spectra of argon ions at different pressures:1,P = 12 × 10–5 Torr;2,10 × 10–5Torr;3,9 × 10–5Torr;4,8 × 10–5Torr;5,7 × 10–5Torr.(a)BrA ≈ 3.12 × 10–2T;z = 18 mm;(b)BrA ≈ 3.78 × 10–2T;z = 60 mm.

Figure 3.The energy distributions of helium ions:1,P = 7 × 10–5 Torr;2,8 × 10–5Torr;3,10 × 10–5Torr;4,12 × 10–5Torr;Vd =1100 V;BrA ≈ 3.12 × 10–2T;z = 14 mm;grounded cathode 1,VC1 = 0.

Figure 4.The energy distributions of argon ions:1,Id = 90 mA;2,Id=170 mA;3,Id=247 mA;Vd = 1100 V;BrA ≈ 3.78 × 10–2T;z = 14 mm;grounded cathode 1(VC1 = 0);[31].

The experiments with argon as the working gas showed super-accelerated　ions　in　the　two　discharge　voltages( figure 5).When Vd2 - Vd1 = 200 V,the difference Wm2 -Wm1 = 1508 - 136 = 146eV ≠ 200 eV.As is evident from the spectra shown in figures 2–5,the entire distribution function is evolving with a change in pressure of neutral,discharge voltages and current.

Figure 5.The energy distributions of argon ions in the second chamber of PAAL(z = 35 mm):1,Vd= 900 V,P = 10 × 10–5Torr;2,Vd=900 V,P=12 × 10–5Torr;3,Vd=1100 V,P=10 × 10-5Torr;4,Vd=1100 V,P=12 × 10-5Torr;VC1 = V fl.

The results of experiments with different working gases are of considerable interest:argon,a relatively readily(also doubly)ionizable gas,and helium and nitrogen with a large double ionization potential,with the ionization of nitrogen producing molecularas well as atomic N+ions.In this case,the contribution fromis as large as 0.78 with an increase in pressure to P = 1.5 × 10–4Torr[34];thus the molecular nitrogen ions in the ion flow make a determining contribution.Recall that the ion/initial neutral ionization potentials I for the aforementioned gases are:14.534eV;I(N+2/N) = 29.60eV;I(Ar+/Ar) =　15.76eV;I(Ar+2/Ar) = 27.63eV;I(He+/He) = 14.53eV and I(He+2/He) =54.42eV[35].Figure 6 presents the spectra of singly charged nitrogen,helium and argon ions for the largest pressure used in the experiment.

3.2.Influence of the external magnetic field

The dependence of Wmon the value of magnetic field induction in PAAL is plotted in figure 7(z = 113 mm).It is seen that there are no energetic ions present in the case of small magnetic fields because the burning of a discharge is difficult in this range of fields.With induction on the anode BrA ≥ 4 × 10–2T,the generation of ions with Wm ＞ eVdis suppressed by the magnetic field.The range of fields 2.8 × 10–2 ≤ BrA ≤ 4 × 10–2T turned out to be the ‘working’range for anomalously accelerated ions.It should be noted that the longitudinal component of the magnetic field BzA ≤ 0.1BrAfor small BrAis about BrA/3 under optimal conditions for the generation of anomalously accelerated ions,and approaches 0.8BrA.When BrA ≥ 4 × 10–2T,there no super-accelerated ions occur.

Figure 6.The distributions of the nitrogen argon(curve 2)and helium(curve 3)ions at P = 12 × 10–5Torr;Vd= 900 V;BrA ≈ 3.12 × 10–2T;z = 14 mm;VC1 = 0.

Figure 7.Dependence of the most probable energy of argon ions on the value of magnetic field induction on the anode(a polynomial approximation by the experimental curve);P = 12 × 10–5Torr.The heavy horizontal line corresponds to eVd.

4.The analytical model

It should be noted that the turbulent nature of ion acceleration implies that the increase in energy is realized for only a certain fraction of the total number of particles that will be captured by the wave and will resonate with it(Vph ≥ Vi,where Vphis the phase velocity of the wave and Viis the velocity of the ions).In our case,as shown in the previous section,most of the ions are accelerated.Oscillations of the ion current have the character of packets separated by about 10–5s and with a time length of about 2 × 10–6s.The oscillations in the packet are amplitude modulated with a frequency of about 10MHz.The relative amplitude of ion current oscillations is ΔIi/Ii ≤ 0.05(here Iiis the total ion current per collector RPA).The nonstationarity of high-frequency oscillations and their relatively small amplitude are not grounds for discussing the possibility that the bulk of ions can be additionally accelerated by plasma waves.From the standpoint of our experiment,anomalous acceleration of ions must occur on a macroscopic potential jump.Therefore we now outline our idea and demonstrate in principle the possible existence of a virtual anode in the PAAL anodic layer.

4.1.Generation of the anodic layer—a virtual anode

A possible generation of a ‘negative anode fall’potential jump Δφ in the near-anode region was shown experimentally in[36].The total voltage drop in the anodic layer is Vd + Δφ in this case.The measured value of Δφ ≤ 0.025Vd.The more serious changes in the plasma potential distribution in the near-anode region of SPT occurred with a change in the width of the discharge channel from 15 to 25 mm[37].For a wide channel,the near-anode potential layer is nearer to the anode and the maximum value of the potential in it exceeds the discharge voltage by about 20%.

A distinct(tens of volts)positive potential plasma jump in the anode region is observed for sources of ions with a hollow cathode[38,39].Our experimental data require higher values of an additional potential jump in the anodic layer.

In the theory of Hall accelerators,the motion of electrons from the cathode to the anode is usually considered within the diffusion　approximation.For a low-pressure discharge,however,the dynamics of electrons can be described in the collisionless approximation.In[40],this approach was realized for the flow model of the E × B discharge with no neutral gas and,accordingly,with no ionization process.The anode and cathode serve as the reservoirs for ions and electrons,and the medium between the anode and cathode was formed as a combination of a stationary ion flow from the anode to the cathode and a stationary electron flow from the cathode to the anode.The potential difference between the anode and the cathode in[40]was assumed such that the energy gained by electrons was comparable to the rest energy of electrons mc2(m is the electron mass and c is the velocity of light).In this case,the dynamics of electrons was described within the relativistic approximation.Based on the Poisson equation for the potential distribution between the anode and cathode,it was shown that the quasi-neutrality of the medium is not realized in such a model and that the maximum density of ionic current can only exceed a few times the current density corresponding to the Child–Langmuir limit jCL.For the flow model of a discharge with eVd/mc2 ≪ 1 relativistic equations from[40]were used in[41,42].They demonstrated the operation of almost quasi-linear modes of the medium between the anode and the cathode with the ionic current density far exceeding jCL.To achieve this,the authors of[41–43]had to use a special procedure in order to choose appropriate conditions for the Poisson equation.

If the plasma in the near-cathode region is quasi-neutral and if the ionic current density is much larger than jCL,then in the flow model for the E × B discharge throughout the gap between the anode and the cathode it can be assumed that ni= ne= n by discarding the Poisson equation.It is this approximation that will be used in this paper by complementing the flow model with a narrow ionization region near the anode,i.e.the anodic layer(a virtual anode)in which quasi-neutrality breaks down.Here,the flow of electrons traveling in a self-consistent longitudinal electric field and a self-consistent transverse magnetic field is decelerated to an almost zero longitudinal velocity,gaining a drift velocity in this case.The electron energy,determined from the drift velocity,will be approximately equal to eφL(φLis the value of the potential in the anodic layer),while the electron density in the anodic layer for a stationary discharge theoretically increases to infinity.A high electron density and energy～eφL,exceeding the ionization potential I,ensure ionization in the anodic layer by electron impact from the bulk of the neutral atoms of the working gas arriving from the anode,and an increase in ionic current density in the discharge.The stationary one-dimensional modes of the E × B discharge will operate within the approximation of an in finitely narrow anodic layer.The primary goal of our examination will involve determining a dependence of the coordinate of the position of the anodic layer and the value of the anodic layer potential φLon the density of the flow of neutral atoms to the anodic layer.It will be demonstrated that the modes with φL＞ Vdare possible,in which the cathode receives anomalously accelerated ions with an energy W = eφL ＞ eVd.

where

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where vex(x),and vey(x)are the components of the electron velocity.

The equation for the ion dynamics has the form

From this relation we have thatand that the discharge current density becomes equal to

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In view of the fact that the ion velocity vixcin the anodic layer is zero,it follows from(2)that

when x = d(on the cathode),we have

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Let us simplify our examination by neglecting the inertia of electrons in the first equation of the system(1).We take into consideration that vey(x = d) = veyk.We then find from equations(1)–(3)that the y-component of the electron velocity and the ion velocity are related thus:

and as the y-component of the electron velocity approaches the anodic layer,it reaches the value of In a stationary discharge,we determine the electron velocity along the X-axis from the condition of constancy(along this coordinate)of the density of the ionic jiand electronic current jex,as well as from the quasi-neutrality condition:

The ionic current density ji = jn = ennvn,where jnis the total ionic current density arising in the case of a total ionization of neutrals in the anodic layer,and nnand vnare the density and the velocity of neutral atoms.It is assumed that the electron flow is responsible for a total ionization of the flow of neutrals:jexφL = jnI.Then we have jex/jn = ξφA/φL,where ξ = I/φA.The density of the discharge current jp,which is the sum of the densities of the ionic and electronic current,becomes:

The electronic Hall current density jey = –enveywill change the external magnetic field B0,so that

We now use the relation n = jn/evixand introduce the y-component of the vector potential Ay = A(x),so that Bz(x) = dA/dx.Then(7)becomes

From the second equation of the system(1),in view of the fact that when x = d the value of veyk = 0 and A = 0,we can develop the relation

Upon substituting(9)into(8)and using(3),we obtain a differential equation:

In equation(10),we designate andη=Then

where we introduce the quantity

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Integrating equation(11),and using the boundary condition Bz(η = 1) = B0,we obtain the expression for magnetic field induction:

Using(13),we obtain the relation

integration of which gives the expression for determining the position of the anodic layer:

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where F(ε)is the function which is defined by formula(15)and,when ε → 1,tends to infinity:

the dimensionless parameter ε is represented as

whereand εAis a parameter which is defined thus:

where f = φL/Vd.

The current density jeAis the analog of the Child–Langmuir current density with consideration for the transverse magnetic field,which we defined as the function jeA(ΔφAL,dL,B0);this will be discussed in greater detail elsewhere.Upon obtaining this function,equations(14)and(17)can be used to de fine the quantities dLand φLwhich,at fixed B0and d,respectively,tend to decrease and increase with an increase of jn.Furthermore,when jntends to reach some critical value jcr,we have dL → 0,φL → φA.That is,the critical density of the total current is determined from equations(14)–(16)when dL = 0:

where εcris determined from the relation

Assume that the E × B discharge modes are realized,where the total density of ionic current jnexceeds the values of jcr.Let us show that in such stationary discharges the anodic layer again moves away from the anode to a finite distance dLand that the potential in the anodic layer exceeds the anode potential:φL ＞ Vd.

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In these modes,the ions that originate in the anodic layer travel toward the anode as well as the cathode,and the total ionic current density will be determined as the sum

where j0is the current density of the ions traveling toward the cathode and jAis the current density of the ions traveling toward the anode.

Let the density of ionic current jAbe represented as

综上所述，铣床上加工多孔专用夹具的设计与研究是在铣弧形面结构设计夹具的基础上对此前夹具的重点与难点和不足做了更加详细的分析，本夹具在原有夹具基础上对螺杆作中间传力元件的夹紧机构时、定位夹紧可靠、实用性能好。其螺旋升角不大，螺旋定位夹紧机构的自锁性能和可操作性能好，夹紧力和夹紧行程都符合设计要求，是手动专用夹具上最常用的一种设计方法，该设计对其它同类专用夹具设计具有一定的参考价值。

where viA(x) ＜ 0 is the ion velocity andis the share of the total ionic current density.The density of the ionic current directed toward the anode is j0 = (1 - ω)jn.The density of the discharge current is identical on either side of the anodic layer;therefore,jp = j0 + jex = jA + jexA,where jexAis the component of the electron current density along the X-axis between the anode and the anodic layer,which is jexA = jn + jex.

A natural feature in a plasma thruster is an uneven distribution of the plasma potential in the radius as well the length of the discharge channel[44].By using the method of laser-induced fluorescence for plasma diagnostics in the CHN output zone,it was possible to record ions with energies W ＜ 10 eV traveling along the thruster axis toward the anode[45],which we describe by formula(21),and gaining energy e(φL - φA)from the anode.The authors of[45]explain the nature of such particles by their acceleration on the ambipolar potential,the value of which exceeds the discharge voltage.

Similar to formula(5),we obtain the expressionwhere provided that viA(x = dL) = 0,and vexA(x = dL) = 0.For the electrons to be able to reach the potential hump e(φL - φA),it is essential that they should arrive from the anodic layer and enter the region x ＜ dL,having a kinetic energyHere veyA(x)is the electron velocity component along the Y-axis.Furthermore,in order for the motion of the electrons in the region x ＜ dLto have vexA(x) ＜ 0,it is essential that veyA(x = dL) ＞ 0.To　minimize　energy　expenditure　itis　assumed　thatIn the modes with φL＞ φA,the energy balance of the electrons in the anodic layer is

where M and vix(x)are the mass and the x-component of the ion velocity.

First it is assumed that the potential of the anodic layer is smaller than the discharge voltage:φL ＜ Vd.As in[40–43],we neglect the proper magnetic field of the discharge current.But the electron Hall current flowing along the Y-axis will change the magnetic field directed along the Z-axis from a constant external field B0to the field Bz(x).It will also be assumed that the magnetic field only influences the motion of electrons.The　electric field　hasonly　the　component Ex = –dφ/dx,where φ(x)is the potential distribution.The dynamics of the electron flow between the cathode and the anodic layer is described by the following equations:

Taking into consideration the Hall current density jeyA and by doing calculations similar to those done previously for the region between the anodic layer and the cathode in the φL＜ Vdmode for the condition Bz(x = 0) = B0,we obtain:

Relation(14)relates two quantities to be determined:dLand φL.The necessary second relation can be obtained in the following manner.When φL ＜ Vd = φA,in the region 0 ≤ x ≤ dL there exists only the electronic current jeAwhich is equal to the discharge current jpdefined by formula(6);therefore,the second relation for determining dLand φLtakes the form

For the φL ＞ Vdmode in the region between the anodic layer and the cathode,the preceding expressions remain the same,including(14);however,BL,magnetic induction in the anodic layer,should be used instead of B0,and ε should be replaced by the value determined by

For the position of the anodic layer we then have the expression

Upon equating(24)and(27),we obtain

By expressing BLin terms of B0in equations(25),(26)and(28)and introducing,instead of jnand ω,the quantities J = jn/jcr(the amount by which the total density of ionic current exceeds a critical density)and h = J(1 – ω) = j0/jcr,we obtain the equation

We now solve the problem for the geometry of the E × B discharge where the anode with potential φA= Vdis located at x = 0,the cathode with a zero potential at the point x = d,and the anodic layer when x = dL.

and

Furthermore,by introducing the expression for how much the discharge current exceeds a critical density,Jp=jp/jcr,from(23)we obtain

Figure 8.Dependence of a critical total current density of argon ions on magnetic induction on the anode.

At fixed J and h the value of　Φ(　f, J, h)is a function of f,and when J ＞ 1 it is only determined for h ＜ 1/εcr.Furthermore,when　hmax ＜ h ＜ 1/εcr(herehmax ＞ 1)the value of　Φ(　f, J, h )when f ≥ 1 is larger than zero,and when h = hmaxthe function　　Φ(　f, J, hmax )reaches zero when f = fm.If h ＜ hmax,　Φ(f　, J, h)is such that the equation Φ　(　f, J, h　)　=0has two roots:f1 ＜ fm,and f2 ＞ fm.When h ＜ 1　there　remains　one　rootwith　f2 ＞ fm.When 1 ＜ h ＜ hmax,the second of these two roots,according to(30),corresponds to a larger discharge current which is energetically unfavorable;therefore,only the root f1should be used.When h → hmax,the values of f1and f2converge to fm,and there arises the maximum possible density of the ionic current flowing from the anodic layer to the cathode.It is therefore assumed that the mode with h = hmaxmust be at work.Next,at a fixed J ＞ 1 it will be assumed that f = fm.Upon determining fmfor different J,we obtain the relationf　=　G( J).

每到夏季，尤其是高温持续时节，南部高山冰雪消融，流水顺冲洪积扇裙而下，扇地下游则满滩为水，无处不有，水深≥50cm，积水成灾，水流夹杂淤泥，覆田淹草，淤泥厚度有时达到15cm，作物难以安全生产。

5.Results of numerical modeling

The algorithm for calculating plasma parameters from analytical expressions presented in section 4 is as follows.For each fixed J ＞ 1 and h from the range 1 ＜ h ＜ 1/εcrwe calculate the value of　Φ(f　, J, h )as a function of f.We find hmaxfor which we determine the value of fm.It is this value of fmthat will be a necessary value of f at a fixed J.Next,we construct the function f　=　G( J).

We calculated the dependences of a critical total current density of argon ions on magnetic induction on the anode,the function f = G(J)and the value of the anodic layer potential for the following conditions:argon gas(Ar+),Vd= 1100 V,and an anode–cathode separation d = 8mm.Results are presented in figures 8–10.

The mode of anomalous acceleration of ions sets in when the total ionic current density exceeds a critical value.More specifically,the smaller the value of the field,the smaller the value of the critical total ionic current density(see figure 8,which also shows that when the value of magnetic field induction is smaller than 0.014 T,there is no super-acceleration of ions).

Figure 9.Relationship of the anodic layer potential and total ionic current density for different values of magnetic field induction on the anode.

Figure 10.The value of the anodic layer potential as a function of magnetic field induction on the anode for different values of total ionic current density.

Relative changes in energy Wmproceeded much faster than the increase in pressure of the working gas.The growth rates of Wmand of the discharge current Idturned out to be almost identical.Information embedded in figure 4 suggests that when Id2/Id1 ≈ 1.9 the energy ratio Wm2/Wm1 ≈ 1696/944 ≈ 1.8;when Id3/Id2 ≈ 1.45 the energy ratio Wm3/Wm2 ≈ 1962/1696 ≈ 1.45.

If the total current density of ions is fixed and if magnetic field induction is taken as a variable,we obtain a family of curves as shown in figure 10.Functions of the form φc/Vd= G(BA,ji = const)have a number of characteristic features.First,they have maxima.Second,when the value of the magnetic field is smaller than 0.014T it is also seen that there is no anomalous acceleration of ions.With an increase in the value of the magnetic field,the suppression of the ion super-acceleration effect is observed,which may be explained by the fact that the value of the critical current density of ions increases with increasing magnetic field,and at a fixed ion current density the ratio ji/jcrdecreases,which,according to data shown in figure 9,leads to a decrease in the potential of the anode current.

It is also evident from figure 10 that the anomalous acceleration effect depends on the total ionic current density:the larger the density,the larger the potential of the anodic layer and the more clearly pronounced the maximum of the function φc/Vd = G(BA,ji = const).Furthermore,with an increase in the total ionic current density,the position of the maximum is somewhat shifted toward an increase in magnetic field induction.

6.Discussion

Analysis of the data in section 2 suggests a number of generalizations.As expected,the ion energy spectra shifted toward high energies in the range determined by the discharge voltage.This is known to be associated with the movement of the zones of ionization and acceleration in the PAAL anode–cathode space(spectra 1 and 2 in figure 2 and spectra 1–3 in figure 3).Furthermore,the authors of[23–26]reported recording energetic ions with W ≥ eVdwhich generate a high-energy tail of the distribution function.The same effect was also observed in mathematical modeling for SPT plasma:the mode of pulsations of the discharge current(voltage)also showed a broadening of the ion distribution function toward low as well as high energies,with a shift of the maximum of the distribution function toward high energies[22,46].It was thought that the generation of energetic ions was due to multiply ionized ions which again became singly charged in the process of charge exchange in the SPT plume[27].It was only observed experimentally[26]‘in the mode of a large current’that the maximum of the distribution function begins to shift beyond eVd.

The influence of the magnetic field on the potential distribution in SPT was pointed out in[47],based on the observation of an additional positive potential jump of the anodic layer when the magnetic field con figuration in the discharge current differed greatly from the traditional configuration for SPT:it included the region with zero magnetic field in which dBr(z)/dz changed sign.

The type of gas is an important parameter for the discussion of anomalous acceleration.A comparison of the spectra shown in figures 2 and 3 reveals that helium ions gain energy in a ‘more complicated manner’:an excess of eVdwas only observed for the case where P ≈ 12 × 10–5Torr.In a helium discharge plasma,the temperature Te ≈ IHe+2/2;therefore in such a situation there would not be an appreciable number of doubly charged helium ions.

It was found that molecular nitrogen is difficult to accelerate to energies W ＞ eVd.Such energy can only be characteristic for nitrogen ions residing in the tail of the distribution function.Conceivably this is due to the smaller(compared with argon and helium)rate of increase in the ionization cross-section for molecular nitrogen with a change of Te,and to the lower Tein nitrogen plasma.Under such conditions,the zone of ionization in the PAAL will be broad and shifted toward the cathode;therefore,the ion energies cannot exceed eVd.For doubly charged argon ions and atomic nitrogen the electron energies in the mode of anomalous acceleration are limiting energies approximately equal to the double ionization potentials IAr+2and IN+2;hence,no such ions can occur.

An analysis of the acceleration processes in the E × B discharge of the PAAL reveals some features common to vacuum and plasma diodes and in plasma sources with a hollow cathode in which the ions with energies Wm ≥ eVd have long been an integral part[1–5,16–18].A similar influence of the magnetic field on energy gain by ions(as was the case in our experiments)was observed for pulsed discharges in diodes and for discharges with a hollow cathode.In the plasma diode,Wmof the magnesium ions decreased by 25%with an increase in the longitudinal magnetic field from 0 to 1.7 × 10–2T[48].In the source of ions with a hollow cathode and with a cylindrical anode with no magnetic field,ions with W ＞ eVdwere recorded;on introducing the field B ～ 10–2T,anomalously accelerated ions disappeared[15].In[21],the ion spectra measured along the axis of the discharge chamber had the form of a beam,with a maximum Wm near eVd;radial spectra included ions with higher energies,Wm ≈ 2eVd.With increasing magnetic field induction the spectra shifted toward lower energies.

Among the ion acceleration mechanisms considered in[1–5,16–18],a well-working scheme with a virtual cathode is notable[6–10].The plasma E × B discharges,on the other hand,showed positive potential jumps with Δφ ＜ Vd[36–39,45].

The appearance of the zone in which there is an additional positive potential jump can be understood if the following factors are taken into consideration.Based on probe measurements,we found that Tein the PAAL increases with increasing discharge current.When P = (6–9) × 10–5Torr,Teapproaches threshold values equal to the single ionization potentials　of　the　gases　we　use.　At　pressures　of(10–12) × 10–5Torr,Teincreases to ≤ 30 eV,which leads to an exponential increase in the ionization cross-sections and,hence,to an ‘explosive’increase in the density of electrons and singly charged ions in the plasma.Furthermore,it is known that the effectiveness of ionization in the E × B layer becomes high,starting with some critical value of the flow rate of the working material:an increase in the density of neutrals decreases the mean free path of electrons with respect to ionizing collisions with working gas atoms.It is estimated that the density of neutrals can differ by one to two orders of magnitude according to the size of the anode–cathode 2(azimuthator)output,which increases the density of charged particles in the local near-anode region of discharge.

The array of experimental and computational results(sections 2–4)suggests that the E × B discharge of a plasma accelerator with an anodic layer allows for a mode of operation where the bulk of ions are accelerated along the direction　of　the　external　electric field　to　energies Wm ≥ 1.5eVd.Such an effect was not observed in previous experiments　with　plasma　accelerators　(thrusters)(e.g.[23–26]).In this study,we presented the idea of forming a virtual anode with the potential φL ＞ Vdin the E × B discharge,thereby illustrating a general approach for gas discharges which implies that acceleration of the bulk of charged particles occurs in a macroscopic electric field.

7.Conclusion

In the process of experimental investigations,we detected the effect of‘super-acceleration’of the bulk of the ions to energies Wm ≥ 1.5eVdwhich is important for the theory and practice of using plasma accelerators with a closed drift of electrons.Such ions were recorded where the pressure of the working gas and the discharge current were increased above a certain limit.The following gases were used:readily ionizable atomic argon and atomic helium,that has a large ionization potential(especially double ionization),as well as molecular nitrogen whose ionization produces mainly N2+ions.The process of energy gain was most effective for singly charged argon ions at the highest discharge currents,and the rate of energy gain corresponded to the growth rate of Idin the pressure range 9 × 10–5 ≤ P ≤ 12 × 10–5Torr.Helium ions with W ＞ eVdonly arrived at the collector of the energy analyzer when P ≈ 12 × 10–5Torr.Molecular nitrogen was not accelerated to energies W ＞ eVd.The most probable,corresponding to the maximum of the distribution function,the ion energy unambiguously followed the change in the discharge voltage.

It turned out that the anomalous acceleration of ions is not realized for all magnetic fields(drift velocities of electrons).The most likely energy Wm ≥ eVdof ions was recorded in the case where 2.8 × 10–2 ≤ BrA ≤ 4 × 10–2T.For smaller fields the discharge burning becomes unstable and the generation of energetic ions is suppressed on the upper boundary of the field;this is likely to occur due to destruction of the virtual anode.

We considered analytically the stationary one-dimensional modes of E × B discharge and determined the relationship of the position coordinate of the anodic layer and the value of the anodic layer potential to the ion current density in the discharge.It was shown that when the total ion current density exceeds a critical value,the anodic layer moves a finite distance from the anode and the potential in the anodic layer exceeds the anode potential.In these modes,the cathode receives anomalously　accelerated　ions with　an　energy W ＞ eVd.Numerical modeling that was carried out according to the developed theory showed a qualitative agreement between modeling data and measurements.

This work was partly supported by a grant‘Organization of the conduct of research’code 82 of the Ministry of Education and Science of the Russian Federation.We are grateful to V G Mikhalkovsky for his assistance in preparing the English version of the manuscript.

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