Electron Preheaths: The Outsized Influence of Positive Boundaries on Plasmas
Benjamin T. Yee, Brett Scheiner, Scott D. Baalrud, Edward V. Barnat, Matthew M. Hopkins
EElectron Presheaths: The Outsized Influence of Positive Boundaries on Plasmas
B.T. Yee, ∗ B. Scheiner, S.D. Baalrud, E.V. Barnat, and M.M. Hopkins Sandia National Laboratories, Albuquerque, New Mexico 87185, USA Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242, USA (Dated: October 15, 2018)Electron sheaths form near the surface of objects biased more positive than the plasma potential,such as a Langmuir probe collecting electron saturation current. Generally, the formation of electronsheaths requires that the electron-collecting area be sufficiently smaller ( (cid:112) . m e /M times) thanthe ion-collecting area. They are commonly thought to be local phenomena that collect the randomthermal electron current, but do not otherwise perturb a plasma. Here, using experiments on anelectrode embedded in a wall in a helium discharge, particle-in-cell simulations, and theory it isshown that under low temperature plasma conditions ( T e (cid:29) T i ) electron sheaths are far from local.Instead, a long presheath region (27 mm, approximately an electron’s mean free path) extends intothe plasma where electrons are accelerated via a pressure gradient to a flow speed exceeding theelectron thermal speed at the sheath edge. This fast flow is found to excite instabilities, causingstrong fluctuations near the sheath edge. INTRODUCTION
A sheath is the space charge region found near thephysical boundaries of most plasmas. The vast major-ity of sheaths are ion rich because this is what natu-rally forms as highly mobile electrons charge a surfacenegatively.Comparatively little is known about electronssheaths, although they are routinely produced when ob-jects are biased positive with respect to the plasma po-tential [1].The most common situation is the electron satura-tion region region of a Langmuir probe sweep, but theyarise in many other situations including negative ionsources [2] and electron sources [3], positive electrodesemployed for blob control [4], particle circulation in dustyplasmas [5], and turbulence-induced particle fluxes [6].Electron sheaths are also common in several other sit-uations, including: near highly emitting surfaces [7], inmicrodischarges [8], during the high potential phase ofthe rf cycle in processing discharges [9], around electro-dynamic tethers [10], the lunar photosheath [11], aroundwire arrays used for electron temperature control [12],and in scrape off layer control [13].In this Letter, results of experiments, particle-in-cell(PIC) simulations and theory are provided showing thatelectron sheaths form a long electron presheath extendingwell into the plasma. Furthermore, electrons are acceler-ated to high velocities in this region, obtaining a distribu-tion that is flow-shifted to a speed exceeding the electronthermal velocity at the sheath edge. This may be consid-ered an electron sheath analog of the Bohm criterion [14].The fast differential streaming between electrons and ionsis found to excite streaming instabilities that give rise tostrong fluctuations of the boundary layer region.Electron presheaths are found to differ in their essen-tial properties from ion presheaths. In particular: (1) thedifferential potential (∆ φ ) is much smaller, nominally bya factor of T i /T e , (2) it is much longer in extent, nomi- nally by a factor of (cid:112) m i /m e , (3) electrons are acceler-ated by a pressure gradient, in contrast to direct electricfield acceleration of ions in an ion presheath, and (4)the differential streaming excites instabilities and strongfluctuations. These results promise new insights into theapplications mentioned above. Advances in the funda-mental physics of electron sheaths may also lead to newapplications. EXPERIMENTS
Experiments were conducted in a cylindrical vacuumchamber in which an electrode with a diameter of 19mm was embedded into one face. The walls and facesof the vacuum chamber were maintained at ground po-tential. The area ratio of the chamber (1 . × mm )to the electrode (1 . × mm ) was chosen to ensurethe formation of an electron sheath above the auxiliaryelectrode when biased above the plasma potential [15].A plasma was generated in 20 mTorr of helium with abarium-impregnated tungsten thermionic emitter (Heat-wave Labs model 101117) located 10 cm from the elec-trode. The emitter was operated at a temperature ofapproximately 1 . × ◦ C. A cross section of the exper-imental setup can be seen in Fig. 1.The discharge current was held constant at 300 mAwhich resulted in a discharge voltage which varied from49 to 54 V over the course of the measurements. Theauxiliary electrode was biased to -50 V, 0 V and 15 V,forming an ion sheath, a weak ion sheath, and electronsheath respectively. An emissive probe was operated ata height of 35 mm and at a radial position of 70 mm andwas used to measure the plasma potential via the float-ing point technique. The plasma potential was 6.2 V, 5.3V, and 6.6 V for the three aforementioned cases. Two-dimensional maps of the electron densities above the elec-trode were generated using the laser-collisional induced a r X i v : . [ phy s i c s . p l a s m - ph ] O c t Therminonic emitterLaser sheet Electrode Guard ringDielectric spacerInterrogation AreaFIG. 1. Sketch of the experimental setup used to measure theelectron densities above an embedded electrode. n e / n -50 V0 V15 V FIG. 2. Measured electron density profiles above the electrodealong the center axis at -50 V (solid line), 0 V (dotted line),and 15 V (dashed line). The profiles are normalized by theirvalues at 35 mm: 3 . × cm − at -50 V, 3 . × cm − at0 V, and 3 . × cm − at 15 V. Also included is a gray linerepresenting the constant density gradient found at z > fluorescence ( lcif ) diagnostic [16]. High energy electronscould potentially cause secondary electron emission, how-ever the uncollided electron density (high energy tail) isestimated to be two orders of magnitude below the mea-sured densities. Given a secondary emission coefficient of γ ≈ .
1, the density of electrons from secondary emissionshould be about three orders of magnitudes below themeasured densities.Figure 2 shows the axial density profiles from lcif measurements (relative uncertainty estimated to be7% [17]), normalized by their density at 35 mm, for theion sheath (-50 V, solid line), a weak ion sheath (0 V,dotted line), and the electron sheath (15 V, dashed line).In all three cases the same linear density gradient is ob-served far from the electrode ( >
30 mm).This gradient is likely the result of the nonuniform gen-eration of plasma in the volume. Attenuation of the pri-mary electrons and their falloff with distance from thesmall source will lead to a locally peaked density distri-bution. This non-uniformity will lead to diffusion to thewalls of the chamber and an associated ambipolar field.Based on Langmuir probe measurements of the electrontemperature and observed density gradients this ambipo-lar field will be of the order 0.6 V/cm and will tend toinhibit the transport of electrons toward the embedded electrode.At -50 V and 0 V, the electron depletion region asso-ciated with an ion sheath can clearly be seen extendingfrom 0–4 mm and 0–3 mm respectively. The presheathis estimated to be the region at which the electron den-sity gradient deviates from the roughly constant slope( >
30 mm). The estimated location of the presheathedge occurs near 12 mm. This can be compared to thecalculated ion-neutral mean free path which is expectedto determine the presheath length scale [18]. Estimatingion energies of 0.1 eV and using the isotropic scatteringcross sections of Phelps [19] ( σ = 2 . × − m ), wefind a mean free path of 6.5 mm which is on the order ofthe estimated location of the presheath edge.The electron sheath (15 V) also features a region ofelectron depletion from about 0–2 mm from the electrode.This is caused by the acceleration of the electron fluidand its consequent rarefaction. The salient feature of theelectron sheath case is that its density gradient is essen-tially the same as the ion sheath’s far from the electrode( >
27 mm), but closer to the electrode it bears a dis-tinctly different gradient. Assuming that this deviationis caused by collisional effects similar to the ion sheathcase, a calculation can be made for electrons based ontheir momentum transfer cross section. Assuming an en-ergy of 4.0 eV ( σ = 6 . × − m per Phelps [19]), theelectron mean free path is found to be 24 mm.This suggests that the positively biased electrode is af-fecting the plasma far from the electron sheath region, onthe order of an electron mean free path, and suggests thepresence of an electron presheath . The Debye length atthe sheath edge is estimated to be 0.38 mm based on the lcif measurements and Langmuir probe measurementsof the electron temperature. Thus this presheath extendsapproximately 70 Debye lengths from the boundary. Inboth the case of the electron presheath and ion presheath,the length scales are significantly smaller than the lengthscale of the system. SIMULATIONS
A complementary analysis was carried out using Aleph,a PIC-DSMC code developed at Sandia National Lab-oratories. Aleph is an electrostatic code intended formassively paralllel (over 10k cores) plasma simulations.Fields are solved using finite-element method librariesfrom the Trilinos project [20]. Particles are advancedusing the velocity Verlet algorithm as described by Spre-iter and Walter [21]. Other recent uses of Aleph includethe study of vacuum arc discharges [22] and the onset ofplasma potential locking [23].The simulation was conducted in a rectangular Cartes-tian domain, 75 mm by 50 mm, with an unstructuredmesh formed of triangles. This is the same geometry asthat used in [24] and depicted in Figure 3. The left-hand
Injection RegionElectrode 03691215 P o t e n ti a l ( V ) FIG. 3. Representation of the simulation geometry with over-laid electric field potential contours from electron sheath sim-ulations. The domain is 75 mm by 50 mm with an axis ofsymmetry on the left hand side. side of the domain specularly reflects all particles andpossesses a Neumann boundary condition of ∂V /∂x = 0,thus representing an axis of symmetry. Aside from theelectrode shown in Figure 3, the outer boundaries areheld at V = 0 and outflux all incoming charged particles.The characteristic size of the triangles used to mesh thedomain was 233 µ m which resolved the Debye length ofapproximately 540 µ m. All simulations used a timestepof 25 ps, which is sufficient to meet CFL requirementsfor all particles observed in the simulation and resolvesthe plasma frequency of 0.28 GHz.The simulation domain maintains an area ratio of wallto electrode similar to that in the experiment, chosen toassure the formation of an electron sheath [15, 25]. Asthe formation of the electron sheath only depends on thearea ratio and not the absolute size of the electrode, ex-periment and simulation should be largely comparable.The simulated electrode was biased to -50 V and 15 V tomatch the measured ion and electron sheath. Quasineu-tral plasma ( T e = 4 . T i = 1000 K) was added tothe simulation domain at a constant rate in a rectangu-lar area located 37.5 mm from the anode. Simulationswere run for 30 µ s at which point they were found tobe in equilibrium based on field energy and total parti-cle number. Field and particle properties were averagedover an additional 20 µ s in order to minimize statisticalfluctuation in quantities of interest.The experimental density measurements are comparedto the equivalent simulations in the plots on the left sideof Fig. 4. Overlaid on the simulated electron densitiesare arrows showing the electron flux vectors scaled tothe same value for both cases. The horizontal axes havebeen normalized by the electrode radii, in order to pro-vide a more suitable comparison. The right side of Fig. 4presents maps of the charge density normalized by thedensity of the collected species. Inset in the charge den-sity maps are power spectra of the sheaths’ positions overtime.In the case of the ion sheath, a large region of elec-tron depletion is visible above the face of the electrode. The size and shape of this region is largely consistentbetween simulation and experiment, with the small dif-ferences likely ascribable to a discrepancy in the electrondensities at the sheath edge. An ion sheath is also ob-served above the grounded wall. Electron current is smallthroughout the simulated domain and is consistent withthe fact that ion sheaths tend to confine electrons. Thecharge density shows the formation of a stable sheath.The electron sheath simulation also features a region ofelectron depletion near the face of the electrode, resultingfrom acceleration of the electron fluid. Though the widthof the sheath in the density maps from simulation appearsqualitatively larger in the density maps, the sheath edgeseen in the charge densities oscillates about 2 mm consis-tent with the experimental electron density profile seenin Fig. 2. Further differences are likely attributable to acombination of potential causes: lower electron densitiesin the simulation and the absence of electron-neutral col-lisions. The former would result in larger Debye lengthsand subsequently larger sheaths. The latter would alsotend to increase the sheath width as seen in equation (6)of [26].The electron sheath simulation also features a substan-tial degree of electron current directed toward the elec-trode from at least as far away as 30 mm. The plasmaproperties in this region ( n e = 4 . × cm − , T e = 2 . .
54 mm, 60 times smallerthan the extent of the directed flow. While it is wellknown that ion presheaths can extend throughout theentire plasma for large mean free paths [18], we willshow that (for the same collision process) the electronpresheath is nominally longer. The extent of the flowis consistent with where the density profile from Fig. 2begins to change slope, indicating the presheath. Theelectron presheath is clearly a significant perturbation tothe plasma. Furthermore, as opposed to the ion sheath,the electron sheath edge exhibits significant fluctuationsin its position.The proximity of the grounded wall to the auxiliaryelectrode leads to a funnel-like structure in the electronflow with a notable convergence. We note that there isthe possibility for some overlap of the electron presheathwith the ion presheath from the abutting walls. In thiscase, the radial electric field of the ion presheath will can-cel as a result of the symmetry of the system. The axialelectric field will tend to accelerate electrons away from,rather than toward the anode, therefore the flows ob-served in Fig. 4 are not the result of adjacent ion sheaths.Finally, we note that the ion presheath length scale is es-timated to be 6.5 mm, significantly less than the observedelectron presheath size and the device dimensions. s i m u l a ti on ex p e r i m e n t n e / n f (MHz)PSD ( m /Hz)0 1 2 3 4 5012 f (MHz)PSD ( m /Hz) ( n i , e n e , i )/ n i , e z ( mm ) FIG. 4. Experimental and simulated electron density maps (left) and simulated on-axis charge densities over time (right).The ion sheath results comprise the upper plots, and the electron sheath results comprise the lower ones. Overlaid atop thesimulated electron densities are the electron particle flux, the magnitudes of which range from 0.1–1.2 × m − s − for theelectron sheath, and below 5 × m − s − for the ion sheath. The power spectral densities of the sheath edge position areinset in the charge density plots. THEORY
Traditional Langmuir probe analysis assumes that aprobe in electron saturation collects the random ther-mal flux of electrons incident on the electron sheath. Animplication of this local picture is that the electron ve-locity distribution function ( evdf ) at the edge of theelectron sheath would be a half-Maxwellian with no flowshift (but with a flow moment) [27–29]. The random fluxof electrons flows into the sheath from the quasineutralplasma, and all electrons reaching the sheath are lost tothe boundary. However, our results show that the pres-ence of an electron presheath leads to a vastly differentpicture.In particular, simulations show that the presheath isfound to introduce a substantial flow-shift in the electrondistribution, approaching the electron thermal speed bythe sheath edge; see Fig. 5. Contrary to the conven-tional picture of a highly-kinetic truncated distributionfunction, this figure shows that it is in fact well repre-sented by a flowing Maxwellian. That the distributionis well-described by a Maxwellian despite the absenceof electron-neutral collisions is notable. Confirmation ofthis flow shift in experiment remains an open challenge. v e , z ( m/s)012 N u m b e r ( ) z = 30 mm z = 2. 5 mm z = 0. 5 mm FIG. 5. The electron velocity distribution functions normalto the electrode, along x = 0, at several locations above theelectrode in the simulations: the injection region ( (cid:52) ), thesheath edge ( ◦ ), and inside the sheath ( (cid:5) ). A significant number of electrons flow out of the electronsheath (negative velocities are electrode-directed) despitethe absence of an explicit collision algorithm in the sim-ulations. The sheath is defined as where quasineutralityis sufficiently violated, or ( n e − n i ) /n e = (cid:15) where we havechosen (cid:15) = 0 . z = 2 . (cid:15) below, in view of the strong density fluctuations in thisregion.While PIC simulations can result in numerical ther-malization of nonequilibrium distributions, this is not ex-pected to be a factor in the present results. Estimatesbased on the work of Montgomery and Nielsen [30] sug-gest that the thermalization time for the present systemis of the order 2.6 µ s. This exceeds the 1.1 µ s requiredfor the electron fluid to transit from the source regionto the electrode. Additionally, recent simulation resultsof the ion to electron sheath transition [31] using com-parable simulation parameters possess evdf s exhibitingboth a flow shift and a loss cone due to the presence ofthe boundary. Both factors are found to affect the flowspeed, with the flow shift being at least as important asthe loss cone.A fluid analysis is used to interpret aspects of the ex-periments and simulations and to better understand theorigin of the observed flow shift. Consider the fluid mo-mentum balance u e du e dz = − eEm e − k B T e m e n e dn e dz − u e ( ν c + ν s ) (1)where e is the elementary charge, E is the electric field, ν c is the collision frequency, and ν s represents the source(ionization) frequency. The terms on the right-hand siderepresent the forces due to the electric field, pressure gra-dient, and collisions respectively. The largest of theseterms is the pressure gradient term.The ions are estimated to vary according to the Boltz-mann density relation, eEk B T i = 1 n dndz . (2)This assumption depends on a number of factors whichmay not necessarily be met for all electron sheaths.Multi-dimensional effects, flow, and collisions may all re-duce the applicability of this relation. Indeed, in the sim-ulations conducted, this relation is not strictly applicableas the ions experience substantial collisional and inertialforces as seen in Figure 6. The figure depicts the iner-tial (solid), pressure (dotted), electric field (dash-dotted),and collisional (dotted) fluid terms for the ions along theaxis of symmetry, neglecting perpendicular components.These cases require significantly more complex analysis;an example of the treatment of the multi-dimensionalcase can be found in [26]. Therefore, this relation is as-sumed here strictly as a means of developing an initialestimate of electron sheath and presheath properties inthe ideal case of minimal ion-neutral collisions and neg-ligible flow. We note that no such assumption is made inthe simulations.Substituting this into Eq. (1) shows that the pressuregradient term is T e /T i larger than the electric field. Thisis consistent with previous emissive probe measurementsin a discharge of lower density (10 cm − compared to 10 cm − ) which showed essentially no field past 4 mm from z (mm)6789101112 l og ( a ) ( m / s ) u i du i / dzeE / m i k B T i /( m i n i ) dn i / dzu i FIG. 6. Calculation of the ion fluid terms along the axis ofsymmetry. Two-dimensional effects are excluded. The iner-tial term (RHS Eq. 1) is the solid line, the electric field term(first term on LHS) is the dash-dotted line, the pressure term(second term on LHS) is the dotted line, and collisions (thirdterm on LHS) are shown by the dashed line. the electrode [25]. Although the potential gradient inthe presheath is small (characterized by T i ), the resultingpressure gradient drives a strong electron flow due to thedensity gradient that results. This contrasts with thesituation found in ion presheaths, where the electric fieldterm exceeds the pressure gradient term by T e /T i , andthe presheath potential drop is of the order of T e (ratherthan T i ). In this case, ions are accelerated ballistically bythe electric field to a speed exceeding the sound speed atthe sheath edge. The importance of the pressure gradientterm with respect to the electric field is confirmed inthe simulations. Figure 7(a) plots the pressure gradient(solid line) and electric field (dashed line) accelerationterms from Eq. (1) on a logarithmic scale, as a functionof distance from the electrode.The simulations indicate that the pressure gradient isthe dominant acceleration mechanism from 2–5 mm and z >
11 mm. The drop in pressure gradient between 5mm and 11 mm coincides with a plateau in the simu-lated electron density near the sheath edge (similar tothat seen in Fig. 2) and a plateau in electron tempera-ture (not shown). Several factors may contribute to thisplateau including a stagnation of ions as they approachthe sheath potential barrier or a change in the conver-gence of the electron fluid. Past 11 mm, the pressuregradient continues to dominate the acceleration of theelectron fluid up to the sheath edge.The degree to which the electron fluid is acceleratedcan be calculated via the electron continuity equation ddz ( n e u e ) = ν s n e , (3)and a common sheath criterion, which identifies thesheath edge as the location where (cid:80) q dn/dz ≤ q is the charge of the species. Droppingthe collision terms, Eqs. (1) and (3) yield dn e /dz = l og ( a ) ( m / s ) (a) Pressure GradientElectric Field z (mm)0123 (b) ( n e n i )/ n e V / V p u e , z / v T FIG. 7. (a) Pressure gradient (solid line) and the electricfield (dotted line) accelerations obtained from simulations.The solid gray lines indicate the locations where the evdf sin Fig. 5 were obtained, with the middle line indicating thelocation of the sheath edge. (b) Normalized charge density(dashed line), the electron fluid velocity (solid line), and po-tential (dash-dotted line) in the direction normal to the elec-trode surface, with respect to the distance from the electrodealong x = 0 . en e E/ ( m e u e − k B T e ). From Eq. (2), and assumingquasineutrality, the sheath criterion provides an electron-sheath analog of the Bohm criterion, u e ≥ (cid:115) k B ( T e + T i ) m e ≡ v T . (4)A similar derivation of the electron fluid speed at thesheath edge was previously obtained by Loizu, Dominski,Ricci, and Theiler [32] This criterion demands a regionof electron acceleration outside of the sheath region.Figure 7(b), shows the charge density (dashed line),the electron fluid velocity (solid line), and the local po-tential (dash-dotted line). The electron fluid is foundto reach a velocity of 0 . v T by the sheath edge (aspreviously defined), in fair agreement with Eq. (4). Anumber of factors may contribute to the remaining dis-crepancy including the ambiguity in the definition of aprecise sheath edge location [24], the use of a planar one-dimensional theory in describing a converging flow, thepresence of adjacent sheaths, and the substantial fluctu-ations observed in the simulations near the sheath edge.The fast electron flow creates a large differentialstreaming between electrons and ions that is expectedto lead to electron-ion streaming instabilities [33] in theelectron presheath and sheath. Indeed, Fig. 4 shows sub-stantial fluctuations of the electron sheath edge, but notin the case of the ion sheath simulation. The frequencyof these fluctuations is observed to peak around 0.8 MHzcomparable to the most unstable mode of ion-acoustic in-stability (ion plasma frequency) which would be 1.4 MHz for n i = 1 . × cm − ( z = 2 . z = 2 . z = 1.5 mmis obtained [38]).The length scale of an ion presheath is typically de-termined by collisional processes, and is estimated asthe ratio of the ion flow speed to the collision fre-quency l i (cid:39) c s /ν for a specific process such as ionization.The electron presheath length would be estimated to be l e (cid:39) v T /ν . This implies that, for the same collision pro-cess, the electron presheath is much longer than the ionpresheath, by a factor of l e l i = v T c s = (cid:114) T e + T i m e m i T e ≈ (cid:114) m i m e . (5)As an example, l e /l i ≈
270 in argon or l e /l i ≈
85 in he-lium. However, this ratio is only applicable in the caseof the same collision process. In the present case, it isbelieved that the ion and electron presheath lengths arelikely governed by different processes, namely ion-neutraland electron-neutral collisions. This suggests l e /l i ≈ CONCLUSIONS
These novel properties of the electron sheath are sur-prising both because of how they differ from ion sheathsand because of their influence on the bulk plasma. Theperturbations in electron density and flow caused by whatwould otherwise be considered a small electrode suggeststhat conventional models of electron sheaths need to berevisited. These fundamental physics results may alsolead to useful new applications.This work was supported by the Office of Fusion En-ergy Science at the U.S. Department of Energy undercontracts DE-AC04-94SL85000 and DE-SC0001939. Oneof the authors, BS, is supported by the U.S. Departmentof Energy, Office of Science, Office of Workforce Devel-opment for Teachers and Scients, Office of Science Grad-uate Student Research Program ( scgsr ). The scgsr program is administered by the Oak Ridge Institute forScience and Education for the the DOE under ContractNo. DE-AC05-06OR23100. ∗ [email protected][1] I. Langmuir, Phys. Rev. , 954 (1929).[2] M. Bacal, Nucl. Fusion , S250 (2006).[3] B. Longmier, S. D. Baalrud, and N. Hershkowitz, RevSci. Instrum. (2006), 10.1063/1.2393164.[4] C. Theiler, I. Furno, J. Loizu, and A. Fasoli, Phys. Rev.Lett. , 065005 (2012).[5] D. A. Law, W. H. Steel, B. M. Annaratone, and J. E.Allen, Phys. Rev. Lett. , 4189 (1998).[6] B. Richards, T. Uckan, A. J. Wootton, B. A. Carreras,R. D. Bengtson, P. Hurwitz, G. X. Li, H. Lin, W. L.Rowan, H. Y. W. Tsui, A. K. Sen, and J. Uglum, Phys.Plasmas , 1606 (1994).[7] M. D. Campanell, Phys. Rev. E , 033103 (2013).[8] Q. Wang, D. J. Economou, and V. M. Donnelly, J. Appl.Phys. (2006), 10.1063/1.2214591.[9] C. M. O. Mahony, R. Al Wazzan, and W. G. Graham,Appl. Phys. Lett. , 608 (1997).[10] J. R. Sanmartin, M. Mart´ınez-S´anchez, and E. Ahedo,J. Propuls. Power , 353 (1993).[11] A. Poppe, J. S. Halekas, and M. Hor¨anyi, Geophy. Res.Lett. , L02103 (2011).[12] C.-S. Yip and N. Hershkowitz, Plasma Sources Sci. Tech-nol. , 034004 (2015).[13] S. J. Zweben, R. J. Maqueda, A. L. Roquemore, C. E.Bush, R. Kaita, R. J. Marsala, Y. Raitses, R. H. Cohen,and D. D. Ryutov, Plasma Phys. Control. Fusion ,105012 (2009).[14] D. Bohm, in The Characteristics of Electrical Dischargesin Magnetic Fields , edited by A. Guthrie and R. K.Wakerling (McGraw-Hill, New York, NY, 1949) Chap. 3,p. 77. [15] S. D. Baalrud, N. Hershkowitz, and B. Longmier, Phys.Plasmas , 042109 (2007).[16] E. V. Barnat and K. Frederickson, Plasma Sources Sci.T. , 055015 (2010).[17] B. R. Weatherford, E. V. Barnat, and J. E. Foster,Plasma Sources Sci. Technol. , 055030 (2012).[18] K.-U. Riemann, J. Phys. D , 493 (1991).[19] A. V. Phelps, “He+ in He Cross Sections,” (2002).[20] M. A. Heroux, E. T. Phipps, A. G. Salinger, H. K. Thorn-quist, R. S. Tuminaro, J. M. Willenbring, A. Williams,K. S. Stanley, R. A. Bartlett, V. E. Howle, R. J. Hoek-stra, J. J. Hu, T. G. Kolda, R. B. Lehoucq, K. R. Long,and R. P. Pawlowski, ACM T. Math. Software , 397(2005).[21] Q. Spreiter and M. Walter, J. Comp. Phys. , 102(1999).[22] H. Timko, P. S. Crozier, M. M. Hopkins, K. Matyash,and R. Schneider, Rev. Sci. Instrum. , 140 (2012).[23] M. M. Hopkins, B. T. Yee, S. D. Baalrud, and E. V.Barnat, Phys. Plasmas , 063519 (2016).[24] S. D. Baalrud, B. Scheiner, B. T. Yee, M. M. Hopkins,and E. V. Barnat, Plasma Phys. Contr. F. , 044003(2015).[25] E. V. Barnat, G. R. Laity, and S. D. Baalrud, Phys.Plasmas , 103512 (2014).[26] B. Scheiner, S. D. Baalrud, B. T. Yee, M. M. Hop-kins, and E. V. Barnat, Phys. Plasmas (2015),10.1063/1.4939024.[27] H. Mott-Smith and I. Langmuir, Phys. Rev. , 727(1926).[28] G. Medicus, J. Appl. Phys. , 2512 (1961).[29] N. Hershkowitz, Phys. Plasmas , 055502 (2005).[30] D. Montgomery and C. W. Nielsen, Phys. Fluids , 1405(1970).[31] B. Scheiner, S. D. Baalrud, M. M. Hopkins, B. T. Yee,and E. V. Barnat, , 1 (2016), arXiv:1604.08251.[32] J. Loizu, J. Dominski, P. Ricci, and C. Theiler, Phys.Plasmas , 083507 (2012).[33] P. Bellan, in Fundamentals of Plasma Physics (Cam-bridge University Press, Cambridge, UK, 2006) p. 177.[34] J. Glanz and N. Hershkowitz, Plasma Phys. , 325(1981).[35] R. L. Stenzel, Phys. Rev. Lett. , 704 (1988).[36] R. L. Stenzel, J. Gruenwald, C. Ionita, and R. Schrit-twieser, Phys. Plasmas , 062112 (2011).[37] R. L. Stenzel, J. Gruenwald, C. Ionita, and R. Schrit-twieser, Phys. Plasmas18