Space-charge-limited bipolar current flow betweenconcentric spherical electrodes

C.B. Wheeler, D.Sc.

Indexing terms: Ionisation, Breakdown and gas discharges

Abstract: Poisson's equation in spherical symmetry is solved numerically assuming that at each electrode there isan unlimited supply of charges of zero energy; positive charges at the positive electrode and negative chargesat the negative electrode. A solution is obtained corresponding to zero electric field at each electrode surface,this defines the maximum possible current flow of each charge polarity. The numerical technique is based ona perturbation of the solution appropriate to parallel plane geometry and is carried out for a ratio of electroderadii ranging from 1 to 400. In the case of the external emitter configuration the increase in charged-particlecurrent afforded by bipolar flow, in comparison to unipolar flow, can be several orders of magnitude. Appli-cation to fusion fuel-pellet compression, field-emission microscopy and thermionic-energy conversion is brieflydiscussed.

1 Introduction

The unipolar flow of current in an evacuated parallel-planediode has an upper limit dictated by the effects of space charge,Child [1] and Langmuir [2]. One technique of overcomingthis limit is to neutralise the space charge by injecting chargesof opposite polarity at the plane that collects the originalcharge species. Langmuir [3] considered such bipolar currentflow and showed by numerical integration that the limitingcurrent of the original charge species was increased by a factorof 1.860. This problem was later solved analytically by Muller-Lubeck [4]. Howes [5] extended the parallel-plane bipolaranalysis by including the effects of finite initial velocities ofthe charged species, and also considered potentials sufficientlyhigh to warrant a relativistic treatment [6]. There are severalsituations in spherical geometry where the increase in limitingcurrent of a charged species afforded by bipolar current flow isbeneficial. One application is to the problem of achieving con-trolled thermonuclear reactions through fuel pellet compressionusing intense beams of charged particles. Yonas et al. [7] en-visaged the compression and heating of a small sphere of soliddeuterium/tritium mixture by isotropic bombardment of itssurface by high-energy electrons. Clauser [8] showed that theelectron current required to achieve fusion is much reduced ifthe fuel is in the gaseous state and contained in a sphericalshell of heavy metal such as gold. There is a more efficientconversion of charged-particle energy to compressional energyin the metal shell if ions are used rather than electrons, Clauser[9]. In both of these situations, where the target pellet is theinner collector electrode, the following analysis shows that anincrease in limiting particle current of several orders of magni-tude can be obtained if charges of opposite polarity are emittedfrom the collector. Another possible application is found infield-emission microscopy. If attempts are made to increaseimage brightness by raising the acceleration potential, it isfound that the field-emission current eventually falls below thevalue indicated by the Fowler-Nordheim equation. Barbour etal. [10] have shown that this is due to space-charge effects andthat the emission current is approaching a Child-Langmuirlimit appropriate to the radius of curvature of the pointemitter. In this case, where the inner electrode is the emitterof charged particles, as in both electron and ion field-emissionmicroscopy, the following analysis shows that some increase inlimiting particle current is obtained if charges of opposite pol-arity are emitted from the outer collector. The analysis can

Paper 2032A, first received 2nd November 1981 and in revised form10th May 1982The author is with the Plasma Physics Group, The Blackett Laboratory,Imperial College of Science & Technology, Prince Consort Road, LondonSW7 2BZ, England

also be applied to the thermionic-energy convertor where en-hancement of the limiting electron current raises the efficiencyof the device. In its simplest form, Wilson [11], the convertorconsists of a thermionic cathode emitting electrons to ananode whose Fermi level is more negative than that of thecathode. Useful power can be obtained by connecting a resistiveload between cathode and anode. To obtain an acceptableefficiency in a vacuum diode it is necessary to employ infinit-esimal electrode spacings, Houston [12], in order that electronspace-charge effects should not limit the Richardson-Dushmanemission current. In a spherical geometry this limiting electroncurrent is increased if the anode is a source of ions, particularlyif the cathode is external to the anode. Consequently, more'practicable electrode spacings can be used in the diode con-struction.

2 Mathematical formulation

Consider a concentric spherical geometry comprising an innerelectrode of radius rx at a potential Vx surrounded by an outerelectrode of radius r2 at a potential V2. Suppose that the innerelectrode emits singly charged particles, mass m1} at zeroenergy, and the outer electrode emits singly charged particlesof the opposite polarity, mass m2, also at zero energy. It isassumed that both these emissions are limited only by theeffects of space charge and that the potential difference is suchas to accelerate the charges away from their source electrode.Let Kbe the potential at radiusr where rx <r<r2. The radialvelocities of these charges, vx and v2, respectively, at theradius r are given by

= e{V-Vx),\m2v22 =e(V2-V) 0)

It is assumed here that the potentials are sufficiently low towarrant a nonrelativistic treatment and that the particlecurrents are sufficiently low for the associated magnetic fieldsnot to influence the particle motion. The densities of the op-posing charges, px and p 2 , respectively, at the radius r, areexpressible in terms of the total currents Ix and I2 of theindividual species:

p, = Ix/4nr2vx p2 = I2/4nr2v2 (2)

In the steady state the potential and total charge density in theregion rx < r < r2 are related through Poisson's equation:

1 d I dV, .- — \r2 — = - (Pi -r1 dr \ dr

(3)

Eliminating charges and velocities between eqns. 1, 2 and 3and introducing the dimensionless potential variabley = (V —

IEEPROC, Vol. 129, Pt. A, No. 6, AUGUST 1982 0143-702X/82/060387' + 04 $01.50/0 387

^i)/(^2 - K i ) , leads to

.2 dyd_

dr dr 4ne ,1/2 y 3/2I V 2X

(4)

where V2l = I V2 — Vx\. Introduce the dimensionless spacevariable x = In (r/rx) and define the currents Ix and / 2 interms of the currents I xo and /20 where

4£o9

4eo9

3 / 2 - 2

3/2 „ - 2"l

Thus Iw 14 nr\ is the Child-Langmuir space-charge-limited cur-rent density for unipolar flow of charges, mass mx, betweenparallel-plane electrodes separated by distance rx and at apotential difference V2l. I20 is correspondingly defined. Withthese introductions eqn. 4 can be written

d2v

<2x2

dy

(5)

Multiply eqn. 5 by 2e2x (dy/dx) and integrate, introducing theboundary condition (dy/dx)l = 0. This condition correspondsto zero field at the inner electrode, radius rx, and to space-charge-limited emission from that electrode. Eqn. 5 becomes

dx

,-1/2

(6)

The requirement of space-charge-limited emission from theouter electrode, radius r2, introduces the boundary condition(dy/dx)2 = 0. From eqn. 6 this condition defines the ratio ofthe space-charge-limited particle currents:

(7){ e2x(l-y'yU2 dy'J o

Solution of eqn. 6 subject to eqn. 7 yields simultaneously themaximum current of charged particles that can flow fromeither electrode to the other when there is an unlimited supplyof charges of appropriate polarity at each electrode.

3 Numerical solution

Integration of eqn. 6 gives

, - 1/2= U) Le*L-(A//2o)(l - ^ T1/2 ] 4 / ] ~1/2 4 / ' (8)

For small values of x, i.e. for x2l = In (r2/rx) < 1, the ex-ponentials can be approximated to unity. Eqn. 7 then reducesto / 2 / / 2 0 = I\ll\o a n d ectn- 8 can be integrated to give

x < 1 (9)

The integral here has been evaluated by Langmuir [3] in his

x/X2i

Fig. 1 The interelectrode voltage distribution

y = (V - Vt)/(V2 - Vt), x -•= In (/•//-,), x7l = In (r2/rt). From theleft-hand side, the curves correspond to x2i = 6, x2t = 4,x21 = 2 and

0

treatment of the parallel-plane geometry. Setting y — 1 ineqn. 9, corresponding to x = x2l (i.e. to r = r2), determinesthe current ratio Ii/lio for that value ofx2 1. Consequentlythe interelectrode potential distribution y = f(x/x2l) isdetermined. A larger value of x2l is then chosen and theintegrals of eqns. 7 and 8 performed numerically using thepotential distribution y =f(x/x2l) appropriate to the previoussmaller value of x2l. Setting x = x2l at y = 1 in eqn. 8 thengives a first estimate of the parameter Ix //10 and also yields animproved potential distribution y = f(x/x21). The cycle of in-tegration is then carried out again with this improved distri-bution, and so on until successive cycles are consistent. Thewhole procedure is then repeated for larger values of x21. Inthis manner solutions were evaluated over the range 0 < x 2 l < 6,i.e. Kr2/r1 <400 .

Fig. 1 shows the potential distribution obtained for selectedvalues of x2l. The curve for x21 -*• 0 corresponds to planegeometry (r2 -+ rx), and in this case the abscissax/x2] reducesto the distance from one plane electrode divided by the elec-trode separation. The close proximity of the successive curvesimplies that the numerical technique of solution is highly con-vergent. In fact only two successive cycles of integration wererequired to produce an accuracy of ± 0.5% in Ijl 10 and I2/I20 using an increment in ^2i as coarse as 0.5. The most con-cise way of presenting these two parameters is in terms of asimple analytic function multiplied by a factor near unity thatis a weak function of the electrode geometry. Over the rangeof radii embraced by the calculations, a good fit was obtainedwith the expressions

IJIio = 1.860j3x2r2 exp(0.28*2i)

/2//2O = 1.8607*21~2 e x p ( - 0 . 1 3 x 2 1 ) (10)

Fig. 2 shows how the factors j3 and y vary with the ratio ofelectrode radii.

4 Discussion

Two previous publications have, in part, dealt with the presentproblem. Amemiya [13] considered internal and externalemitter configurations with 1 < r2/rx < 30. Donega et al.[14] considered just the external emitter configuration with\<r1lrl <500.

Both treatments used the Runge-Kutta-Merson technique forsolving the differential equation unlike the present techniquethat is a perturbation of the parallel-plane solution. The crossedco-ordinates in Fig. 2 represent the values of 0 from Amemiya's

388 IEEPROC, Vol. 129, Pt. A, No. 6, AUGUST 1982

1.2

Fig. 2 The parameters (3 and y of eqn. 10 for the internal emitter andthe external emitter, respectively

The crossed co-ordinates are taken from the results of Amemiya [13],for the internal emitter, and the circled co-ordinates from the resultsof Donega [ 14 ], for the external emitter configuration.

[13] results for the internal emitter, they are up to 12%greaterthan the values calculated here. His calculations for the ex-ternal emitter were not pursued far enough for the current tobe fully space-charge limited. The circled co-ordinates in Fig. 2represent the values of y from Donega's [14] results for theexternal emitter; they agree with the present calculations towithin ±1%.

For many applications it is useful to determine whether bi-polar current flow results in a significant increase in currentover the unipolar case. The limited unipolar flow of totalcurrent between concentric spheres has been evaluated byLangmuir and Blodgett [15], and is expressible as

' • • -2 /a7

167re0 / 2e

^™2

where a2 and (-a)2 are tabulated functions of r2/r1 for aninternal emitter and external emitter, respectively. (This para-meter should not be confused with the a used by Langmuir[3] in his parallel-plane analysis). Fig. 3 shows that the ratio11 jlu! for the internal emitter is always less than the factor of1.860 appropriate to parallel-plane geometry. For the external

flOOO

Fig. 3 Ratio of the bipolar current to the unipolar current

Subscripts 1 and 2 indicate the internal and external emitters, respect-ively.

emitter the ratio I2/Iu2 always exceeds 1.860 and can assumevery high values. This important trend was appreciated byDonega et al. [14] but was unnoticed by Amemiya [13].

5 Conclusions

In field-emission microscopy, where the internal electrode isthe emitter, and for the thermionic-energy convertor, wherethe electrode separation is very small, bipolar flow can increasethe current by no more than a factor of 1.860 in comparisonto unipolar flow. However, in the case of pellet compression,where the internal electrode is the collector and the ratio ofelectrode radii is very large, bipolar current flow can produce acurrent increase of several orders of magnitude. In this appli-cation, it is interesting to determine whether the chargedparticle beam I2 responsible for compression can generate thecounterflow lx of charges of opposite polarity, through inter-action with the collector surface. From eqn. 10 it follows thatthe current Ix required to maximise the current I2 is of theorder Ix — (m2/m x)

1/2 72. If the compressing beam I2 is com-posed of ions then the required electron current is of the order/j — 10 2 / 2 . Electron emissions approaching this value can beobtained by ion impact on cold metal surfaces. For example,Billet aL [16] observed 14 electrons emitted from molybdenumper incident helium ion in the energy range 100 — 400 kV. Inpractice, the surface of the pellet cathode will be raised to itsboiling point by ion bombardment and the process of therm-ionic emission alone should provide ample electrons. If, onthe other hand, the compressing beam I2 is composed of elec-trons then the required ion current Ix is of the order Ix —10~2/2. Ion production by direct electron impact desorptionat a metal surface cannot meet this requirement. The greatestefficiency appears to be about 10"5 ion per electron observedby Young [17] for the production of oxygen ions at an oxid-ised metal surface. It is unlikely that the strong heating of thepellet anode by the electron beam would raise the ion emissionsufficiently. However ion production by indirect desorptiondoes appear to be a possibility. Humphries et al. [18] havefound that there is copious ion production in diodes operatedaround lOOkV if the anode is coated with hydrogenous mat-erial such as varnish. The impinging electrons produce a plasmaat the anode from which ions are extracted by the electricfield.

6 References

1 CHILD, CD.: 'Discharge from hot CaO\ Phys. Rev., 1911, 32,pp. 492-511

2 LANGMUIR, I.: 'The effect of space charge and residual gases onthermionic currents in high vacuum', ibid., 1913, 2, pp. 450-486

3 LANGMUIR, I.: 'The interaction of electron and positive ion spacecharges in cathode sheaths', ibid., 1929, 33, pp. 954-989

4 MULLER-LUBECK, K.: 'Uber die ambipolare Raumladungsstromungbei ebenen Elektroden', Z. Angew. Phys., 1951, 3, pp. 409-415

5 HOWES, W.L.: 'Effect of initial velocity on one-dimensional bipolarspace-charge currents', /. Appl Phys., 1965, 36, pp. 2039-2045

6 HOWES, W.L.: 'One-dimensional space-charge theory', ibid.,1966, 37, pp. 437-439

7 YONAS, G., POUKEY, J.W., PRESTWICH, K.R., FREEMAN, J.R.,TEOPFER, A.J., and CLAUSER, M.J.: 'Electron beam focussingand application to pulsed fusion',Nucl. Fusion, 1974, 14, pp. 731 —740

8 CLAUSER, M.J.: 'Targets for electron beam fusion\Phys. Rev. Lett.,1975, 34, pp. 570-574

9 CLAUSER, M.J.: 'Ion beam implosion of fusion targets', ibid.,1975, 35, pp. 848-851

10 BARBOUR, J.P., DOLAN, W.W., TROLAN, J.K., MARTIN, E.E.,and DYKE, W.P.: 'Space charge effects in field emission', Phys. Rev.,1953,92, pp. 45-51

11 WILSON, V.C.: 'Conversion of heat to electricity by thermionicemission',/. Appl. Phys., 1959, 30, pp. 475-481

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12 HOUSTON, J.M.: 'Theoretical efficiency of the thermionic energyconverter?, /. Appl. Phys., 1959, 30, pp. 481-487

13 AMEMIYA, H.: 'Bipolar space charge limited current between co-axial cylinders and concentric spheres', 5c/. Pap. Inst. Phys. & Chem.Res. (Japan), 1969, 63, pp. 1-6

14 DONEGA, A.I., KAMUNIN, A.A., and TIMOFEEVA, G.F.: 'Bipolarcurrent in the system of spherical electrodes' in FRANKLIN, R.N.(Ed.): '10th international conference on ionisation phenomena ingases' (Donald Parsons, Oxford, 1971), p. 132

15 LANGMUIR, I., and BLODGETT, K.B.: 'Currents limited by space

charge between concentric spheres', Phys. Rev., 1924, 24, pp. 49 -59

16 HILL, A.G., BUECHNER, W.W., CLARK, J.S., and FISK, J.B.: 'Theemission of secondary electrons under high-energy positive ion-bombardment', ibid., 1939, 55, pp. 463-470

17 YOUNG, J.R.: 'Evolution of gases and ions from different anodesunder electron bombardment', J. Appl. Phys., 1960, 31, pp. 921-923

18 HUMPHRIES, S., LEE, J.J., and SUDAN, R.N.: 'Generation of in-tense pulsed ion beams', Appl. Phys. Lett., 1974, 25, pp. 20-22

Book reviewThe theory of magnetisim — ID.C. MattisSpringer Verlag, Series in Sol id-State Sciences, Vol. 17, 1981,300pp.ISBN: 3-540-10611-1

In 1965 this author published a book with the same title(Harper and Row) and this has had a considerable impact onthe magnetism community world wide. A new version is nowbeing published in two volumes; the second volume will beconcerned with finite temperature effects. As this reviewer'srecords show, he regarded the earlier book with a divided mindand it now appears that not much has been done to changethis state of affairs. The earlier book, as well as this newerrevision, give a good account of several of the mathematicalaspects of this subject, but lack a full coverage of the realmotivation behind any piece of theoretical physics; i.e. theexperimental facts which the theory is designed to explain.The subject of theoretical magnetism is a complicated manybody problem, and it is only if all the evidence is brought tobear on this matter that real advances can be made. Toforeshadow Volume 2, the major breakthrough in renormal-isation group theory was paralleled to great effect by high-precision measurements of critical indices. It is thus good tohave at least a few experimental curves in the present book,but one should note with interest that these are concernedin the main only with dilute alloys.

Although this book thus lacks the breadth and sophisti-cation which full reference to the real world would have givenit, there is, nevertheless, sufficient excellence here to deservethe wholesome praise which others will no doubt shower on itswell known author. The historical introduction, which wasco-authored by Mme. Mattis, is as full of excitement now as itwas in 1965. However, why only discuss magnetic bubblesamong so many other recent industrial developments? Andwhy is it noted on page 38 that L. Neel joins the other mag-neticians listed there as a Nobel Prize winner? N^el gave

sufficient evidence that theoretical advances, when motivatedby the facts of the real world, can often be made with aminimum of mathematical sophistication. Nevertheless,this is clearly not always the case, and it is then that thepresent book and its companion, Volume 2, will be found tobe most useful and commendable.

The historical introduction is then followed by a chapter onexchange, that great concept so basic to theoretical magnetismand here discussed in depth for two and three hydrogen atoms,leading to the Heisenberg Hamiltonian. There follows a briefsummary of the quantum theory of angular momentum and anextensive chapter on many electron-wave functions. Theclimax of this discussion is the Lieb-Mattis theorem, for whichthe author is best known, and which concerns the impossi-bility of ferromagnetisim in one dimension. The most strikingmanifestations of spin dynamics are the elementary excitationscalled spin waves; and these are also fully treated, both forferromagnetic and antiferromagnetic situations. A climax ofthis chapter is a discussion of solitons. It would seem to besymptomatic of what is lacking in the book that no mentionis made of the fact that the Bloch walls shown on page 31 arein fact solitons, as so learnedly discussed on pages 198 andfollowing. A great deal of work, since 1965, has been con-cerned with magnetism in metals and this subject forms a largepart of the present book. The tight-binding approximation,paramagnetism and diamagnetism, exchange in solids, theHeisenberg Hamiltonian, the RKKY interaction, the Kondoeffect, spin glasses, the itinerant electron model, spin wavesin metals and the impurity problem are among the subjectsdiscussed. There is enough of value in this final chapter toovercome one's personal regrets at some inadequacies andomissions (e.g., Ni-Cu alloys can no longer be discussed bythe rigid-band model and the discussion of iron is at bestdebatable).

This book thus has many attractive features, someeccentricities and some drawbacks. It will be read by a widevariety of scientists and technologists, not all of whom willagree with the critique of this reviewer.

E.P. WOHLFARTH

390 IEEPROC, Vol. 129, Pt. A, No. 6, AUGUST 1982