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adjustable or arbitrary constants. The onlyparameters entering the experiments are thedirectly measurable dimensions of the sliders andthe distance of the pivot line from the trailingedge. With these predetermined, no further adjustment is possible for bringing the experimental data and theoretical predictions into

agreement. Yet without such adjustment themeasurements confirm in every detail the predictions of the theory.

The authors wish to express their appreciationto Dr. Paul D. Foote, Executive Vice Presidentof the Gulf Research & Development Company,for permission to publish this paper.

Rectilinear Electron Flow in BeamsJ. R. PIERCEBell Telephone Laboratories, New York, New York

(Received March 25, 1940)Electrodes are devised by means of which rectilinear electron flow according to well-known

space charge equations can be realized in beams surrounded by charge-free space. It is shown howthese electrodes can be used in the design of electron guns having desirable characteristics.

1. INTRODUCTIONT HE purpose of this paper is to show howrectilinear electron flow according to certain known space charge equations can be realizedin regions more limited than those usually associated with the equations. Thus it will be shownhow parallel electron flow can be realized in theform of a beam surrounded by charge-free space.This beam, or region containing electron flow,may be of constant thickness perpendicular tothe flow, with the boundaries between the beamand the charge-free region in the form of planes,or it may be of constant circular cross section,the boundary being cylindrical. In a similarmanner, rectilinear electron flow may exist inwhich the electrons move radially in the regionbetween segments of cylinders or spheres, thebeam occupying a wedge-shaped or conicalregion. By the means proposed, edge effects invacuum tubes can be eliminated and electronguns of predictable performance and desirablecharacteristics can be constructed.

I I . PARALLEL ELECTRON FLOWThe assumptions for all sorts of flow areessentially those made by Childl in deriving

1 C. D. Child, "Discharge from ho t CaO," Phys. Rev.32, 492 (1911).548

equations which are used to describe parallelmotion of electrons. According to these assumptions, electrostatic forces only are considered,the magnetic effects of the electron current andof currents in return circuits being disregarded.Thermal velocities are likewise disregarded, theelectron velocity at any point being specified bya potential cp, taken as zero at zero velocity. Thecharge density is regarded as continuous, and itsmagnitude is obtained by dividing the curren tdensity j (which is constant throughout the flow)by the velocity as specified by cpo Assuming theparallel flow to be in the x direction, Poisson'sequation may be written

d2cp/dx2 = jj o(2ecp/m)!,acp/ay = acpjaz= O.

(1)(2)

In Eq. (1), j is in amperes/cm2, ip is involts, elm in coulombs/gram seven (e/m= 1.768X 1015), and Po is the permittivity of vacuum(Po = 8.85 X 10-14) .

Child solved this equation with the boundarycondition of zero gradient at zero potential.Others have supplied complete solutions. 2 3

2 C. E. Fay, A. L. Samuel, and W. Shockley, "On thetheory of space charge between parallel plane electrodes,"Bell Sys. Tech. J. 17,49-79 (1938).3 B. Salzberg and A. V. Haef, "Effects of space chargein th e grid-anode region of vacuum tubes," RCA Review2, 336-374 (1938).

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III. PARALLEL FLOW WITH PLANEBOUNDARIES

The solutions of (1) have generally beenthought of as describing electron flow in theregion between infinite parallel plane equipotential surfaces. Such flow can, however, exist ina region bounded by planes parallel to the flow,the space outside this region being free of charge.In such a case, certain fields must exist in thecharge-free region outside of the flow, and thesefields are determined by conditions at the bound-ary between the charge-free region and the regionoccupied by the flow. The nature of the fieldsoutside the flow will now be investigated.

Imagine electron flow according to (1) existingin the region y 0 ischarge-free. Figure 1 illustrates conditions at theboundary of such flow. As (1) holds for yO, the potentialmust satisfy Laplace's equation subject toboundary conditions (2) and (3) at y = O. Failureof the field outside of the beam to satisfy (2) atthe beam boundary would require a sheet ofcharge at the boundary. Failure to satisfy (3)would require a dipole layer at the boundary.

Both the real and the imaginary parts of an yanalytic function of x+iy are solutions of Laplace's equation. In (3) suppose that x is replacedby x+iy, giving

cp+i'll=!(x+iy) , (3a)where cp and 'l I are real. cp as given by (3a) is thena solution of Laplace's equation which satisfies(3) at y=O.

y {: : ~ ( O x ) oy -

f ioy2+ax2 =ox EDGE ot BEAMo . ~ _

BEAM

FIG. 1.

VOLUME 11, AUGUST, 1940

As lex) is a real function, writing (x-iy) instead of (x+iy) gives the conjugate of (3a), or

cp-i'll=!(x-iy).Thus the potential cp as given by (3a) is the samefor positive values of y as for negative values,and the field is symmetrical about the x axis. Thismeans that at y = 0, the y-derivative of cp is zero,satisfying relation (2). Thus cp as given by (3a)is the potential in the region y>O.

1.61.61.4

::IE 1.2OJ\3 1.0wg 0. 8w:::; 0.6

~ 0 . 4 V:i 0.2>-!!l'" 0 E g ~ ! ~ ~

/ 'V

-0. 8

IIIb I I/W I 11 7/ III I 7I ;III / / II1/1 r; 1 1/ J~ ~ i I /... 0 I~ 0 ~ ~ ~ eLI /o o 7I ?"// e ?/

/v o ~ ~ 1 / rl 67'flV

0. 4 0 0.4 0. 8DISTANCE FROM CATHODE (x ) 1.2

FIG. 2.The practical interpretation of this is that

parallel electron flow according to (1) can existadjacent to a charge-free region, the boundariesbetween the flow and the charge-free region be ingin the form of planes. To realize such flow in acertain region, electrons may enter and leave theregion through two parallel grids, to which arejoined electrodes corresponding to equipotentialsurfaces obtained from (3a).

An example of particular interest is the case inwhich the electrons entering the region comefrom a cathode forming one boundary, and givingspace-charge limited emission. For this case, thegradient is zero at zero potential.* Taking zero

*Smith and Hartman (J. App. Phys. 11,3 (1940 haveproposed to attain rectilinear flow under these conditions,for beams of small cross section, by making th e potentialvary as distance to th e 4/3 power along the axis of anelectrode system. This will give approximately correctelectrode shapes away from the cathode, bu t cannot beapplied to attain rectilinear flow in the vicinity of th ecathode.549

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FIG. 3.

potential as at the origin, (1) may be integratedto give

cp=f(x)=Ax4/3, (4)9' IA = ( 'J ) = 5.69X 1O'lj1. (5)4po(2e/m)iIn the charge-free region outside of the beam,

according to (3a) the potential is given by cp inthe expression

cp+i'l1= f(x+iy) =A (X+iy)4/3. (6)Figure 2 shows a plot of equipotential surfaces

according to (6). The shape of the equipotentialsis independent of the absolute magnitude of thepotentials involved and of the units in whichdistance is measured. Thus, in Fig. 2 potential isindicated in terms of an arbitrary potential, CPo,assigned to one equipotential surface. The potential of the cathode is taken as zero and distance is measured in arbitrary units, with theorigin located at the edge of the cathode. It willbe seen that the zero potential surface is a planewhich meets the edge of the cathode (x=y=O)550

FIG. 4.making an angle of 67.5 0 with the normal to thecathode (the x axis). Figure 3 shows how a planecathode and the zero-potential electrode and apositive-potential electrode from Fig. 2 may becombined to give a parallel beam of electrons.The dashed lines indicate the boundaries of thebeam.

In Fig. 4, both negative and positive equipotential electrodes are used in producing thebeam. In the structure shown, the beam leavesthe second electrode through a slit rather thanthrough a grid. When the width of the slit isconsiderably less than the cathode anode spacing,the electron flow between cathode and anode willbe practically the same as though a grid werepresent. However, the beam emerging from theslit will be divergent because of the lens actionat the slit. The focal distance will be practicallyequal to the focal length of the lens formed bythe slit.*,4

Equipotentials for other solutions2, 3 of (1)* j=2",/(E 1-E 2) where", is the potential and El and E2are the fields at the two sides of th e slit.4 C. J. Davisson an d C. J. Calbick, Phys Rev 42 580(1932). . ., ,

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may be obtained in the same manner, andcorresponding electrodes may be used in constructing vacuum tubes in which edge effects areeliminated.

IV. PARALLEL FLOW WITH CYLINDRICALBOUNDARIES

Electron flow according to (1) may also beobtained in a beam of constant circular crosssection, surrounded by a charge-free region inwhich an appropriate axially symmetrical fieldexists. The conditions which must be met overthe cylindrical boundary between the beam andthe charge-free region are

iJcpjiJr=O,cp=f(z),

(7)(8)

where r is measured normal to the beam surfaceand z is distance from the cathode.

The problem of finding an axially symmetricalfield outside of a cylinder which satisfies Laplace's equation subject to boundary conditions(6) and (7) at the cylinder may be solved experimentally by means of an electrolytic tank. As thebeam and equipotentials are figures of revolution, any sector of the field cu t off by planesthrough the axis of symmetry contains completeinformation about the entire field. Such a sector

FIG. 5.VOLUME 11, AUGUST, 1940

of field may easily be represented by current flowin an electrolytic tank.

A suitable tank construction is shown in Fig. 5.The bottom is a tilted plane of nonconductingmaterial. Above the bottom is a wedge-shapedbody of electrolyte, bounded by the tiltedbottom and the liquid surface. No electricalgradients can exist normal to the bottom or tothe surface, so the wedge-shaped electrolyte canrepresent a sector of an axially symmetricalsystem. The thin edge of the wedge of electrolyteis the axis of symmetry. Suitable electrodes toestablish symmetrical equipotentials would beportions of figures of revolution about this axis.When the angle of the wedge is small, these canbe replaced by singly curved electrodes, such ascan be bent from flat sheets, with little error. The

: ' ~ ~ T ; : 0 ~ . m r 7 m 7 m m m C t ~ 7 o I 2 3 4 5 6 7 6 9 10DISTANCE FROM CATHODE ( ~ o )

FIG. 6.

use of such a tank has been suggested by Manifold and NicolJ.5In use, a piece of insulating material is insertedin the electrolyte to represent the edge of the

beam. This insures that the field normal to thebeam boundary will be zero as required by (7).Usually, in obtaining the field two electrodes areused. These electrodes extend from the insulatingstrip representing the edge of the beam out into

M. B. Manifold and F. N. Nicoll, Nature 142, 39(1938).551

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FIG. 7.

the electrolyte. The electrodes are connected tothe terminals of an oscillator. A calibrated potentiometer makes it possible to measure potentials along the insulating strip which represents the edge of the beam.

As the condition of zero field normal to thebeam boundary is automatically taken care of,the problem of finding suitable electrode shapesis merely that of adjusting the shapes andpositions of the electrodes in the tank until thepotential along the strip varies as the appropriatefunction of z as expressed by (8).

Figure 6 shows electrode shapes which wereobtained for parallel flow from a space chargelimited cathode in a cylindrical beam, accordingto (5). In Fig. 6, ro is the cathode radius, r is thedistance from the axis of the beam, and z is thedistance along the beam measured from thecathode. In Table I, measured potentials for theelectrode system giving the equipotentials shownin Fig. 6 are compared with the desired potential,which varies as Z4/3. It may be seen that the fitis good.

Figure 7 shows an electron gun design based onthe equipotentials in Fig. 6. The focal distanceto the right of the aperture in the anode can becalculated by means of the formula for the focallength of a lens formed by a circular aperture.*. 4

552

V. RADIAL FLOW BETWEEN CYLINDERSAND SPHERES

Besides that described by (1), there are othersorts of rectilinear electron flow in the presenceof space charge. Langmuir and Blodgett6 7 havepresented equations governing radial flow inwhich the equipotentials are concentric cylindersor spheres. The assumptions made in treatingsuch flow are the same as those used in deriving(1). Langmuir and Blodgett give solutions inseries form for the cases in which the gradient iszero at zero potential (similar to Child's solutionof (1)).

Such radial flow is usually associated withcomplete concentric cylindrical and sphericalelectrodes. However, it can also be realized inregions of limited angular extent between segments of cylinders and spheres, provided thatthe proper electrodes external to the electronflow are employed.

VI. RADIAL FLOW BETWEEN SEGMENTSOF CYLINDERS

Converging radial flow in a wedge-shapedregion between segments of cylinders is of par-ticular interest. This case may be treated byap-proximate methods.t In the region ro>r>0.25ro,

TABLE I.z/8ro (z/8ro)'/' 'PI Po MEASURED0.1 0.046 0.048.2 .116 .122.3 .201 .201.4 .294 .292.5 .397 .401.6 .506 .503.7 .622 .618.8 .743 .739.9 .869 .870

where ro is cathode radius and r is distance fromthe center of curvature of the cathode, the following expression gives to a fair approximation the

6 Irving Langmuir and Katherine Blodgett, "Currentslimited by space charge between coaxial cylinders," Phys.Rev. 22, 374 (1923).7 Irving Langmuir and Katherine Blodgett, "Currentslimited by space charge between concentric spheres, " Phys.Rev. 24, 49 (1924).t L. A. MacColl has obtained some equipotentials forth e exact solution of this case. Th e equipotentials aresimilar in shape to those obtained by the approximatemethod here presented.

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O.B

#0"'" / 'V '? '0.7

0.6,........:::nIl''- ' 0.5::;wID 0.4IL

,,--:: / ' VL /./oWg 0.3w

0.2wUZ o. Io o / '

v:~ ~ V ,'

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In the spherical case there is a differen t se tof equipotentials for each angle of the coneoccupied by the flow. A gun design using a par-ticular pair of electrodes for flow in a conewhose sides have a slope of 0.25 with respect tothe axis is shown in Fig. 10. The electrons leavethe cathode, which is in the form of a circularspherical cap, and converge along radii to theanode, where the beam is t as wide as at thecathode. The beam passes through an aperturein the anode electrode, suffering a calculabledivergence. A small aperture limits the size ofbeam at the crossover.

VIII. FURTHER CONSIDERATIONI t will be noted that in all the equipotential

and electrode shapes (Figs. 2, 6, 8, 10) a zeroequipotential meets the cathode edge making anangle of 67.5 with the normal to the cathodesurface. This is because in a very small regionnear the edge of the cathode, the curvatures ofthe cathode become negligible in all cases andthe equations approach (4) and (8) which repre-sent parallel motion between parallel planes.

The absolutely correct electrode shapes, forwhich the appropriate boundary conditions aresatisfied exactly, are unique. There is, however,an infinity of quite dissimilar electrode shapeswith which the boundary conditions can be satis-fied to a good approximation. An approximatesolution may be better in a practical sense thanan exact solution, for the approximate solution

FIG. 9.

554

FIG. 10.

allows a certain amount of adjustment of theelectrode shapes to meet exigencies of con-struction.

The equations on which the work is based werederived using certain simplifying assumptions.Thus the idealized electron flow which has beenconsidered cannot be attained exactly, eventhough the "correct" electrodes are used. Per-haps the most serious divergence of the equa-tions from fact lies in the neglect of thermalvelocities. This is most serious in the case ofconverging flow, as in electron guns, for thermalvelocities limit the current density which can beattained according to known relations. 8 9 Elec-tron guns designed according to the methodsoutlined have good focusing qualities, however,and with such guns current densities of over one-half of the limiting value imposed by thermalvelocities have been attained.

Other practical advantages of these guns arethe uniform cathode current density, whichmakes it possible to take advantage of themaximum allowable emission of the cathode overits entire surface, and the good focusing of elec-trons emitted from the edge of the cathode, whichmakes it possible to utilize a large portion of thecathode current in the beam. Thus, guns havebeen constructed which bring currents of severalhundred milliamperes to a sharp crossover, withonly a few milliamperes lost \ on the anodeelectrode.

8 David B. Langmuir, "Theor etical limitations of cathoderay tubes," Proc. I. R. E. 25, Aug. (1937). J. R. Pierce, "Limiting current densities in electronbeams," J. App. Phys. 10, 715 (1939).

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