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Electron velocity fluctuations intwo-dimensional systemsP A. LINDSAY a & D. DIRMIKIS‡ aa Department of Electronic and ElectricalEngineering , King's College , LondonPublished online: 29 Oct 2007.

To cite this article: P A. LINDSAY & D. DIRMIKIS‡ (1971) Electron velocity fluctuationsin two-dimensional systems , International Journal of Electronics, 30:5, 401-435, DOI:10.1080/00207217108900340

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IST. J. ELECTROSICS, 1971: VOL. 30, S O . 5, 401-435

Electron velocity fluctuations in two-dimensional systems?

Departn~ent of lilectronic RIKI Elcctrical E~~gi~lccring; King's Collcgc, Im~don

[Receivcrl 5 Dcccmber I%O]

1. Introduction 'I'he 11iai11 object of tlic paper is to present a method of calc~~lnt ing electron

vclocity Huetuatior~s and t l ~ c correspondi~~g volume and current de~isit,y distri- butions in systems which w e basically two-tlimensionnl in character.

It has been know~i for some time that , in principle, such calculntions arc bcst performed in terms of Hamiltoninn nieclionics, the fundamental concept in this case being tha t of 11 sis-dimensional phase space (Gabor 1945: 1.i1idsny I!NO a). This applies p;~rticulnrly when all t l ~ l r e components of thc initinl electron velocities have to be taken into account ill the presence of n miig~ietic field, as is oftell the case in con~iection with tho design of crossed-field devices. For p l n ~ ~ e magnetrons ( I h d s n y I ! lW h, l9fi4) this problem has Lee11 solved n~~n~er i cn l ly both for irnlx~scd fields (Lindsay I!)fi:i, 1!)70) and those obtililled by self-consistent field c;r lc~~l;~t ions (Lindsay i!)fi?, Lindsay and Coodcll I!)65) ; self-consistent solutions have also beeti obtained for cylindrical magnetrons although only general dycbraic expressions linvc becn published so far (1.i11dsey 1960 c). I n all s ~ ~ c l ~ cases, however, the geometry of the system wils funda- mentally one-dimensional in character, requiring u. single position rarinblc only (z or r), even thongh two velocity variables (u,? u,, or v,, us) liad to be takcn into account in setting up thc corresponding limits of integration in the velocity sub-space.

!I?hese calculations Iit~ve now bccn e s t c ~ ~ d e d to less symmetrical systcms where four indepcndcnt vi~rit~bles, e.g. z , y, u,, n,, arc squ i r ed for a eornplotc description of the problem. After a brief i ~ ~ t r o c l u c t i o ~ ~ the method is dis- cussed in terms of two co~icrcte examples : t,he volume and current density distributions of c l cc t ro~~s emitted by a thermio~~ic ; semi-infinite, plane cathode im~nersed in a magnetic field which is either parallel or perpendicular to its sorfnce.

t Co~nrnunicatcrl By Dr. I' A. Lindsay. f. Xon in the D C ~ I L ~ ~ I I I C I I ~ of Electrical Engineering, University of Sl~cllicld

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2. General description of the method As before, we are limiting our problem to tha t of elcctro~i velocity fiuct~ra-

tions gencratctl by the process of thermionic emission. 'Clrc corresponding pliase-spwc density of the emitted electrons a t :my point in the i~rterelectrode space is given by (Lindsay 1963, eqn. (9))

nntl

?n = clcct,ron inass: r , I = tc~npcrat,i~rc of tlrc enlitter,

I&= work function of the enlitter, k = l3oltzrn;~nn's constant: h = I'lnnck's constant.

~n,.=electron volume density at the surf;wc of a tc~npcmture limited emitter.

' h e two v ~ ~ r i ; ~ b l e s in eqn. (2.1). viz. the reduced clcctrost;~tic potc~itii~l 7 and the rctluccd clectron velocity w, are respectively dctined by the following cqu a t' ~olis :

nlicrc - e is the charge of the electron, the subscript c referring to the condi- tions a t the emitter (cathode). With the lrclp of eqns. (2.3) and (2.4) we ti~rd it convenient t,o express tlie potential a d kinetic energies of the electron in units of k7'.

, ? Slie volu~iie and current density distril~utions of tlie electrons can now IIC cs~~rcsscrl in terms of the phase-space density fwiction n(r,w),

where rlw = r h , du:,, dloZ and a= (2kT/m)Il2. In spite of their relntively simple algebraic form eqns. (2.5) and (7.6) in fact conceal the innin tlifiici~lty of the lwoble~ii, viz. thc correct choice of the corrcspondi~ig limits of integrnt,ion.

Substituting e q ~ r . (2.1) in eqns. (2.5) and (2.6) we now obtain :

~(r) /rr . =- exp 7 cxp ( - w2) dw, J (2.7)

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Electron uelocity Jl~cctuations in two-dimensional sysle~ns 403

9 . l ( r ) / l = e x 7 w exp (-w2) dw, (2.8)

9,

whcrc J,=nTm/nl/2= rrT(2k7'/n~n)'/2 is tlic usual expression for tlic saturation current tlensity in tcrms of the mean velocity of emission (2!d'/nw1)'/~ (e.g. 1,indsay 1063, cqn. (YS)), thc volume density being normalizetl with respect to 71, and the current tle~isity with respect to J , . Equations (2.i) and (2.8) clcpend on the position vector r=(z , y, z,) tliror~gli the potential function ? = ? ( r ) and, more tlircctly, through the corresponding limits of integration wl~icli may vary from [mint to point, depending on the geomctry of the system. 111 solving, the electrostatic potential 7 is either assumed to be give11 in advance, wlum eqns. (2. i ) antl (2.8) directly providc tlic required volume and current density distributions, or eqn. (2.7) can be looked on as tllc right-hand side of the corresponding I?oisson equation, the solution then being obtained by s~~ccessive approximations-a common procedure when space-charge effects h a w to be taken into account.

In all cases tlic mail^ difficulty is associated with the calculation of the correct limits of integration in eqns. (2.7) and (2.8). It is fairly clear that a t rliffere~~t points r d i t f c r c~~ t parts of the velocity subspace will ccnsc to bc avidithlc to thc cmittcd clcctrons. For exarnplc, in the case of ;t tcmpcrature- limited, plane, pnrirllcl diode thcre will be no low velocity electrons near the anode! since even thosc electrons which liavc left the cathode with very small velocities will havc bccn accclcrated on their way to tlic anodc. 'I'lius, in general, the velocity s~lbspacc will be separated by a surface

into two distinct p~rts-one accessible to the electrons and the othcr onc which is not. N\'atnl.;~lly: the integration indicated in eqns. (2.7) alitl (2.8) must only extend over the first part of the velocity subspace: sincc otherwise wc would be counthg nonexistent electrons in the integration process. Siucc the corresponding contlitions may change however from point to point, the surface F is in generid :I fimction of tlic position coordinates z , y, z. St is important to note ho\vcver tha t the position coordinates enter hcrc as para- meters which are i~ulcl~cndent of thc propcr integration variitbles II:,, lu,, and "'*.

'I!he algebraic form of the separation (or limiting) surface Y 11s givcn by cqn. (2.9) depends entirely on the geometry of the system under co~~sidcration 1~1id the corresponding boundary conditions. In the simple case of n tempera- ture-limited plane, p u ~ l l e l diode me would have wz-{1(z)}1/2= 0: wliicli is the usual expression for the conservation of energy for an electron leaving the cathode with zero initial velocity. Thus, a t any given point z the lower limit of ilitegratio~i is given by z~ . ,=ql /~ rather than zero, the surface F now being a plane p;rr;illcl t o u:,, and u~, axes and cutting w, axis a t the point {v(z)}1/2, z k i n g ill taliis case the only significant position variable. The velocity sobspace has now been subdivided by the plane into that part which is accessible to the electrons (i.c. t o the right to the plane where r)"2< 1cx< CQ) antl the rest; which is not. This fact must be carcfi~lly taken into account when the integrations indicated in eqns. (2.7) und (2.8) are

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being pcrfortned. Naturally, this is n ~:~articularly sitiiple example and, in general, the shape of the P ' surface can be surprisingly complex, especially wlien a magtietic field is prescnt. I n fact, ns we shall see later, it is the actual dependence of tlie surface .F oti tlic position coordinates (x, y, z ) which is pt~rticuli~rly difficult to'foresee.

3. Discussion of the model Computational details of tlie method described in the preceding section

depend a. great deal on tlie actual geometry of the system under consideration and i t is best, a t this stage, to continue our discussiot~ in terms of a simple tiiodel, shown in fig. I ; wliere we havc a thermionic cathode in the fortn of a sc~iii-infinite half-plane wliicli is totally immersed in a magnetic field parallel to its surface, no electric field being present. We have been influenced in the choicc of this pcwticuli~r model by the conditions which may obtain in the ci~se of a low-noise gun (Currie I!)58, Currie and Forster 1958, 1959, hlueller I!)(iL), viz. tiegligiblc electric field immediately in front of a cylindrical catliode i ~ n d a substatltial amount of emission from its side walls. I n practice, the i~xial tnngnctic field of a Currie gun, usually of tlie order of 2000 gauss, is sulficient to make the radios of the cathode ( - 2 mm) much larger than the corresponding average radius of electro~i trajectories, which are helical in nature (for an average initial velocity of 0.04 ev, which corresponds to a cnthodc temperature of T = I O U O O K , tlie average radius R z 0.004 mm). It is tllercforc reasonablc to ignore in our calculations tlie curvature of the cathode : L I ~ C ~ to assunic that arc liavc itistend a semi-infinite plane, au shown i ~ i fig. I. For the sake of completeness a coniplcmenta~.y system. shown in fig. 12, will be disc~tsscd in 10 of the paper.

Y J

Modcl discrtssed in § 3.

4. Electron trajectory calculations 'I.'llc cq~li~t ions of motion for the system shown in fig. 1 are given by

X+w,l '=o, (4.1) '> 1 -w ,x=o, (4.2)

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Electron velocit!~ jli~cttcations in Iroo-dimeclsionul syslems 406

and I

is, ias usual, the cyclotron frequency. When integrated, eclns. (4.1)-(4.3) leap to the following expressioils for the velocity and position of a ]>article :

w, = w,, cos O - tou, sin 0 ,

IU,, = wZc sin 0 t- IU,,, cos 0,

and to, = to,',, (4.10)

X =to,, sin 0 -tuU,(l - cos 0 ) , (4.1 1 )

I 1'- Y , =tu,,(I - cos 0) + wUc sin 0, (4.12)

Z - Z,=tu,,O, (4.13) w L r e

O=w,t (4.14)

is 'the transit angle, the initial values of J', Z, w, and w , being respectively indicated by the subscript c. Equations (4.11) and (4.12) f111ly describe the required electron trajectories which all Ilave the shape of a helix.

5. Limits of integration in the velocity subspace , I n order to evaluate both volume and current density integrals, eqns. (2.7) aitd (2 .8) , i t is necessary to derive an appropriate expression for the limits of integration in the velocity suhspacc. I n the case of a cathode of infinite edtent in the Z direction i t is only necessary to consider the X dependence of tl;e corresponding electron velocity distribution ; for example, under the influence of the magnetic field some electrons may fk~il to reaclr the plane ,Y(=X,, their energy being too low (Litidsay l9GO a ) . When the cathode extends only part of the way in the Z direct,ion, as is sliown, for example, in fik. 1 , i t is necessary to remetnber that a t a point Z =Z1 some velocities may not be represented because the corresponding parts of the cathode are actually

I. . mtsslng. I t is now necessary t o determine under what conditions the elec- t(otis not merely reach the plane X=X, but, in f i~ct , pass through the line q = X 1 . Z=Zl . Fortunately in our case this condition can readily be obtained by eliminating the phase angle O het\veen cqns. (4.11) and (4.13) :

. 2,-Z, XI = tU.z,; SIII - - w., (1 - cos - tozu

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406 P . A . Lindsny and D. Dirmikis

Equation (5.1) provides tlic required relationship between the position vari- ables S,, Z, and the initial condit,ions determined by Z,: tc,, and !I:,,. I n onler to obtain tlie act l~al velocity distribution a t X = S , : Z = Z , it is now necessary t,o express w, in ternis of w by substitnting from cqns. (4.8)-(4.10) :

Sinoc the system is syn~~netricnl iu tlrc l' directioiil the v:wi;rblc Y docs not I L I ~ I ~ L ~ in eqn. (5.2) and the proble~ii becomes two-dimeusionnl in cl~aractcr, ( S , , Z,) now defining an arbitmry point in the (X: Z) plane.

It sliould he noted i n connectiot~ with cqu. (3.2) that due Lo the geometry of the system sho~vn in fig. 1, the values of Z, must all he negative, i.e. - co < Z,<c): the equation representing a family of straight lints in t,hc ( I t o ) pane. By altering the value of Z, and keeping XI: %,: constant it is possible to sweep out that part of tlie my) plane 1vhic11 is accessible to nl l electrons passing t,hrougIi the point (S,, 2,) ; the integri~tio~is inrlicnted in eqns. (2.7) and (2.8) nlust not extend bcyond those limits. Sincel in gcnelxl, the curve representing the very edge of the accessible area must bc IL fnnction of I&, it will genernte a surface which separntcs tllc :iccessil~lc and iut~ccessible parts of the velocity snbspixce (uI,, w,, LC=), as intlicirtcd in cqn. ( ! I ) T h e sl~irpa of this surfiiec can be quite involved and e\.cn in our simple msc it is best to investigate its propertics hy considering the cross scctions m,= const ; cqn. (5.2) can the11 be arittcn in the following fos~n, for c l ~ ~ r i t y :

Hcre tc,, I;,, arc the variables and S,; Z,; Z, and wZ am all treated ;IS porn- meters.

It should be noted for future reference that in the plane of the cathode, i.c. for S,=O, the straight lines given by eqn. (5.3) all pass through the origin. :[?or S, f 0, on the other hand, they have an envelope given by

E q ~ ~ n t i o n (5.4) represents a symrnetric.zl parabola pointing i~pwirrds for X I < O u i d downwards for X, > O . For X,=O t,he p r a l d a clegcncratos into IL ~ t~ r i~ ig l i t line coinciding with the to,, axis. The point of contact l~ctwcen the struight lines and tlic parabola is givcn by

Sinlil;~rlp, substituting oqn. (4.11) in cqn. (4.9) we find that

w, = to,,+ X. (5.7)

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Eleclron velocity jltrclunlioas in / ~ ~ o - d i ? ~ t o ~ s i o ~ ~ n ~ sysle?~ts 407

k:lirninatlng to,, between eqn. (5 . i ) and nn expression for the conservatio~~ of energy,

tczz + m,,= = wZc2 + w ue J (5 .8)

we obtain

luZ2 = + S(X - ?toU). (5.9)

Since eqos. (5.!)) a ~ l d (5.4) are identical when zc;,,=O, the parabola mpresents the velocities of those electrons which leave the oathorlo tnngentially.

At this point i t is convenient to define the following regions i l l fib'. I :

I : region to the right and below the cathode (S,,<O: Z , > 0). I1 : region t o the right and above the cathode (X, 2 0: Z, > 0).

I[[ : region directly above the cathode (XI > 0, Z, ,< I)). ' I h limits of i n k g r ; ~ t i o ~ ~ in each region will now be considered in turn

Region I I n thin region toz > 0 for all electro~ls since ill order to reach a point 2, > 0

tlicy must initially travel in the + Z direction (u::,> 0). However, for n given value of w, the point of emission Z, may lie anywhere between - DJ and zero ; this will generate a family of straight lines, ns indicated in fig. " eenh line corresponding to a different transit angle O = (Z, - Z,)/IG~, the shortest tnmsit angle heing given by Z,=O or O,,,,,=Z,/I~, (see e r p . (4.13) ;rind (5 .3 ) ) . !I'lic rest of the ( I O , ~ YO,,) plane rem:rins inuccessihle t,o thc electrons.

Fig. 2

Electtnn rolocity ilist,l.ih~~t,ion in i.cgio11 I.

As an example let us consider O,,i,=n ; now for Z,=0 we have Z,/IU,= s and eqn. (5.3) gives a horizontal liue cutting the vertical axis a t do,- +S, < 0. However, since w,,. must always be positive in an e~nission process we find front eqn. (4.8) that w,<O-this means that only the negative half of the

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405 1'. A . Lindsay and I). 1)irmikis

straight line ic.,=~S, can actually rol~rescnt the allowed electron velocities a t thc point (XI, Z,). If we move the point of emission awly from the edge of thc c n l l d o Lo, sny, Z,= -fZ, , eqn. (5.8) becomes zo,=w,+X,, where agnin X I < 0. Wow 0 = Rn/2 and we find from eqns. (4.!1) that for w,, > 0 we must linve rc, g 0 ; this is o slight nvcrestimate, however, since the electrons satisfying the conditiol~ S , < to" G O a t the point (XI, Z,) would have to descend bclow the ylanc of the catl~orlc bcfore thcy reach its edge a t % = 0 ; this incnns LhaL instead of reaching the point (9,; 2,) t , l q would return t o tlw o;~t,llode. Similnr rc:~soning, when iq~l~lictl to other wrlues of Z,:, shows tha t for the sl~aded part. of t,Iw (la,, w,) plane is limitcd, after nll, by the contiuuetion of tlie straight linc lu,=bS, t o the riglit of the vertical nsis. In general, for X < O , the sl~aded portion of the (toz, ~i,,) plane is limited from above by ;L straight line \vhicli is obtained from eqn. (:i.3) by potting Z,=0, as is indicatccl, svniewlrat schc~naticnlly, in fig. '7.

Regioit / I As in region I! all electrons reaching the point (S,, Z,) must be travelling

in tlie +% direction, so thttt again 0 4 toc < CQ ; however, thc shaded portion of the ( I L . ~ , LC") plane no\v extends from one of the stmight lines given by eqn. (5.3) to tho parabola, eqn. (5.4), Llw lat,ter representing thc velocity of those e l ec t ro~~s \vhich leavc the cathode tangentially, as shown in fig. 3. The l~oints inside tlie zwnis of the parabol;~ correspond tn the velocity of non- existent clectrons wliich would have to leave the cathode with a negative cotnponcut of the emission velocity w,,. Si~nihrly, t,he points situated above the 8traiglit linc Z,=0 cormspond to tlie velocity of those electrons which conld only lhave been emitter1 a t Z,> 0, wllcre there is no cathode. The other straiglit lines in fig. 8, which is plotted for a fisccl \ d u e of ic,, corres- poud to elcctrons ernittcd a t steadily decrc:ising values of %,-they gradually

Fig. 3

Elect,ron velocity distribution in region 11,

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Electron t;elocily J7ricleatio?1s rn two-di~~ro~sional s!/slen&s 400

sweep out the whole of tlie shaded portion of tlie (1%. rr;,,) plane. As in fig. " only portions of tlie straight lines can be considered, since some electrons would f d to clcnr t l ~ c cnt,l~otlc even though their cncrgy is adequate to reach the point (XI . Z,).

Region 111 I n region 111 the velocity distrihntion is soniewhnt more complicated

than in tlie other two regions. First of all we note tlint. for all Z , < Z , the existence of the edge a t Z = 0 is quite imniaterial-for d l tcz> 0 the whole of the (u:,, w,,) plnne is accessible t o t l ~ c electrons, except for the area between the arms of the l~arnbola, ;IS sliowi i l l fig. 4 ( 1 1 : ~ ~ must be always positive). For the emission points lying between Z, and the edge of tlie cathode, on the other hand, the situntion is somewhat different. When Z, / t r<u: ,< 0 , the corresponding minimum transit. ;~ngle takes all possible values in the interval 3 r s O,,,,,, > co. Since for Z, < 0 thc transit angle O cannot exceed I n mithont the electron hitting t,he cntliode, t,lic velocity distribution in this case will be unaffected by the presence of the cathode edge a t Z = 0 , the

Fig. 4

Electron velocity distribution in region 1 I I ; Z , / h < IG, c w.

conditions being the same as those for 0 < 70, < oo, as shown in fig. 4. How- ever, when - oo < q 4 Z , / t n tlie corrcspontling ~ninimuln transit angle must be less tlian ? n and tlie existence of tlie cnt.liode edge no\v affects the velocity distribution by robbing us of those electrons which would otherwise be emitted a t points Z,>O. The shaded portion of the (u:,, lo,) plane is now shown in fig. 5-it is limited pnrtly by t.lie parubola, eqn. (5.4) and partly by the appropriate straight line (Z,=O) given by eqn. (5.3).

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Electron velocity distribution in region llT ; - m < w,iZ1/7n.

6. Electron volume density The volnn~c density of the electrons a t any given point (XI, Z,) can be

ohtaincd by substituting the corrcot limits of i~itegrat~ion in oqn. (2.7). How- ever, in order to avoid vcry amkrvarrl combinations of fnnctions of the form exp (:+)(I - e r f z ) it is advisable to use polar instead of cartesinn coordinates i n the (toz, 10") pli~ne :

ph zZ2 + ~f:,,~, (6.1)

tan 4 = q,/wZ. (8.2)

I t is worth noting tha t p and 4 diffcr, in general, from the respct ive radial and tnngentid components to, and I,:, of the reduced vclocity vector w. Jn terms of the polar coordinates the straight lincs of eqn. (5.3) and the parabola of cqn. (5.4) heoon~e~ respectively :

Z, -z , Z, - Z,: p , = iXI cosec -

7n,

and

((i.4)

the angle of slope of the straight lines being given by

Sitnihdy, the + coordinate of tho point at which the Z,=0 straight line tonchcs tho pi~rd)ola is givcn, from cqns. (5.5) and (.i.6), by

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It is now convenient to evaluate the limits of integr;ition and the corres- ponding elcutcon volurnc de~rsity separately in the three regions.

Region I Snbstitr~Ling the limits of integration shown in fig. 2 by pnttirrg Zc=O in

cqns. (6.3) and (6.5) we obtain, from eqn. (2.7) :

The second integcnl can be simplified by putting $=4-2,/2w2 and noting the symmetry properties of pal($) :

The integrnnd becomes zero \\,henever Z l /hu ; ,= "7, the deitnsy of snch points increasing witlront limit as we approitch the origin aB= 0. Since the integral rnnst be cvalnnted numerically snch a sitnation is quite unncceptahlc. Fortnnately the problem can be r ~ s o l \ ~ e d By introdncing a new variable 5=Z1/210. which ensnrcs that the zeros of the integnintl are evenly spilced along the new axis :

$5 . 5 - 2 ~ x 1 1 (-Z12/4p) exp { - :XI2 OOSCC~ 5 see2 $1 d $

((i.8)

, I l b e double integral can be fnrtber simplified by expressing i L partly in terms of t l ~ c error firnction. \Vriting r i = tan $. I + u2=sec2 $ and r l u / ( l + 112)=d$ we obtain :

" exp { - hZ(l + uE)] du,

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412 P. A . Lindsq nnd L). Dirinikis

where A = $ 1 X,/sin { I . Using the identity I = l ( A ) = - if (allah) dX and noting that

we can now write

Substitnting eqn. (6.1 I ) in eqn. (6.8) we finally obtain :

I n the plane of the cathode, i.e. for XI = 0: c q ~ ~ . (ti. 12) reduces to

This resnlt can be explained by noting that the only electrons reaching the point (0, Z,), when Zl>O, lire those with tr:,> 0, i.e. one-half of all the emitted elect.rons.

Finally we have to consider the vi~lue of the integral I, when Z,+oo. Writing 3 =Z,s in eqn. (6.1 2) we obtain :

Since the mean value of t,he fluctuating part of tho integmnd over the period n/Z, is given by

= l J l { l - c r f i n l ~ l ) d s , sin s'

which is n fnnction of XI only, we obtain, substituting eqn. (6.15) in eqn. (0.14) and putting t = 1/25

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Eleclron celocily ~Iucl~mlions in. Iu-dimewional sysle~m 413

Kegion I 1 Substituting the limits of integrat,ion shoaw in fig. 9 we now obtain :

I'nttmg O = J(+-nrJ) in the first integral and, as before. $=+-Z,/2w, in the second : ~ n d noting the syrn~lictry properties of pll= ~ ~ ( 8 ) nod p,\ =p,($), we obtain :

111 the plane of the cathode. XI = O? eqn. (6.17) reduces to

which agrees with the corresponding value obtained from eqn. (ti.12) and given by cqn. ((i.l:l), the two curves ~iieeting along S l=O.

Sirnilarly. for Z,+m we obtain from eyw. (6.16) and (6.17)

Region. 111 The li~nits of integmtioo i l l this rcgio~i are son~eahat more complicated,

being dilfiiwnt for w:> Z1/2n and ?I:,< Z1/2n. With the help of figs. 4 and 5 we can now write

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414 P. A. Lindsay and D. Dirmikis

'1:llere is uo need to change the variable frorn to, to 5 in this case: since the limits of integration of the second integral extend from - co only to u and not to zero, as was the case in eqn. (6.;).

In the [ h i e of the cathode, i.e. for S, =0: eqn. (6.20) reduces to

where the exponential integral is defined as

Along the edge of tlie c;rtl~otlc, i s . for X', = Z , =0, eqn. (6.23) reduces to

?~(r<,)/n, = 8. (6 .25)

Comparing eqns. (6.25) and (fi.18) we find that the volume density of the electrons changes suddenly as we move across the edge of tlie cathode? the difference being equal to n,. This can be explained by the simple fact that for all Z, > 0 we lwk the contribution frorn the electrons which, i l l the presence of the cathode, n.ould have been emitted a t that poiut.

Finally we find from eqn. (6.20) tha t for Z, +- m or a +- a, we ol~tain :

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Eleclron uelocil!/ ,flncl~m/io~~s in two-dimo~sio~~ul system 415

7. Electron current density The three comlronents of the cnrrcnt tlcnsity a t m y given point (XI , Z,)

nrc obtained by substituting in cqn. (1 ,s ) the corresponding limits of intcgra- tion sliow~i i l l figs. "5 ; However, except for the i co~npo~ient, it is now morc convenient to use cartesi;ui rather t1i;in po1;ir coordinates: the i~itegrands of the for111 z c x p ( - a : ? ) 1)cing readily expressed it1 terms of elementary func- tions. I'or clnrity the three regions will again be treatcd sep;rr.ztely.

Kegion I Substituting in cqn. (2,s) the limits of integratim sliow~i in fig. 2 n c

ol)tain the following e x l w c s s i ~ ~ ~ ~ for t,hc z con~poncnt of the wrrent density :

where use has becn m;de of the dcrivnt,ion lcnding to eqn. ((i.7). Trans- forming the second integral following eqn. ((i. l I ) and introducing the variable 5 = Zl/21o, wc obtnin :

For S, = 0: eqn. (7. I ) reduces to

'I'hus, in the plant of t,lic cathode the z conrpo~icnt of the current densit,?. is equal to $ J , ; this is quitc reasonable becnuse for all Z, > 0 wc have 06 q < m, no electrons with ~~ega t ive z components of velocity being present.

For 2, +cn we follow eqns. ( D . 14)-(fi. l (i) by putting [=Z,s a l ~ d averaging over t,he period of fluctuations n/Z,. Substituting i n eqn. ( i . 1 ) wc t1ic11 obt i~i t~ :

which is the sanlc as I,,, eqn. ((i.l(i). TII order to calwlatc t.ho other two conlponents of the current tlcnsity we

now express the limits of integration slio\v11 i n fig. 2 in cnrtcsian coordinates. I'iitting Z,:=O in cqn. (5.3) we obtain :

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P. A . Lindsay and D. Dirrrrikis

z exp -sec2 - ( m + 4 ~ l ) 2 1 dw, i:. { Y,, 1

and . P m I,,."*

. l J r ) / . l e = ~ dto* e sp ( - ug2) J -,dlur J -, a, o s p [ - (roz2 + wy2)] dmU n

is, cxp - wmc2 - + .$x,)z} diuS I:~. 1 2 10:

In the planc of the catIiodcl S , = O , we find tha t cqns. (7.4) and (7.5) reduce, respectively, to

( r ) / = -- sin - exp (-we2) drr:,. "112 jo" (7.7)

Along tlic edge of thc cnthode, S, =Z,= 0, eqns. (7.4) and (7.5) further reduce to

J ,(~, .) /J~= - a , (7.8)

Here t,he x cowponent of the current density is negative because the only electrons which can contribute to it are those returning to the cathode: Sirnilarlj., the component must bc zero for reasons of ~ymrnet ry .

Followilig eqns. ((i.14)-(6.16) we obtain the limiting values of the current density when Z , + m by calculating the mean value, over t,lrc period 2n/Z,,

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of the fluct,uating part of the intcgrimd. In the c;~se of eqn. (i .4) we obtain, putting = Z1s :

1 2"

=- J cos s' exp { - $A'? C O S ~ C ~ S ' ~ ~ S ' = O : (7.10) m ,, (.he s ime being t , r ~ ~ c for eqn. ( i .5 ) ; consequently, when Z,+co,

.I,(rm)/J1=.J,(rm)/.J, = 0. (i.1 I )

Region 11 Substituting t , l~e bonndnly conditions shown in fig. 3 we obtain t,he follo\v-

ing cxprer5io11 for the z cotnponent of the current, density :

where I, and 1,; are respectively defined 11y cqtls. ( f i . l i ) and (i .1). For X I = O. c t p (7.1" red11ccs to

.lz(rc)/.lq = I - $ = &, (7. I :3)

in agreement wit,h cqn. (i .2). For Zl+oo, on the other hand, we obtni~i :

w l i c ~ ~ I,+ is defined by cqn. (i.3). I t is I I O W more convenient to use ci~rtesi t r~~ coordi~iitt,es for calcr~lnting the

x and y conqronents of the current dellsky. By inspection of tigs. 2 and :3 we find tlil~t the x conil~onent must have the same algebraic form, eqn. ( i . J ) , in regions L and 11 becawe the parabol:~ i n fig. 3 is sytiilnetrical with lospect to the la,, axis a~i t l will not t~ffect the x directed flow. Thc snnie docs not apply, however: to the y cot~iponent a t~ t l the effect of the parabola niust now he taken into ;rccoilnt :

=l, , ,+Iu, (i. 15)

where I,, is tlefined by eqn. (7.5). I n the plane of the cxtlrode eqn. (i. 15) reduces t,o e q n (7. i ) sil~cc I U =(I when S, = O . Along the edge of the cathode,

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418 P . A . Lindsay and D. Dirmikis

X,=Z,=O, we have from eqn. (7.9), J,(r,)/J,=O. Similarly, for Z,+w we ohtain :

Jv(rm)/Je = I n , (7.10)

since, from eqn. (7.11), the first integral becomes equal to zero, the second being independent of Z, by definition.

Region 111 The z component of the current density is again closely related to the

corresponding expression for n(r)/n,, eqn. (G.20), and is given by

J ~ ( ~ ) / J ~ = ? r du~, . ?cis exp ( - wZ2) = - m

j:r,2 exp - *xl2 sec4 01

where a and a are respectively defined by eqns. (6.21) and (G.22). At the surface of the cathode, X,=O, eqn. (7.17) reduces to

JZ, exp (-wS2) dwS+ we exp ( - we2) dw, + exp ( - a2)

or, for Zl < 0,

I J:(r,)/J,- - exp (-19). (7.18a)

Ja2

Along the edge of the cathode, X I =Z,=O, we obtain from eqn. (7.18); .I,(r,)/J,=+, which agrees with eqn. (7.2).

Siniilarly, for Z, + - a, or a+ - oo, we obtain :

The x and y components of the current density can now be obtained by snbstituting the limits of integration shown in figs. 4 and 5 . Since fig. 4 is symmetrical with respect to the w, axis, there is no contribution to the x component of the current density from electrons in the velocity range Z,/2nS1u,< w, the only contribution coming from the remaining electrons in the velocity range - a, < w, < ZJ2n :

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Electron velocity fluctuatio1i8 in two-dimensional sys tem

J."' dwu J J ~ ~ / J ~ = ! I' dw, exp ( - ~ 2 ) wz exp [ - (wZZ+ ww2)1 dwz

n -, - rn "-sm

dw, exp ( - w,2) I:- exp (-%? rexp (-%**)

- exp ( - W , B ~ ) ] dwy

dw, exp (-202) [r exp [ - (wu - dwV - m - ,

+ X cot Z1/2wrl + exp {- *xlz cmec2 5) 2w, cos 2 (1 + erf sin Z1/2w, j .] dw.. (7.20)

Similarly, for the y component of the current density we obtain :

, dw, exp ( - wz2) dwZ exp ( -wZ2) wy ex11 (-w>) dwy

- , I,. dw, exp ( -wZ2) wy exp (-w,Z) dw,,

-rn I,. ,

+! JI dwz exp ( -wez) dw, exp ( - wZz) wy exp ( - w,z) dw, n 1%.

dw, exp ( - wZ2) exp ( - wZz- wVAZ) dw,

+ ( I -erf a) - exp ( - wZ2- wyB2) dw,

+A J dw, exp ( - we2) 1:; exp {- t g+ d-. n -,

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44 0 .1'. A . Lindsq and 1). 1)innikis

ZI exp ( - we2) sin - exp 21c;

*XI cut Z1/2ulz l+e r f I dw,

sin Z,/3wz,

+?J' dlc, exp ( - roc2) 1- exp { - * (5 + x .toz - m . 0.1

+ (1 -erf u) I,. (7.21)

At the surface of the cathode, .Y1=0! eqns. (i.20) and (7.") reduce to

Along the edge of the cathode, XI = Z, =O: we now obtain :

.lz(rc)/J3 = 4., (7.24)

,Jl,(rc)/Js =O, (7.25)

the differe~lce hetrveen e q m (7.24) I (7.8) being eqnixl to tho en~ission currenL Js.

For Zl+- m or u+- m, on the other hand, we obtirin, since crf ( - m) = - 1,

8. Discussion of results It is convenient to discuss the results nnder four separate headings.

Volume dcnsily The rcsults of nnmcrical calculations of the normalized volume density of

the electrons n(r)/uT are shown in fig. 6. Not surprisingly the volume density is different from zero everywhere, except in the region immediately below the cathode, since no combination of initial velocity components would allow the electrons to reach tha t area. I n region I.' the volume density remains fairly constant with Z,, apart from a sn~a l l fluct,ui~tion of increasing amplitnde, until we reach 2,- 1.0 ; from this point onwards the volume density rapidly

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Elcclron aeloeil!/ jactuations in ~~~~~~dimcnsio?ml syslems

Fig. G

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P. A . Lindsay and D. Dirmikis

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Electron velocily Jluctlcalions in two-dime~~sio?inl systems 423

collapses to zero. It is worth noting tha t , for the cathode temperature T = 1000'~, we obtain from eqns. (4.4)-(4.7), for example :

where B and z are expressed in m.k.s. units. Since in practice Bx0 .2 T, w e find tha t El= l corresponds t o z ~ 0 . 0 2 mm, which is the typical distance of a potential minimum in front of a therrnionic cathodc. Similarly in region I11 the volume density remains fairly constant with 2, until we reach the distance Zl z - 3.0 from the edge of the cathode when again the density rapidly begins to decrease. We find from eqns. (6.17) and (6.20) that the volume density remains continuous over t,he edge of the cathode, the regions I1 and I11 being separated by the curve I, shown in fig. 7 . This does not apply however to the plane of the cathode, X,=O, where a discontinuity occurs due to the abrupt edge of the cathode, the vulurne density suddcnly changing from 1.5 for 2, = 0- to 0.5 for 2, = 0+, the value of I, heiug equal to 1.0 at. X, = 0.

Fig. 7

Auxiliary functions I , , I g , till and 1 , - t M .

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424 P. A . Lindsay and 1). nirmikis

The most interesting results are grohably those obtained for region 11. Here again for Z ,> 1 . 0 , tho volume density remaills f:rirly const,ant, apart from n gent,le ripplc with a rapidly decl.e;ising aniplitndc, bu t for 2, < 0.8 we have a trough and B ridge as we move away from the plane of the cathode. Tlris effect can be explained as follows. For X, = 0 the parabola of fig. 3 de- generates into a double straight line coinciding xvith the ncgwtive p a r t of the u~,, axis, t,he area. between the a r m of the parabola becoming equal t,o zero ; a t the salne time the straight line descends onbil i t !lasses t,hroogh the origin, its slope 'being given by tlre coefficient cot (ZJZu:,). As X I increases the arms of the parabola open up and the vertex begins to travel along t h 20, axis, as sllown in fig. 3. This reduces the shaded area of the (u:,, u:,,) plme more and more and, in cotljunction with the integrand of the form exp (-~9), rapidly rcrluccs the volumc dcnsity of the elcoLro~ls, the strnight line rollirrg over the parabola faster and faster as rcz decreases from w to zero. When 2, is rela- tively small, however, thc straight line in fig. R remains almost. vert.icnl until l o ; i~ equally small. 'l'llis exposes the important area of the (w,, w,,) plane new tire origin t,o the electrons !until exp ( - w z 2 ) almost reaches unity. I n view of tlic lnrgc slope of the straight l im the additiolls to the shaded sre:r due to tlre increase of X I can t l m r outweiglr t,he losses; due to the opening "11 of the parahoh, and, for a while, the volume density timrls t,o increase with X I : this :rlrpcars as a ridge near the origin in fig. 0.

Winally we have to consider the limiting values of the v o l ~ m e rlensit,y whcn Z ,+* m. \VC find from eqn. (lj.26) that in region 111 the limiting distzil~utiol~ is given b,y ?I, (see fig. i ) , and is st,rongly reminiscent of the ' t ,riang~~l:u ' dist.ribut,ions discussed elsewhere (Sindsay l9li2; fig. 2 and l!)li5: figs. 2 and 4). It call also be seen from eqns. (6.16) all11 ( f , . l ! ) ) tha t in regions 1 and 11 tlrc limiting distributions are respectively given by I,,= ;ill and I,- I,, the corrcsponding functio~rs :~gaiii bring shown in fig. 7.

Z component of tlic current density In regions 1 and I1 the z component of' the current density bears close

simil:rrity t o the corresponding volume density, as can be seen from eqns. (7.1), ('7.12); (t i . l2) , (( i .17) and figs. ti! 8, the ditterancc being limited to the

Fig. 6

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expression 2,2/4c3 in place of %,/2n1'2(2 in the respective integrands. Con- sequently, the z component of the current density changes even more rapidly with Z, than doc8 the corresponding volume density, the ridge in region I1 alrcady appearing for values of Z, < 0.7, instead of Z, < 0.4. 'rlie situation is somewhat different however in region 111. Since a t large distances the electron ~ I L S must not bc inflnenced by the edge of the cntliode, Jl(r)J , tends to zero as A1+m (see eqn. (7.19) and fig. 8) ; near the edge of the cathode, on the other hand, for -0.7 <A, < 0.0, u ridge, similar to that in region 11, is forrncd near the plane of the cathode. There is now no discontinuity in the valnc of the function in the plane S l = O as we cross the edge of the cathode, the f~~nct ion being equal to 4 for all Z, 0 and rapidly decreasing to zero for Z, < 0 (see eqns. (7.2) and (7. 1 although there is a singularity there, since thc fnnction I, separating regions I1 nnd I11 is equal to 1.0 st X , = 0 (see eqn. (7.17) and fig. 7). Since for ot,lier values of X I the fnnction rapidly de- creases on both sides of the plane Z,=0, the curve I,, sepanrting the two regions, fmms a s1i;rrp ridge which degcneratcs into a sharp point a t the origin, S, = 0. \\'e find from eqn. (7. I8 a ) that . I , (r , ) /J, < 0.01 vhen a< - 1.5 or Z l < - 10.0.

I t can he seen from cqns. (7.3) and (7.14) that in thc limit, Zl+co, the fnnction tends respectively in regions 1 and 11 to fill and I,- f~ll, the two fnnctions bcing shown in fig. 7 ; these limiting fnnctions are identical to those obtained for volunic density of the electrons, cqns. ((i.lG) and (0.19).

X compon~e7rl bf the c~irre?i/, d e n s i l ~ I . fnnction rcpresenting the normalized x con~poncut of the cnrrent

density J,(r)/J, is shown in fig. 9. We can see from figs. 1 and 3 that in this

Fig. 9

I

I I I 1 2 0 $3.0 f 4 0 ?:

x, (a) and (b)

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Fig. 9 (conlinaed)

(4 Sortnalized z colnponent of the current dcnsity J,(r)/J, in regions I, 11 and I11

case the same expression, which is given by eqn. (7.4), applies t o hoth regions I and 11, the symmetrical part of the limits of integration making no eontri- bution to the valne of t,he integral. The x component of the current densit,y decrenses quite rapidly as cither X I + + co or Z l+ m ; this equally applies to all three regions. In the plane of the cathode, X,=07 the behavionr of the function is shown in fig. 10 ; for Z, > 0 the function oscillates wit.h a rapidly decreasing amplitude starting from -0.5 a t 0 +(see eqn. (7.6)) and for Z, < 0 i t rapidly appronches zero, starting from 0.5 a t 0- (see cqn. (7.18)). The discontinuity aL Zl=O is again due to thc abruptness of the cat.hode edge. lmlnediately t,o the right the x component of the current density is solely dne to electrons ret~rrning to the cat,hode (wz<O): and endowed with a positive z component of velocity, w,zO, the corresponding valuc of J,(r,)/J, being therefore equal to -1. Immediately to the left of the cathode edgc

Fig. 10

I

The fnnctions J,(r)lJ8 and J,(r)/J, for X,=O.

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42 8 P. A. Lindsay and U. Uirrnikis

in addition to the electrons returning to the cathode (.J,(r,)/J,= - +) we also have a contribution from the emitted electrons (J , ( r , ) /J ,= I), the two groups travelling in the opposite directions, the resultant value of the current density being equal to J , ( r , ) /J , = &. It should be noted however that for 2, = 0 and X I > 0, we obtain J , ( r ) /J ,=O. This is due to the fact that for Z,=O the straight line of fig. 3 moves to the left and becomes vertical, making fig. 2 indistinguishable from fig. 4 ; since fig. 4 is symmetrical with respect to the w, axis, any expression of the form

must be equal to zero, i.e. the net flow in the x direction must disappear. Consequently, in this case the curve separating regions I1 and I11 is very singular in character, being equal to zero everywhere except a t the origin, X , = 0 , where it is equal to + a. Y component of the current density

Figure 1 1 shows the normalized y component of the current density, J , ( r ) / J , as a function of X , and 2,. I n region I the function monotonically decreases with the magnitude of X I but in region I1 i t has a pronounced maximum in the interval 0.8 < X I < 1.0, the exact position of the minimum depending on the value of 2 , . We find from eqns. (7 .5) and (7 .15) that the transition between regions I1 and 111 is continuous, the function assuming the value of I, along the edge of the cathode, Z,=O. The behaviour of the function when Z,+ + oo is rather interesting. I n region I the function tends

Fig. 11

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Elcclron velocity jhcluationu in two-dimensional syslems 429

Fig. 11 (continued)

1

Normelieed y component of the current density J,(r) /Jg in regions I, I1 and 111

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430 P. A. Lindsay and U . Dirmikis

to zero as Z,+w, eqn. (7.!)) ; in region I1 it tends to I", eqn. (7.lti) and region 111 it tends to 21,, eqn. (7.27). 'I'he presence of a pronounced milxi- mum in regions 11 and I11 can be explained by the existence of the magnetic field which destroys by its action the symmetry of the system in the y direction and introduces strong vorticity in the motion of the electrons.

9. Numerical calculations 'Hie integrals, eqns. (2.5) and (Lo), turn out to be of such a complicated

form for this problem that they have to be evaluated numerically for various combinations of the space coordinates X I and Z,. I t was found that a satis- factory range for X, is 0.0 to 5.0, the integrals being negligibly small for most purposes beyond the upper limit of the range. Evaluations were per- formed a t intervals of 0.1 for both positive and negative S,. Also, since the system considered is inconipletely symmetrical nnd its most important yeomctricnl feature is the boundary along t h e y axis in fig. 1, one wonld cspect the integrals to change rapidly as one approached the edge by making Z, tend to zero. 'I'his was borne out by preliminary investigations and hence irregularly spaced values of Z, were chosen, with more values towards the Z,=0 end of the range. A typical sot of values considered was 0.05, 0.1, 0.2, 5.0, 10.0 and 50.0, agnin both positive and negative. It was interesting to note that for Z,=50.0 the numerical method used for the integral evalua- tions gave, to five deci~nal place accuracy, identical results with the exact expressions, eqns. (ti.lfi), (6.1!1), ((i.W), (7.0), (7.14), (7.16) and (7.27) for the values of the integrals as Z,+co. The value Z,=50.0 can therefore be used instead of infinity for numerical purposes. , As can be seen frorn the previous sections of this paper the integrals to be evalnated are quite complex, the integrand sometimes varying in an extremely irregular fashion. The computer time required to evaluate such integrals numerically to five decimal places can be considerable, especially for small values of X I when the integrands are large con~pared with those for high X I values. An accurate and above all very efficient integrating sub- programme has to be used. Of the various numerical integration methods investigated a Simpson-Kule-type subroutine originally written in Algol was selected (ilrgelo). It was nsed in this work as a Fortmu FUNCTl'IOR subprogramme.

The method utilizes a five-point Newton-Cotes formula, with some modi- fications, as the basic integrating formula and uses the well-known nested subdivision technique to evaluate the integral over the specified rangc. The particular advantage of the above technique in o w case is that the integrand is only evaluated once a t each point and also that the subprogramme ' con- centrates ' so to speak on the parts of the interwl that are most difficnlt to integrate, i.e. where the integrnnd changes rapidly, as it often does in this case, and does the 'flatter ' pa& fairly quickly. The subprogramme was also very nuccessfully used in evaluating double integrals such as eqn. ((i.20). Since Yortran precludes the recursive use of a function, the latter was used to calculate the inner integral as a function of the variable of the outer one, and then the same integrating function, but with a different name used to evaluate the outer integral. Apart from compiling u,hat was essentially the same subroutine twice, no other waste of computer time was involved.

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Electron uelocily $ucluations in two-dinlemiot~al splems 431

The principal numerical difficulty encountered in evaluating most of the integrals of this paper can be illustrated by referriug to eqns. (6.7)-(&I"), where the original triple integral has been transformed, eqn. ((;.I?), iuto a single one and the zeroes of the integrand are 11ow equally spaced aloug the 5 axis, occurring aL the zeroes of sin 5 . One could now evaluate eqn. (6.12) by c~~lculating the integral between the successive zeroes, i.e. 0 to n, T to 2n etc., and summing the terms fouud until they start I~ccon~ing negligible corn- pared with the accuracy to which the entire integral iu to be evaluilted. If one does this, however, the number of terms to be su~ntned becomes increas- ingly larger as the v~clue of Z, grows. For n nominal accuracy of four decimal places in the find result one has to snm up to 300 terms for Z, = 50.0. Apart from the great length of computer time required to do 300 iutegrations for just one combination of S, and Z,, the suln of truncation errors from etlch term builds up quite rapidly and the final result is considerably in error. The problem here is quite clearly one of finding the sum of a slowly convergent infinite series of positive terms. The method used to find the limit of the series is one due to Salzer (1954). This method is prticularly simple in that only the first few terms of the series have to be determined: the sum to infinity being evaluated by extmpolation using Lagrange polynomials. The parti- cular advantage of the method is tbnt, it uses a purcly numerical device and can be employed without any special analytic work upon the scries.

I n usiug Salzer's method, the nnmber 7a of terms of the series and the order m of the Langmnge polyno~uinl formula to be used in order to evaluate the integral of equ. (6.12) tto five decinlal places for a partic~tlar combinatio~l of X , and Z, was detern~ined empirically. I t was found that 71 and m depended only on 2,. Typically, it was found that the first ten terms and a four-point formula were adequate for 8, = 0.05 while the first 20 terms a d a seven-point forrnula \ \we required for Z,=riO.O. A very convenient criterion for check- ing the accuracy attninable with a particular (n, m) combinat.ion was :~vailnble by considering the values for XI = 0 which, as call be seen from eqn. (fi.13), should all have t,he exact value 03 . The only disadvant:rge of Snlzer's method is that for the very high-order formulae used in evaluating the integral for say Zl=50,0, it is advisable that each integral term of the series be cal- culated to 12 decimal places, which is very time-consuming and, for a small computer, son~ct~ilnes impossible. However, for the machine used for these calculations, a CDC 6600, this ditficulty did not arise.

Finally, since the error function erfx appears quite frequently insidc the integrand of the integrals in this paper a particularly fast and efficient way of calculating it was required. A polynornial npproximation due to Hastings (1055) wt~s chosen, which is pnrt.icnlarly suitable for nnn~erical computation.

erfx = I - (a, l+a21Z+o,t3+(~J4+a5t5) OX]) (-x2), (!I . I )

where the a's are given numerical constants and t n given function of x.

10. Complementary model Tt is of some interest to consider a t this point a complementary model

shown in fig. I ? , where the magnetic field is nt right a,ngles to a sem-infinite, plane cathode. Equations of motion are still given, in this case, by eqns.

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1'. A . Lindsuy u?ld 1). Dirmik is

Fig. 12

hlodel discussed in 9 10.

(4.1 I)-(4.13); excelk that now Z ,=0 througliout and the left-ll:~nd side of eqn. (4. I I ) is eqnnl to S -S,:.

,111 order to determine the range of velocities a t a given point (S,: %,) i t is necessary to clirnili;~te the phase angle O bctuww the modified versions of cqns. (4. I I ) and (4.13) :

Snbstitnting from cqns. (4.8)-(4.10) and replacing the initial velocity w, by its value w at the lmint (S,, Z,) we obtain :

or, rcarr;lnging terms,

IExccpt for the presence of X, and the absence of Z, e lp . (10.3) is inclistin- yoishable from eqn. ( 5 . 8 ) . Equation (10.3) represents :L family of parallel straight lines wl~icli sweep out thnt p&rt of the (ro,: w,,) pl;ule which is accessible to the electrons. 'Illre limits of integration in the (loz, a,,) plane cnn now be o11t.uincd by putting .Y,=O, tllc pattern being quite similar to that shown in fig. 2, except that t.he shacled portion extends above lather than helow tl?e limiting at,r;right line, the values of Y, being all negative, as c;ln bc seen from fig. I.'.

Su1)stitnting eqn. (10.3) in cqn. ( 2 . 7 ) we find that i l l region I < 0 ) the limiting strt~iglit linc cuts the w:, axis below the origin and t,l~e correspouding norinr~lizetl volume densit,y distribution is given by

where I, is defined in eqli. (6.1'7)

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h'ledron oelocily jhc /uat ions 17s two-diw~ensional systems 433

Similarly, in regiou I1 (XI> 0) the limiting straight line cuts the u,, axis above the origin and we obtain :

Since no electrons call reach rcgion 111 the corresponding volume deusity must bc equtd to zero.

Sin~ilnrly, substituting the new bolllldary conditious in eqn. ( 2 3 ) and bearing i u mind the integrations indicated in 8 7 we obtain the following exmessions for the threc comrionents of t,hc uormalized current density

Region 11

where the functions I,,, I , , and I,, are respectively given by eqns. ( i . l ) , (7.4) aud ( i .6) . Again, in view of the geometry of the system shown in fig. 12 thc current density J(r)/J, is identically equal to zero in region 111.

11. Conclusions Having considered the general method of approach and thc two systems

shown in figs. 1 nod 12 i t is of some interest to discuss, vely briefly, a com- posite geometry, i.e. an emitting edge cousisting of two semi-infinite plane cathodes a t right angles aud immersed iu a constant magnetic field which is perpendicular to the edge and parallel to one of tho planes. Such n con- figuration could well arise in practice, either by design or accident, and the r:orresponding volume and current density distributions would be too difficult to guess in view of the complexities of clectron trajectories in the presence of a ~nagnetic field.

Adding the results of $8 G, i and 10 we obtaiu the following expressions for the volume and cnrreut densities in regions I and I I, region 111 being unaffected by thc addition of a cathode a t right anglcs to the ~uaguetic field.

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434 P . A. Lindsay and D . Dirmikis

Region I1 n(r) /nT=Io-I l+I l=Io, (11.5)

J,(r)IJ,=Io-~l,+II,=~o, (11.6)

J,(r)/JB =II, -II, = 0, (11.7)

J,(r)/J,=I,, +IB -Ily =IB. (11.8)

The above equations indicate an interesting degree of complementarity which exists between the two systems. For example, the drop in volume density of the electrons as we approach the edge of the vertical cathode is exactly compensated in region I by the corresponding contribution from the horizontal cathode, the same being true for the z component of the current density. In the case of the x and y components of the current density a similar com- pensating effect reduces their value to zero throughout the region and again nullifies the effect of the cathode edge. This result has practical implications of some interest. If, instead of a plane system, we consider a cathode in the form of a cylinder which is closed at one end then, according to our model, in order to obtain full current density right up to the edge of the cathode i t is only necessary to extend the emissive coating for a short distance round the edge of the cathode and along its cylindrical surface. We assume here that the cathode is immersed in an axial magnetic field, which is frequently the case. This observation throws some interesting light on the old argument concerning the role played by edge emission in the performance of low noise guns.

ACKNOWLEDGMENTB

The early stages of this work were carried out under the U S . Air Force contract No. AF 33(657)-11003 when one of the authors (P.A.L.) was f i s t on the staff and then consulting for Raytheon Company, Waltham, Mass. Some time later the numerical results were obtained by the other author (D.D.) as part of his MSc(Eng.) thesis a t King's College, University of London. Both authors wish to express their sincere thanks to Mr. A. Reddish of the Hirst Research Centre, The General Electric Company, Wembley, Middlesex, for his kind interest and advice. The authors also wish to thank Dr. Radley of the Department of Electrical Engineering, University of Sheffield, for kindly suggesting a method for calculating the limiting value of I,, eqn. (6.16) and Mr. Graham England of the Atlas Computer Laboratory, Didcot, Berks., for his untiring advice and assistance.

REFERENCES ARGIELO, S. M., I' Four Algol Procedures for Integration ", Rijkwaterstraat Nota

MFA 6401 (Computer Programming Circular). CURRIE, M. R., 1958, Proc. Inst. Radio E?zgrs, 46, 911. CURRIE, M. R., and Forster, D. C., 1958, Proc. Inst. Radio Engrs, 46, 88 ; 1959,

J . appl. Phys., 30, 94. GABOR, D., 1945, Proc. R. Soc. A, 183, 436. H ~ s ~ w a s , C. Jr., 1955, Approximations for Digital Computers (Princeton University

Press).

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Blectrm velocity fluctuations i n two-dimensional systems 435

LINDSAY, P. A,, 1960 a, Adv. Electron. electron Phys., 13, 181 ; 1960 b, J. Electron. Control. 8, 177 ; 1960 0, ibid., 9, 241 ; 1962, J. appl. Phys., 33, 3298 ; 1963, Proc. I .E.E.E. , 51, 1710; 1964, J. Electron. Control, 17, 67; 1965, Proc. R. Soc. A, 287, 183 ; 1970, Zbid., 315, 479.

LINDSAY, P. A,, and GOODELL, R. S., 1965, J . appl. Phys., 36,411. MUELLER, W . M., 1961, Proc. Inst. Radio Engrs, 49, 642. SUZER, H. E., 1954, J . Math. Phys., 83, 356.

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