The Bell System Technical Journal : devoted to the Scientific and Engineering aspects of Electrical Communication, Vol. 24, No. 3-4

fest A â Q j i ï M f j K .

xxiv JULY-OCTOBER,

P. ZS~¡IjS

THE BELL SYSTEM

TECHNICAL JOURNAL

DEVOTED TO THE SCIENTIFIC AND ENGINEERING ASPECTS OF ELECTRICAL COMMUNICATION

& % /

Physical Limitations in Electron Ballistics . / . R. Pierce 305

Electron Ballistics in High-Frequency Fields A. L. Samuel 322

Dynamics of Package Cushioning Raymond D. Mindlin 353

Abstracts of Technical Articles by Bell System Authors 462

Contributors to this Issue . 467

f t , j

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EDITORS

R. W. King J. O. Perrine

EDITORIAL BOARD

W. H. Harrison O. E. Buckley

O. B. Blackwell M. J. Kelly

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Copyright, 1945

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P R I N T E D IN U . S . A .

C O R R E C T IO N S F O R IS S U E O F A P R IL , 1945 P ag e 207: A bscissa for Fig. 12.29, P = M

1 5T Ct/Co should be P — M

1 + Ct/Co

P ag e 240: E q u a tio n (15.60), 1F0 —

should be W 0 = | 0

a y

1 ’

I

7 , 7

8 ~

a

+ r

j

i_

i

+ Î

_8 8 *

O

T h e Bell System Technical Journal

Vol. X X I V July-O ctober, 1945 N os. 3 - 4

Physical Limitations in Electron Ballistics*

By J. R. PIERCE In t r o d u c t i o n

T

listics” . I t is p le a sa n t to h av e a chance to ta lk a b o u t such physical lim ita tio n s, because th e re is so little we can do a b o u t th e m . A nd, alth o u g h these lim ita tio n s are a p t to be discouraging, a know ledge of th e m is v e ry v alu ab le , for it keeps us from spending tim e try in g , like th e in ­ v e n to rs of p e rp e tu a l m o tio n m achines, to do th e im possible.

As electro n ballistics is p a rtic u la rly su b je ct to physical lim ita tio n s, th e re are so m a n y th a t it is im possible to discuss all of th e m th o ro u g h ly a t th is tim e. Also, m a n y of th e lim ita tio n s are of a ra th e r com plicated n a tu re , a n d to deduce th e m from basic principles in a q u a n tita tiv e w ay requires m u ch th o u g h t a n d p atience. I th in k th e b est I can do is to t r y to m e n tio n m ost of th e chief lim ita tio n s, as a w arning to th e u n in itia te d th a t rocks lie ah e ad in c e rta in directions, b u t to co n c en tra te a tte n tio n on only a few of th e m . I h av e chosen th is evening to devote p a rtic u la r a tte n tio n to lim ­ ita tio n s th a t b e a r on th e p ro d u ctio n a n d use of elec tro n beam s in w hich considerable c u rre n t is req u ired , such as those used in cath o d e ra y tu b e s an d high-frequency oscillators, a n d to m e n tio n only briefly as a so rt of in tro d u c tio n problem s p erta in in g m ore closely to low -current devices such as electron m icroscopes.

Th e Wa v e Na t u r e o f t h e El e c t r o n

One of th e m o st im p o rta n t lim ita tio n s in electro n m icroscopy is th e du al n a tu re , w ave a n d corpuscular, of th e electron. W ith o u t m ak in g an y a tte m p t to ju stify o r explain th e co m b in atio n of w ave a n d p article con­

cepts w hich is ch a racteristic of m o d ern physics, we m a y describe its con­

sequence a t once; v e ry sm all o bjects d o n ’t cast d istin ct shadow s. T h is ca n n o t be ex p lain ed m erely in te rm s of th e physical size of th e electro n an d th e object. W hen a n electro n beam is reflected from a surface of regularly

* A lecture given under the auspices of the Basic Science Group of the New Y ork Secr tion of the A .I.E .E ., as a p a rt of a n E lectron B allistics Symposium, Columbia U niversity, M arch 21, 1945.

305

spaced obstacles (the ato m s in a c ry sta l la ttic e , for in sta n c e ) diffraction p a tte r n s are o b ta in ed , sim ilar to those w hich m a y be o b ta in e d w ith w aves of X -ra y s or light. I t a p p e ars t h a t elec tro n s get a ro u n d sufficiently sm all o bjects ju s t as sound w aves g et a ro u n d telephone poles, au to m o b iles, an d e v e n houses, a n d if th e o b jects are sufficiently sm all th e ir effect on the electro n flow will e ith e r be a b se n t or w ill consist of a few rip p les w hich are m eaningless in disclosing th e shape or size of th e o b je ct.

T he elec tro n w ave-length, w hich v aries in v ersely as th e m o m e n tu m of th e electro n , m a y be sim ply expressed in te rm s of th e en erg y V in ele c tro n volts.

A sim ple n o n -re lativ istic expression w hich is only 5 % in e rro r a t 100,000 v o lts (a high v oltage for elec tro n m icroscopes), is*

X = V l W v

x

T h u s for 30,000-volt electro n s th e w av e -len g th is 7 X 10- 1 0 cm or ab o u t 1.4 X 10- 7 tim es th e d ia m e te r of a h a ir a n d 1.2 X 10- 5 tim es th e len g th of a w ave of yellow light.

I n te rm s of th is w av e -len g th X a n d th e half angle of th e cone of ray s ac ce p te d b y th e objectiv e, a, we can express th e d istan c e d betw een p o in t o b jects w hich can ju s t be d istin g u ish e d in a n elec tro n m icroscope. T his distan ce is

d = ,61X/sin a (2)

F o r sm all valu es of a

2 a = 1 /f (3)

where / is the well know n p h o to g rap h ic / n u m b er, th e ra tio of th e focal le n g th to the lens d iam eter. W e see th a t, ju s t as w ith cam eras, th e sm aller the / n u m b e r the b e tte r. In electron m icroscopes a sm all / enables us to d istin g u ish sm aller objects.

Ab e r r a t i o n s

J u s t as in cam eras, th e lim ita tio n to th e / n u m b e r is im posed b y lens ab e rra tio n s. B u t in elec tro n lenses th e a b e rra tio n s are m u c h m ore severe.

W hy is th is so? B ecause w ith electro n lenses we h av e less freedom of design th a n w ith o p tical lenses.

C onsider a n electric lens. T he q u a n tity analogous to th e index of refra ctio n for lig h t is th e sq u are ro o t of th e p o te n tia l w ith resp e c t to th e ca th o d e . N ow suppose th a t w ith a lig h t lens we know th e index of re ­ fra ctio n a t ev e ry p o in t along th e axis. Suppose, for in stan c e, t h a t th e in ­ dex of refra ctio n is 1 everyw here along th e axis ex cep t for a space L long

* T he relativistic expression is

X = ( V l 5 0 / V / V i + -98 X 10-» F ) X lO“ 8 cm

P H Y S I C A L L I M I T A T I O N S I N E L E C T R O N B A L L I S T I C S 307

w here it is 2, as in Fig. 1. O ur lens m a y be converging or diverging; stro n g or w eak. I n th e analogous electric case, how ever, th e p o te n tia l th ro u g h o u t the lens space m u st sa tisfy L a p la c e ’s eq u a tio n , a n d th is m eans t h a t if it is specified along th e axis it is know n everyw here. W e can easily see th is by w riting dow n L a p lace’s e q u a tio n for a n axially sy m m etrical field.

i d / d V \ , ô V _

r dr \ dr ) dz2 (4)

LIGHT-CONDITIONS OFF AXIS NOT FIXED BY CONDITIONS ON AXIS

ELECTRIC FIELD-FIELD OFF AXIS SPECIFIED BY POTENTIAL ON AXIS

r-Tt

V =1T f (5 + iL COS e)dL9

Fig. 1— C ontrast betw een optical and electric focussing conditions.

T he field n e a r th e axis m a y be ex p an d ed in pow ers of /

d V ,

—— = ar T ■ • 1

dr (5)

S u b stitu tin g th is in to (4),

l d , 2. I - d 2 V - V (ar ) = 2a ==; ■

r dr dz2

d V - 1 52 V

dr 2 dz2

(

)

As a m a tte r of fac t, th e p o te n tia l V (z,r) rem ote from th e axis can be expressed in te rm s of th e p o te n tia l V 0(z) on th e axis as

V = - f, Vo(z + ir cos 6) dd

7T Jo (7)

If we could in troduce charges in to our lens, L a p lace’s e q u a tio n w ould no longer h o ld a n d we w ould have m ore freedom of design. T he m ethods proposed for th e in tro d u c tio n of charges com prise th e use of free charges (space charge) w hich are largely u ncontrollable, a n d th e use of cu rv e d grids, w hich do m ore dam age th a n good. I n o th e r w ords, th e cures are worse th a n th e disease.

Sim ilar lim ita tio n s ap p ly to m a g n etic lenses, a n d in th e e n d we find th a t because of th e sim plest form of a b e rra tio n , sp h e ric al a b e rra tio n , b e st def­

in itio n is ach iev ed in elec tro n m icroscopes w i t h / n u m b e rs of 1 0 0 o r greater, w hile th e / n u m b e r of a lig h t m icroscope o b jectiv e c o rrec ted for spherical a b e rra tio n a n d o th e r defects as w ell m a y be a ro u n d u n ity . T h u s th e elec­

tr o n m icroscope is severely h a n d ic a p p e d , a n d th is h a n d ic a p is overcom e o n ly because elec tro n w aves are m u c h less th a n 1 / 1 0 0 th e le n g th of lig h t w aves.

W = 2 L 0 t ' 0,W= e2w2

O r V y ^ L

Fig. 2—A pproxim ate relation betw een beam size a n d angular spread.

Th e r m a l Ve l o c i t i e s o e El e c t r o n s

I n m a n y e le c tro n -o p tical sy stem s, a n d p a rtic u la rly in su c h devices as ca th o d e ra y tu b e s, it is desirable to focus a n e le c tro n b ea m in to a sm all are a, so as to produce a v e ry sm all sp o t on a fluorescent screen, or to pass a considerable c u rre n t th ro u g h a sm all a p e rtu re . W e m ig h t th in k a t first t h a t if our focusing sy stem were good en ough, t h a t is, if it h a d v e ry sm all ab e rra tio n s, we could focus a c u rre n t from a ca th o d e of given a re a in to as sm all a space as we desired. T h is, u n fo rtu n a te ly , is n o t so. T h e obstacle is th e th e rm a l velocities of th e elec tro n s e m itte d b y th e c a th o d e .

A sim ple exam ple will show th e so rt of th in g we sh o u ld ex p e c t to ta k e place. F igure 2 show s a p lane ca th o d e a n d n e a r to it a po sitiv e g rid so fine as to cause no ap preciable deflections of th e elec tro n s w hich pass th ro u g h it. F a r th e r on we h av e a n ab e rra tio n le ss elec tro n lens designed to focus

P H Y S I C A L L I M I T A T I O N S I N E L E C T R O N B A L L I S T I C S 309

th e elec tro n stre a m a t a sp o t a distance L beyond it. T he electrons will leave th e ca th o d e w ith some slight sidewise velocity com ponents; so, elec­

tro n p a th s will p ass a t several angles th ro u g h a given p o in t 011 th e lens. T he lens will b end these p a th s app ro x im ately equally, a n d hence we can see th a t a t th e p o in t w here th e beam is narrow est it will still h ave some appreciable diam eter W2.

N ow consider th e beam a t th e lens. Suppose th a t th ro u g h a given p o in t all th e p a th s lie w ith in a cone of half angle d. T h en th e w idth W 2 is app ro x im ately

W 2 = 2 Ldy (8)

We can also see th a t the p a th s a t W 2 will lie w ith in an angle approxim ately

02 = W i/2 L (9)

H ence we see th a t approxim ately

O 1W1 = e2w2 (10)

In o th er w ords, we can h av e a sm all sp o t th ro u g h w hich electrons pass over a wide an g u lar range, or we can h av e a b ro ad beam in w hich all p a th s are n early p arallel, b u t we c a n ’t h ave a narrow spot a n d n early parallel rays.

W e see th a t th e a c tu a l w id th of spot will d epend on th e th e rm a l veloc­

ities, w hich are p ro p o rtio n al to th e square root of th e cathode te m p e ra tu re , an d on th e fo rw ard velocity, w hich is p ro p o rtio n al to th e square ro o t of th e accelerating voltage. B y using m ore involved arg u m e n ts we discover th a t for a n y p o in t in a n electro n stream , w here th e beam is wide, narrow , or in term ed iate, th e c u rre n t in a n a r b itra ry direction chosen as th e x direction can be expressed4'*

d j = ^ ^ l j oVxe(-llkT',<-er~mvil2) dvx dvy dv2 (11) -Km

v J vl # Vy + J

whenz>> y /2 .e V /m ; (12)

or d j = 0 (13)

w hen v < V 2 e V ¡m (14)

Here j 0 is th e cathode c u rre n t den sity , V is voltage w ith respect to th e cathode, T is th e absolute te m p e ra tu re of th e cathode in degrees K elvin, an d vx, vy, a n d vz are th e th ree velocity co m ponents; d j is th e ^ e le m e n t

* T his expression neglects the effects o'f electron collisions, which m ay actually make the cu rren t density smaller.

of c u rre n t d e n s ity c a rrie d b y electro n s w hich h av e v elo city com ponents a b o u t vx, vy, vz, lying in th e little range of v elo city dvx dvv dvz.

T h e reason for re stric tio n (12) is t h a t if a n elec tro n s ta rts w ith zero th e rm a l v elo city from th e c a th o d e , it will a t ta i n th e v elo city given b y the rig h t side of (12) b y falling th ro u g h th e p o te n tia l d ro p V . As electrons c a n n o t h av e velocities sm aller th a n th is, we h av e (13) a n d (14).

B y in te g ra tin g (11) w ith a p p ro p ria te lim its we o b ta in a m ore specialized b u t v e ry useful expression

( - 1 . A , 1 1 6 0 0 V \ . 2 . g J

j < ]m = Jo ( 1 + — T J Sin29 (15) F o r u su a l valu es of v o lta g e, u n ity in th e p a re n th e se s is negligible, a n d we can sa y t h a t if all th e electro n p a th s a p p ro a ch in g a given p o in t in a n electro n beam lie w ith in a cone of h alf angle 6, th e c u rre n t d e n s ity j a t th a t p o int c a n n o t be g re a te r th a n a lim itin g value j m w hich is p ro p o rtio n a l to the

Fig. 3— P aram eters im p o rtan t in determ ining spot size in a cathode ray tube.

ELECTRON DEFLECTING

CATHODE

c a th o d e c u rre n t d en sity , to th e voltag e, to sin2 0, a n d in v ersely p ro p o rtio n a l to th e ca th o d e te m p e ra tu re .

L e t us see w h at th is m ean s in some p ra c tic a l cases. F ig u re 3 shows a

ca th o d e ra y tu b e . T h e elec tro n strea m h as a w id th W a t th e final electron lens, a n d is focused on a screen a distan ce L b e y o n d th e lens. T h e half

angle of th e cone of ray s reach in g th e screen c a n n o t be g re a te r th a n

sin 0 = e = W / 2 L (16)

Suppose th e sp o t m u st h ave a d ia m e te r n o t g re a te r th a n d. L e t th e spot c u rre n t be i. T h e n from (15),

* < T * ( ' + iV ' r >V) , w t'1 L '- (17)

P H Y S I C A L L I M I T A T I O N S I N E L E C T R O N B A L L I S T I C S 311

T h u s if for a given spot size we w ant to increase th e spot cu rre n t, an d if we are lim ited to a given cath o d e c u rre n t d en sity because of cathode life, we m u st m ake V larger, IF larger or L sm aller.

M aking W larger increases b o th lens a n d deflection ab e rra tio n s. M aking L sm aller m eans th a t for a given linear deflection we m u st increase th e ang u lar deflection, a n d th is too te n d s to defocus th e spot. Because of these lim itations, it is necessary to av a il ourselves of th e rem aining variab le an d raise th e o p era tin g voltage V.

A no th er illu stratio n , p erh a p s a little m ore subtle, of th e effect of th e rm a l velocities, lies in th e analysis of th e pro p erties of a ty p e of vacuum tu b e am plifier know n as th e “ deflection tu b e ” . I n such a device, illu stra te d in Fig. 4, an electro n strea m from a cathode is accelerated an d focused b y a lens a n d deflected b y a p a ir of deflecting electrodes so as to h it or m iss an o u t­

p u t electrode. Such a device m a y be used as an am plifier.

N ow it is obvious t h a t as th e o u tp u t electrode on w hich th e beam is focused is m oved fa rth e r aw ay from th e deflecting p lates, a given deflecting voltage will produce a g rea ter linear deflection of th e beam a t the o u tp u t.

As th is a t first sight seems desirable; it has been seriously suggested no t only t h a t th is be done, b u t t h a t an elab o rate electro n o p tical system be interposed betw een the deflecting p la tes a n d th e o u tp u t electrode to am plify the deflection.

The m e rit of a deflection tu b e is roughly m easured b y th e deflecting voltage req u ired to m ove th e beam from en tirely m issing th e o u tp u t elec­

trode to en tirely h ittin g th e o u tp u t electrode, an d , of course, m oving the o u tp u t electrode fa rth e r aw ay or p u ttin g lenses betw een th e deflecting plates a n d th e o u tp u t electrode doesn’t reduce th is voltage a t all. As we im prove the deflection se n sitiv ity b y these m eans, we sim ply increase the spot size a t th e sam e tim e. F ocusing our a tte n tio n on th e beam betw een the deflecting p lates, we ap p reciate a t once th a t th e electron p a th s th ro u g h each p o in t will be sp read over some cone of half angle 6, a n d th a t to change from a clean m iss to a clean h it we m u st deflect the electrons th ro u g h an angle of a t least 26, regardless of w h at we do to th e beam afterw ards.

R e tu rn in g for a m om ent to e q u a tio n (15), we see th a t it says th e cu rren t d en sity can be less th a n a c e rtain lim iting value depending on 6. Y et

CAT

/ ' ' ■ l l D OUTPUT

[\

E L E C T R O N ^ E L E C T R O D E

LENS DEFLECTING

PLATES

Fig. 4— Amplifying tube making use of electron deflection.

expression (15) w as o b ta in e d b y in te g ra tin g a su p p o sed ly e x a c t expression.

W h a t does th is in e q u a lity m ean?

T h e answ er is th a t for th e c u rre n t to h av e th e lim itin g v alu e , electrons of all allowable velocities m u s t a p p ro a c h each part of th e sp o t from all angles ly ing w ith in th e cone of h alf angle 6. W h en th e av e rag e c u rre n t d e n s ity in th e sp o t is less th a n th e lim itin g c u rre n t d en sity , th e p ossibilities are

(a) E le ctro n s are ap p ro a ch in g ea ch p o in t in th e b eam from all angles, b u t along some angles only electro n s w hich le ft th e c a th o d e w ith g re a te r th a n zero v elo city c a n rea ch th e sp o t.

(b) E le ctro n s leaving th e c a th o d e w ith all v elocities c a n reach th e spot, b u t a t some p o rtio n s of th e sp o t elec tro n s d o n ’t com e in a t all angles w ithin th e cone angle 9.

Fig. 5— R elation betw een nearness of approach to lim iting c u rren t density an d fraction of cu rre n t utilized.

T h u s, we ca n h av e less th a n th e lim itin g c u rre n t e ith e r because electrons do n o t rea ch th e sp o t w ith all allow able velocities or fro m all allow able angles. Of course b o th fa c to rs m a y o p era te.

W e ca n easily see how lens a b e rra tio n s, w hich we kn o w are p re se n t in all elec tro n -o p tical sy stem s, c a n p re v e n t o u r a tta in in g th e lim itin g cu rren t d en sity . T h ere is a m ore fu n d a m e n ta l lim ita tio n , how ever. I t can be show n th a t ev en w ith p erfect focusing, we m u s t so rt o u t a n d th ro w aw ay p a r t of th e c u rre n t in ord er to ap p ro a c h th e lim itin g c u rre n t d e n sity , and we c a n ev en derive a th e o re tic a l cu rv e for th e case of p e rfe c t focusing re­

la tin g th e fra ctio n of th e lim itin g c u rre n t d e n s ity w hich is a tta in e d to the fra c tio n of th e c a th o d e c u rre n t w hich ca n rea ch th e sp o t. F igure 5 shows such a curve w hich applies for v o lta g es h ig h e r th a n , sa y , 1 0 vo lts.

U sually, th e failure to ap p ro a c h th e lim itin g c u rre n t d e n s ity is chiefly ca u se d b y ab e rra tio n s, a n d in o rd in a ry ca th o d e r a y tu b e s th e c u rre n t d e n s ity in th e sp o t m a y be only a sm all fra c tio n of th e lim itin g v alu e. A v e ry close a p p ro a ch to th e lim itin g c u rre n t d e n s ity h a s b een ach ie v ed in a

P H Y S I C A L L I M I T A T I O N S I N E L E C T R O N B A L L I S T I C S 313

special cath o d e ra y tu b e designed b y D r. C. J. D avisson of th e Bell T ele­

phone L ab o rato ries.

W hen we becom e th o ro u g h ly convinced th a t these eq u a tio n s expressing the effects of th e rm a l velocities v e ry m uch cram p our style in designing electron-optical devices, as good engineers we w onder if th e re isn ’t , a fte r all, some w ay of g e ttin g a ro u n d them . I d o n ’t th in k th e re is. T he suggestion illu strate d in Fig. 6 is a ty p ic a l exam ple of such a n a tte m p t. We know th a t in a stro n g m agnetic field electrons te n d to follow th e lines of force.

W hy n o t use a v e ry stro n g m agnetic field w ith lines of force approaching th e axis a t a gentle angle to d rag th e electron strea m to w a rd th e axis?

An electro n off axis tra v e lin g p ara llel to th e axis ce rtain ly will be dragged inw ard b y such a field. T he ca tc h is t h a t th e field pulls th e electro n in because it m akes th e electro n sp iral a ro u n d th e axis. As th e beam con­

verges a n d th e field becom es stronger, th e p itc h of each spiral decreases an d the an g u lar speed of each electro n increases. F inally, if th e field is strong enough, all th e k inetic energy of th e electro n is co n v erted from forw ard

ELECTRON PATH

m a g n e t ic' LINES OF FORCE

Fig. 6—Reflection of an electron by a m agnetic field w ith strongly converging lines of force.

m otion to rev o lu tio n a b o u t th e axis; th e electro n ceases to m ove in to th e field a n d bounces b ack o u t. I t m a y be some sm all consolation to know th a t v ery h ig h -cu rren t densities can be achieved b y th is m eans, b u t only because in th e ir flat spiralling th e electrons ap p ro a ch a sp o t a t m u ch w ider angles w ith th e axis th a n th e sm all inclination of th e lines of force.

Sp a c e Ch a r g e Li m i t a t i o n s

In electron beam devices using reaso n ab ly large c u rren ts, th e space charge of th e electrons is a v ery serious source of tro u b le b o th in com pli­

catin g design of th e devices a n d in lim iting th e ir perform ance.

L et us begin our consideration rig h t a t th e electro n gun, th e source of electron flow in m a n y devices such as cath o d e ra y tu b e s a n d ce rta in high- frequency tu b es. E le ctro n guns are som etim es designed on th e basis of rad ia l space charge lim ited electron flow betw een a cath o d e in th e form of a spherical cap of rad iu s ra a n d a concentric spherical anode a distance d from th e cathode. I t can be show n t h a t b y use of suitable electrodes ex tern al to the beam , ra d ia l m otion can be m a in ta in ed betw een cathode an d anode along

s tra ig h t lines norm al to the cathode surface. A hole in th e anode electrode will allow th e beam to em erge from the gun. B ecause of th e change in

CATHODE AN ODE S P A C IN G , d / r Q

Fig. 8—R elation betw een perveance, angle of cone of flow, an d cathode-anode spacing.

held n ea r th e hole, the hole ac ts as a diverging electro n le n s. 11 F igure 7 illu strate s such a g u n. 15 T h e curves show n in Fig. 8 re la te to th is so rt of

P H Y S I C A L L I M I T A T I O N S I N E L E C T R O N B A L L I S T I C S 315

electron gun. T h e y are p lo ts of a fac to r called the perveance, w hich is defined as

p = i / y w (18)

(th a t is, c u rre n t divided b y voltage to the 3 /2 pow er) as a function of 6, the half angle of the cone of flow, an d d /r 0, the ra tio of cathode-anode spacing to cathode radius. In g ettin g an idea of the m eaning of the curves, we m a y note th a t a perveance of 1 0 ~ 6 m eans a c u rren t of 1 m illiam pere a t 1 0 0 vo lts.

I t is obvious from the curves th a t to g et v ery high values of perveance, th a t is, high c u rren t a t a given voltage, 6 m u st be large an d the cathode-anode spacing m u st be sm all. M aking 6 large m eans th a t electrons ap p ro ach the axis a t steep angles; ab e rra tio n s are b a d an d the beam tends to diverge rapidly beyond crossover. M oving the anode n ea r to the cathode m eans th a t the hole w hich m u s t be c u t in the anode to allow the beam to pass through m u st be large, a n d c u ttin g such a large hole in the anode defeats our aim of g e ttin g higher p e rv e a n c e ; we c a n ’t p u ll electrons aw ay from the cathode w ith an electrode w hich isn ’t there. F u rth e r, for ratio s of spacing to cathode rad iu s less th a n a b o u t .29, the lens actio n of the hole in the anode causes th e em erging beam to diverge, w hich w ould m ake the g u n u nsuitable for m a n y applications.

W hen we build guns for sm all cu rren ts a t high voltages, such as cathode ray tu b e guns, space charge causes little tro u b le ; w hen we t r y to o b ta in large c u rren ts a t lower voltages, we find ourselves seriously em barrassed.

Suppose we now tu r n our a tte n tio n to th e effect of space charge in beam s w hen the beam tra v e ls a distance m a n y tim es its own w id th . C onsider, for instance, th e case of a circular disk form ing a space charge lim ited cathode. Suppose we place opposite th is a fine grid, a n d shoot an electron stream ou t in to a co nducting box, as illu stra te d in Fig. 9a. W e im m ediately realize t h a t th ere will be a p o te n tia l g rad ie n t aw ay from th e charge form ing the beam . I n th is case, th e g rad ie n t will be to w a rd th e n ea rest co n d u cto r;

th a t is o utw ards, a n d th e electro n beam will diverge.

How can we overcom e such divergence? One w ay w ould be to arrange th e b o u n d a ry conditions in such a fashion th a t all th e field w ould be d i­

rected along th e beam in stea d of o u tw ard s; th is m ig h t be done b y su r­

rounding th e beam b y a series of con d u ctin g rings a n d apply in g to th e m successively higher voltages as in 9b, th e voltages w hich w ould occur in electro n flow betw een infinite p ara llel planes w ith th e sam e cu rre n t density.

A no th er w ay in w hich th e sam e effect m a y be achieved is th ro u g h use of specially sh ap ed electrodes outside of th e beam , as show n in Fig. 9c.11 In m a in ta in in g p a ra llel flow b y these m eans, th e electric field due to th e elec­

tro n s acts along th e beam , a n d increases co n tin u ally in m ag n itu d e w ith

d istan c e from th e c a th o d e . W e ca n in fa c t calcu late th e p o te n tia l a t a n y d istan c e along th e b ea m b y th e w ell know n C h ild ’s law e q u a tio n

I = 2.33 X 10~6A V m / x 2 V = 5,690xil3I 2ls/ A 213 a

(19)

CATHODE

■M'l'hT

V f e J C 4/ 3

I I I J I I I I 1 1 .L

„ A .A A -fv A b v V \ d . '\ \ • l vCv'X 'vA.Al . V 'i - W v l y X \ Ai v V ' / \ Y —I—

— ---111 ■ i1 1 f lH ---

Fig. 9—Avoiding beam divergence by m eans of a longitudinal electric field.

H ere V is th e anode voltag e, x th e ca th o d e-an o d e spacing, I th e c u rre n t in am peres a n d A th e c a th o d e area.

S uppose we ta k e as an exam ple

A = 1 cm2

I — . 0 1 am p.

x = 1 0 cm

P H Y S I C A L L I M I T A T I O N S I N E L E C T R O N B A L L I S T I C S 317

T hen

V = 5,700 volts

T h u s to m a in ta in p ara llel m o tio n of th e m o d est c u rre n t of 10 m illiam peres spread over a n are a of one square c e n tim ete r requires 5,700 volts. M o re­

over, th e req u irem en t of d istrib u tin g th is voltage sm oothly along th e beam would m ake it v e ry difficult to p u t th e beam to a n y use.

One m eans for m itig a tin g th e situ a tio n is to use a n electro n lens a n d direct th e beam inw ard. Of course, th e beam will ev e n tu a lly becom e p a r ­ allel an d th e n diverge again, b u t b y this m eans a fairly large c u rre n t can be m ade to tra v e l a considerable distance. Some calculations m ade b y T h o m p ­ son an d H ead rick12 cover this ty p e of m otion, w ith a n especial em phasis on the problem in cathode ra y tubes, in w hich the cu rren ts are m oderate.

I n order to coniine large c u rre n ts in to beam s, a n axial m agnetic field is som etim es used, as show n in Fig. 10 H ere a cathode-grid com bination shoots a beam of electrons in to a long co nducting tu b e . A long coil aro u n d th e tu b e produces a n axial m agnetic field in te n d ed to confine th e electron

p a th s in a ro ughly p ara llel beam . T he ra d ia l electric field due to space charge will cause th e beam to exp an d som ew hat a n d to ro ta te a b o u t th e axis. As th e m agnetic field is m ade stro n g er a n d stronger, th e electrons will follow p a th s m ore a n d m ore n ea rly stra ig h t a n d parallel to th e axis.

F o r a given c u rre n t a n d voltage, th e re is one so rt of physical lim ita tio n in th e s tre n g th of m agnetic field we need to get a sa tisfac to ry beam . I t is an o th er effect th a t I w ish to discuss.

Suppose we h av e a v ery stro n g m agnetic field, in w hich th e electrons tra v e l alm o st in stra ig h t lines. W e know , of course, t h a t th e ra d ia l electric field is still p rese n t, a n d th is m eans t h a t th e p o te n tia l to w ard th e cen ter of th e beam is depressed; th is in tu r n m eans t h a t th e ce n te r electrons are slowed dow n. T h is slowing dow n of course increases th e d en sity of electrons in th e ce n te r of th e beam . T he resu lt is t h a t if for some critica l voltage or speed of in jectio n we increase c u rre n t beyond a c e rtain value, th e process ru n s aw ay, th e p o te n tia l a t th e ce n te r of th e beam drops to zero, a n d an o th er ty p e of elec tro n flow w ith a “ v irtu a l c a th o d e ” of zero electro n velocity a t th e ce n te r of th e beam is established. T h u s, alth o u g h th e m agnetic field

CONDUCTING

CATHODER

m m x x x yyyxxxvx zzxzzz +

Fig. 10—Avoiding beam divergence by m eans of a longitudinal magnetic field.

h a s overcom e th e d iverging effect of th e space charge, we still h a v e a space charge lim ita tio n of th e b ea m c u rre n t. C. J . C albick h a s c a lc u la te d th e value of th is lim itin g c u r r e n t.13 If th e b ea m co m p letely fills a co n d u c tin g tu b e a t a p o te n tia l V w ith resp e ct to th e c a th o d e , th e lim itin g b eam c u rre n t is in d e p en d e n t of th e d ia m e te r of th e b ea m a n d is

I = 29.3 X 10“ V /2 (20)

If th e b ea m d ia m e te r is less th a n t h a t of th e c o n d u c tin g tu b e , th e lim iting c u rre n t is lower.

B u t p e rh a p s we ca n com p letely overcom e th e effects of space charge.

Suppose we p u t a v e ry little gas in th e d ischarge space. T h e n p ositive ions will be form ed. A n y te n d e n c y of th e electro n ic space charge to low er the p o te n tia l a n d slow u p th e elec tro n s will tr a p p o sitiv e ions in th e p o te n tia l m in im u m a n d so raise th e p o te n tia l. T h u s th e gas en ab les u s to get rid of th e th e slow ing u p effect of th e space charge as w ell as its diverging effect.

Before we c o n g ra tu la te ourselves u n d u ly , it m ig h t be w ell to m ake sure a b o u t th e s ta b ility of a n elec tro n b ea m in w hich th e electro n ic space charge is n eu tra liz ed b y h e a v y p ositive ions. L an g m u ir a n d T o n k s, in th e ir w ork on p lasm a oscillations, in tro d u c e d a co n cep t, e x te n d e d la te r b y H a h n an d R am o, w hich enables us to in v e stig a te th is problem . T h e co n c ep t is th a t of space charge w aves. I t is fo u n d th a t in a cloud of elec tro n s w hose n et space charge is n eu tra liz e d b y h ea v y , re la tiv e ly im m obile po sitiv e ions, sm all d istu rb a n ce s of th e electro n charge d e n s ity pro d u ce a lin e ar resto rin g force; a n d th is, to g e th e r w ith th e m ass of th e electrons, m a k es possible a ty p e of space charge w ave w hich m a y be co m p ared ro u g h ly w ith sound w aves, alth o u g h m u ch of th e d etailed b eh a v io r of space charge w aves is q uite different from t h a t of sound w aves. We m a y express a d istu rb a n c e in an elec tro n beam in te rm s of these space charge w aves a n d th e n exam ine the su b se q u en t h isto ry of th e d istu rb a n ce as a fu n c tio n of tim e . T h is h as been d one14 a n d th e p erh a p s su rp risin g re su lt is th a t e v e n w hen th e electronic space charge is n eu tra liz e d b y h e a v y po sitiv e ions, th e flow te n d s to collapse if th e c u rre n t is raised above a lim itin g value

I = 190 X l ( r V /2 (21)

I t is tru e t h a t th is c u rre n t is 6.5 tim es th e lim itin g c u rre n t in th e absence of ions, b u t it is a lim it n evertheless.

If th is lim it in th e presence of ions seem s u n n a tu ra l, p e rh a p s we should recall a m echanical analogy. C onsider a v e rtic a l long colum n su b je c te d to a load F . If we su b je ct it to a sidewise force a F p ro p o rtio n a l to F , as shown in Fig. 11a, th e b eh a v io r on increasing F will be a g ra d u a l defo rm atio n (analogous to th e space charge low ering of p o te n tia l in th e absence of ions)

P H Y S I C A L L I M I T A T I O N S I N E L E C T R O N B A L L I S T I C S 319

ending in collapse. H ow ever, ev en if, as in l i b , th e re is no sidewise loading an d no bending d u rin g loading, we know from E u le r’s form ula t h a t beyond a certain loading th e colum n will still collapse. T h is b ehavior is analogous to th a t of an electron beam in w hich th e electronic space charge is neu tralized by positive ions a n d th e re is no depression of p o te n tia l in th e beam .

T his space charge lim ita tio n e ith e r in th e presence or absence of ions allows th e passage of q u ite a large c u rren t th ro u g h a tu b e , as th e ta b le below will show :

Voltage Current, amperes, no ions Current, amperes, ions

1000 .9 2 7 6 .0 1

100 .0 2 9 .1 9 0

10 .0 0 9 .0 6 0

We m ight therefore feel th a t th e space charge is disposed of in a p rac tica l sense, and so it is in m a n y cases.*

-chF

r f f r

1 = 29.3 x |0 ~ V /2 1= 190 xlO~V/2 a

Fig. 11— Comparison of lim iting stable beam currents w ith and w ithout positive ions.

Po w e r Di s s i p a t i o n Lim i t a t i o n s

H aving ta lk e d ab o u t various lim itations im posed b y w ave effects, a b e r­

rations, th e rm a l velocities a n d space charge on th e electro n flow in th e beam itself, I w an t to close b y discussing briefly a topic w hich seems h a rd ly included in electron ballistics b u t y e t is v ita l to a n y ap p licatio n in th a t field. I refer to th e problem s associated w ith pow er dissipation w hen electrons strike som ething a n d stop. T h is is a good deal like th e problem im posed b y sud d en ly com ing dow n to e a rth while stu d y in g th e sensations of a free fall. I t is in evitable a n d m a y be fa ta l unless sa tisfac to ry provision is m ade for th e d issipation of k inetic energy.

W hat I w a n t chiefly to bring o u t are the consequences of scaling a given electronic device dow n in size. If we change th e size of each p a r t of an

* I t appears th a t in m any gas discharges, including those in which plasma oscillations are observed, the current is too high to allow persistence of the homogeneous flow upon which the plasm a oscillation equations are based.

elec tro n device in th e ra tio R , if we keep all v o ltag es th e sam e, a n d if we change all m ag n etic fields in th e ra tio 1/i? , elec tro n c u rre n t w ill rem a in th e sam e (p rovided th e c a th o d e is still cap ab le of giving space ch arg e lim ited em ission). E le c tro n p a th s will rem a in ex a c tly sim ilar, th o u g h sm aller;

th e pow er in to th e electro n beam will re m a in th e sam e, b u t w h a t will h ap p e n to th e pow er d issip atio n ca p ab ilities of th e device a n d w h a t will h a p p e n to th e te m p e ra tu re ?

I n a device cooled b y ra d ia tio n alone a n d w ith cool su rro u n d in g s, the r a d ia tin g a re a v aries as R 2, a n d since th e ra d ia tio n p e r u n it a re a v aries as T \ th e te m p e ra tu re will v a r y as R \

I n considering a case of cooling b y c o n d u c tio n alone, th in k of a rod carrying a ce rta in a m o u n t of pow er aw ay. If all th e dim ensions of a rod are changed b y a fac to r R , th e le n g th will be changed b y a fa c to r R , th e cross sectional are a will change b y a fa c to r R 2, a n d if th e th e rm a l co n d u c tiv ity

Fig.’

rem ains c o n s ta n t the te m p e ra tu re w ill v a ry as R ~ l. T h is is a fa ste r ra te of v a ria tio n th a n in th e case of cooling b y ra d ia tio n , a n d hence as th e system is scaled to a sm aller a n d sm aller size, cooling b y con d u ctio n w ill become negligible a n d ra d ia tio n cooling only will rem a in effective a n d will determ ine th e : te m p eratu re .

F igure 12 gives a n idea of th e v a ria tio n of v ario u s q u a n titie s discussed.

W e w a n t to m ake electronic devices sm aller for a n u m b e r of reasons;

p e rh a p s chiefly to reduce tr a n s it tim e a n d so to secure o p e ra tio n a t higher frequencies. I n doing th is, we en c o u n te r th e fu n d a m e n ta l lim ita tio n of red u c ed pow er dissip atio n cap ab ilities a n d in creased te m p e ra tu re . W h a t is th e tro u b le ? W e h av e scaled ev e ry th in g . O r h av e w e? T h e answ er is, we h av e n o t. T h e electrons, ato m s, a n d q u a n ta are still th e sam e size.

H a d we b een able to scale th e se, we should h av e in c re ase d th e h e a t con­

d u c tiv ity a n d th e ra d ia tin g pow er of our device, a n d all w ould h av e been CURRENT,

TAGE, POWER

ATURE ION CO O LING

________________ EMPERATURE,

0 R LINEAR DIMENSION ^ c O O L d 't G ^

MAGNETIC FIELD

12—V ariation of m agnetic field an d tem perature in scaling an electronic device.

P H Y S I C A L L I M I T A T I O N S I N E L E C T R O N B A L L I S T I C S 321

well. As it is, if we m ake a tu b e for given pow er sm aller a n d sm aller, using th e m ost re fra c to ry m a teria ls available we e v e n tu a lly reach a size of tu b e which will, despite our b est efforts, m e lt, th a w , a n d resolve itself in to a dew.

CONCLUSION

P erh ap s a fte r these som ew hat gloom y w ords concerning physical lim ­ ita tio n s in electro n ballistics, y ou m a y w onder how it is a t all possible to surm ount th e difficulties m entioned. I t ce rtain ly is n o t ea sy ; all electronic devices rep resen t com prom ises of one so rt or a n o th e r betw een fu n d am e n tal physical lim ita tio n s of electro n flow on th e one h a n d an d s tru c tu ra l com ­ plications on th e o th er. I n w orking w ith v ac u u m tu b e s one is p erh a p s tro u b led m ore b y physical lim ita tio n s, difficulties of co n stru ctio n , in ad e­

quacy of m aterials an d the lack of q u a n tita tiv e a g re em e n t betw een com pli­

cated phenom ena an d rela tiv e ly sim ple theories th a n in a n y o th e r p a r t of the electric a r t. I t is for th is reason th a t a friend of m ine tw isted a n old aphorism in to a new one a n d said, “N a tu re abhors a vacu u m tu b e ” .

Re f e r e n c e s

Electron Microscopes

1. Jam es Hillier and A. W. Vance: “R ecent Developm ents in the Electron M icro­

scope,” Proc. I.R .E . 29, pp. 167-176, April, 1941.

2. L. M arton and R. G. E. H u tte r: “T he Transmission Type of Electron Microscope and I ts O ptics,” Proc. I.R .E . 32, pp. 3-11, Jan. 1944.

Thermal Velocities

3. D. B. L angm uir: “T heoretical L im itations of Cathode-R ay T ubes,” Proc. I.R .E 25, pp. 977-991, Aug., 1937.

4. J. R. Pierce: “Lim iting C urrent D ensities in Electron Beam s,” Jour. A pp. Phys., 10, pp. 715-724, Oct., 1939.

5. J. R. Pierce: “A fter Acceleration and Deflection,” Proc. I.R .E . 29, pp. 28-31, Jan ., 1941.

6. R. R. Law : “ Factors Governing the Perform ance of Electron Guns in Cathode- R a y T u b es,” Proc. I.R .E . 30, pp. 103-105, Feb., 1942.

7. J. R. Pierce: “T heoretical L im itation to T ransconductance in Certain T ypes of Vacuum T ub es,” Proc. I.R .E . 31, pp. 657-663, Dec., 1943.

Space Charge

8. C. E. Fay, A. L. Samuel and W. Shockley: “On the Theory of Space Charge Between Parallel Plane E lectrodes,” Bell Sys. Tech. Jour. 17, pp. 49-79, 1938.

9. I. L angm uir and K. Blodgett: “ C urrents Lim ited by Space Chargeb etween Coaxial Cylinders,” Phys. Rev. 22, pp. 347-356, 1923.

10. I. Langm uir and K. B. B lodgett: “ C urrents Lim ited by Space Charge between Concentric Spheres,” Phys. Rev. 24, pp. 49-59, 1924.

11. J. R. Pierce: “Rectilinear Electron Flow in Beam s,” Jour, of A p p . Phys. 11, pp.

548-554, Aug., 1940.

12. B. J. Thompson an d L. B. H eadrick: “Space Charge L im itations on the Focus of Electron Beam s,” Proc. I.R .E .. 28, pp. 318-324, July, 1940.

J. Calbick: “ Energy D istribution of Electrons w ithin Dense Electron Beam s,”

Bull. A m . Phys. Soc., 19, No. 2, p. 14 (April 28, 1944).

R. Pierce: “Lim iting Stable C urrent in Electron Beams in the Presence of Ions,”

Jour. A p p . Phys. 15, No. 10, pp. 721-726 (1944).

L. Samuel, “ Some N otes on the Design of E lectron G uns,” Proc. IR E 33, pp. 233- 240, April, 1945.

By A. L. SAMUEL

H IS , th e final le cture of a series on E le ctro n B allistics, is n o t a su m m ary of th e m a te ria l w hich h as been prev io u sly p re se n te d b u t ra th e r it is an a tte m p t to show how th e b allistic a p p ro a ch can be ex ten d ed to th e analysis of high-frequency devices. M u c h t h a t m ig h t otherw ise be said a b o u t u ltra - high frequencies c a n n o t be said because of secrecy req u irem en ts. H ow ever, th e re is considerable m a te ria l w hich can be p rese n ted , w ith in th e lim its of th e necessary se cu rity regulations, w hich m a y be of in te re st to those w ho are n o t a lre ad y well a c q u a in te d w ith th e su b ject. I will, perforce, n o t be able to say a n y th in g specific a b o u t a c tu a l devices u tilizing th e p rinciples to be discussed.

M a n y of th e u ltra -h ig h -fre q u en c y devices w hich h av e come in to use d u rin g th e la s t few y ea rs h av e em ployed electron b eam s of one so rt or an o th er. T h ese devices can be an a ly se d in a n y one of a n u m b e r of ways.

F o r exam ple, we can w rite th e e q u a tio n of space-charge flow. T h is a p ­ p ro ach considers th e electric charge as a co ntinuous fluid su b je ct to P oisson’s eq u a tio n . T h e sm all-signal th e o ry of P e te rso n a n d L lew ellyn is a n exam ple of th is ty p e of analysis. O r if we wish we can consider th e v ario u s ty p e s of w ave m o tio n w hich can exist in a space-charge region. T h e space-charge- w ave analysis of H a h n a n d R am o as applied to v elo city -v a riatio n tu b e s is an exam ple of this. I n a d d itio n th e re is a n electro n -b allistic a p p ro a c h to the problem a n d it is w ith th is m e th o d t h a t we will be concerned in th e present lecture.

Before we becom e involved in th e d etails of th e analysis, we should perhaps spend a few m o m en ts considering th e rela tio n sh ip b etw een these various m ethods. If we h av e a n in te ractio n ta k in g p lace betw een electric fields an d m oving charges, we know a t once from N e w to n ’s second law t h a t the forces ac tin g on th e electrons m u st of necessity be e q u a l a n d opposite to those ac tin g on th e fields. I t is therefore a m a tte r of sm all concern w h eth er we consider th e forces ac tin g on th e electrons a n d th e effects of these forces on th e electron m otion or w h eth e r we consider th e a lte ra tio n in fields which th e electron m otion produces. W e can, if we wish, co m p u te th e energy tra n sfe r to a n electric field b y th e m o tio n of a n electric charge or we can com pute th e change in energy of th e electro n w hich accom panies th is tran s-

* Originally presented on April 11, 1945 as the concluding lecture of a sym posium on E lectron Ballistics sponsored by the Basic Science Group of the A m erican In stitu te of E lectrical Engineers.

322

E L E C T R O N B A L L I S T I C S I N H I G H - F R E Q U E N C Y F I E L D S 323

fer. I w as te m p te d to say “ which results from this tra n sfe r” b u t th is implies a cause a n d an effect, a notio n w hich has no place in th e p rese n t discussion.

T he d u al aspect of a n y en ergy-transfer problem m u st alw ays be k e p t in m ind. M uch needless discussion freq u en tly arises betw een p ro p o n en ts of one p o in t of view an d those preferring the o th e r w hen the only difference is one of language an d b o th groups are really saying th e sam e thing. T h e electron-ballistic ap p ro ach yields a sim ple physical pictu re; it is capable of being applied to w idely differing situations, b u t it is n o t well suited for a determ ination of th e reactive con trib u tio n s of an electron stream .

Ba s ic Co n c e p t s

T here are several concepts w hich we will find useful in our analysis.

These concepts are extrem ely sim ple, so sim ple in fact th a t one is te m p te d to assum e th a t th e y are well know n. H ow ever, these concepts are so basic to the subject, a n d th e ir results so far reaching th a t we m u st pause to consider them .

T he first is th e concept of to ta l cu rren t, as distinguished from its com ­ ponents. One w ay of w riting K irchhoff’s second law is

D i v . / = 0 ( 1 )

T h is sim ply says th a t th e to ta l cu rren t entering or leaving a n y differential region in space is zero. T h is expression m u st of course be generalized by including displacem ent cu rren ts as proposed b y M axw ell if applied to altern atin g currents. T he c u rren t / is th e to ta l c u rren t d en sity as here defined. A n im p o rta n t consequence of eq u atio n (1), ac tu a lly only an altern ate w ay of sta tin g it, is th a t the to ta l c u rren t alw ays exists in closed paths. L e t us ta k e a sim ple case of a tw o-elem ent therm ionic vacuum tube connected to a b a tte ry . Visualize th e situ atio n existing if b u t a single electron leaves th e cathode an d trav e ls to th e p late. T h e electron ta k es a finite tim e to cross from the cathode to th e p late. D u rin g th is tim e a cu rre n t exists, the m ag n itu d e being given b y th e relationship

I ⁼ ev

and according to our prem ise th is c u rren t is th e sam e in every p a r t of th e circuit. T h e c u rren t begins a t th e in sta n t th a t th e electron leaves the cathode an d it ceases w hen th e electron arrives a t th e p late. I n th e a p p a r­

en tly em p ty region ah ead of th e electron there m u st exist a displacem ent com ponent, num erically equal to th e conduction, or p erhaps we should say convection com ponent accounted for by the m oving electron. A n am m eter, were there one sufficiently sensitive an d fast, connected in th e extern al leads would read a c u rren t du rin g th is sam e in te rv al of tim e.

I have chosen to ta lk a b o u t b u t a single electron to em phasize the electron-

b allistic asp ec t; how ever, th e concept is m uch b ro a d e r th a n th is since it is n o t a t all d ep e n d en t u p o n a co rpuscular concept of th e electron. A s a resu lt of th is p ro p e rty of th e to ta l cu rre n t, th e c u rre n t to a n y electrode w ith in a v ac u u m tu b e does n o t necessarily b e a r a n y relatio n sh ip to th e n u m b e r of electrons w hich e n te r or leave it. O bviously th e n , c u rre n ts can exist in the grid circu it of a th ree-elem en t tu b e even th o u g h none of th e electrons are a c tu a lly in te rcep ted b y th e grid. T h is c u rre n t m a y h av e a n y p h ase rela­

tio n sh ip to an im pressed voltage on th e grid so t h a t th e g rid m a y d raw pow er from th e ex tern al circuit, or it m a y deliver p ow er to th e ex tern al circuit, all w ith o u t ac tu a lly in te rcep tin g a n y electronic cu rre n t. T h e g rid -cu rren t com ponent resulting from th e electronic flow b etw een ca th o d e a n d p la te m a y equally well b e a r a q u a d ra tu re rela tio n sh ip to th e im pressed voltage, in w hich case it will e ith e r increase o r decrease th e a p p a re n t interelectrode cap acitance. I f these effects seem qu eer it is because one is still confusing th e electronic com ponent w ith th e to ta l cu rren t.

A second basic concept once s ta te d becom es self-evident. T h is is to the effect t h a t th e only one th in g w hich we can do to a n electro n is to change its velocity, t h a t is, if we are to confine ourselves to th e classical concept of an electron. W e can change its lo n g itu d in al velocity, th a t is, a lte r its speed b u t n o t its d irection o th e r th a n possibly to reverse it, or we can in tro d u c e a tran sv erse com ponent to its velocity, th a t is, a lte r its d irec tio n as well as its speed. T h o u g h t of in th is lig h t all electronic devices in w hich a control is exercised over a n electron stre a m are v elo city -m o d u late d devices. I t m ight be arg u ed t h a t one could equally well sa y th a t all we can do is to change the electron’s acceleration (derivative of velocity) or its position (integral o f velocity).

T h e singling o u t of velocity is in a sense a rb itra ry . I t does, how ever, have some v ery in te restin g ram ifications.

I m ig h t digress for a m o m e n t to elab o rate on th is idea. Since som e of th e new er devices h av e been labeled v elo city -m o d u latio n tu b e s, th e re is a perfectly u n d ersta n d ab le te n d en c y on th e p a r t of th e u n in itia te d to assum e th a t these tu b e s differ from earlier know n devices, such as, for exam ple, the space-charge-control tubes, th e B a rk h a u se n tu b e or th e m a g n etro n in the fa c t th a t th e y em ploy velocity m odulatio n . T h e rea l difference lies else­

w here as we shall see in a few m om ents. A t th e sam e tim e th a t th e se new er devices were intro d u ced , th e re w as in tro d u c ed a new w ay of looking a t som ething w hich is v e ry old in th e a r t. T h is new er v iew point, to m y w ay of thin k in g , co n stitu te s a fa r g re a te r fu n d a m e n ta l c o n trib u tio n th a n do the specific devices w hich h ave received so m u ch a tte n tio n . T h e p ioneers in this new a p p ro a ch : H eil a n d H eil, B ru ch e a n d R ecknagel, th e V aria n B ro th ers, H a h n a n d M etcalf, to m e n tio n a few, a n d th e m a n y o th e r w o rk ers w ho lost in th e race to publish th e ir in d e p en d e n t c o n trib u tio n s in th is field— all of these people deserve th e g re a te st of p raise for th e ir stim u la tin g c o n trib u tio n s

to o u r thin k in g . M y only p o in t in all th is discussion is to em phasize th a t th e basic m e th o d of ac tin g on th e electron strea m has n o t really been changed a t all. T h e en tire m a tte r is sum m arized in the original sta te m e n t th a t th e only th in g w hich we can do to an electron is to change its velocity.

Before going on to th e n e x t asp ect of th e problem th e re is a closely rela ted concept w hich should be m entioned. T h is concept is th a t a change in th e com ponent of th e velocity of a n electron along one space coordinate does n o t introduce com ponents of velocity in directions orthogonal to th e first. F o r example, if a n electron beam is deflected b y a tran sv erse electric field, th ere will be no accom panying change in th e longitudinal velocity. T h e difficulty in the w ay of doing th is in a p rac tica l case h as n o th in g to do w ith th e concept b u t only w ith th e problem of producing u n idirectional fields. A nalyses of deflecting field problem s w hich ignore th e longitudinal com ponents of th e fringing fields are a p t to be wrong. T h e problem of high-frequency deflect­

ing fields h as been tre a te d in g re a t d etail in th e lite ra tu re a n d fre q u en tly w ith m ore acrim ony th a n accuracy.

One fu rth e r n o te should be ad d ed a t th is po in t. I n a n earlier lecture it was p o in te d o u t th a t th e m agnetic effects of a n electrom agnetic field are in general v ery m uch sm aller th a n th e electric effects. W e will n o t sto p to prove th a t th is is still tru e a t th e frequencies w hich now in te re st us b u t will accept it w ith o u t fu rth e r discussion.

F o r our n ex t concept we leave electron flow for a m om ent a n d consider the fields w ithin a reso n a n t cavity. Y ou m a y v e ry pro p erly o bject t h a t this has n o th in g to do w ith electron ballistics, an d indeed it does n o t. H ow ever, we will find it necessary to discuss problem s involving ca v ity resonators, an d a failure to u n d e rsta n d some of th e p roperties of these circu it elem ents can cause a great deal of trouble. T h ere are tw o conflicting approaches to th is problem w hich I will a tte m p t to reconcile.

T h e physicist w hen first presen ted w ith th e problem of a reso n a n t ca v ity is inclined to say : T h is is a boundary value problem. The solution consists in w riting M a xw ell’s equations subject to the conditions that the tangential com­

ponent of E m ust be zero along the conducting walls. W hile a scalar and a mag­

netic vector potential can be defined, the field is not related to the form er in the simple m anner used in electrostatic problems.

T he engineer, on the o th e r hand, is inclined to say : T h is looks like an extension o f the u sual resonant circuit. A capacitance exists between the top and bottom walls of the cavity; charging currents w ill flow through the single turn toroidal inductance form ed by the side walls. I would like to know what voltage difference exists between the top and bottom walls, and what currents exists in the side walls.

N ow , actu ally , I am m aligning b o th th e p hysicist a n d th e engineer b y m y sta te m en ts; nevertheless, th e re are these tw o approaches. W hich is cor­

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rect? W ell, th e y b o th are. I t is n o t correct to speak of a n elec tro sta tic p o te n tia l w ith in a reso n a n t c a v ity ; nevertheless, we m a y a n d do ta lk ab o u t th e vo ltag e b etw een th e to p a n d b o tto m of a re so n a n t ca v ity . W h a t do we m ean? Sim ply th e m axim um in sta n ta n e o u s line in te g ra l of th e electric field ta k e n along some specified p a th . I n a n y p ra c tic a l device utilizing electron beam s we are n a tu ra lly in te re ste d in th e p a th ta k e n b y th e elec­

trons. T h e fa c t th a t th e line in te g ral is different for different p a th s is of no g re a t concern. W e are in te re ste d in b u t one of these p a th s. W e shall therefore h av e occasion to ta lk a b o u t voltages in cavities b u t we m u st alw ays rem em ber w h a t is m e a n t, an d we m u st n ev e r for one in s ta n t forget th a t this v oltage is n o t u n ique b u t t h a t it depends up o n some assum ed p a th .

T h e second p ec u lia rity of th is vo ltag e m u st also be em phasized. T h e line in te g ral m u st be ta k e n a t a specified in s ta n t in tim e. I n effect one ta k es a p h o to g ra p h of th e field a t som e in s ta n t in tim e a n d th e n a t o n e’s leisure perform s th e in te g ratio n .

N ow , of course, an electron w hen p ro jec ted th ro u g h such a c a v ity will p erfo rm y e t a n o th e r ty p e of in te g ratio n . T h e change in sq u a re d velocity of th e electron as expressed in v o lts will be given b y th e line in te g ra l of the field en co u n tered b y th e electron; th a t is, in te g ra te d n o t in stan ta n eo u sly b u t w ith th e electron velocity. T h is is n o t a sim ple process, because the electron velocity is co n tin u o u sly being changed b y th e field in te ra c tio n and therefore th e velocity w ith w hich th e in te g ra tio n is p erfo rm ed depends up o n th e in te g ra te d v alue of th e field u p to th e p o in t in question. T his h as n o th in g to do w ith th e concept of v oltage in a re so n a n t ca v ity . The c a v ity v oltage can, how ever, be considered as th e m ax im u m change in sq u a re d velocity expressed in v o lts w hich a n elec tro n could receive if its en tra n ce velocity w as v e ry large so th a t th e tra n s it tim e w as sm all com pared w ith th e period of th e c a v ity field.

T h e fo u r basic concepts w hich I h av e chosen to recall to y o u r m in d are, b y w ay of su m m ary : (1) th e to ta l c u rre n t is th e sam e in all p a r ts of a circuit, t h a t is div. J = 0; (2) th e only w ay we can a c t on a n electro n is to change its v elocity; (3) th e changes in th e v elo city com ponent of a n electro n along a n y one rec tan g u la r coordinate h av e no effect on th e v elo city com ponents along a n y o th e r c o o rd in a te ; a n d (4) for convenience, a v o ltag e can be defined in a reso n a n t circuit as th e line in te g ral of th e electric field ta k e n along some p rescribed p a th .

Tr a n s i t An g l e

Since we are to d eal w ith th e in te ra c tio n of electrons a n d h igh-frequency fields, we fre q u en tly find it convenient to m easure elec tro n v elo city no t d irec tly b u t in term s of th e e q u iv ale n t p o te n tia l difference th ro u g h w hich an electron m u st fall to o b ta in th e v elo city in question, a n d th e u n it of m easure

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will be a v o lt. In s te a d of m easuring th e tim e required for an electron to traverse a n y given distance in seconds, it is also convenient to use, as a u n it of tim e, one ra d ia n of angle a t th e o p erating frequency. W e freq u en tly refer to th e tra n s it angle of an electron ra th e r th a n th e tra n s it tim e, alth o u g h b o th term s are used. I n fact, we m a y on occasion m easure d istances in term s of tra n s it angle, a n d th is usage is extended to m easure dim ensions transverse to th e direction of tra v e l of th e electron beam . W hen used in th is fashion, we m ean th a t th e dim ension in question is such t h a t were an electron to be p ro jec ted in th is direction w ith a velocity equal to t h a t of th e electrons in th e m ain beam , th e high-frequency field w ould change th ro u g h the sta te d n u m b e r of rad ia n s during th e tra n s it tim e.

Th e Fi v e Fu n c t i o n s i n a n El e c t r o n i c De v i c e

W ith th is prelim in ary discussion o u t of th e w ay wre can now answ er th e question w hich h as p ro b a b ly been tro u b lin g q uite a few of you. If th e only thing we can do to an electron is to change its velocity, th e n in w h a t basic w ay does th e velocity-m odulation tu b e differ from th e conventional negative grid tu b e or from th e m agnetron?

W ell, th is is a n involved story. If we are to m ake a n y use a t all of an electron b ea m we m u st in general p erfo rm five d istin c t o perations or func­

tions. F irs t w’e m u s t produce th e beam . T h e n we m u st im press a signal of some sort onto th e beam . F ro m w h a t I h av e ju s t said th is can be done only b y vary in g th e velocities of th e electrons contained in th e beam . T he th ird operation consists in converting th is v a ria tio n in to a usable form.

I t is in th is w ay t h a t th e diverse form s of electronic devices differ to th e g reatest degree. W e will go in to th is m a tte r in m ore d etail shortly. T he fo u rth operation consists in a b s tra c tin g energy from th e beam , an d th e final operation consists in collecting th e sp e n t electrons. W hile these operations are d istin ct from a n a n a ly tica l p o in t of view, in m a n y a c tu a l devices th e y are perform ed m ore or less sim ultaneously an d m ore th a n one operation m ay be perform ed b y certain portio n s of th e tu b e stru c tu re . I n fact, in some devices, for exam ple in th e space-charge-control tu b e, th e confusion is so g rea t as to m ake th e se p aratio n seem ra th e r forced. T h is v e ry confu­

sion m a y p a r tly explain wTh y v acu u m -tu b e engineers who were steeped in the a r t were so slow to realize th e ad v a n ta g es of th is new w'ay of looking a t things w hich I will call th e velocity-m odulation concept.

B y wTa y of m e n ta l exercise in th is new w ay of th in k in g le t us see how we can analyze a sim ple space-charge-control triode. W ell, first of all we have to id en tify th e electron gun wThich produces th e beam . T h e electrons m ost ce rtain ly come from th e cathode, b u t w here is th e first accelerating electrode? A ctu ally th ere isn’t a n y unless wre th in k of th e com bined d-c field resulting from th e d-c p o te n tia ls on th e grid a n d p la te as assisted b y