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A study of electron beams and their formation in electrostatic electron guns Goud, Paulus Arie 1964

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The University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY B.Sc, The University of Alberta, 1959 M.A.Sc., The University of B r i t i s h Columbia, 1961 TUESDAY, NOVEMBER 24, 1964, at 3:45 P.M. IN ROOM 208, MacLEOD BUILDING COMMITTEE IN CHARGE Chairman: I. McT. Cowan External Examiner: G. S. Kino of PAULUS ARI.E GOUD E. V. Bohn R. E. Burgess M. Kharadly F. Noakes A. C. Soudack C. A. Swanson Microwave Stanford Stanford, Laboratory University C a l i f o r n i a A STUDY OF ELECTRON BEAMS AND THEIR FORMATION IN ELECTROSTATIC ELECTRON GUNS ABSTRACT A t h e o r e t i c a l study i s made of the formation of electron beams i n e l e c t r o s t a t i c fields„ The electron motion i s assumed to be normal, congruent and regular, so that the equations governing the motion can be set up i n terms of the action function. By assuming con-venient functional forms of the action function, of the p o t e n t i a l and of the metrical c o e f f i c i e n t s , some new solutions are then found by the method of sepa-r a t i o n of v a r i a b l e s . These solutions are studied i n d e t a i l , and are shown to have some desirable properties. In order to employ a given space-charge flow solu-t i o n i n electron gun design, a method i s developed to take into account the d i s t o r t i o n of the f i e l d due to the anode aperture. In t h i s method, the gun i s con-sidered to be made up of two regions, separated outside the beam by an a u x i l i a r y anode. The desired space-charge flow i s assumed to exist i n the cathode region, while i n the anode region the e f f e c t of space-charge on the e l e c t r o s t a t i c f i e l d i s assumed to be n e g l i g i b l e . An estimate i s made of the accuracy of these assumptions. The f i e l d s about four i d e a l i z e d anode geometries are obtained by using Schwarz-Christoffel transformations, and a study i s made of the relevant properties of these f i e l d s . One of these f i e l d s , which has been c a l l e d the "wrap-around f i e l d " , i s shown to have properties that are very desirable for convergent electron guns. The above design method i s i l l u s t r a t e d by two exam-ples; namely, a gun producing a beam that i s i n i t i a l l y p a r a l l e l and r e c t i l i n e a r , and a gun producing a beam that i s i n i t i a l l y r a d i a l and convergent; the l a t t e r incorporates the wrap-around f i e l d i n the anode region. Physical considerations involved i n the determination of the electrodes to maintain a given beam are b r i e f l y discussed, and i t i s shown that the s e n s i t i v i t y of the f i e l d conditions at the beam boundary to errors i n the f i e l d at other locations decreases at an exponential rate with distance. A method i s suggested for determining beam-forming electrodes that avoids the need for an a u x i l i a r y anode to maintain the beam. GRADUATE STUDIES F i e l d of Study; E l e c t r i c a l Engineering Applied Electromagnetic Theory E l e c t r o n i c Instrumentation Network Theory Servomechanisms Communication Theory D i g i t a l Computers Electron Dynamics G„B„ Walker F. Ko Bowers AD. Moore E„V. Bohn A.D. Moore E.Vo Bohn G. B. Walker Related Studies; Noise i n Physical Systems Computational Methods Computer Programming Theory and Applications of D i f f e r e n t i a l Equations D i f f e r e n t i a l Equations R„E, Burgess C„ Froese Co Froese C A , Swans on C.A. Swanson A STUDY OF ELECTRON BEAMS AND THEIR FORMATION IN ELECTROSTATIC ELECTRON GUNS by PAULUS ARIE GOUD B«Sc» t University of Alberta, 1959 M»A«Sc»f University of B r i t i s h Columbia^ 1961 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering Ye accept this thesis as conforming to the required standard Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA October, 1964 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia,, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study, I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that, copying or p u b l i -c a t i o n , of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission* Department of E l e c t r i c a l Engineering The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8, Canada Date ZU November , 196^ ABSTRACT A theoretical study i s made of the formation of electron beams i n e l e c t r o s t a t i c f i e l d s . The electron motion i s assumed to be normal, congruent and regular, so that the equations governing the motion can be set up i n terms of the action function. By assuming convenient functional forms of the action function, of the potential and of the metrical c o e f f i c i e n t s , some new solutions are then found by the method of separation of variables. These solutions are studied i n d e t a i l , and are shown to have some desirable properties. In order to employ a given space-charge flow solution i n electron gun design^ a method i s developed to take into account the d i s t o r t i o n of the f i e l d due to the anode aperture* In this method, the gun i s considered to be made up of two regions, separated outside the beam by an au x i l i a r y anode. The desired space-charge flow i s assumed to exist i n the cathode region, while i n the a,node region the effect of space-charge on the el e c t r o s t a t i c f i e l d i s assumed to be ne g l i g i b l e . An estimate i s made of the accuracy of these assumptions* The f i e l d s about four idealized anode geometries are obtained by using Schwarz-Christoffel transformations $ and a study i s made of the relevant properties of these f i e l d s * One of these f i e l d s * which has been called the "wrap-around f i e l d " , i s shown to have properties that are very desirable f o r con-vergent electron guns* The above design method i s i l l u s t r a t e d by two examples; namely, a gun producing a beam that i s i n i t i a l l y p a r a l l e l and r e c t i l i n e a r , and a gun producing a beam that i s i n i t i a l l y r a d i a l and convergent; the l a t t e r incorporates the wrap-around f i e l d i n the anode region* Physical considerations involved i n the determination of the electrodes to maintain a given beam are b r i e f l y discussed, and i t i s shown that the s e n s i t i v i t y of the f i e l d conditions at the beam boundary to errors i n the f i e l d at other locations decreases at an exponential rate with distance. A method i s suggested for determining beam-forming electrodes that avoids the need for an a u x i l i a r y anode to maintain the beam. TABLE OF CONTENTS Page L i s t of I l l u s t r a t i o n s »»».. «»«»««>»•»»•>»><><>•••»» v i i L i S t O f Tab le S • ( • » % o a © o » f t o o » * o o o c & » f t f c © o * e o o f t & « * * » ^ Acknowledgemen"ts • • » • » • < ) • . « e « e « e « s < > i > e o » « « « a » i > » e 0 » x i 1 * INTRODUCTION * « « . » e o o e Q O O « * o o o o » a o o o * « e e f t ( > * * » 1 1 o 1 Introduction « < > » » < > » e * o « » « < > e e < > » o o » * » o » » » » 1 1:2 Objectives and procedure ....»<><,<>«.<>•»•• 9 2. ELECTRON PLOW IN ELECTROSTATIC FIELDS 11 2• 1 Introduction • • o » » « » o o » < o a « » < > > » e o « » e » * » » 11 2:2 Fundamental Theory »<,.„<><, <,<>».. ...»«»<>••• 12 2:3 Methods of Solution , < > < > o , > o c e o o o o < > . o o o « > » » 17 2:4 Solutions i n Cartesian Coordinates by the Method of Separation of Variables »•»».. 21 2:4sl Electron Motion v i t h Negligible , Space-Charge Effects »<><>»<>.<>o<><>o. 21 2:4i2 Electron Motion under Space-Charge CondxtXOnS • • • • o » e t > 0 » » * » a & e » o i > Q « » 30 2:5 Solutions i n Plane Curvilinear Coordinates by the Method of Separation of Variables 36 2:5§1 Action Function of the Form " V-^  ( cj^ ) + ( )" •••••••••••••• 36 (a) Conditions for separation of "V 3/1* 1 ctt> 1 G S 0 0 « 0 « « 0 0 0 0 0 « 6 B C O C > 0 0 » » 36 (ID ) EX ailip 1 © S fr6 6 o o o » o o 4 > o o i » o o o a o » » * » 39 2 * 5 3 2 Potential Function of the Form (a) Conditions for separation of varxQi\)les » o o « o f t t > * » « o » o o o o o » « » » 59 (t) ) Ex. aitip l e S o o f t » e o o » ( > » o f t o » o » o o o » » f t 61 2 $ 6 Dxscussion » » « o & o « & o « » o » o o o o o & o o ( > ( t o « o 9 « 68 XV Page 3. THE ELECTROSTATIC FIELD OF IDEALIZED ANODE STRUCTURES » o & 4 t o O * * 6 b o O « © 6 o d 6 t > e o o & 6 o o 6 o & 6 o o © Q » * t 0 f e 71 3*1 Ill"fcrOCLtlC"biOIl e » » * * « > e » o e o o o » f t o o o e o o o o o 6 o o « * a 71 3s2 E l e c t r o s t a t i c F i e l d about a Plane with a S l i t » 9 e « ' £ ( > * 9 « * t t » f t f r o o o o o 6 D O & o « > & o o o o e b o o o ' » » » o 72 3:3 Electrostatic. F i e l d about Two Right-Angled 3s4 E l e c t r o s t a t i c F i e l d about Two Semi-Infinite P ar a l l e l PI at eS • tt©ofttt6©oooe.©oo©<>o.»o6oo«0» 82 3 $ 5 The "Wrap-Around F i e l d " .«....<,.•,•<,.<>«• .<..«. 82 4. USE OF SPACE-GHARGE-FREE FIELDS IN ELECTRON GUN DESIGN » o o < r o « 0 0 » « o o « 0 0 « o o o o o o o o o o o o o o o o o o o o o o » o * » 89 4 8 1 I lit T o|dUC t iOIl t L t t o o & O b e o o e o o a o & O Q O O O o b o o e o o f r e 89 4:2 Error Estimate for a Space-Charge-Free F i e l d Approximation i n the Anode Region o e o o o * <•<>*• 91 4:2:1 Flow between Two P a r a l l e l Plates «... 91 4:2:2 Convergent Flow between Two Con-,cM3ntric Cylinders . « ..........« 95 4:3 I n i t i a l l y P a r a l l e l , Rectilinear Flow to an Apertured Anode » . . . . o « » - . » . . . » . » o o o o o <> o . » o o • 100 4:4 I n i t i a l l y Radial, Convergent Flow to an Apertured Anode « < , . . « o . o o o . o o o o o o o o 6 ( . ° 6 o a o o i > 109 4:4:1 Analysis of Anode Fields » . . 0 o o . « > » * » 109 4:4:2 E l e c t r o s t a t i c F i e l d i n the Anode Region Approximated by a Wrap-Around FX B 1 d e « « o o « o » » o « o o * o t > < > o o o o e o 6 0 « « Q O ^ » 119 4 5 5 DiSCUSSion « « e « o « o o e o o o o o o a 4 0 o e o o < > » o « o < > « Q « « * 126. 5. THE DETERMINATION OF BEAM-FORMING ELECTRODES . 0 0 0 127 5 si Physical Considerations • » o . « » » » o o o o o * o o » « * i 127 5*2 Design Procedure o o o o o o o o « o o o o o o » * o o o o « o o o « o 130 5*3 DiSCUSSXOn « * e Q O O O o o C f O o « « o o o t > o t > o o o c > o c o o o 6 0 b o 136 6 o CONCIfUS ION a < r « « » o o a » O ; « o o o o o o o o o o o o o o o o D o o e « ) f r 6 & « » 0 138 Page Appendix A ESTIMATE OP SELF-MAGNETIC FORCES AND RELATIVISTIC EFFECTS 139 Appendix B DEMONSTRATION OP THE EXISTENCE OF W(q 1,q 2,q 3) = c± WHEN THE CONDITION V x i T = O IS SATISFIED 141 Appendix C NUMERICAL METHOD FOR OBTAINING ELECTRON TRAJECTORIES IN ELECTROSTATIC FIELDS .... 142 Cs l Space-Charge Effects Neglected 142 C:2 Correction for Space-Charge Forces ......... 150 Appendix D ANALYSIS OP THE CURVATURE OF AN EQUI-POTENTIAL 153 Appendix E ELECTROSTATIC FIELD REQUIRED TO MAINTAIN TWO PARALLEL, SPACE-CHARGE-LIMITED STRIP BEAMS 157 Appendix F ON THE STABILITY OF THE PIERCE-CAUCHY PROBLEM 160 References »» 166 v i LIST OF ILLUSTRATIONS Figure Page 2-la Sketch of a trajectory and of the surface (J) = 0 when the l a t t e r i s a hyperboloid of one sheet* For, this case C,>0, C 9>0, c±> < 0 .. . 26 ^o 2-lb Sketch of a trajectory and of the surface Cj> = 0 when the l a t t e r i s a hyperboloid of two sheets. For this case C,<0, C 0< 0, <£>•< 0 . 27 o 2-2a Sketch of electron motion from a right-angled cathode 28 ,2-2b Sketch of the electron motion of Figure (2-2a) when the sheets are unfolded 29 2-3 Hyperbolic space-charge flow 34 2-4 Equipotentials of $ = l n r + ^ s i n ( 2 e) • 4 2 r 2-5a Phase plot of the r a d i a l v e l o c i t y component of equation (2.68) 45 2-5b Phase plot of the O-component of v e l o c i t y of equation (2.68) 45 2-6 Logarithmic s p i r a l coordinates: u+v r = , 9 = v-u 51 / T 2-7 Phase plot of electron motion according to equations (2.77) 58 2-8 Electron motion between two inclined plane electrodes. The f i e l d l i n e s , which are arcs of c i r c l e s , are shown as dashed lines .... 63 2- 9 Electron t r a j e c t o r i e s between two equi-angular s p i r a l electrodes. The f i e l d l i n e s , which are equiangular s p i r a l s , are shown as dashed lines 65 3- 1 Mapping the p r o f i l e of an i n f i n i t e plane with a s l i t onto the u-axis of the w-plane .... 72 3-2 E l e c t r o s t a t i c f i e l d about a plane with a s l i t 74 v i i Figure Page 3 - 3 Mapping a degenerate rectangle onto the v—p1ane • • . . . . * * . . » . o * » . . . . . . . » » « . . « * . . » 7 5 3 - 4 E l e c t r o s t a t i c f i e l d about two right-angled 3r*5 Mapping the p r o f i l e of a plate of thickness y^, with a s l i t of half-width x^, onto the w—plane *..»« . » * . . « « . . » . » . . « . . « • » . « . * * » * » • » 78 3 - 6 Plot of equation (3*9 ) . . . . . . . . . . . . . . . . . . . 8 1 / 3 - 7 E l e c t r o s t a t i c f i e l d about two semi-infinite p a r a l l e l plates . . . . . . . . . f t . . . . . . . . . . . . . . 8 3 3 - 8 The "wrap—around f i e l d " . . . . . . < , . . . . . . » . . . . 8 6 3 - 9 Variation of potential along the plane of symmetry of four electrode shapes ........ 8 7 4 - 1 Hypothetical electron gun »»....<,..»»...»• 8 9 4 - 2 Variation of the potential, e l e c t r i c i n tensity* electron v e l o c i t y and space-charge density versus distance from the cathode In a p a r a l l e l plane diode. Also shown is. an approximation of the potential i n the anode region by Cj)^ = a Y + b »..»»» 93 4 - 3 Error i n potential e ^ , and i n electron v e l o c i t y £j> at the anode of a planar diode when the potential i s approximated by $ T = a l + b over the in t e r v a l T < Y< 1 . 0 9 4 h O 4 - 4 Variation of the charge density i n r e c t i -linear, convergent electron motion from a c y l i n d r i c a l cathode 9 6 4—5 Variation of the potential and e l e c t r i c i n t e n s i t y versus distance from'the cathode i n a concentric—cylinder, convergent-flow diode* Also shown i s an approximation of the potential i n the anode region by ^X-^  — a In R *^  b . o . » . . « . « « . o . . . . . . . . . . . . * 9 7 4—6 Error i n potential e ^ , and i n electron v e l o c i t y e^j at the anode of the concentrie-cylinder* convergent-flow diode when the potential i s approximated b y = a l n R + b over the i n t e r v a l R > R>R , where R = 0 . 2 5 9 9 o a a v i i i Figure Page 4-7 Planar diode with an anode aperture .»..•»• 101 4-8 Potential v a r i a t i o n along the plane of symmetry of an i n i t i a l l y p a r a l l e l , r e c t i — 1 ine ar flow *»•»»»•<>••»••«<>«»«»»••«•* 106 4-9a Electron t r a j e c t o r i e s neglecting space-charge e£'fects » i » e o t i < > » 0 i f r o e o o » o f r o 6 » e 6 » » o * 6 o o o f t o * » » * 107 4-9b Electron t r a j e c t o r i e s when space-charge forces are taken into account ..»<><>.<,.<>»<.*• 107 4-10 Cathode region of an i n i t i a l l y r a d i a l , Convergent flOW » • e b o e o o o o o o o o o o o o o o o t a o . o e 109 4-11 Radius of curvature of equipotentials i n the f i e l d about two semi-infinite p a r a l l e l 4-12 Position of the centre curvature of equi-potentials i n the f i e l d about two semi-i n f i n i t e p a r a l l e l plates •» 112 4-13 Radius of curvature of equipotentials i n the wrap—around f i e l d ....<, »<> 115 4-14 Centre of curvature of equipotentials i n the wrap—around f i e l d o < , c , o . o s . . » o . . » . o o o . < . < > . . • » 116 4—15 Variation of the potential gradient along equipotentials of the wrap—around f i e l d 118 4- 16 Potential v a r i a t i o n along the plane of symmetry of an i n i t i a l l y r a d i a l , convergent electron be am « « e e ^ o » » » » » o * o « » » o « e » a « * » » « » » 124 5— 1 E l e c t r o l y t i c tank model of an i n i t i a l l y p a r a l l e l ^ r e c t i l i n e a r - f l o w electron gun «»» 132 5-2 I n i t i a l l y r a d i a l j convergent-flow electron gun with an a u x i l i a r y anode and a two-Po"fcentx3.X main anode » » o o o o o « o « o o o « * o « e o « o » X33 5-3 E l e c t r o l y t i c tank model of an i n i t i a l l y r a d i a l j convergent-flow electron gun ••»»•» 135 C-l Motion of an electron i n a uniform e l e c t r i c f i e l d » 4 < r * O f t O f r o » o o e » < » « o f t c o c o o o e c > o » o « e o o o f t » f t X 4 2 C-2 Electron path i n the j 1 t h in t e r v a l of a non-uniform f i e l d , showing the effect of a uni-form-field approximation. The interval size i s greatly exaggerated ........••»• 144 ix Figure C-3 C-4 D-l E - l F - l F-2 Prediction of " ^ ( j ) i n the i t e r a t i v e pro cess » » ' a » » » » » 6 f t o o » * f r f r « b o » e » t > » » o * » o * f t * e » Space—charge effects i n the anode region «, Centre of curvature of v^ at u^ <,.<>....»» • Electrodes and r e s i s t i v e s t r i p to maintain two p a r a l l e l s t r i p beams • • • » » o e » » » o » o 6 » * » Square lattice (Ar = Az) used for solving equat1on (F»3) • . • • • « * « o e » e o » o e f t * o o » o * » « » » Ratio of the adjacent central column coe f f i c i e n t s a versus distance from the m-^, n Page 149 151 153 158 162 beam boundary 165 LIST OF TABLES Table F - l Coefficients of e,, in equation (F»5) •< Page 164 x ACKNOWLEDGEMENTS I wish to record my thanks to Dr. G. B. Walker* the supervisor of this project*for his encouragement and guidance throughout the course of the work. I wish also to express my appreciation to Dr» G» G. Eng l e f i e l d , Dr. C» R» James, Mrs. W» L. Magar, and Mr* D. R. McDiarmid for helpful discussions. The encouragement and help of my wife, Miriam* i s value greatly. Acknowledgement i s grat e f u l l y given to the National Research Council for the avwiard of Studentships i n 1961 and '62 and for a Research Assistantship, made available through the National Research Council, Block Grant to the Department of E l e c t r i c a l Engineering, U.B.C., during the remainder of this project. x i CHAPTER I - INTRODUCTION l 8 l Introduction U n t i l the invention of the klystron twenty-five years ago, the major devices employing electron beams were cathode-ray tubes and electron microscopes. The current requirements of the l a t t e r devices are modest, being of the order of a few [ia, to several ma. New and d i f f i c u l t electron gun design problems were posed by the advent of beam-type microwave tubes, which require a high-current-density beam at a comparatively low voltage. Valuable indicators of electron gun performance are provided by the concepts of perveance and area-compression r a t i o . For an idealized model of space-charge-limited flow, i n which physical considerations such as i n i t i a l thermal v e l o c i t i e s and va r i a t i o n of the work function, of the conductivity and of the contact potential at the cathode are ignored, the perveance K i s described by K = ( l . l ) V2 where V i s the potential difference between the cathode and anode •— both of which may be of arbitrary shape — and I i s the to t a l current « Equation ( l . l ) i s commonly used to specify the perveance of an electron gun or of an electron beam, even i n cases where th i s r e l a t i o n does not s t r i c t l y apply-. The area-For a b r i e f informative history of equation ( l . l ) and the term "perveance", see reference #1. compression r a t i o of an electron beam i s defined as the ratio of the cathode area to the ultimate beam cross—sectional area* Electron guns that produce beams with an area-compression ratio greater than one are called convergent guns. Despite the simplicity of equation ( l . l ) . the theoretical evaluation of K i s , i n general, very d i f f i c u l t . This can be appreciated i f i t i s considered that the e l e c t r o s t a t i c f i e l d i n the cathode-anode region i s determined by the space-charge d i s t r i b u t i o n i n the beam and by the shape of, and potential difference between, the electrodes. The space-charge d i s t r i -bution i s , however, dependent on the dynamic properties of the electron flow, and these properties are, i n turn* prescribed by the e l e c t r o s t a t i c f i e l d . A mutual interdependence, known as the self—consistency condition, thus exists between the space—charge d i s t r i b u t i o n and the e l e c t r o s t a t i c f i e l d . U n t i l 1949, space—charge flow solutions were known for r e c t i l i n e a r flow only. In r e c t i l i n e a r flow, the tr a j e c t o r i e s coincide with l i n e s of force of the e l e c t r o s t a t i c f i e l d , simplifying the problem greatly. There are three kmawn cases (2) of r e c t i l i n e a r flow; namely, lines of flow (a) p a r a l l e l , (3 ) (b) radiating normally from (or converging to) an axis , and (c) radiating from (or converging to) a p o i n t T h e s e three cases correspond to flow between two i n f i n i t e p a r a l l e l plates, two concentric cylinders, and two concentric spheres, re spectively. Within a year after the invention of the klystron, Pierce published a method for the design of electron guns that was based on the r e c t i l i n e a r space-charge flow solutions. In this 3 design method the electron beam i n the cathode-anode region of the gun i s taken to be a section of a space-charge—limited r e c t i l i n e a r flow* The potential v a r i a t i o n along the lines of flow i s therefore known, and i t i s i m p l i c i t l y assumed that ttie anode i s perfectly gridded; i . e . , the anode allows the beam to pass through i t * but maintains the prescribed p o t e n t i a l . Since the flow i s assumed to be: r e c t i l i n e a r , the potential v a r i a t i o n perpendicular to the lines of flow must be zero both inside the beam and at the beam boundary. The e l e c t r o s t a t i c f i e l d outside a r e c t i l i n e a r - f l o w beam must, therefore* be such that at the beam boundary (a) the potential v a r i a t i o n i s as prescribed by the r e c t i l i n e a r flow, and (b) the normal potential gradient i s zero. The electrodes that produce the desired e l e c t r o s t a t i c f i e l d are called beam—forming electrodes. The general problem of determining the e l e c t r o s t a t i c f i e l d outside a curvilinear electron beam on the surface of which the potential v a r i a t i o n and the normal derivative of potential are prescribed i s termed the Pierce-Cauchy problem. It i s the f i r s t known physical problem involving an e l l i p t i c d i f f e r e n t i a l equation* Laplace's equation, with Cauchy-type boundary conditions on an open boundary. The solution to this problem i s unstable i n the sense that an i n f i n i t e s i m a l change i n the boundary conditions causes a large change i n potential some distance from the beam^^o The physical significance of this i n s t a b i l i t y i s that (a) the boundary conditions can be s a t i s f i e d within f i n i t e , but a r b i t r a r i l y small, l i m i t s by beam-*forming electrodes that are quite d i f f e r e n t i n shape* and (b) the electrodes do not need to extend an i n f i n i t e distance away from ( 7 ) the beam, but can be truncated » The Pierce-Cauchy problem can be solved by a n a l y t i c a l , numerical, or analogue methods. Analogue methods, such as e l e c t r o l y t i c - t a n k models, are often used i n preference to the other two methods because, by th e i r use, beam-forming electrodes of convenient shapes can generally be determined more easily*; In electron guns producing beams with high power densities i t often i s not possible to place a g r i d at the anode aperture (for thermal Considerations)* . The gr i d at the anode of a Pierce gun may be dispensed with, without greatly affecting the beam i n the cathode--anode region, i f the width of the beam at the anode i s small w*r»t s the cathode-anode distance. The defocusing action may then be calculated by means of the wellr-known (8 ) Davisson—Galbiek equation . It i s assumed i n this calculation that the electron t r a j e c t o r i e s remain r e c t i l i n e a r u n t i l they reach the anode aperture* The aperture i s represented 1 by a thin lens, and at the p r i n c i p a l plane of t h i s lens the tr a j e c t o r i e s are assumed to undergo a discontinuous change i n slope* The accuracy ©f the Davisspn-Calbick formula can be improved by , . , „ (9,10,11) applying a spaee^eharge correction * * '. As the perveance i s increased, the f i e l d d i s t o r t i o n due to the anode aperture becomes progressively more severe, and the thin lens model of the anode aperture rapidly becomes inadequate -6 3/2 ( 1 2 ) beyond a perveance of about 0 e l x 10 amp/volt 1 „ I n i t i a l l y the Pierce theory can be extended by treating the anode as a atio] (14) (13) modified thin lens , or by a perturbation analysis of the r e c t i l i n e a r space-*charge flow solutions Danielson et a l * (15) made a very complete study of the divergent effeet of the anode aperture, including the effect of thermal v e l o c i t i e s ^ v a l i d for perveances up to 0*7 x 10"°^* They determined the f i e l d i n the anode region by two dif f e r e n t methods. In the f i r s t method the pr i n c i p l e of superposition i s used; the actual potential d i s t r i b u t i o n i s approximated by the sum of a space-charge—free potential d i s t r i b u t i o n , obtained from the e l e c t r o l y t i c tank, and the potential d i s t r i b u t i o n as prescribed by the rectilinear'—flow solution. In the second method i t i s assumed that the eff e c t of space-charge on the potential d i s t r i -bution can be neglected i n the anode region, A t h i r d electrode, of a shape and at a potential as prescribed by the re c t i l i n e a r — f l o w solution* i s placed between the cathode and the anode i n the e l e c t r o l y t i c tank* and the f i e l d i n the anode region i s then probed. The information from either of these methods i s then used to modify the Davisson-Calbick formula. The second method for obtaining the potential d i s t r i b u t i o n i s e s s e n t i a l l y the same (16) as one described e a r l i e r by Brown and Siisskind » In the extended Pierce theory i t i s assumed that conditions i n the cathode region are r e l a t i v e l y unaffected by the anode aperture* This assumption i s generally considered to be —6 satisfactory up to a perveance of about 1,0 x 10"" * For higher— perveance Pierce guns, the f i e l d d i s t o r t i o n extends to the cathode* reducing the off—cathode gradient, and hence the emission, i n a non—uniform manner. For these guns the actual perveance i s thus lower than the design value, and the current density i s non-uniform across the beam. The l a t t e r condition i s aggravated by the spheriisal aberration of the anode f i e l d , (17) Muller found an approximate r e l a t i o n between the actual value and the design value of the perveance of a conical—flow (18) Pierce gun from ele c t r o l y t i c - t a n k studies, Ambpss carried 6 out a f i r s t ~ o r d e r perturbation analysis for guns of t h i s type, and obtained expressions for the change i n current density across the cathode and elsewhere, for the loss i n perveance, and for several other variables. In his analysis he assumed that the potential d i s t r i b u t i o n i n the anode region could be obtained from the space-charge-free potential* Experimental measurements on a Pierce gun with a design perveance of 3,25 x —6 10 gave good agreement with his theory. If a non—optimum gun design i s acceptable, a Pierce gun of a desired perveance can (17) thus be designed by applying the perveance correction of M\iller v 1 (18) (18) or Amboss % and the gun performance predicted from Amboss' work, (17) (12) Miiller and Brewer developed quite similar electro-l y t i c — t a n k methods, by means of which uniformity of cathode emission and i n i t i a l l y r e c t i l i n e a r flow can be p a r t i a l l y restored. In these methods the off-cathode potential gradient i s made more uniform by reshaping the beam-forming electrode (from the Pierce shapes) i n such a way that the f i e l d along the i n i t i a l part of the beam edge i s weakened. Guns with a perveance of 1,58 x 10~ and an area-compression ratio up to 30 have been made by Mi i l l e r t s (17) (12) method « Brewer appears to have obtained gun perveances of 2.2 x 10"16 by his method. To study the electron t r a j e c t o r i e s i n a proposed high— perveance gunf use i s often made of analogue equipment such as an e l e c t r o l y t i c tank, a resistance network, or a rubber membrane, i n which space^Charge i s simulated. An analogue of the gun structure set up on one of these devices provides e l e c t r o s t a t i c f i e l d data^ which are used by an analogue or d i g i t a l computer c o u p l e d i n t o the system t o s o l v e the e l e c t r o n — d y n a m i c a l e q u a t i o n s and to t r a c e out the t r a j e c t o r i e s • More r e c e n t l y * n u m e r i c a l methods have been used t o o b t a i n , by means of a d i g i t a l computer* b o t h the e l e c t r o s t a t i c f i e l d d a t a and the (26) t r a j e c t o r i e s * from a mathematical model of an e l e c t r o n gun v , Wi t h the a i d of the above-mentioned equipment*^ the i n f l u e n c e of the shape of the e l e c t r o d e s on the t r a j e c t o r i e s of a proposed gun can be i n v e s t i g a t e d e m p i r i c a l l y p r i o r t o the c o n s t r u c t i o n and t e s t i n g of one or more gun p r o t o t y p e s * The t e s t i n g o f the l a t t e r i s g e n e r a l l y c a r r i e d out i n a demountable vacuum system. The v a r i a t i o n of c u r r e n t d e n s i t y a c r o s s the e l e c t r o n beam emerging from a gun p r o t o t y p e may be s t u d i e d by i n t e r c e p t i n g the beam w i t h a f l u o r e s c e n t s c r e e n , or by moving a c r o s s i t an i n t e r c e p t i n g anode c o n t a i n i n g a p i n -h o l e . I n the l a t t e r c a s e , i f the c u r r e n t c o l l e c t o r b e h i n d the p i n h o l e i s a s p l i t Faraday cage, i n f o r m a t i o n about the v a r i a t i o n a c r o s s the beam o f the t r a n s v e r s e e l e c t r o n v e l o c i t i e s can be (27) o b t a i n e d s i m u l t a n e o u s l y * T h i s i n f o r m a t i o n can a l s o be o b t a i n e d by r e p e a t i n g the measurement of c u r r e n t d e n s i t y a c r o s s the beam a t v a r i o u s d i s t a n c e s from the gun. (28) M a t h i a s and K i n g o b t a i n e d a gun w i t h a perveance of 2 x 10 ^ from an e x p e r i m e n t a l i n v e s t i g a t i o n of gun p r o t o t y p e s (17) based on a M u l l e r d e s i g n v ', A knowledge of the v a r i a t i o n of e m i s s i o n a c r o s s the cathode, and of the a b e r r a t i o n of the anode, was i n g e n i o u s l y o b t a i n e d by l e a v i n g v a r i o u s s e c t i o n s of the cathode uncoated i n some of the p r o t o t y p e s * and by n o t i n g the r e s u l t a n t changes i n the c u r r e n t d e n s i t y p a t t e r n s of the emerging '— ~ ; ! — : — ~ A comprehensive a r t i c l e d i s c u s s i n g these methods was w r i t t e n by S i i s s k i n d v 1 ^ ) ±n 1956, Papers r e p r e s e n t a t i v e of more r e c e n t work are l i s t e d i n r e f e r e n c e s (20 - 2 6 ) , beams• (27) Frost et a l * developed an elaborate design method that resulted i n the successful construction of a gun with a g perveance of 2*2 x 10 and a compression ratio of 300* and also —6 a gun with perveance 5 x 10~ and compression r a t i o 6* Starting (17) (12) with a design based on Miiller's or Brewer's methods, a gun was then b u i l t using the cathode and anode as designed, but with the beam—forming electrode replaced by about f i v e annular disc electrodes* The potentials of these discs were adjusted experimentally u n t i l the desired beam was obtained* By the use of an e l e c t r o l y t i c tank a beam-forming electrode shape was then obtained which gave approximately the same f i e l d conditions i n the region of the beam. In empirical design methods, cathode shapes other than those required by the known space-charge flow solutions can be used. A very successful gun resulting from empirical design i s (29 30) the Heil gun v * « The cathode of this gun i s part of an (29) e l l i p s o i d of rotation. Heil ' obtained a design with a —6 perveance of 4*4 x 10 and a compression ratio of 230, while R e e d ^ ^ b u i l t a Heil gun for a 5 mm klystron with a perveance of 3 x 10 ^ and a compression r a t i o of 75* Kawamura^'''^ has designed high pervea,nce guns with oblate spheroidal cathodes. The success of these cathode shapes i s due to the fact that they tend to correct for the spherical aberration of the anode by starting the electrons off on tr a j e c t o r i e s that have a different (32) centre of curvature* depending on the starting point* Lucken corrected for the spherical aberration by d i s t o r t i n g the shape of a spherical -'cap cathode i n such a manner t h a t i t s centre 9 of rotation lay on a c i r c l e . It i s clear that the designing of high perveance electron guns i s s t i l l to a large extent a t r i a l and error process. There have been many s i g n i f i c a n t advances i n the analysis of proposed gun designs* but* when proceeding to improve the design on the basis of these analyses, the gun designer s t i l l needs to rel y on his i n t u i t i o n to decide how the proposed design should be changed, lt2 Objectives and Procedure This study of the formation of electron beams i n el e c t r o s t a t i c f i e l d s has been divided into four sub-problems; namely, (l) electron flow i n e l e c t r o s t a t i c f i e l d s , (2) space—charge-free e l e c t r o s t a t i c f i e l d s i n idealized anode geometries* (3) design of electron beams based on ( l ) and (2), and (4) the Pierce-Cauchy problem* In Chapter I I , the theory of electron flow i n electro-s t a t i c f i e l d s i s derived, and the underlying physical assumptions are discussed* Past methods of solution are noted* New solutions are then found by the method of separation of var i a b l e s . These solutions are studied i n d e t a i l , and are shown to have desirable ch a r a c t e r i s t i c s * To adapt a given space-charge-flow solution for use i n the design of high—perveance electron guns with anode apertures, a knowledge of the form of the e l e c t r o s t a t i c f i e l d s of various apertured anodes i s highly desirable. In Chapter I I I * use i s made of the Schwarz-Ghristoffel transformation to compute and plot the space-charge—free f i e l d s of three di f f e r e n t idealized anode 10 geometries. In Chapter IV, the characteristics of these f i e l d s relevant to electron—beam design are analysed. The information obtained from t h i s analysis i s used to design an anode f i e l d with improved c h a r a c t e r i s t i c s . This design involves a lengthy Schwarz-Christoffel transformation, so that for the sake of continuity the mathematical derivation i s included i n Chapter II I , although i t s real importance does not become clear u n t i l Chapter IV, In Chapter IV, a new gun design method i s also formulated, showing how known space-charge-flow solutions can be matched to the above—mentioned space-charge-free f i e l d s . The procedure i s i l l u s t r a t e d by two examples, and i t i s shown how the electron t r a j e c t o r i e s may be computed i n the anode region, taking space—charge into account, A further adaptation of the present space—charge-flow solutions for electron beam applications i s necessitated by the fact that these solutions involve unbounded flows; i . e . , space-charge occupies the entire space between two equipotential surfaces. This adaptation* the Pierce-Cauchy problem, i s the subject of Chapter V. The error growth i n potential i s evaluated for a c y l i n d r i c a l beam. The results of this investigation are summarized i n Chapter VI. CHAPTER II - ELECTRON FLOW IN ELECTROSTATIC FIELDS 2:1 Introduction In discussing the self-consistent flow of electrons, i t is convenient to disregard the discrete nature of the electron. Instead, the flow i s treated as a continuous compressible f l u i d . This approach w i l l be used here to formulate the theory of space-charge flow i n the absence of externally applied magnetic f i e l d s . It w i l l be assumed that the flow i s congruent, normal, regular and laminar, R e l a t i v i s t i c effects and the effects of self—magnetic forces are neglected. This places an upper l i m i t of about 20 keV on the electron energy, as i s discussed i n Appendix A* (39) The term "congruent flow" means that the ve l o c i t y i s i n general a single-valued function of position, so that only a single flow l i n e passes through any point . The mathematical significance of congruence i s that the flow i s d i f f e r e n t i a b l e . By normal flow two s l i g h t l y d i f f e r e n t concepts are (33) implied, Meltzer defined normal flow (as opposed to abnormal flow) as flow i n which the sum of the kin e t i c and potential energy i s constant for any point i n the flow, Meltzer showed that this requires a unipotential cathode. A necessary and s u f f i c i e n t condition for this normal flow i s that C\7x1?) = 0, The second interpretation of normal flow i s a geometric one* For normal congruent flow i n this sense to occur, i t i s necessary and s u f f i c i e n t that 1T^7xW= 0. When this i s the case* then there exists a one—parameter family of surfaces W(q^, Q.3) = CJL ortho-Thrs i s called "single streaming" by some workers* (34) 1 2 gonal to the flow l i n e s . The assumption of laminar or i r r o t a t i o n a l flow requires that V x 1 ? = 0 throughout the flow. If flow i s laminar, both normality c r i t e r i a are therefore s a t i s f i e d , and the flow originat from a unipotential cathode. Conversely, i f S7xT7 = 0 at the cathode, i t w i l l be zero throughout the flow. This i s true by Lagrange's Invariant theorem, which states that V x T T = constant throughout a flow. (35) The term "regular flow" i s due to Gabor v , and refers to the assumption that the electrons are emitted from the cathode with zero v e l o c i t y . The theory of space-charge flow can also be derived (35) under more general conditions. Gabor showed that skew congruent flow i s possible i n the presence of an externally applied magnetic f i e l d , provided that at the cathode the magnetic (36) f i e l d has no normal component. K i r s t e i n v ' derived the theory f this case* and found some new solutions, which were b a s i c a l l y (37) simple extensions of e l e c t r o s t a t i c ones. Pease ' extended the theory to include time—dependent flow. These more general formulations are not needed for our purposes, since i t i s our ultimate aim to study the applications of the theory to electro-s t a t i c electron guns. 2;2 Fundamental Theory For an e l e c t r o s t a t i c f i e l d Maxwell's equations are \7x E = 0 (2.1) V . D = p (2.2) and D = eE . (2.3) 13 These equations are solved by E = -V<£> (2.4) where $ i s the e l e c t r i c p o t e ntial. Substituting expressions (2.3 and 4) into (2.2), Poisson's equation i s obtained* namely V 2 < £ = -p/z . (2.5) The current density J i s given by "J = piT (2.6) and the time independent form of the continuity equation i s V.J = 0 . (2.7) By considering the Newton force on each electron, i t follows that where Tj = , the charge-to-mass ratio of an electron, a positive quantity. Equation (2.8) has time as a parameter, and this w i l l be eliminated next. The complete time d i f f e r e n t i a l d/dt applied to a dynamical variable X i s | f = (W7)X +|| . (2.9) Since the electron motion i s taken to be steady—state congruent flow, X i n equation (2.9) can represent the v e l o c i t y of an electron, and a t ~ u and hence | f = ( W 7 ) ? r . (2.10) 14 But 6r.V)ir = jsjfav) - x <®xv) . (2.11) From (2.10 and l l ) therefore dt ~ 2 Combining this r e s u l t with (2.4 and 8), | ( V U 2 ) - IT x (Vx5) =TjS7<$> or tf[§ir2 -770]= ^ ( y x ^ ) . ( 2 . i i ) If the electrons start with equal energy from a uni-potential cathode)i then VxlT= 0 and equation (2.12) can be integrated to give \/V2 -T)<&= a constant (2.13) which i s independent of the trajectory chosen. If the electrons start from rest at a zero-potential cathode, this constant i s zero » A step of fundamental importance was taken i n going from equation (2.12) to (2.13). Equat ion (2.12) was s t i l l concerned with the trajectory traced out by a single electron, whereas equation (2.13) applies throughout the flow, because the constant of equation (2*13) i s independent of the trajectory chosen,. If we do not choose V x l ^ ^ 0, i t i s found that \V2 -Tj<&= C , (2.14) 15 where C i s a constant for the motion of the pa r t i c u l a r electron considered • This result i s readily obtained by taking the dot product of V and equation ( 2 , 8 ) , and substituting (2,4) nr* fjf= +r)Gr&)<i> . (2.15) In a frame of reference moving with the electron, the potential v a r i a t i o n i s found from equation (2»9) to be 1^= (TrA7)<£ . (2.16) I f , f i n a l l y , equations (2.15 and 16) are combined, and the result i s integrated, the r e l a t i o n (2.14) i s obtained for the electron considered. For the case V x l T — 0 , i t i s possible to express the v e l o c i t y as the gradient of a scalar potential function or= w ( 2 . 1 7 ) where ¥ i s called the action function. It i s apparent that sur-faces of constant action are orthogonal to the l i n e s of flow. The equations to be s a t i s f i e d by the flow are thus Poisson's equation i n free space V 2 0 = ~p/e (2.5) D e f i n i t i o n of current density J =pV (2,6) Continuity equation V.J = 0 (2.7) Conservation of energy |"7T2 -7]<3>= 0 (2.13) Action function r e l a t i o n IX =^1 . (2*>17) These equations w i l l be referred to as the space—charge—flow In fact C i s the Hamiltonian, the t o t a l energy of a p a r t i c l e , which would be di f f e r e n t for p a r t i c l e s released from rest at di f f e r e n t equipotentials. 16 equations. Equations (2.13 and 17) may be combined to give the Hamilton-Jacobi equation | ( W . V V ) -T)& = 0 . (2.18) In beams i n which the space-charge causes a negligible perturbation of the e l e c t r o s t a t i c f i e l d , p can be set equal to zero. The potential now i s determined only by the boundary conditions, and to obtain an electron flow solution i t i s necessary to s a t i s f y equations (2.5 and 18) only. Yhen p ^ 0, i t i s possible to combine the required equations so that an equation i n V or V alone r e s u l t s . From (2.5 and 18) p = - s Q V 2 |^ VW. V W If this i s combined with (2.6, 7 and 17)? then V.[(V¥)V 2(7¥.V¥)] =0 . (2.19) The space-charge flow must s a t i s f y this fourth-order, third-degree equation i n W. Once a solution for W obeying (2.19) has been found, the other variables of the flow are also defined and can be obtained from the space-charge-flow equations. By combining the f i r s t four of the space-charge-flow equationsj there results V . ^ ^ C z f o i r ) ] = o which can be rewritten as ( V 2 i r 2 ) V . 7 T + v. v ( v 2 v-2) = 0 or V,7T = - \ o ^ ^ ( V 2 ^ 2 ) . (2,20) Equation (2«20) can be expressed as Vi - i r = -nr* V ( i n v2/w2) . (2.-2.1) Since equation (2«17) i s equivalent to the condition V x l T = 0, the vector i d e n t i t y V x (Vxir) =v(v.ir) -v 2 ;zr becomes S7{V.V) = S7 21T . (2.22) Substitution of (2«2l) into (2.22) gives the desired r e s u l t , (2.23) y 2^r= _ y |or. V ( i n y 2 i r 2 ) This equation does not apply i n the absence of space—charge« To obtain realizable space-charge-limited flows, equations (2»19.or 23) must be solved under boundary conditions <& = 0* § r ^ = 0, and — - oo (2.24) 3 n 3 n 2 at the cathode* where " ^ — " indicates d i f f e r e n t i a t i o n normal * 9n to the cathode surface. 2s3 Methods of Solution The Complexity of equations (2.19 and 23) has so far precluded t h e i r being solved d i r e c t l y except for the simplest (38 ) of cases. For instance, Spangenberg solved equation (2.19) for p a r a l l e l r e c t i l i n e a r flow. T r i a l and error approaches are not 18 l i k e l y to produce useful solutions on account of the s e l f -consistency requirement and the boundary conditions. Thus, i f a pa r t i c u l a r form of *Vor ¥ i s assumed, the potential i s immediately defined by the equation for conservation of energy (2.13). However* the potential has to s a t i s f y Poisson*s equation also* and with t r i a l and error procedures i t i s d i f f i c u l t to s a t i s f y both requirements at once. Realizable space-charge-limited flow solutiahs have previously been obtained by using coordinate systems that made one of the variables of the flow a function of only one coordinate. ¥ a l k e r ^ ^ , M e l t z e r ^ ^ , and R o s e n b l a t t s e t up the equations so that the lines of flow lay along one coordinate. (39) ¥alker ' also found solutions for space-charge-limxted flow between two i n c l i n e d planes and between two cones with coinciding v e r t i c e s . In these l a t t e r solutions the potential i s a function of only one Variable* M e l t z e r ^ ^ found a realizable solution i n which the l i n e s of flow are concentric c i r c l e s * K i r s t e i n ejt al_. assumed an action function of the form 3 ¥ = V i(q.) , i = 1 where the q^ represent curvilinear coordinates. It then became possible to use the method of separation of variables to solve the space-charge flow equations i n Cartesian^ c y l i n d r i c a l polar, spherical polar, and equiangular s p i r a l coordinates* These solutions correspond to a x i a l l y symmetric curvilinear flows originating from c y l i n d r i c a l and conical cathodes* and also 19 to planar cu r v i l i n e a r flows from an equiangular s p i r a l cathode, from two i n c l i n e d planes, and from a c i r c u l a r - s e c t i o n cathode, Harker and G o l b u r n ^ ^ devised a stable numerical method to obtain flows with axial symmetry; this method i s also applicable to planar flows. In this method, an analytic form of cathode shape and cathode current density are assumed. This then makes i t possible to set up the space-charge-flow equations i n hyperbolic formf by making an analytic continuation into the complex domain. Since hyperbolic d i f f e r e n t i a l equations are mathematically stable when solved by f i n i t e - d i f f e r e n c e methods, the space-charge—flow equations can thus be numerically integrated away from the cathode i n discrete steps. The problem of electron motion i n e l e c t r o s t a t i c f i e l d s and with negligible space-charge effects has been studied extensively. Goursat^-^ showed that i f the potential <3? has the functional form 3 <P = S Cp^ O.U.) (2.25) i = 1 the Hamilton—Jacobi equation i s integrable by separation of variables. The termCp 1 1 i s the f i r s t row of a matrix calle d a Staeckel matrix. The elements of a Staeckel matrix are functions of the coordinates q^ alone. I w a t a ^ ^ , assuming the functional form (2.25), found e l e c t r o s t a t i c f i e l d s s a t i s f y i n g the Laplace equation for the eleven coordinate systems of Staeckel. In this chapter the approach to the problem of determining soluti@ns of the space-charge-flow equations i s to assume an action function or a potential function that i s the 20 sum of terms. Two cases are considered: Case I: Action function assumed to be of the form V = V ]_(q 1) + V 2 ( q 2 ) and the potential of the form f U i * ^ A 1 1 ( q 1 ) A ] _ 2 ( q 2 ) + A 2 1 ( q 1 ) A 2 2 ( q 2 ) (2.26) Case l i s Potential assumed to be of the form * = * ! < « . ! > + * 2 ( 4 2 ) and the action function of the form »(2,27) V = B 1(q 1) B 2(q 2) where (q^» q_^) are orthogonal curvilinear coordinates. Motion with negligible space-charge effects i s treated f i r s t . With W and <£of the form assumed i n either Case I or I I c o o r d i n a t e systems are then found for which the Hamilton-Jacobi and Laplace equations are separable. The solution of the motion i s then extended to the space-charge domain by assuming that one term of the action function and the potential function remain as determined previously i n the absence of space-charge. When this i s possible, the second term of <$> and W i s then determined by a complete d i f f e r e n t i a l equation. In Section 2:4 the theory i s f i r s t formulated i n Cartesian coordinates. In this coordinate system the functional forms of W and<$> are assumed to be those of equations (2.26a and 27a) . In Cartesian coordinates, equation (2»26b) reduces to the simpler form (2.27a), 21 In Section 2s5 the separability conditions are obtained for two-dimensional flow with negligible space-charge effects* These conditions are i n terms of assumed functional forms of the metrical c o e f f i c i e n t s "h^" of general orthogonal curvilinear coordinates. Case I i s treated f i r s t and i s i l l u s t r a t e d by two examples, formulations i n logarithmic s p i r a l , and i n polar coordinates. It was not found possible to extend these two examples to include space-charge effects by the methods mentioned. The functional forms of Case II are studied next. Two more examples follow* i n which logarithmic s p i r a l , and polar coordinates are again used. These l a t t e r solutions are extended to the space-charge domain. Certain of the solutions obtained by the Case I and II formulations have been obtained by other investigators by independent means, and these w i l l be indicated. 2 84 Solutions, i n .Cartesian Coordinates by the Method of Separation  of Variables 2s4sl Electron Motion with Negligible Space—Charge Effects It was shown i n Section 2s2 that i n the absence of space-charge e f f e c t s , electron motion i n e l e c t r o s t a t i c f i e l d s i s described by the Ha.mi.lton-Jacobi equation ( W ) 2 - 2T)&= 0 (2,18) where the potential has to s a t i s f y the Laplace equation V 2 < £ = 0 . (2.5) We s h a l l assume a solution of these equations of the functional forms and ¥ = ¥ 1(x') + ¥ 2(y') + ¥ 3(z') 22 (2*28) 45 =^1U\) + «3 > 2 ( y t ) + Cj>3(^*) , (2.29) ¥ith <$>of the form (2,29), the Laplace equation (2.5) separates into three ordinary d i f f e r e n t i a l equations* ¥hen these are solved, i s found to be ^ + G 2y' 2 - ( C ^ C 2): }2 + d,x' + d 0y* + d^z* + d 4 * (2.30) In general, equipotential surfaces are thus hyperboloids* It i s convenient t© change the o r i g i n of coordinates to the centre of symmetry of the f i e l d * Equation (2.30) then becomes C l X 2 + C 2 y 2 - (C1 + C 2)z 2^| + CJ)Q . (2*31) The Cartesian coordinate system i s one of the eleven coordinate systems of Staeckel, and the separable form of the potential function described by equation (2.3l) was noted by I w a t a ^ ^ . The motion of an electron released from rest at an arbitrary point ( X q , y Q , Z q ) on the surface <$> = 0 w i l l next be obtained. If equations (2.28 and 3l) are substituted into the Hamilton-Jaeobi equation (2.18), we obtain dx d¥ 2(y) N T)C2y< 'd¥ 3(z) dz + 77(C 1+C 2)z 2 ^ ^ o 0 23 This equation can be separated into the three equations 12 dx -7} + K 1 + C l X 2 ) = 0 dV~(y) - , 9 A - 1 + K 2 + C 2 y 2 0 =0 dy > (2.32) d V 3 ( z ) - 7 ^ - (K, where ^o 2 + K 2 ) - ( C l + C 2)i 3 = 0 i = 1 and and K 2 are separation constants. Each of the three separate equations (2*32) constitutes a conservation theorem of the motion. Equations (2.32) can be immediately integrated between the l i m i t s X q and x, y Q and y, and Z q and z respectively* where ( X Q 9 y Q , ZQ) i s an arbitrary starting point on the zero equi-po t e n t i a l , and (x* y> z) i s any point on the trajectory of an electron released from (x , y , z )* However, as we are 0 0 0 7 interested i n obtaining the t r a j e c t o r i e s as well as the action function, i t i s more convenient f i r s t to eliminate the separation constants i n (2»32). The result w i l l i n any case be the same. It w i l l be recalled that the condition for writing the Hamilton-Jacobi equation (2.18) was that VxQX = Oy and that this condition i s r e a l i z e d for a regular beam; that i s . when the electrons are emitted by the cathode at zero v e l o c i t y . Prom equation (2»3l) we see that the cathode surface, Cj>= 0, i s described by 1 2 G i x o 2 + V o 2 - <ci + C 2 K : 3> 0 (2.33) The condition of zero i n i t i a l v e l o c i t y i s 24 dV 1(x) d¥ 2(y) dV 3(z) dx dz x o y 0 0 ¥hen these i n i t i a l conditions are applied to (2»32), we obtain 0 <£> + K, + C nx 2 1 1 1 o ^2 + K2 + V e " = 0 % - ( K 1 + K 2) - ( C 1 + C 2 ) z o 2 = 0 , and when these equations are substituted back into (2.32), there results 2 'dW^x) dx f p $ z r ) - ^ c 2 ( y 2 - y o 2 ) = 0 > (2.34) ^d¥3(zj dz + C 2 ) ( z Q 2 - z 2) = 0 Equations (2.34) describe the action ¥ at an arbitrary point (x* y, z) on the trajectory originating from a point ( X q , y Q , Z Q ) on the cathode. The electron v e l o c i t y i s , from equations (2.17 and 34 ) , i V V ^ x 2 - x Q 2 ) , ± \ ^ C 2 ( y 2 - y Q 2 ) , +V^((\+ G 2 ) ( z o 2 - z 2) (2.35) The signs of the v e l o c i t y components depend on the f i e l d constants C-^  and C 2, and on the instantaneous position of the electron on i t s trajectory. The trajectory equations can be obtained d i r e c t l y from equation (2.35); 25 dx ~2 27 x - x Q ) dz (2.36) Equations (2«36) have the solutions x = X q cosh ( ]/7) C 1 t) y = y Q cosh ( \J7) C 2 t) > (2.37) z = Z q cos (^(C 1+ C2)^ t ) v i t h time t as a parameter. This may be eliminated to give 1 z = z cos o cosh-"'" j * — cosh -1 is-O/J (2.38) The motion i s seen to be o s c i l l a t o r y , with an amplitude equal to the i n i t i a l coordinate. A sketch of the cathode surface and a trajectory for two characteristic cases i s shown i n Figures (2-la and b). The action ¥ along the tr a j e c t o r i e s described by equations (2.37 or 38) can be obtained d i r e c t l y upon integration of equations (2.34) between the l i m i t s X q and x* y Q and y, and z ( and z respectively, and taking their sums 2 2 2 ,-1 /x \ x - x - x cosh (—) o o vx / o + lv¥^|y\/y 2 - y02 - y02 c o s h - 1 ^ ) • iWci+ c 2 } \z " f T 1 2 2 r - l / Z A z - z cos (•—) (2.39) 26 Because of the o s c i l l a t o r y nature of the electron motion, trajectory cross—over occurs. The function ¥ i s therefore multivalued. It might be supposed that on account of the cross-over of t r a j e c t o r i e s i t i s not possible to construct a one-parameter family of surfaces of constant action perpendicular to the flow l i n e s . This* however, i s not so5 because the electron motion s a t i s f i e s the condition Vx17 = 0, i t i s i n p r i n c i p l e s t i l l possible to construct this orthogonal family of surfaces (see Appendix B)* Gare must be taken, however, to associate the correct branch of ¥ with the appropriate t r a j e c t o r i e s (or sections of t r a j e c t o r i e s ) . This point i s i l l u s t r a t e d i n Figure (2-la)» Sketch of a trajectory and of the surfaceO = 0 when the l a t t e r i s a hyperboloid of one sheet. For t h i s case C 1>0, G 2>0, <$> < 0 Electron Trajectory 27 Figure (2-lb). Sketch of a trajectory and of the surface<$>= 0 when the l a t t e r i s a hyperboloid of two sheets. For t h i s case C,< 0, C n< 0,<P < 0 1 £ o Figures (2-2a and b) for a two-dimensional case, electron motion from a right-angled cathode. This motion results for the special case when the f i e l d constants <J>o, and or are set equal to zero i n the solution just obtained. From equations (2,37), the trajectory equations for this case are x = X q cosh (\J'J]C1 t) z = z Q cos (\/7]C^  t) 28 and the slope along the tr a j e c t o r i e s i s dz = (ZJ\ s i n ( y ^ i t ) d x VV sinh(^7C^ t ) In Figure (2-2a) are sketched several t r a j e c t o r i e s originating from the lower half of the cathode. An envelope which i s tangential to these t r a j e c t o r i e s prior to th e i r f i r s t downward deflection i s also shown. Similar envelopes occur for the second, t h i r d and subsequent r e f l e c t i o n s , and the second and t h i r d envelopes are indicated. These envelopes are straight l i n e s , as may be readily observed from the trajectory equations. It i s further apparent, from symmetry considerations, that conditions which are a mirror image of those just discussed w i l l p revail for electron t r a j e c t o r i e s originating from the upper half of the cathode. Figure (2-2a). Sketch of electron motion from a right-angled cathode 29 F i r s t Envelope Second Envelope ^ ^ o e c o n d . an Third Envelope Axis of W Subsequent Q„ryirr,„ + w r r Envelopes gyjnraetry \ * c — Third Envelope Second Envelope F i r s t Envelope Jonstant jower Half Cathode Figure (2-2b)» Sketch of the electron motion of Figure (2-2a) when the sheets are unfolded Since the t r a j e c t o r i e s are tangential to the trajectory envelopes, action surfaces must be perpendicular to the l a t t e r . A l i n e of constant action i s sketched i n Figure (2-2a) } i l l u s t r a t i n g the multivalued nature of ¥. Although l i n e s of flow and lines of constant action are orthogonal families of curves, they cannot be represented i n this example by the l e v e l l ines of (39) conjugate harmonic functions v '. The "complex v e l o c i t y p o t e n t i a l " W(x,z) + i A l / ( x , z ) , where ^(x,z) i s the stream function, i s there-fore not analyt i c . Nevertheless, the Riemann-surface concept of generalizing the (x,z) plane to a surface of more than one sheet, so that the multivalued "complex v e l o c i t y potential" has only one value corresponding to each point on that surface, may p r o f i t a b l y 30 be used here. This i s i l l u s t r a t e d i n Figure (2—2b)» Consider the t r a j e c t o r i e s that originate from the lower half flf the cathode The f i r s t sheet contains the section of these t r a j e c t o r i e s between the cathode and their point of tangency to the f i r s t trajectory envelope^ and i t also contains the f i r s t branch of ¥. The second sheet contains the section of the t r a j e c t o r i e s between the f i r s t and seeond /envelopes and the second branch of ¥, etc., resulting i n a surface on which the flow i s single—valued. The surface s i m i l a r l y obtained for electron motion originating from the upper half of the cathode can be conjoined to the f i r s t sur-face as shown i n Figure (2-2b). 2:4.2 Electron Motion under Space-Charge Conditions The approach of Sub-section 2:4:1 w i l l now be extended to the space—charge domain. Solutions of the action function ¥ and the potential <£> w i l l once again be sought of the form ¥ = ¥ x(x) + ¥ 2(y) + ¥ 3(z) O = ^ ( x ) + <£>2(y) + 0 3 ( z ) . > (2.40) It w i l l be assumed that ¥^(x) and Y 2(y) are unaltered by the presence of space—charge, so that from equation (2*39) x - x - x cosh (—) o o xx ' o _ W 2(y) = ^ J y ^ 2 - y/ - y^cosa-1^) > (2.41) The charge density p i s therefore allowed to be only z-dependent* From equations (2*41) the v e l o c i t y i s 3 1 When thi s i s substituted into the Hamilton-Jacobi equation, we obtain for the potential CP = ^ ( V W ) 2 = | c i (x 2- x Q 2) + | c 2 ( y 2 - y Q 2 ) + (2.43) which i s seen to be of the desired form (2.40b). Substituting this r e s u l t i n turn into the Poisson equation produces i d2/aw 3(z)\2-p ° 1 + ° 2 + ^ 7 d z 2 V d» Using the above equations for p and 'W , the remaining space-charge flow equation to be s a t i s f i e d , the continuity equation, becomes V . ( p ^ ) = -e ox7. „ ±]/rjC2(y2- y Q 2 ) , dW3(z) dz X = 0 which can be rewritten i n the form 21 C,+ C„+ x d2/dW 3(z) N 1 2 2 7 ? d z 2 V d z (x - x Q ) 2 1 + •V lW3(z)N dz 1 + C 2 2 W l d z , S = 0 . (2.44) For t h i s equation to be an equation i n z only, i t i s necessary that x = y = 0 . o J o (2.45) The equation that must be s a t i s f i e d by W^(z) i s therefore, from (2.44 and 45), 32 T)(C1+ C 2) 4- [ \ I + ' 3 + T7(C 1 + C 2) dz' d 2V 3(z) d z 2 dz dz" + ftp. /av3(z)\ /d 2v 3(z)\ /d 3v 3( z )> 3 2 ^d 2¥ 3(z)\ //d¥ 3(z) N^ /^d 4¥ 3(z)^ dz' dz dz = 0 . (2*46) The dependent variable i s missing i n (2.46); i f we set d¥-(z) nr ( z) = —2-— z dz the order of (2.46) can be reduced by one, resulting i n 7?( C l+ C 2) + 4 nr(zY dz 4 2 i r (z)N dz' a. / i r(z) + ^ C l + °2> - d t — 4i£(z)( ^ ( z ^ ^ z ^ / d / i r ( z ) \ 3 dz dz' dz 4 3 O T ( z ) N dz" 0 . (2.47.) For space—charge flow to be possible according to our assumptions.- equation (2.47) must be s a t i s f i e d . Let us try a solution of the form 1T(z) = Dz m . (2.48) z Then (2.47) beeomes (TjG^ 7]C2)()frjcl +/77<^) + (?7C1 +?7C2)mJ)zm-1 * ( V ^ + ^ 2 ) m ( 2 m l ) D 2 z 2 ( m " l ) + D3m(2m - l)(3m *. 2 ) z 3 ( m " ; L ) = 0 (2.49) 33 Equation (2.49) i s s a t i s f i e d under the following conditions! 2 2 m = 1-j. Cj ==. , C 2 = Sj- , and D = -(A + B) . (2.50) The z-component of v e l o c i t y i s therefore, from (2.48 and 50) 0X(z) = -(A + B)z . (2.51) z Combining this r e s u l t with equations (2.42, 45 and 49), the ve l o c i t y of the flow i s or Ax, By, -(A + B)z (2.52) The e l e c t r o s t a t i c potential of the flow i s obtained from equations (2,43, 45, 50 and 51), and i s A 2x 2+ B 2y 2+ (A 2+ B 2 ) z 2 . (2.53) Equipotential surfaces are seen to be concentric e l l i p s o i d s . The charge density of the flow i s , from Poisson's equation and (2.53), 2e p = - -j~ (A 2 + B 2 + AB) . (2.54) The charge density i s thus constant throughout the motion. The action function V i s readily found to be * = \ Ax 2 + By 2 - (A + B ) z 2 which i s harmonic. The equations of the traj e c t o r i e s are easily obtained from equation (2.52)t t r>x ny dy_ 1 y - (A + B) and hence the tr a j e c t o r i e s are at the intersections of the two families of surfaces 1 1 " K l B y 1 „A 34 "^ T~ = K2 A+B The t r a j e c t o r i e s l i e i n the surface of the rectangular hyper-boloidal family xyz x y z = constant o o o When either A or B i s set equal to zero, or B = -~A. the motion becomes planar* The equipptential surfaces become concentric cylinders, and the tr a j e c t o r i e s become plane rectangular hyper-bolae. A sketch of this flow i s shown i n Figure (2-3). Figure (2-3). Hyperbolic space-charge flow 35 The space—charge flow according to equations (2.52^54) has (33) also been discovered by Meltzer ', by a d i f f e r e n t method. Another solution of the d i f f e r e n t i a l equation (2*47) can be obtained for the special case = = 0. If equation (2,48) again be substituted into (2.47), there results m(2m - l)(3m - 2 ) z 3 ( m - l ) = 0, which i s s a t i s f i e d for m = 0, l/2 and 2/3. It i s readily v e r i f i e d that these three values correspond to p a r a l l e l r e c t i l i n e a r electron motion (a) i n the absence of an e l e c t r o s t a t i c f i e l d * (b) with negligible space—charge effects from the plane z = 0, and (c) that i s space-charge^limited and originates from a cathode at z = 0, respectively* a l l known cases. No further solutions of equation (2*47) have been obtained. The hyperbolic flow solution of equations (2.52—54) brings out an important point about the i n i t i a l conditions that are required for space—charge-limited flow from a zero^poteniial; (36) cathode. K i r s t e i n v ' states: "If we require that the motion be physically realizable under spacer-icharge—limited conditions from a zero-potential cathode K, then i t i s required that* on K, 3^? and i?-^be zero, where n i s the unit vector normal on y to K.» Prom equation (2.53) we see that for hyperbolic flow the conditions mentioned by K i r s t e i n are s a t i s f i e d at the o r i g i n . Yet the flow i s not due to a cathode at the o r i g i n . The reason for this d i s p a r i t y i s that the charge density p for this solution i s constant throughout the motion, whereas for space-charge-limited electron flow originating from a zero potential cathode D —v -oo 3 6 at the cathode* Prom Poisson's equation the equivalent condition for <S> i s —~—y--OO at the cathode* dn2 This i n i t i a l condition thus needs to be applied at the cathode i n addition to the conditions ^ an In this section electron-flow solutions were found by formulating the theory i(h Cartesian coordinates under the assumption that ¥ a n d ^ could be represented as the sum of terms, each term being a function of only one coordinate. In the next section we w i l l adapt this approach to a formulation of the theory i n orthogonal curvilinear coordinates, and determine the conditions for separation of variables to occur. 2%5 Solutions i n Plane Curvilinear Coordinates by the Methodof,  Separation of Variables 2.5:1 Action Function of the Form "W-^q^ + ¥ 2 ( q 2 ) " (a) Conditions for separation of variables Let (q-^, q 2) be an orthogonal curvilinear coordinate system i n which an i n f i n i t e s i m a l l i n e element "d^L " i s described by 2 d l 2 = ^ \ 2 ^ ± 2 » 1 = 1 where the h^ are the metrical c o e f f i c i e n t s , defined by 2 U \2 1 V 8 q i / ' V 3 c l i/ 37 The gradient of the action function V can then be written as 2 VW i = 1 L i h. 3 q . ' and the Hamilton-^Jacobi equation i s therefore 1 / 3¥ h. 2 V a ( l i i - 1 " The Laplace equation becomes h l h 2 9<Ii V h i 3 _ r i ai.N 9 q 2 l h 2 9 q 0 Let i t be assumed that W i s of the form V = V 1 ( q 1 ) + V 2 ( q 2 ) . Equation (2.55) can therefore be rewritten as (2.55) (2.56) (2.26a) (2.57) Ve must next determine the form of the potential <$> for which equations (2*56 and 57) w i l l separate. If we assume that the hu are of the functional form h A = f ( q l f q 2 ) f i l ( q 1 ) f . 2 ( q 2 ) (2.58) then i t i s clear that the left-hand side o# equation (2.57) can be separated by multiplying i t by ^ f ( q 1 ? q 2 ) f 1 2 ( q 2 ) f 2 1 ( q 1 ) j 2 . If this i s done* there results ^ f2iS^V d M 2 f 1 2 ( q 2 ) \ 2 / d V 9 f _ ( d q 2 ) = 2 ^ [ f < * l » * 2 > W * 2) f 2 1 ( * l > ] (2.59) 38 To complete the separation of the Hamilton-Jacobi equation, i t i s required that the right-hand side of equation (2.59) be of the form g ^ ) + g 2(q 2) = 2 ? 7 ^ ( q 1 , q 2 ) f 1 2 ( q 2 ) f 2 1 ( q 1 ) <P . (2.60) The potential<$> must s a t i s f y (2.56); l e t the solution be of the form A n ( q 1 ) A 1 2 ( q 2 ) A 2 1 ( q 1 ) A 2 2 ( q 2 ) 2 2 * f (^r q.2) f (<!!» <12) (2.26b) From equations (2*26b and 60) the conditions for separating the variables such that the l e f t side of equation (2.60) i s s a t i s f i e d are therefore l l (a) f122(<l2) = A 1 2 ( q 2 ) ° r / a n d ( b ) f 2 1 2 ( ^ l ) = 2t (a) f212(ci1) = A ' 2 1 ( q i y o r / a n d (»>) f 1 2 2 ( ^ 2 ) = A 2 2 ( q 2 ) (2.61) where the f. . and the A. . are as defined by equations (2.58 and 1» 3 1»1 26b) respectively. At least one part of each of these two conditions must be s a t i s f i e d for the Hamilton-Jacobi equation to be separable when W i s of the form (2.26a) and lu of the form (2.58). I f , f o r example,the solution of the Laplace equation i n a p a r t i c u l a r coordinate system i s of the form of equation (2.26b) and s a t i s f i e s conditions l(a) and 2(a)» then the potential i s of the form 3 9 A n ( q i ) f 1 2 2 ( q 2 ) f 2 ( g i , d 2) f 2 1 2 ( q i ) f 2 ( q i > : q 2) and the Hamilton-nJacobi equation i s reduced to quadratures. (b) . Examples Example #1t Solution i n Plane Polar Coordinates For plane polar coordinates the metrical c o e f f i c i e n t s are h l = X> h2 = r • Referring to equation (2*58)* i t follows that and f(r,©) = f n ( r ) = f 1 2(°) = f 2 2 ( 0 ) f 2 1 ( r ) = r . = 1 > The separability conditions for the Hamilton-Jacobi equation are thus seen to be, from equation (2.6l), Is (a) A 1 2 ( 0 ) = 1 or/and (b) A - ^ r ) = ^ r 2s (a) A 2 1 ( r ) = ~ or/and (b) A 2 2.(0) = 1 • ¥ith<$• of the assumed functional form of equation (2.26b) the Laplace equation becomes, i n plane polar coordinates, A 1 2 ( 0 ) dA i ; L(r) d < 5A 1 1(r) ' +' r ~—' dr dr + A 2 2(G) d A 9 1 ( r ) d % (r) • + r ^ dr dr' , A l l ( r ) d \ 2 ( 9 ) , A 2 1 ( r ) * \ 2 ^ Q d©' ¥hen conditions l ( a ) and 2(a) are t r i e d i n this equation*- separation 40 of variables i s achieved, and there results 3 / f r A n ( r ) d ^ j X r j N r I — — + r 1 = a and dr cTA o o(0) dr' 22 dO 2 + 4 A 2 2 ( 0 ) = -a . (2.62) here--"a" i s a constant. i s of the form <p = A , , ( r ) + ±^ A „ ( © ) The solutions of the separated d i f f e r e n t i a l equations (2*62) are 1 —2 A i ; L ( r ) = 4 ar" + b } l n r + ^ A--(©) = - T a + b 0 sin (20) + c, cos(2©) , "22x"' ~ 4 a r "2 " " ' "2 where a, b^ and c^ are constants. The potential ^ i s therefore ($> = c 1+ b 1 l n r + r~ 2|^b 2 sin(20) + c 2 cos(20)j (2.63) The e l e c t r o s t a t i c f i e l d described by this equation i s due to a l i n e charge and a double doublet at the o r i g i n . The orientation of the double doublet i s determined by the r e l a t i v e magnitudes of b 2 and c 2 ; this i s apparent i f i t i s considered that the l a s t two terms of equation (2*63) can be rewritten i n the form - 2 , f 2 2 r |/b2 + c 2 cos 2© - tan H — VC2> The constant c^ i n equation (2.63) allows us to assign a reference potential = 0 i n our case) to any equipotential of the f i e l d . The electron motion may thus be i n i t i a t e d at any of the equi-potentials of the f i e l d determined by the values of by* b 2 and c 2 by suitably adjusting the constant c-^ . Equation (2*63) was also 41 derived by I w a t a ^ ^ i by the method of G o u r s a t ^ ^ , The e l e c t r o s t a t i c f i e l d for the case when Cp = l n r + sin(2©) i s i l l u s t r a t e d i n Figure (2-4), A saddle point of potential i s seen to occur at the points r = y2, 0 = — . The saddle point of potential would not have occurred i f the f i e l d had been due to either the li n e charge or the double doublet alone. The motion of electrons i n this f i e l d has some very interesting properties, and these w i l l now be studied. If we l e t ¥ = ¥ r(r) + ¥ Q(0) , then the Hamilton—Jacobi equation (2.59) separates into two ordinary d i f f e r e n t i a l equations when (2.63) i s substituted? dr, 27?r 2(c 1+ b x l n r) = 2 Tj K and ^(2.64) 'd¥, 9. d© - 2Tj b 0 sin(2©) + c 0 cos (20) -277K where K i s the separation constant. Each of these two equations constitutes a conservation theorem of the motion, and they are i n this respect analogous to equations (2.32). From equation (2.63) i t i s seen that the coordinates (r Q,© o) of an arbitrary point on the zero equipotential are related by r o 2 ( e 1 + b 1 l n 1 r ( j ) + (^>2 s i n ( 2 9 0 ) + C 2 C O S ( 2 © q ) J =0 . At (r Q,© o) the v e l o c i t y i s zero; therefore aw r dr 1 **© (r o,© Q) = r d© = 0 42 43 If these i n i t i a l conditions are substituted into equations (2»64), we obtain ~ K + r 2(c,+ b-,ln r ) = 0 o 1 1 o K - ^>2 sin (2© Q) + c 2 cos (2© Q)^ = 0 . The d i f f e r e n t i a l equations (2»64) therefore become 2 dr J -*V .2 W c x ( r 2 - r Q 2 ) + b 1 ( r 2 l n r - r Q 2 l n r Q ) = 0 <d¥, © ,d© - 2 rj b 2 ( s i n ( 2 0 ) - sin (2©^)) + c 2 (cos (2©) - cos ( 2 © o ) | . (2.65) For an electron starting from rest at a point (r Q,© o) on the cathode* the v e l o c i t y at the point (r>©) on i t s trajectory i s , from equations ( 2 * 1 7 and 65)» i r 2 - r Q 2 ) + b ^ A n r - r / l n , ^|/(b 2(sin 2© - s i n 2© Q) + c 2(cos 2© - cos 2© ) o 1 (2.66) If we l e t and R © = © - \ t a n ^ ^ 2 V2J (2.67) then equation (2*66) can be rewritten i n the form 4 4 nx = R / T P _o_ i p ^ if4T](b 2 4- c 2 ) 2 , 2 /In R * — * l n R Q . + ^ \/sm ^ - s m ® R (b 1>0) . To permit the electron motion i n four-dimensional phase-space to be represented i n two dimensions independently of the magnitude of the constants b^, b 2 , ci a n < 1 c2» ^ *-s convenient to write the v e l o c i t y components (nT i n the form r w R ^ r = + l/ln R R <L o /47J ( b 2 2 + c 2 2 ) 2 = + 1/sin 2 ® - s i n 2 ® > (2.68) Equations (2*68) are i l l u s t r a t e d i n Figures (2-5a and b). It i s seen that an electron released from rest at a point ( R Q t © Q ) on the cathode o s c i l l a t e s about the line n% as the r a d i a l v e l o c i t y component increases i n magnitude. If the i n i t i a l r a d i a l coordinate R Q > 0»60653» the electron w i l l move away from the coordinate centre, while for R less than th i s value the electron x o w i l l move towards the centre. This d i v i s i o n of the tra j e c t o r i e s * For b^<0 the r a d i a l v e l o c i t y component can be written as nrT = ±f^j\ \/^2 l n R 0 " l n B * Figure (2-5a). Phase plot of the r a d i a l v e l o c i t y component of equation (2.68) ' 1 V i i f I 11 i l l -1% [ r 1 i.o L i / 4 7 7 ( b 2 + c2 ) 2 . -0.5 \ \ 1 1 1 X / ® ^ I I 1 1 1 1 i 1 T 1 J\ l i r i T I 1 1 ' \ v-i-o v y -o.5 y / / - 1 . 0 Figure (2-5b). Phase plot of the ©-component of v e l o c i t y of equation (2.68) 46 i s due to the saddle-shaped nature of the e l e c t r o s t a t i c f i e l d . For the e l e c t r o s t a t i c f i e l d i l l u s t r a t e d i n Figure (2-4) the normalized coordinates are R = r © = 0 - \ , J For this case the l i m i t i n g cathode radius dividing inward and outward motion i s r = 0.60653. and the electron o s c i l l a t i o n o occurs about the l i n e y = x» When the f i e l d i s due only to a li n e charge, so that l>2= c 2= 0, the equipotentials w i l l be concentric c i r c l e s * and the t r a j e c t o r i e s l i e along r a d i a l l i n e s . For thi s case* i t i s seen from the i n i t i a l conditions that the constant K = 0* so the ra d i a l v e l o c i t y i s simply 1% = y ^ T ^ l n R . For the special case when the e l e c t r o s t a t i c f i e l d i s due only to a double doublet, so that b^ = 0, the r a d i a l v e l o c i t y component of the motion i s , from (2.66), Thus when the f i e l d constant c^ i s greater than or less than zer the r a d i a l v e l o c i t y w i l l be outward or inward respectively, independent of the starting point on the cathode. When c^ i s ze however, the r a d i a l v e l o c i t y i s zero along the trajectory; an electron released from rest from any point on the zero equi-potential thus has the remarkable property of t r a v e l l i n g along an arc of a c i r c l e * From equation (2.63) the zero equipotential 47 of t h i s f i e l d can be deduced to be the straight lines © = + rm + t a n - 1 ^ ) ) Again for this special case the separation constant K i s zero, and the electron v e l o c i t y i s therefore % = ± 1/2 77 ( b 2 2 + c 2 2 ) 2 r o " 2 cos (2©) For the general case, the tr a j e c t o r i e s are described by dr nrg ~ rd© c-, (r 2^- r 2) + b, ( r 2 l n r - r 2 l n r ) b 2 ( s i n 2© - sin 2© Q) + c 2(cos 2© - cos 2© Q) which i n integral form i s dr c 1 ( r 2 - r o 2 ) + b 1 ( r 2 l n r ^ r Q 2 l n r Q ) 2 d© £b 2(sin 2©-sin 2© Q)+c 2(cos 2©-cos 2©Q)j 2 If use i s again made of the normalized variables of equations (2»67), these integrals can be rewritten as dR Ik R ^ 2 l n R - R Q 2 l n R Q \[2{\>22+. c 2 ) d © 2 y r ^ ^ s i n 2 ® where k = — - — - , b1 > 0 „ * sin ® 48 If R = I* the substitution o T R = eJ changes the integral i n r to: V 1 \IS ex*\hr + In r L where erf (^) i s the well-known error function, defined by erf A 2 / -x" e dx » In the integral i n © , |k| ^> 1. The physical significance of the amplitude of k being greater than unity i s that i t r e s t r i c t s the motions i n the ©-direction to the l i b r a t i o n type discussed e a r l i e r * The behaviour of the the ©-component of the electron v e l o c i t y i s quite similar to the well-known motion of a When b^ = 0 the integral i n r, 1^, integrates d i r e c t l y to give 1 V 0 c o f l cos -1 /ro^ For b^ •< 0 the substitution R = e r can again be used* 49 simple pendulum, discussed, for example, by Goldstein (47) Example #2t Solution i n Equiangular Spiral Coordinates The metrical c o e f f i c i e n t of an equiangular s p i r a l (or logarithmic s p i r a l ) coordinate system i s h = h, b^u + b 2v (2,69) where b^ and b 2 are arbitrary constants. If z = x + i y and y = u + iv , then z and w are related by the transformation (b1 - ib 2)w + b 2 V This equation i s rea d i l y expressed i n polar coordinates? b^u + b 2v 2 2 1 + b2 , 0 = b^v - b 2u (2.70) From the polar coordinates the multivalued nature of the (u,v) coordinate system i s readily apparent. Thus the coordinates and V = Vo +1 , 2, , 2 2nn 2n% b l + b2 (2.71) where (U O,V q) are any i n i t i a l coordinates and n i s any integer, refer to the same point i i i space. The range of (u*v) therefore needs to be r e s t r i c t e d so that the mapping of (u>v) i n the z-plane 50 i s single-valued. This w i l l insure that the potential expressed i n (u,v) coordinates w i l l be single-valued i n space* In Figure (2—6) i s shown a sheet of the equiangular s p i r a l coordinate system for which b^ = t>2 = 1, From equations (2,58 and 69) i t i s found that f(u,v) = 1 b, u f u ( u ) = f 2 1 ( u ) = e and f 1 2 ( v ) = f 2 2 ( v ) = e b 2v From equations (2.26b and 61), a possible form of therefore i s -2b-v -2b,u <$> = A i ; L(u)e ^ + e 1 A 2 2 ( v ) . (2.72) ¥hen t h i s functional form of <$> i s substituted into the Laplace equation (2.56), there results 9 2 < £ 9 2 0 -2b 2v • o + — = e 3 u 2 8v 2 d A,, (u) 0 2 + 4 b 2 A l l ( u ) du' +e -2b 1u d % 2 ( v ) dv' + 4 b l A 2 2 ( v ) = 0 This d i f f e r e n t i a l equation i s of separable form, and separates into the equations d 2A,,(u) . -2b, u U2 + 4b 2 2A i ; L(u) - ae 1 = 0 du a'A22(v) -2b„v dv 2— + 4b-L A 2 2 ( v ) + ae 0 H r-Figure (2-6). Logarithmic s p i r a l coordinates: u + v r = , 9 = v - u 52 where "a" i s a c o n s t a n t * The s o l u t i o n s of these e q u a t i o n s are A-^Cu) = c-^sin ( 2 b 2 u ) + d-^cos (2b 2u) + A 2 2 ( v ) = c 2 s i n ( 2 b 1 v ) + d 2 c o s ( 2 b 1 v ) -ae - 2 b 1 u 4 ( b 1 2 + b 2 2 ) ae - 2 b 2 v 4 ( b 1 2 + b 2 2 ) S u b s t i t u t i n g these r e s u l t s i n t o e q u a t i o n ( 2 . 7 2 ) , the e l e c t r o s t a t i c f i e l d i s -2b j r c-^sin ( 2 b 2 u ) + d 1 c o s (2b 2u) + e - 2 b 1 u c 2 s i n ( 2 b 1 v ) + d 2 c o s (2b^v) where c^, c 2» d^ and d 2 are c o n s t a n t s determined by the boundary c o n d i t i o n s . I t i s c o n v e n i e n t to r e w r i t e t h i s e q u a t i o n i n the form 3> o ^ -2b-v / , e,\ c, + d, e ^ cos f (2b 2 u ) - t a n " 1 ( ^ ) J '1 "1 + , / c 2 2 + d 2 2 e \ -2b., u 1 cos ((2\v) - t a n ~ 1 ( ^ - ) ) . (2.73) To u n d e r s t a n d the p h y s i c a l b e h a v i o u r of t h i s e l e c t r o s t a t i c f i e l d , c o n s i d e r f i r s t the f i e l d Cj> d. u e to the f i r s t term o f (2.73) a l o n e . That i s, f = \/ C ; L 2 + e 2 2 cos ( 2 b 2 u - tan"'1(^ A . The zero e q u i p o t e n t i a l s of are the l o g a r i t h m i c s p i r a l s b 2 U = 2 -1 C l ^ + nn + t a n (^—) where n i s any i n t e g e r * Thus, i n the coordinate system i l l u s t r a t e d i n F i g u r e (2—6) f o r which b^ = t>2 = 1? i f ve set c^= 0 the zero e q u i p o t e n t i a l s of are the s p i r a l s For a g i v e n v, when (2*74) i s a maximum* I t can a l s o be observed from equations (2*70) that when b2v—=>-—oo*- while u i s kept constant, r — > • 0* I f we l e t b,>v — > - oo along a u - s p i r a l d e s c r i b e d by equation (2*74), t h e n * ^ - + oo » For a gi v e n v^ when i s a minimum, and f u r t h e r when b,,v—>- - oo along a u - s p i r a l d e s c r i b e d by t h i s equation, then —oo o The behaviour of as b 2 v— > - oo i s seen to be s i m i l a r to the behaviour of the p o t e n t i a l i n the v i c i n i t y of a double doublet. We note from equations (2»7l)* however* that as b 2 u v a r i e s over an i n t e r v a l 2n» the coordinates (u*v) e n c i r c l e the z—plane N times, where Thus* f o r the coordinate S y s t e m of Fig u r e (2-6)* f o r which b, = b0 = 1* the Biemann surface covers two sheets. 54 The e l e c t r o s t a t i c f i e l d described by the second term of equation (2,73) i s <4>2 = \jc22 + d 2 2 e 2 b l U cos ^ b - j V - t a n _ 1 ( ^ . The functional form of i s seen to be the same as that of when the variables u and v are interchanged. Thus, analogously to zero equipotentials of ^J-l, are the constant v - s p i r a l s V = l ( l + ™ + t a n" 1 (a7 }) and <$>2—> + 0 0 when b^u —>• - 0 0 along the s p i r a l b ^ = \ ^ 2nn + tan" 1!^ 2-)) i while <$>2—> - 0 0 when b^u—>• -00 along the s p i r a l 1 / -1 c 2 ^ b^v = 2 f ix + 2mx + tan (^~)J » As b^v varies over an in t e r v a l 2%, the coordinates (u»v) encircle the z—plane M times, where 2 2 V + b2 b l Since u and v are orthogonal coordinates, the sp i r a l s of constant u and of constant v rotate about the or i g i n of the z—plane i n opposite d i r e c t i o n s . The potential ^  described by equation (2,73) i s thus the sum of two potential functions and l$ > 2» whose general properties are the same but are oppositely directed, A section of the equipotential ($>=() i s shown i n Figure (2-6) for 55 the case i n which the constants of the e l e c t r o s t a t i c f i e l d are d-^  = c 2 = 1 and c^ = d^ = 0 . The electron motion i n this e l e c t r o s t a t i c f i e l d w i l l be studied next. If an action function of the form ¥ = ¥ (u) + ¥ (v) u v i s assumed, the Hamilton—Jacobi equation (2.57) can be written as and u ."du" ~dv 2b u / \ - 27] e ( c i s i n ( 2 b 2 u ) + d^os (2b 2u)J + K = 0 - 2 V 2b 2v ^ c 2 s i n (2b 1v) + d 2cos (2b 1v)^ - K = 0 (2.75) where K i s the separation constant. The coordinates ( U O , V q ) of an arbitrary point on the zero equipotential are found from equation (2.73) to be related by 2b u / \ 2b pv / e ° (c 1sin(2b 2u o)-Hi 1cos(2b 2u o)) +e j c 2 s i n ( 2b ]v o )+d 2cos (.21^ = Q . Since by hypothesis the electron v e l o c i t y i s zero at ( U O » V q ) , we have • (bju+bgv) d¥ u du _ e .(b l U+b 2v) d¥^ . ( u o ' V o ) dv = 0 (u i.v ) v O r 0 ' If these i n i t i a l conditions are substituted into equations (2.75), we obtain the following relations for the separation constants 56 2b, u 1 o ^ ^ s i n (2b 2u Q) + d-j^ cos (2b 2u o)^ + K = 0 and 2 o c 2 s i n (2b - j V o ) + d 2cos ( 2 1 ^ )) - K = G , •(2.76) The v e l o c i t y of an electron starting from an arbitrary point (u ,v ) on the cathode i s x o * o or = + yg / ^ s i n (2b2u) + d-^os (2b2u) + Ke J , ( c 2 s i n (2b 1v) + d 2cos (2b 1v) - Ke -2b/ 1 2 If the substitutions a = 2b 2u — tan 1 ( ^ ~ ) |3 = 2b,v - tan (^) 1 a2 are made, the v e l o c i t y equation becomes or = 1/277(0 2 + d , 2 ) * / + - ——r- ( Cos a - e • co-s a — b„v \ o /27)<. 2 2 + d 2 2>V V ^ ' - — b u [cos p - e 1 cos p\ 1 2 In order to permit the motion i n four-dimensional phase-space to be represented i n two dimensions for a par t i c u l a r coordinate system (i,e», b, and b_ given), i t i s convenient to 57 write the v e l o c i t y components (u,v) i n the form u and = + ( cos a -b l I e cos a , o/ L. (2.77) = + (cos (3 -Because the functional forms of equations (2.77) are i d e n t i c a l , only one phase p o r t r a i t i s required. The phase p o r t r a i t of equations (2.77) i s shown i n Figure (2-7). It i s apparent from the phase p o r t r a i t that the electron motion can be either o s c i l l a t o r y or rotational i n either coordinate. Also, i f the cathode passes through the points (a Q,P o) = (- , 2L) or (^ y - —) , a d i v i s i o n of t r a j e c t o r i e s i s seen to occur; t h i s i s due to a saddle point of po t e n t i a l . In Example #1 i t was found that, as special cases of the general solution, electron motion along either of the coordinates could be obtained. This w i l l now be shown to be true also for the present formulation. If the el e c t r o s t a t i c f i e l d i s due only to O-j^ * so that c 2 — cL, = 0, then i t follows from equation (2.76b) that K = 0. If an electron i s now released from rest at a s u r f a c e = 0, that i s . a s p i r a l | + n% + tan 1 ^ p j Uo = W, then i t can be seen from equation (2.77) that the v-component of 59 v e l o c i t y remains zero* The trajectory i s therefore a s p i r a l v = V q » and the v e l o c i t y along the s p i r a l i s u = + e 2 °^ 27] ( c ^ + d ^ ) 2 co! 2b 2u + tan" 1 (-^ 1 2 Conversely* i f the e l e c t r o s t a t i c f i e l d i s due only to <&2§ S G that c^= d^= 0, then the u-component of v e l o c i t y of an electron released from rest from the cathode w i l l remain zero* The cathode surfaces i a -.this case are the s p i r a l s ' C . V o ~ 2b, | + n* + tan"1 and the electron t r a j e c t o r i e s are the s p i r a l s u = u , the Velocity along these s p i r a l s being ^b,u v = ± e lli° j^ 2 7?(c 2 2+ d 2 2 ) 2 cos jj^v - tan ^'(^j 2*5:2 Potential Function of the Form " ^ ( q - ^ +^^2^" (a) Conditions for separation of variables A d i f f e r e n t class of solutions i s obtained i f i t i s assumed that the potential i s the sum of a function of one coordinate alone and a function of the second coordinate alone. Let us assume that the metrical c o e f f i c i e n t s h^ are of the functional form h i - f i l ( * l ) f i 2 ( * 2 > and the action function W i s of the form (2.78) ¥ = B 1 ( q 1 ) B 2 ( q 2 ) . (2.79) 60 Ve must next determine under what conditions separation of variables of the Laplace and Hamilton-Jacobi equations i s possible. To be able to write the potential i n the separated form '•<* - • * 2 ( q 2 ) i t i s necessary that h^ = h 2 within the p o s s i b i l i t y of a scale change. Since h^ = h 2 for a l l conformal transformations from Cartesian coordinates* i t i s possible to write Cj) i n separated form for any member of the class of orthogonal coordinate systems obtained by qonformal transform methods With h, W and<£ of the above assumed functional forms, the Hamilton-Jacobi equation (2.18) becomes 2 2 B 2 2 ( q 2 ) A V ^ A B ^ U i ) /&B 2 (*2^ f l l 2 ( * l ) f 1 2 2 ^ 2 ) V d < 1 l / f 2 1 2 ( ( l l ) f 2 2 2 ( < l 2 ) \ d*2 = 277 ^ ( q ^ + C j > 2 ( q 2 ) ) • ( 2 * 8 0 ) The f i r s t term of this equation i s a function of one variable i f dB, (q,) l : (a) B 2 ( q 2 ) = C l f 1 2 ( q 2 ) , or (b) = V l l ^ ' The condition for the second term i s dB (q ) 21 (a) B x ( q i ) = c 3 f 2 1 ( q 1 ) , or (b) dq 2 = C4 f22 (^2 ) • If conditions l(a) and 2(b) are chosen, the requirement for separation of variables i s thus d f , 0 ( q 9 ) cA IL ° i f f22<*2> * < 2' 8 1 a> 61 If conditions l(b) and 2(a) are chosen, the requirement i s 1 w3 where the c^ are arbitrary constants. d f 0 , (q, ) c 0 An action function of the form ¥ = B l l ( ^ l ) B 1 2 ( < l 2 ) + B 2 1 ( q l ) B 2 2 ( , l 2 ) could also have been assumed for this formulation. It i s readily shown, however, that when separation of variables i s possible with V of this l a t t e r form, the second term becomes redundant, so that V reduces to the form of equation (2,79), (b) Examples Example^ #3; Solution i n Plane Polar Coordinates In plane polar coordinates the function f. . o f the metrical c o e f f i c i e n t s are, from equation (2.78), f n ( r ) = f 1 2 ( 0 ) = f 2 2 ( 0 ) = 1 and f2 1 < r > These functions s a t i s f y equation (2.81b), so i t i s possible to separate the Hamilton—Jacobi equation when V i s of the form (from condition (2a) and equation (2,79)): ¥ = rB 2(Q) . (2.82) The p o t e n t i a l ^ i s of the assumed form <$> = <t>r(r) + %(Q) , 62 which, when substituted into the Laplace equation, results i n ^ r ( r ) = <£>ro + a l n r + b(ln r ) 2 and %W = * e o * CO - bO 2 where <t>. <t> a. b, and c are constants, ro' ©o' 7 1 The Hamilton-Jacobi equation (2.80) becomes under these conditions 2 B 2 2 ( ° ) +(~"a©~"~) = 2 7 ? ^ o + a 1 x 1 r + b * l n r ^ 2 + c Q - b Q^) • The l e f t side of th i s equation i s a function of ©only; the right side also must therefore be a function of © only, requiring that a = b = 0. The potential i s therefore simply ••<$= <t>0 + c © (2.83) and the equation to be s a t i s f i e d by the motion i s 9 /dB.(©)\ 2 _ B2 ( 0 ) + ( d© J = 2 7 ? ( C ^ o + c 9 ) * ( 2 - 8 4 ) By a rotation of coordinates can be eliminated, so that 0 = 0 at © = 0. Equations (2»83 and 84) describe electron motion between two inclined-plane electrodes. This motion has been discovered previously by Walker^ 3^» The solution of equation (2»84)» when = 0, that was obtained by Walker i s oo B 2(0) = ^ a n© (4n-l) 2 where n = 1 2 10 a l = 3 \/2Vc > a2 = ~ 21 a l » a 3 = 2079 V e t c ' T h e 63 electron motion i s sketched i n Figure (2-8)» Valker found that* with V of the functional form of equation (2.82), i t i s also possible to take space-charge into account* The extended solution he thus obtained, namely and n = 1 where ocn and 0 n are constants determined by the space-charge-flow equations, represents space-charge-limited flow between two inc l i n e d planes* Example #4t Solution i n Equiangular Spiral Coordinates For equiangular s p i r a l coordinates the functions f. . o f the metrical c o e f f i c i e n t s are, from equations (2.69 and 78)» c i r c l e s * are shown as dashed l i n e s . 64 b, u f x l ( u ) = f 1 2 ( u ) = e f 1 2 ( v ) = f 2 2 ( v ) = e b 2 v Both conditional equations (2.81a and b) are observed to be s a t i s f i e d by the t. .d and i t i s thus possible to obtain separatio x * J of variables vhen ¥ i s either of the two functional forms b,u ¥ = e B 2(v) or b.v ¥ = e ' B 1(u) . If we choose the f i r s t of these, and assume that the potential <3> i s of the form 4> = O ( u) +<& (v) , u V the Hamilton-Jacobi equation (2.80) becomes -2b 2v e 'dB 0(v)\ 2" = 2T)<$> (2.85) where 4> i s found from the Laplace equation to be <J> = a ( u 2 - v 2 ) + bu + cv + <|> and a. b* c. and <J>^  are constants. The l e f t side of equation (2.85) i s a function of V only; the right side must therefore be a function of v only, as well. It i s thus necessary that a = b = For convenience we w i l l also set Cj>Q = 0, so that the potential is cv and equation (2.85) becomes 65 0 0 /dB (v)\ 2b 2v These two equations describe electron motion between two equi-angular s p i r a l electrodes* The Hamilton-Jacobi equation i s solved B 2(v) = co n .= 1 a v n (2n + l ) 2 2 / 1 3 2 / 9 2 2\ where ^ = J y^Tjc* &2 ~ 5 b 2 a l * a3 = 21 I 4 b2 " b l j a l * 2 /3 2 1 2\ a^ = ^ ^2\4~ b2 ~~ 3~ b I ) a l * e"kc» ^he electron motion i s sketched i n Figure (2-9). The v e l o c i t y of the electrons i s b,u + b 0v , 1 2 du dt ' e b l u + b 2 V dv dt which can be rewritten i n terms of the action function as i r = -b 0v -b 0v dB 0(v) V B 2 ( v ) ' 6 -IV-<3> = Figure (2-9). Cp= 0 Electron t r a j e c t o r i e s between two equiangular s p i r a l electrodes. The f i e l d l i n e s , which are equiangular s p i r a l s , are shown as dashed lines 66 The t r a j e c t o r i e s are therefore described by the d i f f e r e n t i a l equation du h B 2 ( V ) dv _ D l /dB 2(v) s I dv which has the solution oo ^ n + 1 u = u + b, 7 c v o 1 / n n = 1 where ^ = j, c 2 = - ^ b 2 , C3 = ^ b i " 9 b2 J > E T C ' It w i l l be recalled that there were two possible functional forms for V, and that the f i r s t one was chosen for the present solution. To obtain the solution for the second functional form, i t i s merely necessary to replace u by v, by b 2 , and vice versa, i n the above solution. It i s also possible to extend these solutions to include the effects of space-charge, with the same functional forms of W» Again i t i s only necessary to show the derivation for one case. Let ¥ be of the form b, u V = e 1 B(v) . (2.86) The Hamilton-Jacobi equation i s therefore 2 b l 2 B 2 ( v ) + =2TjcPe 2b.2v * (2.87) The potential <3> i s thus a function of v only, and Poisson'i equation can be written as 0 -2(b u + b v) , 2 ^ r o o , 2 dv (2,88) 67 From equations (2»17 and 86) the v e l o c i t y i s i r = K * b 2 V TX( \ ~°2V dB(v) -b^v (2.89) Equations (2.88 and 89) can be combined with (2.6) to give the current density J = = p - i r = - c o r-2(b1u + b 2v) d 2 c j 5 N dv' -b^v b„v If t h i s equation be substituted into the continuity equation (2*7), there results dv' = 0 (2.90) A l l the space—charge—flow equations have now been used* and are involved i n equations (2.87 and 90). If equation (2»87) i s di f f e r e n t i a t e d three times with respect tp v, the second and t h i r d derivatives of may be substituted into equation (2.90). An ordinary d i f f e r e n t i a l equation i n B(v) only w i l l then r e s u l t . This d i f f e r e n t i a l equation i s very lengthy, so i t w i l l not be given here; i t i s solved by B(v) = oo n = 1 a v n (3n + 2) 3 If this series i s substituted into equation (2.87), i t follows that the p o t e n t i a l & i s of the form 68 <S>(v) = n = 1 Prom th i s series i t i s seen that the potential and the potential gradient are zero at v = 0. The solution thus describes space-charge-limited flow between two equiangular s p i r a l electrodes. 2i6 Discussion The e l e c t r o s t a t i c f i e l d of Example #1 has properties which are very similar to those of the e l e c t r o s t a t i c f i e l d of a convergent electron gun with an anode aperture. This s i m i l a r i t y may be observed by comparing Figures (2-4) and (3—7)* Thus the l i n e x = y i n the f i r s t quadrant of Figure (2-4) may be considered to be the gun axis, while the cathode coincides with an equipotential such as <$>= 1.5. The constants of can be adjusted so that the cathode i s at zero potential, and so that i n i t i a l l y the potential increases from the cathode inwards* It i s then seen that, as we proceed inwards from the cathode, the equipotential curves f i r s t f l a t t e n and then reverse i n curvature, which i s also the case i n convergent guns with an anode aperture. Further, by varying the re l a t i v e magnitudes of the l i n e charge and the double doublet of equation (2.63), the curvature of the equipotentials may be varied over a wide range* In view of the above-mentioned s i m i l a r i t y , further study of Example #1 i s warranted. Of p a r t i c u l a r interest would be the extension of t h i s solution to the space-charge domain, which may be possible by a perturbation method. Since the number of analytic solutions that are known for 69 electron motion i n e l e c t r o s t a t i c f i e l d s i s s t i l l r e l a t i v e l y small, there i s much scope for further work i n this area. Two coordinate systems that appear promising i n this regard are toroidal and bispherical coordinates, since they are closely related to the Staeckel coordinates. The metrical c o e f f i c i e n t s h^ i n the previous section of this chapter were taken to be of product form, as described by equations (2,58 and 78), K i r s t e i n ^ 3 ^ showed tliat the only planar coordinate systems with the bu of the functional form (2,78) are logarithmic s p i r a l , Cartesian and polar coordinates, and also a coordinate system with h^ of the form I 2 2x a(g 1 - g 2 ) h x = h 2 = e This lat t e r coordinate system did not s a t i s f y the separability c r i t e r i a of either Sub—section 2:5:1 or 2, so no solution was obtained. The extension of the solutions to three dimensions i s i n general very d i f f i c u l t , but can i n some cases be accomplished. Thus, i n logarithmic-spiral c y l i n d r i c a l coordinates (Example #2 extended) the h. are l b l u + b 2 V  h l = h2 = 6 , h^ = 1 and i t i s readily shown,,: that the Hamilton-Jacobi and Laplace equations are separable when V and <$> are of the form 70 2 ^ V h 2 * t 2 2 ) (277)(-|k l Z 2+ a 1z+a 2) dz + k 2 k 1 e 2 ( b l u ^ b 2 v : 4 ( b l 2 , b 2 2 ) 1, 2^  2T1 Z a l z 2 * 71 CHAPTER III - THE ELECTROSTATIC PIELD OF IDEALIZED ANODE STRUCTURES 3il Introduction The e l e c t r o s t a t i c f i e l d d istributions of the presently known space—charge—flow solutions are not of the kind that exist i n the v i c i n i t y of apertured anodes. As a res u l t * these solutions need to be adapted for use i n electron gun design. To f a c i l i t a t e t h i s adaptation, i t i s desirable to know the form of the e l e c t r o s t a t i c f i e l d about various apertured anode shapes. For this purpose, certain assumptions must be made regarding the other gun electrodes, as well as the space—charge d i s t r i b u t i o n . In this chapter the el e c t r o s t a t i c f i e l d about four idealized two-dimensional anode geometries w i l l be derived. It w i l l be assumed that the other electrodes are an i n f i n i t e distance away* and that spa;ee-charge effects are ne g l i g i b l e . Under these conditions the el e c t r o s t a t i c f i e l d s can be obtained (49) by a Schwarz-Christoffel transformation 'i r ' % t ^ z = z Q + A / (w - b Q) (w - b.^ ) (w - b,,) . , . dw, (3.1) This transformation maps the upper half of the complex w—plane onto the i n t e r i o r of a polygon i n the z-plane. The real axis v= 0, with points u = b Q, b^, b 2, . . . , transforms into the boundary of the polygon* having exterior angles ^ »^ |\**~P2* • • , at the corresponding v e r t i c e s . The complex constants Z q and A 72 are determined by the o r i g i n , scale, and orientation of the polygon. The boundaries of the polygons to be considered are the projections of the anode surfaces i n the zr-plane. The anode apertures are of unit width, and provide the reference dimension of distance i n th i s chapter and the next. The potential and flux w i l l be taken to correspond to v and u respectively i n this chapter. By a s h i f t of o r i g i n and a change of scale of the w-plane, i t i s a simple matter to adjust the potential and gradient as needed i n Chapter 4. 3s2 E l e c t r o s t a t i c F i e l d about a Plane with a S l i t The f i r s t and simplest idealized anode to be considered w-plane Figure ( 3 - l ) . Mapping the p r o f i l e of,an i n f i n i t e plane with a s l i t onto the u-axis of the w-plane i s an i n f i n i t e plane with a s l i t . The Sphwarz-Christoffel (49) transformation for this case i s well knownv y, and i s z = z + A(w + -) . (3.2) 0 w The transformation i s i l l u s t r a t e d in.Figure ( 3 - l ) . To evaluate Z q and A, l e t w = + 1« Then i t i s seen from Figure (3—l) that z = 4- -j^ respectively, and i t follows from equation (3.2) that z =0 and A = - T . If the l a t t e r values for z and A o 4 o are substituted into equation (3.2), and this equation i s separated into i t s rea l and imaginary parts, there results 1 [ A u X = - T U + 4" \ 2 ^ 2 u + v y = - 4 v - ~~T~ 2 L u 4- v > (3.3) The e l e c t r o s t a t i c f i e l d described by equations (3.3) i s i l l u s t r a t e d i n Figure (3-2). In Figure (3-9) i s shown the potential v a r i a t i o n along the axis of symmetry of the anode. 383 E l e c t r o s t a t i c F i e l d about Two Right-Angled Plates The polygons i n the z-plane projected by two right-angled plates, two p a r a l l e l semi—infinite plates, and the f i c t i t i o u s "negative thickness" anode of Section 3s5, are degenerate rectangles. A f i n i t e embodiment of these rectangles i s of the general form as shown i n Figure (3-3)» Since the vertices a^ and a^ are symmetrically located about &2 i u the z—plane, this symmetry must also pertain to the w—plane for the corresponding points b-^  and b^ with respect to b_. Therefore, i f the transformed point b i s placed at 2 * o * 74 v 0 141 0-16\ I 0 18 020 I 0-24 0 3 0 4\ 05 0 71-0 0 0 5 7 0 /•25l-75 20 V 2 5 30\ 3 5 40] i — , — — i i .i 9> CD <N W> <N C5 <*> C5 ">» f ) »o m io o <o t o t< IO N. O 03 IO do 55 S-0 Ii Figure (3-2). E l e c t r o s t a t i c f i e l d about a plane with a s l i t as i n f i n i t y , and b 1 = -1 while b 2 = 0, then b ? i s at u = 1* shown i n Figure (3.3). Under these conditions, equation (3.1) becomes n _ -3. TC 71 TC z = z Q + A / (w + 1) w (w - l ) dw o (3. When the anode consists of two right-angled plates* the w-plane Figure (3-3). Mapping a degenerate rectangle onto the w-plane 76 angles of the rectangle are CpQ = 2%f cp^ = Qp^ = T i • Equation (3.4) therefore becomes IT - and or + A o V^2 - 1 - A cos 1 {^j . (3.5) Prom Figure (3-3), when w = 1* z = - ^ , so from equation (3.5) i t follows that z = - i - . Similarly* when w = -1, z = ~k? , and from (3»5) we have A = - — . The transformation i s therefore z = 71 2 + \A2 - 1 % - c o s" 1(w When thi s transformation i s separated into i t s real and imaginary parts* there results ! + B. 2 cos (^ ) — tan-"*" B B' 1 71 R2 sin(|) - l n A2 + 1 > (3.6) where = ( ( u 2 - v 2 H l ) 2 + (2uv) 2 1 2 0 = tan ->1 2uv 2 2 , u - v - 1, A = B k " ( I - a 2 ~ b' ') + ^ ( a 2 + b 2 - l ) 2 + 4b2|] |^(1 _ a 2 - b 2) + ^ ( a 2 + b 2 - l ) 2 + 4b: 1 2 1 2 78 and The e l e c t r o s t a t i c f i e l d described by equations (3.6) and i t s defining relations i s shown i n Figure (3-4). The potential v a r i a t i o n along the axis of symmetry of the right-angled anode i s shown i n Figure (3-9)* The anode shapes just discussed, the anode consisting of two right-angled plates and the anode consisting of a plane with a s l i t , represent two extreme cases of a physically more important anode geometry, an anode consisting of a plate of thick-ness y^ and having a s l i t Of half-width x 1 (see Figure (3-5))» The computation of the e l e c t r o s t a t i c f i e l d about an apertured plate of f i n i t e thickness i s more involved than i s the case when the plate i s of semi-infinite or of vanishing thickness. Fortunately, the f i e l d i n the region of greatest int e r e s t , the region near the anode aperture i n the lower-half z-plane, very rapidly approaches the f i e l d i n the corresponding region near two right-angled plates as the anode thickness i s increased. There-fore, the f i e l d i n the region of interest near an anode of f i n i t e thickness can generally be represented accurately by the f i e l d about an anode of semi-infinite thickness. This w i l l now be demonstrated. Consider Figure (3-5), The exterior angles of the polygon i n the z—plane are seen to be % = % = 2". 9l = ?2 - <fc = % ~ " I • The transforms of the vertices of the polygon to the w-plane, b^, are located as shown, the location of b 2 and b^ being as yet undetermined except for the fact that they are symmetric about the o r i g i n , and l i e a distance 0 ^ k 1 from i t . Under these conditions equation (3.1) becomes This equation has been integrated by D a v y ^ ^ f who used the substitution 1 w = sn (-u,) thus making k the modulus of the e l l i p t i c function used. The transformation then becomes 80 z = z - A o (3*7) 2 2 where k* = 1 - k , K and E are* respectively, e l l i p t i c integrals of the f i r s t and second kind, and Z(M, ) i s the Jacobian Zeta-function. The constants z and A can be evaluated by substituting o the coordinates of b^ and b,. into (3,7), giving z = 0 o A = Kk5*2 - 2E (3.8) If the coordinates of b^ and b^ are substituted into (3.7 and 8), then i t i s found that _ (K'k* 2 ~2K* + 2E 1) 2E - Kk ,2 (3.9) where K 1 and E* are, respectively* the e l l i p t i c integrals of the f i r s t and second kind, i n terms of k f» From equation (3.9) i t i s observed that the distance k of the points b^ and from the o r i g i n depends solely on the ratio of the height to the half-width of the anode aperture. Equation (3.9) has been plotted i n Figure (3—6). When y^ = 0, then k = 1, which represents the case of a plane with a s l i t . It i s seen from Figure (3-6) that when the thickness y^ i s increased, k i n i t i a l l y decreases very rapidly; i . e . * i t rapidly approaches the case of an anode consisting of two right-angled plates. For example, when the thickness of the plate i s equal to half the 81 width of the anode aperture, k has already decreased to about JTJ • From Gauss T law i t i s ea s i l y deduced that k i s the ra t i o of the charge on the upper surface a 2 - a^ (or a^ - a^) to the charge on the surfaces a^ — a 2 — (or a^ - a^ - a^)» The physical significance of the rapid decrease of k when y^ i s increased i s thus that the proportion of the lines of force entering the anode aperture that terminate on the upper surfaces a 2 - a-j and a^ - a^ decreases rapidly when the anode thickness i s increased. Consequently, the f i e l d i n the lower-half z—plane converges rapidly to that for an anode consisting of two right—angled plates when the anode thickness y^ i s increased, 1,0.1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1— 0 0,1 0.2 0.3 0,4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Figure (3-^6). Plot of equation (3.9) 82 3 ;.4 E l e c t r o s t a t i c F i e l d about Two Semi-Infinite P a r a l l e l Plates For an idealized anode consisting of two semi-infinite p a r a l l e l plates, i t i s seen from Figure (3-3) that the polygon angles are 0pQ = 3TC* (p^ =^3 = i -7tr ahd. (f^ = T C . The trans-formation i s , therefore, from equation (3.4), or Z q + A I 4- l n w (3.10) By r e f e r r i n g to Figure (3-3), the constants Z q and A can be evaluated as before* giving o A . 1 1 — TC Equation (3.10) then becomes 1 + i n 4- w - 21n w) or x 2uv + 2 tan' v 2 4- u 2 - In (v 2 4- u 2)J - (3.11) Figure (3^7) shows the e l e c t r o s t a t i c f i e l d described by equations (3.11)* while the potential v a r i a t i o n along the axis of symmetry of the anode i s plotted i n Figure (3-9). 3 85 The "Wrap-Around F i e l d " The properties relevant to electron gun design of the three e l e c t r o s t a t i c f i e l d s derived above are studied i n the next Figure (3-7). E l e c t r o s t a t i c f i e l d about two semi-infinite p a r a l l e l plates 84 chapter, and i t i s inferred there that the f i e l d characteristics improve as the angles Cp^ and Cp^ of Figure (3-3) are made increasingly negative. The l i m i t of physical r e a l i z a b i l i t y of the anode i s attained when = cp^ = -.-JI, as then the anode i s of vanishing thickness. Nevertheless, i n a Riemann surface the projections a - a, and a 0 - a* of the outside anode surfaces can be conceived o 1 3 o of as continuing past the respective projections and &2 — &3 °f "the inside anode surfaces i f C{)^  and Cj^ are reduced beyond The e l e c t r o s t a t i c f i e l d of the resulting f i c t i t i o u s negative thickness electrode w i l l be called the "wrap-around f i e l d " . The transformation for the case when Cp^ ..= Cp-j = - ^ = 7t» a n < i QP0=2^i; w i l l n o w he derived. Under these conditions, equation (3,4) becomes z = z + AI o (3.12) where I = or dw I =4 1/ 2 5<v - ir - - 1 ) (w2 - l ) 2 + /T(w2 ~ l ) 4 + 1 + — — In + S t a n " / ^ ~ - 1)4".'+ ly .1- (v' Equation (3,12) can be separated into i t s real and imaginary components, giving x = Xo + AR XR y = y o + A J Z R + AR*D i (3.13) 85 where z = x + i v o o J o A = AJJ + iA«) I = I R + i l 0 and where 1 1 •1^4__,5. N «T_._/ax . 1 B ( i ) S 4 c o s ( J a ) - S 4cos(£) +• 4 | F l n i/E*' + F + t a n " 2G 1 _ C 2 - D 2 j I 3 = (^)S 4sin(|a) - S 4sin(|) + -± 4 / T t a n - H f ) + ^ l n f C I + ( D \CT+ (D - l ) ' and where S = ( u^ _ ^ _ ^ + ( 2uv)' . - l / 2uv n 2 a = t a n ,u 2 - v 2 - lj C = /2Vcos - S 2) D = E = [g 2 - 2 + i . r i ,2 (-) 2 c o s v 2 1 + 1 S + 2 S 2 [ l + cos (§)] - 2 ^ T S 4C09(|)(1 + S 2) + 1 F = / ? S 4 s i n f f i ( l - S 2) S + 2 S 2 [ l + cos (§)] - 2 / 2 " s 4 c o s ( | ) ( l + S 2) + 1 86 Figure (3-9). Variation of potential along the plane of symmetry of four electrode shapes 88 The constants z and A can be evaluated as before by o J setting v = 1 and z = - >^ and w = -1 and z = .ijj-. Equations (3.13) then become y = - # ( l R + I ^ * (3.14) the desired transformation* The e l e c t r o s t a t i c f i e l d i n one half of the f i r s t sheet of the Riemann surface i s shown i n Figure (3^8) The potential v a r i a t i o n along the axis of symmetry i s plotted i n Figure (3-9). For electron beam design, the region of interest of t h i s e l e c t r o s t a t i c f i e l d l i e s between the flux l i n e s u = 1 and u = -1» and from about v = 2.75 to V == 0, The f i e l d i n this region could be approximated by placing electrode sheets along several equi-potentials just outside t h i s region, and by maintaining these electrodes at the required potentials. 89 CHAPTER IV - USE OF SPACE-CHARGE-FREE FIELDS IN ELECTRON GUN DESIGN 4sl Introduction In t h i s chapter, the e l e c t r o s t a t i c f i e l d i n an electron gun is considered i n two partsJ the f i r s t part i s the low—potential region, i n which space—charge effects are appreciable, and the second i s the high-potential region, i n which the effect of space-charge on the e l e c t r o s t a t i c f i e l d i s assumed to be n e g l i g i b l e , Consider the electrode configuration of Figure (4-1), The region from the cathode to the a u x i l i a r y anode i n this figure represents a strip-beam Pierce g u n ^ \ The upper surface of the aux i l i a r y anode i s shaped to coincide with a suitable equipotential of the free-space e l e c t r o s t a t i c f i e l d of the main anode ( i . e , , an equipotential of the e l e c t r o s t a t i c f i e l d that would exist i n an isolated system consisting of the main anode, held at a potential V^, and a charge located an i n f i n i t e distance away). The e l e c t r o -Main Anode r - • • ; a i. i.. ••• : • • • i • • . •. •. *. Beam-Forming Electrode Figure (4-1), Hypothetical electron gun 90 s t a t i c f i e l d i n the auxiliary—main anode region can thus i n principle be obtained by a conformal transformation of the' space exterior to the main-anode surface* such as was carried out i n Chapter III* Assuming that the Pierce-gun section of the electrode configuration of Figure (4—l) i s operating, imagine that the potential of the main anode* V^ ., i s adjusted so that the potential gradients at the upper and lower surface of the thin section a—b of the a u x i l i a r y anode are approximately the same. If section a—b i s now removed^ the f i e l d d i s t r i b u t i o n w i l l therefore be n e g l i g i b l y affected, provided that the potential-depressing effect of the space-charge which w i l l now enter the auxiliary-main anode region can be ignored* If t h i s proviso applies, the e l e c t r o -s t a t i c f i e l d i n the two—anode gun thus i s known. (51) Van Duzer and Brewer\ y obtained equipotential plots for —6 a Milller-type gun with a perveance of 2.3 x 10 by means of an e l e c t r o l y t i c tank with provisions for space-charge simulation. Comparison of an equipotential plot taken i n the absence of space-charge simulation with a plot taken under conditions of space-charge simulation shows that the equipotentials i n the anode region of the second plot have been displaced but have changed l i t t l e i n shape. For the two—anode gun discussed above, the displacement of the equipotentials i n the auxiliary-main anode region has l a r g e l y been taken into account by the gradient-matching procedure used. Van Duzer and Brewer's experimental data thus give an indication of the a p p l i c a b i l i t y of the assumption that the e l e c t r o s t a t i c f i e l d i n the auxiliary-main anode region can be approximated by a space—charge—free f i e l d . This problem i s pursued further i n the next section. In Sections 4:3 and 4 91 the gradient-matching procedure i s applied to the study of two electron gun configurations. The a u x i l i a r y anode i n the hypothetical gun of Figure (4-1) has been inserted only as a temporary measure, to aid the analysis of conditions inside the electron beam. In Chapter 5 i t i s shown for the electron guns of Sections 4s3 and 4 how t h i s anode can be removed, 4s2 Error Estimate for a Space-Charge-Free F i e l d Approximation  i n the Anode Region The error involved i n approximating the e l e c t r o s t a t i c f i e l d i n the anode region by a space-charge-free f i e l d i s readily obtained for two space—charge-limited flows of interest i n this chapters flow between two p a r a l l e l plates and flow between two concentric cylinders, 48231 Flow between Two P a r a l l e l Plates In an ideal planar diode the following relations h o l d ^ Potential E l e c t r i c Intensity Electron V e l o c i t y Charge Density where Y _ y. j_ distance from cathode d cathode-anode distance _ V potential at y V, ""anode potential * I 3 1 t = - i l 3 • T = P = 2 r3 > 2 "3-(4.1) Equations (4.l) are i l l u s t r a t e d i n Figure (4-2). It i s seen that as T increases, P decreases monotonically. As a result<I>becomes more li n e a r with increasing X, tending toward the solution of the Laplace equation, <£>L = a l + b (4.2) where a and b are constants. / Let the potential i n the diode be approximated by (4.2) over the i n t e r v a l I < I <, 1 o by matching the gradients of and at I j as shown i n Li O Figure (4-2). Therefore* at T , <J> = O t and ^ 3 IJ dY dY and from equation (4.1a) i t follows that the constants of equation (4,2) are a - ± Y 3  a ~ 3 o b = - \ J 3 3 o The maximum error incurred by this approximation depends on the value of Y q . Let the error i n po t e n t i a l , e ^ , be defined as 1* and the error i n v e l o c i t y * e^ -* be defined as I - 1/27/ ej = 100 For the example i l l u s t r a t e d i n Figure (4-2), Y Q = 0.75 and thus 93 Figure (4—2)» Variation of the potential, e l e c t r i c i ntensity, electron v e l o c i t y and space—charge density versus distance from the cathode i n a p a r a l l e l plane diode. Also shown i s an approximation of the potential i n the anode region by<$ = a Y + b 94 equation (4.2) i s 3> = 1.21141.- 0.2271 Li while the errors i n potential and v e l o c i t y incurred by this potential approximation^ are, at the anode, e ^ = 1.57$ e y = 0*789$ » The values of e ^ and e^ at the anode for other values of Y q are shown i n Figure (4-3). -% 1-\ ' 1 ' 1 1  11 10 9 8 7 0 O 6 < 5 -p -p a 4 u o u 3 2 1 0 A T • 2 ^  -P\ 0.4 Distance from Cathode at which the Potential Approximation Commences \ O Figure (4-3). Error i n potential e ^ , and i n electron v e l o c i t y e^, L at the anode of a planar diode when the potential i s approximated by CPT = aY + b over the interval I < I < 1*0 o When the anode has an aperture, the errors incurred by approximating the f i e l d i n the anode region are no longer readily calculable. However, i f the flow being considered i s i n i t i a l l y p a r a l l e l and r e c t i l i n e a r , which i s the case i n Section 4 s 3 , then this analysis i s a valuable guide. 4?2s2 Convergent Flow between Two Concentric Cylinders In an ideal concentric-cylinder diode the (3) following relations apply Potential 2 2 \ 3 \R p \ a ra E l e c t r i c Intensity £^ = — ^(—— 2 i ,R p v a a P^ + 2p dB_ d* 1 ,3 RP' ^ 6,ra tron V e l o c i t y R = J^Tjf R ^ J \ B a B a / Elec Charge Density P • " I «. R p ' * a a R R3 1 2 \ 3 iR P v a r a where g r _ radius  — r . ~~ cathode radius c (4.3) potential at r 0 = 5 - = — V potential at r , the anode a r a' P — Langmuir parameter, a function of R r R = S a r c The r a d i a l dependence of the charge-density P i s shown i n Figure (4-4), while <$> and £ are shown i n Figure (4-5) for the case R = 0,25. Near the cathode, the behaviour of the flow parameters for this case and for the planar diode case i s seen be s i m i l a r . However, as the flow converges, the parameter behaviour becomes decidedly d i f f e r e n t . The charge density P attains a minimum and then increases without l i m i t as R —*-0, 1.0 0.9 0.8 0,7 0.6 0.5 0.4 0.3 0.2 0.1 * 3 ° / O Normalized Radius u -p a Figure Variation of the charge density convergent electron motion from cathode i n r e c t i l i n e a r , a c y l i n d r i c a l Figure (4-5)» Var i a t i o n of the potential and e l e c t r i c intensity versus distance from the cathode i n a concentric-cylinder, convergent-flow diode. Also shown i s an approximation of the potential i n the anode region by Cj> = a l n R + b 98 Thus, to accurately approximate the e l e c t r o s t a t i c f i e l d i n the anode region of a c y l i n d r i c a l diode by a space-charge-free f i e l d <£ L = a l n R + b, (4.4«> the anode radius R should preferably not be too small. An anode a radius of R = 0.25 was found to be a convenient value, a Let the f i e l d be approximated by (4.4 over the range R > R > 0.25 o by matching the gradients of •$> and <$> at R , as shown i n Li O Figure (4-5)* Therefore* at R = R Q , ^ = a n d ~dR = I S -so from equations (4.3a and b) i t follows that the constants of equation (4.4) are a R = R b =Cj> + R t R = R l n R o R =F R For the example i l l u s t r a t e d i n Figure (4-5), R = 0.45, so equation (4.4) i s <&L = -0,79421 l n R - 0.12463 Let the error i u p o t e n t i a l , e ^ , and the error i n v e l o c i t y , e^, be defined as i n Sub-section 4s2:l, namely /4> -<3>r\ e ^ = 1001 e^ = 1001 R 99 For a given B , the maximum errors occur at the anode. & o 7 The errors at the anode for the case B 0.45 are = 2.36$ e^ = 1*19$ The errors at the anode incurred by i n i t i a t i n g the f i e l d approximation at other values of B q are shown i n Figure (4-6)* The distance coordinate i n thi s graph i s normalized w.r.t* the eathode-anode distance to permit a comparison to be made with the O N Distance from Cathode (Normalized) at which 1 — R Potential Approximation Commences, o 0.75 S 0) ^ 3 K Figure (4-6). Error i n potential e,^, and i n electron v e l o c i t y e„, at the anode of a concentric-cylinder, convergent flow diode when the potential i s approximated by CP. = a l n B + b over the interval B > B> B * where 1» o fc a* B = 0.25 a 100 planar diode approximation, Figure (4-3). It i s seen that the errors for the c y l i n d r i c a l diode case are higher, as anticipated from the behaviour of P • For example, the f i e l d approximations over the intervals 0.75 <Y< 1 and 0.4375 > R>0.25 for the plana'r diode and c y l i n d r i c a l diode respectively, both cover one quarter of the cathode—anode distance, and the ve l o c i t y errors at the anode are e. = 0.79$ and e^ = 1.03$ respectively. The space-charge-free potential that we w i l l match to an i n i t i a l l y r a d i a l , convergent flow i s one calculated for an anode with an aperture. In the region of the aperture the beam i s def ocused,, due to the combined action of the anode f i e l d and the space-charge forces. As a r e s u l t , the charge density w i l l not follow the theoretical curve of Figure (4-4) beyond the matching radius R q , but w i l l tend to diverge from i t on the low side, which should improve the accuracy of the space-charge—free potential approximation. 483 I n i t i a l l y P a r a l l e l * Rectilinear Flow to an Apertured Anode The ideas presented i n Sections 4sl and 2 w i l l now be applied to the study of spaee^charge-limited flow between a planar cathode and an anode consisting of two right—angled plates, as shown i n Figure (4~7)» I f "the anode aperture i s taken to be of unit width, then the region of pa r t i c u l a r interest i s |x| < 0»5» The problem of providing a bounding e l e c t r o s t a t i c f i e l d for a beam within t h i s region i s taken up i n Chapter V* It w i l l be assumed that the electron motion i n the cathode region, y c < y < y Q » ' i s p a r a l l e l and r e c t i l i n e a r , and that the e l e c t r o s t a t i c f i e l d i n the anode region, y > y , can be 101 represented by the f i e l d derived i n Section 3g3. The accuracy aperture width to the cathode-anode distance because, i f the cathode-anode distance i s reduced rela t i v e to the aperture width the perveance i s increased and the perturbation of the f i e l d due to the anode aperture becomes more severe i n the cathode region. The accuracy with which i t i s desired to s a t i s f y the assumptions thus determines the values of y Q and y c; y Q must be s u f f i c i e n t l y far from the anode that the equipotential Cj>Q can be considered to be planar, and y must be a s u f f i c i e n t distance beyond y so c o that the e l e c t r o s t a t i c f i e l d i n the anode region may be approxi-mated by a space-charge-free f i e l d . From Figure (3-4) i t i s seen that the equipotential surfaces near y = -1 have become almost planar, varying from the planar by less than .01 or 1$ over the distance |x|<C0.5. Accordingly, y Q w i l l be taken to l i e near y = -1, although i t s value w i l l not be specified at this stage. of these assumptions depends primarily on the ratio; of the /////////// y / / / / / / , "Cathode T7TT Figure (4-7). Planar diode with an anode aperture 102 The flow i n the cathode region w i l l be taken to be the same as that i n the region 0 < Y < ^ 0 « 7 5 (notation as i n Sub-section 4:2*1) of a planar diode. If the anode of the structure i n Figure (4-7) had no aperture, the maximum error incurred by-approximating the f i e l d i n the anode region would thus be e ^ = 1,57$. Furthermore, i f i n the l a t t e r case y Q = -1, i t follows that y c = -4, so the cathode-anode distance d = 4, and y and Y are related by y = -4(1 - Y) If this equation i s substituted into equations (4.1a and b) and evaluated for y = y Q = —1, there result <3> = 0.681420 o ° o - " dy =-0.302853 (4.5) In the actual diode under study an aperture i s present. It i s mathematically convenient to treat this case by moving the entire cathode region s l i g h t l y closer to the anode; i . e . , the distance between the cathode, y = y , and the equipotential O c o at y = y Q remains y G - y c = 3 (4.6) and conditions (4,5) must be s a t i s f i e d at y = y . If i n addition we specify that the anode potential ^ = 1, then the value of y , and the space-charge-free f i e l d i n the anode region, can be calculated from equations (3.6); this w i l l now be done. The mapping v = 2(T? - 7 ? k ) (4.7) where 103 w = u 4- i v rescales w by a factor J}*"1* and s h i f t s the o r i g i n of the w-plane a distance T^* ^ n the z-plane, v = 0 coincides with the anode surface. Since the anode potential i s to be unity, we thus must set = 1, Further, l e t = 0. Equation (4.7) then becomes v = Q(c£ - l ) (4*8) When equations (4*8) are substituted into equations (3,6) and the defining relations following the l a t t e r , we obtain TC TC TC , D2 /0\ . -1 f B > 2 4- B cos^) - tan f - j — . R 2sin^|j - ln^A 4- ^ A 2 4- 1 / y (4*9) where R = Q2A •^2 M < cj> _ 1}2 _ ^ l j [ 2 * * " ( ^ - i f 0 = Arctan 2^((p * 1) , - (<£ - l ) 2 i f = TC 4* Arctan Q 71 2^(<£ - 1) -vjr2 - ( q p - D 2 > o ¥ - (<£ - i ) ' ,0/ i f Y - (o - 1 ) 2 - y <0 104 A =• B =X (1 - a 2 - b 2) + »/ ( a 2 + b 2 - l ) 2 + 4b 2 i [ -l 2 b 2) + J ( a 2 + b 2 - l ) 2 + 4b 2 and a = Q [ V 2 + l ) 2 ] ' " p j V 2 + (3> - l ) 2 ] # - ( ^ - 1) Along the plane of symmetry of the anode region N =^ 0, so i t follows from equation (4,9b) and the defining relations that the potential v a r i a t i o n along the plane of symmetry i s 1 .71 x = 0 1 2 Q 2(<£ - l ) 2 + l j " - s i n h - 1 ( Q f c ^ t T T T (4*10) The derivative of y w , r . t . O i is x = 0 1 .71 x = 0 Q<2 + 1 (3> - i ) 2 The potential gradient along the plane of symmetry i n the anode region i s therefore t x = 0 dy_ dc£ x = 0 ft2"* 1 (<3>- i ) 2 which can be solved for Qs Q = -1 -I 2 .71 w (4,11) x = 0' «J> - i ) : 105 Equation (4.11) relates the scaling factor Q to the potential and gradient at an arbitrary point pn the y-axis. Ve wish to f i n d the scaling factor when conditions (4,5) are s a t i s f i e d at y Q» If conditions (4,5) are substituted into equation (4»ll), the scaling factor i s Q = -9.88700 . (4.12) The value of y Q can now be computed by substituting (4,5a and 12) into (4,10)» y Q = -0.952495 . (4.13) The potential i n the anode region i s described by equations (4.9) and t h e i r defining r e l a t i o n s , with Q as given by (4*12). The flow parameters i n the cathode region can be obtained from equations (4.l) since* from equations (4.6 and 13), y and T are related by y = -4(1 - T) + 0.047505 . (4,14) The potential variation, along the plane of symmetry i s shown i n Figure (4.8). If space—charge forces are neglected, electron t r a j e c t o r i e s the anode region can be obtained by a numerical method developed i n Appendix G* Section 1. Using t h i s method, traj e c t o r i e s were computed i n the region of the anode aperture*/ |xj <C0.5» resulting i n the trajectory shapes shown i n Figure (4r-9a). For the case when an electron beam enters the anode region over the i n t e r v a l x < 0.5 only , the electron t r a j e c t o r i e s i n This can be accomplished by* for example, inserting i n the structure of Figure (4—7) an intercepting anode which coincides with the equipotential <$> over the interval |x| > 0.5. Cathode Figure (4-8), Potential variat ion along the plane of symmetry of an i n i t i a l l y p a r a l l e l j-r e c t i l i n e a r flow Figure (4-9a)» Electron trajectories neglecting space-charge effects y 1—• 1 . r o 1 1 - 0 5 1 1 1 1 .-OS 1 r i 2 4 ( 7', » ( •j; 1 K •; — i '0 sau ( T 1 .-10 X 0 0 5 1 0 I—1 o Figure (4-9b). Electron t r a j e c t o r i e s when space-charge forces are taken into account 108 the anode region w i l l be the same as those shown i n Figure ,(4-9a), i f space-charge forces are neglected. Since the transverse electron v e l o c i t y of the beam i s much less than the axial v e l o c i t y , a f i r s t - o r d e r correction can be made for the transverse space-charge forces. This correction i s mad.e i n Appendix C* Section 2t and space—charge—corrected electron t r a j e c t o r i e s obtained by this method are shown i n Figure (4-9b). The difference between the trajectory shapes of Figures (4-9a and b) i s seen to be s l i g h t . The perveance of the beam i n the region |xj < 0,5 i s approximately equal to the perveance of an equivalent planar diode with cathode—anode distance d = 4, because the cathode was moved closer to the anode to maintain the perveance i n the region |xj <C 0.5 when an aperture was cut i n the anode. From equation ( l , l ) and the Langmuir-Child law v ' i t i s readily found that the perveance per unit distance perpendicular to the x-y plane i s v 2k.. x I ~ r <4-15> where k l = I eo {*V Referring to Figure (4~9b)» i t i s observed that a trajectory with coordinate x = 0.4641 i n the i n i t i a l l y p a r a l l e l part of the beam has diverged to x = 0,4752 at the plane y = 0, An electron beam with an i n i t i a l half'-width, of 0,4641 would therefore be suitable for this anode aperture* Setting x = 0,4641 and d = 4 i n equation (4«15)» the perveance of the resultant beam i s thus 3 K 6 2 = 0,14 x 10" amp/volt meter. 109 4g4 I n i t i a l l y Radial,, .Convergent Flow to an Apertured Anode, As a second i l l u s t r a t i o n of the usefulness of the "two-region concept" for electron gun studies, a case of great p r a c t i c a l interest i s considered nexts convergent flow from a cathode that i s a section of a cylinder. It w i l l be assumed that the flow i n the f i e l d i n the anode region can be approximated by a space— charge-free f i e l d , To thi s end a study w i l l f i r s t be made of pertinent characteristics of the anode f i e l d s derived i n the t h i r d chapter to determine t h e i r s u i t a b i l i t y for this electron motion, 4s4 8 l Analysis of Anode Fields Consider Figure (4=10)•»• For electron motion i n the cathode region, r "> r ^ r *.• to be r e c t i l i n e a r and convergent toward a common axis (x ,y„)» i t i s clear that the equipotentials i n the cathode region must be e y l i n d r i c a l sections with a common eentre of the cathode region i s r a d i a l and space-charge limi t e d , and that Cathode Figure (4.-10). Cathode region of an i n i t i a l l y radial*, convergent flow 110 curvature (x , y )• The basic conditions that must thus be met by c c the equipotential Q> at r = r Q are that over the width of the beam (a) ^ be a section of a cylinder and (b) ihe gradient at ^ be independent of 0* In addition, i t i s desirable that J^) o r 1 o be close to the anode aperture and that Cj> have a short radius o of curvature, because th i s improves the beam perveance* It i s clear from Figures (3-2 and 4.) that an anode consisting of a plate with a s l i t , or consisting of two r i g h t -angled p l a t e s , i s not suitable for an i n i t i a l l y r a d i a l * convergent flow, because no equipotential i n front of t h e i r anode apertures s a t i s f i e s the basic requirements for ^PQ» However, the f i e l d surrounding an anode consisting of two semi-infinite p a r a l l e l planes, which i s i l l u s t r a t e d i n Figure (3—7), i s seen to exhibit the required behaviour* The v a r i a t i o n of the radius of curvature and of the centre of curvature along the equipotentials of this f i e l d were obtained by a method presented i n Appendix D, and are shown i n Figures (4-11 and 12). For example, the radius of curvature of the equipotential v = 2.50 i n Figure (3-7) at u = 1.0 i s , from Figure ( 4 — l l ) , r = 3.!164, and the angle between the radius of curvature and the plane of symmetry i s 9 = 16*85°; the centre of curvature of the above equipotential at u 1*0 i s , from Figure (4-12), (x&i y ) = (-7.496 x 10~ 4, 1.718). Also shown i n Figure (4-rll) i s a curye that i s indicative of the maximum half—angle 0 that can be occupied by the electron beam* This curve, which w i l l be c a l l e d the "maximum l i n e " , was obtained by determiningj for each of several equipotentials, the point on the equipotential from which an electron, t r a v e l l i n g along the radius of curvature and continuing i n a straight l i n e * w i l l graze 4.0t 0 8 12° 16" Wedge-Half Angle 20l 24' 28' Figure (4-11). Radius of curva+nr^ . • about two " S ^ I . X ^ S t ' . i " f l e l d 112 Figure (4-12)^ Position of the centre of curvature of equi-potentials i n the f i e l d about two semi-infinite p a r a l l e l plates 113 the mouth of the anode. For example* the maximum half-angle determined by the maximum li n e c r i t e r i o n for the equipotential v .« 2*50 i s © = 16.25°. From Figures (4-11 and 12) i t i s seen that the equipotentia v = 2«T5 i n Figure (3-7) has a radius of curvature and a centre of curvature that are almost constant over the interval |u| = 0*6* This equipotential crosses the y—axis at y = -1.6848 and the h a l f -angle at u = 0*6 i s © = 11*6 * Therefore, although the shape of t h i s equipotential makes i t ideal for use as the matching equipotential (pQ f the r e l a t i v e l y large distance of t h i s equi-potential from the anode aperture and the r e l a t i v e l y small h a l f -angle would result i n a flow that would have neither a high perveance nor a high convergence* The equipotentials v < 2*75 have r a d i i of curvature that f i r s t decrease and then increase again with increasing |u|; the centres of curvature simultaneously move towards the equipotential when the r a d i i of curvature decrease, and move away from them again when the r a d i i of curvature increase. When these variations i n the curvature are not excessive, the equipotentials can be approximated accurately over the region of interest by sections of a cylinder. For example, a section of a cylinder with centre of curvature at ( x c , y c ) = (0* 1*51714) and having a radius of curvature r = 2.5343, coincides with the equipotential v = 2.00 at |uj - 1*0 and deviates from this equi-potential by less than 0*0008* or 0*032$ of the radius. Since th i s equipotential crosses the y—axis at y = -1.0164, and since the maximum half-angle determined by the maximum li n e c r i t e r i o n fo 114 this equipotential i s 0 = l8»9°* this equipotential i s obviously much more desirable for use as the matching equipotential ^ than the equipotential v = 2*75* For equipotentials v < 2 the va r i a t i o n i n curvature and of the gradient along the equipotentials increase rapidly for decreasing v, which makes them unsatisfactory for use as The question that arises next i s i f the outside surfaces of a semi-infinite parallel^plane anode can be changed i n shape i n such a way that the desired f i e l d characteristics are improved^ i*e*» the equipotential that i s suitable for use as the matching equipotential occurs closer to the anode aperture and has a shorter radius of curvature* It i s apparent from Figure (3-7) that to e f f e c t this improvement i t i s necessary to make the lines of force that issue from the aperture (the curves u —constant* where |u|<l) spread out more rapidly. Now the semi-infinite parallel-plane anode i s a special case of the anode geometry of Figure (3-3), with Cp^  = Cp^ == -^ .* Furthermore, the li n e s of force u = + 1 that leave the anode at the intersection of the inside and outside surfaces bisect the exterior angles between these surfaces. The magnitude of the angle that the flux lines u = + 1 make | c p | + n i n i t i a l l y with both surfaces i s thus 1 — ^ . Clearly, the way to increase this angle, and hence the spreading of the flux l i n e s , i s by making Cp^ and CJD^  more negative. However, the case of an anode consisting of two semi-infinite p a r a l l e l planes i s already the l i m i t i n g ease of physical r e a l i z a b i l i t y and any attempt to.make the angles and Cp^ more negative would require the outside surfaces to pass through the inside surfaces, creating electrodes of "negative thickness", a physical i m p o s s i b i l i t y . Mathematically this i s s t i l l possible, and has been discussed i n Section 3 s 5 for the case Cp^ = (p-j = - ^ * Since the region of interest i n t h i s "wrap-around f i e l d " i s the anode-aperture region, i n which the potential i s single valued, the f i e l d i n th i s region can be re a l i z e d by providing the required potential d i s t r i b u t i o n at i t s boundary* The v a r i a t i o n of the radius of curvature and of the centre of curvature along the equipotentials of this wrap-around f i e l d were obtained by the method of Appendix D, and are sl^own i n 2.00 V = 1.75 v = v = 1.2 0.4 0.2--0 -0.5 r-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0*4 0*5 Figure (4-14). Centre of curvature of equipotentials i n the 5 " wrap—around f i e l d 117 Figures (4-13 and 14). The v a r i a t i o n of the potential gradient along the equipotentials i s shown i n Figure (4-15)• For equi-potentials far from the anode, the magnitude of the potential gradient i s seen to decrease with increasing distance from the axis of symmetry. For equipotentials closer to the anode, the anode aperture causes an i n i t i a l drop i n the potential gradient near the axis of symmetry. As a r e s u l t , the potential gradient 4 at the surface v = 1,60 i s almost constant, to |u| = 0.9, varying by less than ^ from the average gradient over this i n t e r v a l . Comparison of the characteristics of the wrap-around f i e l d with those of the semi-infinite parallel-plane f i e l d reveals that the r a d i i of curvature of the equipotentials of the wraparound f i e l d are shorter and more constant. For example, the equi-potentials v = 1,75 i n the wrap-around f i e l d and v = 2.,00 i n the parallel-plane f i e l d , which are approximately the same distance from the i r respective anode apertures, have r a d i i of curvature that compare as followss Wrap-Around F i e l d Semi-Infinite P a r a l l e l Plane F i e l d v = 1,75 r q y = 2.074 max r . = 1.816 mm Ar = 0.258 v = 2.00 r = 2.651 max r . = 2.298 min Ar = 0.353 The above change i n radius for the wrap-around f i e l d occurs over a half—angle of © = 29.5% whereas the larger change of radius for the p a r a l l e l plane f i e l d occurs over a half-angle of only © .= 18.9°. Clearly, the desired characteristics of the wrap-around Both equipotentials pass through the points (x,y) = ( + 0.860, -0.866), 1.35 118 1.30-1.25, v=l .225 1.20+ 1.15+ 0.95+ 0.90+ 0.8-54— 1.0 |u| Figure (4-151. Variation of the potential gradient along equi-potentials of the wrap-around f i e l d 119 f i e l d are superior to those of the f i e l d about two semi—infinite p a r a l l e l planes. 4?4:2 E l e c t r o s t a t i c F i e l d i n the Anode. Region Approximated  by a Wrap-Around F i e l d A study w i l l now be made of the i n i t i a l l y r a d i a l j convergent flow of Figure (4*-10) f o r the case when the f i e l d i n the beam part of the anode region can be represented by the wrap-around f i e l d shown i n Figure (3-8), The procedure to be used p a r a l l e l s that of Section (4s3)« The i n i t i a l task i n th i s study i s to determine i n the wrap-around f i e l d an equipotential surface that i s suitable for use as the matching equipotential surface at potential The equipotential v = 1,60 i s seen to be suitable for this purpose, because the potential gradient along the equipotential i s almost constant and because the equipotential surface can be approxi-mated accurately over the region of interest by a section of a cylinder with centre of curvature at (x c,y c) = (0* 0.900000) and with radius r - 1.806283* This c y l i n d r i c a l section coincides o ^ with the equipotential v = 1*60 at |u| = 0.9, and deviates from thi s equipotential by less than 0*0032, or about ^ of the radius* The c y l i n d r i c a l section thus crosses the plane of symmetry x = 0 at ~f0 = Yc ^ *"-0? i*e», "yo = -0.906283. The equi-potential v = 1*60 crosses the plane of symmetry at y Q = -0,903119* The flow i n the cathode region w i l l be taken to be the same as that i n the region 1^ R> 0.45 of a concentric cylinder diode with R = 0.25 (notation as i n Sub-section 4s2:2); i . e . , R = 0*45* 120 From the d e f i n i t i o n of B i t f o l l o w s that the cathode rad i u s r can now be evaluated? r c = = 4.013962 o and hence the anode r a d i u s r of the e q u i v a l e n t c o n c e n t r i c c y l i n d e r diode i s r = B r = 1.003491 a a c Using these values of r and r , the parameters of the e l e c t r o n motion i n t h e cathode r e g i o n * equations ( 4 . 3 ) 7 become P o t e n t i a l <$> = 0.300150(rp 2) 3 E l e c t r i c I n t e n s i t y ctl - -.200100 2 M P^ + 23 dY <r3 2) 3 r (4.16) E l e c t r o n V e l o c i t y r = 3.2494 x 1 0 5 ( r p 2 ) 3 Charge D e n s i t y P = -L.18115 x 10 1 1 r ( r p 2 ) 3 where 4.013962> r > 1.806283. When equations (4.16a and b) are evaluated at r r e s u l t r . there o * r = r = 0.509549 o and (4.17) t = = 0. 439692 , r = r o 121 Equations (3.13), describing the potential i n the anode region, must next be rescaled so that they att a i n the potential <$> and gradient 1 at r = r . For this purpose the mapping * = Q( T) - ??k) (4.7) i s again required. Let = 0 as before; equation (4.7) then becomes u = v = Q ( ^ - O k ) (4.18) ¥hen equations (4.18) are substituted into equations (3.11) and the defining relations of the l a t t e r , i t i s seen that only the .parameters S and a are affected: Q2^ ^2 _ (<J>_ C|>k)2 2m - o>k) a = tan -1 2 W(<p _Cj>k) •V2 - ( < £ - CJX) 2 - ( ^ .Q. (4.19) Along the plane of symmetry of the anode region 'ty = 0, so i t follows from equations (3.13b and 4.19) that the potential Variation along the plane of symmetry i s TC x=0 |- S 4 ( f + 1) + J tan -1. / i i \ 2 S 4 ( r - s 2 ) / i I \ s 2 ( s 2 - 2) +iy +tanh -1 2S 4(l+S 2) ,2/„2 +2)+l)i (4.20) where S = Q2(<£> - C$> ) 2 + l . The derivative of y w.r*t, cJ 5 it x = 0 d v d<£ [ Q 2 ( Q - <4\)2 + 1 x = 0 T t ( <3> - Cj?, ) 122 The potential gradient along the plane of symmetry i n the anode region i s therefore x = 0 1 d£_ dcj> x = 0 [ Q 2 ( ^ - c £ k ) 2 + i ] 4 which becomes, when solved e x p l i c i t l y for Q, Q = - TC 4 5 1 2 (4*21) At t h i s point the procedure used to obtain the potential i n the anode region of the flow d i f f e r s from that used i n Section 4s3. In Section 4s3, the anode potential was set equal to unity by l e t t i n g Cf^ = 1 i n equation (4.1l); the analogous equation i n the present problem i s equation (4.2l). Upon substitution of £^ and into equation (4.1l), the scaling constant Q, and subsequently y Q were obtained. In the present case we cannot follow t h i s procedure, because i t was necessary to specify r Q , and hence y Q» i n order to determine from the cylindrical-diode equation (4.3b). Therefore, i t i s not permissible to specify i n equation (4.21), otherwise too many 123 constraints are placed on the anode f i e l d . The two quantities to be solved for i n the present case are thus Q and ^ J ^ * If conditions (4.17) are substituted into equation-(4.21) r there results 6 '5 1 2 Q = -^4.821709(0.509549 -4^) - (0.509549 - ^ J " 2 ] (4.22) Further* i f the values of y and Cj) are substituted for y and o o 17 <$> respectively i n equation (4.20), we obtain 0 = 1.4031189 +,JJ-S 4 ( f+ l )+ \ TC tan -1 / 1 1 . 1 1 1 1 \ 2S 4(1-S 2) +tanh~"'' 2S 4(l+S 2) 1 1 1 1 \S 2(S 2-2)+lj lS 2(S 2+2)+l/ (4.23) where S = Q2(0.509549 -Q^)2 + 1. The solution of simultaneous equations (4.22 and 23) i s Q = -2.337807 ^ k = 1.193951 (4*24) The potential i n the anode region of the electron beam i s described by equations (3.13) and their defining relations (4.19), with Q and <$>k as given by (4.24). The potential v a r i a t i o n along the plane of symmetry of the gun has been calculated by using equation (4*16a) for the cathode region and using (4*20 and 24) for the anode region; t h i s potential v a r i a t i o n i s shown i n Figure (4-16). The maximum Value of the wedge half-angle 0 of the beam i n the cathode region i s lim i t e d by considerations of beam - <PQ = 0.509549 -3.5 / Cathode — • 2 . 5 - 2 . 0 -1.5 y. = -3.11396 k -1.0\^ -0 .5 0 y = 0.906283 ^ o 0.5 / 1.0 1.5 y Figure (4-16). Potential variation along the plane of symmetry of an i n i t i a l l y r a d i a l , ; convergent electron beam to interception by the anode* If the t r a j e c t o r i e s near the beam boundary continued onward i n a straight l i n e from the equipotential ^ 0 t the angle at which electrons would graze the mouth of the anode would be 29*05°, In an actual electron gun, the beam boundary w i l l be i n d i s t i n c t due to i n i t i a l thermal v e l o c i t i e s of the electrons* The beam boundary i n the anode region i s also broadened by the beam space-charge. The maximum value that can be used for the half—angle © of the cathode i s thus a few degrees less than 29°; the actual value of © i s probably best found by experimentation, although valuable guidance i n this regard can be obtained from a preliminary study of the t r a j e c t o r i e s by analogue or by numerical methods. Let © a r b i t r a r i l y be taken at 25°, so that an estimate of the gun perveance Can be made. Since the flow i n the cathode region of the gun i s , by hypothesis, the same as that occurring i n the corresponding region of an equivalent concentric-~eylinder diode with r = 4*01396 and r = 1,00349, the beam current per c a * (3) unit length i s v I k 2 V r 3 ' (l.8o) (4.25) where k 2 -and where V i s the anode potential of the equivalent diode* El To maintain the o r i g i n a l conditions i n the cathode region when a wrap—around f i e l d was used i n the anode region* a normalized anode potential greater than unity was found to be required* = 1.19395 = — . On combining this with equations ( l . l and 4,25), the perveance per unit length i s 126 K _ k 2 ° L ~ 2 180 r 3 2 Q 2 a a k or 3 K —6 2 = 0*26 x 10 amp/volt meter. 4s5 Discussion The method employed i n thi s chapter to adapt spaee**charge free anode f i e l d s aud space—charge—limited flow solutions to electron gun design can be used for other anode f i e l d s and flow solutions. For example* the solution for space—charge—limited (36) flow from a cathode consisting of two in c l i n e d plates ' could be used; th i s solution has properties that make i t desirable for use i n electron gun design. With appropriate modifications* the method of thi s chapter could also be used for the design of axially-symmetrie guns* 1 2 7 CHAPTER V - THE DETERMINATION OF BEAM-FORMING ELECTRODES L i t t l e has been said so far about the electrodes required to produce the desired f i e l d conditions at beam boundaries. In pa r t i c u l a r , i t i s desired to determine beam-forming electrodes for the two examples i n the previous chapter. To this end, a b r i e f account w i l l f i r s t be given of the physical considerations which apply to this problem. 5:1 Physical Considerations The "engineering problem" of determining the shape of electrodes that w i l l produce prescribed f i e l d conditions at the boundary of an electron beam i s not a problem that arises i n a natural or a direc t way? rather* i t i s an inverse problem. Implicit i n thi s problem are the assumptions that (a) such physically realizable electrode shapes exist, and (b) the electron beam i s i n a stable configuration. Neither assumption ( 5 2 53} i s necessarily warranted* For example, i t has been shownv *J~'> that a hollow beam confined by an axial magnetic f i e l d i s i n an unstable configuration. An example of a case where the prescribed boundary conditions cannot be met by beam—forming electrodes i s the case of two p a r a l l e l , space—charge—limited s t r i p beams (see Appendix E ) . If analogue equipment, such as an e l e c t r o l y t i c tank, i s used to obtain the shape of the beam-forming electrodes, then the procedure involved i s generally one of t r i a l and error, with some guidance from theory? i»e., the electrode shapes of the analogue model are adjusted u n t i l the boundary conditions are 128 s a t i s f i e d to a s u f f i c i e n t accuracy. If analytic methods or i f numerical methods are employed to obtain the shape of the beam-forming electrodes* then the procedure generally followed i s : (a) Laplace's equation i s solved exterior to the beam surf ace ~ an open boundary — subject to Cauchy-type boundary conditions, and (b) beam-f ormisng electrodes are arranged along equipotential surfaces exterior to the beam* generally and preferably only along the equi-potentials at cathode and at anode potential* Suecess i n part (b) of the procedure i s contingent upon the solution being w e l l -behaved i n the v i c i n i t y of the beam. The remainder of t h i s section w i l l be devoted to part (a) of the procedure* the nature of the solution. A boundary—value problem i s properly set " i f and only i f i t s solution exists* i s unique and depends continuously on the data assigned"^ 4^» A Gauchy problem for Laplace's equation on an open boundary i s an improperly set problem, because the solution does not depend continuously on the boundary conditions^ That this i s so can be readi l y demonstrated with an example contrived by Hadamard^ 4^* Consider the Laplace equation i n two dimensions, o x d y which i s to be solved subject to the Cauchy conditions O (0,y) = 0* S £ ^ sin(ny) (5*2) 0 129 where n i s a large number* Equations (5*1 and 2) are solved by <J>(x,y) - ~£ sin(ny)sinh(nx) * (5.3) n It i s seen that by increasing the value of n, the Cauchy conditions (5*2) can be made as close to zero as desired* The solution of (5*1) when the Cauchy conditions are i d e n t i c a l l y zero i s <$> = 0* However, the solution (5*3) i s by no means i d e n t i c a l l y zero; for large x, sinh(nx) grows as e n x * so that (5*3) o s c i l l a t e s with an amplitude that increases i n d e f i n i t e l y * Any attempt to approximate zero Cauchy conditions more closely by increasing the value of n w i l l increase the amplitude ®f the o s c i l l a t i o n * The solution exhibits the same discontinuous dependence on the boundary conditions when, instead of the Cauchy conditions (5.2) eh®sen by Hadamard, the Cauchy conditions 0>(0ry) - i sin(ny) , n o x = 0 (5.4) x = 0 are s p e c i f i e d . In this ease (5.1 and 4) are solved by ^ (x,y) = i - sin(ny)cosh(nx) (5*5) which* for large x-j again grows as e » The above examples i l l u s t r a t e that i f the potential or the gradient or both are not prescribed exactly ( i . e . , are prescribed by analytic functions) on the boundary surface, then the e r r t r i n the data* no matter htw small i t may be, can cause an error i n the f i e l d which increases exponentially with distance fr®m the boundary. The rate of error growth i s discussed i n mitre 1 3 0 d e t a i l i n Appendix F« Conversely* the s e n s i t i v i t y of the f i e l d conditions at the beam boundary to errors i n the f i e l d conditions elsewhere decreases at an exponential rate with increasing distance from the beam boundary* The converse result has two most important consequences already mentioned; namely, that the electrodes can be truncated at a reasonable distance from the beam** and that some degree of v a r i a t i o n i s possible i n the electrode shapes* 5 82 Design Procedure An e l e c t r o l y t i c — t a n k procedure for obtaining beam—forming electrodes for the i n i t i a l l y r a d i a l , convergent flow of Section 4 § 4 and for the i n i t i a l l y p a r a l l e l , r e c t i l i n e a r flow of Section 483 w i l l now be e©nsidered» The l a t t e r case w i l l be taken f i r s t o It w i l l be Recalled that the electron motion i n the cathode region was assumed, to be the same as that occurring i n a strip-beam Pierce gun* The shapes of the beam-forming electrode at cathode potential and of the equipotential surface O 0 are thus prescribed (the outward analytic continuation of equations (E.2 and 3))* In Chapter 4* the potential and the potential gradient of the f i e l d s i n the cathode and i n the anode region were matched inside the beam at the equipotential Cj> Q* The f i e l d conditions outside the beam were assumed to be s a t i s f i e d by the use of an auxi l i a r y anode* as shown i n Figure ( 4 — 1 . ) * It i s now desired In some exceptional cases* the beam-forming electrodes may completely enclose the space exterior to the beam, making truncation of the electrodes unnecessary(56) e 131 to eliminate the need for this a u x i l i a r y anode by a l t e r i n g the main anode surface i n such a way that the potential and the potential gradient are s u f f i c i e n t l y well-matched along the equi-potential surface so that the f i e l d i n the beam region i s neg l i g i b l y affected by the removal of the aux i l i a r y anode. Prom the e a r l i e r treatment of the f i e l d at the anode i t can be shown that an adjustment of the angle X alone,.keeping the anode faces plane, could not be expected to provide the desired r e s u l t . A more promising approach would be to keep the angle X fixed at ^  > whereby i t i s already known that a good match occurs i n the beam region at the surface By curving the anode away from the cathode i t should, be possible to improve the match at points on further^removed from the beam without seriously affecting conditions at the beam. It is a d i f f i c u l t problem to determine the required anode shape a n a l y t i c a l l y , but an analogue method* u t i l i z i n g an e l e c t r o l y t i c tank or a r e s i s t o r network could be used. An e l e c t r o l y t i c tank model of the electron gun i s shown i n Figure (5—1). An electrode which coincides with the equipotential O Q has been inserted. The required anode shape i s obtained by adjusting the contour of the electrode representing the anode u n t i l the potential gradient on either., side of ? measured by probing points as indicated, becomes approximately equal, and the potential gradient inside the beam boundary at ($> remains approximately o constant. The determination of beam-forming electrodes f o r the r a d i a l l y convergent flow i s more d i f f i c u l t a n a l y t i c a l l y , since electrode shapes must f i r s t be found which w i l l r e a l i z e the wrap—around f i e l d inside the beam* The main problem to be solved i n thi s regard' involves the termination of the f i e l d before i t becomes multi-valued. The determination of the beam-forming electrodes can be carried out i n two steps; namely, (a) by designing a gun with a two—potential main anode and an a u x i l i a r y anode, and (b) by subsequent a l t e r a t i o n of the shape of the main anode to eliminate the a u x i l i a r y anode. (a) Since the electron motion i n the cathode region i s by hypothesis the same as the convergent electron motion i n a concentric—cylinder diode, the shape of the beam-forming electrode at cathode potential and that of the lower surface of Electrode Figure (5-1). E l e c t r o l y t i c tank model of an i n i t i a l l y p a r a l l e l , r e c t i l i n e a r - f l o w electron gun 1 3 3 the a u x i l i a r y anode are thus prescribed . The shape of the upper surface of the au x i l i a r y anode i s also known, since by hypothesis i t coincides with the equipotential v = 1,60 of the wrap-around f i e l d . The determination of the shape of the main anode i s more d i f f i c u l t , involving some approximations. Figure (5-2). I n i t i a l l y r a d i a l , convergent-flow electron gun with an a u x i l i a r y anode and a two-potential main anode 7V The f i e l d outside a wedge beam has.been found by the use of an e l e c t r o l y t i c tank analogue by P i e r c e a n d by the use of an analytic method by Radley(7). 134 It can be seen by refe r r i n g to Figure (3-8) that the f i e l d at the mouth of the anode aperture i s very intense. To reduce the danger of arcing to other electrodes, and to aid the physical r e a l i z a t i o n of the wrap-around f i e l d i n the beam region, i t i s desirable to choose for the anode surface* i n the region of the mouth of the aperture, an equipotential other than v = 0. The equipotential surface v to be chosen for t h i s purpose i s a compromise because, although equipotentials with a larger value of v have more rounded corners at the mouth of the anode aperture, these equipotentials are also more prone to intercept the electron beam. The equipotential v = 0.05 appears to be a . suitable compromise* By refe r r i n g to Figure (3-8) i t i s observed that nine—tenths of the lines of force that enter the anode aperture intersect the equipotential v = 0.05 i n the i n t e r v a l between the mouth of the anode aperture and y = 0.5* For values of y greater than 0*5* the f i e l d i s thus very weak* and the f i e l d w i l l be r e l a t i v e l y unaffected i f the anode wall deviates from this equipotential as shown i n Figure (5-2), so that electron-beam interception i s avoided. It remains to s a t i s f y the f i e l d conditions on the outside of the main anode* This can be accomplished to a good approxi-mation by inserting a second anode along an equipotential of the wrap—around f i e l d that l i e s close to the mouth of the main anode, yet i s far enough away that at the upper end this equipotential does not cross oVer into the multivalued region u n t i l i t i s well away from the mouth of the main anode. From Figure (3-?8) i t i s seen that the equipotential v = 0.75 i s well suited for this purpose. The complete electrode configuration i s shown i n 135 Figure (5-2). (b) To permit the a u x i l i a r y anode to be removed, i t would be necessary to reshape the contour of the main anode. An e l e c t r o l y t i c tank procedure similar to that described for the p a r a l l e l beam case could be used for this purpose. Such a model i s shown i n Figure (5-3). Figure (5-3) E l e c t r o l y t i c tank model of an i n i t i a l l y r a d i a l , convergent-flow electron gun 136 58 3 Discussion There i s scope for further work on this approach to gun design. The consideration of a x i a l l y symmetric guns* p a r t i c u l a r l y convergent ones i n which the f i e l d i n the anode region i s similar to the wrap-around f i e l d , i s very desirable. To optimize the electrode shapes* an experimental investigation of prototypes of these gun structures i s essential; i n thi s way also, guns with higher perveances than the theoretical figures obtained above can no doubt be attained. In the i n i t i a l l y i r a d i a l , convergent flow gun described above, the use of the wrap-around f i e l d i n the anode region has the effect of reducing the f i e l d d i s t o r t i o n , due to the anode aperture, i n the cathode region. This effect has been obtained (17) by the use of an intensifying electrode by Muller and (12) Brewer , and by al t e r i n g the shape of the cathode by other investigators. Other means that could be employed to reduce or compensate for the f i e l d d i s t o r t i o n i n the cathode region ares (a) d i e l e c t r i c s between the beam-forming electrodes, and (b) non-unipotential cathodes. A d i e l e c t r i c block of suitable shape and d i e l e c t r i c constant, placed near the beam-forming electrode at cathode po t e n t i a l , could be used to a l t e r the shape of the equipotentials i n the cathode region i n such a way as to neutralize the f i e l d d i s t o r t i o n due to the anode aperture. A more exotic but impractical example i s the problem of bounding the f i e l d i n a s t r i p beam by d i e l e c t r i c material placed outside the beam. Since the potential along the tr a j e c t o r i e s of this beam varies 4 3 as y and since the normal gradient along them i s zero* the 1 3 7 d i e l e c t r i c material would need to have an anisotropic d i e l e c t r i c constant varying as y i n the y-di r e c t i o n , and remaining constant at e i n the x-direction. For current-type analogues of electro-s t a t i c f i e l d s the r e s i s t i v i t y at a pa r t i c u l a r location i n the model i s proportional to the d i e l e c t r i c constant of the actual medium i n which the. f i e l d i s being studied. An analogue study of the effect of the shape and d i e l e c t r i c constant of a d i e l e c t r i c block between the beam-forming electrodes on the f i e l d i n a proposed gun structure i s thus quite f e a s i b l e . If a network analogue i s used* the d i e l e c t r i c block can be represented by suitably changing the values of the r e s i s t o r s ; i f Teledeltos paper i s usedf the r e s i s t i v i t y can be adjusted by using several layers of paper. By the use of a non-unipotential cathode, the off—cathode gradient can be made more uniform, thus improving the uniformity of emission when the f i e l d d i s t o r t i o n due to the anode aperture i s severe. Although a laminated cathode with insulated laminae could be used f o r t h i s purpose, a more p r a c t i c a l embodiment would probably be a cathode made of a r e s i s t i v e material. By appropriately adjusting the thickness of the cathode, the desired potential v a r i a t i o n along the cathode surface could then be attained. CHAPTER VI - CONCLUSION 138 By choosing certain convenient functional forms O f the action function* potential and metrical c o e f f i c i e n t s * some new solutions have been obtained for electron motion i n el e c t r o s t a t i c f i e l d s * by the method of separation of variables. A study of the f i e l d s and of the electron t r a j e c t o r i e s of these solutions has revealed some interesting properties. The e l e c t r o s t a t i c f i e l d about three idealized two-dimensional anode geometries has been derived. These geometries ares (a) a plane with a s l i t , (b) two right-angled plates, and (c) two semi^-inf i n i t e p a r a l l e l plates. The wrap-around f i e l d , an anode f i e l d with improved charac t e r i s t i c s * has resulted from a study of the characteristics of the above three f i e l d s . I t has been shown how use may be made of the above space-charge-free anode f i e l d s i n the design of electron guns* An estimate has been made of the error introduced by approximating the f i e l d i n the anode region by a space-charge—free f i e l d . An i n i t i a l l y p a r a l l e l * r e c t i l i n e a r flow gun and an i n i t i a l l y r a d i a l , convergent flow gun have been designed as examples. The i n s t a b i l i t y of the Pierce-Cauchy problem has been discussed, and an estimate has been made of the rate at which errors i n the Cauchy data are propagated when the beam boundaries are a x i a l l y symmetric. An e l e c t r o l y t i c tank method has been suggested for the determination of beam—forming electrodes for the two above guns that obviates the need for an au x i l i a r y anode. 139 APPENDIX A - ESTIMATE QF SELF MAGNETIC FORCES AND BELATIVISTIC EFFECTS Both self-magnetic forces and r e l a t i v i s t i c effects depencl on the electron v e l o c i t y . The theory of Chapter II assumes that the electron mass i s constant; however, according to r e l a t i v i t y theory i t i s given by 9 where mQ i s the rest mass of the electron, and c i s the v e l o c i t y of l i g h t . An electron that, starting from r e s t , has been accelerated through 20 kV w i l l have a v e l o c i t y of 0*28cV From the above equation* the increase i n electron mass i s thus 3.93$, which generally can be ignored. To study the v a l i d i t y of the neglect of self—magnetic forces* consider the case of a p a r a l l e l c y l i n d r i c a l electron beam of radius R, constant charge density p , and t r a v e l l i n g at constant v e l o c i t y The e l e c t r i c i ntensity E r due to the space charge of the beam i s directed r a d i a l l y inward and at a radius r<R i t i s "' 0 The e l e c t r o s t a t i c force F experienced by an electron at radius re J r i s thus » « - - « r — ( 2 ^ P r • and i s directed r a d i a l l y outward. The magnetic induction BQ due to the beam current i s , at a radius r R, 1 4 0 The magnetic force F r m experienced by an electron at radius r < R i s thus and i s directed r a d i a l l y inward. The ratio of the magnitudes of the two forces i s 2 2 % = ^o £o "*£ = — T • c At 20 kV the r a t i o of the forces i s 0.078. It i s apparent that for electron energies greater than 20 keV the self—magnetic forces generally need to be taken into account. At 10 kV tb.e ratio of the forces i s TTF » 25 In evaluating the self-magnetic force^ i n electron beams of actual electron guns, other considerations are involved besides the one obtained for the above idealized beam model. Some of these considerations are discussed i n reference (57). F rm re 141 APPENDIX B - DEMONSTRATION OF THE EXISTENCE OF V(q l f. q 2 , q_3) = c 1 ¥HEN THE CONDITION V x f = 0 IS SATISFIED For electron motion occurring i n an e l e c t r o s t a t i c f i e l d and starting from rest at a zero-potential cathode* the condition ( 3 5 3 9 ) V x V = 0 i s s a t i s f i e d , by Lagrange's invariant theorem * /, Therefore we may write or dW = ^.ds It i s seen that ¥ i s constant when ds i s normal to I T » Let us c a l l t h i s normal d i f f e r e n t i a l vector dn. ¥e then have d¥ = 0 for ar.dn = 0 But the P f a f f i a n d i f f e r e n t i a l equation 'TXdn = 0 i s integrable because by hypothesis "XT s a t i s f i e s the condition v^7x V" = 0* and therefore i t follows that there exists between the space-coordinates q^* q 2 j and q^ a one-parameter family of surfaces q 2 » q 3) = c]_ (34) normal to the t r a j e c t o r i e s 142 APPENDIX C - NUMERICAL METHOD FOR OBTAINING ELECTRON TRAJECTORIES IN ELECTROSTATIC FIELDS C t l Space-Charge. Effects Neglected Consider the motion of an electron entering a uniform e l e c t r i c f i e l d E at (x Q>y o) with i n i t i a l v e l o c i t y (* o,y Q), as i l l u s t r a t e d i n Figure ( C - l ) . From equation (2.8) the equations of motion are * = - V E x '--Vfy Integration of these equations w.r.t. time results i n J and x = - 71E t + x / x o y = - T? E y t + y o E = - 7 7 t 2 + x t + x 1 2 o o E + y. ( c i ) (C.2) (C.3) Figure ( C - l ) . Motion of an electron i n a uniform e l e c t r i c f i e l d 143 The parameter t i n equations (G.3) can be eliminated by the use of equations (C.2); taking the sum of the squares of the l a t t e r , we obtain ' U 2 = 7 ? 2 E 2 t 2 - 2 7 7(E x* o + E y y Q ) t + O T 2 . (C.4) I t w i l l be assumed that the t o t a l electron energy i s zero; therefore, from (2.13), n r 2 = 2T)<& . (C*5) Upon substitution of equation (C»5) into (C.4) and solving the l a t t e r for t, there results t = - ± 7]E2 ( E x + E y ) + i f 2 7 ) C £ E 2 - (E y - E x ) 2  v x o yJ o' u / N x J o y o' (C.6) Equations (G.3 and 6) describe the parabolic trajectory i n a uniform f i e l d i n terms of the parameter . of the e l e c t r i c i n t e n s i t y components, and of the i n i t i a l v e l o c i t y components and position coordinates. In e l e c t r o s t a t i c f i e l d s that are not uniform but that are slowly varying i n spacer electron t r a j e c t o r i e s can be obtained by considering the e l e c t r o s t a t i c f i e l d to be uniform over short i n t e r v a l s , and applying equations (C.2, 3 and 6) to plot the trajectory i n each i n t e r v a l * The size of i n t e r v a l to be used i n a p a r t i c u l a r region of a f i e l d depends on the desired accuracy of the t r a j e c t o r i e s , on the degree of non-uniformity of the f i e l d , and on the round—off errors i n the computations. Let j be an index denoting the interval number along an electron trajectory. Thus ( x ( j ) , y ( j ) ) are the coordinates of 1 4 4 the j ' t h point on this trajectory. The equipotential passing through the j ' t h point i s OCj). If the e l e c t r o s t a t i c f i e l d over the j ' t h i n t e r v a l of the trajectory i s approximated by a uniform f i e l d , conditions w i l l be as i l l u s t r a t e d i n Figure (G~2)» In p a r t i c u l a r , i t i s seen that .the equipotentials ^ ( j - l ) and O(j) are distorted to the p a r a l l e l straight lines O(j-l) and respectively* so that the trajectory obtained by using the uniform-field assumption crosses the li n e ^ ( j ) u at (j)u» A f i r s t - o r d e r correction w i l l be made by extrapolating the trajectory to the point where i t intersects the equipotential 3>(j); this point w i l l be taken to be the j ' t h point on the trajectory. 3) ¥ ( 3 - 1 ) y ( j - i x ( j - l ) x(j) x(¥(j-i),<£(jj) Figure (C-2). Electron path i n the j ' t h i n t e r v a l of a non-uniform f i e l d , showing the effect of a uniform-f i e l d approximation. The i n t e r v a l size i s greatly exaggerated,. 145 The v e l o c i t y components and the coordinates of the electron at the point (j) are, from equations (C.2, 3 and 6) * ( j ) u = - 7 / B x ( j ) t ( j ) + x ( j - l ) y ( j ) u = 7 7 E y ( j ) t ( j ) 4- y ( j - l ) (G.7) x ( j ) u = t($j) [- ^ E x ( j ) t ( j ) + x ( j - l ) ] 4- x ( 3 - D y ( j ) u = t ( j ) [- ^ E y ( j ) t ( j ) + y ( j - l ) ] + y ( ^ . l ) (G,8) where t ( j ) = 7?E 2(j) J E x ( j ) x ( j - l ) 4- E y ( j ) y ( j - l ) [2770(j)E 2(j) - ( E x ( j ) y ( j ~ l ) - E y ( j ) x ( j ~ l ) ) 2 ( C 9 ) and where the e l e c t r i c intensity components are as described below i n equations ( C . l l and 14). The e l e c t r o s t a t i c f i e l d s i n which i t i s desired to obtain electron t r a j e c t o r i e s are described by equations of the form x = xOJ>-,4>) y = y(^,Cf>) where the curves constant are lines of force. The magnitude of the e l e c t r i c intensity i n the j'th int e r v a l can therefore conveniently be taken as (G.10) E(j) = ^ ( . i ) - 3>(,i-i)l ( c .n) where D(j) i s the distance along the secant lin e joining the point ( j - l ) and the point marking the intersection of " ^ ( j - l ) and " ^ ( j ) , as shown i n Figure (G-2), It follows that 146 D( j) = \ / A X 2 ( J ) + Ay 2 ( j ) (C.12) where Ax( j) = x ( ^ ( j - l ) , < £ ( j ) ) - x ( ¥ ( j-l)» <2>(.j-l)) Ay( j) = y ( T ( j - l ) , <3>(j)) - x ( ¥ ( j - l ) , 4>( j-l)) From equations (G»ll—13) i t i s readily seen that (C13) (C14) for <$>U)> ^ ( j - 1 ) . To extrapolate the trajectory from the point ( j ) u to i t s intersection with the equipotential ^ ( j ) ve shall approximate the trajectory between these two points by a straight l i n e whose slope i s the same as the slope of the trajectory at (j)u» The str a i g h t - l i n e trajectory i s thus ax + b (C.15) where y ( j ) a u and b u y( j ) y ( j ) „ - jTjy- X U ) u u u The remaining problem i s to determine the coordinates of the intersection of equation (C.15) with the equipotential (^j); the l a t t e r i s seen from equations (C.10) to be described by equations of the form x = x(^,0(j) y=y(V,3>(jj) (C.16) 147 The f o l l o w i n g i t e r a t i v e procedure can be used t o determine the c o o r d i n a t e s of the i n t e r s e c t i o n : ( l ) Solve e q u a t i o n s (C.16) w i t h a s u i t a b l e i n i t i a l guess f o r ^ P ( j ) . For the f i r s t i n t e r v a l i n a t r a j e c t o r y a s u i t a b l e v a l u e f o r ^ ( j ) i s g e n e r a l l y the v a l u e of x - i / ( j - l ) ( i . e . , when j = l , use the v a l u e of ^/"(O) as an i n i t i a l guess f o r vK(l))« For subsequent i n t e r v a l s , b e t t e r i n i t i a l guesses of ^ ( j ) can be made by n o t i n g how ^ v a r i e s a l o n g the t r a j e c t o r y from i n t e r v a l to i n t e r v a l , I f the i n i t i a l guess i s denoted by 4 ^ ( j ) , e q u a t i o n s (C.16) become x T ( j ) = x ( ^ ( j ) , r P ( j ) ) yj(3-) = y ( r ( j ) ) (C.17) where- ( x ^ ( j ) , y - ^ ( j ) ) are the corresponditngi i n i t i a l 'guesses of the ' ; d e ^ s i r e d i : c o o r d i n a t e s — ( x ( j )y y( j ))\, (2) • S u b s t i t u t e (C.17a) i n t o (\Ga 5 ) : t o : dete rmi-ne:. the-value, of •the - i n i t i a l e s t i m a t e of -'the y - c o o r d i n a t e , '.y^Cj:")',.''-o'nv^the-^str.aight' l i n e t r a j e c t o r y ; i . e . , y"~ ( j ) = ,ax-L (j ) + b (3) D e f i n e an e r r o r f u n c t i o n as e i = ^ i ^ ^ ~ y 2 (J ) (4) I f e .< e whe re e i s a p r e d e t e r m i n e d e r r o r bounds then ( x2 (3 ) > y^ (3 ) ) are • •tafcem-'tw"be* the •  c o o r d i n a t e s of the j 1 t h p o i n t on the; t r a j e c t o r y , and the computations f o r ; the' j ' t h i n t e r v a l are completed. (5) I f e-, >^ e Q , s u b t r a c t an i n c r e m e n t a l / from ^ ( j ) » 148 obtaining and repeat steps (l) - (4) above; (a) x 2 ( j ) = x( <J>(j)) y 2 ( j ) = y( %(j)f <J>(j)) (b) T2^) = a x 2 ^ + b (e) s 0 =y" 2(j) - y 2 ( j ) ? 2 1 ^ ' 2 ~ J2 (<*) e J ^ e (6) If e 2 >|e Q > the i t e r a t i o n must be repeated. Since from the f i r s t two it e r a t i o n s two values on the curve of e vs. "^"(j) are known, the t h i r d value of "^(j) to be used for the i t e r a t i o n can be predicted from a straight l i n e approximation (see Figure (C-3))* Thus e 2 ^ ( d ) - e i ¥ 2 ( j ) ( e 2 - e l) (7) In general* after n i t e r a t i o n s , the curve ^ ( j ) = f(e) can be approximated by a (n- l ) ' t h degree polynomials ¥(3) 2 3 0 1 2 3 n-l n-1 The co e f f i c i e n t s a. can be determined from the set of equations: 149 -p a •H O P4 CD Predicted Point ( £2 'V 2 (d) Figure (C-3). Prediction of Nfc(j) i n the i t e r a t i v e process 2 3 e^a^ + e-^  a 2 + e-^  a^ + . . e 2 a l + e 2 2 a 2 + e 2 3 a 3 + 2 3 e 3 a l * e3 a2 + e3 a3 + * * + e i n " l a n - l + ao = \ M + e 2 n " l a n - l + ao = % M + E 3n - 1 a n _ 1 + a Q = ¥ 3 ( 3 ) E n a l + e n 2 a 2 + £ n 2 a 3 + * * + e n _ 1 a , + a = ¥ (j) n n-1 o n d ' and thus 0 n n n-1 n-1 n-1 n-1 n where A i s the determinant of the coef f i c i e n t s a The t r a j e c t o r i e s shown i n Figure (4-9a) were obtained by 150 the above method* It vas found that i n most cases only four —6 ite r a t i o n s were required for an error bound of e = 10~ . . ': o C:2 Correction for Space-Charge Forces Some useful approximations can be made for the beam discussed i n Section 4:3 to take into account spreading of the beam due to space-charge forces. This beam and the electro-s t a t i c f i e l d are symmetric about the plane x = 0, the transverse v e l o c i t y of the beam i s small compared to i t s a x i a l v e l o c i t y ( x<<y) and, i n the cathode region, conditions i n the beam are assumed to be independent of the transverse direction* If image charge effects due to the electrodes are ignored, i t i s there-fore not unreasonable to assume that i n the anode.region the flux due to the beam i t s e l f i s transversely directed, whence by Gauss' law o where E = e l e c t r i c i n t e n s i t y due to the electrons only, at a s point (x,y) i n the beam X = charge/unit length/unit depth contained by an incremental beam section of width 2x at (x,y)» It follows from the d e f i n i t i o n of X, and equation (2*6) that In the beam under study i n Section 4:3, we shall not be far wrong i f i t i s assumed that the electron v e l o c i t y i s constant across the beam because, by the time the f i e l d becomes markedly 1.51 non-uniform, the electrons are a l l t r a v e l l i n g near terminal v e l o c i t y . Rewriting equation (C.19) and combining i t with (C»18) results i n x J(y ) n / \ o w o ' l — 2 (C20) where (see Figure (C-^4)) E (x,y) = e l e c t r i c intensity due to the electrons only, at s a point (x,y) on the trajectory that had coordinates (x Q,y o) when i t passed through Cj)^ J(y ) = current density of the beam at the equipotential, ^ ( x j y ) = ^"{^(Ojy) + 0(x,y)J , an approximate average of the potential across a beam cross-section of width 2x at (x,y). x Beam Boundary Figure (C-4)» Space-charge effects in,the anode region 152 The contribution of space-charge to the e l e c t r o s t a t i c f i e l d i n the anode region can be incorporated into the trajectory-c a l c u l a t i o n procedure of Section Csl by evaluating equation (C*20) at each trajectory interval and adding i t to E (j) i n equations (C.7-^9)» The tr a j e c t o r i e s shown i n Figure (4-9b) were obtained by th i s modified method. 153 APPENDIX D - ANALYSIS OF THE CURVATURE OF AN EQUIPOTENTIAL Consider a continuous curve v^ which i s described i n terms of the parameter u by the equations x = x(u) y = y(u) (D.l) We wish to determine the radius of curvature and the centre of curvature of v^ at u^, as shown i n Figure (D-l). For this purpose, consider i n addition two nearby points u^_^ a n < i u±+± o n v^» The centre of curvature of v^ at u i i s then obtained by the following procedure! (l.) Join the points u^ ^ and u^, and u^ and u^ +^ on v^ by two secant l i n e s , (2) Bisect the secant lines and erect lines perpendicular to them passing through the bis ect ;. points. (x .,y •) v c i ' J c i Secant Lines •9*- x Figure (D-l). Centre of curvature of v, at u. 154 (3) The intersection of the two perpendiculars i s then, i n the l i m i t as u. , and u. ,,-l - l l+l at the point u^. u^, at the centre of curvature of v^ Algebraically this i s accomplished as follows: The slope of the secant l i n e passing through u^_^ and u^ i s y± - y i _ i m. = l x, — x. (D.2) i-1 while the slope of the l i n e passing through u^ and u^ +^ i s m _ y i + l y i i +1 ~ x. ,, - x. l+l l (D.3) Thus the slopes of the li n e s perpendicular to these are and "i+1 ' x i + l x .> l i+1 - y< (D.4) (D*-5.) respectively. The bisecting point on the l i n e u^ ^ - u^ has coordinates x, . = x. ^ + b i l — l y ^ i y i - l \ b i i-1 (D.6) The bisecting point on the l i n e u^ - u ^ + 1 has coordinates x. + i X b(i+1) y b ( i + D = y± + l+l i> y i + l ~ y i (D*7) 155 Thus the equation of the perpendicular bisector of - u^ i s (y - y b i ) = s,(x - x w, ) bi' (D.8) and the equation of the perpendicular bisector of u^ - u ^ + ^ i s ( y - y b ( i + i ) = s i + i ( s i + i l x - x b ( i + i (D.9) Substituting equations (D.4 and 6) into (D.8), and (D.5 and 7) into (D.9). we obtain y + A ix = B^  (D.10) and y • A 1 + 1 X = B 1 + 1 (D.ll) respectively, where x. r* x. , . _ l i - i 1 " y± - y i - i and B i = 2 (x 2 - x. 2 Equations (D.10 and 11) intersect at x . = c i c i Bi+1 - B i  A i + 1 " A i B i A i + l " B i + l A i A i + 1 " A i J (D.12) and i n the l i m i t , as u. , and u. , >u., (x . ,y .) i s the 1*"*J. 1T1 1 C l cx centre of curvature of the curve v^ at (x^,y^). For computational purposes, we need to ret a i n the small increments u^ - u^ and 156 - the size of the increments depending on the behaviour of v,, and the accuracy to which (x . ,y .) i s desired. Prom Figure (D-l) the radius of curvature i s readily seen to be r i = " x c i ) 2 + ^1 - y c i ) 2 (D.13) while the angle with which r^ intersects the y-axis (the slope of v^ at u^) i s = tan' Equations (D.12, 13 and 14) were used to compute the centre of curvature (x .,y . ) * the radius of curvature r., and the angle C1 C1 X 0- for a series of points u^ on equipotentials i n the f i e l d about two semi-infinite p a r a l l e l planes and i n the wrap-around f i e l d (see Figures (4-11 to 15)). (D.14) 157 APPENDIX E - ELECTROSTATIC FIELD REQUIRED TO MAINTAIN T¥Q PARALLEL, SPACE-CHARGE-LIMITED STRIP BEAMS For simplicity the s t r i p beams w i l l be assumed to be of vanishing thickness; no generality i s l o s t by this assumption, since the f i e l d required outside two beams of f i n i t e thickness i s the same as that required for the case to be discussed. In Figure (E-l) are shown two s t r i p beams, which are a distance 2h apart. The potential v a r i a t i o n along these beams (2) i s , from the Langmuir—Child law , V(h,y) = A xy V(-h,y)= k0y where A i - \jq (E.l) and and ^  are the current densities i n the right and i n the l e f t beam, respectively. If the Laplace equation i s solved subject to '(E.l) by analytic continuation, i n the manner of Pierce i t i s found that the e l e c t r o s t a t i c f i e l d required to maintain the right beam i s V(x,y) = A x (x - h ) 2 + y 2 n 3 cos 4 . -1 /x -•Tr tan 3 \ y (E.2) while the f i e l d required to maintain the l e f t beam i s 2 ,2 . 2 n 3 V(x,y) = A2[^(x + h)* + y' cos 4 + -lfx + h 3 a n \~T (E.3) If J 1 = J 2 , then the f i e l d s described by (E.2) and (E.3) 158 are observed to be the same and to be sp a t i a l l y displaced from each other a distance 2h; further, the potential v a r i a t i o n along the y-axis described by (E.2) and (E.3) i s then the same. Since the f i e l d s described by equations (E.2) and (E.3) overlap, they cannot simultaneously be realized p h y s i c a l l y , as t h i s would require multivalued potentials i n the overlapping regions. However, by suitably terminating the f i e l d s i n the region between the two beams, this overlapping of the f i e l d s can be avoided. The f i e l d s can be terminated by various means. For example, for the case when = , a r e s i s t i v e s t r i p of thickness 2s could be inserted, midway between the two beams, as shown i n Figure ( E - l ) . If the r e s i s t i v i t y of this s t r i p varies as 2 Figure ( E - l ) . Electrodes and r e s i s t i v e s t r i p to maintain two p a r a l l e l strip-beams 159 and the ends of the s t r i p are e l e c t r i c a l l y connected to the beam-forming electrodes, the desired f i e l d conditions are attained, because the r e s i s t i v e s t r i p divides the region between the beam-forming electrodes — which t h e o r e t i c a l l y shduld extend to i n f i n i t y — i n t o two closed regions (the potential i n a closed region i s specified uniquely once the potential i s known on the boundary)* 160 APPENDIX F - ON THE STABILITY OP THE PIERCE-CAUCHY PROBLEM; If the beam boundary as v e i l as the Cauchy boundary conditions are prescribed by analytic functions, then* by the (58) Cauchy-Kowalewski theorem v , a unique solution of the electro-s t a t i c f i e l d exists* at least i n the neighbourhood of the beam surface. Several exact, stable methods, based on solving the Laplace equation by analytic continuation, have been developed (5) i n the past. Pierce used analytic continuation to determine the plane e l e c t r o s t a t i c f i e l d required outside a s t r i p beam with a r e c t i l i n e a r boundary* Lomax^^^ and K i r s t e i n ^ ^ used this process for plane f i e l d s outside planar flows with curvilinear boundaries, and H a r k e r ^ ^ applied i t to the a x i a l l y symmetric case for beams with curvilinear boundaries. When "marching-i.type" numerical procedures are used to solve the Laplace equation outside the beam boundary* errors are inevitably introduced by the f i n i t e - d i f f e r e n c e approximation of the Laplace equation and by the limited precision of the numerical computations. It has already been shown i n Section 5sl (61) that these errors can grow at an exponential rate. Sugai (62) and Meltzer found an upper l i m i t of 5.828 for the growth of the error per step when the five-point star formula = 4 3> - 3> 1 - ^ -<2>, , (p.i) k+l,n k*n k-l,n k,n+l k,n-l- ' i s used to solve (5»l), the Laplace equation for plane electro-s t a t i c f i e l d s . For a x i a l l y symmetric f i e l d s , the Laplace equation to be solved i s 161 so the five-point star formula i s Cp = fa - ^  Cp - fm ~ A Cp _ Cp - Cp m+l,n \ m / m * n \ m / m _ l ? n m,n+l m,n—1 (F.3) where the notation i s as i l l u s t r a t e d i n Figure ( F - l ) . The upper l i m i t of the growth rate of the error per step for equation (F«3) i s the same as that for ( F , l ) j this i s clear i f i t i s considered that when "m" i s large i n equation (F.3), the l a t t e r approaches the functional form of (F»l)* For f i n i t e "m", however, the maximum growth rate of the error of (F»3) i s always somewhat smaller than that of the error of (F«2)* as w i l l be demonstrated. Let e be the error i n the value of the potential Cp a t m,n * (r,z) = (mArj nAz). Since equation (F.3) i s l i n e a r , i t can be used to describe the propagation of i n i t i a l errors i n potential, as well as to compute the potential i t s e l f . Therefore, em+l ,n y m J m,n \ m J m-l,n m,n+l em,n-l For s i m p l i c i t y , l e t i t be assumed that the beam boundary coincides with the li n e m = mQ, so that the Cauchy conditions specify the values of potential at mesh points of the lines m = mQ and m = m^  (see Figure (F—1 ) ) . It w i l l further be convenient to define a new index "k", which i s zero at the beam boundary, so that k = m - m o 162 As an example, the case mQ = 10 w i l l be taken* Using equation (F.4). errors i n potential at the mesh points of the lines mQ = 10 and m^  = 11 are found to propagate as follows: 12 13 ,n [" ,n - [" 10 11 e10,n fcll,n-l. 11 e l l , n Ell,n+1 + 10 r T 37 ,10 11 e10,n-l " J 66 e10,n + 11 e10,n+l _ 7 109 l f i 13 - 109 _H»n-2 ~ ' 132 e l l j n - l + i b 33 £ l l , n ~ 1 132 ell,.n+l + ell,n+2 etc. In general, p = n+(m-12) e q = n+(m-ll) m,n a10,p e10,p + a l l , q E l l , q p = n->(m—l2) q = n-(m-ll) m:, .+• 1 m m - 1 3 2 1 Axis of < ¥ i P——% i ! kr + Az -«—* k k = 1 k = 0 Symmetry o 1 2 3 4 n-1 n n+1 Figure ( F - l ) . Square l a t t i c e (Ar = Az) used for solving equation (P.3) 163 where a n ^  and a,, are c o e f f i c i e n t s . Since the potentials 10, p 1.1 *q at the mesh points of the line m^  = 1.1 depend both on the prescription of the potential at mesh points of mQ = 10 and on a f i n i t e - d i f f e r e n c e approximation of the normal gradient* the errors on the l i n e m^  = l l w i l l i n general be greater than the errors on the l i n e m = 10, The second summation of equation o ^ (F»5) i s thus the dominant part of e . Values of the * m,n coe f f i c i e n t s a ^ ^ ^ a r e shown i n Table (F-l) for values of m up to 17j t h i s table i l l u s t r a t e s the rapid rate of growth ©f the i n i t i a l errors. The worst case occurs when the potentials at each mesh point on the two starting lines m and m, have the r o 1 maximum allowed i n i t i a l error, and this error alternates i n sign for consecutive mesh points. The rate of growth of e, n w i l l & l l * q then be the sum of the magnitudes of the co e f f i c i e n t s ; i t i s seen from Table (F—l) that upon reaching a mesh on the line k = 7 the i n i t i a l errors E ^ J already w i l l have grown to 37,700 times t h e i r o r i g i n a l value. Since the dominant c o e f f i c i e n t i s a,, » i t s rate of 11, n y growth has been taken to give a more v a l i d indication of the rate of growth of the i n i t i a l errors to be expected i n the (6l) general case. This procedure was also followed by Sugai . In Figure (F-2) have been plotted the ratios of adjacent central column co e f f i c i e n t s of e for the cases when m = 10 and m^  , q o when mQ = 0, The curve obtained by Sugai for plane electro-s t a t i c f i e l d s * when equation (F,l) applies, i s also shown* As mQ i s increased* the growth of the i n i t i a l error per step i s seen to approach the curve for the plane case* 164 m 11 12 13 14 15 16 17 all,n+6 1 all,n+5 -1 -23.548 all,n+4 1 19.611 232.41 all,n+3 -1 -15.677 -155.13 -1,254.0 all-*~n+2 1 11 .749 9 3 . 3 9 9 638.24 4,055,8 a l l , n + l -1 -7.8257 -47.172 -266.15 -1,47.5.2 -8,148.1 a l l , n 1 3.90 16 .39 76.358 378.89 1,951.4 10.-27.9 Ratio of adj acent central—co eoefficien 3.< luran Lts 30 4.1 94 4 .658 4. 962 5. 150 5 ,267 Table ( F - l ) . Coefficients of e,-, i n equation (F.5) i i ,q In view of the high i n s t a b i l i t y of these marching-type methods* the best procedure to follow i s , i n general* to start with a fine l a t t i c e between mQ and m^  (to keep the i n i t i a l errors low), but to enlarge the l a t t i c e as rapidly as possible when working away from the beam boundary. For example* for plane f i e l d s * to obtain the value of n? the equation 14*n ^12,n 12,n-2 12*n+2 10,n would be preferable to the equation <£> = 4 <J> _ d b _ cb ^ db 14 *n 13,n 13,n-l 13,n+l 12,n * 165 rH + s s a rH rH S S a 5.828 Rectangular/,-, \ Coordinates Figure (F-2). k = m - m o Ratio of the adjacent central-column coefficients a m n versus distance from the beam boundary REFERENCES 166 1. O'Neill* G,D.* "The Unit of Perveance", Proc. IRE* 37sll«1295, (November 1949) 2. Langmuir* I»* "The Effe c t of Space-Charge and Residual Gases on Thermionic Currents i n High Vacuum"* Phys. Rev., 2s450-486, (1913) 3. 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