ENGINEERING RESEARCH INSTITUTE LUNVERSITY OF MICHIGAN ANN ARBOR THE PROPAGATION OF ELECTRDOLAGNETIC WAVES IN A MAGNETRON-TYPE SPACE CHARGE Teohnioal Report No. 8 Eleotron Tube Laboratory Department of Eleotrioal Engineering BY, GEORGE Re BREWER Approved by: W. G. DOW G. HOK ProJeat M921 CONTRACT NO. DA.S36-.039 so-5423 SIGNAL CORPS, DEPAR1EEiT OF THE AM Y DEPARIENT OF AMEY PROJECT NO* 3-99-13-022 SIGNAL CORPS PROJECT 27-112B-0 Submitted in partial fulfillmant of the requirements for the Degree of Dootor of Philosophy in the University of Miohigan. July, 1951

ABSTRACT The propagation of eleotromagnetio waves in a magnetron type spaoe oharge is studied by using small signals non-relativid;ie approximations. The following oases are analyzed: 1. Plane Magnetron - ao Propagation of a plane eleotromagnetio wave in a direction parallel to the applied magnetic field. b. Propagation of a plane eleetromagnetic wave in a direction perpendicular to the applied magnetio field and normal to the anode and oathode, o. Propagation of a plane electromagnetie wave of phase relocity slow compared to that of light in a direcetion perpendicular to the applied magnetic field and parallel to the electron drift motion. 2. Cylindrioal Magnetron - a. Propagation of a TEl-type eleotromagnetio wave in a oylindrical space ohargo in a-direction parallel to an axially applied magnetio field. b. Radial propagation of a cylindrical electromagnetic wave in a cylindrical spaoe charge. The analysis yields values for the propagation constant of the wave in the space charge, expressed in terms of an effeoive dielectric constant, which depends on the ratio of the signal radian frequency c to the oyolotron radian frequency Co ( = eBe/m ). It is i

found that this effective dielectric oonstant oan assum any real value, positive or negative. For given ao/o, this knowledge of the effective dieleoctrio oonstant makes possible the determination of the reactive effects of the spaoe charge on a confining circuit. The influence of the space oharge on the frequency of a multi-sanode magnetron is discussed qualitatively, as is the possibility of amplification of an electromagnetic wave along the plane magnetron space charge. Several experiments, conducted to determine the validity of the theory,, are described. The results of these experiments appear to confirm certain critical parts of the theory. iii

ACKNOWIEDGIENTS The author wishes to aknowledge with gratitude the assistance, during the oourse of the research which is reported herein, of all of the members of the University of M1iohigan Electron Tube Iaboratory. He is most particularly indebted to Mr. Gunnar Hok for the numerous valuable suggestions obtained in the course of many hours of patient consultation. He is also appreciative of Mr. H. 1XW. Welch's enoouragement and advioe, and of Mr. John W. Van Natter's consoientious efforts in the design and construction of the experimental tube models. Persomnel of the Laboratory H.la,. Welch Research Engineers J. Re Black G. Hok J. S. Needle Instructor of Electrioal Engineering S. Ruthberg Research Associate V. R. Burris Machine Shop Foreman R. F. Steiner Assembly Technicians J. W, Van Natter R. F. Denning Laboratory Machinists Do L. MoCormiok T. G. Keith E. A. Kayser N. Navarre Draftsman S. Spiegelman Stenographers J. tong iv

TABLE OF CONTENTS Page ABSTRACT ii ACKNONIEDGEMENTS iv LIST OF ILLUSTRATIONS vii I, INTRODUCTION 1 II. THE BASIC REIATIONS 4 1. The Equation of Motion 6 2. The Spaoe-Charge Distributions 13 III. DETErfINATION OF THE CCOMPLE INDEX OF REFRACTION 23 1. Propagation in the Direction of the Applied Magnetio Field A. Plane Magnetron 28 Be Cylindricoal Magnetron 30 C. Disoussion of Variation of &e with r. 35 2. Propagation in the Direction Normal to Anode and Cathode A. Plane Magnetron 38 Bo Cylindrioal Magnetron 40 3. Propagation in the Direction Parallel to the Steady Electron Motion A. Plane Magnetron 45 1V. DISCUSSION OF THE RESULTS OF CHAPTER III 64 1. Discussion of Electron-Wave Interaction 65 2. Illustration of Dieleatrio Constant by its Effect on Resonant Cirouit 68 3. Discussion of the &e vs o/coo Curves 71 4. EleotronmWave Energy Exchange 78 5. Discussion of Loss in the Space Charge 80 V. BOUNDARY CONDITIONS AND EFFECT ON RESONANT CIRCUIT 91 1. Boundary Conditions at Edge of Space Charge 91 2. Effect of Space Charge on its Associated Circuit 95 VI * EXPERaLENTAL RESULTS 98 1. Propagation in the Direction of the Applied Magnetic Field 98 2. Propagation in the Direction Normal to Anode and Cathode 108 3. Effeat of Space Charge on Resonant Wavelength of Multianode Magnetron 115

TABLE OF OONTENTS (Cont 'd) Page VII. 0ONCLUSIONS 123 1. Conclusions - Agreement between Experiment and Theory 124 2. ResumA of Assumptions 127 3. Possible Applioations of the Type of Space Charge 129 4. Suggested Topics for Fut;re Investigation 132 APPENDICES APPENDIX 1- - Derivations from the Boltzmann Transport Equations 134 2 Influenoe of the Pressure-Gradient Term in the Euler Equation 141 3 -Effect of Eleotron-Ion Collisions 147 4 - Effect of Eleotron-Eleotron Collisions 154 5 - Conditions Under heioh the Second Order Terms in the Equations of Motion can be Neglected 156 6 - Calculation of the Shift Due to Space Charge in Resonant Wavelengbh of a Coaxial Cavity 158 7 - Caloulation of the Shift in Resonant Wavelength of a Cavity in the TE011 Mode, Due to Spaoe Charge 162 BIBLIOGRAPHY 164 SYMBOLS USED IN THE TEXT 169

LIST OF ILLUSTRATIONS Fig. Page 2.1 Coordinate System and Field Vectors of Cylindrioal Magnetron 5 2.2 Coordinate System and Field Vectors of Plane Magnetron 5 2.3 Comparison of Space Charge Density Distributions Obtained by Various Workers: Plane Magnetron 15 2,4 Comparison of Space Charge Density Distributions Obtained by Various Workers: Cylindrical Magnetron 17 2.5 Comparison of Space Charge Density Distributions Obtained by Various Workerst Cylindrical Magnetron 17 2.6 Rotating Space Charge Current vs Anode Voltage 21 3.1 Orientation of Field Vectors Assumed for Development of Wave Propagation in Plane Magnetron 26 3.2 Orientation of Field Vectors Assumed for Development of Wave Propagation in Cylindrical Magnetron 26 3.3 s for Plane Magnetron 27 3.4 se for Cylindrical Magnetron: Propagation in z Direction 34 3*5 Effect of Cloud Radius on Critioal Values of o/o - Approximate Interpolation: Propagation in z Direction 37 3.6 6e fbr Cylindrical Magnetrons Propagation in r Direction 43 3.7 Effect of Cloud Radius on Critical Values of /ao - Approximate Interpolation: Propagation in Radial Direction 44 3.8 Idealized Space Charge in Plane Magnetron with Periodic Anode 49 3,9 X Directed Electric Field Distribution in Space Charge 54 3.10 Y Directed Electrio Field Distribution in Space Charge 57 3*11 Susoeptance of Electron Stream as Seen from Anode. 60 4.1 Resonant Wavelength vs Dielectric Constant 69 5.1 Perturbed Surface of Plane Magnetron Space Charge 92 5.2 ( Qualitative Configuration of Electric Field Lines in Interaction Space of Multianode Magnetron 97 vii

LIST OF ILLUSTRATIONS (Cont'd) Fig. Page 6.1 10 CM Magnetron Diode (Experimental) Model 3 99 6.2 Photograph of 10 CM Experimental Diode 100 6.3 Wavelength Shift in Coaxial Cavity Predicted from Theory-Using Hull-Brillouin Value of Space-Charge Density 100 6.4 Change in Resonant Wavelength of 10 CM Cavity vs m0/0D 103 6*5 Cutx;off Curves 10 CM Coaxial Cavity Magnetron Diode-Low Voltage 106 6,6 Change in Resonant Wavelength of 10 CM Cavity vs Magnetio Field-Showing the Cyclotron Resonance 107 6.7 TE1l Resonant Cavity for Spaoe Charge Study 109 6.8 Photograph of Experimental Tube 111 6.9 Magnetron Spaoe Charge Diode 112 6.10 Effect of Space Charge Cloud in TEol Cavity 114 6.11a 'A and Go of Hot Magnetron as Function of Plate Voltage 119 6.llb Xo and Go of Hot Magnetron as Funotion of Plate Voltage 120 6*11o Wavelength Shift in Interdigital Magaetron Due to Expanding Space Charge Cloud 121 6.12 Change in Resonant Wavelength of Multi-Anode Magnetron vs Magnetio Field 122 viii

I. INTRODUCTION The propagation of electromagnetic waves in ionized media has been treated in a large number of papers, particularly with reference to the ionosphere. However, only a very small number of these papers are applicable to the type of space oharge region which is presumed to exist in a magnetron. In this paper the propagation of eleotromagnetie waves in the magnetron space charge is studied, together with the effeot of the spaoe charge on an r-f circuit in which it is placed. It is well known that the problem of the interaction between the fields in an oscillating multi-anode magnetron and the rotating space charge oloud is suffioiently complex to have allowed, to data, only solutions containing several restrictive approximations. In order to obviate mathematical entanglements as much as possible (and thus avoid the type of solutions requiring numerical integration), this analysis will be concerned with the small signal interaction of waves and electrons in a specified space charge cloud with certain uniform and simple types of electromagnetic fields. It is hoped that this presentation will allow the desired physical principles to be brought out with. out requiring extended mathematical treatment. The results are believed See for examples Welch, H. W., Jr., "Space Charge Effects and Frequenoy Charaoter. istics of CW Magnetrons", Univ. of Mich. Electron Tube Laboratory Technical Report No. 1, November 15, 1948. Blewett, J. P. and Ramo, S., "High Frequency Behavior of a Space Charge Rotating in a Magaetio Field", Phys. Rev., V57, pages 655-641, April, 1940. Lamb, 1i. E. and Phillips, M., "Space Charge Frequenoy Dependence of a Magnetron Cavity", J, Appl Phys., V18, pages 230-238, February, 1947. Welch, H. WI., Jr., nEffecots of Space Charge on Frequenoy Charaoteristics of Magnetrons", Proo. I.R.E., page 1434, Deoember, 1950. j ~ eI

to be applicable, insofar as the small signal analysis will allow, in the case of space charge clouds used for frequency modulation whlich are usually placed in a structure of such geometry that the simple field analysis is valid. It is also hoped that from these results one may be able to deduoe qualitatively or ssmi-quantitativaely the effects in the case of the more complioated fields of a multi-anode magnetron and other structures in hiioh this type of space charge cloud could be used. This analysis is an extension of that reported by Wrelohl and is speoifically an attempt to determine the effective index of refrac. tion of the space charge region as experifenoed by an electromagnetio wave propagating into or through this region. A knowledge of the index of refraction, and thus the dielectric constant, as a function of the frequency of the wave and the magnetic field, will enable the caloulation of the reactive (and in some cases also resistive) effects of the space charge on the miorowave oirouit. Weloh treats this problem under the assumption that the space charge swarm moves with oonstant linear velocity independent of position, so that the second term on the left side of Eq. II-1 below was not included. The present work is an extension and refinement on the previous treatment in that the variation of the electron velocity with position in the magnetron is considered. Since the most general type of plane (or cylindrical) wave can be considered to be resolved into plane (or cylindrioaJ) waves travelling along the coordinate axes, in this report the propagation of eleotromagnetio Waves in the magnetron space charge will be idealized by considering the wave to be plane (or oylindrioal), propagating along one W~elch, H. W,, Jr., Loc. cit.

of the coordinate axes. The electric field of the propagating wave will cause the electrons to undergo perturbations about their steady or equilibrium paths. The electrons will be acted upon by other forces also, including the applied magnetio field, and the motion of the eleotrons subJeot to these forces represents a current associated with the propagating wave so that its velocity of propagation is affected. In addition the electrons oan collide with other particles in the space or with the electrodes, thus losing some' of their energy; as a result, the wave will be diminished in amplitude as it progresses through the medium. These effects resulting from the electron motion are the subject of this study. The results of this analysis are presented in a form enabling predictions to be made of the effeots of the space charge cloud on an r-f circuit. From this information, one oould design structures for frequency modulation, amplitude modulation, et.,, of a microwave signal, using this type of space charge. In what follows, the basic equations to be used in the analysis are discussed in Chapter II, followed in Chapter III by a derivation of the index of refraction of the plane and cylindrical space charges. Chapter IV contains interpretations of the results of Chapter III; Chapter V the effect of the space charge on its associated rof circuit. In Chapter VI, the results of. experiments oonducted to verify the theory are presented. The more detailed mathematical treatments are included in the appendices so that this material can be omitted in reading with no loss of oontinuity. The MLKS rationalized system of units is used throughout.

II. TIHE BASIC RELATIONS A magnetron space charge is created between two parallel plane or conoentric cylindrical elootrodes, one an electron emitter, by the application of a d-o electric potential between the electrodes and a steady magnetic field parallel to the electrodes in the plane case and along the axis in the cylindrical caseo The electrons will possess a drift velocity normal to both the magnetic and electric fields. In this section the equation of motion of the electrons in this spaoe oharge, under the influence of the electric and magnetic fields, is derived using perturbation methods, and is discussed briefly. The formn of the space charge density distribution in the static magnetron is not as yet known with certainty; therefore the various distribum tions obtained by several workers are presented and discussed. The shape of the magnetron space charges for the plane and cylindrical geometries are represented in Figs. 2-1 and 2-2. These figures also show the coordinate systems and field vectors to be used in the analysis which is to follow.

5 Ee FIG. 2.1 COORDINATE SYSTEM AND FIELD VECTORS OF CYLINDRICAL MAGNETRON,.. * "...:. FIG. 2.2 COORDINATE SYSTEM AND FIELD VECTORS OF PLANE MAGNETRON

The Equa ation of Motion. The equation of motion of the electrons, subject to the foroes of the applied magnetio field and the eleotric field of the elsctromagnetio wave will be disoussed initially in terms of the Euler Hydrodynamical equation. In order to provide a firm basis for consideration of the Euler equation as the equation of motion in this non-relativistio treatment of electron-wave interaoction, this equation is shown, in Appendix 1, to be derivable from the Boltzmann Transport equation af -ALif 1a f (Of' with no knowledge of the exact form of the velocity distribution funotion. In this equation f represents the electron velocity distribution function, r is the position veotor of the group of electrons under consideration, o the vector describing the velocity and F the vector force field aoting on the electrons. The equation relates the change in the occupation of a oell of velocity space due to the action of the fields eto. to the change due to encounters with other electrons of the gas. However, sinoe the Boltzmann equation is valid only under conditions of approximate thermal equilibrium in the gas, the a priori assumption must be made that the wave propagating through the eleorron gas produces only a small perturbation on this equilibrium. That is, the energy of the random motion tending to maintain thermal equilibrium is assumedl large in comparison with the energy imparted to the particles by the wave. This thrmal equilibrium does not exist near the boundaries In order to obtain some idea of the conditions imposed by this assumption, consider that the mean random energy of the electrons is 3/2 1X per electron. Then this assumption can be written as W<<3/2 kT where Ww is the mean vibrational energy imparted to the electrons by the wave. Since the applied magnetio field does not affect the energy of the electrons, Wwg can be written for purposes of illustration ass

of the space charge, unless rather artificial boundary conditions are im. posed, possibly raising a question as to the validity of the application of the Boltzmann equation to such a medium. Consistent with the non-relativistvistio case, the Lorentz force due to the mtgnetio field of the wave is neglected, so that the Euler Hydrodynamical equation is: a~ m. e_ E vx Bo VP II-S m ~ Pnm where v = linear velocity of a group of electrons oontained in elemental volume dr1 Bo = applied oonstant magnetic field p m eleotron gas pressure n a number density of electrons In a number of papers treating the same general subject as this report, the pressure gradient term in the equation of motion is negleoted. In so doing, no account is taken of any effects due to the random or thermal motions of the electrons. In order to be able to give a complete treatment of the effect of random motions, the form of the 1 mp~Z~ie E2 Ww~ 2 m- mI; so that E << -- - e If T = l04EK and co x 6 O x109 it is found that the field strength of the wave must be mucoh less than 230 volt/om. The volume dr must be of dimensions very much less than a wavelength of the propagating wave but must contain a sufficiently large number of electrons that a statistical mean value of their behavior can be obtained. Due to the high electron density, both of these conditions can be satisfied. There will be random fluctuations with time in the number of electrons in dr, causing fluctuations in the effects of these electrons but this will cause no apprec.iable chane in the prop. agating characteristics of the wave.

electron velocity distribution function would be required. This determination usually involves mathematical complications unjustified for the purpose of this report.1 However, a first-order approximation can be made by considering the electron gas as exhibiting ideal behavior so that p = nkT. This is admittedly a rather severe assumption but is believed to be at least somewhat closer to the actual state of affairs than the complete neglect of the pressure term. The temperature T referred to here is a measure of the mean random energy of translation of the electrons. There is reason to believe2 that by some mechanism of electron interaction this electron temperature can aohieve values as high as 1050K, greatly in excess of the oathode temperature. In Appendix 2, this substitution p = nkT is made into Eq. II-1. The solution for the wave velocity for a typical case is carried through in the same manner as in Chapter III. The resulting equations show that the inclusion of the pressure gradient term does not affect the propagation characteristics of the electronagnetio wave but causes to appear another rave similar to the plasma oscillations found in gases. The pressure gradient term will therefore be omitted and See for example - Cohen, Spitzer, and Routley, "The Electrical Conductivity of an Ionized Gas", Phys. Rev. 80, 2, October 15, 1950. 2 The reasons for the belief in the existence of this large electron temperature are: (a) the relatively large current collected by the anode in a cutoff smooth-bore magnetron, and (b) the experimental measurements of Linder. Linder, E. G., "Excess Energy Electrons and Electron Motion in High Vacuum Tubes", Proc, I.R.E. 26, page 346, 1938. Linder, E. G., "Effect of High Energy Electron Random Motion upon the Shape of the Magnetron Cutoff Curve", J. Appl, Phys. 9, page 331, 1938.

Eq. 11-1 becomes at+ v - E + v x Be II-2 which is the usual equation describing electron motions in a field in which the velocity varies with both position and time. It is shown in Appendix 3 that the effect of collisions between electrons and atoms or ions in the space charge, in which the eleotrons lose some of their translational energy, oan be represented by a frictional type force proportional to the velocity, i.e. by gv, where g is the inverse mean time between collisions. Another effect of the oollisions between electrons and ions or moleoules, can be seen from a oonsideration of the force aoting on the electrons due to the positive ions present in the space charge. Lorentzl, in his treatment of wave propagation in material media, uses as the total force on a charged particle F E + P where P is the polarization of the medium and E the electrio field of the electromagnetio wave. Darwin2 shows, by a consideration of the electron orbits near idealized positive charges, that the average effect of electrom-ion collisions is to produce an acceleration -1 NE where C is the perturbed electron position. The equation of motion of such an electron is then, since Ne C = Pt eF 1 8 m q m Lorentz, H. A., n"The Theory of Eleotrons", B. G. Teubner, Leipzig - 1909, Chapter IV. 2 Darwin, Chas., "The RPfrative Index of an Ionized Medium" II, Proo. oy. Soo London 82, page 152, 1944.

and, using 1 F E+ + P m E He conoludes that the "process of collision produces dynamically a depolarizing effect, reducing the effective average force on an electron from F to E". The force on an electron due to the eleetric field is then -eE. It is shown in Appendix 4 that eleotron-electron collisions in the gas do not change the total dipole moment of the space charge and thus do not affect the propagation of waves in the medium. Eq. II2 becomes, with the addition of the frictional force representing collisions: 0 + gv + V = E + v x Bo I 13 As mentioned before, in the absence of the wave the electrons are presumed to pursue steady orbits under the influence of the applied magnetic field and d-c electric field. In the presence of the wave the electrons will be periodioally perturbed from this steady or time invariant motion. (The word perturbed as used here is meant to imply smallness of the magnitude of the deviation from the steady orbit except where specifically noted.) The total electron velocity will then be represented by an ordered or d-c term and a perturbed or a-c term as v =v o + v II-4a Likerwise small volumes of space oharge will be moved periodically from their mean positions, so let the total space charge density likewise be represented by a d-c term and a perturbed term ass P Po + P1 I14b

11 Substituting these relations into Eq. l..3 and keeping only perturbation terms: a2. + gv~C+ +vl + (vo'V)vl - e + x Bo II5 Since this report is concerned with the steady state propagation of waves in a particular type of mediums assuming the wave will propagate, it appears reasonable to suppose that the state of the medium will be perturbed by the moving wave. Consequently the velocity and space charge density will be assumed to vary as rv Ie~ ' ys Pl eiCet ' YS where co is the angular frequency of the impressed wave, Y the propagSation oonstant, and s a lengbt imit along the direction of propagation. The equations of motion will be linear only if the term (vl*V)vl vanishes, either as a result of other assumptions or by speoi-e fioation that the magnitude of v1 is so small that this product is corm paratively negligible. This small signal assumption will not always be necessary and this will be pointed out in each of the several cases to be treated in the next chapter. The convection ourrent density, when represented in the form, J = (po + Pj) (vo + vl) can be separated into zero and first order terms, the perturbation or a-o part being J1 =" pv1 + vl II-6 where the term Plvl has been neglected. Again this may be due to the fact that Pl W 0 or it may be necessary to assume the smallness of both vi and pj. *Which of these reasons is responsible for the vanishing of

the Plvl term will be pointed out in each case separately. Eq. II-5 will now be reduced to its form appropriate to the plane and cylindrical oases to be considered. ae Plane Manetron- In this case the steady velocity is entirely in the x directionl so that Eq. II-5 becoomes in oomponent form: x8 +gvx:+ VO +y Vy + gVX + VYVo+ vz Ge Ex -Co Vx y: + VY + vof + e +r + + +Z m deEy o +c vx Yx x e+ a+gv+ a a ylindri agnetron In this ase the steady e letron V. glindrical ~aM e ron. In this oase the steady eleotron velooity is considered as entirely in the e direction so that Eq. II-5 becomes in component forms rav + gr + MO. + r a + a II8 O ry 1r r8 z - Er - (O ve a z 'In the ease of the so-called "double stream" solution for electron motions, for a magnetron in the cub-off condition, that is no net electron ourrenrt touward the anode, there will be as many eleetrons passing through the volume dr toward the anode as away from it so that the net steady eleotron motion is still tangential.

13: +; + 8_ ve av +ve 0: a ~ s e, Vr~7 +7+ 11r8 z'.Z~ + go + Y as + vr +e a + V E (t zr 0r r O e z z In the next chapter the Eqs. II-7 and II-8 will be combined with the Maxwell field equations to enable a determination of the complex index of refraction of eleotromagnetio waves through this space charge medium. In these next sections the propagating wave will be con-.sidered as either plane or cylindrical, whose phase is invariant in the plane (or cylinder) perpendicular to the direction of propagation, thus neglecting consideration of any boundary conditions imposed by the circuit in which the space charge is placed. In later sections the effect of the boundary conditions imposed by the circuit will' be treated. 2. The Space Charge Distributions. In the next chapter the space charge density will be kept as an unknown parameter, until the final relation for the index of refraction is obtained. However for illustrative purposes and for numerical calculations, the EHull-Brillouin values of charge density will be used. Therefore, in order to justify this substitution, the various solutions of the statio magnetron space charge will be examined briefly. A number of papers have been published describing the theoretical steady-state space charge distribution in plane and oyrlindrical non-oscillating magnetrons. Figs. 2.3, 2.4 and 2.5 allow a comparison of the distributions obtained by various workers.

Fig. 2.3 shows the distribution of space charge density for the plane magaetron as given by Hulll and Brillouin2, Page and Adams3, and Twiss4. It is seen that the work of Twiss (when he considers only initial norml velocities of emission), Hull and Brillouin agree very closely, -while that of Page and Adams is lower by as much as a factor of two. The curve reported by Twiss when both initial tangential and normal emission velocities are considered is also showxi, the solid curve being derived from his equations and the dashed ourve from the results of his qualitative reasoning. The Page and Adams solution is seen to differ from the other cases. The sharp rises at the cathode and at the edge of the space charge cloud are due to the assumption of zero velocity of emission and zero escape current from the cloud boundary. For the actual case of finite initial electron velocity and non.. zero esoape current these singularities will disappear, the ourves intersecting the cathode and space charge outer boundary with finite values. Thus if the Page and Adams solution contained these boundary conditions the resulting curves would appear in somewhat better agreement with the others. In the Page and Adams distribution the electrons are presumd to execute approximately cyoloidal orbits from the cathode to the edge Hull, A. N., "The Effeot of a Uniform Magnetic Field on the Motion of Electrons Between Coaxial Cylinders", Phys. Rev. V18, page 31, 1921. 2 Brillouin, L., oural de Pysiqe, 1940. 3 Page and Adams, "Space Charge in Plane Magnetron", Phys. Rev. V69, page 492, 1946. Tyiss, R. Q. "On the Steady State and Noise Propertis o Liear and Cylindrical Magaetrons", M.I.T. - Ph.D. Thesis, 1950.

15 of the cloud and return, the so-called double stream motion. Brillouin considers that the electrons move parallel to the electrodes, the socalled single stream motion. FIG. 2.3 COMPARISON OF SPACE CHARGE DENSITY DISTRIBUTIONS OBTAINED BY VARIOUS WORKERS PLANE MAGNETRON 44 h =.5x 12m. 40 36 32 24 t ~1 PAGE B ADAMS BRILLOUIN TWISS (CONSIDERING ONLY NORMAL EMISSION VELOCITIE -TWISS (CONSIDERING BOTH - -,- -~ NORMAL AND TANGENTIAL EMISSION VELOCITIES) O.2.4.6.8 1.0 Y/h In an unpublished reportl Brillouin discusses in detail the types of distributions possible in a plane magnetron. He points out that either a single stream or double stream electron motion is possible; in either case the total charge within the cloud will be the same, as Brillouin, Lo, "Electronic Theory of the Plane Magnetron", Columbia University, AEP Report 129.1iR - OSRD 4510, to be published in part in the third volume of "Advances in Electronics".

16 will be the total electron energy. Thus, one cannot make a choice between the two possible solutions on the basis of energy, but he showes that the double stream motion leads to a vsolt-ampere characteristic with a negative resistance region so that this distribution should be unstable, with a tendency to degenerate into sustained oscillations. He also shows that for a space charge limited cathode, the electron trajeotories do not crosst indicating single stream motion. Brillouin conoludes from this that the stable steady-estate space charge distribultion in a space charge limited plane magnetron is that corresponding to single stream motion. The Page and Adams double stream motion and the Brillouin single stream motion are the limiting eases of the possible types of electron orbit possible in the magnetron space charge. YTore generally, the electrons can be thought of as executing approximately cycloidal orbits whose maximum excursion is less than the thickness of the space charge cloud so that a number of those swarms can lie parallel to each other, each farther from the oathode; thus there appear a number of "virtual cathodes" between each of which exists one swarm. This is the so-oalled "multiple swarm motion". For a single swarm the Page and Adams solution is valid and for an infinite number of swarms the Brillouin distribution is more nearly correct. It would therefore be expected tlhat the space charge density distribution corresponding to the actual electron motion lie between those corresponding to these twoo limiting cases. In Figs. 2.4 anld 2.5 are shiovm the space charge density distributions for the cylindrioal magnetron, due to Brillouin1, Brillouin, L., "Theory of the Eagnetron - I"t hys. Rev. VGO, page 385, 1941.

FIG. 2.4 FIG. 2.5 COMPARISON OF SPACE CHARGE DENSITY DISTRIBUTIONS COMPARISON OF SPACE CHARGE DENSITY DISTRIBUTIONS OBTAINED BY VARIOUS WORKERS OBTAINED BY VARIOUS WORKERS CYLINDRICAL MAGNETRON CYLINDRICAL MAGNETRON 44 44 r0 ra 25....!CATHODE ANODE 5 40. 40 36 - - - - - - - -~- 3 32 MOELLER 28 ~~~~24- ------- - — ~~~~24 20- ~~BRILLOUIN BRILLOUIN 16 881~ ~,~.- e i RiI I II I I II II I` C-MOELLER PAGE 81ADAMS ~0.~~ ~ ~0.2 4 6.8 1.0 o.2.4.6.8 1.0 r/ra r ra

18 Page and Adams1, and lIoellor2. These latter two are seen to differ rather markedly from the distribution of Brillouin. The Brillouin solution, it is remembered, results from the assumption of zero radial electron acceleration, that is, the electrons are considered as travelling in ciroles concentric with the cathode (single stream tnotion), The Page and Adams and the Moeller solutions are obtained by the use of mathematical series and involve no such assumption. As in the plane case, tAhe infinite values of space charge density at the cathode and boundary, obtained by these latter workers, will be reduced to finite values upon inclusion of the initial velocity of emission and escape current. Glagolev3 has succeeded in integrating the equation of motion, by a method of successive approximations, to obtain the potential dlstribution of the cylindrical magnetron space charge. He finds that the potential differs only slightly from the Langmuir distribution in a space chlarge limited diode without mag:netio field; the maximum deviation occuring at the edge of the space charge where it is 9% greater than the Langmuir value. From this one can conclude that the space charge density is only slightly greater than that derived from the Langmuir potential distribution. This latter yields a space charge density distribution olosely resembling the. Hull.-Brillouin case. Page and Adams, "Space Charge in Cylindrical Magnetron", Phys. Rev. V69, page 494, 1946. -ryoeller, TI. 0., "EleLtronenbannen und Mechanismus der Schwingungserregung in Schlit zanodenmagnetron", Hochfrequenztecra nik und Elak., V47, page 115, 1936. Glagolev, V. 1l., "The Passage of Steady Current in a Cylindrical Non-slit Magnotron" Zhzur. TeE iZ. iz TJSSR - 19 - page 943, August 1949 - Translated by Naval Research Laboratory,?;ashington, D. C.?ItRL Translation No. 318.

19 With regard to the choice between these various space charge distributions; the problem is such in the cylindrical case as to make this more difficult than in the plane case. Allis maintains that the double stream solution is the stable one, based on the observation that since anode current is observed in a nonl-osoilla;ting cut-ofr magnetron, and in the single stream case (his Bo solution) radial current is impossible, the double stream motion must take place. However, he seems to neglect the fact that the ratio of rotating spaoe oharge current to anode current in a out-off magnetron is large, suggesting the possibility that most of the electrons are moving more or less in circles con.r centrio with the cathode and the anode ourrent be due to those relatively few electrons which, by a process not yet known, have lost some of their ordered or rotational energy. Allis shows that the single stream motion is possible for any radius of the space charge aloud. However for a ratio of space charge aloud radius to cathode radius greater than 2.023 the double-stream motion is also possible, and in view of the previous reasoning, Allis believes this latter tyrpe of motion probable. Brillouin2$3 presents a criterion to enable a decision to be made between double and single stream motion, based on whether the eleotron trajectories do or do not cross. He finds that under space charge limited conditions, for ratios of space charge aloud radius to cathode radius less than 2.273 (for small oscillations) the trajectories do not cross, indicating single-stream motion. However, for ratios greater Allis, W. P., "Electronic Orbits in the Cylindrical Magnetron with Static Fields', Radiation Laboratory Special Report 9S, Seation V, R.L. Report 122, October 1941. Brillouin, L., loo. cit. Brillouin, L., "The Influence of Spaoe Charge on Electron Bunaching", Phys. Rev. V70, page 187, August 1946.

20 than this, the trajectories can cross so that one cannot decide, for this case, between the possible motions on this basis. On the basis of statistical considerations of electron motion, Hok has recently demonstrated that the space charge distribution may be appreciably modified by the random interaction, however weak, between the discrete electrons forming the space charge. Wasserman2 has conducted an experiment in an attempt to determine which of the distributions, Brillouin or Moeller, is more nearly correct. His method involved measurement of the magnetic flux assooiated with the total space charge current rotating around the cathode. Fig. 2.6 shows his results, in which curve 1 represents the rotating current calculated from Moeller, and curve 2 the rotating current deduced from experimental measurement of the magnetic flux. Wasserman claims agreement within fifteen per cent of the Brillouin case and fortyfive per cent writh the Moeller case, from which he concludes that the "steady state cylindrical space charge distribution is best represented by the Brillouin relation". It is believed that his experimental method is somewhat inaccurate, however, and should yield values of rotating current lower than the actual values so that perhaps the agreement is better than shown by Fig. 2.6. Similar measurements have been made by Mtllerz3who claims Hok, G., To appear in a forthcoming report from the University of Michigan Electron Tube Laboratory. 2 Wasserman, I. I*, "nRotating Space Charge in a Magnetron with Solid Anode", Jo Tech. Phys. (USSR) V18, page 785, 1948. MIbller, J., "Measurement of the Circulating Electron Current in a Magnetron",.Ioohfrequenzteohik und Elak., V47, page 141, July 1936.

21 agreement within nine per cent of the value of magnetic flux determined from the Hull-Brillouin space charge distribution. However, this writer believes M8ller's experimental technique to be susceptible to inaccuracies also. 1.8! i.i i 1.4 1.2 02 z 0. 800 900 1000 1100 ANODE VOLTAGE (VOLTS) FIG. 2.6 The proper ohoioe of the distribution of space charge will influence the results to be derived here to a certain erxtent; however, as will be seen later, the character of the solutions is determined mainly by thie functional variation of electron velocity with distance from the cathode. Fortunately the angular velocity in the oylindrical case and the steady linear velocity in the plane case result from integrals of the equations of motion without resort to any assumptions as to the trajectories. That is, ~hile one must make an assumption as to the electron orbits (as did Brillouin) or use a series method (as M81ller or Page and Adams) to obtain a solution for the potential and space oharge variation, the velocity is independent of these difficulties. The Hull-Brillouin expressions for the velocity and space

22 charge distribubtion in the plane and oylindrioal magnetrons are given be low. Plane Magnetron Cylindrioal Magnetron e e 2 II-9 II-10 VoxC'-tY v r 2 [=i] where vo is the steady velooity in the x and e directions and $ is the angular velocity. coo = eB is the cyclotron angular velocity and re is the oathode radiuse

III. DETERMINATION OF THE CCMPLEX INDEX OF REFRACTION In this chapter the complex index of refraction of a magnetron spaoe charge will be determined for various directions of propagation of the wave. In the case of a plane magnetron the waves are assumed to be plane; that is, invariant in phase in the plane normal to the direction of propagation1 The following cases are considered: (a) propagation in the direction of the applied magnetic field (z), (b) propagation in the direction normal to anode and cathode (y), and (o) propagation parallel to the steady electron motion (x). In the case of the cylindrical magnetron the waves are in each case assumed invariant in phase with angle around the cathode (8) and the following oases are oonsideredt (a) a plane TEM wave propagating. in the direction of the applied magnetic field(z), and (b) propagation of a cylindrical wave in the direction normal to anode and cathode (r). An electromagnetic vave of specified characteristics is con. sidered to impinge on the space charge along the desired direction. Part of this wave will be transmitted into the space charge and set the electrons in motion. The magnetic field will cause the electrons to have additional components of velocity than those given them by the elec. trio field of the impinging wave, necessitating additional field Insofar as the space charge can be oonsidered as a linear medium, waves which do not fulfill this assumption can be formed by suitable superposition of these plane waves. 23

components associated with the wave in the space charge. This can be expressed by thinking of the vwave in the space charge as made up of an inducing field, and a scattered or radiation field due to the motion of the electrons. This electron motion set up by the inducing wave is equivalent to an elementary current ev so that the coherent scattering or radiation due to the motion of all charges is equivalent to a current density distribution J = nev. This equivalent current distribution is the only radiation source in the space charge and since it is due to the inducing fields this and the radiation field must be identical and satisfy the field equations: V x H = J + ico toE / xE - ini.o V H = O from which can be derived the wave equation: V BE + ' 2 E = 0O 1),the steady state complex index of refraction of the medium,is the quantity sought in this analysis. The index of refraction is a function of the ratio CO/Coc of the signal frequency to the cyclotron frequency. The space charge can, of course, be thought of as an electronic plasma and this relation between 7 and co as analogous to the usual dispersion equation for waves in a dispersive medium. Since to the extent that the space charge can be considered as a linear medium, any wave motion in the magnetron can be made up of monochromtio waves propagating along the specified coordinate axes, this chapter will be concerned only with waves propagating along these axes and varying with time and space as eiS -Y S where s is the unit of length measured along the direction of propagation.

25 The electron velocities are obtained in terms of the field oomponents by the use of the equations of motion Eqs. II17 and II-8. From the velocity equations the current is determined, which when substituted into the field equations allows solution for the propagation constant r and therefore 7. In the cylindrioal case explicit solutions for i7 as function of o/A0 are obtained only for the limiting ease of no variation of p0 with r; corresponding, for the Hull-Brillouin space charge distribution, to a vanishingly small cathode. Some qualitative arguments are advanced allowing an approximate interpolation of the properties of the space charge to be made between the limiting eases.of plane magnetron and cylindrical magnetron with very small cathode. The orientation of the field vectors to be used in the following developments are shown in Figs. 3.1 and 3.2e

'.". '."'' /o, ". ".':'*... '. ' <.......'.'*.'.... ~ **0**wu@.*~ ~'. ORIENTATION OF FIELD VECTORS --..'=o'.'7. ~ ~ - '_- - o'. FIG.3.1 ORIENTATION OF FIELD VECTORS WAV'PROPAGATION IN L A NE MAGNETR O N ~ ~ ~ ~~~~~ ~ ~ ~~~~~~~~~~~~~~~:..? ~~~ ~ ~ ~~~~~~~~~~~~~~',~ I:. "~:'1...... /,: _~...... ~~~~~~ '..":'.'"::'"'':

5 4 7 2 Ee r -I -2 -3 -5 -6 ___ ___ 7 0.2.4.6.8 1.0 1.2 1.4 1.6 1.8 2.0 CAC FIG. 3.3 Ee FOR PLANE MAGNETRON

28 1. Propagatiion in the Direction of the Applied Magnetic Field. A. Plane Magnetron. The propagation in the z direction of a plane wave (a/ = a/ay = 0) with field and velocity oomponents vx Vy Ex Ey ll I. varying with z as e."Z is considered in this section. Then Eqs. Il-7 becomes Ui VX + g vx - Wo vy O/me - EX c Vy III-l 10 + g VY G e/m Ey +o3 o It is noted that in this oase, since Ez 0, vz = 0 and the above equations are exact, requiring no assumption as to the magnitude of the velocities to remove the non~linear (v1 ~ V ) v1 term. Solving for the velocities: _e/m Ev c-e/m[Ey + g E] w u +g V 3 Jz Since V J =i =P1 0 the currents are: Jx Po VX Jy pvy Po V Combining these expressions with the field equations one obtainst: I II-2 -m. ~ — + (a + i p Ey - (b) Ey E M- -Wo H (o) YEx = -i~0 Ey (o)

29 These equations oan be solved by evaluating the determinant of the ooeffioients of Ex and Ey to yield +G +k -co/m +- -40 & 0 So 2 2 y. = Lo o+ + and, sinoe g << ~o: 2 + 1+ 2 + i &b e = se CO Naking the substitution Po e/m - 2 eo from Eqs. II-9: and Oz= ~$o)2 p where ' is the complex index of refraction 17 = qr + i7i and se the effective dielectric constant, and oe the effective conduotivity of the eleotron gas. It can be shown from the well knomvn equation that: tjr.[v l l+. 2 +1 1 e 2 w2 The quantity %e is plotted in Fig. 3.3 as function of o/co. It can be shown that wave polarized with its eleotrio field component in the z direction will experionce the same effective dieleotric constant as found above.

30 B Cylindrical aetron. Propagation in the z direction of a plane wave ( O 0 = 0) with field and velocity components vr ve Er Ee Hr He varying as e-Yz is considered in this section. This corresponds to a TM wave propagating along a coaxial line, the components Ee IH being necessary to account for the tangential electron motions in the space charge. The equations of motion Eq. II8 become: i vr + g r + r — a = - e/m Er - oc ve I II-4 Or Or In order to linearize the equations it is necessary to assume the perturbation velocities so small as to make any cross product term negligible in comparison with the other terms in Eq. III-4. Solving these equations for the velocity components, after dropping the terms vr r and vr avr (assuming the derivatives to be continuous and bounded): Or e /m [ESB iO+ Er] cOrVr (ia+ g) a O e/m [Er+ -t- Fg- Ee] From Eq. II-10: aa OVo (1 + ro) so that the above equations become:

c er [r0/ Ee Er (a) oo' + (o+ g)2 III-. V _ El (b) (~ + g)2 -O where r Sincein this case V E = 6 1 0 the currents are Jr i PoVr Jeo PO ve The field equations then becomes +me/m ~Jo + gP Oo e/mcooe r] He = iH eo Er + (a) aH. ~ /,PO / Er +' E+] -Y EHr - ~'Oe 4CoO + (io+g)2 (b) a (r He)'o (o) 1 (. a~F (r E ) ='ico Hz (H) 5 7F~~~~~~~~~~~~~~

32 Differentiating Eq. III-6(f) with respect to r: a.Hz 1 a 1 a2 - ~ ro - -= ar (r Eq) + 1 F (r ) From Eqs. III-6(b) and (d) above: iY2 a 1 1 12 e q - (r Ee) + (r e) = Ee Er (iM + g)x wl-ere X = + 2 7 code + ~k D,O and from Eqs. II1-6 (a) and (c): r2 -- Er = Ai o Er + cO X Ee - (i +g) X Er From this last equation Ee =A Er where r 2/0 i Co + (40 + g) X 3C X Therefore: r;(r Ee) =' (r Er) = rEr rA Lr + + Era and ad (r E) 2 = r E + 2r +aA 2a] Er + r E (3r ar L Substituting these expressions into Eq. III:-6(g): a2 7~ + 1 [2S ak + 8 aE [III-7 a Ma r + r ar +A 2- r 6rlr

33 The general solution to the differential equation III-7 would be rather formidable, unless X does not vary with r. In the case of the Hull-Brillouin space-charge solution this condition can be satisfied if the cathode is infinitely thin. In this special case Eq. III-7 reduces tos r2 r ar r2 ( E which is simply the Bessel equation. Comparing Eq. III-8 with the similar equations for wave transmission in a cylindrical dieleotric-filled waveguide it is sera that r2- m,2/02 ijoILo (cAX + ) (a) or II1-9 + g)2 crx p1 (b) Letting y 2 = -c2/cZ I2 and using the Hull-Brillouin value for the space oharge density (with vanishingly small oathode) from Eq. I_-10, 2 COO m 6 o. _~ and the corresponding angular velocity 4 This equation then beoomes: 1 4" = pe2 = 1 + 1 _ 2 2 2 (a) 1III-10 1 +2w2/2 = - I 2 22/c2 (b) Eqs. III-10 are plotted in Fig. 3.4. It is seen that, depending on the value of O/6oc, se can take on all real values; positive and negative. The oonduotivity resulting from energy loss by the electrons due to oollisions with the heavy partioles in the space charge is seen

89 ~~~~~~~~~~~~FIG. 3.4 8 Ee FOR CYLINDRICAL MAGNETRON PROPAGATION IN *Z DIRECTION ~.(I) I 7~ ~ ~~~T- 70 x IO7 6 60 j ~~~~~Hr 5 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~rc; ~ 50 4'. I I I 40 3~ ~ ~~~~~ V3 Ee 0~~~~~~~~~~~~~~~~~~~~~~~~~1 0 -1 I I I I II I \ I I I I L -L - - - - - 30 - ' 1 i II _____ __ __ ______ I I I I i I 2 i~~~~~i Ee~~~~~~~/ ~o _ G - I. 0.2 ~~/4.6 1.0 12 1.16 18 20 i) __ -2 -3~~~~~~~~~~~~~~~~~~-3 55 -6 -7 0.2 4.6.8 1.0 1.2 1.4 16.8.. ~ ~~~~~~~~~~G

35 to reach a maximum value for CO//o = 1/V as oould be predicted from the velocity Eqs. III-5 since the velocities are maximum at this value of C. Discussion of Variation of.e with r. The effective dieleotric constant of a magnetron space charge has been determined exactly, (for the small signal case) for the plane magnetron and the cylindrical magnetron with vanishingly rsmall filament,in the above sections (A) and (B). Since neither of these structures is used in practice, it becomes very desirable to be able to determine the dielectric properties of the space charge as a function of the radius. Unfortunately the complete solution of Eq. III*7 is very laborious, even if the Hull-Brillouin relation for po is used. Perturbation, or approximate methods do not appear to simplify the problem appreciably. Therefore until such time as the need for a complete solution to Eq. IiI-7 arises, thus justifying the time required for its solution, one must be content with an interpolation based on the limiting solutions already obtained and a lmowtledge of the variation of the space charge density po and angular velocity as functions of re 1 dA d hci Eq. III7 is seen to contain terms = which is, A dr except for a constant, equal to d Zn X. For rrv greater than about three, the quantity en X does not vary more than 25% from its value at rH/ro O a). For this reason it does not appear to be a prohibitively bad approximation to negleot the terms in A-, in Eq. III-7 for rH/r > 3. This equation is then reduced to Eq. III-7. The critical points of the first term in the brackets in Eq. III-8 will be examined qualitatively as a ~unotion of r to attempt an interpolarion between the values r.H/rc = 1 and r/re = 0.

36 Eq. II-9 (b) can be written in the form m~ [1 kj] se = 1 - c I -- Upon, substitution of the relations Eqs. II-10, it can be shown that the variation of the above fraction with r is negligible for rH/re >2 so that for larger values of rH/rO the value of o/,o at vhich se = 0 does not change appreciably from its value for rHI/ro = o0. This slow variation with r results from the mathematioal form of the various funotions of r involved in the expressions for po and. However in the case of the infinite singularity, the value of r 2 cm//c is determined almost completely by 2 = 1 which is a fairly rapidly varying funotion of r for rij/ro < 5. Therefore this oritical point can be expected to vary more, for large values of rE/re, than the point se = O. From these qualitative considerations the curves of Fig. 3.5 were interpolated from the values determined from the analysis (shown as heavy dots on the ordinates). It is to be emphasized that these curves are only the result of some judicious guessing and do not result direotly from the solution of Eq. III-7. In a strict mathematioal sense one cannot draw such conolusions as these, since they necessitate the dependence on r of the propagation constant in Eq. III8, which violates the assumption made there. However it is believed that these approximations are reasonably valid so long as the radial propagation constant is only a slowly varying function of r, i.e. for rH/rO > 3. A partial justification for these approximations, made in violation of the condition that r be independent of r, lies in the faot that the energy density aries lgarihmialy with r, so that in vte region where po and t vary most rapidly, there is relatively low energy density.

37 FIG. 3.5 EFFECT OF CLOUD RADIUS ON CRITICAL VALUES OF GCO//c - APPROXIMATE INTERPOLATION PROPAGATION IN Z DIRECTION 1.4 O< Ee <1 I 1.2 Eeff:'O -U- -~~-MO I. _ _ _ _ _ _ __ _ _ _ _ _ Ee <0O 3 0 INFINITE SINGULARITY -..2 -J 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 _ _ _ _ _ -t- - -~,,,~ 12 34 5 678 9.10 rH rc

2z Propagation in the Direotion Normal to the Anode and Cathode. A. Plane Magnetron. In this section propagation in the + y direction of a plane wave (ax = a/az 0) with field components Ex Ey Hz vx Vy varying as e')~ will be considered. The force equations in this ease are identioal to Eqs. IIIll so that the velocity oomponents are ex e/m --- s m ~ i vX 3 - x=i +- g IIIl where it has been necessary to assume the perturbation velocities small so that their produot and therefore the non-linear term (V1 V)Vl can be neglected. The perturbed charge density is found, using the continuity relations v-J. P1 Z Y = _. Jy so: P1= & Jy, but there is assumed to be no oomponent of steady eleotron velocity in the y direction; so thatsinoe P does not vary with xthe ourrents are given bys J r Po Vx JJy Po vy. Use of the velocity equations above yields: ~'mJ x ' J-y + = -1+11 E III1 Substituting these relations into the field equations, there result: -yr'= -[_ E + +i40&J E: (a) III-13

39 O =[Pm e + ix )e] E y P E: (b) III-13 rEx - HZ0 EZ (O) Eliminating Hz from the first two by the use of Eq. III-13(c): [Fr2 wC O- 1 Ex + O = O [/3 Exc + [ - i&o] E O so that: [+<~ i2: [72 e/m - i &o = I I1-14 and Using Eq. II-9: e' 1 — (-U-~)2, ~2 )2 = whioh are the same as Eqs. III-3. These relations are plotted in cons idered above.

B. Cylindrical Magnetron. In the oase of the cylindrical nmgnetron, propagation normal to the anode is oonsidered as in the rad_ ial direction, with field components Er Ee Hz vr ve and with aoe ' a/z = 0. The foroo equations are then identical to Eqs. III-4 so that the velooity components are: e [e + vr = -r +ct + g)2 =-.C[Er + 0 Coc + (ic + g)2 where again it has been necessary to assume the perturbation velocities small so that the non-linear product term oan be neglected. The oharge density P1 can be found froms V-J = Jr ipl However, since we have assumed the steady eleotron flow to be entirely tangential, the currents are simply Jr = Po Vr Je = Po vI The field equations are then: [ PO e/m (icE +.i..... Er +- PO2. Ee = / + (icc + g) + ice 2 (a) II *17 HZ e/m (ie5 + g) a- Er a F (r EEe) - -. /O Hz (o)

41 Differentiating Eq. III1l7(o) with respeot to r and substituting into Eq. III-17(b): a2E + _ E _ _ ar2 r ar r [ jL e-& /m (ilo +) + a sei ~ Po e/m c4 ( +g)2 + tC + (o + g)2 By solving Eq. III-17(a) for Er -00 /m Ee ', - ~,'o e/m;t~ + -+;; o [~' + 0'a + ' Er U If TT+3 j ' and substituting into the differential equation for E0 one obtains: + 1 E ar2 r ar r2 E -COIL [0 C~O Fo 40 +3 +g - CCo - iC) e/m ( ) + g) 2 -4Pb e/m (:4 + g) + lo eo Goat + (0 + g)2 E, So long as the part in braokets is independent of r, it oan be shown to be equal to the square of the radial propagation oonstant, y 2. Separating the bracketed expression into real and imaginary parts and substituting 712 02/co2 r2 mRe 1 - r a larif ion of his analogy and a general discussion of ylindrioal waves, see e.g. Sohelkunoff, "Eleotromagnetic Waves", D. Van Nostrand and Co., page 406, 1943.

42 Imp 2 p po mi 7 2 + om+ (ccIm 12POat[PO - + IE (03a2 III-19(b) where as before terms in g2 are negleoted in comparison with CO2 or o?2. These expressions are plotted in Fig. 3.6. Using the Hull-Brillouin relations Eq. II-lO,for a vanishtngly small cathode,the above equations beoome: 2 /o~o - 1 -se /12 4/8.2 ale = II < 1 Le 2o By using the same type of qualitative arguments as mentioned in Section 14e above, the variation in the oritioal points of Eq. III-20 can be interpolated between the rH/r 1 and rH/ro = ao values. These ourves are shown dashed in Fig. 37.?

~~~81~~~~~~~~ 1 I I ( I I I IFIG. 3.6 7 Ee FOR CYLINDRICAL MAGNETRON 6 PROPAGATION IN r DIRECTION 4:-5~ - 2 ~ _ " -3 7 0.2 4.6.8 1.0 1.2 1.4 1.6 1.8

44 FIG. 3.7 EFFECT OF CLOUD RADIUS ON CRITICAL VALUES OF.)/WC -APPROXIMATE INTERPOLATION PROPAGATION IN RADIAL DIRECTION 1.4 O< Ee 1 1.2 Eeff 0 I1~~~~~~~~~f. - INFINITE SINGULARITY _' - I 3. \ '~LLO~~ I``~~ ~~Eeff = I -.6 O<Se C II I -~4~iL I / / Eeff o Ee <1.2 2 2 3 4 5 6 7 8910 rH rc

3. Propa ition in the Direction Parallel to the Steady Electron Motion. I. Plane. Magnetron. The propagation of eleotromagnetio waves in the direction parallel to the steady electron velocity in the magnetron should prove of interest from two points of view. While this analysis is carried out only for the plane magnetron, the general results should be applicable at least qualitatively to considerations involving the oylindrioa' magnetron, and might be of assistance in explaining certain frequency effects of the space charge on the resonant circuit, such as frequency pushing and voltage tuning. Secondly, it is believed (as has been suggested by a number of people) that a magnetron structure should be capable of providing amplification of electromagnetic waves. This possibility was suggested earlier in the course of this work, the idea being based partly on an analysis carried out at that time and partly on intuitive oonsiderations, extrapolating from the successful Electron Wave Tube2 due to Haeff. The analysis presented in the aforementioned report (reference 1 on this page) has since been found to be incorrect, however the possibility of amplification has not been abandoned. The problem of wave propagation in the direction of electron motion in a plane magnetron spaoe oharge has been treated in a Quarterly Progress Report No. 2, Electron Tube Laboratory, University of Miohigan, July 1950. Iaeff, A. Val "The Electron rWave Tube - A Novel Method of Generation and Amplifioation of Miorowave Energy", Proc. I.R.E 37, pages 4-10, 1949. Labus, J., "High Frequenoy Amplifioation by Means of the Interaction Effeot between Electron Streams", rh. Elekt, bertragung 4, pages 353-360, 1950.

46 comprehensive manner by Macfarlane and Hayl. However, while they apparently found suitable mathematical expansions enabling rather general solution of the equations, some of their most interesting results are not presented for the case of the magnetron (their case a = 1). That is, they consider interaction with a beam of electrons injected between two parallel structures, the electron velocity varying linearly with distance normal to these structures, but not necessarily vanishing at one of them as in the magnetron with a cahnode as one element of the delay line. This problem of amplification in a plane magnetron structure has also been mentioned in a note by Buneman2. Therefore, while the present treatmenb of this problem will necessarily be of more limited scope than that of Macfarlane and Hay, it is hoped that the results can be applied profitably to the magnetron. For consideration of electromagnetic wave propagation in the direotion parallel to the steady or drift velocity of the electrons, two types of waves must be distinguished. The first wave will propagate with a velocity near that of light, being determined by the dielectric properties of the space charge, as well as the boundary conditions imposed by the confining circuit. A possible example of this case would be wave propagation along a plane parallel (without loading) transmission line, one of whose elements is an emitter, giving rise to a space charge with drift velocity along the length of the line. Since this case is of Maofarlane, G. G. and Hay, H. G., "WIfave Propagation in a Slipping Stream of Electronss Small Amplitude Theor;y", Proc. Phys. Soo., Lond. B, IXIII, pages 409-427, 1950. Buneman, 0., "Generation and Amplification of Waves in Dense Charged Beams under Crossed Fields" Nature, V165, page 474, March 1950.

47 little praotioal intorest, it will not be treated here. The second type of wave involves propagation along a strucoture (usually periodic in space) suoh that the phase velocity of the wave is oonsiderably less than the velocity of light. This struoture is usually made as some type of periodically loaded transmission line, such as a loaded waveguide, (the side opposite the "slow wave" struoture being an emitter) and the wave velooity can be made (within reason) to oonform to the designer's wisthes, usually one-texrth or less of the velocity of light. This low value of wave velooity allows certain simplifioations to be made in the equations, as will be seen later. The essential differences between the "field wave" and the "spaoe charge wave" oan then be summarized as follows. In the field wave, which is propagated with a phase velocity near to that of light, the space charge density exerts relatively little influence on the fields; that is, there is relatively little wave energy stcared in the electron motions, so that this wave is oharacterized by V E = O. The wave energy in this case is stored alternately in the electric and magnetio fields. The space charge wave, propagating with a phase velooity small compared with that of light, is influenced to a great extent by the space charge; in fact the wave energy is stored alternately in the electric field of the wave and in the kinetio energy of the electrons. Since in this case there is relatively little energy stored in the magnetio fields, this wave type can be oharacterized by V X E = 0

ProR ation along a "Slow Wave" Structure A schematic drawing of the structure to be considered in the following analysis is shown in Fig. 3.8. As in other sections of this report it is assumed that the electron velocity is slow compared to the velocity of light so that the treatment is non-relativistio and also the usual small signal method is used. In addition it is assumed that the wave velocity is small compared to the velocity of light so that the time rate of change of magnetic field can be neglected, and the electric field derived from a potential function. Then V x E = 01 and the field components present are Ex, Ey, and Hz. The equations of motion are, from Eq. II-7t it vx - YVa vX 8/m Ex I -21 ~ 8r - y v vy = - e/m Ey - COo The degree of approximation of this customary simplificoation can be seen by writing the electric field in terms of the vector potential A and the scalar potential V ass and using the supplementary condition on As Then: VE a2 at2 Considering the potent.a to represent a wave motion in the x direction so that = o e 'Yx E 2 2/ + cOn so that when 2/02 < Y2 this last term can be neglected and E = - V pi

49 FI:G. 3.8 IDEALIZED SPACE CHARGE IN PLANE MAGNETRON WITH PERIODIC ANODE and from these the eleotron velocities are: VX )+= + YOYY + roY ' - + r ')Y 1 11-22 where the substitution v. = -<ocy has been made. By this substitution it is implicitly assumed that the cathode is located at y = 0. The field equations are: -~~ ~~r sy w aE7 ~(a) a =Po vx + vo P1 + ic Ex (b) 7az = po +y+ m ~o Ey (o) p1]Z S o[4 s Ec I+N.] (d)

50 Substituting Eq. III-22 into Eq. III-23, the follovring equations are obta ined: ~a.l = y,) (a) ay (A- r Eo vo + Eso) Ex + Eo Vo ay III-24 and YHz= (A+iLo s)Ey CEPx (b) where A Po e C o + r C0y)2 Differentiating Eq. III24(b), it follows thats ~ayE = A,. A E a +V s A E, C aY r au r ay r u ay Equating this to Eq. III-24(a): aA [ + (k +yr cOy) o] + yb C III-5 _ L [ + A + (i + rCOY J From Eq. III-23: E -L a~x 1 d~i~E Y r yY au r so that Eq. III-25 becomes; 1 aZE, r 1 aA 1~C [A + (im +y oYy) Eo] 7 r aY + aY I II26 +l aau t- +A+ (i +yo y) o]r = o The equations will be simplified by substitution of the new variable:

51 Then Eq. III-2 (b) becomes: + a# 2 [e3 e 6 ~ 0 11I-28 This is the basic differential equation representing the electrio field in the space charge region and presupposes only that the velocity of the wave in the space charge is small compared with the velocity of light. In what follows the exact nature of the external cirouit will not be specified, it being presumed that a circuit of the characteristics desired to achieve certain performance can be constructed. It is seen from Eq. III-27 that Re e = O corresponds to synchronism betwveen a layer of electrons at distance y above the cathode and a wave travelling in the -x direction. Since the most interesting interaction effects take place for velocities near this synchronism condition, in what follow;s the attention will be confined to small values of Z. Therefore the solution to Eq. III-28 will be found in terms of a power series expansion in 4. However, first the equation will be examined briefly to demonstrate that this expansion is possible. Eq. III28 can be written in the form: as + ftow." [-iso'- I ] ix = o r me 1 FZ -e2 + e4 1 2 2- L2 + L4 The terms a2 and -e2 e are both analytic at Z 0 and the equation has a regular singular point at Z = O. A power series expansion is therefore possiblel about Z = O (at least in the region -1< Z<1). 1 See e.g. Rainville, E. D., "Intermediate Differential Equations", John Wiley and Sons, page 90, 1943.

52 Therefore letting: 00 Ex Z ne o I II-29 it follows that: o n-I X~ = nan'and, = Z n(n-l) ane lation is fbounds - [n (n - 3) + 2] ans + En (n -1) + 1] a1 n-I 0o ~ ~ ~ ~ ~~0 o - z n +1+l = O so that the first few ooeffioiasts are: aO = 0 a1 arbitrary a2 = arbitrary a3 = al/2 a4 a 5/2 a5 ' 5/24a1 a6 = 11/40 a2 a7 93/720 al a8 = 321/1680 a2 a9 3848/40320 aL alO a *1475 a2 all =.0760 a1 The series solution fbr Ex is thens al j n 5 a., t5 A.11 e6 + 1 2 =2 a2 2 24" a2 IIIO30

53 It is seen that two independent series are obtained, one includ ing the even and the other the odd potwers of the variable quantity. The boundary condition to be imposed on the electric field EX, in the space charge is that it must vanish at the cathode, i.e. Ex = 0 when y = ( = _). Under this condition Eq. III-30 becomess 0,~. [... 1 + —2 --- + + 0.275 + 8a2 cog [1 O0.Zlin4 i0 c02 [I 2 ~c2 " so that for co/"c < 0.8 the relation al/a2 - o/w II 1/3l is valid and is a very close approximation fbr /0%, < 1. Eq. III-30 then becomes: a [1. + e 2 + 0.2 p4 [+ 0.129 6 + 0.0966 8 + III-b2 + 2 [1 +;2 + 0.275 +4 + 0.191 Z6 +..] The electrio field Ex in the space charge is plotted in Fig. 3.9, vs Re Z from Eq. III-32 using the boundary condition Eq. III-31. Certain qualitative information concerning the space oharge can be obtained from a study of these tlrves. The oathode of the magnetron is, of course, represented by the right hand interseotion with the absoissa of the ourve corresponding to the particular value of co/mc under consideration, The intersection at S = O oorresponds t;o synchronism between the layer of elecbrons at a given value of y and the travelling electromagnetic wave.

FIG. 3.9 X DIRECTED ELECTRIC FIELD DISTRIBUTION IN SPACE CHARGE 3.5 WC I w~~~~~~~~~~~~~~~~~~~~ w~~l ywcyi 3.0I( 2.5! \ I WC Ex 2.0 - 02 ' i~~~~~~~~.5 ~~ 4 ~'X~~~~~~~~~~~~~~~~~~~~~~~~~ c~ -I dop~rl 1.5 I 3, 6 /I / IOJI~~~~~~~~~~~~~~~~~~~~~ __~~~~~~~~~~~~~~~.,L I' / I -:05 1~~.0 I _I_.aI / ~/ / *.3 4.5 Re ~~~~Q

It is seen that this synchronous layer of electrons beoonms an inf inite admittance sheet in the spaoe charge. The portion of the ourves to the rift of ~ = 0 represent the field in the region in hich the electrons are moving slower than thle wave and the part to the left of Z = O the region in which the electrons are moving faster than the wave. Fromi this it appears that the interaction space in a magnetron is di_ vided into two regions by this admittance sheet; the region between cathode and the infinite admittance sheet and the region between this sheet and the anode. As the electrons are caused to increase in velocity (e.g. by increasing the magnetic field) this sheet will be displaced to. ward the cathode. Examination of Eq. III-28 reveals that in addition to the regular singular point at = 0, this equation contains a second regular singular point at C = ~ 1. It will prove interesting to examine breifly the physical nature of these singularities also. For the value X = -1, the real part of Eq. III-27 becomes: ~o=~ (P~ -1) where vp is vhe phaside velooiy of thhe ave propagaing with the elentron stream, i.e. vp = - mm/t. The right side of this relation is seen to be the frequency of the wave whose forces are acting on the electrons, as seen by the moving electrons. That is, while a stationary observer (an electron) experiences a force due to the fields of frequency co, an observer moving with velocity vo experiences a force of frequency mc (vo/Vp - 1). Therefore at the value of vo/vp for which this Doppler frequency is equal to the cyclotron frequenoy, the layer of electrons for which Re X t 1 experienoes a resonance effect between the wave

56 and the applied magnetio field. From this it would be expected that the electrio fields of the wave have a singularity at the value Re $ - - 1. That this is indeed the oase can be seen from Figs. 3.9 and 3*10. The singularity at the point Z + 1 is of less interest since this inrteraotion (for c/co < 1) takes place below the oathode. However this singularity corresponds to the same type of phenomenon, with eleotrons moving in the + x directior interacting with fields of Doppler frequency co(l - Z- ) of a wave travelling in the + x direction. VP Evaluation of the Electronic Admittanoe Using the series expression (Eq. III-32) for the x directed electrio field in the spaoe charge region, together with Eq. III-23-a, the y directed eleotrio field distribution can be determined. 1 (Ex Ey 7 y T i -' = ia2 - 1 + t2 + 1.05 + 0.9 6 0.878 +....] III-33 + [2 4 + 23 + 1.65 + 153 7 + 1l.47 9 +....] This series converges even less rapidly than Eq. III-*32 so that values were not obtained above $ = 0.9; the ourves representing this series solution are shown in Fig. 3.10. It is seen from this figure that the y directed eleetrio field also appears to inorease without limit near S - + 1. It can be shown (Whittaker and Watson, "A Course of Miodern Analysis", Cambridge, page 201, 1946) that in such a case as considered here, the function will be 6ither analytic or have a logarithmic singularity at the regular singular point. In view of the above-mentioned aocount of the nature of the interaotion at the singularity, the latter possibility appears more probable.

-6 I I I I I I I I I I I I I I @ -0 /, w wi1~~~~~~~l~/. sOmSIC Y DIRECTED ELECTRIC FIELD __1 __t~~~~ 1v DISTRIBUTION IN SPACE CHARGE -8 __""", - 10 -- 1.0 -.8 -6 -4 -.2 0.2 4.6.8 1.0 Re d

58 Reference is now made to section V-1 where the boundary oonditions at the edge of the spaoe oharge cloud are discussed. Eq. V-8 of that section can be used to evaluate the admittanoe Y. seen looking into the space charge cloud from the anode. Applying Eq. III-27, Eq. V-8 can be written in the form: Ye(h), = =z - c 2 i /| III-34 1 Ey(h) The fraction ri E can be determined from the quotient of Eqs. III-33 and III-32 However since 4 will in general be complex, this expression will become quite long. Therefore a simplification will be effeted by restricting the consideration to values of eleetron velooity near the synohronism condition. That is, by restricting ~ to small values ( e < 0.4) a simple equation for Ey(h)/Ex(h) oan be obtained, allowing an analytioal expression for Ye to be found. By neglecting powers of Z(h) higher than the first in Eqs. III-33 and III-34 there resultst 1 ~E (h) 2 - (h) --- e(h) so that the eleotronio admittanoe beoomes: ( i @ ohrF 1m,4 \1_ 21 Ye(h) '(-)7 a- 2 + /2C III5 Letting y= a + ii J

59 Eq. III-27 shows that the phase oonstant of the travelling wave is given by: 1 = y (r C-/Co) and the attenuation constant by: y The real and imaginary parts of the electronio admittanee are therefore given by: Im Ye(h) - O [(t C-mo)~2 ++ ei2 X ( 4 ) 2+ [2 1 o + 4_,i +4 -2 2) L +r2 ei +\ / ) (2r24i2) + 4 ei2 r2i + +/O +4 (Z_2), i2) In the above equations it is understood that the values of 4r and Ei to be used are those at the boundary y = h. In order to illustrate the nanner of variation of Im Ye(h), Eq. III-36-a is shown plotted in Fig. 3.11 for the simple case of no time average energy exohange between electrons ard fields (ei = 0).

60 200 FIG. 3.11 SUSCEPTANCE OF ELECTRON STREAM AS SEEN FROM 160 ANODE. Im =0O UZ0.5 120 80__ - Iw -40 4 -40 -120.. -160 -200 it -.8 -.6 -.4 -2 0.2 4.6.8 10 Rei.

61 These ourves show the existance of the zero reaotance sheet at 5 = O. mentioned previously. They also show that the space charge appears inductive to the anode oirouit for Z> r 1 0 and capacitive for S < a. The value of electron velocity at vwhioh the electronio susoepeanoe vanishes thus oorresponds to one-half of the phase velocity of the wave, and the (Doppler) frequency of the force experienced by the electrons due to the travelling wave corresponds to the Larmor frequency (%c/2). Eleotrons with velocity greater than that of the travelling wave appear capaoitive but not so much so as the synchronous electrons. Therefore as the space charge cloud is expanded by increasing the d-o anode voltage, the boundary of the cloud appears induotive for 9 > co/2% and oapaoitive for 4 < o/2%X,. These curves or axy of the other theoretical results derived here are,of course, not valid under conditions at or near osoillation, in which the cloud surface and the motions of the electrons can be greatly changed by the large fields present in the interaction space. It is seen from Eq. III-*36-b that for r.< 0, corresponding to eleotrons moving faster than the travelling wave (3<O0), i>O0 results in a negative electronic conduotanoe. Since the wave has been assumed to vary as eihhhhYX, positive 4i corresponds to a growing wave. This result is at least oonsistant and it appears that the magnetron spaoe oharge is capable of delivering energy to the fields of an anode structure of the proper characteristics. Of course, a definite answer to this question of amplification can be obtained only from a complete solution inoluding the boundary conditions imposed by the confining

62 circuit1. That is, one must demand that Ye + Y =0 where Yo is the cirouit admittance. Lack of time prevents the completion of this study. Amplification by means of the magnetron space charge should be characterized by greater efficiency and power output capabilities than the travelling wave or electron wave tube. This follows from con. sideration of the phase focussing effect of the magnetic field (that is, removal from the interaction space of out-of-phase electrons) and from the fact that an electron bunch after having given energy to the Ahields of the wave, can move toward the anode, thus abstracting additional energy from the d-c field which can in turn be transferred to the r-f wave. As has been suggested before2, the existance of such amplification due to a magnetron space charge may aid in the explanation of the large observed noise output of magnetrons. That is, most considerations of possible noise mechanisms in the magnetron result in the conclusion that some amplifioation of the noise must take place after its generation (probably in the immediate vicinity of the cathode) until it is delivered to the fields of the resonators. Twiss3 proposes a mechanism of radial amplification, involving the coherent constructive interferencoe of the radially moving electrons in the multiple stream type space Maacfarlane and Hay found that amplifioationx along the stream was in fact possible, but it is believed that the boundary conditions they used are open to question. 2 This possibility has been suggested by a nuniber of authors, one of the first was probably Haeff, Phys. Rev. 75, page 1546, 1949. 3 Twiss, R. Q., L~oc. cit.

63 oharge. However it is at least conceivable that all or part of the amplification take place in the tangential direction, the waves represenbting the noise energy travelling with the electrons and irnteracting with the fields of the resonators.

IV. DISCUSSION OF THE RESULTS OF CHAPTER III In the previous chapter the complex index of refraction of the magnetron space charge was determined as a function of the applied magnetic field and the frequency of the wave. By considering the medium as a nearly perfect dielectrio, the index of refraction was expressed in terms of an effective relative dielectric constant. The results of this analysis are contained in the curves of ee vs Co/c. Considerable care must be exercised in the interpretation of the dielectric constant A. The only safe interpretation that can be given to &e is as an expression of the phase velooity of the travelling wave. That is, a wave propagating through the space charge region which exhibits dielectric constant &e will travel with the same phase velocity as a wave in a real perfect dielectric of relative dielectric constant Se. In the region &e> 1 the space charge can be thought of as behaving as a real dielectric. In the region e(< 0 the space charge can be thought of as analogous to a conducting material in which the wave is attenuated. However ii the region O< so < 1 no such comparisons can be made since real dielectrics exhibiting this property are nob known. Similar concepts to this are encountered in ionospheric propagation problems. In this chapter an attempt will be made to explain the mechanism and results of the electron-wave interaction and to give as mod a physical picture of these various values of dielectric constant as possible. 64

65 1. Disoussion of the Eleotron-meave Interaotion. An eleotromagnetio wave, polarized along one of the coordinate axes, impinging from free space onto a magaetron space charge will, in general, be partially transmitted into the medium. In the space charge the. electrons will be aooelerated by the electric field of the impinging wave; however due to the applied magnetic field the electrons will have a velocity oomponent normal to the direction of polarization of this wave, necessitating an additional component of electric field (and therefore magnetic field). The electrons thus set into motion by this "inducing field" will radiate (as small doublets) and can be thought of as coherent sources of a "radiation field" in the space charge. The inducing and radiation fields must be one and the same and constitute the wave whose propagation characteristics were found in Chapter III. In the present analysis, the space charge is thought of as exhibiting medium.like behavior. That is, the motions of the electrons are considered to be influenced by and interlocked with the motions of the surrounding electrons. This type of interaction would be characteristic of a region of high electron density. This behavior is to be oon. trasted with that believed to exist in a region of relatively low eleotron density in which the effect of the electrons is just the summation of the effects of the individual electrons aoting independently. That is, while the r-f fields in the electron cloud, in the latter case, will be essentially the same as those in the absence of the electrons; in the former case, in which the space charge exhibits medium-like properties, the total fields will be those dictated by the electron motions in the spaoe charge.

The electrons in the space charge will be accelerated, thus abstraoting energy from the wave, during one-half cycle and will, if they suffer no energy loss, return exactly this amount of energy to the wave during the next half cycle. This is shown in Section 4 of this chapter. The electrons therefore contribute primarily a reactive effect. If this reactive electron current leads, in time phase, the electric field of the wave the effect will be capacitive (&e >1) and the wave will be propagated Writh phase velocity less than that; of light in vacuo. If the electron current lags the field, the effect will be analogous to an induotance (0< E< 1) and the phase velocity of the wave will exceed that of light. At the value of 6/oo at which e= 0 the wave will be totally reflected from the space charge boundary. The behavior of the electrons at the boundary of the space charge at the value of t/o, at which e 0= has been discussed in a brief note by Forsterling and 1Wusteri in connection with the ionosphere. It is noted that only those solutions have been obtained, for wave propagation in the space charge, for which the electric fields of the wave are not parallel to the applied magnetic field. The solutions for those oases in which the electrio field is polarized in the direotion of the applied magnetic field are immediately separable from those found above. These latter solutions are well known2 and are trivial from the point of view of this analysis since the magnetic field does not in. fluenoe the motion of the electrons, its effect on wave propagation being Forsterling, KI. and Wuster, H.I 0,, "On the Origin of Harmonics in the Ionosphere - At Points where the Dieleotrio Constant is Zero", Comptes Rendus - Aoadamie des Sciences - 231 - No. 17, page 831, Otober 23, See for example: Nichols, H. W. and Sohelleng, J. C., "On the Propagation of Electromagnetic Waves Through an Atmosphere Containing Free Electrons" Bell Sys. Tech. J. - 4, page 215, 1925.

67 felt only through the space charge density. It is the motion of the electrons due to the various forces (including the applied magnetic field) that causes the interesting effects of the space charge on wave propagation found here.

68 2. Illustration of Dieleotric Constant by its Effect on a Resonant Circuit. A magnetron space charge placed in a resonant circuit in such a way that it can interact with the electric fields, will have an effect on the resonant frequency of the circuit. Reference to the curves of ee as a function of c/coo will show the nature of this effect. It is seen that ee can assume values which are positive and greater than unity, in which case its effect on the circuit will be similar to that of an ordinary dielectric; it can be negative, in which case the effect will be similar to that of a conducting material. Finally, values of ea positive but less than one can be obtained; in this ease the phase velocity of the wave will exceed the velocity of light and in certain simple cases this effeot on the oirouit can be considered as similar to that of an inductance though this concept must be used with caution. In order to illustrate the effect of a dielectric on an associated cirouit, the resonant wavelength of a coaxial line one-acuarer wvelength long, containing a dielectrio materials will be computed below. Consider a coaxial cavity containing a oylindrically symmetri dielectric material located in one end of the cavity. The electric fields of the T3IN mode are in air, region ls Erl Alex + Ble "c- IV-1 in dielectrio, region 2: Er2 l A2e' + B2e v2 IV-2 subject to the boundary conditions (see Fig. 4*1)s Erl = Er2 x a Erl = x 0O ~ O0 x m b

69 Therefore co(b.a) CIXI1~ ~oot z1 =tatn Z2 0 z2 The oondition for resonance is then.of Zn (b,.a Z ta oot < tan w = (s tan, V The curve of Fig. 4.1 shows the resonant wavelength of this cavity as a fumotion of ere 58 56 52 50 w 48 -j 6 00074 XLX 0 9 00 CM 44~ 42 40 FIG. 4.1 RESONANT WAVELENGTH VS. /38 - - - - - - DIELECTRIC CONSTANT )X/4 COAXIAL CAVITY 36 0 1 2 3 4 5 6 7 8 DIELECTRIC CONSTANT- Er

70 It is seen that when er = 1 the cavity is merely a X/4 open line with resonant wavelength X = 4b = 40 am. An increase in er has the effeot of increasing a shunt capacitance across the open end, thus increasing the resonant wavelength. A decrease in er below unity has the effect of decreasing the normal shunt capacitance of that portion of the line occupied by the dielectric, thus reducing the resonant wavelength, From Eq. IV-3 above, as erO -0 t4a = 36 cm, so that the line is effectively opened at the boundary (a) of the dielectric. This is, of course, because the waves cannot penetrate the dielectric viwose =r 0 and are thereby totally reflected.

71 3. Disoussion of the se vs. m Curves. An attempt will be made in this section to explain certain features of the curves showing the variation in the effective dielectric constant. It has been mentioned previously that the effective dielectrio constant can take on values which are positive or negative. The effect on the propagating wave of the space charge, exhibiting properties in these various regions of sea will be discussed together with a possible physical explanation of the zeros and infinities in these Ocurves The region se >! needs no explanation since here the wave be. haves as if it were propagating through a real, perfect dielectric with relative dielectric constant &e. Waves impinging from free space onto the space charge will be partially reflected from the boundary of the space charge, due to the discontinuity in properties of the media. In the region 0 < se < 1I the wave travels with a phase velocity greater than that of light in vacuo. For a space charge cloud of dimensions small compared with the wavelength of the impinging wave, the properties of the cloud can be explained with the aid of lumped oircuit analogies. Thus in the region 0 < ce < 1, the capacitance between boundary planes (normal to the electric field) of the space charge will be less than the corresponding capacitance with only free space betwveen the planes. This decreased value of capacitance can be thought of as the parallel combination of a capacitor (er = 1) and an inductance. The inductance seems to represent the effect of the space charge coloud, thus leading to the consideration of the space charge as exhibiting inductive properties in the range 0 <e <1.c This concept must be used with caution howevere

At the oritioal value of o/ao for which se 0, it is seen that at the boundary of the cloud (or at the surfaoe in the oloud at whioh se = 0) the normal oomponent of eleotrio field of the impinging wave must vanish. However, the remaining fields will oompletely penetrate the space charge, and because their phase velocities are infinite, the electrio field will be in time phase at all points throughout the spaoe oharge region. In the region of negative effective dielectric constant, it is seen that the wave is attenuated in the spaoe oharge (no propagation occurs). This attenuation is similar to that experienced by a wave impinging upon a oonduoting material, except that in the space oharge there is no energy loss as a result of the attenuation, i.e. it is purely reactive in nature. A better analogy for the purpose of explain. ing the phenomenon of wave reflecotion in this region if that of a wave guide beyond outoff. In this latter case, the propagation oonstant becomes imaginary so that the magnitude of the Poynting vector decreases exponentially along the direotion of propagation, the energy of the wave being reflecated continuously from various poinrts along the line. This same type of oontinuous refleotion prooess is believed to occur in the spaoe charge with negative dielectrio constant. The oonoept of a skin depth can be applied to the space charge. In th~e region in which ee< 0, the wave will vary along its direction of propagation as exp - 2,lj s so that the skin depth oan be written; = c~~

where X is the free space wavelength of the propagating wave. This skin depth relation will be used later to assist in an explanation of experimentally observed results of tests on this space charge. Some additional information on the behavior of the electromagnetio waves propagating in the spaoe charge can be obtained from oorv putation of the group velocity in this medium. The most oonvenient expression for the group velooity for this purpose iss vg - where k C O/vpP circular wave number. The phase velooity of a wave propagating in any direction normal to the steady electron motion in a plane magnetron is given by (Eq. III-S or III-6): 02 c2k2 2 so vg.kco2 ic } 2 1P..c o, 9O t V a k2 =co -~ It is seen that in this case r7 g = 1 where 77 and17 g are the phase and group refractive indices respectively, defined as 77 = 7 = '.. VP Vg This relation 71 s7g U 1 is normally satisfied in dielectric media and in regions contaixning electrons not under the influence of a magnetic field. It is not ordinarily satisfied in an electron atmosphere under the influence of a magnetic field. However it is seen that in the case of the plane magnetron space oharge this relation is satisfied even See e.g. Wale, H. A. and Stanley, J. P., "Group and Phase Velooities from the Magnefto-Ionic Theory", J. Atmos. Terr. Phys. VI - No. 2, page 82, 1950.

74 though a magnetio field is present. This is a result of the singular variation of steady electron velocity with distance from the cathode, which just counterbalances the x directed Lorentz force. In the case of z-directed propagation in the cylindrical space charge (Eq. III.*10): 2 02k2 50 = c 2 = 1+1 1, 1 d2 2=[+ 2 + ] vg + _______2 so that the relations = 1 is not satisfied in this case. For a radially propagated wave in the cylindrical space charge c2k2 2 cL2. coI 2 1 1 $ [ (of ' 6:2 In this latter case also o77X / 1. It is seen that in the above three cases the group velocity is equal to thle product of the phase velocity and a positive quantity whioh depends on CO/)c. The group velocity therefore behaves similarly to the phase velocity insofar as the regions of, negative dielectric cone stant are concerned. That is, the group velocity is imaginary where the phase velocity is imaginary (de 0), ndicating no propagation in these regions of o/o.

75 In order to attempt a physical explanation of the singularities in e curves, examination of the equations of motion Eqs. II-7 and II-8 reveal that a group of electrons in the space charge is subjected to three forces; the eleotrio field of the travelling wave, the Lorentz force due to the perturbed velocity, and the acceleration suffered by an electron moving to a point in space at vhich the steady electron velocity is different from the velocity at the point; from whioch the olectron came. For example, in the plane magnetron the equations of motion are given below and illustrated by the diagram. vx- c vy " _ e/m E; - ic ty. vy Cv e/m vy - V VY = - /m Ey + ~Y O o vy Wc vy VY / Bo It is seen that teUe steady electi;ron velooity is just such a function of the y coordinate that the x directed acceleration due to the motion of an elecotron from one place to another (vr V) vO is just balanoed by the Lorentz force o)cY. Solution for the electron velocities then shows no re sonance offect.

76 In the cylindrical cases oonsidering for example propagation in the z direction so that the fields are invariant in phase in an re plane, the equations are (for r/r>> l) e/m Er -izo vr = - eAn r coo0 v -1 e + Vr Cio/2 -e/me + aoVr 2r + e/m Ee LO0V9 It is seen from these that the aoceleration (vr.V) vo does not exactly oounterbalance the Lorentz force coo vr so that solution for tle radial electron velocity shows a resonance effect. yr [ico g ] = / - c e/mrm Fo O /mEe That is, for values of o2 near to %c2/2, vr can take on very large values with finite fields. This "resonance" between the forces acting on the.eleotron accounts for the singular value of 5e observed for example at c0/eoo l/f2of Fig. 3.4. One additional point, which should be mentioned before oonsideration of the effeotive dieleotrio oonstant curves is completed, is in relation to the existance of double refraction of the waves propagating through the space charge. In most treatments of the interaction of an electromagnetio wave with an electron stream under the influence of a static magnetic field, it is found that the wave exhibits double refrac. tion. It has been showi in Chapter III that no suoh phenomenon occurs in the ease of propagation in the magnetron space charge. In the plane nmagnetron space charge this is a result of the singular manner of

varlation of steady electron velocity with distanoe from the cathode, which just counter-balances the effect of the x directed, Lorentz foroe& The absence of double refraction in the cylindrical spaoe oharge must be explained separately for the two directions of propagation. In an eleotron atmosphere under the influence of a magnetic field, such as the ionosphere, the eleotric field of an electromagnetic wave propagating in a direction normal to the magnetio field will be split into two components. The wave whose eleotrio field is parallel to the magnetio field (ordinary wave) will travel with a different velocity from the wave whose electric field is normal to the magnetic field (extraordinary wave). In the present treatment there was considered to be no eleotric field parallel to the magnetic field so that in section III-2 only the propagation oharaoteristics of the wave oorresponding to the extraordinary ray were found. In the cylindrical space charge no double refraction was found for a wave propagating along the magnetio field because of the geometry chosen. That is, only propagation of a TBM wave in a oaxial structure was os considered, thus allowing no possibility of ciroular polarizations of different rotstions.

78 4 EleotronWave Energy Exohange. The time rate of absorption of energy by the electrons from the electromagnetic field of the wave is: W /=J-E dV IV-ZW where the integration is extended over the spaoe charge. For illustras tion, consider the oase of propagation in a plane magnetron space charge in the y or z direction. Then from Eq. ITI.ls -Jx Po ~+gIV-5 so that the energy absorption rate is: Re W -*-po e/m -Re E T *] d IV_6 12o+g Ex gj dV IV-*6 which beoomes, since g2 <<2: Re W -o e/ +[ E._, Ex dV, IV-7 However, it can be shomv, for example from Eqs. III-2, that Ex and Ey are 900 out of time phase so that when the oonsideration of the energy absorption is extended over a oomplete oycle this last tena vanishes, leaving: Re VT~2m [Ex 2+ I EyI dV. IV-$ 2m Y12 In the oase of no oollisional energy loss by the electrons g = O and it is seen that there is no net ierohange of eaergy between the electrons and the wave. Sinoe the fields vary as ei* the electrons gain energy from the waave during one-half cycle and give energy to the

79 wave during the other half cycle so that the net energy ohange is zero. That the effeat is therefore entirely reactive in the case of no collisional loss can be seen from Eq. IV-9. Another interesting result can be derived from these energy considerations. The Poynting theorem can be expressed in the forml V-S = J.E + i[ [o IE12 i-EolH2] IV-9 where S is the complex Poynting vector. Then - V'Re S = Re J.E* IV-l10 which is the relation used above. Also.VeIm S = Im J-E* - i[*oElZ - anoI 2] IV-1l The collision-free space charge is a non-conducting medium, so that Im S = 0, from whioh it follows that Im JE* represents the difference between the mean values of electrio and magnetic energy densities in the wave, From Eqs. IV-4 and IV-5: Ifm k L2..x. Evl dV Im.E* dV Therefore, sinee IE12 - IEx12 + IEy121 if one takes the time average of the energy density the following expression is obtained: O = +a [k IE.i2 +lEyl2]dV +- j [ioxit2 + ~OlEyl2 -LoIi]dv Using Eq. II-9 to eliminate po; this beoomes:. sL [iE:i2a + dV + 1/2 ~ 0(jiiExl~ 12 + yl2) - ol:,r1l a dV This equation becomes, since the integrations are extended over the same volume: [_! +1] ho[2l + i;,y2] - Ato j 2 IV-12 Stratton, J. A., "Eleotromagnetic Theory", M1oGraw-Hill, page 137, 1941.

80 From this relation it is seen that the eleotrons oontribute.the term -~ +x2 + IEyl2] to the eleotrio energy storage. The effeot of the space oharge can therefore be oonsidered as equivalent to a ohange in the relative dieleotrio oonstant by the faotor 1 - co2/o2 which is the same as derived in a previous section. Eq. IV-12 illustrates the oondition existing when o/cn < 1. In this region of o/o0 Eq. IV-12 oannot be satisfied by real E and H fields so that wave propagation per se oannot ooour.

81 5. Disoussion of Energy LOss in the Space Charge. In general, as an electromagnetic wave is propagated through the magnetron space charge, it will be diminished in amplitude as the result of a real power loss by the wave to the surroundings. In the analysis of Chapter III the only source of loss menrtioned was that due to collisions between the electrons and the heavy particles present in the sp&ce charge. In this seotion several additional sources of energy loss by the electrons will be considered. This loss of energy by the perturbed electrons is the same as energy loss by the propagating wave, since an eleotron will return to the wave all of the energy imparted to it during one-half oyole only if it is has lost no energy during the cycle. This diminution of the amplitude of the wave with distanoe is due to a real energy loss and is entirely different from the corresponding phenomenon ooouring when the effective dielectric oonstant is negative. In this latter case, as mentioned before, the amplitude is diminished because of continuous reflection as it progresses into the space charge. The souroes of energy loss by the electrons to the surroundings to be discussed here are: (a) loss due to collisions with gas particles (b) loss due to energy stored and dissipated in harmonic fields (c) wadiation loss (d) loss due to collisions with the oathode (e) loss due to collisions near a region where e = 0.

82 a. Loss Due to Collisions with Gas Partioles. As mentioned previously, when the amplitude of the propagating wave is small, the electrons in the space charge are linearly perturbed from their steady motion and during these perturbations can collide with heavy, relatively fixed atoms or ions. In the process of these collisions the electrons will transfer energy to the atoms, thus inereasing the temperature of the gas atoms present, at the expense of the energy of the wave. The gross effect of these collisions was shown to be qualitatively similar to a frictional force proportional to the perturbed electron velooity. As shown, e.g. in Fig. 3.4, the effective conductivity result. ing from these collisions is of the order of 1016 mhos/meter in the neighborhood of the singularity (i.e. near the value of co/o corresponds ing to maximum perturbed velocity), for typical gas pressures. This is of the same order as the conductivity of, e.g. Balmlite. This mechanism of energy loss can explain a decrease in Q of a high Q circuit in which the space olarge is placed, but will not account for any magnitude of power loss such as might be observed at high r-f voltage levels. b. Lss Due to Energy Stored and Dissipted in Harmonic Fields. Reference to the equations of motion Eqs. II7 and II-8 or to Appendix 5 will show that when the term(vl.V)vl is included in the equations, the perturbed electron motion will no longer be sinusoidal with time but will contain oomponents at harmonies of the fundamental frequenoy. These harmonios in the electron current will, of course, necessitate corresponding harmonic components of electric and magnetic fields, so that there will be additional waves propgated in the space charge. Some energy from the fundamental frequency wave will be transferred into

83 these harmonic fields, which will in turn beat together giving rise to additional harmonics plus a wave of fundamental frequency. Two cases can be considered. If the circuit in which the space charge is placed can absorb energy at any of these harmonic frequencies, there w1ill be a net transfer of energy from the fundamental wave to this harmonic and will be lost from the system. If the circuit cannot absorb power from any of the harmonics (such as efg. an idealized case of wave propagation in a space charge of infinite extent) there must exist a type of equilibrium between the rate of energy transfer from the fundamental to the harmonic fields and by means of the boating phenomena from the harmonic fields back to the fundamental wave. In a cavity not especially designed for the purpose, it would be highly coincidental if a resonance occurred at any of the lover harmonics, so that in general it would not be expected that appreciable energy could be transferred from these harmonic fields to a surrounding circuit; since little energy could be absorbed if the electrons had no large electric fields with which to interact. o. Radiation Loss. The electrons, in their periodic motion, will be accelerated by the electric field of the propagating waves It is well lkown that an accelerated electron will radiate energy. The total energy so radiated per second from n electrons per unit volume isl R 2 e2 a2 ers 20 e2 a2n oules 3 c2 cm3seo c2 m3see IV-13 where e is the electronic charge, a the acceleration and o the velocity of light. Neglecting for the moment the effect of the magnetic field, 1 See e.g. Page and Adams, "Electrodynamics", Van Nostrand, page 328, 1940. Alfyen, II., "Cosmioal Eleorodyna mis", Oford page 35, 1950.

84 the acceleration oan be written a = (e/m) E. The energy stored in the field of the wave per unit volume is 2 joules Thus the "Q" or 2w times the ratio of peak energy stored to average energy dissipated per unit volume per oyole, due to radiation will be: Q = ' = 2t & 02 E2 2w 8 02 m2 R x 20 n e2a2r 20n 6300'4 for n = 5 x 1010 electrons per cm. (Corresponding to a magnetic field of about 1000 gauss.) These considerations of the energy radiated by a periodically accelerated,eleotron assume, of course, that each electron can radiate independently of the surrounding electrons and that all of the energy so radiated is absorbed by the surroundings. These assumptions are believed to be fulfilled in practice. The Q value of 6300 found above, for a space charge density corresponding to a magnetic field of 1000 gauss, will be increased or decreased as the space charge density is increased or decreased. This rate of energy loss by the space charge is not considered appreciable, so that it can be neglected while considering loss effects at large values of r-f signal strength. d. Loss Due to Collisions with the Cathode. In this section the electrons will be considered to execute a double stream type of motion in the space charge, so that there is ourrent both toward and away from the oathode. Then, in the absence of any r-f fields, an electron leaving the cathode will move out to a maximum distance from the cathode

85 and return to the cathode, arriving with just zero energy. If during its path, an electron loses some of its ordered energy, eg. to noise, it will not be able to return to the cathode. On the other hand if an electron acquires additional energy during its exoursion, e.g. from r-f fields in the region, it oan arrive at the cathode with non-zero energy. In t;his way some of the energy of the r-f wave present in the spaoe oharge can be lost to dissipation at the oathode. It would be desirable to determine the magnitude of this loss as a function of the applied fields and frequency, etc. A comoprehensive treatment of the oathode bombardment energy loss by the r-f wave in the space charge would, unfortunately, be quite laborious. Therefore this seation will be restricted to a consideration of a planes, temperature limited magnetron spae chaorge and the order of magnitude of the loss determined for these conditions. It is hoped that from this calculation some useful information as to the loss in the spaoe oharge limited cylindrical spaoe oharge (with small r1/ro) oan be inferred. Using the notation of Fig. 2.2, for the temperature limited ease the equation of motion of an electron in the inberaction space of a plane magnetron oan be writtens IV.14 The anode voltage is consideredas the sum of a d-o and an a-o term so that the fields ares Ey "EO + Em Sin (as +4') where Eo = VoA and Em = VYa, Vo and Vm representing the d-c and peak a-c anode voltages respeotively.

86 Then if the x directed velocity is zero at the cathode x = -ooy and y= - e/o Eo e/m E Sin (ot ++) - o 2y This equation can be solved very easily for y as function of t by the use of the Laplaoe Transform to yield the following expression (the electron being assumed to leave the cathode with zero y directed velooity at t = O)s y (t, ~V) =eE0 Cti (1-os 000o ) c co02 -..COO o —15 'CO0 e 2 [Cos cot - Cos ot The y directed velocity is then: y (to, ) = -(e/m) (Eo/oo) Sin coot Emo (Cos t Cos coot) Co2 - Co IV-16 + Em Sin (co Sin ot - coo Sin cot) coo 2.-c02 The total electron velocity is v2 = x2 + y2 but at the cathode X 0 so that for y 0= v = y and the energy of an electron upon reaching the cathode is: 1 Em 2 2 Cii2 CO Sin co -b [(ob Ctl _- Cos wtl) Co cos* IV-17 + Sin (co Sin Al - coo Sin cotl)]2 where tl is the time at whioh the electron strikes the cathode. That is, t1 is such as to satisfy the equation, from Eq. IV-15: Y (t1,. ') "o IV-18

87 The energy of arrival of an electron at the cathode is considered as energy which has been imparted to it by the r-f fields. In the absence of r-f fields the electron arrives at the oathode with zero energy. In the presence of the r-f fields the electron, in its complete trajectory from emission to capture by the cathode, will effect a zero net energy exohange with the d-o fields so that the "excess" energy at the cathode must be due to the r-f fields. This assumes, of course, no capture of the ele otrons by the anode. Eq. IV-18 was solved for the special case Eo = Ems, C/Co 1/2, for several values of 4 from 0 to 2n,allowing determination of tl. These values were in turn used in Eq. IV.17 to calculate the energy of arrival at the cathode. This latter information was plotted vs 4 and integrated graphically. The total loss can then be found, by letting the loss due to dR electrons striking the oathode be dw- 1/2 m y2 (tlu+) dR and doll f d4+ where f is the number of electrons emitted (and therefore captured) per cycle (c) per unit angle (4I), per unit cathode area. Then if / is the number of electrons emitted per cycle (Xo) per unit cathode areas 3 = 2ffd4 0 Assuming f to be independent of 4, (which should be realized in practice for this temperature limited case)s 3 =z2trf and 2 Walky2 dw

88 but y2 (tlo 2) x Area under ourve of -2 V This area was found to be approximately 4T so that vJF ~[8m io]p mBm Z m22 oules V9 4,u ' coo_ c m2 oyole The value of ocan be found from the Dushmann equation: 'e " AOT2 e 11600 whioh in the case of Tungsten at 25000K is: Rp 2 x 0.75 = 4 electrons eo m2 oyole If the magnetron is visualized as composed of two parallel plates aoting as a transmission line, the energy stored in the region between plates is, per unit areas L/2C o0 B2m h 1/2 s i where h is the distance between plates. The Q of the region between plates is then: Stored Energy Density Dissipated Energy Density per cyole 1/2 o V/h _2n xh e V 10 2 I Using the aforementioned values Vo V Ym o - 2co and letting X - 10 om. h 1 l om. this beoomess 4mo 3 h 1.5 e= 900

89 To anyone familiar with the baokbombardmern capabilities of an oscoillating multioavity magnetron this value of Q for the interaotion spaoe at first probably appears astonishingly high. owever, it is in-* teresting to oompute a corresponding value of Q for such an osoillating tube. Typioal values will be taksn as follows: QL = 60 Baok-bombardmenr Power a 5% Power Output Energy stored in resonator = QL Power Output Then QBB w " 1t200 vhioh agrees surprisingly we1ll with the value previously oaloulated. Sinoe the energy density stored in a transmission line (in which might be plaoed a spaoeoharge cloud for frequenoy-modulation purposes) is muoh less than in the multioavity magnetron, it appears that the energy loss in the spaoe charge is not a very serious oonsideration. Of oourse, a spae-coharge oloud intended Iobr switohing or modulation purposes would naet; be subjected to the tangential r-+f eleotric fields whioh oontribute substantially to the electron energy at the oathode of an osoil. lating magnetron. That is, while the baok-bombardment; losses in an oscillating tube are much greater than those obtained in a simple modulation struoture, the energy stored in the former is oorrespondingly larger so that the Q values appear in agreement as to order of magnitude. This would appear to lend a greater feeling of oonfidenoe in the value 900 obtained above.

90 e. Loss due to collisions near a region where. = O. There is another possible source of energy loss by the eleotrons, which involves collisions with the heavy particles in the space, but which was considered a bit too speculative to be included under the previous discussion of oollisional losses. Let it be supposed that the boundary or arny other surface in the space charge cloud has the property that along this surface e 0o The contfinuity of the normal electric displacement across this surface is, of course, demanded. In the case of a wave impinging normally onto such a surface it would be expeted that this boundary condition would result in dissappearance of the normal electric field at this surface (se = 0). However other conditions can be imagined in ihioh this situation might not prevail. For example, for a TEM wave striking the cylindrioal space charge in a coaxial line in which the surface se O0 would probably be a cylinder coaxial with the line, the electric field on the outside of this surface might be reluctant to vanish. Under this condition the other possibility arises, upon appli. cation of the oontinuous normal displaoemnnt boundary condition; namely, the possibility of extremely high (theoretically infinite) electric field at the surface Se = 0. If such a condition could exist, electrons traversing this surface might well acquire a very considerable energy which would be dissipated upon collision (either elastic or inelastic) with a gas particle. In this way it is at least conceivable that an appreciable amount of energy could be lost by the propagating wave in the neighborhood of a surface at which.e 0.r

V. BOUNDARY CONDITIONS AND EFFECT ON RESONAITT CIRCUIT In Chapter III the properties of a magnetron type space charge as a medium for the propagation of electromagnetic waves were determined under the condition that the fields are invariant in the plane normal to the direction of propagation. This condition can be met in some very simple practical cases, however in general it is -invalid, so that the results of Chapter III will be re-evaluated in a qualitative vmy. The solution of the force and field equations becomes considerably more dif. fioult under these more general conditions and will not be solved here, In arny event the results of Chapter III provide at least qualitative indication of the properties of the space charge and are particularly applicable then the space oharge is of dimensions small compared with a wavelength so that the fields can be considered invariant over the swarm. In order to determine the effect of the space charge cloud on a microwave cirouit, the appropriate field equations must be solved, subject to the boundary conditions of the confining circuit and the edge of the space charge. Since this solution obviously depends on the shape of the confining circuit, and the possibilities of variations in this shape are praatically limitless, no attempt will be made here to obtain a solution for the exact effect of the space charge. However, using the equations from Chapter III the influence of the space charge on twCo typic cal circuits will be determined qualitatively. 1. Boundary Conditions at the Edge of the Space Charge. The conditions to be met by the field components of an electromagnetic wave at the boundary between the space charge and free space 91

92 are, with one exception, identiocal to the boundary conditions at any surface of discontinuity. That is, the continuity of the tangential E and H fields and of the normal B and D fields must be insured, If the surfacoe of the space charge is perturbed periodically in space by the propagating wave, and if there is a steady electron vel. ocity parallel to the direotion of propagation, an additional boundary condition becomes necessary. Fig. 5-1 shows a oross section, parallel to the direction of propagation, of such a perturbed surface, This co0. dition will prevail in the case of propagation parallel to the steady electron motion. It is seen that these "ripples" constitute an r-f surface current. Therefore in addition to the usual boundary conditions, the discontinuity in the Iz fields at the boundary must be equal to this surface current. This boundary condition can be evaluated in the fobllowing approximate manner. Ei/y ~~y h Ey Vo < — x X" FIG. 5.1 PERTURBED SURFACE OF PLANE MAGNETRON SPACE CHARGE

93 The magnetic field, of a wave propagating in the x direction, at the surfacoe of the space charge of a plane magnetron cn be onsidered as replaced by the value of the magetic field at the munperturbed bound ary plus that due to a surfaoe ourrentl. That is, referring to Fig. 5.1, at the boundarys IV x dx A T/. dS JXs dA V-1 where Js is the surfaoe ourrent due to the -x direocted motion of ithe perturbations in the surfaoe of the spaoe oharge. Then if the boundary is y =hs Hz (h + a) - Hz (h - ) =+ P | VohI i V-2 where Pvohl is the magnitude of the steady eleotron velooity at the boundary and s is a vanishingly small quantity. The plus sign corresponds to wave propagation in the -x direetion and the minus sign to propagation in the +x direction. The equation of this perturbed surface iss F(x,y,t) = (y-h) - 8oe0-x P. V0 3 Sinoe the surface of any moving fluid ooincides with a stream line, and along this line D/Dt O, if F O represents the surfaoe, the following relation is valid: DFP/t = O This becomes, with the substitution of V-3: 0.t +(v V)FV -S OI +Voh +Vy 0 v =4 so that Eani, W. C., "Small Signal Theory of Veloity Modulated Eleotron Beams", G. E.e ReV 2, No. 6, page 258, June 1939.

Therefore from Eq. V-2: RHz(h + ) + Vo +H (h) V The magneti f ield at the boundary can be found from the relations valid inside the space charge: (from Eq. III-a): -= i= co Ey + Po vy so that Hz (h-) = h) + V() From Eqs. V-6 and III-22: Hz(h + 8) = _Y -1 + _-v,-+ e/ E + - ' h [tjoL +r wLa 2t{2h 1 + %r The admittanoe, at the boundary, of the wave propagating in the-x direction is, using Ivohl= o0oh: H(h) o+ 2 hCb P hn +e - &1 EI(/h) 7wYL7t7sWh ) L[ Fr;~,;Y-th - X + rT O yh V-8 - + 2Y0h Xn O/ For a direation of propagation normal to the steady electron motion there will be no perturbation as considered above. For example, in the case of propagation in the y direotion, from Eq. III-13 the admittance at the boundary is: H _ '1 r e/m

95 2. Effeot of Space Charge on Its Associated Circuit. A. In a Situation Suitable for Frequenoy Modulation of a CouPled Cirouit. One of the principal applications of a magnetron type spaoe charge (other than in an oscillating tube) is in its use as a frequency modulating element, The space charge can be incorporated, for example, in a coaxial line which is coupled into a resonant oirouit, the frequency of which it is desired to change. Under proper conditions of applied magnetic field and sigaal wavelength the space charge can be made to exhibit the desired dielectric properties. Then a change in the size of the space charge cloud will result in a change in the reaotanoe coupled into the resonant circuit, and therefore a change in its frequency. (See Section IV-2) In this case, since the axial length of the cloud will usually be small compared with a wavelength, if o/oo is adjusted, using Figs. 3.4 and 3.5 to yield a value se<l or ee >1, the effect of the modulating structure will, in first approximation, be the same as if a capaoc itanoe were connected across the line at the midpoint of the space charge. The value of this capacitance will change with diameter of the space charge. This principle was used as the basis of a frequency modulated magnetron in the University of Michigan Electron Tube Laboratory. B. Space Charge in a Multicavity Magnetron. The elecotrio flux lines in the interaction space of a multicarity magnetron are shown (at a fixed instant of time) schematically in Fig. 5.2. The electric and magnetic fields in the interaction space are represented matheematically by equations of the following form: Mipowave Magnetrons, RTdiation Laboratory Series No. 6,1, MGraw-ill, page 65, 1948.

96 NEe Z Sin Y' z, ( jYe EG- = o 2 5 r9! Zy (kcr) or Y k Z 5 Ukra) m =-o 17 sin Y Z,_(r) Er = -a wzrzy- am e mswim" where Zry Jy (kr) Jr (kr)) - y~.kro) Ny(~) rY n +inT m = summation index, an integer n = mode number N number of anode segments 2 P angle subtended by the gap between anode segmnets ra anode radius ra ocathode radius It is seen that the eleotrio fields in the interaction space have both r and e components while the magnetio field is entirely z direooted. These fields may be thought of as due to waves propagating in the + r and +edirections. One of these waves is qualitatively similar to the type of field considered in Section III-b_2, namely that due to a radially propagating transverse magnetic wave with field components Er, Ee * Therefore the characteristics of the inner sub-synchronous (socalled Hull-sfrillo electron cloud should be given approximately by the ourve in Figs. 3.63 or 3.7 with suitable modification for hie effsect of the radius of the cloud. That is, for example, witeh /Co, <.4 the

97 FIG. 5.2 QUALITATIVE CONFIGURATION OF ELECTRIC FIELD LINES IN INTERACTION SPACE OF MULTIANODE MAGNETRON oloud should appear as a conducting surface, so that as it is expanded the result will be an increased capaoitance between vanes with oorresponding reduotion in resonant wavelength. Experimental investigations conducted to oheok these points are reported in Section VI-3. The other wave type, involving propagation in the + 0 direction are similar to those used in the analysis in seotion III-3 above. These effects of the inner space barge oloud on the resonanr frequenoy of a magnetron are present even in a non-osoillating tube and are not to be confused with synchronous effects of the spaoe charge in an osoillating magnetron, such as pushing and voltage tuning. These effects have been oonsidered in more detail by Weloh2. Pushing is the change in resonant frequecoy with anode current and is believed to be at least partly due to a change in the phase between the rotating synchronous spaae oharge "spokes" and the interaction fields. Voltage tuning is an effeot notied in a magnetron with an extremely low Q resonant oirouit having very low r-f voltage between segments. Under these oonditions the frequency of osoillation is affected very considerably by the ohange in d-o voltage. Univ, of Mioh. Eleotron Tube Lab. Tech. Report No. 5. Also to be treated in a forthooming report by H. W. Wlelch.

VI., V iERI~METAL RESULTS In order to provide verification of the theoretical results and their interpretation presented in previous chapters, several experiments were performed and will be described in this chapter. In each of these experiments a microwave resonant oavity was constructed to include a cylindrical magnetron type space charge in a part of the cavity, so that the space charge could interact with the electric fields. In order to duplicate as closely as possible the conditions under which the analy-_ sis was made, the cavities were designed to present to the space charge only fields of a simple and symmetric geometry. That is, the fields corresponded to one of the fundamental modes of the cavity, as contrasted with the relatively more complex fields in the interaotion space of a multicavity magnetron. In addition, the effect of the space charge on the resonant wavelength of a multicavity magnetron was studied. The space charge, presenting a reactive impedance to the cavity, will affect the resonant wavelength of the cavity. This change in resonant wavelength as a function of t;he magnetic field is used as an indication of the dielectric properties of the space charge in relation to o/,1. Propagation Parallel to the Applied Magnetic Field. To duplicate the field configuration considered in section 1.b of Chapter III for propagation parallel to the applied magnetic field; namely, an electric field with cylindrical symmetry, of the TIN1 type, a coaxial cavity was constructed as shovmn in Fig. 6.1. This cavity has a filament as part of its center conductor so that a space charge cloud is created in the center (longitudinally), where the electrio fields 98

8 'ON -9Ma CERAMIC COUPLING SYSTEM INSULATOR FILAMENT RESONANT 8 CAVITY (a SPRING END HATS POLE PIECE- d ( EXHAUST TUBULATION FIG. 6.1 10~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1 ALL DIMENSIONS UNLESS OTHERWISE SPECIFIED MUST BE HELD TO A TOLERANCE - FRACTIONAL ~ I/., DECIMAL ~.005' ANGULAR ~ N ENGINEERING RESEARCH INSTITUTE BDESIGNEDBY SCAOED UNIVERSITY OF MICHIGAN CHECKED BY V28 6 DATE 4-21-50 ANN ARBOR MICHIGAN PROJECT 10 CM MAGNETRON DID PROJECT M-762 (EXPERIMENTAL) MODEL 3 U ATL CLASSIFICATION B- 11,003

100 are of maximum amplitude. The resonant wavelength of this tube is about 10 cm. The cavity is provided with two coupling loops so that the cavity resonance can be located easily be detecting the r-f sigaal transmitted through the oavity. A photograph of this tube is shown in Fig. 6.2. I — 0.2 4.6.8 1.0 12 1.4 1.6 1.8 2.0 2.2 Cr.CI ~C/wc FIG. 6.3 WAVELENGTH SHIFT IN COAXIAL CAVITY PREDICTED FROM THEORY- USING HULLBRILLOUIN VALUE OF SPACE- CHARGE DENSITY

101 Before considering the experimental data obtained with this tube, an examination of Fig. 3.4 and the interpolated ourves of Fig. 3.5 for the value of rH/r0 used in the experiment, will enable a qualitative prediction to be made of the expected resonant wavelength shift of the cavity. The experiment was conducted maintaining Va/B2 constant, as BO is varied, so that the cloud radius was presumably constant with a value rH = 3 rc. Then as the applied magnetic field is increased from zero (decrease in o./CO) Figs. 3.4 and 3.5 show that the cloud should exhibit a positive dielectric constant less than unity, so that the resonant wavelength would be expected to decrease slightly from the value for B,= Va = 0. When the value C/mc = 1.12 is reached, Co = 0, the space charge oloud begins to behave as a conducting material, increasing the resonant wavelength. That is, as &e becomes negative, and the cloud begins to behave more and more as a conductor, its effect on the resonant wavelength should be similar to that of a solid conductor of approximately equal volume. Therefore, after a first abrupt increase in wavelength at m/oo = 1.12,?o should continue to increase1 slowly with deoreasing Co/CO. For co/co, slightly less than the singular value (.68) se The reason for the continued increase in Xo as se becomes more negative lies in the existance of a "skin depth" associated with the space charge, as mentioned previously. That is, when e < 0 the waves will not be completely reflected from the boundary of the swarm but will diminish in amplitude at an exponential rate, being reduced to l/e of their value at the boundary after penetrating a distance V/2z v1il. Therefore as &e becomes more negative, the "virtual boundary" approaches the actual boundary of the space charge so that the effective volume of the space charge is increased. This increases the resonant wavelength. For example, with X = 10 om, ee " - 3 (cO/Coo =.82), the "skin depth" is approximately equal to the length of the space charge cloud used in this experiment. In the region O < e< 1, the waves will not be totally reflected from the space charge; however, that reflection which does occur will do so from the physical boundaries of the space charge.

102 changes to a very large positive value but no change in resonant wavelength should be noted until ae begins to decrease sharply (around o/Ac = 0.6) when it should decorease. The resonant mavelength in the region 0.1 < o/0<O.55 should be relatively constant, less than the value in the range 0.6<c/Oc,< 1.12 but greater than the value for co/o>1.12. For o/o,<0.1 there should again be an abrupt increase in o0. These expecbed wavelength shifts are shovn in Fig. 6.3* The data obtained frcr this tube are shown in Fig. 6.4. These data were obtained by measuring the shift in resonant wavelength of the oavity, (the resonant wavelength was determined as that which gave maxim mum signal amplitude transmitted through the cavity) due to the presence of the space charge, as a fuRntion of the strength of the applied magnetic field. It is seen from Fig. 6.4 that the resonant wavelength of the cavity remains constant as the applied magnetic field is increased (c/oc decreased) until m/&o = 1.4, when it rises; one mode reaching a reasonably constant value and the other continuing to increase, both dropping again very abruptly at C/oc 0.63. The two curves shown are believed to be due to two resonances in the cavity, probably because of a longitudinal asymmetry in the formation of the cloud. However both curves exhibit the same general behavior as far as the discontinuties are concerned. The sharp drop in?o at /Ctoo = 0.63 is regarded as oonfirmation of the analytical method followed. This follows from the fact that thne angular velocity of an electron in a maegnetron results from the integral of the angular equation of motion with only the assumption of zero angular velocity at the cathode. The value of the angular velocity, unlike the potential or space charge density, does not depend on any

FIG. 6.4 CHANGE IN RESONANT WAVELENGTH OF 10 CM. CAVITY vs w/wc.10.08 BIB o<.06 ci ___ X0:9.65 CM. x A x. —.04.02.~~~~~~ ~ZERO 0.2.4.6.8 i1. 0 2 1.4 1.6 1.8 2.0 2.2 2.4 SINGULAR POINT CYCLOTRON RESONANCE WI wc PG1 89 BK. 6 G. R. B

104 choice or assumption regarding the electronic orbit, or on a series solution. Therefore, the angular velocity is believed to be invariant under any changes in emission, charge density, voltage, etc., and sinoe the position of the singular value of c/co is, from the above theory, governed entirely by the functional dependence of the angular velocity on radius, this point of agreement between Figs. 6.4 and 6.3 is thought to be signifioaxrt. The situation regarding the increase inAX at C/oo - 1.4 does not agree with that predicted from Fig. 3.4. This will be discussed below and some possible explanations of the disagreement advanced. From Eq. III-9 it is seen that the value of co/o for which e= 0 is a function and the space charge density at the edge of the cloud. Thus, if the abrupt increase in o0 at CO/C = 1.4 is interpreted as the point e = 0, Eq. III-10 can be solved for po(rE) to obtain (co/c 1.4) whereas the Hull-Brillouin relation would give This value, greater than that given by the IIull-Brillouin solution, is slightly surprising and reminds one of the solutions of Page and Adams and Moeller2 whose theoretioal analyses indicated that po increases abruptly at the edge of the space charge swarm. Also the measurements of Reverdin3 yielded a space charge distribution (No. 1) with a peak at Page and Adams, loc. oit Moeller, H. G., loc. oit 3 Reverdin, D. L., "Electron Optioal Exploration of Space Charge in a Cut-off Magnaetron", J App. Phys. 22, page 257, Mlarch 1951.

105 the boundary of the cloud, However this peak was shown as lower than the Hull-Brillouin value. The above interpretation of the increase inAX at 0/0 =- 1.4 of Fig. 6.4 should not be considered as final, however, since the ex.perimental oonditions of the space oharge oloud were far from ideal, The principal possible source of error lies in the formation of the spaoe charge cloud. The magnetio circuit used apparently made the pro-. duction of an absolutely uniform magnetic field impossible, so that there will be a longitudinal force1 Fz ac aM proportional to the z component of V H. Thus electrons can move axially from the filament under the influence of this force. In operation, suach ourrent was observed, oonstituting up to one-tenth of the Allis ourrent for high anode voltages,as shown in in the outoff curve of Fig. 6.5. It would be expected that this drain of current would affeot the oomposition of the space charge cloud. A calculation of the wavelength shift2 for the case e =8 (o,/ = 0.4) yielded a value greater than the observed by a factor of four, indicating that the cloud is probably much smaller (probably shorter) than believed, but the sharp rise and fall of the AX oharacteristic is an indioation that at least part of the cloud is of the form expected. The shift in resonant wavelength of the cavity was observed as a function of magnetic field in the immediate vicinity of the cyclotron field (calculated 1060 gauss) and is shown in Figs. 6.4 and 6.6. I See, for example, Alfven, "Cosmioal Electrodynamics", Oxford, page 19, 1950. 2 See Appendix 6

106 1.4 -r- -. ~FIG. 6.5 - - - - ~CUTOFF CURVE 10 CM COAXIAL CAVITY MAGNETRON DIODE 1.2 F LOW VOLTAGE I-!I - Io Bco If 8.2 Vo 200 0 I 2 3 4 5 Bco This observation was made with the positive side of the filament con. neoted to the anode, no shift in wavelength being observed when the negative side of the filament vwas connected to the anode. It is noted that the wavelength shift maximum occurs at a value of C/c, about seven percent greater than unity, indicating a probable error in magnet calibration or residual magnetism effect in the cyclotron resonance test. A search was made to try to detect any shift in the negative direction (decrease in So) without success. The tube on which the data shotz. in Figs. 6,4, 6.5, and 6.6 were taken was the fourth tube constructed for this purpose. The first of these proved unsatisfactory due to low Q caused by improper design of the resonant system. The second was accidentally melted in process of assemblry. The third tube rwas constructed without end hats on the filament, so that the longitudinal leakage current was so large as to

107 prevent electrons from reaching the anode cylinder surrounding the cathode. Under this condition no out-off could be observed and a satisfactory space charge cloud could not be formed. FIG. 6.6 CHANGE IN RESONANT WAVELENGTH OF 10 CM. CAVITY vs. MAGNETIC FIELD- SHOWING THE CYCLOTRON RESONANCE POSITIVE SIDE OF FILAMENT CONNECTED TO ANODE Xo=9.65 CM..06 970 990 1010 1030 1050 1070 B GAUSS PG. 91 BK. 6 G.R.B

108 2. Pro.pag9aition in the Direction Normal to Anode and Cathode. As a further study of the r-f properties of the magnetron space charge and to provide additional information as to the validity of the theoretical analysis in Chapter III, a suitable electron tube allowing experimental verification of Eq. III-20 was sought. This experimental electron tube would be placed in a resonant cavity of such configuration that the r-f eleotric and magnetic fields with which the electrons could interact would be as similar as possible to those assumed in the analysis of section III-2-a a That is, a tangential electric field (E8) and longitudinal magnetic field (1z) were desired, with ab8e = 0. The only feasible oirouit capable of producing these field configurations is a cylindrical cavity resonating in the TEOL, mode. The space charge was to be plaoed along the axis of such a cavity. A tube and cavity of this design were constructed for the purpose of investigating the r-f properties of the space charge as seen by a radially propagating wave. A schematic drawing of this tube and resonant cavity is shown in Fig. 6.7. A photograph of the tube is shown in Fig. 6.8 and an assembly drawing of the tube in Fig. 6.9. However, a cavity designed to resonate at 10 centimeters in this mode will have a radius of approximately 7 centimeters, far too great to allow the outer wall to be used as the anode for the space charge. That is, for magnetic fields in the desired range (2000 gauss) an astronomical d-e voltage would be required to expand the space charge to a small fraction of this anode radius. Therefore a series of longitudinal rods spaced on a circle concentric with the cathode were used as the d-o anode. If the rods are small enough and few in number, there would be relatively little metallio surface parallel to the e

109 directed electric field so that there should be only small perturbation of the r.f fields at the cathode. This arrangement is shown schematically in Figs. 6*7. The question naturally arises as to the ability of suoh an anode to produce d-c equipotential lines which are approximately ciroles concentrio with the cathode. A field map of a section of such a (space charge free) structure was made by Mr. J. S. Needle, from which it was8 found that this anode approximates a solid oylindrical anode very closely for any radius less than about 0.7 of the anode radius. This type of experimental tube has the advantage that only the longitudinal bars and the cathode need be included in the vacuum envelope; the cavity may be external to the vacuum. These bars should interfere very little with the r-f fields in the oavity so that inside the anode bars the electron-wave interaction can take place. \GLA5~~~ W ~-ANODE RODS GLASS ENVELOPE SPACE CHARGE CLOUD RESONANT CAVITY (TEoll MODE) FILAMENTUDY FIG. 6.7 TEotl RESONANT CAVITY FOR SPACE CHARGE STrUDY '

110 The shift in resonant wavelength of this cavity due to the presence of the space charge was calculated (Appendix 7). This calculation showed that a shifrt of the order of two peroent could be expected. To cheok this, a conducting rod, of approximately the diameter of the space charge, was inserted in the oenter of the cavity; the wavelength shift was observed to be approximately one and one-half percent. To test the shieldingeeaffet of the rods in isolating the center from the outer region of the cavity, the conducting rod was inserted with the small rods in place to duplicate the rods in the actual tube (see Fig. 6.7). In this latter case the wavelength was increased by about threetenths of a percent, still a measurable amount. However, with the tube in its place in the cavity, no shift in resonant wavelength, due to the space charge, could be detected, even with the aid of several elaborate methods of detection. Two possible explanations can be advanced for this lack of effect of the space charge. In the first place, since the electric field must vanish at the cathode surfaceG, the proportion of the total r-f energy stored in the region near the cathode, that is in the region occupied by the space charge cloud, is quite small. On this basis alone it would be expeoted that the wavelength shift be small, and it was in fact calculated to be at most two percent (in the absence of the longitudinal bars). However since a wavelength shift was observed in the preliminary test using a conducting rod to represent the space charge, it can be concluded only that the presence of the tube in the cavity so disturbed the fields as to reduce the effect of the space charge. The presence of the tube (with its Eovar-glass seals, eto.) would undoubtedly reduoe the QO of the cavity (usually very high in the TEO1 mode),

111 possibly enough to cause the bandwidth to exceed the wavelength shift oaused by the space charge. A second possibility is that there might exist in the cavity certain extraneous fields which can interact with the eleotrons in suoh a mnner as to reduce the effeot of the electrons on the fields of the desired mode. FIG. 6- 8 PHOTOGRAPH OF EXPERIMENTAL TUE One effect of the space charge which could be detected was the change in the amplitude of the transmitted signal vs frequency oharac. teristic of the cavity as a function of the applied magnetic field. The space charge cloud radius was mintained constant and the relative change:5.. in transmitted signal amplitude (at resonance) measured as a function of the applied magnetic field. This data is shown in Fig. 6.10 where

9 'ON 9M I-I 5, POIE: PIECE 7 A MENA/vT LEAD 8C ANODE BAR FIG. 6.9 9. FlA_ AVfENT /O. GLALSJ ENVcLOPE c- "-H ALL DIMEENSIONS UNLESS OTHERWISE SMPCIFIED MUST BE HELD TO A TOLERANCE - FRACTIONAL * C." DECIMAL.05. ANSULR * M DIES ~ APPROVE ENGINEERING RESEARCH INSTITUTE z::oD my UNIVERSITY OF MICHIGAN CHECKED NY DATE 3 DA - ANN ARSOR MICHIGAN TITLE PROJECT TlT.K 114GIETRON P C M- 9ZI CHA0RGE DIODE CLASSIFICATION ISSUE DATE DWG.Ho. B- lI1oo.5

113 cyclotron value of magnetio field. This is as predicted by the theory (Eq. 111-20) but it is not suggested here that the experimentally ob. served change in cavity Q necessarily be due to the electron-atom collision mechanism proposed in Chapter III. However as shown in Eqs. III-17 the perturbed velocities pass through a maximum value at C/ C 1 so that any mechranism by ihioh the electrons can lose an amount of energy proportional to their velocity would produce the effect shown in Fig. 6.10. About the most that can be claimed for this data, however, is that it does not contradict the theory.

40 FIG. 6.10 EFFECT OF SPACE CHARGE CLOUD IN TEoii CAVITY (n cr Z 0 H 13:: I — w 0 _rJ C') cn >0 -- 2O3t O') z 2 cr o -~J 0 0.2 4.6.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 w/wc

115 3. Effect of Space Charge on the Resonant Wavelengh of a Multioavity agnet ron. It was mentioned in Section V-2 above that the space charge in a magnetron will change the resonant wavelength (\X) of the oirouit. Since the electric and magnetic fields 'in the interactifon space of a multioavity magnetron are qualitatively similar to those considered in the analysis of radial wave propagation in Section III-2, it would be expected that the variation in se vs o/(o) be represented by a curve similar to Fig. 3.6. The analysis leading to the curves of se vs o/0c such as Fig. 3.6, was carried out under the condition /es8 = o. Havoever, while this condition is not satisfied in the multi-anode magnetron, these theoretical results should-be at least qualitatively core rect for this case, especially in the vicinity of the cathode. In order to check this, the resonant ivamelength of a multicavity magnetron was measured as a function of the applied magnetic field. These measurements were made by determining the Q value of the resonator, maintaining the space charge cloud radius constant by keeping Va/B,2 invariant. The radius of the space charge was adjusted so that no electrons attained synchronism with the rotating wave on the anode, thus avoiding any effects due to beginning of oscillation. Despite this precautions it is usually somewhat uncertailn whether the wavelength shifi observed is due to the "bulk" properties of the cloud, analyzed in Chapter III, or to motional effects.l This is partioularly true in the vicinity of the cyclotron field. For this reason, resonant cavities possessing simpler field configurations (and which cannot oscillate) such as those described in the preceding sections of this Lamb and Phillips, Loc. cit.

chapter are more advantageous for the study of the spaoe charge oharaoteristics. Using Fig. 3.7, an estimate of the variation in ~e vs O/Co can be found for the value of rH/r, used in the test (rq/rc = 1.1). The data obtained from these tests are shown in Figs. 6.lla, 641b, 6.llo, and 6412. The first three of these are typical curves showing the variation in?0 as the space charge cloud is expanded by increasing the d-c anode voltage at constant magnetio field. Go is.the conductance seen looking into the output coupling system of the tube, its value is determined from measurement of the input standing wave ratio at resonance; Ia is the anode current. It is seen in Fig. 6.11a that for B = 210 gauss, CO/o = 3, and the resonant wavelength decreases as the space charge is expanded. At this value of magnetic field, the space charge appears as a region with dielectrio constant positive and less than unity. Similarly, in Fig. 6.11b, B ='1700 gauss, Co/oo.37 and the expansion of the space charge increases the resonant wavelength. At this value of magnetio field, the space charge is believed to have a negative value of dielectric constant. The sharp rise in 'o at high voltages seen in Fig. 6.1lb is the result of the electrons approaohing syncohronism with the rotating wave of the magnetron. Fig. 6.11o shows the change in resonant wavelength of a multi-anode magnetron operated with magnetic field below that necessary for osoillation. Under this condition the space charge can be expanded to the anode and since in this region of CO/COO O< Le<1, a considerable reduction in resonant wavelength is obtained.

117 Fig. 6.12 shows A X as a function of c/co (curve A) and can be compared with the A X predicted (cure B) from this theory, and that predicted by the Lamb and Phillips theory (curve C). The main points to be considered in comparing experimental data of this type with that predicted from the theory, are the values of CO/c, at which the A X curve shows a discontinuity. The absolute value of A X for any o/c, region is of less importance, since this cannot be ecalculated accurately fbr a multicavity magnetron without an unreasonable amount of mathematical labors. Examination of Fig. 6.12 reveals the expected sharp rise beginning near co/co = 1.1, but the expeoted decrease in?0 near o/co =.9 was not observed. It must be remembered that the predicted curve B results from use of Fig. 3.7, which is merely an approximate interpolation based on purely qualitative considerations. If the more exact theory of the plane magnetron (Fig. 3.3) is used, the predicted curve of Fig. 6.12 would be as shown except that there would be no rise between CO/0c =.85 and mo/c = 1.07. This would not improve the agreement with experiment in the region.7< /c/,,<l however. The theory of Lamb and Phillips indicates a wavelength shift of the form 12/2-1 (ourve C) which seems to be in reasonable agreement with the experimental points for o)/Oo>L but not for /oc< 1. Here the direct effeot of the cyolotron resonance of the electrons in the space charge tends to obscure the "bulk" effect of the space charge in the resonant circuit. The wavelength variation described in Fig. 6.12 probably represents a come bination of these two effects.

118 It is seen that in general the expected wavelength shiftst predicted from the result of the analysis in Chapter III, are observed except in the neighborhood of oa/c, 1 where the synchronism effects are predominant.

119 -.o Go I -.. CM MHOS MA 16.90- - - B = 210 GAUSS Wf/tc =3 LOW RF VOLTAGE:MIGH. MAGNETRON MODEL 3 SERIAL NO.8 300 1685.020.016 200 l _.012 1!6.80,o COLD __.008 -100 Go.004. ~. Go COLD i,~_ o7 000o / 0 5F 10 l 150 200 250 Ea (VOLTS) HULL VOLTAGE FIG. 6.11a Xo AND Go OF HOT MAGNETRON AS FUNCTION OF PLATE VOLTAGE

120 X.o Go CM MH B = 1700 GAUSS C/C0C-.37 LOW RF VOLTAGE MICH. MAGNETRON MODEL 3 _ / SERIAL NO. 8 16.90 Xo 16.85.005. Go COLD Go.003 _t 16.80 ~'Xo COLD 002.001x 16.75 000~ 0 500 1000 1500 E a (VOLTS) FIG. 6.11b Xo AND Go OF HOT MAGNETRON AS FUNCTION OF PLATE VOLTAGE

121 17.80.78 1~35 WC.76.74.72 17.70.6 8_ _-__ - i~ TUBE NOT OSCILLATING.66 0[ 4 I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.64 cl.62 t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.58.56.54~ 52 1 Z.50 0 100 200 300 400 500 600 ANODE VOLTAGE - VOLTS FIG. 6.11c WAVELENGTH SHIFT IN INTERDIGITAL MAGNETRON DUE TO EXPANDING SPACE CHARGE CLOUD.

.16 FIG- 6.12 CHANGE IN RESONANT WAVELENGTH OF.12 MULTI-ANODE MAGNETRON VS. MAGNETIC FIELD EX PERIM ENTAL ---.08.04 -— PREDICTED FROM THEORY ot 0 B r -.04 L LAMB a PHILLIPS -08 ORDINATE SCALE FOR CURVES -.12 Ba80C IN ARBITRARY UNiTS MODEL 46 NO. 20 0.2..6.8 1.0 1.2 1.4 1.6 18 2.0 22 Wc

VII. CONCLUS IONS In this chapter the considerations of the magnetron space charge will be concluded by a discussion of the results of the experiments performed in an attempt to check the theory, a resume of the assumptions underlying the analysis, a brief list of the possible applications of a space charge of this type and a few topics suggested for future investigation. Unfortunately this last section contains some rather important points, indicating that this present work is by no means a complete exposition on the subject. It is hoped however that a few of the physical principles have been brought out and that a basis for, future investigations has been established. The magnetron space charge is somewhat unique among electron atmospheres under the Anfluence of a magnetic field (at least the plane magnetron and cylindrical magnetron with propagation along the applied magnetic field) in that no double refraction of the waves vwas found. It is believed that this type of space charge cloud could be used with profit more generally for such purposes as frequency modulation. 123

124 1. Conclusions - Agreement between Theory and Experiment. Three experiments have been performed in an effort to determine the validity of the analytical expressions for the dielectric properties of the magnetron space charge, determined in Chapter III. In order to duplicate experimentally the case of wave propagation in the direction of the magnetic field, a coaxial cavity was construoted so that the travelling waves in the cavity propagate through a cylindrical space charge cloud surrounding a filament serving as part of the center conductor. The results of tests on this tube seem to con. firm the theoretical predictions as to values of magnetic field at whioh certain disoontinuities in the properties of the space charge cloud are expected. The magnitude of the wavelength shift observed did not oheck well with that oalculated, but this is believed to be of less importance. In oomparing the theoretical singular values of co/coc with those observed, it is remembered that one value should be independent of the potential or spaoe charge distributions, depending only on /c. This point was checked quite well by the experiment. The other singular value of /Coc to be sought is dependent on the space oharge distribution and therefore cannot be predicted with certainty. The value obtained experimentally appears to correspond to a value of space charge density at the edge of the cloud approxinately twice the Hull-Brillouin value. This suggests the existance of a space charge distribution with a peak at the boundary of the cloud. In general, the results of this experiment are considered as satisfactory confirmaation of the theory The part of the theory concerned with wave propagation in the

125 radial direction was investigated in two experiments. One of these in.. volved measurements on a multianode magnetron in which the results are less conclusive, because the synchronous effects of the electrons at the cyclotron magnetic field tends to obscure the "bulk" effects of the space charge sought here. However it was observed that at values of magnetic field well above the cyclotron field the space charge cloud still exhibited a positive effett on the resonant wavelength of the magnetron, as predicted by the theory. This effect for B.o> B0 would definitely be due to the "bulk" properties of the cloud since the synchronous effects would be noticeable only in the neighborhood Bo= Bo. The second of these two latter experiments was performed by inserting a space charge cloud into a cylindrical cavity in the TEO11 mode. While no effect of the space charge on resonant frequency could be detectedt the space charge appeared to abstract net energy from the fields near the cyclotron frequency. This behavior is at least not at variance with the theory. This analysis of the propagation of electromagnetic waves in the magnetron space charge considered the electron cloud as a medium whose notional behavior can be explained with the aid of the hydrodynamical equation. This treatment is contrasted with that which considers the motion of the individual electrons as obeying Newton's law. The results of the analysis were presented in terms of an equivalent or effective dielectric constant depending only on the ratio of the fre. quenoy of the wave to the cyclotron resonance frequency. Since, in the tubes built to test this theory the space charge occupied a region small compared with the wavelength, the space oharge cloud was considered as a lumped cirouit element and its effect on the associated circuit

126 predicted, by certain qualitative arguments, from the se vs Co/c curves. Agreement between these theoretical predictions and experiments can be interpreted as indicating that the consideration of the space charge as a medium is valid. Unfortunately only one set of experimental data is considered sufficiently reliable to serve as a check on the validity of the theory and its interpretation; namely the data in Fig. 6.4. It is believed that this data does provide reasonable confirmation of this analytical treatment of the nmgnetron space charge.

127 2. Resume of Assumptions. A brief resume of the priori assumptions, on which this analysis is based, will be given in this section. This will provide a basis for consideration of the validity of the analyrtical results obtained. a. It is assumed that the electron space charge can be treated as a gas, the motion of whose particles (although they interact with only inverse square law forces) are mutually interlocked to such an extent that the gas exhibits medium-elike behavior. The particles of this gas are considered to possess a random motion which is sufficiently large in comparison with the applied r-f signal voltage so that thermal equilibrium is maintained in the gas. In such a case the equation of motion of the electrons can be derived from the Boltzmann Transport equation. Certain properties (such as thermal conductivity, mean free path, etc.) of such an electronic gas cannot be determined without evaluation of the electron velocity distribution funotion. This evaluation would entail, in view of the necessity for consideration of both near (binary) and distant encounters, considerable mathematical difficulty. Fortunately the desired dielectric properties of the space charge can be determined without knowledge of the distribution function. b. In general it is assumed that the amplitude of the perturbed electron velocity (that is, the part of the velocity due to the applied r-f field) is small enough that terms involving its square oan be neglected in comparison with terms in the equation involving only the first power of the velocity. That is, the usual small signal theory is used throughout.

128 o. Certain limitations have been placed on the form of the waves considered to be propagating in the space charge. That is, in general the electromagnetic waves are considered invariant in phase in directions normal to the direction of propagation. In the cases of very simple geometrical r-f structures, this assumption will be fulfilledo The result can be applied to r-f fields of more complex oonfiguration by suitable superposition of waves of the type considered here. d. The complete solution of the problem in the cases with cylindrical geometry was found to involve lengthy mathematiocal treatment. Therefore the solutions for the propagation constants in the cylindrical oases were obtained only for the limiting case of no variation with radius of space charge density and angular velocity. This condition would generally be satisfied by use of a vanishingly small cathode. A rigorous solution is obtained therefore only in the two limiting oases of small cathode on the one hand and plane electrodes on the other. It is believed however that the results of the cylindrical analysis provide a useful approximation for rHj/rO 3. e. In this report, including the considerations of the possible space charge density distributions in Chapter II, the effect of interactions between the discrete electrons is completely neglected. This is not meant to imply that these effects are unimportant, but merely that their consideration is the subject of a separate study6

129 3. Possible Applications of This Type of Space Charpe. A consideration of the waver propagation characteristics of the magnetron-type space charge suggests possible applications for such a medium other than in an oscillating magnetron. There are, of course, many more possible applications than those listed here, these being merely the moat obvious examples. a. The use of a cylindrical magnetron space charge in a coaxial structure coupled to a resonant oscillator circuit has been suggested before as a means for obtaining frequency modulation. This type of space charge should be capable of producing relatively large changes in reactance with little energy loss by the fields. b. It has been shown that this space charge can be made to have such oharacteristics as to prevent wave propagation through this medium. These results were derived on the assumption of a small signal, however it does not appear unreasonable that the behavior, in the presence of a signal of considerable amplitude, will be qualitatively simie. lar. Therefore it is suggested that this space charge can be used in a transmission line as a switch of microwave energy for such purposes as antenna lobe switching, etc. o. Since the space. charge can be made to reflect almost any desired proportion of the incident wave energy, it would seem that a microwave signal could be amplitude modulated without necessity of modulating the generator, by using the space charge as a variable switch in the transmission line. An engineer seeking to use the magnetron type space charge for one of the above or any similar application naturally requires a knowledge of a number of characteristics of this medium other than its

130 dieleotric properties, sto. For example he would be interested in the rate of r-f energy loss in the space charge. In this connection it is believed that this space charge has anadvantage over many other forms of electronic modulation, such as the spiral beam, in that its properties are not so critical as to frequency and magnetic fieldsand a loss (due to an in-phase component of electron current) is not inherent. For a large signal applioation this space oharge would undoubtedly be more lossy than in the small signal case but from the results of section IV-5 the energy loss would not be expected to be prohibitivee In any very small signal applicatimn such as modulation of a local oscillator the question of noise naturally arises. There is relatively little published, experimental information concerning the noise power output of a non-osoillating magnetron space charge. Riekel presents the results of some measurements on a 10 om cOw multi-anode magnetron with output power do the order of 100 watts. The noise power output was measured on this tube while in the outoff condition. The noise power increases as the space charge cloud is expanded; for an anode voltage of about three-fourbhs of that required for initiation of oscillations, the noise power was observed to be about3 60 to -70 db below one watt. For comparison, the shot noise power from a space charge limited triode in a microwave cavity can be calculated from the relation Pn r 2.5 kToAf. Using 4f = f/Q = 3 x 109/100 this power is of the order of -120 db below one watt. Comparing this value to that quoted Rieke, F. F,, in Miroraave Magnetrons", Radiation Laboratory Series No.6, McGraw-Hill, page 390, 1948.

131 above for the magnetron, this latter devioe appears very unfavorably. However this comparison is not quite fair since the magnetron noise was measured on a device whose power capabilities and therefore size and rotating space charge current are considerably larger than would be used in a modulator structure. If this noise figure of -60 db for the magnetron is scaled down with output powver, maintaining signal to noise ratio oonstant, a noise power of the order of -80 db would be obtained. This latter figure is slightly more favorable in comparison with the triode oscillator tube. From this it appears that the use of a magnetron space charge for low level modulation purposes is not out of the question.

132 4. Suggested Topics for Future Investigation. The experiments described in this report, undertaken to obtain some indication of the validity of the theoretical results, by no means exhaust the possibilities of experimental investigations of a magm netron-type space charge. Exploration of a magnetron space charge by means of high frequency waves oan have three objectives: (a) by assume ing a knowledge of the statio space-charge density, experimental data could be used to check the theory developed, (b) to determine purely experimentally the properties of certain oonfigurations~ of space charge, and (o) assuming the theory to be oorreot,experimenral data could be used to obtain some information on the space charge density. Since the tools with which an experimental study of the static magnetron space charge can be made are extremely limited, this last objective seems to be at least worthy of more consideration. It is the author's opinion that the major source of error in making measurements of this kind on a space charge lies in the problem of obtaining a space oharge of such configuration that it duplioates closely the idealized form considered in the analysis. It is very difficult to obtain an absolutely uniform and laminar magnetic field, so0 that some leakage of electrons from the ends of the cloud is inevitable unless suitable precautions in the form of shields etc. are taken. One specific investigation which might increase the understanding and possible applications of this type of space charge involves the application (b) in section 3 above. The measurement of the refleotion of microrave energy from a space charge in a coaxial line should Povinde useful information relative to its use as a microrave suitah in a transmission 1ine, This information would be particularly valuable

133 if the signal power used is high (0.1 - 1 bk. or more). Also it would be desirable to ascertain if the space charge in a plane magnetron could be made to amplify a microwave signal. Along the line of continued theoretioal investigation, very valuable information as far as applications are concerned could be de. rived from a large signal study of the space charge. If this were done using'electrio and magnetic fields of simple geometry, the problem may admit of a solution with a reasonable amount of labor and the results might aid in the understanding of the role of the space charge in an oscillating magnetron. Also a solution for the dielectric properties of the cylindrical space charge for various values of rH/rc would yield useful design information. Finally, continuation of the study of wave propagation parallel to the steady electron velocity and its extension to the cylindrical mnetron should yield results which can be applied profitably to a further understanding of magnetron characteristics.

APPEIDICES Appendix 1 - Derivations from the Boltzmann Equation. In order to attempt a justification for the treatment of an eleotronio space charge as a gas obeying the Euler hydrodynamioal equation, this latter and other pertinent equations will be derived briefly from the Boltzmann transport equation. This equation considers the gas to be nearly in thermodynamic equilibrium, so that the applied fields cause only a small perturbation on this equilibrium. Therefore it must be assumed a priori that this condition is satisfied. The Boltzmann equation for the behavior of a gas in an applied vector field of foroe whioh is independent of the particle velocity, can be written ~ ~-~+ * + F 0ol (as l where f is the (unsnormalized) velocity distribution fluntion, 0 the 2 particle velocity. and r the position vector of a partiole with referenoe to some chosen coordinate system. The term O(~f^)coll denotes the time rate of change of the number density of particles at ir, t with velocities c, dt due to collisions with other particles of the gas, and 2 is an operator representing the differential operations on the left side of the Eq. Al-l. It is assumed that the collisions occupy only a small part of the lifetime of a particle. Chapman and Cowling - nThe Mathematical Theory of Non-Uniform Gases" - Cambridge, 1839, Chapters 3 and 18, 2 o is used for particle velocity in this section only, in other sections of this report v will denote partiole velocity while a will represent the veloc~ity of light in vacouo. 134

135 In the electronic space charge in a magnetron, the particles of the gas are under the influence of the additional (Lorentz) force due to the magnetic field B. This applied force depends on the particle velocity through the relation -eo x B. Then Eq. Al-1 becomes: f b -A. ea bf fo Al-2 + [-m a B F ( )oi To save space in writing equations, let P be any property of the particles, suoh as their kinetic energy, momentum, etc. Then, multiplying Eq. Al-l by Vda and integrating throughout velocity space (assuming the integrals to be oonvergent), also specoifying that pf tends to zero as o becomes infinitely large, the result can be written f do n A a= do Al-3 where nAp is the change, due to collisions, of the mean value of the sum of the property p' over all particles. If, from all of the electrons represented in velocity space is chosen a number occupying volume r to r + dr, whose position is given by the vector r at time t and whose velocity is in the range c to I + do, and the sum of the particle property V is taken over this group, the result is written i I. Then (( A) do is a measure of the rate of change, due to collisions, of this sumZ V. ThereforeJ. Ft() d;; is the rate of change of IV.* due to collisions, of all particles in unit volume of real space. By definition of p': Zba'nt a

136 where the superscript bar denotes the mean value over the group of particles considered and n is the number density of partioles in real space. By the meaning of the mean value nV = Jtfd A1-4 so that bo t g do- a/~J%~o_ do -gdc (n7)-n8 Al-.5 and similarly do 0 a udCU do = (nu)-n( Ux A1-6 where r represents any coordinate direction x y z. Substituting Eq. Al-2 into Eq. Al-3 and using Eqs. A1-5, A1-6 and A-?: du dv d do A1-8 If the particles of the gas have a mean or drift velocity c~o the uarT. re the components of velocity along the oordinate syss.em oving with velocity cO, this "peouliar velocity" or particle velocity referred

137 to the moving system is C = o- 0 Then P' oan be expressed as a funotion of C instead of c', ax ~.VY~0 o.v~- o.rcV - aj. C~ -V)oo Substituting these relations into Eq. A1-8: f do +'4 v+ (np tt"o + )) -n (0j+O + c)-V Using the notation D +a -. Eq. A1-9 reduoes to: f C - 4" =. nBC + nVIo _,,[ + c V- oo x B -D Al-10 - ~ V.(oo~ - ~O' (. -. o_ e CxB Three equations important in gas analysis oan be deduced from this equation with no prior kcowledge of the form of the distribution function f. This reduotion oan be aooomplished by making use of oertain properties of the gas, which are functions of the particle velocity0 and whose sum over the partioles partioipating in a oollision is oonserved, so that ap' = O. These properties are: (a) aonservation of number

138 density of particles, (b) conservation of momentum, (o) oonservation of energy. These so-called "summational invariants" will be applied to Eq. Al-10 separately. A. Since the number density of particles is conserved in collisions; letting /= 1 so that = 1, it is seen that = O. Then 78O= 0 (sinoe C = O by definition), = V = = and Eq. Al-10 reduces to nV. co + Dn 0 l-ll which is just the equation of continuity of numbers (or density) of particles. B. Using the law of conservation of momentum; letting ' = mU (where U is the x component of C), then = O so that tV = O. Also n= nm Un = p, 1 (p is the hydrostatic pressure of the gas), ( m, C =0 Dt and Eq. Al-10 becomes for the x component of momentum: To veriLty this, consider that as each particle in G to ~ + do crosses a unit surface area it carries with it a quantity p' so that the total contribution of this group to the flow of ' across the area is, per unit timCn f d where Cn is the component of C normal to the unit area, Cn = C nl;u is a unit normal vector. The total net flux of ' is then JCn f dC n CnwV In this case =- mU and the rate of transport of momentum across the area is by definition the hydrostatic pressure of the gas acting on this surface. Thus qn " em (cnu) ~ nl - -o CU

139 VPc- nm [F - J + ne u x B = Al-12 where uo is the x component of ~o. In general the equation of momentum s Vp- nm [F g + ne o x B - 0 Al-13 which is the Euler equation of hydrodynamics, Eq. II-1. C. From the conservation of energy relation, let p' E, the energy of translation of a partiole. Then for the type of gas considered here ' = 3/2 kT and from considerations similar to that in the- footnote to paragraph (B) above, n a = nEC = q the thermal flux vector (the rate of transfer of heat across unit area). Then V7 = 2 = and since from which it follows that "- O; n ) C = nm CC = p Then Eq. Al-10 becomes: D N N V. wjF N nkT + N2 nkT Va o + q V + P eO A1-14 n.+po Al-14 N number of degrees of freedom This reduces to: DT + 2 DTjF fi[P + [+ V-' = 0 Al-15 which is the equation of thermal energy of the gas. The Boltzmann Eq. (2) can be solved in general by a series of suoocessive approximations allowing determination of the pressure, thermal flux vector, coefficients of diffusion anrd thermal conductivity, etc. Chapman and Cowlng, 7.3 - 7.5.

140 In first approximation q = 0 and in seoond approximation q =-i7 VT A1-16 where * is the coefficient of thermal conductivity. Then Eq. AI-15 beoomes s DT + 2 [ V o -kV2Tl] A O -i7 In the case of two constituents of the gas, suach as electrons and ions., Eqs. Al-ll, A1-1, and Al-17 beoome:s Dnl +n1V'o0 + V'(nlC) = 0 A1-18 +nzV o+ V'(n2 c)-o (nlm + n2m2) RL nlm1 F1 +n2m2 F + (nle1 n2e2) -o X B Al-19 + (niel Ci + n2e2 C2) x Bf- A-p j k(nl + n2) DT.N kTV-(nlC1 + n2C2) + nlml F1 C1 + n2m2 F2 C2 A1-20 +(nle1 1 +ne2 e2) ' (Co xB) - p. To + * 72 T Chapman and Cowling, page 332.

APpendix 2- Influence of tle Pressure Gradient Term in the Euler Equation ~ In this section the effect of the random or thermal motions of the electrons on the propagation of electromagnetio waves in the magnetron space charge will be considered, using an approximate method. In the equation of motion, the effect of these random motions is represented by the pressure gradient term V_ * A comprehensive treatment of this probe nm leam would require a knowledge of the velocity distribution function. This distribution function is unknown and the labor involved in its determination is considered unjustified for the purposes of this treatment. Therefore as a first approximation it will be assumed that the electron gas obeys the ideal gas law so that p = nlT. Partly because of the long range forces (Coulomb) existing between particles of the gas, a more exact treatment is difficult. In order for the thermal motions of the electrons to have an influence on electromagnetic wave propagation, it would seem reasonable to suppose that some perturbation of the thermal velocities must progress with the wave. In his treatment of the effect of Hydrostatic Pressure on the Operation of Travelling Wave Tubes, Parzen1 considers that both temperature and charge density are perturbed in the same manner as the travelling wave, that is T To + T1 e ts A2-1 n + n+ 1 ei A - Is.I- I. Parzen, P. - "The Effect of Hydrostatic Pressure in an Electron Beam on the Operation of Travelling Wave Devicesn - Tech. Memo. 391, Federal Telecommunioations Lab, March 1950, Recently published in Jour. App. Phys., April 1951.

142 Using these substitutions in Eqs. A1-13 and Al-17 the propagation in the y direction in a plane magnetron (Section III-l-a) is evaluated. It is found that the effective dielectric constant is given by Eq. III-3-a. That is, the temperature motion has no effect on this wave propagation. However there appears an additional wave, which, if the electron gas is considered isothermal (T1 0 ), is given by ),2=. m 2 [ 1 kT To Wo4 A2-2 and if the gas is adiabatic (K = 0): r2= ram2 [ 2A23 2 _; [ 9.- 1 2-3 This method of accounting for the electron temperature may be useful in the case of the Travelling Wave Tube. However in the magnetron space charge, where the temperature To is known to be high, it is not apparent how this temperatture can be perturbed in a wave-like fashion, certainly it cannot be considered that the fields of the propagating wave accaoomplish this since it is fundamental to this analysis that the random energy be large compared with the energy imparted to the electrons by the wave. Therefore it is believed that a more realistic oonsideration would be that the temperature To not exhibit any vwave-like properties but the random energy of small volumes of space charge varies with the wave through the variation in space charge density nl eit - s. One oan obtain a rough indication as to whether -the electron gas is isothermal or adiabatic by oomparing the mean time between collisions of electrons and the period of the propagating wave. Referring to Appendix 3 for notation, if 2/co >> T, a large number of eleotronelectron collisions take place during a cycle so that energy can be conducted away from any given volume of the gas; this condition will describe an isothermal gas. On the other hand, if 2/c < r T the gas will be adiabatic. Considering the electron-electron collisions as binary encounters (which condition is not satisfied) Eq. A3-7 can be used to determine the order of magnitude of r. If the electron diameter is taken as 10-12 cm, 1/t Y 10-4/sec so that r >) 27/o and the electron gas is probably adiabatic.

143 Then p (no + n1) kTo A2-4 so that nm nom nom A25 From this it is seen that in the case of wave propagation in the z direction, for both plane and cylindrical geometries, since n1 O p 0= and the thermal motions have no effect on the wave propagation. In the case of propagation in the y direction: VP_= Q L + bn To =M~ no. aj Y. a Y A2-6 since n1 varies only in the direction of wave propagation, and it is improbable that To varies with any dimension except distance from the cathode. From the continuity equation, so that 2 nm' non U U nom by A2-7 Then the velocity equations are: m EX Ey + Ex mY krT Ito; rra To) let: 2 aMb~

144 Combining these with the field Eqs. II-.13: L+- O &o ox + 00 -2A2-8 ILO +z I L E + ~0 s=0 with solutions 2 o O 2 r2= / p0 9 +' s which is the same as Eq. III-15. Also cm+ o so f+2 + = 0 so that 1 bT0 + P hlE 7 ~ t4T2 y2T( bMe A2. It is seme that in this oase also, the inclusion of the pressure gradisrr term does not influence the electromagnetio rave propagation but leads to a nevw vave,.whio'nch if 0 O is given bys y 2y m (CO2 _ 2) A2-1 which is, of course, the same as Eq. A2-2 found by the other method and is the same as that found by Linder for a plane wave in a simple gas (no magnetic field). Linder, E. G., Phys T ev 49 - 753, 1936.

145 Unfortunately, in the case of radial wave propagation the in.. elusion of the pressure gradient term so complicates the equation corresponding to Eq. III-18 as to make the solution impractioal, These plasma waves (Eq. A2.10) are important in this space charge analysis only by virtue of their being separate from and independent of the propagating electromagnetic waves (this statement is applicable, of course, only in those eases which have been solved). This subject will therefore not be pursued further except to note that these are propagating waves (the group velocity is not zero). Linderl has shown the existence of suoh waves from a wave equation derivable from the Euler equation. For a comprehensive treatment of plasma oscillations see Bohm and Gross.2' Eq. A2-9 is seen to be identical to the corresponding expression for the propagation oonstant obtained in Chapter III, Section 2-a, where the pressure gradient term was not included. 7hile this does not justify rigorously the neglect of the pressure gradient term, it would seem quite reasonable that it have no effeot on electromagnetic wave propagation. This follows from the fact the mechanism of the influence of the electron motions on wave propagation is different in the two types of waves. That is, the electromagnetic waves (transverse waves) are influenoed princoipally by the motions of the electrons transverse to the direction of propagation while the plasma waves (longitudinal waves) are influenoed by the motions of the eleotrons parallel to the direction of propagation. 2pc. cit loo. cit Gross, E. P. "Plasma Oscillations in a Statio Magnetio Field" - Phys. Rev. V82 - April 15, 1951.

146 Also the tWave motion represented by the pressure gradient term is a longitudinal wave in which the energy is transmitted by physical movement of the particles from one place to another, and is therefore independent of the eleotromagnetio waves.

147 A pendix 3 The Effect of iElectron-Ion Collisions. In this present work the electrons in the magnetron space oharge are thought of as moving along oertain orbits prescribed by the steady fields. The electric field of t'he electromagnetic wave propagating in the space charge will periodically perturb the electrons from their steady orbits. During one-half cycle of the propagating wave the electrons till acquire energy from the fields, if there has been no energy lost by the electrons all of this energy will be returned to the wave during tho nexb half cycle and it will propagate undiminished in amplitude. The ooncept of a wave propagating in a material medium with amplitude undiminished with distance is contrary to our physical experionce, so that some mechanism of energy loss by the electron in its perturbed path must be sought. In even the best obtainable vacuum, the space charge will contain of the order of 10l moleoules/om3 which is of the same order as the electron density. Therefore in their periodio motion the electrons will have a reasonable probability of oolliding with gas molecules. These oollisions will provide one mechanism for energy loss by the electrons. The inclusion of suoh an energy loss term avoids the difficulty of equations for the electron amplitude or velocity becoming infinite at certain values of /mc. Appleton and Chapman1 consider an analogy between the velocity equations of electrons vibrating under the influence of a periodic eleo-ric field, suffering collisions with gas atoms, and electrons vibrating but experiencing a frictional type force proportional to the velocity. They show that the effect is sitnilar if the frictional coeffioient (g) Appleton and Chapman - "The Collisional Friction Experienced by Vibrating Electrons in Ionized Air" Proc, Pys. So o London - XLIV, page 246, 1932.

148 is replaced by the mean inverse time between eleotron-atom collisions. That is, the term g in Eq. II-3 is g = 1 where r is the mean time betw-een collisions of an elooectron and a gas atom. Using a similar line of reasoning to that followed by Appleton and Chapman, this analogy will be extended to inolude the oase in which the electrons are in a magnetic field. It will be shown that the same substitution g = is valid in this case also. r For simplicity a stream of electrons with uniform velocity is considered. Then the equations of motion of an electron under the influenoe of an alternating eleotric field and a steady magnetic field is given, between collisions, by the following: y = -in Ey + oO x A3-1 X W V E oy These may be solved to give e 1 COO 1 mE y - Ex + C A3-2 y*;$ [%~ ~E~+ C2. 1 - o02/C2 Now suppose that at some time t t1, out of the large number of electrons a group can be selected which suffer collisions so that those electron's velooities vanish. That is when t a t, c = = = O

149 so that elW C1; y eS Ex ecot -I 1 + 0a2/C2 A3-3 c2 e= 1 ewl Thus at any time after the oollision! + 0 oJ2/co2 A3-4 - -i 2/CO2 (1 J ejo) Let 0 = t - tl1 Then if the collisions ooour randomly, the number of collisions dNo which ooour in time dt after t1 is equal to the product of dt and the average number of Oollisions per unit time, A. dNo = A dt If r is the average time between collisions and there were No collisions at t tl: A NO/T Then integrating t N0 0 _T so d ~~~~/

150 where dN is the number of electrons colliding in time e to 9 + de Then to find the mean velooity over all times t > tls dc e 1 -v* m eO - eO(. + x = + C.02/C e2 - J- ee - e 0 A3-5 - e;; [e~ Ey -t Ex] 1 + C~o2/0)2 -. and similarly for ihe y +elocity component. Now from Eq. II-7 the equation of motion of an electron subject to a frictional force is: y + gy = -. Er + OX *.. e x +gx - - 0 x+g~ - m zxcY So that the velocities are Ey +) s + g + A3-6 w c g _ 9:O + g Comparing these two expressions for the velocities it is seen that, since g (< c, the velocity equations determined from collision considerations and those found using a frictional force are identical. From this one can conolude that the energy lost by each of these processes is equal when the association g = - is made. WVhile other forms of the electron cloud will change tle equations, it is believed that the

151 qualitative reasoning behind the analogy will be valid in any type of electron stream in a magnetic field. In order to determine a numerical value of the coefficient g, two methods can be useds 1) Appleton and Chapmanl describe an experiment from which the value of g could be obtained. However, their work vwas conducted at a gas pressure of about 0.1 mm Hg, considerably above that existing in the type of electron tubes considered here. If it is assumed that the frictional coefficient is proportional to the molecular density and therefore to the pressure, the experimental value of g can be extrapolated to the value at a pressure of 10'6 mm Hg. This procedure results in the value g = 104/seo. 2) The second method for determination of the numerioal value of g is through the evaluation of the mean time between collisions, assuming only binary encounters. In the case where the electrons and gas molecules are considered as smooth, rigid, elastic spheres, affecting each others motion 2 only at a collision, the number of collisions per unit time per unit volume between electrons of mass m and density n, and gas molecules of mass mg and density ng is: 1/2 12 nng 12 ( 1 m2 t mo ) A3m7 where a12 -= a-a rnd aog being the electron and molecule diameters respectively. hmo m + mg loc. oit. 2 Chapman and Cowling - page 90.

152 Since m (<mg and O << Og this expression reduces to 2 1/2 f_2 _n n_ 2 2 k TA58 The average number of collisions undergone by each electron per unit time is the collision frequency so that 1i = N12/ For a typical magnetron space charge, with applied magnetio field of 1000 gauss, n 5 x 10 electrons 4 x 108 m n 5 x 10 OM;'gz4l o M 11 molecules4 ~E ng~ 10 o ~ 104 K so that 1 lll x 16 x 1016 r 2T x 1.37x 10'16 x 104 ]/2 T 2 9 x 1023 and: g = 4 x 104/seo A3-9 The close agreement between this result and that obtained by the first method is largely fortuitous, since the assumption of binary encounters is not valid, but at least the order of magnitude of this quantity has been determined. The assumption that, during the course of an encounter between two particles, their motion is uninfluenced by the other particles in the region is probably valid for moleoules interacting with forces wvarying rapidly Yrith distance (such as an inverse fifth power law). However for the long range inverse square law forces this is not oorrect. Jeans1 showPs that the effect, on the motion of a given partiole, is Jeans, J. H - Astronomy and Cosmogony - Cambridge, 1929, Chapter XII.

153 greater due to all of ith "distarnt" partieles than those few whioh come very olose. These "distant" encounters are not considered by Chapman and Cowling. It is seen that for eleotromagnetic waves in the microwave region where co 101, the inequality g << co is valid even if the effect of distart encounters increases g by several orders of magnitude. This inequality is used to simplify the equations developed in Chapter III.

154 A entdix 4- The Effeots of Eleotron-Eleotron Collisions. In this section the effect of electron-electron collisions in the spaoe charge on the velocity of propagation of electromagnetio waves in this medium will be examined. By an electron-electron collision is meant the process in which two electrons move close enough to each other that their fields of influence overlap and eaoh of their rospective orbits is affected by the presenoe of the other electron. Using the law of conservation of momentum an attempt will now be made to show that the total dipole moment of the space charge is uninfluenced by collisions between electrons. From this it follows that the dielectric oonstant and therefore th'e wave propagation velocity is similarly uninfluenced. The electrons are thought of as moving along a path under the influence of certain time invariant forces; its path is periodically perturbed by an additional foroe alternating in time as ei. If the electron velocity is given by vo + V1 e A4-e the momentum is m (v +v1 et) Using subscripts b and a to denote before and after collision, and superscript numbers to denote one or the other of the two interacting eleotrons, the oonservation of momentum relation states that in a collision m(vOl +v1leb) + m(vv+viJ2 e) + m ( = ~~~~~b b ~A4-2 m(vo1 + v1j ei') + m(vo2 + v12 eit) aa

155 so that o b+b b 7 1a + to a vl b + 1 2 Vllb + 1t b = 1 a + Vl a These equations show that the sum of the perturbed velocities, and the sum of the unperturbed velocities are conserved in a collision. The electrid dipole moment rv, defined as the product of the charge and its displacement from its mean position, can be represented as TV ' ea a =A emt where A is the maximum amplitude of the electron excursion from its unperturbed position. The velocity of the particle is then T ioa so that: a v= and TV w * A4-4 Using Eqs. A4-3 it follows from this that 1 2 1 2 Tv lb + T v lb T= Tvla + v2la A4.-5 Therefore the sum of the dipole moments of the electrons is conserved in an electron-electron collision. This can be extended to include all electrons in a given region so that the total polarization of the medi um remains unchanged as a result of collisions. The polarization P is related to the dielectric oonstant er by P = &oE (er - 1) so that the dieleotric constant and therefore the velocity of wave propagation is unaffected by the collisions. The above reasoning considers, of course, that the oollisions cause no periodic variation of electron density, which would effeot the wave propagation.

Appendix 5 - Conditions Under Which the Second Order Terms in the Equations of Motion can be Neglected. The equations of motion of the electrons in the space charge have been linearized in all oases by neglecting the seoond order term (vl-V) vl. The complete equation of motion will be examined briefly for a typioal oase to determine the conditions under which this approximation is valid. From Eqs. II7, the equations of motion of the electrons in a plane magnetron space oharge under the influence of a wave propagating in the y direction are: Lo V- x YVxY - e Ex i- 12YVy2 =. X vx-ty2"~m Ex + coo Vx From the first of these, the x directed velocity is given bys e/m Ex X so that: Y2 vy3 + r m Ey_ 2 m (io Ey + coo Ex) O It is seen that the last t1wo terms of this equation represent the y directed velocity relation in the linear approximation. Therefore the remaining two terms represent the nonmlinear effeot and the condition under which these oan be neglected will be found as follows. If it is arbitrarily specified that the magnitude of the y directed velocity derived from the linear equation be changed not more than two percent by the inolusion of the nomelinear terms, the b llowting relationzs are valid: (y2vy2 + yr _Ey)vy <.02 vyc2

157 This oondition will be met if: y2 2 <.ol 2 and mY E.y< *Ol The first of these oonditions beoomess e2 s where i7 is the index of refraction of the space charge. This relation is merely a slightly mre stringent imposition of the non-relativistic assumption and will be satisfied whenever Vy2/02 ~ 1. The second of these conditions beoomess.01 ~cm.7.e whioh assumes the value (for co 6= zx 109/sec, - 1): E < 3 x 105 lts m which would be satisfied under almost all conditions except for the fields in an oscillating high porer magaetron.

158 Appendix 6 - Calculation of the Shift, Due to Space Charge, of the Resonant Vwavelength of a Coaxial Cavity. The change in resonant frequency due to the insertion of a dielectric in the coaxial cavity shown in Fig. 6.1 is oaloulated in this section using perturbation methods. The results of this computation can be compared with the experimental data shaon in Fig. 6.4. Following a procedure similar to that used by Bethe and Sohwinger1' let the fields in the unperturbed cavity (8r = 1) be represented by the subscript 1 and the fields in the cavity with dielectric be denoted by subscript 2. Then Vx H1 = I1 so E1 Vx E1 -;lr 1 D HI A6-1 Vx H2 = i(32 2 &o E2 'Vx E2 = -i2 Fo qH2 By suitable multiplioation and subtraction these become, after integration over the volume of the cavitys [(2 2 1- )l)SoE l' E2 + /Lo(CO21CS l)Hl Hj2 ]dVG,a A6-2 lJ 1' 7X H2-E2'VX H1 - H1 V x E2 + H2Vx El dV Vo The integral on the right beoomes, by a well known vector relation E1 xH2 + E2 x H1 dS A6-3 - - Bethe and Sohwinger l "Jerturbation Theory fbr Cavities" - NDRC Report DI-117 PB 18340, March 4, 1943.

where SO is the boundary wall of the cavity. For a perfectly oonducting wall this surfaoe integral vanishes so that the left side above becomes: o(~)2 2-<l1) E1' E2 - Lo) tor Hlz dV 0 A6-4 Vo which can be written (t-l)j coEl'Ez + /LoHl*a2dV -(2-l)c E E2 dV A6 -If the volume of the inserted dielectric is small compared with the total volume, its presence will cause only small perturbation of the fields so that since the elecotric displacement (D = or &oE) is continuous across the boundary of the dielectric,l in the above expression set c2 E2 a E1 so that: (02-01 __ so/i 26 dV 2 A6-6 we 2 -f [....l...L jvishtoaeni2 _ where W = o [IEli + to i E ]dV is the total energya2stored in the cavity. The integral in the numerator is taken over the volume of the dieleotric. In this case, using the notation of Fig. A6-1 Er =Cos " I'2) where The cavity is assumed to be in resonance in the TEl mode so that only a radial ele otric field component exists in the oavity. In the caloulation of the stored energy it is assumed that the presenoe of the dielectric does not materially affect the magnitude of the stored energy, so that in this expression the equality E1 = E2 is made.

160 ra 41 W *oJr 2(Z -) 2fr dr dz + vo ro r4 42 A6.. fjf2 Scos 2 2(Z+2) 2r dr dr ro k Using the dimensions for this 10 cm cavity W ( 17.1 eo A6.8. t r SPACE CHARGE FILAMENT FIG. A6- 1 SKETCH SHOWING DIMENSIONS USED IN CALCULATIONS ON COAXIAL CAVITY For the volume occupied by the dieleotriot jrE2 dV = 2 r-2 Cos2 Z- i.T) v2r dr dz VD rc O o 1.43 A6-9

161 so 122 2~ x- 143, =-.042 Coo E2 2 Y ~202A610e2 for example if ~2 = 6 (o/co42 0.6) *X -.00 0 O 0 0,33 cm As mentioned before, the magnetron space charge is oonsidered, in certain regions of 3/c00,, to exhibit properties of a dieleotric. From this consideration it would be expected that the resonant wavelength shift of the cavity be the same for the space charge as for a dielectric of the same volume and value of dielectric constant as the space charge. As seen from Fig. 6.4 the maximum shift obtained experimentally was.08 cm which is less by a factor four than the computed value. This discrepancy is probably due at least in part to improper formation of the space charge cloud, i.e. a cloud which does not approximate sufficiently closely that considered in the analysis, in particular the actual space charge cloud is probably shorter than considered in this computation.

162 Appendix 7 - Calculation of the Shift in Resonant WIavelength of a TEol1 Cavity, Due to the Presence of Space Charge. The change in resonant wavelength of the cylindrical cavity, shown in Fig. 6.7, is caloulated in this section for a dielectrio inserted along the axis, This calculation is made using the perturbation method outlined in Appendix s. The electric field varies as E = 2A J( r ol) Sin T a A7_1 so that the energy stored in the cavity can be found by straightforward methods1 to be W =A2H.o Tr2 z a JO (r t1) A7.4 where a and ~ are the radius and length of the cavity respectively, and rtol is the first root of the equation Jot (r) = 0. The integral of Ee in the region occupied by the dielectric is found to be: eofIFU2 dV t 8ffAo ((o ).8 [ T. Sin T X 0 where Zs is the length and rH the radius of the space charge, considered as a dielectric. See for example: Sarbacher and Edson- "Hyper and Ultra-High Frequenoy Engineeri'ng" - John WTiley and Sons, page 383.

163 This integral can be evaluated as rH rJ41(a r Io)dr = {LJ1l'( r'toij +111 ][j( _, Oro J1 l o A7-4 0 With s = 1.5 om rK = 0.5 om a = 7.95 cm 4 = 4.42 om o = 10.5 cm; these relations give: W- 45.5 A2Fi(4t. (;)3 A7_5 EE 1 dV *075 ALo r(7 () ) Since in this case the eleotrio field is everywhere parallel to the surface of the dielectric, the electric field oontinuity across the boundary yields the relation: (Er-l) eo Bel 2d A76 cit 2 seen ha it is seen that 7-;.0157 3 -5 - E5 r A7-7 So that for er 8 $ AV = 1.4% For the values of c/coo such as to yield values of ~eff >l. this calculation of A/i0 should be valid for the magnetron type space charge placed on the axis of the cavity.

BIBLIOGRAPHY REFERENCES WNHICH HAVE APPEARED IN THE TIXT "Space Charge Effects and Frequency Characteristics of cWt Magnetrons", H. W. Arelch, Jr*, Univ. of Michigan Eleotron Tube Laboratory Technical Report No. 1, Novembe 5, 1948. "High Frequency Behavior of a Space Charge Rotating in a Magnetio Field", J. P. Blewett and S. Ramo, Phys. Rev., V57, pages 635-641, April 1940. "Space Charge Frequency Dependence of a Magnetron Cavity", W. E. Lamb and M. Phillips, J. Appl. Phs., V18, pages 230-238, February, 1947. "Effects of Space Charge on Frequency Characteristics of Magnetrons", H. W. Welch, Jr., Proo I.R.E., page 1434, Deoember, 1950. "The Eleotrioal Conductivity of an Ionized Gas", Cohen, Spitzer and Routley, Phys. Rev, 80, 2, October 15, 1950. "Exoess Energy Electrons and Electron Motion in High Vaouum Tubes", E. G. Linder, Proo. IR.E. 26, page 346, 1938. "Effect of' High Energy Electron Random Motion upon the Shape of Magnetron Cutoff Curve", E. G. Linder, J. Appl. Phys. 9, page 331, 1938. "The Theory of Electrons", H. A. Lorentz, B. G. Teubner, Leipzig, 1909, Chapter IV. "The Refractive Index of an Ionized Medium" II, Chas. Darwin, Proc. ROy d Soo, London V182, page 152, 1944. "The Effect of a Uniform Magnetio Field on the Motion of Electrons Between Coaxial Cylinders", A. W. Hull, Phrys. Rev. V18, page 31, 1921. "Space Charge in Plane Magnetron", Page and Adams, Phys. Rev. V69, page 492, 1946. "On the Steady State and Noise Properties of Linear and Cylindrical Magnetrons", R. Q. Twiss, M.I.T. - Ph.D. Thesis, 1950. "Electronic Theory of the Plane Magnetron", L. Brillouin, Columbia University, AJdP Report 129.1R - OSRD 4510, to be published in part in the third volume. of "Advances in Electronics". "Theory of the Magnetron - I", L. Brillouin, Phys. Rev. V60, page 385, 1941. "Space Charge in Cylindrical Magnetron", Page and Adams, Phys. Rev. V69 page 494, 1946.

REFERENCES TIHIC2H HAVE APPEARED IN THE TEXT (Cont 'd) "Elektronenbanen und Meohanismus der Sohwingungserregung in Sohlitzanodenamagnetron", H. G. Moeller, Hoohfrequenztechnik und Elak., V47, page 115, 1936. "The Passage of Steady Current in a Cylindrical Non-slit Magnetron", V. M. Glagolev, Zhur. Tekh. Fiz. USSR - 19, page 943, August 1949. Translated by NaTj Researeh Laboratory, Washington, D. C. NRL Translation No. 318. "Eleotronic Orbits in the Cylindrical Magnetron with Statio Fields" Radiation Laboratory Special RePrt 9S, Section V, R.L. Report 122, Ooto ber, 1941* "The Influence of Spaoe Charge on Eleotron Bunching", L. Brillouin, Phys, Ryev V?0, page 187, -August, 1946. "Rotating Space Charge in a Magnetron with Solid Anode", I. I. Wasserman, J. Tech. Phys. (USSR), V18, page 785, 1949. "Measurement of the Circulating Electron Current in a Magnetron", J. lioller, Hoohfreguenztehnik und Elak., V47, page 141, July, 1936. "Eleotromagnetio Waves, Sohelkunoff, D. Van Nostrand and Co., page 406, 1943. "The Electron Wave Tube - A Novel Method of Generation and Amplification of Microwave Energy", A. V. Haeff, Froc. I.R.E. 37, pages 4-10, 1949. "High Frequency Amplification by Means of the Interaction Effect between Electron Streams", J. Labus, Aroh, Elekb. Ubertragung 4, pages 353-360, "Wave Propagation in a Slipping Stream of Eleotronss Small Amplitude Theory"r, G. G. Macfarlane and H. G. Hay, Proc. Phys. Soc., Lend. B, LXIII, pages. 409427, 1950. "Generation and Amplification of Waves in Dense Charged Beams under Crossed Fields", 0. Buneman, Nature, V165, page 474, March, 1950. "Intermediate Differential Equationst", E. D. Rainville, John Wliley and Sons, page 90, 1943. "A Course of Modern Analysis", Whittaker and Watson, Cambridge, page 201, 1946. "On the Origin of Harmonies in t/he Ionosphere - At Points where the Dielectric Constant is Zero", K. Forsterling and H. 0. Wuster, Comptes Rendus - AAadamie des Soienoes -231 - No. 17, page 831, Ootobe`rF`as;T95o. "On the Propagation of Electromagneltic Waves Through an Atmosphere Containing Free Electrons", H. W. Nichols, and J. C. Sohelleng, Bell Syst. Teoh. J. - 4, page 215, 1925.

REFERENCES WEiICH HAVE APPEARED IN THE TEXT (Cont'd) "Group and Phase Velocities from the Magneto-Ionic Theory", H. A. Wale and J. P. Stanley, J, Atmos. Terr. Phys. VI No. 2, page 82, 1950. "Electromagnetic Theory", J. A. Stratton, McGraw-Hill, page 137, 1941. "Electrodynamics", Page and Adams, Van Nostrand, page 328, 1940. "Cosmical Eleotrodynamios", H. Alryen, Oxford, page 35, 1950. "Small Signal Theory of Velocity Modulated Electron Beams, W. C. Hahn, G. E. Rev. 42, No. 6, page 258, June, 1939. "Electron Optioal Ebploration of Space Charge in a Cutoff Magnetron", D. L. Reverdin, J Phys22, page 257, March, 1951. 'Viorowave Magnetrons", F. F. Rieke, Radiation Laboratory Series No. 6, MlloGraw-EHill, page 390, 1948. "The Mathematical Theory of Non-Uniform Gases", - Chapman and Cowling, Cambridge, 1939, Chapters 3 and 18. "The Effeot of Hydrostatio Pressure in an Electron Beam on the Operation of Travelling Wave Devices", P. Parzen, J, Appl. Phs., April, 19 1. "Plasm Oscillations in a Static Magnetic Field", E. P. Gross, Phys. Rev. V82, April 15, 1951. "The Collisional Friction Experienced by Vibrating Electrons in Ionized Air", Appleton and Chapman, Proo, Phys Soc, of London - XLIV, page 246 1932* "Perturbation Theory for Cavities", Bethe and Sohwinger, NIDRD Report D1-117 PB 18340s, arch 4, 1943. "Hyper and Ultra-eigh Frequeny Engineering", Sarbacher and Edson, John Wiley and Sons, page 383.

ADDITIONAL GE0ERAL REFERENCES "Effect of Space Charge in the Magnetron", E. B. Moullin, Camb. Phil. Soo. 36, page 94, 1940. "Propagation of Electromagnetic Waves in an Atmosphere Containing Free Eleotrons", Huxley, Phil. Mlag. 29, page 313, 1940. "Propagation of Electromagnetic Waves in an Ionized Medium", Rydbeck, Phil Mag. 30, page 282, 1940. "Theory of the Ionosphere" Majundar, Zeitschrift tr Physik 107, page 599, 1937. "Propagation of Electromagnetic Waves in an Ionized Atmosphere", Huxley, Phil. Mag. 25, page 148, 1936. "Motions of Electrons in Magnetic Fields and A-C Electric Fields" Huxley, Phil. Nag. 23, page 442, 1937. "Oscillations in Ionized Gases", Langmuir, Proo. Nat. Acad. Soi. 14, page 627, 1928. "The Propagation of Electromagnetic Waves in a Refracting Medium in a Magnetic Field", D. R. Hartree, Proc. Camb. Phil. Soo. 27, 143, 1930. "Wireless Studies of the Ionosphere", E. V. Appleton, Proc. Inst. Elec. Engineers 71, page 642, 1932. "Collisions Between Electrons and Gas Molecules", Jones and Langmuir, Phys. Rev. 31, page 390, March 1928. "Physics of the Ionosphere" Mimno, Rev, Mod Phys. 9, page 1, January, 1937. nThe Quantitative Study of the Collisions of Electrons with Atoms", Brode, Rev. Mod, Phys. 5, page 257, 1933. "On the Effect of an External Electromagnetic Field on a Split-Anode Magnetron", S. Ya. Braude, Zh. Tekh. Fig. 13 7/8 431, 1943 (Russian). "On the Vibrations of the Electronic Plasma", L. Landau, Journ, Phys. USSR 10, 25, 1946. "Plasma -- Electron Osoillations", E. B Armstrong, Nature (London) 160, 713, November 1947. "Frequences de Resonance de la oharge d'espaoe d'un Magnetron", P. Fechmer, C. R. Academie des Scienoes, page 270, July 24, 1950. Iviesure des frequences de resonance de la charge d'espaoe d'un Magnetron a' oavities", P. Fechner, C. R. Academie des Sciences, page 270, July 24, 1950.

ADDITTONAL GLINERAL IREFERMCES (Contt'd.) "Uber die Eigenschwingung freier Elekcronen in einem Konstanten Magnetfeld" $. Benner - Die Naturwis, 17, 1929, page 120. "Oscillations in a Plane One-Anode Magnetron" L. Brillouin, OSRD Repr No. 5173, AiP Columbia Univ., No. 129.3R, May 1945. "On the Properties of Tuabes in a Constanrt Magnetic Field" I - 0. Doehler, Ann. de Radioeleo. - Till, No. 11, January 1948. nEleotrooMagneto - Ionic Optics" - V. A. Bailey, Jouro Soo, 1948, pages 107-113, V72. "Group and Phase Velocities from the Magneto-Ionic Theory" - H. A. Whale and J. P. Stanley, Jour. Atmos. Terr. Phys. VI - No. 2, 1950, page 82. "Space Charge and Field Waves in an Electron Beam", S. Ramo, Phys. Rev. V56, 1939, page 276. "Energy Conversion in Electronic Devices", D Cabor, J Instn Eleot Engrs., Vol 91, Part III, page 128, 1944.

SYMIBOLS USED IN THE TEXT a = attenuation constant of propagating wave Bo = applied magnetic flux density -- ephase constant of propagating wave y = a + j1 propagation constant of propagating wave o velocity of light (exoept in Appendix 1) = a instantaneous magnitude of space charge surface perturbation D = e E electric displacement vector E = electric field intensity vector,e ' Re72 effective dielectric oonstant of space charge 10"9 farads o =aftoa dielectric constant of free space relative dielectric constant of a dielectric e electronic charge 7 = o/rv indeP of refraction of space charge f~ electron velocity distribution function g = 1 eleetron damping coefficient h = anode cathode distance in plane magnetron e = angular coordinate in cylindrical magnetron = magnetic intensity of propagating wave K = ooefficient of thermal conductivity = unit of length, also used as defined by Eq. III-27 =X wavelength to = resonant wavelength ~,t = 4w x 10'7 O permeabilify of free space m

SY-VIBOLS USED IN THE TEXT (Cont'd) n = electron number density no = time invariant electron number density ( = steady eleotron angular velocity in cylindrical magnetron ra = anode radius of cylindrical magnetron ro * oathode radius of cylindrical magnetron ra = radius of outer edge of cylindrical space charge p = total electronic charge density Po " time invariant electronic charge density Pi = a-o component of electronic charge density a,8 effective conductivity of space charge T mean time between electrornion collisions V total electron velocity Vo " steady (drift) electron motion vi = a-c component of electron velocity p= * phase velocity of propagating wave c = radian frequency of propagating wave c% = fe cyclotron angular velooity m Ye electronic admittance of space charge Ye=

UNIVERSITY OF MICHIGAN 111111111111111111 11111111111111111119 3 9015 02519 7354

ERRATA TO TEE EROPAGATION OF ELECTROMAGNETIC WAVES IN A MAGNETRON-TYPE SPACE CHARGE P. 6 The equation should be written: at br m bc bt coll. P. 7 Eq II-1 'Right side should be: le [- + Bo VP P. 9 Eq II-2 Right side should be: e P. 9 Equation in center of page should be: mF = E+ P e 3Eo P. 9 Replace N by n (electronic number density) P.10 Equation at top of page should be: e P e = E + m P.10 Eq 1I-3 Riight side should be': e -- X -- iE+v x B P.11 Fifth term should be: (vl ' V)Vl P.15 The dotted curve, referred to in text, which is missing from Fig. 2.3, should be the same as the lower curve except that it will show a peak near y/h = 1 and will decrease approximately exponentially for y/h 1. P.18 The last two sentences on this page should be deleted. This result obtained by Glagolev is at variance with other works of a similar nature and leads to a space-charge density distribution considerably different from the Eull-Brillouin value for small1 filament. It is believed that there must be an error in his calculation.

P.37 Fig. 3.5 The lower curve should be deleted for rH/rc '^Z 2. P.44 Fig. 3.7 The lower three curves should be deleted for rH/rc 2. P.99 The scale marked 2" should be changed to 1".

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