ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR SPACE-CHARGE EQUILIBRIUM IN A MAGNETRON A STATISTICAL APPROACH Technical Report No. 10 Electron Tube Laboratory Department of Electrical Engineering BY G. HOK Approved by: W. G. DOW Project M921 CONTRACT NO. DA-36-039 sc-5423 SIGNAL CORPS, DEPARTMENT OF TRE ARMY DEPABTMlENT OF ARMY PROJECT NO. 3-99-13-022 SIGNAL CORPS PROJECT 27-112B-0 July 13, 1951

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TABLE OF CO01TETS Page ABSTRACT iii LIST OF ILLUSTRATIONS ii I. INTRODUCTION 1 II. EQUATIONS FOR ELECTRON MOTION IN A LINEAR MAGNETRON 3 III. EQUILIBRIUM STATES 6 IV. INITIAL STATES AND PHASE-SPACE REPRESENTATION OF ELECTRON EINERGY STATES 11 V. DISCUSSION OF THE MAGNETRON SPACE-CHARGE DISTRIBUTION AS A DIFFUSION PROBLEM 18 BIBLIOGRAPHY 24 LIST OF ILLUSTRATIONS No. Title Page Fig. 1. Coordinate System for Plane Magnetron 3 Fig. 2. Space-Charge Distribution at Thermal Equilibrium 10 Fig. 3. Mapping of Regions in Phase Space Accessible to Electrons Whose Orbits Intersect the Anode or Cathode Plane 12 Fig. 4. Brillouin.Distribution in Phase Space 14 Fig. 5. Double-Stream Distribution in Phase Space 14 Fig. 6. Space-Charge-Limited Conditions 16 Fig. 7. Temperature-Limited Conditions 16 ii

ABSTRACT This report discusses the steady-state space-charge distribution in a d-c magnetron when its anode voltage is lower than its cut-off voltage. It is shown that the discrete electron-to-electron interaction has a cumulative effect on the space-charge distribution that is not negligible. The distributions derived by Brillouin, Slater, and others are not steady-state distributions, since they are obtained without regard to the discrete interaction. The actual distribution and its dependence on the ratio of anode voltage to cut-off voltage are discussed qualitatively, but no attempt is made to calculate the spacecharge distribution explicitly. iii

SPACE-CIARGE EQUILIBRIUM IN A MAGNETRON A STATISTICAL APPROACH I. INTRODUCTIOIJ As an initial state for the oscillating magnetron the non-conducting (or approximately non-conducting) condition of a d-c magnetron has a considerable interest and has received appreciable attention, both of a theoretical and an experimental nature. Nontheless, a satisfactory agreement has not been reached about the shape of the electron orbits and the detailed distribution of potential and electron density in such a space charge. The solutions presented necessarily rest on postulates and assumptions that constitute idealizations of the real conditions. So does, of course, all physical theory, but in this case well-known factors have been neglected with the justification that their effect is judged to be so small that the approximate solution resulting will be close enough to be of value. Furthermore, the mathematical difficulties of a more rigorous approach appeared prohibitive. It is the purpose of this report to reconsider the conventional simplifications on which these solutions are based, to discuss whether or not they introduce appreciable errors, and to investigate the feasibility of a solution from a more realistic set of assumptions. Since the space charge is essentially a gas formed by discrete electrons, classical statistical mechanics offers a logical approach to a fresh study of the problem.

2 We shall first place the problem into the framework of statistical mechanics. The next step is to consider the solutions proposed so far in this new light and to show that they can not possibly represent steady states. This discussion does not directly indicate whether the errors in the distribution of space charge and potential-are large or small.. Finally, the problem of finding a solution consistent with the laws of statistical mechanics will be investigated and the result compared with the previously suggested solutions.

3 II. EQUATIONS FOR ELECTRON MOTION IN A LINEAR MAGNETRON Since we are going to apply the methods of statistical mechanics to the problem of electron motion in a magnetron, it is convenient to write the equations of motion in Hamiltonian form. The total energy of an electron is: W = 2m e (1) where p is the momentum vector, A the vector potential of the constant magnetic field, and E the scalar electric potential. The momentum p is given by p = mv + eA, (2) where v is the velocity vector of the electron. To simplify the mathematical processes, we choose a plane magnetron and a Cartesian coordinate system (Fig. 1). The cathode is represented by Y j -E-Ea / / ///~ ////////'/////// ///// /// d B ~ Z ~///////////////^O////////////7X E=0 FIG. I COORDINATE SYSTEM FOR PLANE MAGNETRON

4 the plane y = o, the anode by the plane y = d. The uniform magnetic flux vector B is parallel to the positive z-axis. The potentialac the cathode is taken to be zero and the potential of the anode positive is equal to Ea. No variations are taking place with x and z. ox a ~. (3) ax az The vector potential A then has only one component, Ax, such that aAx -= B = B, (4) Ax = - B y, (5) since we can choose A as well as E to be zero at the cathode. The Hamiltonian (Eq 1) then is W = 4{(px eBy) + eB y p pz2 eE (6) and the canonical equations of motion d - a = 0 m d + eBvy (7) d= dt, W 1 DE 1 Eo d- = = (Px +eBy)eB eBvx - e (8) dt a= = (9) d' aw = 0, dx = a = p (Px + eB y) = vx (10) dt a Py M d aw5 = vY- (11) dt = aw = Vz. (12) WE dP

The constants of the motion, or the parameters that determine the energy state of each electron, are W, Px and pz. Optionally, we can specify the state of an electron by px, pz and pyo, the last quantity being the value of py at the cathode (y = 0). It should be noted that pyo may be imaginary, since the orbit of an electron may not necessarily reach the cathode. In the study of the space charge in the magnetron we shall be interested in the distribution of the electrons in phase space or p. -space, i.e., a six-dimensional space with the coordinates x, y, z, Px, Py, Pz. Actually, a subspace y, Px, Py will contain all the boundary surfaces and orbit projections necessary for the study of the problem.

6 III. Qc~D-iLI:lTIu-i;J STATES If the cathode of the magnetron is at constant temperature and no -current flows to the anode, no energy is received or lost by the swarm of electrons in the tube. It should, under these hypothetic conditions, be possible to consider the magnetron as a closed system in thermal equilibrium. In this section we shall consider the conditions to be satisfied for thermal equilibrium in a magnetron. When a minute current flows to the anode the problem becomes a transport problem, and the solution is obtained from. the thermal equilibrium by applying a small perturbation to the distribution function for the electrons. As the current is increased, the diffusion of electrons through the space charge from the cathode gradually changes to a steady flow of the whole space charge towards the anode. The intermediate conditions, with severely distorted distribution function but no coherent flow pattern, offer the greatest mathematical difficulties; unfortunately, there are reasons to believe that this is the actual state of affairs in a cut-off magnetron. Since the space charge is assumed to be in equilibrium with the thermionic cathode emitter, the distribution density function at y = 0 is necessarily of the Maxwell-Boltzmann type. P =- A exp [- -(px2 py2 + )] (15) The distribution density function P at an arbitary point approaches its equilibrium value under the influence of two simultaneous processes, convection by electron motion, and exchange of energy and momentum between the electrons because of Coulomb interaction.

7 We shall not include here the complete classical derivation of the distribution density function. It is only necessary to investigate the consequences of the magnetic field and the curvilinear orbits of the electrons. Under equilibrium conditions the convection-current element formed by a certain group of electrons in phase space must be continuous throughout its path through x-y-z space. The expected value of a current element formed by electrons of y-directed momentum between pyo and pyo + dpyo at the cathode is, therefore, equal to the corresponding element at any other value of y. ePo Pyo dpyo = eP py dpy. (AL) However, conservation of energy requires W = {2 + P +2 + 2 = 1{(Px + eB y)2 + Py2 + p2} - eE, (15) W 2m x + Py o and consequently Pyo dpyo = Py dpy. (16) Continuity therefore requires that the, distribution density at y is P dpy = Po dpy. (17) It should be noted that the distribution density function P is given by (17) only for the regions in phase space accessible to electrons emitted from the cathode. We shall later map these regions. The interaction between the electrons during thermal equilibrium must be such that the same number of electrons are removed from a certain cell in phase space as are entering into the cell during the same time interval. Suppose that a certain encounter involves n electrons with the initial t t energy states WT1....Wn, and after the encounter, the energy states W1.....Wn ~ Conservation of energy requires that

8 W1 + W2 +....Wn = W1 + W2+.. n (18) The rates at which such encounters and. the inverse encounters take place should be equal and are proportional to P1iP2 P3...Pn = P1 P2 n.P n' (19) Since the exponential function transforms a sum into a product, it is obvious that the Maxwell-Boltzmann distribution function satisfies the requirements (18) and (19), as well as (13) and (17). Thus we can write: -aW P = A-e, (20) where A is constant related to the total number of electrons in the space charge and a= 1/kIT. It is interesting to note that the presence of the magnetic field and the consequent curvature of the electron orbits does not affect the distribution function. The factors that determine the distribution arise from conservation of energy and isotropy of interaction. In order to find a space-charge distribution compatible with thermal equilibrium, we integrate Eq 20 with respect to px, Py} and Pz from -oDor + oD. The result is P p eCeE (21) This relation is combined with Poisson.'s equation to 2E P = - PoeaeE () dy2 - So o~ 1 2 Po eE (25) ) = _ -e"-eE + c1 (25) (B P~=

9 In order to perform the second integration, we introduce Q = eeE,(24) dQ - -2p eo ay. (25) Since the first term in Eq 23 is always positive, C1 and a must be negative if a potential minimum exists. Changing the sign of a and integrating, we get 2 tan-1 a by + C2 r_ ^tan1 - +c(26) P0*a cos by 1 2 cos2b(y - ) - [cosb (ym (7) lo2 cos bym ~E = IlogCOSe 1a (28) EM ecos b(y- ) ) where b = -2 p e (29) no The boundary conditions are represented by the constants po, a, and ym. For negative values of Ym the solution represents temperature-limited conditions. Fig. 2 indicates the space-charge distribution required for this thermal equilibrium. It obviously does not resemble the distribution in a cut-off magnetron, since it demands emission of electrons from the anode with the same temperature as those emitted from the cathode but with considerably higher density, Pa Po eEa (30)

10 0 Ym d Y FIG. 2 SPACE-CHARGE DISTRIBUTION AT THERMAL EQUILIBRIUM Nontheless this equilibrium has a certain interest to us, since the difference between this distribution and the actual distribution in any particular volume element in real space dictates the amount of diffusion that takes place in that element. A steady state is reached when the diffusion into every energy state is equal to the diffusion out of the same energy state. Before we investigate more closely this diffusion process, we shall in the next section discuss the representation of electron energy states as points and orbits in phase space.

ll IV. INITIAL STATES AND PHASE-SPACE REPRESENTATION OF ELECTRON ENERGY STATES When a magnetron is switched to a source with a voltage E smaller than the cut-off voltage of the magnetron, a charge Q will flow to the magnetron. The source supplies the energy EQ, but the stored energy in the magnetron is only 1/2 EQ. The difference is lost in the circuit resistance and, possibly, in a temporary increase in the electron temperature of the magnetron space charge above the cathode temperature. These transient effects we shall disregard and assume that the initial conditions are consistent with conservation of energy and momentum for each electron emitted from the cathode and with Poisson's law. As stated earlier in this report, the energy state of an electron is characterized by its energy W and two of the components of its momentum, Px and pz, these three quantities being constants of the motion, as long as energy and momentum are conserved. On the other hand, the component py and the potential energy vary during the motion. In order to map the regions in phase space accessible to the electrons emitted from the cathode it is sufficient to consider the space Px, Pyr Y. The accessibility criterion is obtained from 2mW = x2 + Pyo + Pz = (x + eBy) p2 + z - 2m eE or py2 = 2 + 2m eE - e2B2y2 - 2Px eBy (31) py2 = Pyo (31) where py and Pyo have to be real quantities. Fig. 3 shows a sketch of the boundary surface between the accessible and the inaccessible part of px-py-y-space. It is, of course, qualitative only, since E (y) is not known but related to the space-charge distribution by Poisson's

12 Py = Oy ANODE''/ / / / /'''''' //' /'' /I / / /' /',/ // REGION ACCESSIBLE FROM THE CATHODE \ SECULAR \ REGION \ REGION ACCESSIBLE \ \ FROM THE \ \ ANODE A I A,/////////// //////j/j,/////jj / /~ PX CATHODE PLANE- AA Py (y = a) REGION REGION ACCESSIBLE / ACCESSIBLE FROM THE ANODE / \ FROM THE / SECULAR \ CATHODE / REGION i\'/ Y~ _^/Px FIG. 3 REGIONS IN PHASE SPACE ACCESSIBLE TO ELECTRONS WHOSE ORBITS INTERSECT THE ANODE OR CATHODE PLANE.

15 law. Integration of the distribution function (20) over the accessible space with respect to px and py would, give the relation between p, E, and y that, combined with Poisson's equation, determines the initial conditions. The inaccessible region has the shape of a distorted cone with its apex on the axis py = 0, but in general not at px = 0. The axis py = 0 to the left of the apex is a generatrix of the surface. It should be pointed out that the fact that the inaccessible region reaches the axis does not mean that electrons with py - O and Px in this'range can not escape from the cathode. They do, but only tangentially to the px-py-plane, describe a broad elliptic orbit in the y-py-plane and reach their turning point on the upper part of the boundary line of the accessible region or hit the anode. In Fig. 3 the' boundary for interception of the electrons by the anode is also indicated. In other words, any electron to the left of this surface will be removed by irmact on the anode. The space-charge density can therefore be considered zero to the left of this boundary except where this region overlaps the region accessible from the cathode. The volume between the two boundaries contains all the possible energy states whose electron-orbits reach neither the anode nor the cathode. We shall refer to this volume as the secular region since the life of an electron energy state in this region is very long compared with the period of the cyclic motion of the electron. It should be noted that: 1. Only an infinitesimal change in momentum is required for an electron to cross the boundary into this volume. 2. The electron population of this volume will continue to increase until the current flowing to the anode through the opposite boundary exceeds the current entering the volume from the cathode-accessible region. 3. Whether the discrete electron-electron interaction is weak or strong determines primarily the time required to reach a steady state but not necessarily the final space-charge distribution.

14 lY Ly ///////. ////// //////: A A b/ / / / / /PX / / / / / / / / / / / / FIG. 4 BRILLOUIN DISTRIBUTION IN PHASE SPACE y y ///////,///////ZLZ'///PX / / / / / // // / / / / py FIG. 5 DOUBLE-STREAM DISTRIBUTION IN PHASE SPACE

15 Before discussing the possibility of determining at least roughly the final space-charge distribution, we shall give some brief comments on previously proposed solutions to the space-charge distribution in the cut-off magnetron. Connon to them all is that the solution is identified with some form of what we have called initial conditions; sometimes the justification is given that the interaction between discrete electrons is small enough to be neglected. The three underlined conditions above show that this reasoning is not correct because of the effect of the interaction, however weak, is cumulative. 12 The Brillouin or single-stream space-charge distribution in phase space (Fig. 4) is limited to the line AB and independent of y within this range. At first sight this distribution may appear to be a possible equilibrium at 0~K temperature. However, it would be strange indeed if all electrons occupied the same point in momentum space, although lower energy states certainly are possible. But we must consider the possibility that the state be metastable because of the lack of a process whereby the unoccupied energy states could be populated. Obviously such a process does exist. Since the space charge is formed by discrete electrons whose velocity varies with y, even though their orbits do not intersect, the electric field at the edge of the spacecharge cloud necessarily fluctuates in magnitude and direction both in time and space. A diffusion will therefore take place, both out into-the unoccupied space and towards the cathode. The energy required for this random motion is, of course, supplied by the d-c electric field. The double-stream distribution described by Slater,4 Page and Adams,5'6 and others is represented by an approximately elliptic line charge in phase space (Fig. 5). The space-charge density in real space at the cathode and at the edge of the swarm is infinite; a finite minimum is located at an intermediate plane. This is also a 0~K distribution, since all the electrons

y ANODE PxBx=eBye l _____, \\___ -\. _ _ -EDGE OF'-"\ \ "\ ~ SPACE CHARGE CATHODE 0 PX FIG. 6 SPACE CHARGE-LIMITED CONDITIONS ANODE -eBy v -ZeBy- \ -e\__ \_____ \ \ __ \ _ EDGE OF'\ \,\ ^ SPACE CHARGE /// ///////////////// //p./?p CATHODE 0 P FIG. 7 TEMPERATURE- LIMITED CONDITIONS

17 have the same energy although occupying a line rather than a point in momentum space. There can be no question about a metastable state in this case, since the electron orbits intersect with considerable relative velocity so that exchange of energy and momentum is inevitable. Twiss has considered the modification in. the initial space-charge distribution produced by the initial velocities of the electrons. The. result is essentially the initial state mentioned earlier in this section as obtainable by integration of the distribution of function (20) over the accessible region of momentum space. It avoids the discontinuities and singularities of the two distributions mentioned above, but is otherwise subject to the same criticism; it is an initial state but not a steady state.

18 V. DISCUSSION OF THE MA.GMETBRON SPACE-CHARGE DISTRIBUTION AS A DIFFUSION PROBLEM A calculation of the steady state in the cut-off magnetron from the initial state previously indicated is complicated by the following circumstances: 1. The relationships governing the transfer of momentum between the electrons, including close encounters as well as distant encounters, are rather involved even when the state of the space charge differs very little from thermal equilibrium., 2. The actual distribution density is very irregular because of the initially abrupt variation at the accessibility boundaries. 3. The distribution density function is furthermore distorted by the increased electron temperature produced by the diffusion of electrons from the low-potential parts of the space charge to the high-potential parts, so that the energies of electrons returning to the cathode are distributed according to a higher temperature than those leaving the cathode. (We use the term "temperature" for convenience although the distribution is not regular Maxwell-Boltzmann. ) We shall here try to arrive at a qualitative understanding of the way in which a steady state is reached without resolving quantitavely the complications enumerated above. The drift of electrons through phase space is such that if the system were temporarily closed its state would gradually approach thermal equilibrium. During thermal equilibrium the number of electrons that move from one particular volume element in phase space to another such element is equal to the number that moves in the opposite direction. We should expect it to be possible to calculate the instantaneous value of the interaction current from

19 one volume element in the phase space to another by comparing the instantaneous distribution with the equilibrium distribution. Here, however, we should not consider thermal equilibrium, which is determined by the boundary conditions of the whole space charge, but the standard normal distriubtion of (Px + eBy), py and pz that has the same total population, the same energy, and the same center of gravity as the initial distribution. The total interaction current across a certain boundary surface in momentum space would be obtained by a double integration over these two distribution density functions. In addition to a function of the densities of the two volume elements in both distributions the integrand must necessarily contain a weighting function expressing the probability of the required change of momentum between the two volume elements. At least two simultaneous asymptotic processes should be considered that give quite different weighting functions. The first one operates by close binary encounters and is a discrete random process of well-known type. The influx to a certain volume element in phase space because of its density deficiency is largely determined by the distance of the element from the center of gravity of the distribution density functions. In this case the weighting function accounts for the impact parameter and the angle between the relative velocities of the two electrons with respect to their common center of gravity. The second process is the result of interaction between a large number of electrons at considerably larger distances. The weighting function in this case permits only very small changes of momentum. The interaction current density across a certain surface in momentum space is consequently determined chiefly by a relation between the actual density gradient at this surface and the equilibrium density gradient there. It has been shown8 that when the interaction is due to Coulomb forces neither one of these two processes can in general be neglected in comparison with the other.

20 In enumerating the factors that determine the steady state we should begin with the boundary conditions at the cathode. It is natural to assume that the mass and thermal capacity of the cathode are so large that the temperature and distribution of the emitted electrons are independent of the temperature and distribution of the returning electrons, that is to say, the energy and momentum of the electron cloud are not conserved at the cathode. Wemade this assumption previously in order to arrive at the thermal equilibrium distribution, Eq 20. If momentum were at least partly conserved, the center of gravity of the distribution could not fall on the line Px = - eBy but somewhere between this line and px = 0. The assumption made here appears satisfactory as long as the emission is strictly thermionic; if secondary emission is appreciable, the boundary conditions become much more difficult to state. Between the cathode and the potential minimum the inaccessible part of the phase space is likely to be small and not to include dense regions of the distribution. The potential and space-charge density then drops approximately according to Eq 21 until roughly the Brillouin density is reached. The most interesting region in real space is between the potential minimum and the edge of the cloud. The boundary of the secular volume in phase space on the cathode side is likely to be almost parallel to the y-axis here, since the potential distribution is probably not very different from that of the Brillouin solution (see Fig. 6). The space-charge density in the secular volume obviously must be such that the net number of electrons entering from the right equals the net number leaving at the left boundary. This density is obviously smaller than the one required for ideal thermal equilibrium; the center of gravity of the distribution must therefore be expected to be to the right of the line Px = - eBy, that is, closer to the right-hand boundary than to the left-hand one. Thle space-charge density required for a steady state is therefore closer to the density at the right boundary than to the

21 one at the left, which is close to zero. To calculate this space-charge distribution is evidently very difficult, but it should be clear that its density is by no means negligible as far as calculation of space-charge distribution in real space is concerned. Twiss maintains that a double-stream motion with considerably more than thermal energy exists here in the cathode-accessible volume of phase space. The square of the y-directed momentum of an electron can be written (Eq 31) py2 = Pyo2 + 2mn - e2B2y2 - 2px eBy ~ pyo2 - 2px eBy (32) The orbits of the electrons in the py-y-plane are parabolas, and since eBy px, py may be of a different order of magnitude than pyo for negative values of px. The orbits do not extend to the anode, because outside the edge of the space-charge swarm e2B2y2 is considerably larger than 2mE, so that the electrons turn back into the cloud. At first sight these conclusions seem to be inconsistent with the view presented in section IV, that the initial energy distribution at any point is an incomplete Maxwell-Boltzmann distribution of cathode temperature. The answer is that as the electrons move in the positive y-direction and gain kinetic energy from the electric field, they also move towards the high-energy fringes of a distribution that, if it were complete, would have a much higher space-charge density than at the cathode.. The high-energy electrons will be much closer to the center of gravity of the actual, considerably distorted distribution than to the center of gravity of the Maxwell-Boltzmann distribution of which they form a small part. Consequently, the average energy or "temperature" of the actual distribution will be much higher than at the cathode.

22 This increase of the average electron energy with y no doubt contains the clue to a number of magnetron problems. Considering the cut-off magnetron as a resistor, we should. expect the increased electron temperature to increase the noise output. Twiss has attributed the noise level to a "noise amplification causedby the double-stre interaction in the space-charge cloud. The points of view may not be equivalent, but they are certainly related. The difference between the space-charge distribution discussed here and the Brillouin or Triss solution is probably not very large when the volume of the secular region in phase space is small, i.e., when the tube is operated close to the cut-off voltage. The anode current will be appreciably larger than the direct convection current from cathode to anode, however, because of the diffusion through the secular region. When the anode voltage is far below the cut-off voltage, on the other hand, the secular volume is large, and the total space charge there may be large enough to affect the space-charge density in real space and the potential distribution appreciably. So far it has been assumed that space-charge-limited conditions prevail in the magnetron, so that a potential minimum exists a finite distance from the cathode.. Sufficient emission is then available to maintain a spacecharge density in the cloud roughly equal to the Brillouini density. Under temperature-limited conditions this is not the case (Fig. 7), initially at least, although the possibility should be considered that the accumulation of space charge in the secular region could modify the potential distribution in such a way that Fig. 7 changed into something like Fig. 3. This is very likely to be approximately true when the magnetron is operated far below the cut-off voltage, so that the anode current is infinitesimally small. There can be no potential minimum, however, and this part may be significant.- Twriss

23 has described a process whereby the shot noise from the cathode is increased by the space-charge effects at the potential minimum instead of being reduced as in a space-charge-limited diode. Twiss's suggestion that actual oscillations, similiar to plasma oscillations, may take place between the cathode and the potential minimum seems less plausible. The mathematical difficulties discourage anyattempt to predict numerically the space-charge distribution in a cut-off magnetron; we must therefore look to experimental investigations for quantitative information. Reverdin9 has described an interesting electron-optical method to explore the magnetron space charge. Eis results show that the steady-state formation is sensitive to cathode geometry and temperature, but his data are too scanty to yield any definite quantitative information as to the effect of the various parameters of the problem. It is to be hoped that the exploring-beam experiments under way at Columbia Radiation Laboratory and at the University of Michigan will throw more light on the subject.

24 BIBLIOGRAPHY 1. Brillouin, L., "Electronic Theory of the Plane Magnetron," Advances in Electronics, New York, Academic Press, Inc., 1951, v. 3, p. 85 2. Brillouin, L. and Bloch, F., "Electronic Theory of the Cylindrical Magnetron," Advances in Electronics, New York, Academic Press, Inc., 1951, v. 5, p Ti 45 3. Hull, A. W., "Effect of a Uniform Magnetic Field on the Motion of Electrons between Coaxial Cylinders," Phys. R., v. 18, p. 31, July 1921 4. Slater, J. C., Microwave Electronics, New York, Van Nostrand, 1950 5. Page, L. and Adams, N. J., "Space Charge in a Plane Magnetron," Phys. R., v. 69, p. 492, May 1946 6. Page, L. and Adams, N. J., "Space Charge in Cylindrical Magnetron," Phys. R,, v. 69, p. 494, May 1946 7. Twiss, R. Q., On the Steady-State and Noise Properties of Linear and Cylindrical Magnetrons, Doctoral Thesis, M.I.T., 1949 8. Cohen, R. S., Spitzer, L. and Routly, P. M., "The Electrical Conductivity of an Ionized Gas," Phys. R., v. 80, p. 230, October 15, 1950 9. Reverdin, D. L., "Electron-Optical Exploration of Space Charge in a Cutoff Magnetron," J. Ap. Phys., v. 22, p. 257, March 1951

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