NAVAL POSTGRADUATE SCHOOL THESIS

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1 NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA THESIS SPACE CHARGE LIMITED EMISSION STUDIES USING COULOMB S LAW by Chritopher G. Carr June 004 Thei Advior: Second Reader: Ryan Umtattd Chri Frenzen Approved for Public Releae; Ditribution i Unlimited

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3 REPORT DOCUMENTATION PAGE Form Approved OMB No Public reporting burden for thi collection of information i etimated to average hour per repone, including the time for reviewing intruction, earching exiting data ource, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comment regarding thi burden etimate or any other apect of thi collection of information, including uggetion for reducing thi burden, to Wahington headquarter Service, Directorate for Information Operation and Report, 5 Jefferon Davi Highway, Suite 04, Arlington, VA 0-430, and to the Office of Management and Budget, Paperwork Reduction Project ( ) Wahington DC AGENCY USE ONLY (Leave blank). REPORT DATE 3. REPORT TYPE AND DATES COVERED June 004 Mater Thei 4. TITLE AND SUBTITLE: 5. FUNDING NUMBERS Space Charge-Limited Emiion Studie uing Coulomb Law 6. AUTHOR(S) Carr, Chritopher G. 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Naval Potgraduate School Monterey, CA SPONSORING /MONITORING AGENCY NAME(S) AND ADDRESS(ES) N/A 8. PERFORMING ORGANIZATION REPORT NUMBER 0. SPONSORING/MONITORING AGENCY REPORT NUMBER. SUPPLEMENTARY NOTES The view expreed in thi thei are thoe of the author and do not reflect the official policy or poition of the Department of Defene or the U.S. Government. a. DISTRIBUTION / AVAILABILITY STATEMENT b. DISTRIBUTION CODE Approved for public releae; ditribution i unlimited. 3. ABSTRACT (maximum 00 word) Child and Langmuir introduced a olution to pace charge limited emiion in an infinite area planar diode. The olution follow from tarting with Poion equation, and require olving a non-linear differential equation. Thi approach can alo be applied to cylindrical and pherical geometrie, but only for one-dimenional cae. By approaching the problem from Coulomb law and applying the effect of an aumed charge ditribution, it i poible to olve for pace charge limited emiion without olving a non-linear differential equation, and to limit the emiion area to two-dimenional geometrie. Uing a Mathcad workheet to evaluate Coulomb law, it i poible to how correlation between the olution derived by Child and Langmuir and Coulomb law. 4. SUBJECT TERMS Space Charge Limited Emiion, Coulomb Law 7. SECURITY CLASSIFICATION OF REPORT Unclaified 8. SECURITY CLASSIFICATION OF THIS PAGE Unclaified i 9. SECURITY CLASSIFICATION OF ABSTRACT Unclaified 5. NUMBER OF PAGES PRICE CODE 0. LIMITATION OF ABSTRACT NSN Standard Form 98 (Rev. -89) Precribed by ANSI Std UL

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5 Approved for public releae; ditribution i unlimited. SPACE CHARGE LIMITED EMISSION STUDIES USING COULOMB S LAW Chritopher G. Carr Enign, United State Navy B.S., United State Naval Academy, 003 Submitted in partial fulfillment of the requirement for the degree of MASTER OF SCIENCE IN PHYSICS from the NAVAL POSTGRADUATE SCHOOL June 004 Author: Chritopher Carr Approved by: Ryan Umtattd Thei Advior Chri Frenzen Second Reader/Co-Advior Jame H. Lucombe Chairman, Department of Phyic iii

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7 ABSTRACT Child and Langmuir introduced a olution to pace charge limited emiion in an infinite area planar diode. The olution follow from tarting with Poion equation, and require olving a non-linear differential equation. Thi approach can alo be applied to cylindrical and pherical geometrie, but only for one-dimenional cae. By approaching the problem from Coulomb law and applying the effect of an aumed charge ditribution, it i poible to olve for pace charge limited emiion without olving a non-linear differential equation, and to limit the emiion area to two-dimenional geometrie. Uing a Mathcad workheet to evaluate Coulomb law, it i poible to how correlation between the olution derived by Child and Langmuir and Coulomb law. v

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9 TABLE OF CONTENTS I. INTRODUCTION... A. APPLICATIONS OF VACUUM DIODES... B. DEFINITION OF SPACE CHARGE-LIMITED EMISSION... II. CHILD-LANGMUIR LAW FOR SPACE CHARGE-LIMITED EMISSION...3 A. GEOMETRY OF AN INFINITE VACUUM DIODE...3 B. DERIVATION OF CHILD-LANGMUIR LAW...4 C. LIMITATIONS OF CHILD-LANGMUIR DERIVATION...7 III. COULOMB S LAW...9 A. COMPARING POISSON S EQUATION WITH COULOMB S LAW...9 B. COULOMB S LAW GEOMETRY...0 IV. MATHCAD WORKSHEET...3 A. CHILD-LANGMUIR...3 B. COULOMB S LAW...4 C. COMPARING COULOMB S LAW AND CHILD-LANGMUIR...5 D. ADJUSTING THE COULOMB FIELD...6 E. GENERALIZING THE COULOMB INTEGRAL...7 V. CONCLUSIONS AND FUTURE WORK...9 APPENDIX... LIST OF REFERENCES...7 INITIAL DISTRIBUTION LIST...9 vii

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11 LIST OF FIGURES Figure. Diode Geometry...4 Figure. Coulomb Diode Geometry... Figure 3. Child-Langmuir Electric Field...4 Figure 4. Coulomb Electric Field...5 Figure 5. Comparion of Child-Langmuir and Coulomb Law...5 ix

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13 ACKNOWLEDGMENTS I would like to thank CAPT Umtattd and Prof. Frenzen for guiding me on thi thei, epecially given the hort window afforded by the IGEP program. Your excitement for the project wa contagiou and kept me motivated throughout. I would alo like to thank the Combat Sytem Science Department for providing an intriguing and informative equence of coure and leaving me more intereted in Phyic than before. I can never expre my appreciation to my parent for giving me the tart to thi wonderful journey I have begun and continue on. Your guidance and upport ha never failed me. To the Reno Boy, thank for great time and allowing me to keep perpective on the world outide the claroom. xi

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15 I. INTRODUCTION A. APPLICATIONS OF VACUUM DIODES Recent emphai on miile defene and the expanion of electronic warfare ha increaed a need for high-energy electromagnetic weapon. Directed energy microwave weapon require high-energy electron beam to achieve the power level adequate for tandoff utilization, which can reach the gigawatt range for peak power, which i far beyond the power production of a emiconductor device. One method of creating electron beam i through vacuum diode. In a vacuum diode, an electron cloud i created at a cathode and accelerated acro a vacuum gap to a collector anode. Thi can be done through thermionic emiion, field effect emitter, or exploive plama emitter. For the pule power requirement of high power microwave, exploive emitter enure that enough electron will be available to create current neceary for high power device within the deired turn-on time. Experimental work on diode emitter i an ongoing ubject, and i focued on diode geometrie and material. Suppreion of plama effect, cathode longevity, and the ue of micro-fiber cathode are all area of interet in the creation of fieldable high-energy weapon. However, the electromagnetic effect of large current and charge denitie in uch a diode are not entirely undertood. One effect i the creening proce from the vacuum charge of the electric field produced by the diode. B. DEFINITION OF SPACE CHARGE-LIMITED EMISSION At high current denitie, the electric field due to the electron and/or plama in the diode cannot be ignored.

16 Since the operation of thee diode i in the high current regime, thi effect on the total electric field i important. In a parallel-plate vacuum diode, the electric field produced by the diode i a function of gap ditance and the potential difference applied. The vacuum electric field i contant and uniform. In thi cae, the current would be olely limited by the capacity of the cathode to emit charge. However, experiment how that the current produced by a real diode i indeed finite, regardle of how much charge the cathode can emit. The reaon for thi i the decreae of the electric field due to the charge in the diode gap. A electron enter the diode, the charge create a field oppoite to the field applied by the diode. If emiion from the cathode i allowed to continue, the electric field of the charge can eventually cancel the diode field. At thi point, the charge emitted by the cathode are no longer accelerated away from the cathode and current i limited. Thi i pace charge-limited emiion. The current produced by the diode i not limited by the ability of the cathode to produce charge, but by the canceling effect the current ha on the total electric field in the diode. Thi limit ha been explored ince 9. However, no cloed analytic olution exit for finite diode, which of coure, all practical diode are. Since ultimately, the power of an electromagnetic weapon i limited by the power produced by the electron beam, thi limit i an important factor.

17 II. CHILD-LANGMUIR LAW FOR SPACE CHARGE-LIMITED EMISSION A. GEOMETRY OF AN INFINITE VACUUM DIODE The derivation of pace charge-limited emiion i a corner tone of plama phyic. Space charge-limited emiion i defined a driving the electric field at a cathode emitter to zero, preventing any increae in the current denity. The firt form of the well-known Child- Langmuir Law wa publihed in 9. Beginning with Poion equation and the one-dimenional boundary condition of a parallel infinite plate diode, the maximum current denity can be found. The geometry of the parallel infinite plate diode conit of an infinite grounded cathode and an anode held at ome potential, V o. The two plate of the diode are ome ditance, D, apart. Current i then allowed to flow acro the vacuum gap. When the pace charge-limited condition i met at the emiion urface, the electric field produced by the current in the gap will balance the electric field produced by the potential difference of the two plate. Thi geometry i reflected in the following diagram. 3

18 Figure. Diode Geometry B. DERIVATION OF CHILD-LANGMUIR LAW The unknown quantity i the maximum current denity, J(z,r) that will be produced by uch a diode. A reaonable tarting point i the Poion equation, relating the Laplacian of the potential, Φ, to the charge denity, ρ. ρ ( z, r) Φ( z, r) = (a) ε o For the one-dimenional cae, the potential doe not vary in the radial directional, o the Laplacian reduce to: Φ( z) ρ ( z) = z ε o (b) The current denity i alo related to the charge denity and velocity of the charge by definition. Since no 4

19 charge are being created or detroyed in thi teady-tate problem, the relation i imply: J ( z) = ρ( z) * v( z) () Then the charge denity i expreed a: J ( z) ρ ( z ) = (3) v( z) The conervation of energy i ued to expre the velocity in term of known parameter of the diode: PE ( 0) + KE(0) = PE( z) + KE( z) (4) Since the charge i aumed to be emitted with zero velocity, the kinetic energy at the cathode i zero. Since the cathode i grounded, the potential energy at the cathode i alo zero, neglecting gravitation effect. Therefore, when expreion for the potential and kinetic energy of an electron are ubtituted: 0 e * ( z) m v z Solving for velocity yield: = Φ + e * ( ) (5) e v( z) = * Φ( z) (6) m e Subtitution of the expreion for charge denity and velocity into the Poion equation reult in an expreion relating potential and current denity. The expreion i a econd-order non-linear differential equation. Φ( z) J ( z) me = * (7) z ε o e Φ( z) The boundary condition that lead to the pace chargelimited olution are the potential at the anode and the cathode, and the condition that the electric field at the cathode be zero. Φ ( 0) = 0 (8a) 5

20 Φ ( D ) = V o (8b) Φ(0) z = 0 (8c) The lat boundary condition i impoed becaue we are olving for the pace charge-limited emiion cae. Uing the preceding boundary condition, an outline of the olution to the differential equation follow. Multiplying both ide by the derivative of the potential and integrating: Φ Φ J m Φ z z ε o e z Φ e * = * * (9) The integral are of the form: ( du) u * = u * du Solving thee integral yield: (0) Φ J me * = * * Φ z ε e o + C () Here the contant of integration i zero from the boundary condition that the electric field i zero at the cathode urface. Taking the quare root of both ide: Φ z Rearranging: / 4 J me = * Φ e ε o / 4 () J me Φ / 4 Φ = * * * z ε e o / 4 (3) 6

21 Integrating thi differential equation: 4 J me Φ 3 / 4 = * * * z + C 3 ε e o / 4 (4) Again, the boundary condition of the grounded cathode force the contant to zero. By applying the anode boundary condition, one can olve for J, the Child-Langmuir Space Charge Limit for emiion denity: J 3 / o 4 e V = * (5) 9 m D cl ε o e Thi i the maximum current denity that can be produce from a parallel infinite plate diode, given a potential difference, V o, and pacing, D. Given thi current denity, ubtitution yield olution to the following quantitie: Potential: 4 / 3 z Φ( z) = Vo * (6) D Electric Field: E = 4 3 Vo * * D Charge Denity: z D / 3 (7) / 3 4 Vo ( ) * z ρ z = ε o (8) 9 D D C. LIMITATIONS OF CHILD-LANGMUIR DERIVATION Although the Child-Langmuir approach olve for the pace charge limit, it i only valid for the onedimenional cae. It ha been generalized to infinite cylindrical and pherical geometrie, but it ha not been olved for a finite emiion area. Intead of approaching the problem from the Poion equation, which generate a 7

22 non-linear differential equation, if the integral form of Coulomb law i ued, the pace charge limit can be developed by integration. 8

23 III. COULOMB S LAW A. COMPARING POISSON S EQUATION WITH COULOMB S LAW In the diode configuration for the pace charge limited formulation, it i aumed that the diode i in the teady tate condition, and doe not vary with time. With the addition of a background confining magnetic field, thee condition reduce Maxwell Equation for electrodynamic down to Gau Law: ρ ( r) E( r) = (9) ε o One reult of Gau law i the Poion equation, from which the Child-Langmuir olution for pace chargelimited flow i derived. A a definition, the electric field i the negative of the gradient of the calar potential: E( r) = Φ( r) (0) Uing the definition of the Laplacian operator: ( Φ) = Φ () Subtitution into Gau Law give the Poion Equation: ρ ( r) Φ( r) = () ε o Gau Law alo generate the integral form of Coulomb law. Subtituting the integral of a Dirac delta function into the expreion for ρ: ρ( r) 3 = 4πδ ( r r') ρ( r') dτ ' ε 4πε o o One form of the delta function i: (3) 9

24 3 ( r r') 4πδ ( r r') = 3 (4) ( r r') Subtituting for the Dirac delta function: 4πε o 3 4πδ ( r r') ρ( r') dτ ' = 4πε o ( r r') ( r r') 3 ρ( r') dτ ' (5) Since the integral in the above equation i with repect to r ', and the divergence i with repect to r, the divergence can be placed outide the integral, and the contant placed inide. From Gau law, thi i equal to the divergence of the electric field: ( r r') E( r) = ρ( r') dτ ' 3 4πε o ( r r') (6) Thi mut be hold for any r, o the argument mut be equal. Thu, Coulomb law i derived from Gau law: ( r r') E( r) = ρ( r') dτ ' (7) 3 4πε o ( r r') Therefore, Coulomb law and Poion equation are two different, but equally valid form of Gau law. The reult obtained by applying either equation to the condition of the pace charge-limited diode hould be equivalent. B. COULOMB S LAW GEOMETRY Coulomb Law in integral form i an extenion of Coulomb force law for two point charge. Evaluating Coulomb law reult in an expreion for the electric 0

25 field produced by a ditribution of charge. The integral form i: E λ ) = dq( r ) 4πε o λ λ ( rf (8) In thi cae, the electric field can vary depending on the poition where it i meaured, the field point, r f. It i alo dependent on the magnitude and poition of the charge ditribution. The vector λ i the diplacement between the ource point, r, and the field point. λ = r f r ) (9) ( The geometry i reflected in the following diagram. Figure. Coulomb Diode Geometry The differential charge element, dq, can be expreed a the product of the charge denity, ρ, and a differential volume element. In cylindrical coordinate, thi become: dq = ρ ( r ) * r dr dθ dz (30) To implify the analyi of Coulomb law, the beam can be aumed to be confined horizontally by an external

26 magnetic field, thu require only the analyi of the vertical component of the electric field. In thi cae: λ z = ( z z ) z (3) f Although the directionality of λ and E have radial and azimuthal dependence removed by the confinement aumption, it doe not change the geometry of the problem. The magnitude of λ remain: f [ in( θ )in( θ ) + co( θ )co( ] + ( z z ) λ = r + r θ rf r f f ) f (3) Now that we have found expreion for all the component, the ability of oftware uch a Mathcad to numerically olve integral become attractive.

27 IV. MATHCAD WORKSHEET A. CHILD-LANGMUIR Mathcad i a commercially available numerical calculation program. Uing Mathcad ymbolic interface, it i poible et up a workheet that compare the electric field predicted by Child-Langmuir and from the integral form of Coulomb law. If the two correpond, the flexibility of the workheet will allow for the ubtitution for variou charge ditribution geometrie. The firt input into the workheet are the global variable that control the initial geometry of the problem. For the initial purpoe of comparing Child-Langmuir and Coulomb Law, the gap ditance, D, and potential difference, V o, are contant factor, and can be et to unity to implify the analyi. The ame i true of the permittivity of free pace, ε o. The geometry of the Child-Langmuir diode i two infinite plate. The correponding parameter i an infinite radiu, R. For comparion the reult of the Child-Langmuir olution are defined. The functional form of the potential, Φ(z), i given. Then the Child-Langmuir electric field, Ecl(z), i the one-dimenional divergence, or the derivative with repect to z. The vector component of the electric field in the z-direction i denoted by a ubcript z. Ecl z (z) i plotted for comparion with the electric field calculated by Coulomb law. 3

28 0 0 Ecl z ( z) Figure 3. B. COULOMB S LAW z Child-Langmuir Electric Field The charge denity follow a the divergence of Ecl(z). Thi charge denity will be ued in the Coulomb law integral. However, a ditinction mut be made be in thi tranition between the ource point, z, and the field point, z f. In Child-Langmuir, the electric field i evaluated at the ame z a the charge denity. For Coulomb law, the charge denity i integrated over all the ource to give the field at a ditinct point. The Coulomb integral i integrated over all θ, therefore the choice of θ f i arbitrary. Chooing θ f to be zero implifie the expreion for the diplacement vector. The magnitude of the diplacement vector become: [ co( ] + ( z z ) f + r rf r ) f λ = r θ (33) A a firt check to the model, the condition of the Child-Langmuir diode are applied. The ymmetry of the infinite diode allow r f to go to zero. Thi further implifie the diplacement vector, removing azimuthal dependence. ( z z ) + f λ = r (34) The evaluation of the Coulomb integral can be plotted a a function of z. 4 D

29 E z ( z f ) z f Figure 4. Coulomb Electric Field C. COMPARING COULOMB S LAW AND CHILD-LANGMUIR Ecl z ( z) E z ( z f ) ( ) E zcorrected z f z, z f, z f D Figure 5. Comparion of Child-Langmuir and Coulomb Law 5

30 It i evident from the plot that the limit of Coulomb field differ from the limit of the Child-Langmuir field. V Both the upper and lower limit are too high by. 3 D Subtracting V 3 D from the entire Coulomb field, it i een that the Coulomb field differ from the Child-Langmuir field by thi contant. The above plot how that the Coulomb law integral and the Child-Langmuir olution differ only by a contant. D. ADJUSTING THE COULOMB FIELD That the Coulomb integral differ hould not be a urpriing reult. The integral only account for the effect of the charge in the diode gap. However, the total electric field, a predicted by Child-Langmuir, i alo influenced by the potential difference of the conducting plate. One approach to finding the effect of the conducting plate i the method of image. However, the olving for the image of a charge ditribution on two conductor that are held at different potential proved beyond the cope of thi thei. Another approach i to apply the boundary condition of the potential difference acro the gap. V o D = 0 E zcorrected ( z ) dz (35) f f Here, E zcorrected (z f ) i a um of the electric field from the pace charge, and a correction factor, the electric field produced by the conducting cathode and anode. E z ) = E ( z ) + E ( z ) (36) zcorrected ( f z f zplate f Integrating the pace charge electric field: 6

31 3 Subtracting from the total potential difference: 3 plate i: D Ez ( z f ) dz f = Vo (37) 0 D Ezplate ( z f ) dz f = Vo (38) 0 From thi, the correction to electric field from the E zplate Vo ( z f ) = (39) 3 D Thi i exactly the magnitude of the difference of the Coulomb field from Child-Langmuir. Thi formulation how a correpondence between the prediction of Child- Langmuir olving of a differential equation and integration uing Coulomb Law. E. GENERALIZING THE COULOMB INTEGRAL Thi integral formulation ha the ame limitation a the differential equation olution, in that it only hold for an infinite area diode. To be effective for olving for a finite diode, the formulation mut account for offaxi value of r f, introducing an azimuthal dependence. The generalized Coulomb integral doe not have imple ymmetrie to allow for contraction of a dimenion and i: E ( z ) = z f DRπ ρ( z ) * r f r f + r + r r f r ( z f z ) rf r [ co( θ )] + ( z f z ) [ co( θ )] + ( z z ) f * r dθ dr dz (40) In thi form, Mathcad i unable to reliably olve for E z (z f ) over the range 0 < z f < D. 7

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33 V. CONCLUSIONS AND FUTURE WORK The goal of thi thei wa to how a correpondence between the pace charge limited condition predicted from the Child-Langmuir law, and the condition predicted from Coulomb law. In thi repect, the Mathcad workheet wa ucceful. However, it hould be noted, that aumption impoed on the Coulomb integral limit the workheet to the one-dimenional, infinite diode cae. The aumption that limit the Mathcad workheet are auming azimuthal ymmetry of the charge ditribution about the field point. For a finite ditribution, thi contrain the field point to be on the centerline of the ditribution. Thi limitation doe not allow for the exploration of the edge effect of a finite beam, where the maximum current denity exceed the Child-Langmuir limit. Thee wing could vatly affect the performance of practical vacuum diode, a much more power would be produce than that predicted by the imple infinite approximation. Opportunity for future work exit in extending the workheet to allow Mathcad to olve for arbitrary geometrie. Including the azimuthal dependence would generalize the Coulomb integral, making the workheet a powerful tool for exploring the electric field of a vacuum diode. 9

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35 APPENDIX The following page are a rich text copy of the Mathcad workheet ued.

36 Global P R D V o ε o δ.00 Subcript on λ and E repreent vector t Child - Langmuir: Potential: Electric Field: z Φ ( z ) := V o D 4 3 Ecl z ( z ) Charge Denity: d := d z Φ ( z ) Ecl z ( z) 4 3 z 3 ( ):= ε o d Ecl z z z ( ) ρ z Graph of the Electric Field: d ( ) ρ z z Ecl z ( z ) z

37 Coulomb' Law: Infinite Diode Cae: Diplacement Vector: ( ) λ z z f, z, r := λ z f, z, r ( z f z ) ( r ) + ( z f z ) ( ) := ( r ) + ( z f z ) Charge Denity: ( ) ρ z 4 := 3 9z Electric Field: ( ) E z z f D R λ z z f, z, r := ε o λ z f, z, r 0 0 ( ) ρ ( z ) ( ) r dr dz θ dependence removed due to azimuthal ymmetry. E z (.0003) = E z ( z f ) z f 3

38 E zcorrected z f ( ):= E z ( z ) f V 3 D 0.5 Ecl z ( z ) E z ( z f ) ( ) E zcorrected z f z, z f, z f Space Charge Effect on Diode: D V := E z ( z f ) d.00 z f V =

39 Allow for a Finite Diode: Diplacement Vector: ( ) := r f λ r f, z f, z, r, θ ( ) λ z r f, z f, r, z, θ := + r r f r r f + r r f r ( co( θ ) ) ( z f z ) ( co( θ )) + + ( ) z f z ( ) z f z z-component of the Electric Field: E z ( r f, z f ) := ε o D 0 R 0 π 0 ( ) λ z ( r f, z f, r, z, θ ) λ( r f, z f, z, r, θ ) ρ z r dθ dr dz 5

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41 LIST OF REFERENCES. Anderon, Weton. Role of Space Charge in field emiion cathode. Journal of Vacuum Science Technologie. Volume Number. Mar/Apr Barker, Robert and Schamiloglu, Edl. High-Power Microwave Source and Technologie. IEEE Pre, New York Griffith, David. Introduction to Electrodynamic. Third Edition. Prentice Hall, Inc. New Jerey Goldten and Rutherford. Introduction to Plama Phyic. IOP Publihing, Jackon, John David. Claical Electrodynamic. Third Ed. John Wiley and Son, New Jerey, Kirtein, Kino, and Water. Space Charge Flow. McGraw-Hill, New York, Langmuir, Irving and Blodgett, Katherine. Current Limited by Space Charge between Concentric Sphere. Phyical Review Letter. Volume 4, Lau, Y.Y. Simple Theory for the Two-Dimenional Child-Langmuir Law. Phyical Review Letter. Volume 87, Number 7. December Luginland, John et al. Beyond the Child-Langmuir law: A review of recent reult on multidimenional pace-charge limited flow. Phyic of Plama, Vol. 9, Number 9, May Rokhlenko, A. and Lebowitz, J. Space-Charge-Limited D Electron Flow between Two Flat Electrode in a Strong Magnetic Field. Phyical Review Letter. Volume 9, Number 8. Augut

42 . Umtattd, Ryan and Luginland, John. Two-Dimenional Space-Charge-Limited Emiion: Beam-Edge Characteritic and Application. Phyical Review Letter. Volume 87, Number 4. October 00. 8

43 INITIAL DISTRIBUTION LIST. Defene Technical Information Center Ft. Belvoir, Virginia. Dudley Knox Library Naval Potgraduate School Monterey, California 3. Captain Ryan Umtattd, USAF. Department of Phyic Monterey, California 4. Profeor Chri Frenzen Department of Mathematic Monterey, California 5. Chairman, Phyic Department, Code PHMW Naval Potgraduate School Monterey, California 6. Engineering and Technology Curriculum Office, Code 34 Naval Potgraduate School, Monterey, California 9