Space Charge Limited Currents Calculations in Coaxial Cylindrical Diodes Using Particle-in-Cell Simulations

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1 The Open Plsm Physics Journl, 2009, 2, Open Access Spce Chrge Limited Currents Clcultions in Coxil Cylindricl Diodes Using Prticle-in-Cell Simultions S. Mhlingm *, C. Nieter, J. Loverich, D. Smithe nd P. Stoltz Tech-X Corportion, Boulder, Colordo, USA Abstrct: Three dimensionl prticle-in-cell (PIC) simultions re conducted to mesure the spce chrge limited (SCL) currents in coxil cylindricl diodes. New cut-cell emission nd bsorption boundry lgorithms implemented in the electromgnetic PIC code, VORPAL re utilized for this study. Comprisons of VORPAL SCL current results re mde with i) theoreticl SCL current results for the one dimensionl plnr diode, ii) nlyticl solutions of SCL current results for the one dimensionl nd two dimensionl cylindricl diode cses, nd iii) n experimentl cylindricl diode operted s virtul cthode oscilltor. VORPAL SCL current results show greement within few percent with the nlyticl two dimensionl cylindricl diode results for the non-reltivistic operting voltges of kv nd within error brs with the experimentl cylindricl diode results for the diode operting voltge of 340 kv. 1. INTRODUCTION The mesurement of spce chrge limited (SCL) current density for the cylindricl diodes is useful for the development of high power microwve devices, vcuum electronics, virctor nd number of other plsm pplictions. The SCL current hs been theorized [1, 2] nd numericlly studied [3-5] well for the 1-D, nd 2-D plnr diode pplictions. Similrly theoreticl [6, 7], nlyticl [8] nd numericl studies [9, 10] were performed for mesuring the SCL current in coxil cylindricl diodes. But the coxil cylindricl diode SCL current studies were either bsed on the power series ssumption mde by Lngmuir nd Blodgett [6] nd by Pge nd Adms [7] for the 1-D cylindricl diode or bsed on the pproximte nlyticl solutions obtined for the 1-D nd 2-D cylindricl diodes [8]. Also in reference [9], 2-D PIC simultions were conducted to mesure 2-D effects of SCL currents in coxil cylinders with finite emitter width in the low voltge nd low current operting regime. Numericl PIC codes such s KARAT [11], OOPIC [12] nd number of other tools were used in the pst for the SCL current studies in both plnr diodes nd cylindricl diodes in 1-D nd 2-D. In this work, we extend the SCL current mesurement in three dimensions using VORPAL [13], which llows 3-D PIC simultions of coxil cylindricl diodes. These simultions re possible due to new chrgeconserving emission nd bsorption lgorithms, nd prt of the gol of this work is to benchmrk these lgorithms. Presently other PIC codes [12] require seprte divergence correction steps long with the electromgnetic updte steps to mintin chrge conservtion in electromgnetic simultions nd represent the curved geometries using stirstepped boundries tht mke stir-stepped emission profile. The new cut-cell emission nd bsorption lgorithms in VORPAL eliminte the requirement of seprte divergence corrector nd correctly represent the curved surfces so tht *Address correspondence to this uthor t the Tech-X Corportion, Boulder, Colordo, USA; E-mil: sudhkr@txcorp.com ccurte emission profile is obtined in PIC simultions. This SCL current study looks in to the spects of operting conditions such s the vrying gp voltge, the chrcteristic impednce (i.e., by ltering the A-K gp) nd the vrying emitter width under the non-reltivistic regime of the cylindricl diode opertion. The numericl SCL current results re compred for the cylindricl diode with the theoreticl predictions for the 1-D plnr diode, nd the nlyticl predictions for the 1-D nd 2-D cylindricl diodes. In this pper, first we present the bsic governing equtions estblished for the SCL current in plnr nd cylindricl diodes. Then we give the detils of the VORPAL, numericl PIC code tht is used for the SCL simultions, nd discuss new cut-cell emission nd bsorption lgorithms. Next, we discuss the coxil cylindricl geometry set up tht is used in this SCL study nd the computtionl prmeters used in the numericl simultions. Also we present the detils of experimentl coxil cylindricl diode operted s virtul cthode oscilltor (virctor). Then we provide the VORPAL simultion SCL current results for the coxil cylindricl diodes nd compre the VORPAL SCL current results with theoreticl, nlyticl nd experimentl SCL current results. VORPAL SCL current results show greement within few percent with the nlyticl two dimensionl cylindricl diode results for the non-reltivistic operting voltges of kv diode with A-K gp size of m nd within error brs with the experimentl cylindricl diode results for the diode operting voltge of 340 kv. 2. GOVERNING EQUATIONS The 1-D SCL current density in the plnr diode, J 1D, pl, is given by the Child-Lngmuir Lw [2] J 1D, pl = e V (1) m d 2 where 0 is the permittivity of free spce, e is the electron chrge, m is the electron mss, V is the A-K gp voltge, nd d is the A-K gp distnce. Similrly the Lngmuir / Benthm Open

2 64 The Open Plsm Physics Journl, 2009, Volume 2 Mhlingm et l. Blodgett s lw [6] gives the 1-D SCL current density in the cylindricl diode, J 1D, cyl, J 1D,cyl (r) = 4 9 2e V (r) 0 (2) m r 2 2 where is the geometry correction fctor given by the infinite series = μ 2μ μ μ +... (2) ( ), nd r c is the rdius of the cthode Here μ = ln r r c cylinder. In reference [8], Chen et l., provided pproximte nlyticl solutions for the SCL current in 1-D infinitely long cylindricl diodes nd 2-D cylindricl diodes. J 1D,Cyl = nd 2e m V (r) d 1/2 r c 1 ln r rc ( ) for r c < r, (3) J 2 D,Cyl = J 1D,Cyl 1 + 2r 2 c W 2 for W >> r c, (4) Here r is the rdius of the node cylinder nd W is the width of the cthode emission. In reference [8], reserchers found tht Equtions (2) nd (3) give identicl results. Consequently, we ve chosen to use the 1D current density given in Eqution (3) for comprisons with VORPAL cylindricl diode simultions. Also Chen et l. [8] noted tht the 2-D SCL current density solution given in Eqution (4) is vlid for lrge emission widths tht stisfy W>>r c. The chrcteristic impednce [14] of the coxil cylindricl diode is given by, Z = L C =138.0 log 10 r ( rc ) (5) where L is the inductnce per unit length nd C is the cpcitnce per unit length which re functions of the outer nd inner rdii of the coxil cylindricl diode. The unit of chrcteristic impednce is given in Ohms. Also in the SCL current studies, the SCL current density re lso reported in terms of pervence, P, which is the proportionlity coefficient between the SCL current, I, nd the A-K gp voltge V. It is given by [15], P = I V = JA s V (6) where A s is the surfce re nd the unit of pervence is A-V -. Mxwell s equtions for solving the electromgnetic fields re given by [16] E = 0, (7) B = 0, (8) B t E t = E, (9) = c 2 B j 0. (10) Here E is the electric field vector, B is the mgnetic field vector, j is the current density vector, c is the speed of light, nd t is the time. In electromgnetic PIC (EM-PIC) simultions, the mgnetic nd electric fields re obtined by solving the Frdy s eqution (Eqution (9)) nd Ampere- Mxwell s eqution (Eqution (10)). Equtions (7) nd (8) need not be solved in the EM-PIC s they re ssumed to be stisfied by the continuity equtions. In electrosttic PIC (ES-PIC) simultions, the currents re ssumed to be smll so tht no significnt self-induced mgnetic fields exist nd tht the chrge prticle densities do not chnge quickly reltive to the speed of propgtion of electromgnetic wves. So in this sitution, only Guss s lw (Eqution (7)) is solved to get electric fields. In terms of electric potentil,, the electric field is given s E =. (11) Substituting Eqution (11) into Eqution (7) gives the Poisson s eqution 2 = 0. (12) The governing equtions for the prticle motion re given by the clssicl Newton-Lorentz equtions of motion. The chrge prticle equtions of motion in the electric nd mgnetic fields re [16]: m du dt u B = q E+, (13) dx dt = v (14) where q, u, nd x re the prticle chrge, prticle reltivistic velocity vector, nd position vector respectively. The reltivistic prticle velocity is clculted using u = v (15) where the reltivistic fctor,, is = 1 + v2 c VORPAL PIC CODE VORPAL [13] is 3-D prticle-in-cell electromgnetic nd electrosttic simultion tool. VORPAL uses finite difference-time domin to solve Mxwell's equtions, but includes n dvnced technique known s embedded boundries [17] to llow ccurte representtion of curved geometries within rectngulr grids. VORPAL cn solve for

3 Spce Chrge Limited Currents Clcultions The Open Plsm Physics Journl, 2009, Volume 2 65 electromgnetic fields in full 3D, nd the embedded boundry methods mke VORPAL suited to perform PIC simultions of the SCL current study in coxil cylindricl diodes. Fig. (1) illustrtes the schemtic view of the coxil cylindricl diode simulted using VORPAL. Here the electrons re emitted from the inner cthode cylinder nd the outer cylinder is the node. Electrons re colored bsed on their reltive energies. initilized t conducting corner. Using re weighting ll of the chrge is distributed to tht sme corner. The chrge is then moved to the surfce of the conductor nd then into the vcuum in Fig. (2b). This pproch to prticle emission preserves Guss s lw in electromgnetic simultions. On the other hnd, if prticle is initilized t the cut cell surfce chrge is distributed to the four corners of the cell including corner inside the vcuum. When the prticle is finlly moved inside the domin some chrge remins t these vcuum corners resulting in unphysicl electric field build up so tht Guss s lw is no longer preserved. In high emission current simultions the field build up cn be lrge enough to destroy the solution. Fig. (1). Schemtic view of the coxil cylindricl diode simulted using VORPAL. These simultions tke dvntge of new cut cell emission nd bsorption lgorithms, nd prt of the gol of this work is to benchmrk these lgorithms Cut-cell Emission nd Absorption Algorithms New cut-cell emission nd cut-cell bsorption lgorithms [18] re implemented in VORPAL to hndle prticle emission nd prticle kill on embedded boundries. Conforml boundries re importnt for prticle emission for number of resons. First of ll, when field emitters re used it is extremely importnt to compute the correct electric fields t the boundry to obtin the correct emission current. Also, in mny situtions it s importnt to obtin uniform prticle emission from curved surfces, this cn be chllenging nd unnturl using stir stepped boundries since the emitted prticles initilly produce stir-stepped emission profile. Some conforml representtion of the boundry is necessry to produce ccurte emission profiles. Stir step meshes do hve one dvntge since it s very esy to design n emission routine tht conserves Guss s lw. Prticles re simply plced on the edge of the conductor nd moved into the solution domin in the stir step boundry cse when inverse re weighting is used. For conforml, embedded boundries it s necessry to move the prticles from region in the conductor where no chrge is distributed inside the vcuum region initilly. Fig. (2-c) show the prticle move in cut-cell geometry. In PIC codes prticles contined in cell hve their chrge distributed or weighted to computtionl nodes of the grid for the electromgnetic field updte. A very common chrge weighting scheme is re weighting whereby chrge is distributed entirely to single node of the computtionl grid if the prticle lies directly on top of tht node, but is otherwise prtilly distributed to ech of the 4 nodes of the computtionl cell tht contins the prticle (in 2D) description of re weighting cn be found in [19]. In these figures red dots indicte non-zero chrges distributed from the electrons using re weighting while blue dots represent electrons. Also shded region represents conductor while the white region represents vcuum. Fig. (2) shows prticle Fig. (2). Digrm showing prticle move in cut-cell geometry. Fig. (2, b) represent Guss s lw preserving prticle move in cutcell geometry during electromgnetic simultions. The prticle strts t conducting corner () nd is then moved to the surfce, then it s moved from the surfce into the domin (b). Strting the prticle t the cut-cell surfce (c) distributes chrge inside the domin in the beginning nd this chrge cuses unphysicl field buildup t the boundry. The field inside the conductor is 0 so it s OK to leve chrges behind in the conductor since the field equtions re not solved there. The move from the conducting corner into the vcuum llows the fields in the first cell to be updted self consistently so tht the chrge conserving properties of the prticle move re mintined inside the vcuum. On the other hnd, the chrge tht is left behind from move directly from the surfce of the cut-cell boundry is not completely eliminted during the chrge conserving move, resulting in field build up inside the domin. Absorption is performed in the reverse order. Detils of the cut-cell emission nd bsorption lgorithms re given in reference [18]. 4. SIMULATION SET UP VORPAL simultions re performed to benchmrk these new lgorithms by compring SCL currents in coxil cylindricl diode for vrying gp voltges (V ), for vrying chrcteristic cox impednces (Z) nd for vrying emitter width (W) with the theoreticl SCL current of 1-D plnr diode nd with the nlyticl solutions of SCL currents of 1- D cylindricl, nd 2-D cylindricl diodes, nd to benchmrk the VORPAL SCL current results with experimentl SCL current mesurements of coxil cylindricl diode. A cylindricl geometry with the node rdius lrger thn the cthode rdius is chosen. For the vrying voltge simultions, the node rdius is 0.01 m nd the cthode rdius is m tht gives n A-K gp of m. The emitter width is 0.02 m for both vrying A-K gp voltge study nd the vrying chrcteristic impednce study. The electric potentil of the cthode cylinder is mintined t 0

4 66 The Open Plsm Physics Journl, 2009, Volume 2 Mhlingm et l. volts nd the electric potentil of the node cylinder is vried to obtin the desired gp voltges. Seven A-K gp voltge cses 0.1 kv, 0.5 kv, 1 kv, 5 kv, 10 kv, 50 kv, nd 100 kv re considered. For both the vrying chrcteristic impednce nd vrying emitter width SCL current simultions, constnt A-K gp voltge of 1 kv is considered. Five different A-K gp distnces 0.001m, m, 0.005m, m, nd 0.009m re considered by vrying the rdius of the cthode cylinder which gives the following five cox chrcteristic impednce 6.3 Ohms, 17.2 Ohms, 41.5 Ohms, 83.0 Ohms, Ohms cses bsed on Eq. (5). Seven emitter width cses m, m, 0.01 m, 0.02 m, 0.03 m, 0.04 m, nd 0.05 m re considered for the emitter width SCL simultions nd the cthode rdius is kept t m which gives emitter width rtios (W/r c ) of 0.5, 1, 2, 4, 6, 8 nd 10. A uniform computtionl grid is chosen in which the number of grid points in the xil direction is 10, nd the number of grid points in both trnsverse directions re 50. More computtionl grid points re included in the y nd z directions to sufficiently resolve the curved surfces of the coxil cylinders. This computtionl grid hs grid resolution of 2.87x10-4 m, time step size of 0.47 ps nd Dey-Mittr frctionl time step fctor of 0.5. These computtionl prmeters re selected such tht they stisfy the numericl stbility limits such s Cournt-Friedrichs- Lewy (CFL) condition. The number of mcro prticles per computtionl cell in the simultion is 20 nd the mcro prticles re weighted with physicl prticles. Time integrtion of chrge prticle equtions of motion given in Equtions (13)-(14) is hndled using reltivistic Boris dvnce technique [19]. The cthode electron emission current density is sptilly uniform nd ll electron mcro prticles re emitted with zero kinetic energy. The electrons re bsorbed t the xil boundries. The electric fields re self-consistently modeled either using n electrosttic lgorithm or electromgnetic lgorithm, nd we compre the results from the two s n dditionl check on the lgorithms. The EM lgorithm solves the full set of Mxwell s equtions consistently without mking ny ssumptions. It includes the electric fields produced by time vrying mgnetic fields nd the mgnetic fields produced by time vrying electric fields. It lso determines the current vlues. The ES lgorithm models only the dynmic electric fields nd does not model the dynmic mgnetic fields. The ES lgorithm ssumes tht the currents re smll so tht no significnt self-induced mgnetic fields exist. Also, the chrge prticle densities do not chnge fst reltive to the speed of propgtion of electromgnetic wves. So the choice of electric solver lgorithm is bsed on the operting prmeters such s the significnce of self induced mgnetic field effects. When the electromgnetic lgorithm is used for the SCL current study, perfectly mtched lyer (PML) boundry conditions re tken t both ends long the xil direction of coxil cylinders to control the numericl reflection propgting in to the computtionl domin. Also in the electromgnetic simultions, current sources re pplied to crete the desired A-K gp electric field nd feedbck system [18] is enbled to control the pplied voltge between the A-K gp. In these simultions the voltge is clculted by integrting the electric field cross the gp, the voltge is then modified by incresing or decresing the current pplied t the boundries during the simultion. When prticle current exists, the voltge drops so the current source t the boundries compenstes for this drop in voltge nd constnt voltge is mintined throughout the simultion. Adjustment of the current source is ccomplished through feedbck clss implemented in VORPAL. In these simultions the following feedbck fctor clcultion is performed f n+1 = f n + t ( D M ) D + M ( ) f n (16) where f n is the feedbck fctor t time level n. D is the desired voltge cross the gp, M is the mesured voltge cross the gp t time level n, is the relxtion time nd t is the integrtion time step for the simultion. This feedbck fctor is then used to scle the current source t the boundries (not the prticle emission current) so tht J n = f n J 0 where J is the scled current density nd J 0 is the initil current density. An initil guess is used for the SCL current density vlues the vlue from the 1-D Child-Lngmuir's eqution given in Eqution (1). The simultion times re selected to ten times the time required for n electron to cross from the cthode cylinder to the node cylinder. The current density vlues re incresed until the pplied current density exceeds the spce chrge limit. To know whether n pplied current density is exceeding or not exceeding the coxil cylindricl diode SCL current density, ll electron mcro prticles direction of trvel long the rdil direction re checked. When the current density exceeds the spce chrge limit vlue, the creted electrons ner the cthode experience the electric field creted by the electrons before it nd they get decelerted. These decelerted electrons reflect nd return to the cthode. This cretes virtul cthode formtion ner to the cthode surfce. This limits the mount of current emission from surfce only emitting electrons. If none of the electrons re found reversing to the cthode then it cn be tken tht the pplied current density is not exceeding the SCL nd cn be tken s no formtion of virtul cthode. If this is the cse, then the specified current density is incresed nd VORPAL simultion is repeted. If more thn 1% of the totl electrons re found reversing towrds the cthode, then the pplied current density vlue cn be tken s tht it exceeds the SCL current density of the coxil cylinders. In this cse, the pplied current density vlue is lowered nd VORPAL simultion is repeted. This tril nd error process is continued until the SCL current vlue is determined for the coxil cylinders. It should be relized tht this tril nd error process could hve ccurcy error s the finl selected SCL current results could be differing within few percentge (+ or 1-2%) with the ctul SCL currents. Figs. (3, 4) show the prticle rdil phse-spce velocity results long the rdil gp between the coxil cylinders. Ech blue dot in Figs. (3, 4) represents mcro electron prticle. Fig. (3) represents the cse where the pplied current density vlue is below the SCL current of the coxil cylindricl diode. No virtul cthode formtion is seen ner the cthode rdil loction of 5 mm s none of the electrons decelerted bck to the cthode (i.e., negtive rdil velocity vlues). Fig. (4) represents the cse where the pplied current

5 Spce Chrge Limited Currents Clcultions The Open Plsm Physics Journl, 2009, Volume 2 67 density exceeds the SCL current of the coxil cylindricl diode. It is clerly shown in Fig. (4) tht virtul cthode is formed ner the cthode rdil loction of 5 mm s few of the electrons re found to hve negtive rdil velocity vlues. Fig. (3). Electron rdil phse-spce velocity results in m/s cross the rdil gp between the coxil cylindricl diode for the cse when the pplied current density does not exceed the SCL current density. The physicl prmeters re: V = 0.1 kv, d = 5 mm, r = 10 mm, r c = 5 mm, nd j = 185 A/m 2. Fig. (4). Electron rdil phse-spce velocity results in m/s cross the rdil gp between the coxil cylindricl diode for the cse when the pplied current density does exceed the SCL current density. The physicl prmeters re: V = 0.1 kv, d = 5 mm, r = 10 mm, r c = 5 mm, nd j = 200 A/m RESULTS AND DISCUSSIONS In this section, first SCL current density results for the vrying gp voltges nd vrying chrcteristic coxil impednces re presented. The SCL current density results for the coxil cylindricl diode re compred with the theoreticl 1-D plnr diode SCL current density, the nlyticl 1-D cylindricl diode SCL current density nd the nlyticl 2-D cylindricl diode SCL current density results. Then the detils of comprison of simultion with n experimentl coxil cylindricl high power diode operted s virctor re given SCL Current Study Figs. (5-7) show the SCL current density results from VORPAL PIC simultions of different A-K gp voltges, different coxil chrcteristic impednces nd different emitter widths respectively. The SCL current density vlues re incresing with incresing gp voltges nd the trend line fitted to the VORPAL results indictes tht the current density is proportionl to the voltge s J ~ V which mtches well with the SCL current density results given in Equtions (1)-(3). Fig. (6) shows tht SCL current results re decresing for incresing vlues of coxil chrcteristic impednces (up to 41.5 Ohms). But for lrger coxil chrcteristic impednces, the SCL current density vlues re found to be incresing. A smller coxil impednce mens tht the gp between the node nd the cthode cylinders is smller. As the A-K gp becomes smller compred to the cthode rdius, the coxil cylinder problem looks lmost similr to the 1-D prllel plte. The VORPAL results indicte this behvior for the smller cox impednce cse (Z=6.3 Ohms, d=0.001 m) s shown in Fig. (6), where the 1-D theoreticl current density for plnr diode is greeing closely. For the lst three cox impednce cses where the cthode rdii vlues re mde smller for rising the cox impednces hs resulted in the rise of SCL currents. The SCL current density vlues increse for the smller emitter width cses (for W/r c = 0.5 nd 1) nd mostly remin unchnged for ll other emitter width cses tested. Chen et l. [8] hd noted tht the 2-D nlyticl solution for the cylindricl diode given in Eqution (4) is vlid for W >> r c. This condition is observed to be true in Fig. (7) where the 2- D nlyticl solution gives erroneous SCL current results for the smller emitter width cses. VORPAL simultions disgree with the 1-D cylindricl infinite diode of Eq. 3 by 13-36%. This discrepncy between VORPAL nd the 1-D cylindricl infinite diode solution likely due to the finite length of the simultion domin considered. At low emitter width cses, VORPAL results re greeing closer to the 1-D cylindricl infinite diode. In this emitter width rtio study, the A-K gp is m sme s the cthode rdius. Since both the smller emitter width vlues nd the smller A-K gp mke the VORPAL results closely greeing to the 1-D cylindricl infinite diode solution for the low emitter width cses. If lrger A-K gp is picked for the emitter width study, VORPAL SCL current results could be differing from the 1-D cylindricl infinite diode SCL solutions. Fig. (8) shows the rtio of the computed SCL currents nd i) the theoreticl SCL current for the 1-D plnr diode (Eq. 1), ii) n nlyticl solution of SCL current for the 1-D cylindricl diode, (Eq. 3) nd iii) n nlyticl solution of SCL current for the 2-D cylindricl diode (Eq. 4). For this study, the cthode emitter width is tken s 0.02 m nd the A-K gp is tken s m. The rtio of the simulted current results nd the 1-D plnr diode results sty within few percent of 2.0 for ll the A-K gp voltges studied. The 1-D cylindricl infinite diode results re roughly 10-20% below the simulted results over the voltge rnge considered. This discrepncy is likely due to the finite length of the simultion domin, s the rtio of VORPAL result nd

6 68 The Open Plsm Physics Journl, 2009, Volume 2 Mhlingm et l. Fig. (5). VORPAL SCL current density results for the vrying A-K gp voltges of coxil cylindricl diode. Both xes re given in logrthmic scle. VORPAL results indicte tht the SCL current density vlues re incresing with incresing gp voltges (s J ~ V ). Fig. (7). VORPAL SCL current density results for the vrying emitter widths of coxil cylindricl diode. The A-K gp voltge used for this study is 1 kv. VORPAL simultions gree within 1-10% with the 2D cylindricl theory of Eq. 4 for W/r c >4.0. We find tht for W/r c <2.0, VORPAL simultion results nd Eq. 4 begin to disgree by fctors of 2 or more. VORPAL simultions disgree with the 1-D cylindricl infinite diode of Eq. 3 by 13-36%. Fig. (6). VORPAL SCL current density results for the vrying coxil chrcteristic impednces of coxil cylindricl diode. The A-K gp voltge used for this study is 1 kv. As the A-K gp becomes smller compred to the cthode rdius (i.e., smller cox impednce vlues), the coxil cylinder problem looks lmost similr to the 1-D prllel plte. The VORPAL results indicte this behvior for Z=6.3 Ohms cse (d=0.001 m, r c = m) where the 1-D theoreticl plnr diode of Eq. 1 grees within 5%. the nlyticl solution of 2-D cylindricl diode re between The dips observed in Fig. (8) on ll SCL current rtio curves for the regions between kv nd kv could be due to the ccurcy error of the tril nd error process used for the SCL current determintion Experimentl Cylindricl Diode SCL Current Study As finl benchmrk of the simultion results, we compre experimentl results from coxil cylindricl diode [20]. In reference [20], the uthors present experimentl studies on high-power coxil cylindricl diode used for producing intense reltivistic electron flow in virctor regime. They present pervence dt for the diode Fig. (8). SCL current rtio results for vrying A-K gp voltges. VORPAL simultions significntly differ from the 1-D plnr diode theory of Eq. 1 by more thn 100% nd differ from the 1-D cylindricl infinite diode theory of Eq. 3 by %. However they closely gree within 5% with the 2-D cylindricl theory of Eq. 4 tht is likely due to the finite length of the diodes considered in simultions. operting under different input voltges nd different nodecthode (A-K) gp vlues. We simulted this experimentl cylindricl diode set up with geometry of inner dimeter of m (r c = m) nd the copper node mesh cylinder hs dimeter of m (r = 0.043m). This gives n A-K gp of m. The simulted emission width (W) on the cthode cylinder is 0.02 m, nd the diode voltge is 340 kv. A uniform computtionl grid tht hs 20 grid points in the x-direction nd 50 grid points in both y nd z directions is used. This gives 3-D grid spcing vlue of 1.33 mm. The electric fields re self-consistently modeled using the electrosttic lgorithm. The time step size is 0.88 ps, which

7 Spce Chrge Limited Currents Clcultions The Open Plsm Physics Journl, 2009, Volume 2 69 ensures tht the electrons do not cross more thn one computtionl cell in given time step. All other simultion prmeters re sme nd similr procedure is followed to determine the SCL current density s pplied in the previous SCL current simultions. For these prmeters, the SCL current density is found to be 1.2x10 6 A/m 2. This gives pervence result of 4.68x10-5 A/V. The experimentl pervence vlue from the figure given in reference [20] is between 3.5x x10-5 A/V. Thus the VORPAL numericl result is within the experimentlly mesured rnge. 6. CONCLUSIONS In this work 3-D PIC simultions using the VORPAL code re conducted to study the SCL current in coxil cylindricl diodes, using new chrge-conserving emission nd bsorption lgorithms for embedded boundries. The results gree within few percent with the nlyticl solutions of SCL current results given for the 2-D cylindricl diodes for the non-reltivistic diode operting voltges of kv nd A-K gp size of m. Finlly, we show tht results re within experiment error brs when compring with published pervence mesurements for high-power cylindricl diode operting t diode voltge of 340 kv. ACKNOWLEDGEMENTS The uthors would like to thnk Tech-X Corportion for the finncil support of this project. Also the uthors would like to thnk the VORPAL Tem. REFERENCES [1] Child CD. Dischrge from hot CO. Phys Rev 1911; 32: 492. [2] Lngmuir I. The effect of spce chrge nd initil velocities on the potentil distribution thermionic current between prllel plne electrodes. Phys Rev 1923; 21: 419. [3] Wtrous JI, Luginslnd JW, Ssser III GE. Current nd current density of finite-width, spce-chrge-limited electron bem in two-dimensionl, prllel-plte geometry. Phys Plsms 2001; 8: 289. [4] Umstttd RJ, Luginslnd JW. Two-dimensionl spce-chrgelimited emission: bem-edge chrcteristics nd pplictions. Phys Rev Lett 2001; 87: [5] Lu YY. Simple theory for the two-dimensionl Child-Lngmuir lw. Phys Rev Lett 2001; 87: [6] Lngmuir I, Blodgett K. Currents limited by spce chrge between coxil cylinders. Phys Rev 1923; 22: 347. [7] Pge L, Adms NI. Spce Chrge between coxil cylinders. Phys Rev 1945; 68: 126. [8] Chen X, Dickens J, Htfield LL, Choi E, Kristinsen M. Approximte nlyticl solutions for the spce-chrge-limited current in one-dimensionl nd two-dimensionl cylindricl diodes. Phys Plsms 2004; 11: [9] Kostov KG, Brroso JJ. Spce-chrge-limited current in cylindricl diodes with finite-length emitter. Phys Plsms 2002; 9: [10] Wheeler CB. The pproch to spce chrge limited current flow between coxil cylinders. J Phys A Mth Gen 1975; 8: 555. [11] Trknov VP. User s Mnul for Code KARAT. Springfield, VA: Berkeley Reserch Assocites [12] Verboncoeur JP, Lngdon AB, Gldd NT. An object-oriented electromgnetic PIC code. Comp Phys Commun 1995; 87: 199. [13] Nieter C, Cry JR. VORPAL: verstile plsm simultion code. J Comp Phys 2004; 196: [14] Elmore WC, Held MA. Physics of Wves. New York: Courier Dover Publictions [15] Cho AW, Tigner M. Hndbook of Accelertor Physics nd Engineering. World Scientific 1999; p [16] Griffiths DJ. Introduction to Electrodynmics, 2 nd ed. Englewood, NJ: Prentice Hll [17] Dey S, Mittr R. A loclly conforml finite-difference timedomin (FDTD) lgorithm modeling three-dimensionl perfectly conducting objects. IEEE Microw Guid Wve Lett 1997; 7: 273. [18] Loverich J, Nieter C, Mhlingm S, Smithe D, Stoltz P. Chrge conserving emission from conforml boundries in electromgnetic PIC simultions. Comp Phys Commun 2009; (to be submitted). [19] Birdsll CK, Lngdon AB. Plsm physics vi computer simultion. 1 st ed. Bristol: Adm Hilger [20] Roy A, Menon R, Mitr S, et l. Intense reltivistic electron bem genertion nd prepulse effect in high power cylindricl diode. J Appl Phys 2008; 103: Received: Februry 25, 2009 Revised: April 14, 2009 Accepted: April 17, 2009 Mhlingm et l.; Licensee Benthm Open. This is n open ccess rticle licensed under the terms of the Cretive Commons Attribution Non-Commercil License ( which permits unrestricted, non-commercil use, distribution nd reproduction in ny medium, provided the work is properly cited.