Browman, M J

Chan, K C D

Cooper, R K

Loo, J

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Particle Accelerators, 1988, Vol. 23, pp. 279-287Reprints available directly from the publisherPhotocopying permitted by license only© 1988 Gordon and Breach Science Publishers, Inc.Printed in the United States of AmericaNUMERICAL STUDIES OF THE JUNGLE, GYMSLOW-WAVE ACCELERATING STRUCTUREtJ. LOO, M. J. BROWMAN, K. C. D. CHAN, and R. K. COOPERLos Alamos National Laboratory, MS-H829, Los Alamos, NM 87545(Received January 14, 1988)Many structures of interest to accelerator physicists have been inaccessible to analysis because therewas no three-dimensional electromagnetic field-solving code. This report discusses the analysis of onesuch structure, the Jungle Gym, using the recently developed MAFIA codes. The Cornell version ofthe Jungle Gym has been modeled and the dispersion curves for the TMOIo and TMo2o modescalculated and plotted. The shunt impedance (r) and r/Q for the Jr-mode were also calculated andcompared with experimental data. The calculations and experimental results were in good agreement.1. INTRODUCTIONJungle Gym is a bar-loaded traveling-wave structure for use in accelerators. Thisslow-wave structure, used in the Cornell electron synchrotron, was known to beeasy to cool and to have a reasonable shunt impedance. It was also very compactand mechanically rigid. The three-dimensional MAFIA l codes were used tosimulate this structure.2. THE MODELA unit cell begins and ends with half a pair of bars and with another pair of barsrotated 90° separating these end bars. The number of cells to model depends onthe phase shift per cell desired. This requirement is discussed in more detail in thenext section. A three-cell model is plotted by the MAFIA mesh generator in Fig.1. Only one-quarter of the actual transverse physical structure needs to begenerated because of its two-fold symmetry. Table I shows the dimensions of aunit cell.Although circular tapered bars were used in the actual structure, octagonalcylindrical bars were used for the model. An estimated average radius of 1.466 cmwas used for the bars, and the cavity radius was changed until the n-mode was at714.3 MHz; i.e., the phase velocity equaled the particle velocity (see AppendixA). The resulting cavity radius was 12.136 cm, a difference of less than 4% fromthe actual value of 12.625 cm. This small error may result from the coarseness ofthe mesh used.t Work supported by the U.S. Department of Energy.279280 J. LOO ET AL.FIGURE 1 A three-cell Jungle Gym model.3. BOUNDARY CONDITIONS AND THE DISPERSION CURVEBoth Dirichlet (tangential electric field equals zero) and Neumann (tangentialmagnetic field equals zero) boundary conditions were used in solving for thestructure. By placing Neumann boundary conditions on the sides where the cutswere taken in the nonpropagating (x and y) directions to model one-quarter ofthe cavity, the lowest-order TEmodes are eliminated.In the propagating (z) direction, to find the desired phase shifts per cell, cjJ's,for the lowes.t mode, one chooses the number of cells (n) and the boundaryconditions according to the following formulas:Desired (j)'sk:Jr/nk:Jr/nk:Jr/(2n)k0,1, , 2n-11,2, ,2n1,3, , 2n-1Boundary ConditionDirichlet- DirichletNeumann-NeumannDirichlet- NeumannFor example, in a three-cell model, the first two sets of boundary conditions(Dirichlet-Dirichlet or Neumann-Neumann) gave the modes with phase shiftsbeing multiples ofJr/3 whereas the last set (Dirichlet-Neumann) gave modeswith phase shifts being multiples of Jr/6. These phase shifts, along with theircorresponding frequencies, are shown in Table II for the lowest band.The smooth curves formed from the plot of resonant frequency versus phaseshift per cell in Fig. 2 show that the boundary conditions used appear to becorrect in modeling the slow-wave rf structure. Modes with phase shifts beingmultiples of Jr/12 were also found using six cells, and these fit perfectly on theTABLE IDimensions of the Jungle Gym ModelRadius of barsLength of bars (& radius of cavity)Spacing between barsLongitudinal length per cell1.466 em12.136 em10.493 em20.865 emSLOW-WAVE ACCELERATING STRUCTURE 281TABLE IIPhase Shift per Cell from a Three-Cell ModelFrequency(MHz)Phase Shiftper Cell Boundary Conditions617.9620.8629.4643.5662.7686.5714.3745.8780.4817.3855.6891.9911.52nIln/65n/33n/241t/37n/6n5n/62n/3n/2n/3n/6oNeumann-NeumannDirichlet- NeumannDirichlet- Dirichlet or Neumann-NeumannDirichlet- NeumannDirichlet-Dirichlet or Neumann-NeumannDirichlet- NeumannDirichlet-Dirichlet or Neumann-NeumannDirichlet- NeumannDirichlet-Dirichlet or Neumann-NeumannDirichlet-NeumannDirichlet- Dirichlet or Neumann-NeumannDirichlet- NeumannDirichlet- Dirichletcurves in this figure. The lowest band corresponds to the TMolo, whereas the nextband of modes corresponds to the TMo2o mode. The dashed curve is theTMolomode found by Tigner and is given for comparison.2According to Tigner, the upper frequency of the lowest band is givenapproximately by theTMol cut-off of an empty pipe of the same diameter. l Thiscutoff frequency for a radius of 12.136 cm is 946.1 MHz, which is approximatelythe calculated frequency of 911.5 MHz.• TM (010) CALCULATED MODE~ TM (020) CALCULATED MODE+ TM (010) MODE FROM TIGNER8508007507006506000 rr/2 rr 3rr/2 2rr1450 '--'---'-"'---'~-'--r----T--'--"---r--T"~-r-r--r----'--"""""'r--T'"--'--r---I-'14001350130012501200N-1150I~ 1100[) 1050dj 1000:::>a 950wff: 900PHASE SHIFT PER UNIT CELL (radians)FIGURE 2 The Jungle Gym dispersion curve.282 J. LOO·ET AL._Lop,ctureconLenLs,' ceLL JungLe g~" .. /round bors03/01/66 12:15:23loloLrong8pLott,ngrong8::1 -0.63 0.631:1 -0.63 0.63L,nepLoL81-f,eLdfrequenc~662.66.- 0.000~- 0.0006.008-01z/m....Lop,cturecot'lenl.,.ceLL JungL.g!j_ .. /roundborll03/01/68 12:15:23lOloLrongepLott,ngrong81:1 -0.63 0.63l:l -0.63 0.63L,ne pLotel-f,.Ld-ode no. 6frequenc~760.36.- 0.000!I- 0.000z/mFIGURE 3 The Ez line plots for the six-cell structure with Dirichlet boundary conditions.SLOW-WAVE ACCELERATING STRUCTURE 283pcctur-econtentSc. ceLL Jungle g~"' w/r-ound bor-s03/01/88 12:15:23lOlolr-ong8plotlcngr-ong8z:1 -0.63 0.63z:1 -0.63 0.63lcneplot",Od9 no. 6rr-equenc~711.31.- 0.000~- 0.0006.00_-01 8.00_-01z/m_ Lapcctur-econtentsi.. cell Jungle g~"' w/r-ound bar-s03/01/88 12:15:23lololr-ong_plOll,ngr-ong_z:1 -0.63 0.63z:1 -0.63 0.63lcneplatez-fceld",ode no. 10rr-equenc~855.60.- 0.000~. 0.000z/mFIGURE 3 (contd.)284 J. LOO ET AL.4. PHASE SHIFTS PER CELLThe E z line plots generated by the MAFIA postprocessor verified the phase shiftsper cell. Figure 3 shows some of these plots for the six-cell structure withDirichlet boundary conditions. These plots clearly indicate the phase shift of theE z field per cell. For example, in the lower left plot, there are two complete fieldpatterns or 2(2n) = 4n in six cells; therefore, this is the (4n/6) = (2n/3) mode.The values of these phase shifts per cell were also verified by having thepostprocessor do electric-field arrow plots at selected x or y planes. Figure 4shows the arrow plots of different modes at the x = 0 plane in the length of thestructure with Dirichlet boundary conditions on the ends. The darkened octagonsrepresent the ends of the loading bars for the one-quarter Jungle Gym structure,whereas ¢ indicates the phase shift per cell.Looking at one end of the structure and comparing it to the other end, we seethat for the odd multiples of n/3, the fields are completely reversed; for the evenmultiples of :It/3, the fields are pointing in the same direction, as expected.5. SHUNT-IMPEDANCE CALCULATIONSThe Jungle Gym studied was designed to operate with particle velocities at thespeed of light, essentially, and with a phase shift per cell of :It radians. TheI.. .. \ , . .1- ....... ·l-_--._, f ,,_ ........ _1__--=' • ___~ =- :J.. - I.. \ , , .. I1- - - - - - - - - -- - - - - 1r-= e__-_ - -. ~=_:::: e__~.-7tI.. , .. \ t I .. • , , , • I1- , .- - ~ " t .. - _ :::._ ~ _Il- - - - - - • - - -- - .:J- -.:21U3- - - -.. , I II.. '." - - - -.- - - ..... , , • Il- -.: -.::::: e-:...__-..::~ .:... :J•• KI3FIGURE 4 Electric-field arrow plots from the MAFIA postprocessor.SLOW-WAVE ACCELERATING STRUCTURE 285MAFIA postprocessor computes the force integrals needed to calculate the shuntimpedance.The formula used for shunt impedance per unit length for a traveling wave· (seederivation in Appendix B) isrTW=2{[J Ez cos (kz) dzf+ [J Ez sin (kz) dzfl(1)where~z is the distance along the structure over which the integration takes place,V is the maximum potential difference,(~oss)sw is the power loss through the walls, as calculated for the standing waveEz is the electric field along the z-direction,k is w/v, where w = 2n (frequency), andv is the speed of the particle.The power loss and the two integrals on the right side of the equation arecomputed by the postprocessor. The Q is also given by the postprocessor, so thatthe r/ Q values could be calculated.The values of rand r / Q calculated for each of the phase shifts per cell betweeno and 2n (Table III) came from the six-cell structure with a short on each end.This setup did not include the 2n-mode. As shown in Table III, the shuntimpedance calculated at the·n-mode was 25.4 MQ/m. To two significant digits,this is the same value measured by Tigner. The r/Q found was 1370 Q/m, whichis within 1.4% of Tigner's value of 1389 Q/m.TABLE IIIShunt Impedance (r) and r/Q from a Six-Cell ModelFrequency Phase Shift r r/Q(MHz) per Cell (M'l/m) ('l/m)617.9 2Jt620.8 11Jt/6 0.00272 0.177629.4 5Jt/3 0.0506 3.23643.5 3Jt/2 0.152 9.42662.7 4Jt/3 0.309 18.4686.5 7Jt/6 8.05 459.714.3 Jt 25.4 1370.745.8 5Jt/6 15.4 786.780.4 2Jt/3 1.90 90.2817.3 Jt/2 0.181 7.94855.6 Jt/3 1.61 63.6891.9 Jt/6 1.93 64.2911.5 0 0.349 9.562866. CONCLUSIONJ. LOO ET AL.It can be seen from this study on the Jungle Gym that by taking a simplestanding-wave model for a traveling-wave structure and using the MAFIA codeswith the correct boundary conditions, one obtains the desired traveling-waveproperties. In particular, the shunt-impedance calculations for the n-mode andthe dispersion curve for the TMo1o mode compared well with experiment.ACKNOWLEDGEMENTSWe would like to thank Dr. Maury Tigner for giving us his experimental results.His work was done at Cornell and was funded by the National ScienceFoundation.REFERENCES1. R. Klatt, F. Krawczyk, W.-R. Novender, C. Palm, T. Weiland, B. Steffen. T. Barts, M. J.Browman, R. Cooper, C. T. Mottershead, G. Rodenz, S. G. Wipf, "MAFIA-A Three-Dimensional Electromagnetic CAD System for Magnets, RF Structures, and Transient Wake-fieldCalculations," 1986 Linear· Accelerator Conference Proc., Stanford Lin~ar Accelerator CenterReport SLAC-303, 276-278 (1986).2. M. Tigner, Bar Loaded Waveguide for Accelerator Service, IEEE Transactions on NuclearScience, NS-18, No.3 (1971).APPENDIX ADerivation of the Frequency at the n-ModeThe phase velocity is defined to bewvp = {3,where w (= 2nf) is the radian frequency of the wave, and f3 is the propagationconstant equal to 2njA. For the Jt-mode, A== 2L, where L is the distance betweencells. Because we would like the phase velocity to be the same as the particlevelocity, which is essentially at the speed of light c, we set vp = c.The equation above then becomes (for the n-mode)2nfc = (2;r/A) = fA = f(2L)orCf=2L·For the Jungle Gym modeled, L = 0.2098 m; this gives f = 714.3 MHz.SLOW-WAVE ACCELERATING STRUCTUREAPPENDIX BDerivation of the Shunt-Impedance FormulaIn general, for a standing wave, the shunt impedance per unit length isR !VIz IfEzeikZdZlzrsw=-= =-----~z (~oss)sw~z (~oss)sw~z[fE z cos (kz) dzr+[fE z sin (kz) dzr(~oss)sw ~zFor a traveling wave, in general,(~oss)TW = !(~oss)sw.Therefore,2UfE z cos (kz) dzr+ [fE z sin (kz) dzr}rTW=287

Numerical Studies Of The Jungle Gym Slow Wave Accelerating structure

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