Development of a Charge-Conserving Particle-in-Cell Code for Klystron Modeling

1. Jean-Yves Raguin :: Paul Scherrer Institut A fast 1D charge-conserving particle-in-cell code for the modeling of klystrons ARIES Workshop on Energy Efficient RF 18-20 June 2019, Uppsala University, Uppsala, Sweden

2. Klystrons - Overview Electrongun Input cavity Idlers cavities Output cavity Collector (*) Solenoid Typical arrangement of a high-power klystron (*) Adapted from Sprehn D. et al, Performance of a 150-MW S-Band Klystron, AIP. Conf. Proc. 337, 1995, p. 44

3. Distribution function and Vlasov equation with: and • Electrons emittedfromthegunarecharacterizedby a 7Ddistributionfunction that corresponds to a statisticalmeanofrepartitionof e- in phasespace • Properties: • Electronic density givenby • Meanvelocityverifies • Distribution functionevolvesaccordingtotherelativisticVlasov equation: • Charge and currentdensities, , sourcestermsforMaxwell’sequations

4. Macroparticles and reduction to 1D • Approximation ofthedistributionfunctionfor point-charge macroparticles: whereisthenumberof e- per macroparticle • Withthemacroparticlescharge, theparticledensityandthechargeandcurrentdensities and currentarethen: • Define linear chargeandcurrentdensities:

5. Macroparticles equations of motion and • Model assumptions: • Electronflowconfinedto longitudinal directionby infinite longitudinal focusingmagneticfieldpropagatingalongdrifttubeof radius i.e. neglectsanytransversemotion • Beam currentlow enoughtoneglect beam self-magnetic field • Beam radius constant alongthe longitudinal direction • TakingthemomentsoftherelativisticVlasov equation withtheapproximateddistributionfunction, itcanbeshownthatthemotionofthemacroparticlesisruledbytheequationsofmotion: correspondstotherfcavityfields, istheaveragedspace-charge fieldinducedby all macroparticles , mass of a macroparticle such that:

6. Normalizations, particleschargeand RF e-field particles per RF cyclewhenthereisnobunching and isthe beam current Unbunched e- beam propagates at thevelocity and RF signaltobeamplifiedhasfrequency. Normalization ofdistancestoi.e. new longitudinal coordinate: Normalization oftimesto RF period Defineparticles charge by: whereis thenumberof Normalization ofthevelocitiesto: Normalization oftheRF cavity e-fields: whereisthecavityinstantaneousvoltagenormalizedtothedepressedgunvoltage and isthenormalizedcavityprofilealongtheaxis i.e.:

7. Normalizationsandspace-charge e-field • Normalization ofthelinear electronic density: • Normalization ofthelinear charge density: • Normalization ofthecurrent density: • KnowingtheGreen’sfunctionof a unit-chargedsourcediskof radius confined in a cylindricaltubeof radius , thenormalizedspace-charge e-fieldis: with , , beingthenormalizeddrifttube radius.

8. Spatialdiscretizationandparticleshape 2 3 -2 -1 -1 Cell n° 1 -1 × × × × × × × × × with for otherwise Discretization oftheklystroninteractionspace: Celledgesandcenters: and , So far, eachmacroparticlesisconsidered as a point charge. Todefinethe linear charge at any, onereplacestheDirac distributionbyintegrableshapefunctions so that: Choice of: 1st order B-spline

9. Charge projectionto 1D numericalgrid for for otherwise forany and Let theparticlespositionsbeknownat w Deposition ofparticles charge at cellcentersand at timesleads tocell-averaged charge density: whereisthe 2nd order B-spline: with - generalsplineproperties:

10. Charge conservingcurrentdepositionprinciple Cell n° -1 -1 × × × Choice: computationofcurrent density at celledgesand at times Lettheparticlesvelocitiesbeknown at Bycomputingthedensitiesofcurrentas: thecontinuity equation discretized as isenforced

11. Charge conservingcurrentdepositionexample Depending on initial positionandvelocity, thereareno, 1 or 2 cellborderscrossedby 1 particuleduringone, each associated with different currents

12. Grid e-fields, back-interpolation, andleap-frog and and Space-charge e-field on numericalgrid at obtained bysolvingconvolution integral withdensityofcharge RF e-fields on numericalgrid at calculated from tabulated normalizedcavityfieldsandcomputationoftheinducedvoltage E-fieldsacting on eachparticlescomputedbyinterpolating e-fields on grid at locationsofparticles: Normalizedequationsofmotions: discretized in time withtheclassical leap-frog scheme:

13. Cavity model - General e-beam Idler cavities Output cavity Input cavity Klystron cavitiesmodeledby parallel RLC circuitthecurrentsourcebeinginducedby e-beam current Input cavitysources: currentgeneratorandunbunchedincoming e-beam (beam-loading) Output cavityterminatedbymatchedloadcurrent-drivenbybunched e-beam

14. Cavity model – System of differential equations withfor input cavity for input and outputcavities otherwise Time-dependentcavityvoltageand inductancecurrentruledby: or, functionsofcavity RF parameters, with Withnormalizedcavityvoltageand currentsnormalizedto with: , and , beingthestatic beam admittance

15. Convergenceacceleration and To acceleratetheconvergencetosteadystate, changethe transient behavioroftheabovesystem. Set replacementsystem: Steady-stateof initial systemgivenby:and Choosingand - ortheeigenvaluesof - anddefiningthecomplexquantity: theconstraints on ’s and’s forthesteady-stateofthereplacementsystemtobeidenticaltothe original oneare:

16. Final replacementsystemand leap-frog The abovesystemissolvedtoobtain real and Foreachcavity, voltageandcurrentsolved withtheleap-frog scheme: Inducedcurrent in cavitydefinedas: with normalized cavitye-field at Currentgeneratornormalizedtothe beam currentchosenas a sin time-domain variationand an amplitudefunctionofinput power

17. SLAC 5045 S-band klystroninputdata (*) (*) (*)from A. Jensen with AJDISK, priv.com. Jan. 2012(**)at radius 0.707𝑏

18. SLAC 5045 klystron numerical experiment (no space charge) Cavity 1

19. Cavity 2 Cavity 3

20. Cavity 4 Cavity 5

21. Cavity 6 Beam currentharmonics

22. SLAC 5045 klystron numerical experiment Charge conservation check

23. References - Numerical methods Yonezawa, H. andOkazaki, Y., A One-dimensional Disk Model Simulation For Klystron Design, SLAC-TN-84-005, 1984 - FORTRAN codeancestorof AJDISK Zambre, Y. B., Numerical simulation of transient and steady state nonlinear beam-cavity dynamics in high power klystrons, doctoral dissertation, Stanford University, 1987 Hockney R. W., and Eastwood, J. W., Computer Simulation UsingParticles, Taylor & Francis, 1988 Birdsall, C. K. andLangdon, A. B., Plasma Physics Via Computer Simulation, Adam Hilger, 1991 BouchutF., On the discrete conservation of the Gauss–Poisson equation of plasma physics, Commun. Numer. Meth. Engng., 1998 , pp. 23-34 Barthelmé, R. andParzani, C., Numerical chargeconservation in particle-in-cellcodes, in Cordier et al. Eds., Numerical Methods for Hyperbolic and Kinetic Problems, 2005, pp. 7-28 – and references within Shalaby M. et al., SHARP: A Spatially Higher-order, Relativistic Particle-in-Cell Code, Astrophys. J., 2017, pp. 1-26

24. References - Klystrons and RF cavities Warnecke R. and Guénard P., Les tubes électroniques à commande par modulation de vitesse, Gauthier-Villars, 1951 T. Lee et al., A fifty megawatt klystron for the Stanford Linear Collider, International Electron Devices Meeting, pp. 144–147, 1983 Konrad, G. T. , High Power RF Klystrons for Linear Accelerators, SLAC-PUB-3324, 1984 Vaughan, J. R. M., The input gap voltage of a klystron, IEEE Trans. Electron Devices, 1985, pp. 1172-1180 Collin, R. E., Field TheoryofGuidedWaves, 2nd Ed., IEEE Press, 1990 SprehnD. et al, Performance of a 150-MW S-Band Klystron, AIP. Conf. Proc. 337, 1995, pp. 43-49 Whittum D. H., Introduction to Electrodynamics for Microwave Linear Accelerators, in Kurokawa S. I. et al. Eds., Frontiers of Accelerator Technology, pp. 1-135, 1999 CaryotakisG., High Power Klystrons: Theory and Practice at the Stanford Linear Accelerator Center, SLAC-PUB-10620, 2004

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