We describe breakdown in rf accelerator cavities in terms of a number of mechanisms. We divide the breakdown process into three stages: 1) we model surface failure using molecular dynamics of fracture caused by electrostatic tensile stress, 2) the ionization and plasma growth is modeled using a particle in cell code, 3) we model surface damage by assuming unipolar arcing. Although unipolar arcs are strictly defined with equipotential boundaries, we find that the cold, dense plasma in contact with the surface produces very small Debye lengths and very high electric fields over a large area, and these high fields produce strong erosion mechanisms, primarily self sputtering, compatible with crater formation.We compare this model with arcs in tokamaks, plasma ablation, electron beam welding, micrometeorite impacts, and other examples

Huang, D.

Insepov, Z.

Mahalingam, S.

Norem, J.

Veitzer, S.

English

Nazarbayev University Repository

The Problem of RF Gradient LimitsJ. Norem, Z. Insepov, D. Huang, S. Mahalingam, and S. VeitzerCitation: AIP Conference Proceedings 1222, 348 (2010); doi: 10.1063/1.3399340View online: http://dx.doi.org/10.1063/1.3399340View Table of Contents: http://aip.scitation.org/toc/apc/1222/1Published by the American Institute of PhysicsThe Problem of RF Gradient LimitsJ. Norem, Z. Insepov∗, D. Huang†, S. Mahalingam∗∗ and S. Veitzer∗∗∗Argonne national Laboratory, Argonne, IL 60439, USA†Illinois Institute of Technology, Chicago, IL 60616, USA∗∗Tech-X Corp, Boulder, CO, USAAbstract. We describe breakdown in rf accelerator cavities in terms of a number of mechanisms. We divide the breakdownprocess into three stages: 1) we model surface failure using molecular dynamics of fracture caused by electrostatic tensilestress, 2) the ionization and plasma growth is modeled using a particle in cell code, 3) we model surface damage by assumingunipolar arcing. Although unipolar arcs are strictly defined with equipotential boundaries, we find that the cold, dense plasmain contact with the surface produces very small Debye lengths and very high electric fields over a large area, and these highfields produce strong erosion mechanisms, primarily self sputtering, compatible with crater formation. We compare this modelwith arcs in tokamaks, plasma ablation, electron beam welding, micrometeorite impacts, and other examples.Keywords: gradient limits, rf breakdown, arcs, unipolar arcsPACS: 52.50.Nr, 52.80.Pi, 87.56.bdINTRODUCTIONAlthough the mechanisms of arcing have been understudy for over 100 years, and are vital to the performanceof high gradient accelerators and may other applications,the field of arcing and gradient limits is not well under-stood and the basic mechanisms are still debated [1, 2, 3].Linear colliders and other energy frontier collidersrequire a level of stability and emittance growth thatseem to be provided only by weak focusing acceleratorsystems using focusing produced by iron quadrupoleyokes. This argument seems to require metal acceleratorstructures and their limitations [4].In general, gradient limits are set by a number of fac-tors. Metal structures can be limited by multipactor, butare primarily limited by arcing. Superconducting sys-tems are limited by multipactor and arcing, but, becauseof the delicate quantum mechanical nature of supercon-ductivity, many more factors such as quenching, externalnoise, imperfections in the surface, magnetic oxides andother effects can also produce gradient limits. This paperis concerned only with arcing in copper structures.We find that our breakdown model is quite general andshould also apply to other types of arcs, for example,arcing between the plasma and wall in tokamaks, laserablation, micrometeorite impacts, electron beam weldingand small gap arcs.THE BREAKDOWN MODELWe work with a model that assumes that arcing in accel-erators and most other systems consists of three stages:a) the trigger of the arc is a failure of the surface thatArc: r ~ microns        P ~ kWe Beam 1 - 10 AMWFIGURE 1. The components of the arc in cavities.may be due to a number of causes, b) field emission (FE)ionization of the fragments produced by the failure of thesurface, and c) formation of a unipolar arc that damagesthe surface [5]. We find that the arcing phenomenon issimilar in a very wide range of contexts, and may not de-pend strongly on surface polarity, frequency (DC or rf),geometry, etc. If a common arc mechanism is assumed,modeling is more highly constrained.In rf cavities arcs the pre-breakdown state can beunderstood by looking at dark currents, which can easilybe detected using x ray fluxes if the dark currents cannotbe measured directly. We have found that all cavitiesin which the measurement has been done show an E14behavior indicative of local surface fields at emitters onthe order of 10 GV/m.We assume that breakdown arc, Fig. 1, consists of thefollowing mechanisms:• Breakdown is triggered by Coulomb explosions,most probably aided by fatigue (creep at the atomicscale) and Ohmic heating due to field emitted elec-trons. Mechanical failure of the surface occurs at lo-cal surface fields of ∼ 10 GV/m [6].• Breakdown arcs are initiated by FE ionization of348fracture fragments. The field emitters will continueto operate, and ionize the material that has beenpropelled off the surface during surface failure.• The plasma density rises exponentially, the arc be-coming small, very dense, cold, and charged φ =(50-100) V to surface. The ions are confined iner-tially and electrons tend to diffuse away. The growthphase of these arcs have been extensively modeledusing OOPIC and the exponential increase in the ionand electron densities is well understood.• The small Debye lengths produced by these denseplasmas,λD =√ε0KTneq2e∼ f ew nmyield very high surface fields E = φ/λD ∼GV/m. Atthese surface fields, field emission could occur overlarge areas and ion bombardment would occur withan ion energy equal to the sheath potential, furtherheating both the plasma and the surface.• High electric fields produce micron-sized unipolararc discharges [7, 8]. The plasma requirements forproducing a unipolar arc are not known precisely,however there seems to be a minimum surface fieldand plasma density.• Unipolar arcs seem to be very efficient in convertingarc energy into surface damage (craters).The unipolar arc would continue to burn as long asthe plasma could be maintained by the cavity fields. Webelieve these unipolar arcs are similar to arcs producedin tokamaks, laser ablation, small gap arcs, and perhapsother examples.We have previously published papers on modelingthe trigger using Molecular Dynamics and calculatinggradient limits assuming a spectrum of enhancementfactors produced by arc damage [6]. This paper considersplasma calculations and the physics of the unipolar arc.Molecular DynamicsWe have modeled the initiation phase of the dischargeassuming only the electrostatic tensile stress is needed tofracture the material [6]. We assume that fatigue (creepat the atomic scale) and Ohmic heating would also con-tribute to the mechanical failure of the surface. MolecularDynamics is also used to calculate the self sputtering thatseems to control the unipolar arc parameters.It is difficult to directly couple Molecular Dynamicsand plasma codes together because they operate over dif-ferent ranges of time and space and particle density, thuscoupling the calculations must be done carefully. TheseCopper GasPointed asperitySurfaceE fieldFIGURE 2. The geometry used in OOPIC Pro plasma calcu-lations of ionization (the cone is stepped).calculations are used to evaluate specific processes suchas Coulomb explosions of surfaces and free particles, andsurface processes like self sputtering.OOPIC Pro ModelingThe formation of the plasma has been modeled usingthe 2.5D (cylindrical symmetric) OOPIC Pro [9]. Thegeometry used in the model is shown in Fig. 2, where weassume a generic field emitter is located below a cylinderof neutral copper gas, and the plasma code models thedevelopment of the plasma produced as the field emittedelectrons ionize the gas over a number of rf cycles.The development of the plasma in OOPIC Pro can beused to understand such mechanisms as how the pulsednature of the field emitted beam affects the initial prop-erties of the arc, and the minimum density of materialrequired to produce an arc.Figures 3 and 4 show output from the OOPIC Procode, giving the dependence of the the arc density onthe initial gas density (Fig. 3) and the overall numbersof field emitted electrons ad ion simulated, the ion tem-perature in the arc, and the density of line radiation pro-duced in the center of the arc (Figs. 4a, 4b, and 4c). Thepressure of neutral gas is on the order of 30 mTorr whensuccessful arcs are modeled, corresponding to approxi-mately one half of a monolayer of material injected intothe region above the field emitter. This is consistent withthe dimensions of the gas being larger than the ionizationlength for 100 eV electrons, xi = 1/niσi, where xi,ni, andσi are the ionization length, ion density and ionizationcross section.The evolution of the plasma over the first 7 ns is shownin Fig. 4a, where the density of field emitted electrons,ion density, plasma electrons, are, respectively, green,349Arc densityFIGURE 3. The dependence of arc density on the initialpressure of neutral gas.blue and yellow. The Ion temperature, shown in Fig 4b,is consistent with the very low temperature plasma onewould expect from recently ionized neutral gas in a par-tially ionized environment. Because of the space poten-tial of the plasma, ions are accelerated as they, com-paratively slowly, move away from the region wherethey were ionized, and reach energies comparable to theplasma potential when they hit the wall.The optical radiation produced by the plasma is pri-marily line radiation from the single and doubly ionizedatomic states, and is produced primarily in the visibleand near UV region. The rate of growth of this radiationis essentially proportional to the density squared.Figure 5 shows the plasma potential in the early stageof the discharge. The surface electric field is defined bythe relation E ∼ φ/λD, where φ is the plasma potential.As the density of the arc increases, the Debye length de-creases, continually increasing the surface electric field.OOPIC Pro implies that the plasma density can reach1024− 1025 / m3, where the Debye length is a few nmand the surface electric field is a few GV/m, over the en-tire region covered by the arc. Under these conditions theplasma is continually heating the surface (OOPIC Pro es-timates temperatures of 104 deg C), and both field emis-sion and ion bombardment contribute to a very large lo-cal current. We believe that this environment, primarilydefined by the plasma density, is the trigger for the theunipolar arc formation.UNIPOLAR ARC PHYSICS“The unipolar arc may be expected whenever a plasmaof sufficient density and electron temperature is in con-tact with a metal surface of sufficient area” [8]. Al-though it seems to be the primary mechanism by whicharcs can damage surfaces, there has been little studya)b)c) Line Radiation Power (t)Total Density (t)0                  1                   2                   3                  4                   5 ns                                              Time              1050050z, micronsr,  micronsionsFE electronstrapped electronssecondaryemissionelectronsIon Temperature,   eV25500                  1                 2                  3                4                 5                6 ns                                              Time              FIGURE 4. OOPIC Pro calculations of the initial 7 ns ofthe plasma arc, showing a) particle numbers simulated, b) iontemperature and, c) optical radiation.of this mechanism, theoretical or experimental, in thelast 15 years. Unipolar arcs were first proposed by Rob-son and Thonemann, but more thoroughly described bySchwirzke (Fig. 6) in the context of laser ablation andtokamak plasmas arcing to the wall, where they left char-acteristic “chicken tracks” of arc pits [7].Unipolar arcs are driven by the sheath potential andsurface electric field between the plasma and the surfaceand simulations imply that the field can be very high,producing very high self-sputtering rates and potentiallylarge field emitted currents. Sheath potentials of 70 V,combined with Debye lengths of 7 nm, give surface elec-350λDφFIGURE 5. Plasma potential calculated by OOPIC Pro in thefirst ns of the arc, the surface electric field is E ∼ φ/λD.tric fields of 10 GV/m, combined with surface tempera-ture above the melting point of copper produce an en-vironment in which no surface can survive. The envi-ronment is extreme, capable of producing MG magneticfields circulating around the the central ion and electroncurrents. At these parameters, the Debye length is nolonger thick enough to shield the sheath potential.We have calculated the self sputtering yields of cop-per for high temperature surfaces and high electric gra-dients using Molecular Dynamics, and found that abovethe melting point the self sputtering coefficient may begreater than 10 (Fig. 7). We also find that at local sur-face fields above E = 3 GV/m, the self sputtering coeffi-cients can be equally high. This, combined with the highflux of ions hitting the surface, producing exponentiallyrising temperatures, also produces both high surface ero-sion and a continuing source of atoms and ionization forfurther plasma heating and density increases. The resultis that once a plasma has been produced near a surface,all processes are strongly exothermic and likely to moveforward as fast as ion motions and processes like ioniza-tion permit. Once a given level of ionization is reached,arcing must proceed.DIFFERENT ARCING ENVIRONMENTSBecause they are rare, unpredictable, and difficult tomeasure, there is limited systematic data on arcs in rfstructures, and it seems desirable to look at arcs in othercontexts, Unipolar arcs have been identified with arcsfrom in a large number of environments: small gaps,large gaps, RF - DC, laser ablation, e beam welding,varying polarities, between a plasma and the wall of atokamak, lightswitches, micrometeorites, etc.. All these0.01.02.03.04.05.06.07.08.09.010.011.012.013.014.015.016.017.018.019.020.021.022.023.00.01.53.04.56.07.5-0.4-0.20.00.20.40.60.81.0length, micronsPotential, a. u.Radius, micronsRRIon cloudElectronCloudElectron motionLFIGURE 6. Potentials and electron motion in the classicunipolar arc as described by Schwirzke.FIGURE 7. Molecular Dynamics estimates of self sputteringfor melted copper.arcs seem to have much in common, in particular, itseems likely that the primary mechanism for damage isunipolar arcing, and differences between arcs may beprimarily differences between pathways leading to theunipolar arc. There are a number of papers exploring theparameters of these arcs in different contexts [7, 8, 10,11, 12].We believe that the behavior of unipolar arcs is cen-tral to the study of arcs wherever they occur, and find thelack of interest in this mechanism somewhat incompati-351ble with their importance. Because the Debye lengths areso short, (a few nm) the number of particles in a Debyelength can be less than or equal to one, violating one ofthe defining properties of “plasmas”, however the meth-ods of plasma physics, numerical methods and atomisticsimulations in particular, remain valid and useful.GRADIENT LIMITSWe assume that the maximum operating gradient of aparticular structure is due to an equilibrium that developsbetween the cavity damage and the arc parameters in thatstructure. An earlier paper has shown that all structuresseem to break down at a fixed value of Elocal and anyvariation in the gradient limits must be due to the maxi-mum enhancement factors in specific structures [14]. Asthe basic processes in all rf arcs follow similar mecha-nisms, and the primary difference between the particularstructures is due to the energy of the arc.Other work has shown that the surface density of as-perities with a given enhancement factor, n(β ), seem tofollow an n(β ) ∼ e−cβ distribution, where β and c arethe enhancement factor and a constant, which seems tobe roughly 0.03 [13]. If it is assumed that n(β ) is pro-portional to the energy in the arc, and the maximum en-hancement factor is determined by the constraint that thetotal number of asperities above this energy is constant,this gives the relation that βmax (and thus the maximumgradient) depends logarithmically on the energy in thearc. A more detailed analysis of gradient limits using thismodel has been published in Ref. [14].SUMMARYBreakdown and the gradient limits that result from thisprocess seem to be due to high local fields at small as-perities on the surface which mechanically fail, ionizeand, within a short time, become dense plasmas in con-tact with the wall. We assume that these plasmas even-tually become unipolar arcs and cause surface damageby melting and eroding the surface of the structure. Thismechanism may be similar to the formation of arcs inmany other environments. The damage produced lim-its the maximum gradient that can be maintained in thestructure.ACKNOWLEDGMENTSWe would like to thank A. Moretti, A. Bross and M.Zisman and the members of the Neutrino Factory andMuon Collider Collaboration (NFMCC) and the staffof Fermilab for their support. The work was supportedby the US DOE / Office of High Energy Physics underArgonne contract DE-AC02-06CH11357.REFERENCES1. R. F. Earhart, Phil. Mag. 1, 147 (1901).2. Lord Kelvin, Phil. Mag. 8, 534 (1904); also, Mathematicaland Physical Papers, Vol. VI, Voltaic theory, Radioactivity,Electrions, Navigation and Tides, Miscellaneous,Cambridge University Press, Cambridge (1911), p. 211.3. L. L. Laurent, High Gradient rf Breakdown Studies,University of California, Davis, PhD Thesis (2002).4. http://public.web.cern.ch/public/en/Research/CLIC-en.html.5. Z. Insepov, J. Norem, D. Huang, S. Mahalingam, S.Veitzer, “Modeling rf Breakdown Arcs”, Proceedingsof the PAC09 Conference, May 4-8, Vancouver,Canada(2009).6. Z. Insepov, J. H. Norem, A. Hassanein, Phys. Rev. STAccel. Beams 7, 122001 (2004).7. F. R. Schwirzke, IEEE Trans. on Plas. Sci. 19, 690 (1991).8. A. E. Robson and P. C. Thonemann, Proc. Phys. Soc. 73508 (1959).9. D. L. Bruhwiler, R. E. Giacone, J. R. Cary, J. P.Verboncoeur, P. Mardahl, E. Esarey, W. P. Leemans andB. A. Shadwick, Phys. Rev. ST Accel. Beams 4, 101302(2001).10. A. Anders, Cathodic Arcs: from fractal spots to energeticcondensation, Springer, New York (2008).11. A. Anders, S. Anders, M. A. Gunderson, and A. M.Martinovskii, IEEE Trans. on Plas. Sci. 23, 275 (1995).12. G. Federici et al., Nucl. Fus. 41, 1967 (2001).13. A. Moretti, Z. Qian, J. Norem, Y. Torun, D. Li, M.Zisman, Phys. Rev. ST Accel. Beams 8, 072001 (2005).14. A. Hassanein, Z. Insepov, J. Norem, A. Moretti, Z. Qian,A. Bross, Y. Torun, R. Rimmer, D. Li, M. Zisman, D. N.Seidman, and K. E. Yoon, Phys. Rev. ST Accel. Beams 9,062001 (2006).352

The problem of RF gradient limits

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The problem of RF gradient limits

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