We present a theory that allows us to accurately calculate the distribution
functions and the emittance growth of a thermal charged-particle beam after it
relaxes to equilibrium. The theory can be used to obtain the fraction of
particles, which will evaporate from the beam to form a halo. The calculated
emittance growth is found to be in excellent agreement with the simulations.Comment: 3 pages, 3 figure

Levin, Yan

Pakter, Renato

Teles, Tarcísio Nunes

English

arXiv

We present a theory that allows us to accurately calculate the distribution functions and the emittance growth of a thermal charged-particle beam after it relaxes to equilibrium. The theory can be used to obtain the fraction of particles, which will evaporate from the beam to form a halo. The calculated emittance growth is found to be in excellent agreement with the simulations

Teles, Tarcisio Nunes

Lume 5.8

Relaxation and emittance growth of a thermal charged-particle beamTarcísio N. Teles, Renato Pakter, and Yan Levin  Citation: Applied Physics Letters 95, 173501 (2009); doi: 10.1063/1.3254245 View online: http://dx.doi.org/10.1063/1.3254245 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/95/17?ver=pdfcov Published by the AIP Publishing  Articles you may be interested in Studies of emittance growth and halo particle production in intense charged particle beams using the PaulTrap Simulator Experimenta) Phys. Plasmas 17, 056707 (2010); 10.1063/1.3354109  Erratum: “Warm-fluid equilibrium theory of a thermal charged-particle beam in a periodic solenoidal focusingfield” [Phys. Plasmas14, 103102 (2007)] Phys. Plasmas 15, 029901 (2008); 10.1063/1.2839353  Warm-fluid equilibrium theory of a thermal charged-particle beam in a periodic solenoidal focusing field Phys. Plasmas 14, 103102 (2007); 10.1063/1.2779281  Streak Camera 101: Visualizing Charged‐Particle Beam Dynamics AIP Conf. Proc. 868, 124 (2006); 10.1063/1.2401398  Transverse distribution of density in charged-particle beams at later times Phys. Plasmas 4, 3692 (1997); 10.1063/1.872264    Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP:  143.54.44.137 On: Thu, 05 May2016 18:58:44Relaxation and emittance growth of a thermal charged-particle beamTarcísio N. Teles,a Renato Pakter,b and Yan LevincInstituto de Física, UFRGS, Caixa Postal 15051, CEP 91501-970 Porto Alegre, Rio Grande do Sul, BrazilReceived 10 July 2009; accepted 6 October 2009; published online 26 October 2009We present a theory that allows us to accurately calculate the distribution functions and theemittance growth of a thermal charged-particle beam after it relaxes to equilibrium. The theory canbe used to obtain the fraction of particles, which will evaporate from the beam to form a halo. Thecalculated emittance growth is found to be in excellent agreement with the simulations. © 2009American Institute of Physics. doi:10.1063/1.3254245The understanding of physics involved in the transportof high-intensity charged-particle beams is of fundamentalimportance in the development of a new generation of accel-erators and electromagnetic wave generators to be used inapplications such as heavy ion fusion, high-energy physics,communication, materials processing, and cancer therapy. Avery detrimental effect that may seriously influence the effi-ciency of such devices is a halo formation and emittancegrowth of the beam.1–5 These not only cause degradation ofthe beam quality but may also be responsible for the activa-tion of accelerator channel wall and pulse shortening in mi-crowave devices. Emittance growth is generally associatedwith the relaxation of initially nonstationary beam toward amore stable stationary configuration. The emittance growthcan be calculated if the final stationary distribution isknown.3,6,7 However, the determination of this distribution isnot an easy task6,8,9 because particles in an intense beaminteract through long-range forces, which prevent the systemfrom relaxing to the true thermodynamic equilibrium.10–12Instead these systems get trapped in metastable states, thelifetime of which diverges with the number of particles. Tounderstand the properties of these states, one cannot use thestandard statistical mechanics, and new nonequilibrium theo-ries must be developed.13In this letter, we will present a theoretical frameworkthat will allow us to accurately calculate the density and thevelocity distributions of particles in the final stationary stateachieved by a space-charge dominated beam focused by auniform external magnetic field. Our approach is based onthe theory of violent relaxation in gravitational systems,14modified so as to explicitly account for the effects of singleparticle resonances1 responsible for the halo formation.13 Thetheory is applicable to arbitrary initial conditions. In this let-ter we will show how the theory can be used to accuratelycalculate the density and the velocity distributions as well asto account for the emittance growth of a charged-particlebeam launched with a thermal Maxwell velocity distribu-tion. The predictions of the theory will be tested against themolecular dynamics simulations.The physical system considered here is an intensecharged-particle beam of perveance K=2q2Nb /b3vz2m—where c is the speed of light in vacuo, and q, m, andb= 1− vz /c2−1/2 are the charge, mass, and the relativisticfactor of the beam particles, respectively—propagating withan axial velocity vzêz through a magnetic focusing channelenclosed by a cylindrical conducting wall located at r=rw.8,15The external focusing magnetic field is given by B=B0êz. Itis convenient to work in the Larmor frame, which rotateswith respect to the laboratory frame with angular velocityL=qB0 /2bvzmc, normalized to vz. In the Larmor frame,the external magnetic field produces a parabolic confiningpotential Ur=zr2 /2, with the focusing field parameter z=L2 /c2. The effective electromagnetic scalar potential be-tween the particles  incorporates both the self-electric andthe self-magnetic fields, Es and Bs. This potential satisfiesthe Poisson equation with the boundary condition rw=0,2 = − 2K/Nbnbr,s , 1where Nb is the number of particles per unit axial length, r isthe position vector in the transverse plane, and nbr ,s=Nbfd2v is the transverse beam density profile, given interms of the one particle distribution function fr ,v ;s. Inthe Larmor frame, the dynamics of the beam reduces to thatof a two dimensional one component plasma with logarith-mic interaction between the particles, confined by a parabolicpotential Ur. The axial coordinate s=z=vzt plays the roleof time for this two dimensional system.We will suppose that the initial transverse distributionof the beam is Gaussian in velocity space and is uniform incross section,f0r,v =1222rm2 rm − re−v2/22, 2where 2 is the initial mean square transverse velocity and rmis the beam radius. The quality of the beam is inverselyproportional to the emittance, defined as 2=4r2v2, for astationary beam. For the distribution 2, the emittance is0=2rm.It will be convenient to discretize Eq. 2 into a p-leveldistributionfp0r,v = j=1p	 j j0r,v , 3where  j0r ,v	v−v j−1v j −vrm−r and v j and 	 jare the maximum velocity and the amplitude of the level j,respectively, with v0=0. For a perfect description of Eq. 2,an infinite number of levels p→ in Eq. 3 will be nec-essary. In practice, however, we find that a small number oflevels are already sufficient to provide a very accurate ap-aElectronic mail: teles@if.ufrgs.br.bElectronic mail: pakter@if.ufrgs.br.cElectronic mail: levin@if.ufrgs.br.APPLIED PHYSICS LETTERS 95, 173501 20090003-6951/2009/9517/173501/3/$25.00 © 2009 American Institute of Physics95, 173501-1 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP:  143.54.44.137 On: Thu, 05 May2016 18:58:44proximation for the beam dynamics. For a given value of p,the optimal values of 	 j and v j can be obtained by minimiz-ing the functional F=f0− fp02d2rd2v, with the constraintson the kinetic energy and normalization,F + 1 v22 fp0d2rd2v − 2 + 2 fp0d2rd2v − 1= 0, 4where 1 and 2 are the two Lagrange multipliers. Minimi-zation of Eq. 4 yields the optimal parameters 	 j and v j.The many-body dynamics of systems with unscreened long-range interaction is governed by the collisionless BoltzmannVlasov equation. The distribution functions that satisfy theVlasov equation evolve in time as the density of an incom-pressible fluid. In particular this means that the p hypervol-umes, 	 j=fp0r ,v−	 jddrddv, of the distribution 3will be preserved by the Vlasov flow.14In Ref. 14 it was argued that the stationary solution ofthe Vlasov equation could be obtained by maximizing thecoarse grained entropy, with the constraints imposed by theconservation of energy and the hypervolumes of the p levelsof the initial distribution function. For matched beams andwater-bag initial conditions, the resulting distribution wasshown to be in excellent agreement with the molecular dy-namics simulations.13 However, for mismatched beams, theplasma oscillations result in parametric resonances, whichlead to a significant particle evaporation. After the relaxationprocess is complete, the stationary beam phase separates intoa cold core, surrounded by a halo of highly energetic par-ticles. For a water-bag initial condition p=1, it was shownthat the core was very well described by a cold Fermi–Diracdistribution with the temperature TTF /40, where TF is the“Fermi temperature” of the beam. The halo was reasonablyapproximated by a step function with energy range of oneparticle resonance. The full distribution function had theform offr,v = fcr,v + fhr,v . 5For a p-level system, which is used to approximate the ther-mal distribution given by Eq. 2, a similar phase separationwill occur. The form of the core distribution function can beobtained, once again, by maximizing the coarse grained en-tropy to yieldfcr,v = j=1p	 j −  jr,v , 6with jr,v =e−	jr,v+ji=1p e−	ir,v+i + 1, 7where the mean particle energy is r ,v=v2 /2+Ur+rand  and  j are the Lagrange multipliers for the energyand the hypervolumes conservation. The oscillations of themismatched beam excite the parametric resonances, resultingin a halo formation.13 The parameter  determines the frac-tion of the particles, which will evaporate to form the halo ofthe beam. The coarse grained distribution can no longer pre-serve all the hypervolumes of the original fine-grained dis-tribution function so that only the lower energy hypervol-umes will be conserved, while the particles from the higherenergy states will evaporate to form a halo. We find that thehalo can be modeled accurately by the distributionfhr,v =  − r,v + r,v − R− r,ve−−. 8The extent of the halo1 is up to one particle resonance energyR. The low energy part of the halo distribution is flat, whilefor energies =R /2, it decays exponentially with expo-nent 8. We can now, in principle, numerically solve Eqs.1, 5, 7, and 8 to calculate the stationary distributionfunction fr ,v of the relaxed beam. There is, however, oneproblem. Equations 7 and 8 contain p+2 parameter: , j, and . The conservation of energy, norm, and lowerenergy hypervolumes gives us p+1 additional equations, d2rd2vr,vfr,v = 0, d2rd2vfr,v = 1, d2rd2v jr,v = d2rd2v j0r,v , 9where 1 j p−1 and 0 is the average energy per particleof the initial thermal distribution,0 = 2 + zrm24+ K18−12ln rmrw . 10There, however, still remains one missing condition neces-sary to uniquely determine the distribution function. Forwater-bag distributions, this condition was provided by therequirement that in the relaxed state, the core temperature isvery low, TTF /40. It is difficult, however, to numericallyimplement this condition for p-level distributions. On theother hand, if we discretize the original thermal distributioninto only one level p=1, the condition TTF /40 is easilyimplemented and allows us to uniquely close all the equa-tions and calculate the relaxed distribution function.13 Wefind that although the core distribution is not well describedby a p=1 system, the halo part of the distribution is found tobe quite accurate. This allows us to fix the value of . Usingthis , we can now improve the description of the core re-gion by including additional levels into the discretizationprocedure. To compare the predictions of the theory with thesimulations, we calculated the number of particles in the in-terval r ,r+dr, Nrdr=2Nbrdrd2vfr ,v, and the num-ber of particles with velocities between v and v+dv,Nvdv=2Nbvdvd2rfr ,v, for various initial conditions.The simulations are based on the Vlasov dynamics in whichparticles interact with the mean-field potential. This avoidsthe collisional effects present in finite size systems but whichmust vanish for one component plasmas in thermodynamiclimit. The simulation code uses the Gauss law to calculatethe mean electric field felt by each particle.7 For axisymmet-ric beams studied in this work, this proves to be very effi-cient since the electric field at a radial coordinate r is deter-mined simply by counting the total number of particles withcoordinates smaller than r. Simulations were performed with20 000 particles. As can be seen from the Figs. 1 and 2, theagreement between the theory and the simulations is excel-173501-2 Teles, Pakter, and Levin Appl. Phys. Lett. 95, 173501 2009 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP:  143.54.44.137 On: Thu, 05 May2016 18:58:44lent. In the figures, the distances are measured in units of0 /L, and the velocities are in units of 0L. We havealso defined a scaled perveance K	K /L0 and the mis-match parameter, 	rm /r0, which measures the deviation ofthe initial beam radius from the corresponding virial valuer0=K+42 /L, for which the oscillations of the beam en-velope are very small. In particular, we find that the discreti-zation of the Gaussian by only four levels already providesus with an almost perfect description of the core region.As a direct application of the theory developed above,we calculate the emittance growth of an originally thermalbeam. This quantity is of fundamental importance for thedesign and development of high-intensity space-chargedominated beams.16 The calculations are performed forbeams of varying scaled perveance K and mismatch param-eter . The results are compared with the molecular dynam-ics simulations. Once again, an excellent agreement is foundbetween the theory and the simulations Fig. 3.To conclude, we have presented a theory that allows usto calculate the density and the velocity distributions of aninitially thermal beam after it relaxes to the final stationarystate. Comparing to the simulations, the theory is found to beextremely accurate without any adjustable parameters. Inparticular, it can be used to calculate the emittance growthand the fraction of particles, which will evaporate as thebeam evolves to its final stationary state.This work was supported by CNPq, FAPERGS, andINCT-FCx of Brazil and by the Air Force Office of ScientificResearch AFOSR, USA, under Grant No. FA9550-09-1-0283.1R. L. Gluckstern, Phys. Rev. Lett. 73, 1247 1994.2S. Banna and L. Schächter, Appl. Phys. Lett. 80, 2842 2002.3C. K. Allen, K. C. D. Chan, P. L. Colestock, K. R. Crandall, R. W. Garnett,J. D. Gilpatrick, W. Lysenko, J. Qiang, J. D. Schneider, M. E. Schulze, R.L. Sheffield, H. V. Smith, and T. P. Wangler, Phys. Rev. Lett. 89, 2148022002.4Yu. Chekh, A. Goncharov, I. Protsenko, and I. G. Brown, Appl. Phys. Lett.86, 041502 2005.5P. Muggli, B. E. Blue, C. E. Clayton, F. J. Decker, M. J. Hogan, C. Huang,C. Joshi, T. C. Katsouleas, W. Lu, W. B. Mori, C. L. O’Connell, R. H.Siemann, D. Walz, and M. Zhou, Phys. Rev. Lett. 101, 055001 2008.6M. Reiser, J. Appl. Phys. 70, 1919 1991.7R. P. Nunes, R. Pakter, and F. B. Rizzato, J. Appl. Phys. 104, 0133022008; Phys. Plasmas 14, 023104 2007.8R. C. Davidson and H. Qin, Physics of Intense Charged Particle Beams inHigh Energy Accelerators World Scientific, Singapore, 2001.9J. Zhou, K. R. Samokhvalova, and C. Chen, Phys. Plasmas 15, 0231022008.10T. Padmanabhan, Phys. Rep. 188, 285 1990.11P.-H. Chavanis, Physica A 359, 177 2006.12R. Bachelard, C. Chandre, D. Fanelli, X. Leoncini, and S. Ruffo, Phys.Rev. Lett. 101, 260603 2008.13Y. Levin, R. Pakter, and T. N. Teles, Phys. Rev. Lett. 100, 040604 2008.14D. Lynden-Bell, Mon. Not. R. Astron. Soc. 136, 101 1967.15M. Reiser, Theory and Design of Charged Particle Beams Wiley, NewYork, 1994.16T. P. Wangler, K. R. Crandall, R. Ryne, and T. S. Wang, Phys. Rev. STAccel. Beams 1, 084201 1998.00.20.40.60.810 1 2 3 4N(r)/Nbr10-410-310-310-210-12 3 4FIG. 1. The relaxed particle density of an initially thermal beam with scaledperveance K=1 and mismatch of 75% =1.75. The points are the resultsof the simulations, and the solid line is the prediction of the theory. Insetshows the exponential decay of the halo close to one particle resonanceenergy.00.20.40.60.810 1 2 3 4N(r)/Nbr(a)00.40.81.21.60 1 2 3 4N(v)/Nbv(b)FIG. 2. The density and velocity distributions: solid line is the prediction ofthe theory, and points are the results of the molecular dynamics simulation.The scaled perveance is K=1 and mismatch is 50% =1.50.0510152025301.4 1.6 1.8 2 2.2 2.4 2.6ε/ε 0µK*=1K*=5K*=10FIG. 3. The final emittance of an initially thermal beam. The points are theresult of molecular dynamics simulation, and the lines are the predictions ofthe theory.173501-3 Teles, Pakter, and Levin Appl. Phys. Lett. 95, 173501 2009 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP:  143.54.44.137 On: Thu, 05 May2016 18:58:44

Applied Physics Letters

Relaxation and emittance growth of a thermal charged-particle beam

Tarcísio N. Teles

Renato Pakter

Yan Levin

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Universidade Federal do Rio Grande do Sul

https://lume.ufrgs.br/bitstream/10183/141745/1/000730295.pdf

Relaxation and Emittance Growth of a Thermal Charged-Particle Beam

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Lume 5.8

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Universidade Federal do Rio Grande do Sul