Comptes Rendus
Theoretical and numerical approaches for Vlasov–Maxwell equations
Case studies in space charge and plasma acceleration of charged beams
Comptes Rendus. Mécanique, Volume 342 (2014) no. 10-11, pp. 647-661.

Plasma acceleration with electron or proton driver beams is a challenging opportunity for high-energy physics. An energy doubling experiment with electron drivers was successfully performed at SLAC and a key experiment AWAKE with proton drivers is on schedule at CERN. Simulations play an important role in choosing the best experimental conditions and in interpreting the results. The Vlasov equation is the theoretical tool to describe the interaction of a driver particle beam or a driver laser pulse with a plasma. Collective effects, such as tune shift and mismatch instabilities, appear in high intensity standard accelerators and are described by the Poisson–Vlasov equation. In the paper, we review the Vlasov equation in the electrostatic and fully electromagnetic cases. The general framework of variational principles is used to derive the equation, the local form of the balance equations and related conservation laws. In the electrostatic case, we remind the analytic Kapchinskij–Vladimirskij (K–V) model and we propose an extension of the adiabatic theory for Hamiltonian systems, which ensures stability for perturbation of size ϵ on times of order 1/ϵ. The variational framework is used to derive the Maxwell–Vlasov equations and related conservation laws and to briefly sketch the particle-in-cell (PIC) approximation schemes. Finally, the proton-driven acceleration is examined in the linear and quasi-linear regime. A PIC simulation with the code ALaDyn developed at Bologna University is presented to illustrate the longitudinal and transverse fields evolution which allow a witness electron bunch to be accelerated with a gradient of a few GeV/m. We also present some remarks on future perspectives.

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Accepté le :
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DOI : 10.1016/j.crme.2014.07.004
Mots clés : Plasma acceleration, Vlasov equation, Stationary distributions, PIC simulations

Armando Bazzani 1, 2 ; Massimo Giovannozzi 3 ; Pasquale Londrillo 2 ; Stefano Sinigardi 1, 2, 4 ; Giorgio Turchetti 1, 2

1 Dipartimento di Fisica e Astronomia, Università di Bologna, Via Irnerio 46, 40126 Bologna (BO), Italy
2 INFN Sezione di Bologna, Via Irnerio 46, 40126 Bologna (BO), Italy
3 BE Department, CERN, CH-1211 Geneva 23, Switzerland
4 Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche (CNR/INO), U.O.S. ‘Adriano Gozzini’, Pisa, Italy
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Armando Bazzani; Massimo Giovannozzi; Pasquale Londrillo; Stefano Sinigardi; Giorgio Turchetti. Case studies in space charge and plasma acceleration of charged beams. Comptes Rendus. Mécanique, Volume 342 (2014) no. 10-11, pp. 647-661. doi : 10.1016/j.crme.2014.07.004. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2014.07.004/

[1] T. Tajima; J.M. Dawson Laser electron accelerator, Phys. Rev. Lett., Volume 43 ( July 1979 ), pp. 267-270

[2] P. Chen; J.M. Dawson; R.W. Huff; T. Katsouleas Acceleration of electrons by the interaction of a bunched electron beam with a plasma, Phys. Rev. Lett., Volume 54 ( Feb. 1985 ), pp. 693-696

[3] P. Chen; J.J. Su; J.M. Dawson; K.L.F. Bane; P.B. Wilson Energy transfer in the plasma wake-field accelerator, Phys. Rev. Lett., Volume 56 ( Mar. 1986 ), pp. 1252-1255

[4] W.P. Leemans; B. Nagler; A.J. Gonsalves; C. Tóth; K. Nakamura; C.G.R. Geddes; E. Esarey; C.B. Schroeder; S.M. Hooker GeV electron beams from a centimetre-scale accelerator, Nat. Phys., Volume 2 ( October 2006 ), pp. 696-699

[5] I. Blumenfeld; C.E. Clayton; F.J. Decker; M.J. Hogan; C. Huang et al. Energy doubling of 42 GeV electrons in a metre-scale plasma wakefield accelerator, Nature, Volume 445 (2007), pp. 741-744

[6] A. Caldwell; K. Lotov; A. Pukhov; F. Simon Proton-driven plasma-wakefield acceleration, Nat. Phys., Volume 5 ( May 2009 ), pp. 363-367

[7] I.M. Kapchinskij; V.V. Vladimirskij Limitations of proton beam current in a strong focusing linear accelerator associated with the beam space charge, HEACC 59 (1959), pp. 274-287

[8] M. Reiser Theory and Design of Charged Particle Beams, Wiley Series in Beam Physics and Accelerator Technology, Wiley, 2008

[9] F.E. Low A Lagrangian formulation of the Boltzmann–Vlasov equation for plasmas, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., Volume 248 (1958) no. 1253, pp. 282-287

[10] P.J. Morrison Variational principle and stability of nonmonotonic Vlasov–Poisson equilibria, Z. Naturforsch., Volume 42a (1987), pp. 1115-1123

[11] G. Turchetti Mathematical models in beam dynamics (P. Cerrai; P. Freguglia; C. Pellegrini, eds.), The Application of Mathematics to the Sciences of Nature, Springer, 2002, pp. 117-136

[12] A. Franchi, M. Comunian, A. Pisent, G. Turchetti, S. Rambaldi, et al., HALODYN: a 3D Poisson–Vlasov code to simulate the space charge effects in the high intensity TRASCO linac, 2002.

[13] F. Bergamini; G. Franchetti; G. Turchetti The micromaps description of a beam with space charge, Il Nuovo Cimento A, Volume 112 (1999) no. 5, pp. 429-435

[14] G. Franchetti; I. Hofmann; S. Machida et al. Towards the description of long term self-consistent effects in space charge induced resonance trapping, 2–6 October 2006, Chamonix Mont-Blanc, France (2006)

[15] G. Franchetti MicroMAP library http://web-docs.gsi.de/~giuliano/micromap_manual/micromap_reference_manual.html

[16] S. Rambaldi; G. Turchetti; C. Benedetti; F. Mattioli; A. Franchi Accuracy analysis of a spectral Poisson solver, Proceedings of the Workshop on High Intensity Beam Dynamics Coulomb05 Workshop on High Intensity Beam Dynamics, Nucl. Instrum. Methods A, Volume 561 (2006) no. 2, pp. 223-229

[17] G. Turchetti Hamiltonian maps and normal forms for intense beams, Proceedings of the Workshop on High Intensity Beam Dynamics Coulomb05 Workshop on High Intensity Beam Dynamics, Nucl. Instrum. Methods A, Volume 561 (2006) no. 2, pp. 151-157

[18] J. Qiang; R.D. Ryne; I. Hofmann Space-charge driven emittance growth in a 3D mismatched anisotropic beam, Phys. Rev. Lett., Volume 92 ( Apr. 2004 ), p. 174801

[19] Y.Y. Yamaguchi; J. Barré; F. Bouchet; T. Dauxois; S. Ruffo Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model, Phys. A, Stat. Mech. Appl., Volume 337 (2004) no. 1–2, pp. 36-66

[20] R. Bachelard; F. Staniscia; T. Dauxois; G. De Ninno; S. Ruffo Stability of inhomogeneous states in mean-field models with an external potential, J. Stat. Mech. Theory Exp., Volume 2011 (2011) no. 03, p. P03022

[21] V.I. Arnold; E. Khukhro; V.V. Kozlov; A.I. Neishtadt Mathematical Aspects of Classical and Celestial Mechanics, Encyclopaedia of Mathematical Sciences, Physica-Verlag, 2007

[22] P.J. Morrison Hamiltonian and Action principle formulations of plasma physics, Phys. Plasmas (1994–present), Volume 12 (2005) no. 5

[23] H. Ye; P.J. Morrison Action principles for the Vlasov equation, Phys. Fluids, B Plasma Phys. (1989–1993), Volume 4 (1992) no. 4, pp. 771-777

[24] A.J. Brizard New variational principle for the Vlasov–Maxwell equations, Phys. Rev. Lett., Volume 84 ( Jun. 2000 ), pp. 5768-5771

[25] A.J. Brizard A new Lagrangian formulation for laser-plasma interactions, Phys. Plasmas (1994–present), Volume 5 (1998) no. 4, pp. 1110-1117

[26] A.B. Stamm, B.A. Shadwick, E.G. Evstatiev, Variational formulation of macro-particle models for electromagnetic plasma simulations, ArXiv e-prints, October 2013.

[27] E.G. Evstatiev; B.A. Shadwick Variational formulation of particle algorithms for kinetic plasma simulations, J. Comput. Phys., Volume 245 (2013), pp. 376-398

[28] R.W. Hockney; J.W. Eastwood Computer Simulation Using Particles, Taylor & Francis, 2010

[29] J.W. Eastwood The virtual particle electromagnetic particle-mesh method, Comput. Phys. Commun., Volume 64 (1991) no. 2, pp. 252-266

[30] C. Benedetti; A. Sgattoni; G. Turchetti; P. Londrillo ALaDyn: a high-accuracy PIC code for the Maxwell–Vlasov equations, IEEE Trans. Plasma Sci., Volume 36 ( Aug. 2008 ) no. 4, pp. 1790-1798

[31] P. Londrillo; C. Benedetti; A. Sgattoni; G. Turchetti Charge preserving high order PIC schemes, Coulomb'09 – Ions Acceleration with High Power Lasers: Physics and Applications, Nucl. Instrum. Methods A, Volume 620 (2010) no. 1, pp. 28-35

[32] F. Rossi; P. Londrillo; A. Sgattoni; S. Sinigardi; G. Turchetti Towards robust algorithms for current deposition and dynamic load-balancing in a GPU particle in cell code, AIP Conf. Proc., Volume 1507 (2012) no. 1, pp. 184-192

[33] F.M. Lee; B.A. Shadwick Modeling asymmetric beams using higher-order phase-space moments, AIP Conf. Proc., Volume 1507 (2012) no. 1, pp. 393-398

[34] B.A. Shadwick; G.M. Tarkenton; E. Esarey; F.M. Lee Hamiltonian reductions for modeling relativistic laser-plasma interactions, Commun. Nonlinear Sci. Numer. Simul., Volume 17 (2012) no. 5, pp. 2153-2160 (Special Issue: Mathematical Structure of Fluids and Plasmas, Dedicated to the 60th birthday of Phil Morrison)

[35] A. Caldwell; E. Gschwendtner; K. Lotov; P. Muggli; M. Wing AWAKE design report: a proton-driven PlasmaWakeField Acceleration experiment at CERN, CERN, Geneva, Apr. 2013 (Technical Report CERN-SPSC-2013-013, SPSC-TDR-003)

[36] K.V. Lotov Simulation of proton driven plasma wakefield acceleration, Phys. Rev. Spec. Top., Accel. Beams, Volume 13 ( Apr. 2010 ), p. 041301

[37] N. Kumar; A. Pukhov; K. Lotov Self-modulation instability of a long proton bunch in plasmas, Phys. Rev. Lett., Volume 104 ( Jun. 2010 ), p. 255003

[38] A. Caldwell; K. Lotov; A. Pukhov; G. Xia Plasma wakefield excitation with a 24 GeV proton beam, Plasma Phys. Control. Fusion, Volume 53 (2011) no. 1, p. 014003

[39] R. Assmann, M. Giovannozzi, Y. Papaphilippou, F. Zimmermann, A. Caldwell, G. Xia, Generation of short proton bunches in the CERN accelerator complex, CERN-ATS-2009-050, 4 p., Sep. 2009.

[40] E. Gschwendtner, C. Bracco, B. Goddard, M. Meddahi, A. Pardons, E. Shaposhnikova, H. Timko, F. Velotti, H. Vincke, Feasibility study of the AWAKE facility at CERN, CERN-ACC-2013-0275, 4 p., May 2013.

[41] C. Bracco; J. Bauche; D. Brethoux; V. Clerc; B. Goddard; E. Gschwendtner; L.K. Jensen; A. Kosmicki; G. Le Godec; M. Meddahi; P. Muggli; C. Mutin; O. Osborne; K. Papastergiou; A. Pardons; F.M. Velotti; H. Vincke Beam transfer line design for a plasma wakefield acceleration experiment AWAKE at the CERN SPS, CERN, Geneva, Jul. 2013 (Technical Report CERN-ACC-2013-0087)

[42] H. Timko; T. Argyropoulos; H. Bartosik; T. Bohl; J. Müller; E. Shaposhnikova; A. Petrenko Short high-intensity bunches for plasma wakefield experiment AWAKE in the CERN SPS, CERN, Geneva, May 2013 (Technical Report CERN-ACC-2013-0208)

[43] J.M. Dawson Nonlinear electron oscillations in a cold plasma, Phys. Rev., Volume 113 ( Jan. 1959 ), pp. 383-387

[44] K.V. Lotov Controlled self-modulation of high energy beams in a plasma, Phys. Plasmas (1994–present), Volume 18 (2011) no. 2

[45] A. Pukhov; N. Kumar; T. Tückmantel; A. Upadhyay; K. Lotov; P. Muggli; V. Khudik; C. Siemon; G. Shvets Phase velocity and particle injection in a self-modulated proton-driven plasma wakefield accelerator, Phys. Rev. Lett., Volume 107 ( Sep. 2011 ), p. 145003

[46] C.B. Schroeder; C. Benedetti; E. Esarey; F.J. Grüner; W.P. Leemans Growth and phase velocity of self-modulated beam-driven plasma waves, Phys. Rev. Lett., Volume 107 ( Sep. 2011 ), p. 145002

[47] D.H. Whittum; W.M. Sharp; S.S. Yu; M. Lampe; G. Joyce Electron-hose instability in the ion-focused regime, Phys. Rev. Lett., Volume 67 ( Aug. 1991 ), pp. 991-994

[48] K.V. Lotov Optimum angle for side injection of electrons into linear plasma wakefields, J. Plasma Phys., Volume 78 (2012), pp. 455-459

[49] AWAKE Collaboration http://awake.web.cern.ch/awake/physics.html

[50] J.B. Rosenzweig; B. Breizman; T. Katsouleas; J.J. Su Acceleration and focusing of electrons in two-dimensional nonlinear plasma wake fields, Phys. Rev. A, Volume 44 ( Nov. 1991 ), p. R6189-R6192

[51] J.B. Rosenzweig; N. Barov; M.C. Thompson; R.B. Yoder Energy loss of a high charge bunched electron beam in plasma: simulations, scaling, and accelerating wakefields, Phys. Rev. Spec. Top., Accel. Beams, Volume 7 ( Jun. 2004 ), p. 061302

[52] W. Lu; C. Huang; M.M. Zhou; W.B. Mori; T. Katsouleas Limits of linear plasma wakefield theory for electron or positron beams, Phys. Plasmas (1994–present), Volume 12 (2005) no. 6

[53] J.B. Rosenzweig; G. Andonian; M. Ferrario; P. Muggli; O. Williams; V. Yakimenko; K. Xuan Plasma wakefields in the quasi-nonlinear regime, AIP Conf. Proc., Volume 1299 (2010) no. 1, pp. 500-504

[54] J.B. Rosenzweig; G. Andonian; S. Barber; M. Ferrario; P. Muggli; B. O'Shea; Y. Sakai; A. Valloni; O. Williams; Y. Xi; V. Yakimenko Plasma wakefields in the quasi-nonlinear regime: experiments at ATF, AIP Conf. Proc., Volume 1507 (2012) no. 1, pp. 612-617

[55] P. Londrillo; C. Gatti; M. Ferrario Numerical investigation of beam-driven PWFA in quasi-nonlinear regime, Proceedings of the First European Advanced Accelerator Concepts Workshop 2013, Nucl. Instrum. Methods A, Volume 740 (2014), pp. 236-241

[56] C. Joshi; B. Blue; C.E. Clayton; E. Dodd; C. Huang; K.A. Marsh; W.B. Mori; S. Wang; M.J. Hogan; C. O'Connell; R. Siemann; D. Watz; P. Muggli; T. Katsouleas; S. Lee High energy density plasma science with an ultrarelativistic electron beam, Phys. Plasmas (1994–present), Volume 9 (2002) no. 5, pp. 1845-1855

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