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Generalized collision operator for fast electrons interacting with partially ionized impurities

  • L. Hesslow (a1), O. Embréus (a1), M. Hoppe (a1), T. C. DuBois (a1), G. Papp (a2), M. Rahm (a3) and T. Fülöp (a1)...


Accurate modelling of the interaction between fast electrons and partially ionized atoms is important for evaluating tokamak disruption mitigation schemes based on material injection. This requires accounting for the effect of screening of the impurity nuclei by the cloud of bound electrons. In this paper, we generalize the Fokker–Planck operator in a fully ionized plasma by accounting for the effect of screening. We detail the derivation of this generalized operator, and calculate the effective ion length scales, needed in the components of the collision operator, for a number of ion species commonly appearing in fusion experiments. We show that for high electric fields, the secondary runaway growth rate can be substantially larger than in a fully ionized plasma with the same effective charge, although the growth rate is significantly reduced at near-critical electric fields. Furthermore, by comparison with the Boltzmann collision operator, we show that the Fokker–Planck formalism is accurate even for large impurity content.


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Adamo, C. & Barone, V. 1999 Toward reliable density functional methods without adjustable parameters: the pbe0 model. J. Chem. Phys. 110 (13), 6158.
Akama, H. 1970 Relativistic Boltzmann equation for plasmas. J. Phys. Soc. Japan 28, 478.
Aleynikov, P. & Breizman, B. N. 2015 Theory of two threshold fields for relativistic runaway electrons. Phys. Rev. Lett. 114, 155001.
Aleynikov, P. & Breizman, B. N. 2017 Generation of runaway electrons during the thermal quench in tokamaks. Nucl. Fusion 57 (4), 046009.
Barysz, M. & Sadlej, A. J. 2001 Two-component methods of relativistic quantum chemistry: from the Douglas–kroll approximation to the exact two-component formalism. J. Mol. Struct: THEOCHEM 573 (1), 181.
Berger, M., Coursey, J., Zucker, M. & Chang, J.2005 ESTAR, PSTAR, and ASTAR: computer programs for calculating stopping-power and range tables for electrons, protons, and helium ions., [accessed: 2018, April 6].
Berger, M. J., Inokuti, M., Anderson, H. H., Bichsel, H., Dennis, J. A., Powers, D., Seltzer, S. M. & Turner, J. E. 1984 4. Selection of mean excitation energies for elements. J. Intl Commission Radiat. Units Measurements os19 (2), 22.
Bethe, H. 1930 Zur theorie des durchgangs schneller korpuskularstrahlen durch materie. Ann. Phys. 397 (3), 325 (in German).
Boozer, A. H. 2015 Theory of runaway electrons in ITER: equations, important parameters, and implications for mitigation. Phys. Plasmas 22 (3), 032504.
Braams, B. J. & Karney, C. F. F. 1989 Conductivity of a relativistic plasma. Phys. Fluids B 1 (7), 1355.
Breizman, B. & Aleynikov, P. 2017 Kinetics of relativistic runaway electrons. Nucl. Fusion 57 (12), 125002.
Cercignani, C. & Kremer, G. M. 2002 Relativistic Boltzmann equation. In The Relativistic Boltzmann Equation: Theory and Applications. Springer.
Chiu, S., Rosenbluth, M., Harvey, R. & Chan, V. 1998 Fokker–Planck simulations mylb of knock-on electron runaway avalanche and bursts in tokamaks. Nucl. Fusion 38 (11), 1711.
Connor, J. & Hastie, R. 1975 Relativistic limitations on runaway electrons. Nucl. Fusion 15, 415.
Douglas, M. & Kroll, N. M. 1974 Quantum electrodynamical corrections to the fine structure of helium. Ann. Phys. 82 (1), 89.
Dreicer, H. 1959 Electron and ion runaway in a fully ionized gas. I. Phys. Rev. 115, 238.
Dwyer, J. R. 2007 Relativistic breakdown in planetary atmospheres. Phys. Plasmas 14, 042901.
Embréus, O., Stahl, A. & Fülöp, T. 2018 On the relativistic large-angle electron collision operator for runaway avalanches in plasmas. J. Plasma Phys. 84 (1), 905840102.
Eriksson, L.-G., Helander, P., Andersson, F., Anderson, D. & Lisak, M. 2004 Current dynamics during disruptions in large tokamaks. Phys. Rev. Lett. 92, 205004.
Finken, K. H., Watkins, J. G., Rusbüldt, D., Corbett, W. J., Dippel, K. H., Goebel, D. M. & Moyer, R. A. 1990 Observation of infrared synchrotron radiation from tokamak runaway electrons in textor. Nucl. Fusion 30 (5), 859.
Frisch, M. J., Trucks, G. W., Schlegel, H. B., Scuseria, G. E., Robb, M. A., Cheeseman, J. R., Scalmani, G., Barone, V., Petersson, G. A., Nakatsuji, H. et al. 2016 Gaussian 16 Revision B.01. Gaussian Inc. Wallingford CT.
Fülöp, T., Pokol, G., Helander, P. & Lisak, M. 2006 Destabilization of magnetosonic-whistler waves by a relativistic runaway beam. Phys. Plasmas 13 (6), 062506.
Gulans, A., Kontur, S., Meisenbichler, C., Nabok, D., Pavone, P., Rigamonti, S., Sagmeister, S., Werner, U. & Draxl, C. 2014 Exciting: a full-potential all-electron package implementing density-functional theory and many-body perturbation theory. J. Phys.: Condens. Matter 26 (36), 363202.
Helander, P., Eriksson, L.-G. & Andersson, F. 2002 Runaway acceleration during magnetic reconnection in tokamaks. Plasma Phys. Control. Fusion 44, B247.
Helander, P. & Sigmar, D. 2005 Collisional Transport in Magnetized Plasmas. Cambridge University Press.
Hess, B. A. 1986 Relativistic electronic-structure calculations employing a two-component no-pair formalism with external-field projection operators. Phys. Rev. A 33 (6), 3742.
Hesslow, L., Embréus, O., Stahl, A., Dubois, T. C., Papp, G., Newton, S. L. & Fülöp, T. 2017 Effect of partially screened nuclei on fast-electron dynamics. Phys. Rev. Lett. 118, 255001.
Hesslow, L., Embréus, O., Wilkie, G. J., Papp, G. & Fülöp, T. 2018 Effect of partially ionized impurities and radiation on the effective critical electric field for runaway generation. Plasma Phys. Control. Fusion 60 (7), 074010.
Hollmann, E. M., Aleynikov, P. B., Fülöp, T., Humphreys, D. A., Izzo, V. A., Lehnen, M., Lukash, V. E., Papp, G., Pautasso, G., Saint-Laurent, F. et al. 2015 Status of research toward the iter disruption mitigation system. Phys. Plasmas 22 (2), 021802.
Hoppe, M., Embréus, O., Paz-Soldan, C., Moyer, R. & Fülöp, T. 2018a Interpretation of runaway electron synchrotron and bremsstrahlung images. Nucl. Fusion 58 (8), 082001.
Hoppe, M., Embréus, O., Tinguely, R., Granetz, R., Stahl, A. & Fülöp, T. 2018b SOFT: a synthetic synchrotron diagnostic for runaway electrons. Nucl. Fusion 58 (2), 026032.
Jackson, J. D. 1999 Classical Electrodynamics. Wiley.
Jayakumar, R., Fleischmann, H. & Zweben, S. 1993 Collisional avalanche exponentiation of runaway electrons in electrified plasmas. Phys. Lett. A 172, 447451.
Kirillov, V. D., Trubnikov, B. A. & Trushin, S. A. 1975 Role of impurities in anomalous plasma resistance. Sov. J. Plasma Phys. 1, 117.
Landau, L. D. & Lifshitz, E. M. 1958 Quantum Mechanics: Non-relativistic Theory. Pergamon Press.
Landreman, M., Stahl, A. & Fülöp, T. 2014 Numerical calculation of the runaway electron distribution function and associated synchrotron emission. Comput. Phys. Commun. 185, 847.
Lehtinen, N. G., Bell, T. F. & Inan, U. S. 1999 Monte Carlo simulation of runaway mev electron breakdown with application to red sprites and terrestrial gamma ray flashes. J. Geophys. Res. Space Phys. 104 (A11), 24699.
Martín-Solís, J. R., Loarte, A. & Lehnen, M. 2015 Runaway electron dynamics in tokamak plasmas with high impurity content. Phys. Plasmas 22, 092512.
Mosher, D. 1975 Interactions of relativistic electron beams with high atomic-number plasmas. Phys. Fluids 18, 846.
Mott, N. F. & Massey, H. S. W. 1965 The Theory of Atomic Collisions, vol. 35. Clarendon Press.
Parks, P. B., Rosenbluth, M. N. & Putvinski, S. V. 1999 Avalanche runaway growth rate from a momentum-space orbit analysis. Phys. Plasmas 6 (6), 2523.
Putvinski, S., Fujisawa, N., Post, D., Putvinskaya, N., Rosenbluth, M. & Wesley, J. 1997 Impurity fueling to terminate tokamak discharges. J. Nucl. Mater. 241, 316.
Reux, C., Plyusnin, V., Alper, B., Alves, D., Bazylev, B., Belonohy, E., Boboc, A., Brezinsek, S., Coffey, I., Decker, J. et al. 2015 Runaway electron beam generation and mitigation during disruptions at JET-ILW. Nucl. Fusion 55 (9), 093013.
Roos, B. O., Lindh, R., Malmqvist, P.-Å., Veryazov, V. & Widmark, P.-O. 2004 Main group atoms and dimers studied with a new relativistic ano basis set. J. Phys. Chem. A 108 (15), 2851.
Roos, B. O., Lindh, R., Malmqvist, P.-Å., Veryazov, V. & Widmark, P.-O. 2005 New relativistic ano basis sets for transition metal atoms. J. Phys. Chem. A 109 (29), 6575.
Rosenbluth, M. & Putvinski, S. 1997 Theory for avalanche of runaway electrons in tokamaks. Nucl. Fusion 37, 13551362.
Rosenbluth, M. N., MacDonald, W. M. & Judd, D. L. 1957 Fokker–Planck equation for an inverse-square force. Phys. Rev. 107, 1.
Sauer, S. P., Oddershede, J. & Sabin, J. R. 2015 Chapter three – the mean excitation energy of atomic ions. In Concepts of Mathematical Physics in Chemistry: A Tribute to Frank E. Harris – Part A, Advances in Quantum Chemistry, 71, p. 29. Academic Press.
Smith, H., Helander, P., Eriksson, L.-G., Anderson, D., Lisak, M. & Andersson, F. 2006 Runaway electrons and the evolution of the plasma current in tokamak disruptions. Phys. Plasmas 13 (10), 102502.
Sokolov, Y. 1979 ‘Multiplication’ of accelerated electrons in a tokamak. JETP Lett. 29, 218221.
Solodov, A. A. & Betti, R. 2008 Stopping power and range of energetic electrons in dense plasmas of fast-ignition fusion targets. Phys. Plasmas 15 (4), 042707.
Stahl, A., Embréus, O., Papp, G., Landreman, M. & Fülöp, T. 2016 Kinetic modelling of runaway electrons in dynamic scenarios. Nucl. Fusion 56 (11), 112009.
Stahl, A., Landreman, M., Papp, G., Hollmann, E. & Fülöp, T. 2013 Synchrotron radiation from a runaway electron distribution in tokamaks. Phys. Plasmas 20 (9), 093302.
Wesson, J. 2011 Tokamaks, 4th edn. Oxford University Press.
Widmark, P.-O., Malmqvist, P.-Å. & Roos, B. O. 1990 Density matrix averaged atomic natural orbital (ano) basis sets for correlated molecular wave functions. Theor. Chim. Acta 77 (5), 291.
Wilson, C. T. R. 1925 The acceleration of $\unicode[STIX]{x1D6FD}$ -particles in strong electric fields such as those of thunderclouds. Math. Proc. Camb. Phil. Soc. 22, 534.
Zhogolev, V. & Konovalov, S. 2014 Characteristics of interaction of energetic electrons with heavy impurity ions in a tokamak plasma. VANT or Problems of Atomic Sci. Tech. Series Thermonuclear Fusion 37, 71 (in Russian).
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Generalized collision operator for fast electrons interacting with partially ionized impurities

  • L. Hesslow (a1), O. Embréus (a1), M. Hoppe (a1), T. C. DuBois (a1), G. Papp (a2), M. Rahm (a3) and T. Fülöp (a1)...


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