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Generalized collision operator for fast electrons interacting with partially ionized impurities

Part of: Focus on Fusion

Published online by Cambridge University Press:  16 November 2018

L. Hesslow Open the ORCID record for L. Hesslow [Opens in a new window] ,
O. Embréus Open the ORCID record for O. Embréus [Opens in a new window] ,
M. Hoppe ,
T. C. DuBois Open the ORCID record for T. C. DuBois [Opens in a new window] ,
G. Papp ,
M. Rahm  and
T. Fülöp Open the ORCID record for T. Fülöp [Opens in a new window]
L. Hesslow*
Affiliation:
Department of Physics, Chalmers University of Technology, SE-41296 Göteborg, Sweden
O. Embréus
Affiliation:
Department of Physics, Chalmers University of Technology, SE-41296 Göteborg, Sweden
M. Hoppe
Affiliation:
Department of Physics, Chalmers University of Technology, SE-41296 Göteborg, Sweden
T. C. DuBois
Affiliation:
Department of Physics, Chalmers University of Technology, SE-41296 Göteborg, Sweden
G. Papp
Affiliation:
Max Planck Institute for Plasma Physics, D-85748 Garching, Germany
M. Rahm
Affiliation:
Department of Chemistry and Chemical Engineering, Chalmers University of Technology, SE-41296 Gothenburg, Sweden
T. Fülöp
Affiliation:
Department of Physics, Chalmers University of Technology, SE-41296 Göteborg, Sweden
*
†Email address for correspondence: [email protected]
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Abstract

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.

Keywords

fusion plasmarunaway electrons
Type
Research Article
Information
Journal of Plasma Physics , Volume 84 , Issue 6 , December 2018 , 905840605
Copyright
© Cambridge University Press 2018 

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References

Adamo, C. & Barone, V. 1999 Toward reliable density functional methods without adjustable parameters: the pbe0 model. J. Chem. Phys. 110 (13), 6158.Google Scholar
Akama, H. 1970 Relativistic Boltzmann equation for plasmas. J. Phys. Soc. Japan 28, 478.Google Scholar
Aleynikov, P. & Breizman, B. N. 2015 Theory of two threshold fields for relativistic runaway electrons. Phys. Rev. Lett. 114, 155001.Google Scholar
Aleynikov, P. & Breizman, B. N. 2017 Generation of runaway electrons during the thermal quench in tokamaks. Nucl. Fusion 57 (4), 046009.Google Scholar
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.Google Scholar
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. http://physics.nist.gov/Star, [accessed: 2018, April 6].Google Scholar
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.Google Scholar
Bethe, H. 1930 Zur theorie des durchgangs schneller korpuskularstrahlen durch materie. Ann. Phys. 397 (3), 325 (in German).Google Scholar
Boozer, A. H. 2015 Theory of runaway electrons in ITER: equations, important parameters, and implications for mitigation. Phys. Plasmas 22 (3), 032504.Google Scholar
Braams, B. J. & Karney, C. F. F. 1989 Conductivity of a relativistic plasma. Phys. Fluids B 1 (7), 1355.Google Scholar
Breizman, B. & Aleynikov, P. 2017 Kinetics of relativistic runaway electrons. Nucl. Fusion 57 (12), 125002.Google Scholar
Cercignani, C. & Kremer, G. M. 2002 Relativistic Boltzmann equation. In The Relativistic Boltzmann Equation: Theory and Applications. Springer.Google Scholar
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.Google Scholar
Connor, J. & Hastie, R. 1975 Relativistic limitations on runaway electrons. Nucl. Fusion 15, 415.Google Scholar
Douglas, M. & Kroll, N. M. 1974 Quantum electrodynamical corrections to the fine structure of helium. Ann. Phys. 82 (1), 89.Google Scholar
Dreicer, H. 1959 Electron and ion runaway in a fully ionized gas. I. Phys. Rev. 115, 238.Google Scholar
Dwyer, J. R. 2007 Relativistic breakdown in planetary atmospheres. Phys. Plasmas 14, 042901.Google Scholar
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.Google Scholar
Eriksson, L.-G., Helander, P., Andersson, F., Anderson, D. & Lisak, M. 2004 Current dynamics during disruptions in large tokamaks. Phys. Rev. Lett. 92, 205004.Google Scholar
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.Google Scholar
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.Google Scholar
Fülöp, T., Pokol, G., Helander, P. & Lisak, M. 2006 Destabilization of magnetosonic-whistler waves by a relativistic runaway beam. Phys. Plasmas 13 (6), 062506.Google Scholar
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.Google Scholar
Helander, P., Eriksson, L.-G. & Andersson, F. 2002 Runaway acceleration during magnetic reconnection in tokamaks. Plasma Phys. Control. Fusion 44, B247.Google Scholar
Helander, P. & Sigmar, D. 2005 Collisional Transport in Magnetized Plasmas. Cambridge University Press.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Jackson, J. D. 1999 Classical Electrodynamics. Wiley.Google Scholar
Jayakumar, R., Fleischmann, H. & Zweben, S. 1993 Collisional avalanche exponentiation of runaway electrons in electrified plasmas. Phys. Lett. A 172, 447–451.Google Scholar
Kirillov, V. D., Trubnikov, B. A. & Trushin, S. A. 1975 Role of impurities in anomalous plasma resistance. Sov. J. Plasma Phys. 1, 117.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1958 Quantum Mechanics: Non-relativistic Theory. Pergamon Press.Google Scholar
Landreman, M., Stahl, A. & Fülöp, T. 2014 Numerical calculation of the runaway electron distribution function and associated synchrotron emission. Comput. Phys. Commun. 185, 847.Google Scholar
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.Google Scholar
Martín-Solís, J. R., Loarte, A. & Lehnen, M. 2015 Runaway electron dynamics in tokamak plasmas with high impurity content. Phys. Plasmas 22, 092512.Google Scholar
Mosher, D. 1975 Interactions of relativistic electron beams with high atomic-number plasmas. Phys. Fluids 18, 846.Google Scholar
Mott, N. F. & Massey, H. S. W. 1965 The Theory of Atomic Collisions, vol. 35. Clarendon Press.Google Scholar
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.Google Scholar
Putvinski, S., Fujisawa, N., Post, D., Putvinskaya, N., Rosenbluth, M. & Wesley, J. 1997 Impurity fueling to terminate tokamak discharges. J. Nucl. Mater. 241, 316.Google Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Rosenbluth, M. & Putvinski, S. 1997 Theory for avalanche of runaway electrons in tokamaks. Nucl. Fusion 37, 1355–1362.Google Scholar
Rosenbluth, M. N., MacDonald, W. M. & Judd, D. L. 1957 Fokker–Planck equation for an inverse-square force. Phys. Rev. 107, 1.Google Scholar
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.Google Scholar
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.Google Scholar
Sokolov, Y. 1979 ‘Multiplication’ of accelerated electrons in a tokamak. JETP Lett. 29, 218–221.Google Scholar
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.Google Scholar
Stahl, A., Embréus, O., Papp, G., Landreman, M. & Fülöp, T. 2016 Kinetic modelling of runaway electrons in dynamic scenarios. Nucl. Fusion 56 (11), 112009.Google Scholar
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.Google Scholar
Wesson, J. 2011 Tokamaks, 4th edn. Oxford University Press.Google Scholar
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.Google Scholar
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.Google Scholar
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).Google Scholar