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Magnetic Mirror? Introduction to Nuclear Fusion Prof. Dr. Yong-Su Na What is magnetic mirror? 2 Open Magnetic Confinement Z pinch θ pinch screw pinch Magnetic mirror http://phys.strath.ac.uk/information/history/photos.php http://www.frascati.enea.it/ProtoSphera/ProtoSphera%202001/6.%20Electrode%20experiment.htm 3 Magnetic Mirror - Suffering from end losses 4 Magnetic Mirror Gamma 10 and Hanbit - To reduce end-leakage by establishing an increasing magnetic field at the two ends - Many of charged particles are trapped due to the imposed constraints on particle motion with regards to conservation of energy and the magnetic moment. - First proposed by Enrico Fermi as a mechanism for the acceleration of cosmic rays 5 Magnetic Mirror What is the motion of particles? 1 mv2 F B B || 2 B || || μ : magnetic moment of the gyrating particle 6 Magnetic Mirror • Single particle picture - As the ion approaches the ends, it is increasingly subjected to drifts due to the inhomogeneity of the mirror field. - Drifts occur in azimuthal directions and the particles are still bound to their magnetic surfaces. mv2 R B v v || 0 0 D,R gc,x 2 2 qB0 R v m r L q B z A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000) 7 Magnetic Mirror • Conservation of kinetic energy and magnetic moment d d 1 2 1 2 E0 mv mv|| 0 dt dt 2 2 2 q 2 c 2 d mv / 2 M ISn r n q r n 0 L L dt B 2 2 B q mv Magnetic moment q n 2m B q mv2 / 2 n B 8 Individual Charge Trajectories • Invariant of motion d d 2 1 2 1 2 mv / 2 dv dB E mv mv 0 || dv|| dB 0 || m mv dt dt 2 2 B || dt v|| dt dt dt dv B B ds dt B ds dB F || B 1 d 1 2 dB || m || mv|| dt s s dt ds s dt v|| v|| dt dt 2 dt d 1 2 1 2 d dB mv mv|| B 0 dt 2 2 dt dt d dB d B B dt dt dt d 0 : adiabatic invariant dt 9 Magnetic Mirror Enrico Fermi (1901-1954) Nobel Laureate in physics in 1938 Cf. Marshall Rosenbluth (Doctoral student) CP-1 (Chicago Pile-1, the world's first human-made nuclear reactor) and Drawings from the Fermi–Szilárd "neutronic reactor" patent 10 Magnetic Mirror 11 1 1 1 E = mv2 = mv2 + mv2 = const. 0 2 2 2 || Magnetic Mirror 1 1 ⊥ mv2 mv2 sin 2 θ 2 2 μ = = = const. B ⊥ B 12 Magnetic Mirror • Fermi as a genuine scientist 13 Magnetic Mirror • Condition for trapping of particles 2 v 0 1 mv || B B F B B r max || 2 B || || 1 2 1 2 mvmin mvr 2 2 1 2 1 2 1 2 1 2 1 2 E mv||min mvmin mv||r mvr mvr Bmin Br 2 2 2 2 2 B 2 2 min Br Bmax vmin Bmin vmin Bmin 2 2 2 vr Br v||min vmin Br 14 Magnetic Mirror • Condition for trapping of particles 2 v 0 1 mv || B B F B B r max || 2 B || || 1 2 1 2 mvmin mvr 2 2 1 2 1 2 1 2 1 2 1 2 E mv||min mvmin mv||r mvr mvr Bmin Br 2 2 2 2 2 v|| 2 2 2 loss cone vmin Bmin vmin Bmin vmin 2 2 2 2 2 sin vr Br v||min vmin Br vmin Condition for trapping of particles Bmin 2 Bmin Br Bmax sin Br Bmax loss cone 15 B B Magnetic Mirror min sin 2 min Br Bmax • Mirror ratio Bmin 0 arcsin 2 0 B f (v) max 3 2 v f (v)d v d sind sind 2 v dv || 4v loss cone double cone 0 0 0 0 floss 2 f (v) f (v)d 3v d sind v2dv 2 0 0 0 0 4v 1- cos0 loss cone mirror ratio: B Bmax f 1 f cos 1 min R Determining the effectiveness trap loss 0 B m Bmax min of confinement 1 ftrap 1 Rm A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000) 16 Stability in mirror? 17 Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field http://www.threes.com/index.php?view=article&id=20:greek-columns&option=com_content&Itemid=39 18 Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field E - Particle picture: due to + - The curvature drift leading curvature of B in Z to azimuthal polarisation to create the E-field resulting in the ExB drift displacing the plasma particles radially outward BB mv2 R B v 1 v || 0 0 D,B vrL 2 D,R 2 2 2 B qB0 R 19 Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field Magnetic well minimum-B field configuration: field lines are (almost) everywhere concave into the plasma 20 Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field x x F. F. Chen, “An Indispensable Truth”, Springer (2011) http://blog.naver.com/PostView.nhn?blogId=ray0620&logNo=150112423635&parentCategoryNo=1&viewDate=&currentPage=1&listtype=0 http://en.wikipedia.org/wiki/File:St_Louis_Gateway_Arch.jpg 21 Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field x x F. F. Chen, “An Indispensable Truth”, Springer (2011) http://blog.naver.com/PostView.nhn?blogId=ray0620&logNo=150112423635&parentCategoryNo=1&viewDate=&currentPage=1&listtype=0 http://en.wikipedia.org/wiki/File:St_Louis_Gateway_Arch.jpg 22 Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field Yin-Yang coil Ioffe bars Magnetic field profiles in a Ioffe-Pritchard trap A. A. Harms et al, “Principles of Fusion Energy”, World Scientific (2000) 23 Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field Baseball coil http://www.wangnmr.com/Copper_magnet_catolog.htm 24 Magnetic Mirror • Instabilities - Flute instability: convex curvature of the magnetic field Flux surfaces Plug/Barrier Baseball coils (potential (MHD stabilising) plugging) ICRF ECRH Gamma 10 Tandem mirror (Univ. of Tsukuba, Japan) 25 Magnetic Mirror • Instabilities - Velocity-space instability: driven by the v|| non-Maxwellian velocity distribution due to loss cone the preferred loss of particles with large v||/v . - Enhancing⊥ the velocity-space diffusion into the loss cone - Observed that such it is less harmful to plasma confinement when the mirror loss cone device is short in dimension 26 Magnetic Mirror • Classical Mirror Confinement - If collisions are neglected, particles are trapped in a mirror when they do not appear in the loss cone. - Collisions can bring them randomly from the confinement region into the loss cone. - Due to their relatively small mass, electrons diffuse more rapidly. → a positive electrostatic potential built up in the confined plasma tending to retain the remaining electrons in the magnetic bottle - Overall plasma confinement time is governed by the ion escape time. 27 Magnetic Mirror • Classical Mirror Confinement - If collisions are neglected, particles are trapped in a mirror when they do not appear in the loss cone. - Collisions can bring them randomly from the confinement region into the loss cone. - Due to their relatively small mass, electrons diffuse more rapidly. → a positive electrostatic potential built up in the confined plasma tending to retain the remaining electrons in the magnetic bottle - Overall plasma confinement time is governed by the ion escape time. ion ion loss loss cone cone ion electron 28 Magnetic Mirror • Classical Mirror Confinement - If collisions are neglected, particles are trapped in a mirror when they do not appear in the loss cone. - Collisions can bring them randomly from the confinement region into the loss cone. - Due to their relatively small mass, electrons diffuse more rapidly. → a positive electrostatic potential built up in the confined plasma tending to retain the remaining electrons in the magnetic bottle - Overall plasma confinement time is governed by the ion escape time. B A 2/1 T 2/3 ln max B i i max R mirror ratio Bmax Bmin m ln C : mirror confinement time B M ii 4 min Bmin ni Zi 1 A 2/1 kT 2/3 i i : ion-ion collision time ii 4 ni vrs n qii vrs - Mirror confinement time does not depend on the actual magnitude of B or the plasma size but on the size of the loss cone. - Higher density enhances the scattering into the loss cone. 29 .
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