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 mvmin mvr 2 2 1 2 1 2 1 2 1 2 1 2    E  mv||min  mvmin  mv||r  mvr  mvr Bmin Br 2 2 2 2 2

B 2 2 min Br Bmax vmin Bmin vmin Bmin 2  2 2  vr Br v||min  vmin 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 mvmin mvr 2 2 1 2 1 2 1 2 1 2 1 2    E  mv||min  mvmin  mv||r  mvr  mvr Bmin Br 2 2 2 2 2

v||

2 2 2 loss cone vmin Bmin vmin Bmin vmin 2 2  2 2   2  sin  vr Br v||min  vmin 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 sind  sind  2 v dv ||      4v loss cone double cone 0  0  0  0 floss     2   f (v) f (v)d 3v d sind v2dv     2 0 0 0 0 4v

1- cos0 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

BB mv2 R B v 1 v || 0 0 D,B   vrL 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 x x

F. F. Chen, “An Indispensable Truth”, Springer (2011)

http://blog.naver.com/PostView.nhn?blogId=ray0620&logNo=150112423635&parentCategoryNo=1&viewDate=¤tPage=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 x x

F. F. Chen, “An Indispensable Truth”, Springer (2011)

http://blog.naver.com/PostView.nhn?blogId=ray0620&logNo=150112423635&parentCategoryNo=1&viewDate=¤tPage=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 device is short in dimension loss cone

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