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4) Magnetic confinement of plasma Due to the shielding in the plasma, there is almost no control with electric fields. A control is possible with magnetic fields, as particles are bound to the field lines. This is called plasma confinement. It is of fundamental interest for the magnetic confinement fusion (Tokamak, Stellarator). Example in nature: Guidance of High energetic particles of the cosmic radiation by the earth magnetic field to the poles generation of polar lights. WS2011/12 4.1 We solely take single particle motion in the magnetic field into account. If we include collective effects, we need magneto hydrodynamik (MHD) theory. Lorentz forcet: m v FL q v B FL B m v|| 0 , As yields the magnetic field has no influence on the motion parallel to the magnetic field. The direction of the velocity changes perpendicular to the magnetic field, hence the kinetic energy of the particles is constant (orbit motion). v2 m m c2 r q v B q c r B r c q B m cyclotron frequency ce Hz 1.76 1011 BT , ci WS2011/12 (4.1) me ce , mi 4.2 Radius on orbit: r m v qB Gyration of electrons Plasma diagnostic Injection of an electromagnetic wave of = ce plasma heating Using 1 2 m v2 kT (2 degrees of freedom perpendicular to B) the Lamor radius rL is Te eV m v 2mkT 6 m rL 3.4 10 qB qB BT (4.2) Example: Te = 1 eV, B = 1 T → rL = 3.4 m Let’s now investigate mv F qv B Assuming an additional force F acting on the particles no closed orbit WS2011/12 4.3 For constant B und F in time and space guiding center-Ansatz The center motion is calculated via mv v B m rc r rg , rg v B 2 q B v B q B The velocity of the guiding center is give by m 1 vc rc r rg v v B v ( F q v B) B 2 2 qB qB with ( v B ) B ( v B ) B B 2 v B 2 v F B vc v|| q B2 the following result of the motion is derived and mv|| F|| (4.3) The second term is called drift velocity. This term does not accelerate and is directed prependicular to B and F|. WS2011/12 4.4 Examples: 1.) elektric field F qE E x B –drift EB vD B2 The drift velocity is not dependent on the charge of the particle. ExB-drift in direction of the x-axis. The gyration radii are not drawn to scale! m g B v D F mg 2.) Gravitation q B2 g The particle gains energy in the direction of || . The drift is charge dependent and leads to a charge g B mi j n e ( v v ) n e m e i e e e separation introducing the current g e B2 Z For sources this drift is negligible. More important are drifts due to gradients. WS2011/12 4.5 3.) B-Drift, if B f ( x, t ) curvature drift B 0 Due to a gradient in B leads to a curvature of the field lines. This results in a 2 v|| ˆ F m r c. centrifugal force Rc 2 m v rˆc B || vD Rc q B2 ˆ ˆ rc dT B Rc ds B The centrifugal force is in direction of B and hence 2 m v || vD B B 3 qB Particle drift in a toroidal magnetic field. The drift results in a charge separation and the resulting ExB-drift moves the plasma outward Example: toroidal magnetic field WS2011/12 4.6 Adiabatic invariance of the magnetic moment A charged particle on a circular orbit creates a current in the magnetic field. I q v 2 r q 2 magnetic dipole moment: q2 2 1 m I dF nF I F nF 2 q c rL nF B rL2 nF 2m F Assuming the Lamor radius v m rL Bq 2 v m q m v2 W m B 2m B q 2B B 2 (4.4) How does the magnetic moment change with a slow drift of the guiding centre? Change of the magnetic flux through the orbit area! 2 W q c and dW W q rL dB dB m dt c c dt dt Gyration period C WS2011/12 d dB q rL2 dt dt if c and rL are introduced. 4.7 On the other hand dW d d m dB ( m B) B m dt dt dt dt d m 0 , m const . hence: dt The magnetic moment is constant under a slow drift of the guiding center (slow in comparison to the orbit motion). Example: adiabatic invariant magnetic mirror If a charged particle moves into a zone with stronger magnetic field, the kinetic energy of the gyration increases due to the invariance of the magnetic moment. As the total kinetic energy in of the particle does not change (no collisions) the kinetic energy in Direction of the guiding center motion must be reduced. In order to reduce v|| completly m ( Bmax B1 ) W W2 W1 W||,1 The particle starts at position 1 between the coils. At Pos. 2 is the maximum of the magnetic field Bmax. WS2011/12 4.8 B2 W2 W 1 B1 W1 W1 B2 W 1 B1 W1 W1 W||,1 2 v1|| cot 2 v1 B2/B1 is called mirror ratio. v1 v1|| The loss con eis characterized by arcsin( Bmin ) Bmax (4.5) This equation demonstrates that a magnetic mirror is not perfect. Particles with small velocity components perpendicular to the magnetic field are not reflected, but slowed down (as their angles towards the axis are smaller then . WS2011/12 4.9 Particle motion within the earth magnetic field. At the polar regions magnetic mirrors exist and the curvature drift lead to an equatorial current. Magnetic confinement is used in the field of ion source in Electron Cyklotron Resonanz Ion Source (ECRIS) (resonance heating of plasms using rf) Confinement of ions in an Electrone Beam Ion Source (EBIS) Multicusp-Ion Sources Motion of ions in an EBIS or a Penning trap WS2011/12 4.10