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Magnetic Confinement of Charged Particles

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Magnetic Confinement of Charged Particles Magnetic confinement of charged particles Thomas Sunn Pedersen Max-Planck Institute for Plasma Physics and University of Greifswald, Germany Overview •Motivation for studying single particle orbits in magnetic fields •Motion in a straight uniform B-field (basic review) •The guiding center approximation •Motion including a static electric field, or gravity •Motion in inhomogeneous and bent B-fields and the first adiabatic invariant •Mirror confinement •Time variation (no derivation) Lecture on guiding center approximation 2 The need for magnetic field confinement •The easiest fusion process to reach is D-T fusion •This requires particle kinetic energies in the range 10-100 keV •Even at the particle energy of peak D-T fusion reactivity, non-fusion collisions (scattering) dominate over the fusion collisions By two orders of magnitude •Must confine plasma at T>10 keV (~120 M Kelvin) for many collisions •Thermal speed of D and T is on the order of 106 m/s at these temperatures (and even higher for electrons) – μs confinement if you have no confining field? •Electric fields alone won’t work: Confine only one species •Magnetic fields may work (must Be Bent!) •Gravity works on the sun, But not on Earth Lecture on guiding center approximation 3 Charged particle motion in a straight magnetic field •A charged particle performs a screw-like path if it is confined by a straight uniform magnetic field and it feels no other forces •Start with Newton’s 2nd law and the Lorentz force: F = ma = qv × B B B zˆ = 0 € Lecture on guiding center approximation 4 Charged particle motion in a straight magnetic field •Newton’s 2nd law written coordinate by coordinate: F = ma = qv × B ⇒ dv qB x = 0 v dt m y dv qB y = − 0 v dt m x dv z = 0 dt € Lecture on guiding center approximation 5 Charged particle motion in a straight magnetic field •Newton’s 2nd law written coordinate by coordinate: •qB0/m is an inverse time scale dv qB x = 0 v dt m y dv qB y = − 0 v dt m x dv z = 0 dt € Lecture on guiding center approximation 6 Charged particle motion in a straight magnetic field •qB0/m is an inverse time scale: give it it’s own symbol dv x = ω v dt c y dv y = −ω v dt c x dv z = 0 dt qB q B ω = 0 - sometimes ω = 0 c m c m Lecture on guiding center approximation 7 € Charged particle motion in a straight magnetic field •Decouple vx and vy equations: d # dv & x = ω v dt $% dt c y '( dv y = −ω v dt c x dv z = 0 dt qB q B ω = 0 - sometimes ω = 0 c m c m Lecture on guiding center approximation 8 € Charged particle motion in a straight magnetic field •Eliminate vy from vx equation by differentiation and substitution •Vz equation is trivial 2 $ d vx dvy 2 2 = ω c & dt dt d vx 2 % ⇒ 2 = −ω cvx dvy dt = −ω v & dt c x ' dv z = 0 ⇒ v = constant = v dt z || qB ω 2 = ( 0 )2 c m Lecture on guiding center approximation 9 € Charged particle motion in a straight magnetic field •Eliminate vy from vx equation by differentiation and substitution: " dvx = ωcvy $ 2 dt $ d vx 2 # ⇒ 2 = −ωc vx dvy $ dt = −ωcvx dt %$ •We recognize the simple harmonic oscillator for vx •Find vy by differentiation •Vz equation is trivial: dv z = 0 ⇒ v = constant=v dt z || 2 qB0 2 ωc = ( ) Lecturem on guiding center approximation 10 Charged particle motion in a straight magnetic field •Eliminate vy from vx equation by differentiation and substitution: dv " d 2v x = ω v $ x = −ω 2v dt c y $ dt 2 c x # ⇒ dvy $ 1 dvx = −ω v vy = c x $ dt % ωc dt •We recognize the simple harmonic oscillator for vx •Find vy by differentiation •Vz equation is trivial: dv z = 0 ⇒ v = constant=v dt z || 2 qB0 2 ωc = ( ) Lecturem on guiding center approximation 11 Charged particle motion in a straight magnetic field •Velocity components: vx = v⊥cos(ω ct +δ) vy = −v⊥sin(ω ct +δ) vz = v|| qB ω 2 = ( 0 )2 c m •Next step: integrate to get position € Lecture on guiding center approximation 12 Charged particle motion in a straight magnetic field •Integrate in time to get position •Define Larmor radius and guiding center: v⊥ vx = v⊥ cos(ωct +δ) ⇒ x = xgc + sin(ωct +δ) ωc v⊥ vy = −v⊥ sin(ωct +δ) ⇒ y = ygc + cos(ωct +δ) ωc vz =v|| ⇒ z = z0 +v||t qB ω = 0 (cyclotron frequency), f = ω / 2π c m c c v⊥ mv⊥ rL = = (Larmor radius, gyroradius) ωc qB0 Lecture on guiding center approximation 13 The guiding center approximation: •If the Larmor radius is on the order of the size of the confining field, then only collisionless orbits are confined: •After a collision, the particle will have a new guiding center, about one Larmor radius away from the original guiding center •If this new Larmor orbit intersects material walls or extends to regions of much lower B-field, the particle is not confined •In order to confine charged particles magnetically for many collision times, the Larmor radius must be small! •When the Larmor radius is small, and the cyclotron frequency is large: • one can derive analytic formulas for the time evolution of the guiding center (xgc, ygc, z), averaging over the gyration ∂B 1 r << B / ∇B and ω >> (*) L c ∂t B Lecture on guiding center approximation 14 The guiding center approximation is often necessary: •For a 1 eV electron in a B=1 T field, the cyclotron frequency is large and the Larmor radius small: 11 −1 ω c ≈1.8 ×10 s −6 rL ≈ 3 ×10 m •Just to follow the electron for one microsecond requires >106 time steps if a simple numerical scheme is used. •Almost the full computational€ effort is spent calculating the circular motion…. •Averaging over the gyromotion allows fast and accurate calculations of the motion of charged particles in a magnetic field, both analytic and numerical •Essentially, the gyrating particle is replaced by a charged (q), massive (m) ring of current (I=eωc/2π) , with its center at the particle’s gyrocenter. •We will do this for a few important cases in the following: Lecture on guiding center approximation 15 ExB drift •Since particles are charged, the electric field naturally enters the equations (collective phenomena, external confining fields, single particle Coulomb interactions) •Electric field component along B gives simple acceleration or deceleration •Electric field component perpendicular to B is more interesting: v⊥ •Example below: Electron rL = ωc Lecture on guiding center approximation 16 ExB drift •Net effect is a motion in the ExB direction which has a steady state component •Let’s calculate this drift: Lecture on guiding center approximation 17 ExB drift •Newton’s 2nd law written coordinate by coordinate (again) •Let’s add an electric field now dv qB x = 0 v dt m y dv qB y = − 0 v dt m x dv z = 0 dt € Lecture on guiding center approximation 18 ExB drift dv qB x = 0 v dt m y dv qB q y = − 0 v + E dt m x m y dv z = 0 dt € Lecture on guiding center approximation 19 ExB drift •Ey/B0 is a velocity, vE; it is constant since we assumed E and B constant. dv qB x = 0 v dt m y dvy qB0 Ey = − (vx − ) dt m B0 dv z = 0 dt Lecture on guiding center approximation 20 ExB drift •Ey/B0 is a velocity, vE; it is constant since we assumed E and B constant. dv qB x = 0 v dt m y dv qB y = − 0 (v − v ) dt m x E dv z = 0 dt Lecture on guiding center approximation 21 ExB drift •E/B is a velocity, vE; it is constant since we assumed E and B constant. •Since vE is constant, we can subtract it inside the d/dt of x-equation d(v − v ) qB x E = 0 v dt m y dv qB y = − 0 (v − v ) dt m x E dv z = 0 dt Lecture on guiding center approximation 22 ExB drift •E/B is a velocity, vE; it is constant since we assumed E and B constant. •Since vE is constant, we can subtract it inside the d/dt of x-equation d(v − v ) qB x E = 0 v dt m y dv qB y = − 0 (v − v ) dt m x E dv z = 0 dt Now € Lecture on guiding center approximation 23 ExB drift •E/B is a velocity, vE; it is constant since we assumed E and B constant. •Since vE is constant, we can subtract it inside the d/dt of x-equation d(vx − vE ) qB0 dvx qB0 = vy = v dt m dt m y dvy qB0 dvy qB0 = − (vx − vE ) = − v dt m dt m x dvz dv = 0 z = 0 dt dt Now Before € Lecture on guiding center approximation 24 ExB drift •It’s clear we have the same equation as before, just replacing vx by vx-vE d(vx − vE ) qB0 dvx qB0 = vy = v dt m dt m y dvy qB0 dvy qB0 = − (vx − vE ) = − v dt m dt m x dvz dv = 0 z = 0 dt dt Now Before € Lecture on guiding center approximation 25 ExB drift •It’s clear we have the same equation as before, just replacing vx by vx-vE vx − vE = v⊥ cos(ωct +δ) vx = v⊥cos(ω ct +δ) vy = −v⊥ sin(ωct +δ) vy = −v⊥sin(ω ct +δ) vz = v|| vz = v|| qB qB ω 2 = ( 0 )2 ω 2 = ( 0 )2 c m c m Now Before Lecture€ on guiding center approximation 26 ExB drift •It’s clear we have the same equation as before, just replacing vx by vx-vE v v cos( t ) vx = vE + v⊥cos(ω ct +δ) x = ⊥ ω c +δ v v sin( t ) vy = −v⊥sin(ω ct +δ) y = − ⊥ ω c +δ vz = v|| vz = v|| 2 qB0 2 2 qB0 2 ω = ( ) ω c = ( ) c m m Now Before € Lecture€ on guiding center approximation 27 ExB drift •It’s clear we have the same equation as before, just replacing vx by vx-vE v v cos( t ) vx = vE + v⊥cos(ω ct +δ) x = ⊥ ω c +δ v v sin( t ) vy = −v⊥sin(ω ct +δ) y = − ⊥ ω c +δ vz = v|| vz = v|| 2 qB0 2 2 qB0 2 ω = ( ) ω c = ( ) c m m Now Before € Lecture€ on guiding center approximation 28 ExB drift •It’s clear we have the same equation as before, just replacing vx by vx-vE Ey vx = + v⊥ cos(ωct +δ) vx = v⊥cos(ω ct +δ) B0 vy = −v⊥sin(ω ct +δ) vy = −v⊥ sin(ωct +δ) vz = v|| vz = v|| 2 qB0 2 qB ω c = ( ) ω 2 = ( 0 )2 m c m Now Before Lecture€ on guiding center approximation 29 ExB drift: Coordinate-free formulation •For situations with constant uniform E and B fields, we can always define a local coordinate system where z is in the B-field direction and y is in the direction of the component of E perpendicular to B; hence, our derivation is valid in any coordinate system.
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