Methods of plasma description (suggested reading D.R. Nicholson, chap., Chen chap., 8.4) Charged particle motion in eternal electromagnetic (elmg) fields Charged particle motion in self-consistent elmg fields Kinetic equations Fluid (hdrodnamic) equations Particle motion in eternal fields A) Homogeneous fields dv a) E = m qv dt = mv = qv mv = qv v q = m v v - fluids - 1 fluid - magnetohdrodnamics = zˆ m v = q = m v ω c z q m cclotron frequenc PČ 1
v = v ep( ± iω t+ δ ) v = v ep( iω t), c, c m iωc v = v = ± iv e q t v i e ω i ct = ωc v i c =± e ω ω c t Larmor radius r L v = = ω c mv q µ when kt v = vt = m 1 r L = ( mkt ) q 1/ 1 q q mv mv µ = r Id l = IS= Sn= π n= n T π m q (,,z) gration center DIAMAGNETIC MEDIUM PČ
b) E dv m qe ( v ) dt = + d v = q E ±ω v dt m c E = ( E,, E ) dv dt z d dt v z = q E m = ωcv v = ω v z c q E =± ω E ± m ω = ω + v c cv c v d E E v v + = ωc + dt c v = v i( t ) e ω + δ E c v =± iv e i( ω t + δ) PČ 3
v gc - station. E+ v = E gc= gration center v gc = v E drift in E field here q >, v E = ω C r L v E < ω C r L v E > ω C r L potential energ - 1 ma, 3 min when v(t=) =, then v E = ω c r L = v (ccloid), otherwise trochoid, often case v v T > v E general force e.g. gravit force 1 F mg v f = v g = q q gravit drift different direction for electrons and ions ( ) g j= nm+ m gravit current PČ 4
) Inhomogeneous mv mv F ˆ cf = r = R R R k curvature drift k k (centrifugal force) v R 1 = = q q R Fcf mv Rk k div = rot = curved field cannot be constant a) grad- drift z = z( ) = + linear approimation of field during the particle motion on the Larmor circle F = qv z( ) = qv cos ct ± rl cos ct ( ω ) ( ω ) PČ 5
= + ( r ) +... z = + ( z ) +... cos ωct = 1 mv 1 F =± qv rl = v =± v rl 1 curvature drift and grad- drift often complement each other Rk m Rk 1 v v v v + R = + R qr k b) Magnetic mirrors clindrical smmetr div = k 1 z ( rr ) + = r r z => in the neighborhood of ais r z r z r = z z r PČ 6
v r θ rl mv Fz = qv r = qv = µ = magnetic moment m µ? µ = J S S = πr = π q Invariance of µ µ v L 1 mv µ = (s trajector along the line of force) F ( µ ) = µ J q qωc q = = = T π πm dv d 1 v d m = µ m = µ v = µ dt s dt s dt d 1 1 d 1 d d dµ m m m ( ) dt dt dt dt dt adiabatic invariant v + v = v + µ = µ + µ = = PČ 7
v v = = sin v v θ Where is particle with v from area reflected? 1 1 mv = mv v = v = v sin 1 θ m = = m Rm where R m is mirror ratio, it defines loss cone for θ, < θm particle is not trapped Adiabatic invariant quantit that is conserved during slow spatial and temporal variations of the sstem Classical mechanics during periodic motion, action J = pdq is conserved. PČ 8
Gration motion p= mv; q = π ω c π mv mπ J = mvd = mv sin ( ω ct) dt = = µ µ = const. ω q When is the adiabatic invariant µ not conserved? a) cclotron heating ω ω c, E, oscillates ω << ωc does not hold µ const. b) magnetic pumping varies sinusoidall in time, the invariance of µ is broken during particle collisions with moving magnetic mirror If compression (field increase) occurs during = collision, then partl v v in epansion v is unchanged c) magnetic cusp in the middle = ω c = µ const. c PČ 9
Second adiabatic invariant a, b turnabout points J b = v ds longitudinal invariant a Third adiabatic invariant v,v R, J R k 3 v d - drift in direction of angle ϕ = dl - 3. adiabatic invariant PČ 1
C) Inhomogeneous E E = ˆ cos k E = z ˆ dv ( m = q E( ) + v ) = ± rlcos ωct dt E v = = ωc v ωc cos k( ± rlcos ωct) 1 cosk 1 k r 4 L E 1 1 E ve = 1 kr 1 L = + rl 4 4 Polarization drift (temporall varing E) E E t = zˆ E() t = Et ˆ PČ 11
mv = qe + qv v= v + v ˆ ˆ E+ vp assumption v p = const. m v +v = qet ˆ+ qv qv ˆ+ qv ˆ ( ) E E p mv = qv cclotron rotation mv v ˆ E = q p v p = polarization drift = q ˆ v ˆ set qs E v E E drift Et E E me 1 m v ˆ ˆ E= = v v E = p = = E q q m d Mi ne de ρm de v = E J = ne(v v ) = m + = q dt Z dt dt p p e pi pe e polarization current PČ 1
PONDEROMOTIVE FORCE = low frequenc force acting on charged particles in inhomogeneous high frequenc electromagnetic field Oscillation energ of charged particles in high frequenc field is given b the particle position so it is some kind of potential energ U and there is force F = U that epels charged particles from the area of strong field. Ponderomotive force acts on an dielectric if its dielectric constant depends on densit (electrostriction)!! Ponderomotive force consists of parts force caused b particle oscillation in inhomogeneous electric field and force due to magnetic field action on oscillating particle First, we derive it force caused b particle oscillation in inhomogeneous electric field E of frequenc ω : E = E( r)cosωt mr r = r = qe = qe ( r )cosωt r = r + r 1 PČ 13
We linearize field variations in oscillation area r1 and equations of motion are mr ( + r1) = q E + ( r1 ) E cosωt qe q ( ) q mr = qe cosωt r = cosωt r r cosωt E ( E ) E mω = m = m ω q F ( ) E = E E So low frequenc force is mω 1 1 1 High frequenc magnetic field acts on oscillating particle 1 q = ( r)sinωt = rot E v = Esinωt ω m ω The force is then given b the following epression q 1 q F = qv = E sin ωt E rot E mω = mω PČ 14
The total ponderomotive force is then given b the sum of the forces 1 q ( ) 1 q 1 q Fp = FE + F = E E + E rot E = E = E mω 4 mω 4 mω We have used a real amplitude during the derivation, however, the field ma be phase shifted in general, thus, the general epression includes absolute value of the field amplitude. Low frequenc force acting on the particle thus reads, as follows F p q = 4mω E F p = W 1 1 qe 1 q Wosc = mv = m t = E m ω 4 mω osc cos ω force equal to -gradient of potential energ There eists also high frequenc force of frequenc ω. For field containing frequencies there forces with sum and difference of ω. PČ 15