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Diffusion and Transport Transport equations •For given volume define flux through surface (number of particles per area and time) • this gives the particle balance: N dA S dV t Source term S :particles that are born in plasma volume, e.g. by ionisation. •Particle balance as differential equation: N dA S dV t n dV dV S dV t n S t • similar equation for energy balance • as usual in hydrodynamics we use the ansatz Dn nv i.e. diffusion and convection separated Diffusion • diffusion coefficients from „random walk ansatz“: red: Binomial distribution x2 green: exp 2 2 N x •Step size : x , time for step: t Average time, to reach point x : For given t, x: confinement time (radius)2 t x2 t x 2 Diffusion • random walk:no net flux •With density gradient:net flux towards smaller densities 1 nAx N 2 t A t A 1 (n n)Ax N 2 t A t A n x 2 1 x n Dn 2 t x 2t x 2 D 2t Particle diffusion „random walk“ ansatz for diffusion coefficient: x: average mean free path t: time in between two collisions (inverse collision time) 2 2 v m D 2 th2 s kT m Particle diffusion and mobility Equation of motion for a particle in a plasma: For stationary plasma we obtain the particle flux: n n x p q n E m m n kT kT n m m Limit low temperature plasma Particle diffusion and mobility n Dn n n n E Diffusion coefficient Mobility constant D kT q Diffusion coefficient agrees with random walk result! Ambipolar Diffusion: assume a low temperature plasma: ions are pulled by the electrons via an electric field against the friction of neutrals Total flux of positive und negative particles from plasma must be equal (quasi neutrality!) e e ne E Dene i i ni E Di ni This is estabilshed by E-field: Ea Di De n i e n (ne=ni) ambipolar particle flux: i De e Di n Da n i e i De e Di Da i e ambipolar diffusion since: and using e >> i D kT q i Da Di De e Te Da Di 1 Ti In low temperature plasmas, we often have: Te>> Ti i i n Ea e e n Ea Den electrons pull ions ions try to ‚hold‘ electrons Ambipolare diffusion The outflux of electrons is substantially reduced e e n Ea Den 0 Ambipolar diffusion e e n Ea Den 0 e n Ea Den n / n Ea e / De eEa / kT Electron density is Boltzmann-distribution: ne ne 0 e e Ea ( x )dx / kT Heat transport heat transport: similar ansatz as for particle transport: Heat transport coefficient (conductivity) per particle qn n Tn n 1 Ws m s m m2 s Electron heat conductivity usually dominates: e v the ee 2 Ce 5/ 2 kTe ne Ci / Ce me / mi Z 2 Electric resistance of plasmas Ohm‘s law : j E bzw. j E j e ne v e e ne E e 2 ne j E m e resistivity: me || stoß 2 ne e Electric resistance of plasmas For ionised plasma: consider only Coulomb collisions: me1/ 2 e1/ 2 Z || K ln 3/ 2 2 T 0 e Te in eV 0,52 10 5 ln Z Te 3/ 2 m • is independent of particle density • decreases stronlgy with increasing electron temperature (~Te3/2) For low ionisation fraction, friction with neutrals has to be considered Neutra lg as me ( ei en ) 2 ne e Summary: diffusion and transport n Dn n n n E ambipolarity constraint gives: i De e Di Da i e Heat conduction qn n Tn 1 Ws m s m e v the 2 ee Ce 5/ 2 kTe ne n Ci / Ce me / mi Z 2 Electrical resistance: j E bzw. j E me || stoß 2 ne e m2 s