Fundamentals of Plasma Physics Transport in weakly ionized plasmas
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1 Fundamentals of Plasma Physics Transport in weakly ionized plasmas APPLAuSE Instituto Superior Técnico Instituto de Plasmas e Fusão Nuclear Luís L Alves (based on Vasco Guerra s original slides) 1
2 As perguntas fundamentais Diffusion and mobility We have studied waves in idealized media, infinite and homogeneous Any more realistic situation involves density gradients The plasma tends to diffuse to the lower density regions We shall start with the case B= and weakly ionized plasmas (δe<10-3 ) 2
3 As perguntas fundamentais Weakly ionized plasmas There is a non-uniform distribution of electrons and ions over a neutral background The plasma diffuses as a result of the pressure and electric forces, in a highly collisional process The collisions of charged particles with neutrals are dominant, thus controlling the collision parameters 3
4 Collision As perguntas parameters fundamentais Cross section All the collision dynamics is contained in the differential collision cross section, σ(v,θ,φ), which can be calculated if we know the interaction potential It usually depends on the collision energy It represents an effective (collision) area and it has dimensions L 2 The integral cross section is obtained upon integration of σ(v,θ,φ) over all angles 4
5 Collision As perguntas parameters fundamentais Cross section In the hard spheres model, σtot=πd 2 We can define a cross section for momentum transfer 5
6 Collision As perguntas parameters fundamentais Mean free path Consider an electron beam with a given speed, arriving at a target of area A and thickness dx, with n atoms per unit volume, each with total cross section σ, placed perpendicularly to the beam -The number of atoms in the target is nadx -The probability that the electrons makes a collision while crossing the slab is (nadx)σ/a 6
7 Collision As perguntas parameters fundamentais Mean free path -If a flux I of electrons hits the target, the flux I crossing it without making a collision is given by I = I (1-nσdx) di = - I nσdx I(x) = I0 exp(-nσx) I0 exp(-x/λ) 7
8 Collision As perguntas parameters fundamentais Mean free path and collision frequency An electron has a reasonable probability of making a collision after traveling a distance equal to the mean free pathλ The average time between collisions is The collision frequency is 8
9 Collision As perguntas parameters fundamentais Mean free path and collision frequency In general We have also The mean frequency is obtained integrating ν(v) over the velocity distribution of the electrons (often a Maxwellian): 9
10 Collision As perguntas parameters fundamentais Diffusion and mobility The force equation is We assume: - ν = constant Mean coll frequency -collisional regime (small v and/or large ν): a fluid element does not move to regions of very different E and P within the collision time 10
11 Collision As perguntas parameters fundamentais Diffusion and mobility -Stationary regime -Isothermal plasma (γ=1) In these conditions, d/dt = 0 and we obtain the flux where the mobility is 11
12 Collision As perguntas parameters fundamentais Diffusion and mobility and the diffusion coefficient is The mobility and diffusion coefficients are related by the Einstein relation This approach corresponds to the drift-diffusion approximation 12
13 Collision As perguntas parameters fundamentais Diffusion and mobility The mobility and diffusion coefficients can be calculated from the kinetic theory, in particular from integrals over the distribution function The characteristic energy, uk=ed/µ, corresponds to k B T in the case of a Maxwellian distribution and to if ν is independent from v 13
14 Diffusion As perguntas parameters fundamentais Fick s law If E = 0 and/or the particles are neutral, we recover Fick s law Diffusion is a random walk process Plasmas can exhibit a collective behavior, and in this case diffusion is not a completely random process 14
15 Decay As of perguntas a plasmafundamentais Ambipolar diffusion We want to know how a limited plasma decays, due to diffusion to the walls of the container Electrons and ions recombine when they reach the walls The density of the charged particles at the walls is approximately zero Assume the drift-diffusion approximation 15
16 Decay As of perguntas a plasmafundamentais Ambipolar diffusion ors s If L λd The plasma is quasi-neutral, ni ne Γi Γe (hypothesis of congruency) The electron and ion diffusion rates adjust to become equal! (consequences for processing) 16
17 Decay As of perguntas a plasmafundamentais Ambipolar diffusion The physical picture: the lighter electrons arrive first at the wall the positive ions are left behind an ambipolar electric field is established, which tends to retard the electron motion and accelerate the ion motion towards the wall 17
18 Decay As of perguntas a plasmafundamentais Ambipolar diffusion We get If Da is constant, 18
19 Decay As of perguntas a plasmafundamentais Ambipolar diffusion Since µi µe If Te=Ti, Da 2Di 19
20 Decay As of perguntas a plasmafundamentais Diffusion eigenmodes Diffusion in a bounded system is made according to several eigenmodes, each with a characteristic decay time The fundamental mode has the longest decay time For a given initial electron density profile, after a while the plasma is diffusing according to the fundamental mode 20
21 Decay As of perguntas a plasmafundamentais Diffusion eigenmodes: slab solution 21
22 Decay As of perguntas a plasmafundamentais Diffusion eigenmodes: cylinder solution n(r,t) = n 0 exp(-t/τ) J 0 (2.405 r/a) 22
23 As perguntas fundamentais Steady-state solutions In order to keep the plasma and reach steady-state there must be ionization sources and/or injection of plasma The continuity equation has to include the term describing the creation of new electrons, S(r): In steady-state, 23
24 Steady-state As perguntas solutions fundamentais Constant ionization frequency In many cases the ionization is produced by energetic electrons in the tail of the (Maxwellian) distribution function The ionization rate is nνi νi is the (mean) ionization frequency, obtained by integrating the corresponding cross section Steady-state is described by 24
25 Steady-state As perguntas solutions fundamentais Constant ionization frequency The steady-state spatial profiles correspond to the profile of the fundamental diffusion mode! NO DECAY, though There must exist an external energy source that keeps the electron temperature constant and compensates the diffusion losses (recall νi = n gas <σ i v>) 25
26 Steady-state As perguntas solutions fundamentais Localized sources For ionization sources localized in space we have to solve the equation which is valid in all points except on the positions of the sources 26
27 Steady-state As perguntas solutions fundamentais Localized sources: plane source Slit-collimated beam of UV light 27
28 Steady-state As perguntas solutions fundamentais Localized sources: line source Beam of energetic electrons 28
29 Steady-state As perguntas solutions fundamentais Volume recombination We may also have charge losses in volume, due to electron-ion recombination In this case there is a loss term proportional to nine = n 2 In the absence of diffusion, the continuity equation is where α [m 3 /s] is the recombination coefficient 29
30 Steady-state As perguntas solutions fundamentais Volume recombination There must be a third particle, to conserve the momentum of the collision, which may be -an emitted photon (radiative recombination) -a neutral particle (3 body recombination) -a dissociation product for molecular gases and ions (dissociative recombination) 30
31 Steady-state As perguntas solutions fundamentais Volume recombination: the recombination coefficient 31
32 Steady-state As perguntas solutions fundamentais Volume recombination The solution is, c1=1/n0 or where n0(r) is the initial radial distribution For times long enough, which allows distinguishing the decay due to diffusion! 32
33 Steady-state As perguntas solutions fundamentais Volume recombination 33
34 Steady-state As perguntas solutions fundamentais Volume recombination 34
35 As perguntas fundamentais Diffusion in a magnetic field Diffusion is modified in the presence of a magnetic field Motion in the direction parallel to B (uz) is not affected Without collisions there is no diffusion in the direction perpendicular to B (the particle gyrates along the same B line) 35
36 As perguntas fundamentais Diffusion in a magnetic field 36
37 As perguntas fundamentais Diffusion in a magnetic field Collisions make the particles migrate across the lines of B! The phase of the rotation movement changes abruptly (and the Larmor radius may also vary), in a random walk -like process The particle diffuses in the direction perpendicular to B in the opposite direction to n The length scale of the random motion is rl, not λ! 37
38 As perguntas fundamentais Diffusion in a magnetic field We can reduce the diffusion across the magnetic field by reducing rl, i.e., by increasing B 38
39 As perguntas fundamentais Diffusion in a magnetic field Assuming (as before) - an isothermal plasma (γ = 1) - high enough collision frequency ν (so that we can neglect dv /dt), 39
40 As perguntas fundamentais Diffusion in a magnetic field The perpendicular velocity has two components: -Drifts parallel to the potential and density gradients, similar to the case B=0 but with mobility and diffusion coefficients reduced by a factor 1+(ω c /ν) 2 -Drifts (ExB and diamagnetic), in the directions perpendicular to the potential and density gradients, slower than in the collisionless case, by a factor 1+(ν/ω c ) 2 40
41 As perguntas fundamentais Diffusion in a magnetic field The factor ω c /ν =ω c τ =µb λ/rl is relevant: -ω c τ 1, B does not affect diffusion -ω c τ 1, B significantly slows down diffusion Increasing B hinders diffusion across the fieldlines Decreasing collisionallity (ν) hinders diffusion across the field-lines 41
42 As perguntas fundamentais Diffusion in a magnetic field For ω c ν 1, we have Comparing with the case B=0 -the dependency with ν is inversed -the dependency with m is inversed 42
43 As perguntas fundamentais Diffusion in a magnetic field Electrons diffuse faster than ions in parallel direction, but diffuse slower in the perpendicular one Finally, which shows we have random walk processes, with space scales λ and rl 43
44 Diffusion As perguntas in a magnetic fundamentais field Ambipolar diffusion Ambipolar diffusion in the presence of a magnetic field... is rather complex :) The total diffusion losses must be ambipolar, but not the losses of each component ( or )! For instance, the ions may be essentially lost by radial diffusion to B and the electrons by diffusion along B The required condition is Γi = Γe 44
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