The ideal Maxwellian plasma

Transcription

1 The ideal Maxwellian plasma Dr. L. Conde Departamento de Física Aplicada. E.T.S. Ingenieros Aeronáuticos Universidad Politécnica de Madrid

2 Plasmas are,... The plasma state of matter may be defined as a mixture of positively charged ions, electrons and neutral atoms which constitutes a macroscopic electrically neutral medium which responds to the electric and magnetic fields in a collective mode Properties Charged particles interact through long distance electromagnetic forces in addition to short range molecular collisions. The density of negative n e and positive n i charged particles are equal, so that on the average the medium is electrically neutral (quasineutrality). The response to external perturbations is collective, large number of charges are involved. We may have multicomponent plasmas, ions with negative charge, dusty plasmas (complex plasmas),...etc.

3 Practical considerations... For simplicity, we will limit to one component classical plasmas with single charged ions. The particles in our plasmas α= a, i, e will be electrons e, ions i and eventually neutral atoms a with number densities nα The particle temperatures equivalent to the average kinetic energy of particles are usually expressed in ev and 1 ev = K, e k B T = , 0 = , K The ion charge is Qi = ez but in most cases we will consider only single charged ions (Z=1). We will use MKSC unit system.

4 The equilibrium state of a neutral gas,... The particle energy distribution function of equilibrium system is Maxwellian with temperature k B T 3/2 ma f a (v) = exp m av 2 2π k B T 2k B T g a (E) = 2 π E (k B T ) 3/2 exp ( E/k BT ) dn a = n ao f a (v) dv dn a = n ao g a (E) de The normalized probability distribution 0 dn a g a (E) de = particles with + f a (v) dv =1 (v, v + dv) (E, E + de)

5 The macroscopic properties,... The macroscopic physical properties are calculated as averages of the particle energy distribution function. The average energy per particle, E = 0 v ave = v v = E i = n ao E = 3 2 n ao k B T The thermal speed redefines the velocity distribution function, 2 kb T 1 v th = f a (v) = ( π v th ) exp v 2 /v 2 3 th m a The velocity averages, Eg a (E) de = 3 2 k BT define the particle flux, v x = v y = v z =0 8kB T π m a v x = 2 kb T π m a Γ a = 1 2 n ao v ave J α = q α Γ α = q α 1 2 n ao v ave

6 The equilibrium of a ideal Maxwellian plasma,... The energy distribution function of charged particles ions and electrons is Maxwellian. All particle groups have the same kinetic temperature K B T e =K B T i =K B T and K B T is the temperature corresponding to the thermodynamic equilibrium. Plasma quasineutrality, ne ni implies that no electric fields exists in the equilibrium plasma bulk; uniform plasma potential. No transport; no particle currents.

7 The Maxwellian plasma in a external electric field... Background equilibrium, n eo n io E α = m αv 2 2 n α = n αo 0 + q α φ(x) g α (E) de eφ n e (x) =n eo exp k B T n i (x) =n io exp eφ k B T In cold plasma k B T 0 the charge separation is complete while in a finite temperature plasma k B T 0 the thermal and electrostatic energies compete. This permits to shield out the small amplitude fluctuations of electric and magnetic fields.

8 The length and time scales,... The damping and/or attenuation of small amplitude fluctuations of the equilibrium takes place over a characteristic length and time scales. Space fluctuations: Debye length Time fluctuations: Plasma frequency Number of charges: Plasma parameter λ De = o k B T e 2 n eo λ Di = o k B T e 2 n io k B T e k B T i λ De λ Di ω pi = e 2 n io o m i ω pe = e 2 n eo o m e ω pi ω pe = me m i

9 The Debye length and shielding,... The initial equilibrium: E o =0 n eo n io = n o k B T e ; k B T i where we introduce an small perturbation in the electric charge, δρ ext = q δ(r) δρ sp = e [n i (r) n e (r)] E(r) E o + E 1 (r) E 1 (r) = ϕ 1 (r) and E1(r) represents the perturbed electric field. The field E1(r) is governed by the Poisson equation, E 1 = q o δ(r)+ e o [n i (r) n e (r)]) where, E = δρ ext + δρ sp n α (r) =n αo exp o ± e ϕ 1(r) k B T α Small amplitude perturbations of the charge/electric field means that the thermal energy e φ(r) dominates over the electrostatic energy kbt α = e, i e ϕ 1 (r) k B T α 1 n α (r) n αo 1 ± e ϕ 1(r) we approximate k B T α

10 2 ϕ 1 (r) = 1 o δρ ext + e 2 n o ϕ 1 (r) k B T e + e 2 n o ϕ 1 (r) k B T i 1 Λ 2 = 1 λ 2 De + 1 λ 2 Di λ Dα = o k B T α α = e, i that introduces the Debye lengths, 2 1Λ 2 e 2 n o ϕ 1 (r) = q o δ(r) ϕ 1 (r) = Setting, q e r/λ 4π o r The perturbations of the electric charge and/or plasma potential decay in space with an exponential rate governed by λd The Debye length accounts for thermal effects, the size λd of the perturbed region increases with kbt This is a linearization only valid whereas, e ϕ 1 (r) k B T α 1 We deal with a non linear Poisson equation otherwise,

11 The plasma frequency,... Again the initial equilibrium: E o =0 n eo n io = n o k B T e ; k B T i and we consider a one small perturbation of the electric charge in one dimension as in the figures, ρ = en o (A δx) E 1 ds = AE 1x = ρ = e n o (A δx) o o S The electric field, E 1x = e n o δx o gives us an equation of motion for ions or electrons d 2 e 2 dt 2 (δx)+ n o (δx) =0 m α o and defines the electron/ion plasma frequencies α = e, i ω pα = e 2 n o m α o

12 The plasma length and time scales,... The Debye length λd (also called debye radius) determines the spatial scale where the small amplitude perturbations of the electric field are shielded out. The plasmas have two time scales; ion motion τpi = 1/fpi the (slow) and the electron response τpe = 1/fpe (fast). The ion motion is frozen over the time scale τpe = 1/fpe the electron time scale represents the faster plasma response, ω ie ω pe = me m i = = = (Argon) Alternatively, the time scale the τpe = 1/fpe is proportional to the time that a thermal electron travels along a Debye length, λ De V th = o k B T e /e 2 n o 2 kb T e /m e = 1/2 o m e 2 n o e 2 = 1 2 ωpe = τ pe 2 π 2

13 Approximate magnitudes in some typical plasmas... Plasma ne (cm -3 ) KBTe (ev) λde (cm) fpe (Hz) neλde 3 Interstellar gas , Solar corona ,2 2, Solar atmosphere Gas discharge , , Tokamak , , Values from NRL Plasma Formulary This practical reference could be downloaded for free at;

14 The plasma and coupling parameters,... In order to shield out the electric field perturbations a minimum number of charges are needed within a sphere of radius λd. This defines the plasma parameter ND for ions and electrons, N De = 4π 3 n o λ 3 De k B T e k B T i λ De λ Di N De N Di The plasma parameter is closely related with the coupling parameter ΓC the ratio between thermal and electrostatic energies. Γ C = r c r d 1/3 r d no average packing of charges rc is the closest colliding approach as in the figure E(r, v α )= m α v 2 α 2 e2 4π o r E(r c,v th )=0 r c = e 2 4π o k B T Γ C = e2 n 1/3 o 4π o k B T

15 For the coupling parameter we also have, no e 2 o k B T Γ 3 c = 1 (4π) 3 1 n 2 o and we obtain, Γ c = 1 36 π N 2 D 3 = 1 (4π) 3 1 n 2 o λ 6 D equivalent to, = 1 (4π 9) 3 4π n o λ 3 D Γ C = e2 n 1/3 o 4π o k B T 2 Description Plasma parameter Limits ΓC >> 1 (ND << 1) ΓC >> 1 (ND >> 1) Coupling Strongly coupled Weakly coupled Debye sphere Sparsely populated Densely populated Electrostatic influence Strong Weak Characteristic Cold and dense Hot and diffuse Examples Laser ablation plasmas Inertial fusion experiments White dwarfs Neutron stars Ionospheric plasmas Magnetic fusion experiments Space plasmas Electric discharge plasmas

16 Magnetized plasmas,... The force experienced by electric charges in this course, r L,α = m α v q α B α = e, i F α = q α n α (E + v α B) this gives the cyclotron frequency, Ω α = q α B /m α and using the perpendicular speed to the magnetic field lines we obtain the Larmor radius a new length is defined using the particle thermal speed R L,α = V th,α Ω α Magnetized electrons and unmagnetized ions R L,e R L,i >L R L,i = mi m e R L,e Magnetized electrons and ions R L,e R L,i <L

17 The plasma skin depth,... dv ey = e E y (x, t) dt m e B z x = eµ o n eo v e B z = E y t x We consider the plasma electrons B = B z (x, t) u z B z (0,t)=B o (t) We also have, B = µ o J e + 1 c 2 2 E y x 2 B t = E E(x, t) =E y (x, t) u y E t en eo v e and the equation of motion for electrons = e2 m e n eo µ o E y = 1 e 2 n eo c 2 o m e E y 2 E y x 2 = ω pe c 2 Ey B z (x, t) =B o (t) e x/λ p E y (x, t) =λ p ( B o / t) e x/λ p λ p = c ω pe The magnetic field exponentially decreases in the plasma along the so called skin depth of London characteristic length

18 Back to plasmas in nature and in the laboratory,... Plasmas are roughly classified according to the temperature and charged particles densities Fusion reactor: KT 10 4 ev, n cm -3 Laser plasmas: KT 10 2 ev, n cm -3 Glow discharge: KT 1-3 ev, n 10 8 cm -3 Ionosphere: KT 0,05 ev, n 10 6 cm -3 Water at room temperature: KT 0,025 ev, n cm -3

19 Hollow Ioniz Plasma stream of Hollow cathode Solar corona KT 100 ev, n 10 9 cm -3 Low pressure discharges KT 1-3 ev, n 10 8 cm -3 Plasma stream of Ioniza Fusion reactor KT 10 4 ev, n cm -3

20 The Earth ionospheric plasma,... The black curves are for neutral particles and the red line is the altitude dependent electron density. The sum of all different ion densities ni equals the electron density ne The ions are produced by the absorption by the neutrals of parts of the solar radiation spectrum. The maximum rate takes place at the F1 and F2 peaks.