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Characteristic Properties of Plasma The study of the behaviour of the plasma is based on the description of the collective behaviour of the particles that compose it. It passes from the microscopic description of the plasma characteristic phenomena to a macroscopic description. One of the main properties of plasma is its tendency to charge neutrality. When in a volume of appreciable dimensions the density of electrons and the density of positive charges (ions) differ, an electrostatic force with a potential remarkably higher than thermal energy, is induced: ∇ ⋅ E = (n i − n e ) e ε0 where n e and n i are the electron density and the ion density. Unless special mechanisms will not support this high electric field, rapidly the charged particles moves to reduce these fields, and rebuild electrical neutrality. 1 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna ΔE ≈ e ε0 n Δx For regions of dimensions of the order of a centimetre and deviation from neutrality with densities of 10 18 m-3, fields of the order of 100 MV/m are created. Usually deviations from neutrality can be created in very small regions, such that the energy required to support the corresponding electric potential is the thermal energy. When perturbations of charge neutrality occur in those regions, immediately the charged particles tend to restore neutrality by crossing again the mentioned regions with thermal speed. The phenomenon goes on up to the constitution o a charged region in the opposite direction. Then the charges travel back again and an oscillatory motion is induced. 2 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 1 Debye length In unperturbed conditions it is n e = n i = n (n e, n i electron and ion densities). For a disturbance causing a charge separation region of size x0 (see figure), the force on to charge - e at the position x within the range (0, x0), is expressed by: The word to move the particle from 0 to x0 is: The Debye length λ D is the dimension of the region of charge separation that can be created using the medium thermal energy. In the x-direction it is: 3 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna In 1929 Langmuir introduced the term plasma to totally or partially ionized gases for which λ D is sufficiently small compared to the other macroscopic lengths of interest (for example, the characteristic length of the electron density variation). In these conditions it is possible to assume charge neutrality for which ne = ni (ne electron density, ni singly ionized ion density). 4 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 2 Plasma frequency A plasma of density n = n e = n i with a one-dimensional geometry is considered. For a displacement x of the electrons for Gauss law it is: (motion law of an electron) This is the equation of a harmonic motion with angular speed ω p, that is said plasma frequency: ω p = e2 n ε 0 me 5 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna Debye length and plasma frequency When the electron density and the ions density differ, electrostatic forces are produced. They induce a motion of the charged particles so as to reduce quickly these forces and reconstitute electrical neutrality. The dimension of the shift from charge neutrality in a plasma is given by λD. The shift from charge neutrality produce a force that tends to restore neutrality. Electrons are pushed backwards to retrace the distance λD going ahead to constitute a shift from neutrality in the opposite direction. Thus an oscillatory motion is originated with a frequency ωp. The speed of the motion through λD is given by: 6 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 3 Shielded Coulomb potential A charge within an ionized gas attracts in its around charged particles of opposite sign. The electric potential which it induces, is given by Poisson's law: In a plasma at thermodynamic equilibrium the electrons and the ions are subject to an energetic distribution that depends on the electric potential. This distribution is given by the Boltzmann distribution of. In an electrically neutral plasma it is n i = n e = n and: Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna Shielded Coulomb potential When it is set g(r) = r f(r), we obtain: For a charged particle radius r0, when position r near the particle given by r-r0 is sufficiently small the shielding effect must be negligible: φ (r) = ( q exp - 2 r/λD 4πε 0 r ) Shielded Coulomb potential Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 4 Sheaths Solid surface 0φ0- ne = ni = n x φ(x) A region where there is not charge neutrality, is in the vicinity of solid surfaces. The thickness of this region is of the order of λD. The region is called sheath or plasma sheath. 9 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna Sheaths In the plasma around a conductive element (an electrode) when is not heated and does not emit charged particles, and when it is floating (it is not referred to ground): T = Te = Th and for x >> λD: 0φ0 - T = Te = Th ne = ne = n x φ(x) n = ni = ne . Because Te = Th , initially the electron flow towards the surface is much bigger than the ion one as <v e> ≫ < v i >. The floating electrode becomes charged negatively and acquires a negative potential in order to reduce the electron flow that must exactly balance the ionic. Indeed, under stationary conditions, there must be no further accumulation of charge. The potential on the floating electrode becomes φ0 and potential distribution in the sheath becomes φ (x). 10 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 5 Sheaths Only the electrons with speed directed toward the surface, with sufficient kinetic energy to overcome the potential barrier given by φ0 (εc = mev2 ex0 /2 = -e φ0 ), come to the electrode. 0φ0 - T = Te = Th ne = ni = n x φ(x) Near the wall, within the sheath, the electron and ion fluxes in the x-direction are: From the condition Γ ex = Γ ix , it follows: 11 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna Sheaths From the Poisson’s equation it is: 0φ0 - T = Te = Th ne = ni = n x φ(x) Therefore it follows: where 12 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 6 Electrical conductivity • • Partially ionized plasma with electrons, ions and neutral particles with densities: n e, n i, n n . elastic collisions with frequencies: νei, νen , νeH = νei+νen • v e ≫ v i, v n; v i ≈ v n ≈ u. The law of motion of an electron accelerated by the electric field E and decelerated by collisions is + d ve = - e E + FC, e (t) dt -eE _ - + me eE FC,e(t) is the force due to the collisions of electrons with heavy particles. 13 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna Electrical conductivity The value of the collisional force FC,e(t) is averaged in a time much higher than the time between two consecutive collisions ( characteristic time Δt ≫1/νeH) . <FC,e(t)> is given by the number of collisions (assumed elastic) that an electron undergoes in the time unit, times the momentum loss per collision (total momentum loss of an electron per unit of time due to collisions): FC,e (t) = By assuming: Δpeh Δt = -m e Q(1) ei Q (e) ei (U e -U i )ν ei - m e Q(1) en Q(e) en (U e -U n )ν en Qei(1) ≈ Qei(e), Qen (1) ≈ Qen (e) , v e ≫ v i, v n; v i ≈ v n = u: FC,e (t) = - m e U e ν eH For t≫ Δt and sufficiently large such that equilibrium is reached between the electric field accelerating force that and the decelerating collisional force: me d ve = - e E + FC,e (t) = 0 dt 14 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 7 Electrical conductivity From the definition of current density due to the electrons (usually the ionic contribution to the current is neglected) it results: where σ e is the electrical conductivity due to the electronic contribution. Hence σ e is given by : σe e2 ne = m e ν eH and, when the ionic contribution to the current is neglected, Ohm’s law is. : J = σE Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna Electrical conductivity In a plasma, dominated by Coulomb collisions, the electron-ion collision frequency νei and the electrical conductivity σe due to the electron flow are: ν ei = 3.64 × 10 -6 lnΛ σe = e2 ne m e ν ei ni Z T3 = 7.739 × 10 -3 ne Z n i lnΛ T3 un plasma ioni ionized ionizzatiions una sola (Z = 1) ed elettricamente In Per a plasma withcon singly (Z =volta 1 ) and electrically neutral neutro (ne ≈ ni): (n e = n i) it is: σ e = 7.739 × 10 -3 1 lnΛ T3 16 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 8 Hall Parameter For the fields E and B uniform and stationary, on a free charge a drift motion perpendicular to E and B is generated. In a plasma, the comulative effect of the drift depends on collisions. Immediately after a collision the particle is accelerated by E along its direction and starts its revolution around B. Therefore the motion in the direction of E depends on the number of collisions which the charge undergoes during its revolution and becomes more important for increases of this number. The radians done two consecutive of collision during the revolution is given by the Hall parameter: ωq βq = νq where ωq is the angular velocity of the revolution and νq is the collision frequency between the particle q and the other particles. z x y B E 17 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna Hall Parameter Electron and ion Hall parameter In a plasma the order of magnitude of the Hall parameter of electrons is much higher than that of ions. This depends on the difference of the masses of these two particles. where βi and βe are the Hall parameters of electrons and ions. The collision frequency of a charged particle νq (electrons or ions) may be with a good approximation by: This is an expression valid for the order of magnitudes where it is assume that <Qe> ≈ <Qi>. 18 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 9 Hall Parameter E = Ey j, B = B k (E and B constant and uniform) Collision dominated plasmas: β e ≪ 1 The drift motion induced by B in x-direction is negligible compared to the motion in y-direction due to E. This is due to the many collisions during one revolution. Hence it follows that: z x y B E Jx ≪ Jy, Jy ≈ σ Ey. Collisionless plasmas: β i ≫ 1 The drift velocities of electron and ions in x-direction ExB/B2 are equal. The velocities of ions and electrons in y-direction is negligible as between two consecutive collisions B induces many revolutions of the charged particle. Hence it follows that: Jx = Jy = 0 19 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna Hall Parameter Ions collision dominated, electrons not: β i ≪ 1, β e ≳ O(1) The average diffusion velocity of the ions will remain in the y-direction, but because of the E × B drift, the average electron diffusion velocity will exhibit a component in the xdirection. Only a fraction of the full E × B drift velocity wD,E, is attained, because of the interruptions caused by collisions. |Jey| βe |Jex| For sufficiently large βe, this fraction is nearly equal to 1. Hence: J ex ! - e n e w D,E en e2ne σ = - e Ey = Ey = - e Ey B m eω e βe βe The current component Jex which flows in the direction mutually perpendicular to both E and B is called the Hall current, after Edwin 20 Herbert Hall who discovered this phenomenon in solid conductors in 1879. 10 Hall Parameter Electrons collisionless, ions not: β e ≫ 1, β i ≲ O(1) The ion current Jiy provides the dominant contribution to the y-component of the current, since Jey → 0. This phenomenon, called ion slip. Ion slip can occur only in a partially ionized gas. From the definition of mean mass velocity: u= ρ eue + ρ iui + ρ n un ρ ρ eU e + ρ iU i + ρ n U n = 0 |Jey| βe |Jex| for a fully ionized gas βe ρ eU e = - ρ iU i Therefore, for a fully ionized gas, as the mass density of electrons ρe is much smaller that the mass density of ions ρi, |Jey | always is much greater than |Jiy |, independently of the driving mechanisms. 21 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna Generalized Ohm’s law • • Partially ionized plasma with electrons, ions and neutral particles with densities: n e, n i, n n . elastic collisions with frequencies: νei, νen , νeH = νei+νen • v e ≫ v i, v n; v i ≈ v n ≈ u. The law of motion of an electron in a region where E and B are present, becomes: for t ≫ 1/νeH and if J ≈ Je Generalized Ohm’s law or Ohm-Hall’s law 22 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 11 Generalized Ohm’s law The generalized Ohm’s law may be written as: where for B = B k: And when defining E* = E + u × B, the generalized Ohm’s law becomes: 23 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna MHD Approximation The Magneto-fluid-dynamics (MHD) studies the behaviour of a conducting fluid in a field of fluid-dynamic and electromagnetic forces. The main forces and the main energetic processes acting on a conducting fluid are: Forces: • Fluid-dynamics: pressure gradients, frictional forces • Electromagnetics: electromagnetic fields, Lorentz forces Energy processes: • Thermal flows, work due to pressure and friction • Ohmic heating (the Joule effect) 24 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 12 MHD Approximation MHD model Fluid dynamic equations • Conservation equations of mass, momentum, and energy (Euler equations or the Navier-Stokes equations) • Equation state of the fluid (for ideal gas: p = RρT) Electromagnetic equations • Maxwell equations • Generalized Ohm's law For the definition of this model it is necessary to establish the conditions at which it has to operate. These conditions are in the field of engineering interest and leads to the determination of a model with a very wide validity field. The name of MHD approximation. 25 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna MHD Approximation Conditions a. Speed of the fluid u is much smaller than the speed of light: u ≪ c. b. Mean free path between two consecutive collisions is much smaller than the characteristic plasma dimension: LC ≫ < v s>/ν (where <vs> average speed of species s and ν is collision frequency of the species). c. Validity of Ohm's law: tC ≫ 1/νeH d. The displacement current is negligible with respect to the conduction current. e. The convection current is negligible with respect to the conduction current f. The electrostatic force is negligible with respect to the Lorentz force. 26 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 13 MHD Approximation Conditions From condition c. through f. it follows: c. tC ≫ 1/νeH d. = = = = tC ≫ ≪1 From c. and d. it follows: tC ≫ 1/ωp 27 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna MHD Approximation Conditions e. ≪ tC ≫ 1/ωp f. ≪ 1 tC ≫ 1/ωp The MHD approximation considers plasmas at speeds much lower than the speed of light, pressures that allow the use of the model of continuum ( LC ≫ <v s>/ν ) and describes the behaviour of plasmas with time resolutions far greater than the period of oscillation of the plasma around the charge neutrality configuration ( tC ≫ 1/ω p ). 28 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 14 Magnetic regimes Equations of Electromagnetism (Collision dominated plasmas approximation - βe ≪ 1 ) Magnetic regime From the above equations and from = the magnetic field equation describing the magnetic field behaviour, results to be: 29 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna A. Regime magnetico diffusivo: Magnetic regimes A. Magnetic diffusive regime (u = 0) For a steady fluid (u = 0), from the equation of the magnetic field behaviour in a medium with an electrical conductivity σ, given by: ∂B 1 = ∇2B ∂t µ 0σ This equation has the form of an equation of diffusion. It shows that in a medium of finite conductivity the magnetic flux density tends to spread in space and to decay in time. It is the equation of magnetic diffusion. 30 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 15 A. Regime magnetico diffusivo: Magnetic regimes A. B. Magnetic convective regime (σ = ∞) For a moving fluid (u ≠ 0) and a medium with infinite electrical conductivity from the equation of the magnetic field behaviour in a medium with an electrical conductivity σ, given by: ∂B = ∇ × (u × B) ∂t This equation is the equation of magnetic convection. It shows that in a conductor of infinite conductivity the magnetic flux density is moving with the fluid. 31 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna Magnetic regimes C(t+Δt) S(t+Δt) A. B. Magnetic convective regime (σ = ∞) uΔt In a time variation Δt the magnetic flux ds S(t) C(t) through the surface S with moving with the fluid. The variation of the magnetic flux ΔΦB in the time interval Δt is given by: The flux ΦB (t) changes for two reasons; first, because B = B(t) is changing, and second, because the area S = S(t) bounded by the curve C = C(t) is changing. In an interval of time Δt, each fluid particle on the curve C(t) moves distance uΔt, so that an element of length ds along C(t) sweeps out a vector element of area [ds x (uΔt)] (the surface vector direction is perpendicular to the surface of it). 32 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 16 Magnetic regimes C(t+Δt) S(t+Δt) A. B. Magnetic convective regime (σ = ∞) uΔt The decrease in the flux associated with the motion of the fluid during the time Δt is therefore: ds C(t) S(t) ∫∫ B(t + Δt) ⋅ n! dS - ∫∫ B(t) ⋅ n! dS Δ ΦB = S(t + Δt) S(t) For the first order Taylor series approximation it is B(t+Δt)≈ B(t)+𝜕B/ 𝜕t. Moreover, as B is solenoidal, its flux through S(t) is equal to its flux through S(t+Δt) plus its flux through the lateral surface obtained by the series of the vector area elements [ds x (uΔt)] along C(t). Hence it is: 33 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna Magnetic regimes C(t+Δt) S(t+Δt) A. B. Magnetic convective regime (σ = ∞) For the Stokes theorem it is: uΔt ds S(t) C(t) = = 34 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 17 Magnetic regimes C(t+Δt) S(t+Δt) A. B. Magnetic convective regime (σ = ∞) For Δt → 0 and from: uΔt ds S(t) C(t) When the equation of the magnetic convection is satisfied, it results: Therefore the magnetic flux through a surface moving with the fluid is equal to zero. Hence the magnetic lines of force move with the fluid. 35 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna Magnetic regimes Magnetic Reynolds number (effect of the flow on B) For a non-stationary fluid of finite electrical conductivity, the magnetic induction will change as a result of both convection with the fluid and diffusion through the fluid. The quantity called the magnetic Reynolds number Rm, evaluates this effect as is given by the ratio of the convective and the diffusive terms in the equation magnetic field equation. If L C denotes the characteristic length of macroscopic change, it is: 36 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 18 Magnetic regimes Magnetic Reynolds number (effect of the flow on B) From the Maxwell equation (𝛻×B = µ0 J), it BC (ind.) indicates the contribution of the magnetic field induced by the currents flowing within the plasma to the total magnetic field BC , it follows: For flows with small Rm, the convection of B lines by the fluid is negligible, and the magnetic induction produced by currents in the fluid/plasma can be neglected. 37 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna Magnetic regimes Magnetic interaction parameter (effect of B on the flow) The magnetic interaction parameter is a measure of the ratio of the J × B force to the inertia force: Magnetic interaction parameter The Hartmann number is a measure of the ratio of the J × B force to the viscous force: H = L C BC (σ C /ηC ) 2 1 38 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 19 Magnetic regimes Magnetic interaction parameter (effect of B on the flow) 39 Department of Electrical, Electronic, and Information Engineering (DEI) - University of Bologna 20