Introduction to Nuclear Engineering, Fall 2018 Plasma and Sheath Fall, 2018 Kyoung-Jae Chung Department of Nuclear Engineering Seoul National University Motions in uniform electric field Equation of motion of a charged particle in fields = ( , ) + × ( , ) , = ( ) Motion in constant electric field For a constant electric field = with = 0, ( ) = + + 2 Electrons are easily accelerated by electric field due to their smaller mass than ions. Electrons (Ions) move against (along) the electric field direction. The charged particles get kinetic energies. 2/38 Introduction to Nuclear Engineering, Fall 2018 Motions in uniform magnetic field Motion in constant magnetic field = × For a constant magnetic field = 0 with = 0, = 0 = −0 = 0 Cyclotron (gyration) frequency = = 2 2 0 2 − 3/38 Introduction to Nuclear Engineering, Fall 2018 Motions in uniform magnetic field Particle velocity = cos + = sin + ⊥ 0 = 0 −⊥ 0 Particle position = sin + + sin = cos + + cos = + 0 0 − 0 0 0 − 0 0 0 Guiding center , , + Larmor0 0(gyration)0 0 radius = = ⊥ ⊥ 0 4/38 Introduction to Nuclear Engineering, Fall 2018 Gyro-frequency and radius The direction of gyration is always such that the magnetic field generated by the charged particle is opposite to the externally imposed field. diamagnetic For electrons = 2.80 × 10 Hz in gauss 6 0 0 3.37 = cm ( in volts) 0 For singly charged ions = 1.52 × 10 / Hz in gauss 3 0 0 144 = cm ( in volts, in amu) Energy gain?0 5/38 Introduction to Nuclear Engineering, Fall 2018 Motions in uniform E and B fields Equation of motion = + × Parallel motion: = 0 and = 0 , = = + 0 Straightforward acceleration along B 6/38 Introduction to Nuclear Engineering, Fall 2018 × drift Transverse motion: = 0 and = 0 , = + = 0 0 −0 Differentiating, = 2 2 2 = − + × 2 = 2 0 2 − 0 Particle velocity = cos + = ⊥ sin +0 0 ⊥ 0 − − 0 7/38 Introduction to Nuclear Engineering, Fall 2018 DC magnetron A magnetron which is widely used in the sputtering system uses the × drift motion for plasma confinement. What is the direction of × drift motion? 8/38 Introduction to Nuclear Engineering, Fall 2018 Motions in gravitational field Generally, the guiding center drift caused by general force 1 × = If is the force of gravity , × = What is the difference between and ? 9/38 Introduction to Nuclear Engineering, Fall 2018 ⊥ : Grad-B drift The gradient in | | causes the Larmor radius to be larger at the bottom of the orbit than at the top, and this should lead to a drift, in opposite directions for ions and electrons, perpendicular to both and . Guiding center motion 1 × = ± 2 ⊥ 10/38 Introduction to Nuclear Engineering, Fall 2018 Curved B: Curvature drift The average centrifugal force = = 2 ∥ 2 � ∥ 2 Curvature drift 1 × × = = 2 1 ∥ 2 ∝ Total drift in a curved vacuum field (curvature + grad-B) × 1 + = + = 2 2 2 ∥ ⊥ 2 2 2 − 11/38 Introduction to Nuclear Engineering, Fall 2018 ∥ : Magnetic mirror Adiabatic invariant: Magnetic moment 1 2 = = 2 ⊥ As the particle moves into regions of stronger or weaker , its Larmor radius changes, but remains invariant. Magnetic mirror 1 1 2 2 2 = 2 ⊥0 ⊥ 0 = + 2 2 2 2 ⊥ ⊥0 ∥0 ≡ 0 = = = sin 2 2 0 ⊥0 ⊥0 2 2 2 ⊥ 0 12/38 Introduction to Nuclear Engineering, Fall 2018 Motions in a dipole magnetic field Trajectories of particles confined in a dipole field Particles experience gyro-, bounce- and drift- motions 13/38 Introduction to Nuclear Engineering, Fall 2018 Boltzmann’s equation & macroscopic quantities Particle density Particle flux Particle energy density 14/38 Introduction to Nuclear Engineering, Fall 2018 Particle conservation + = � − The net number of particles per second generated within either flows across the surface or increases the number of particles withinΩ . Γ Ω For common low-pressure discharges in steady-state: = , 0 ≈ Hence, = � The continuity equation is clearly not sufficient to give the evolution of the density , since it involves another quantity, the mean particle velocity . 15/38 Introduction to Nuclear Engineering, Fall 2018 Momentum conservation Convective derivative Pressure tensor isotropic for weakly ionized plasmas The time rate of momentum transfer per unit volume due to collisions with other species + = � − − 16/38 Introduction to Nuclear Engineering, Fall 2018 Equation of state (EOS) An equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions. = , , = ( , ) Isothermal EOS for slow time variations, where temperatures are allowed to equilibrate. In this case, the fluid can exchange energy with its surroundings. = , = Theenergy conservation equation needs to be solved to determine p and T. Adiabatic EOS for fast time variations, such as in waves, when the fluid does not exchange energy with its surroundings = , = = (specific heat ratio) The energy conservation equation is not required. 2 Specific heat ratio vs degree of freedom ( ) = 1 + 17/38 Introduction to Nuclear Engineering, Fall 2018 Equilibrium: Maxwell-Boltzmann distribution For a single species in thermal equilibrium with itself (e.g., electrons), in the absence of time variation, spatial gradients, and accelerations, the Boltzmann equation reduces to = 0 � Then, we obtain the Maxwell-Boltzmann velocity distribution 2 = 3 2 2 − 2 / 2 = 1 2 − The mean speed / = 1 2 8 ̅ 18/38 Introduction to Nuclear Engineering, Fall 2018 Particle and energy flux The directed particle flux: the number of particles per square meter per second crossing the = 0 surface in the positive direction / = 1 2 푡 1 = 4 Γ ̅ The average energy flux: the amount of energy per square meter per second in the +z direction 1 = = 2 2 2Γ The average kinetic energy per particle crossing = 0 in the positive direction = 2 19/38 Introduction to Nuclear Engineering, Fall 2018 Diffusion and mobility The fluid equation of motion including collisions = + = � − − In steady-state, for isothermal plasmas 1 1 = = − − = = ± − Drift − Diffusion In terms of particle flux = Einstein relation = = ± ∶ − = Mobility = Diffusion coefficient ∶ ∶ 20/38 Introduction to Nuclear Engineering, Fall 2018 Ambipolar diffusion The flux of electrons and ions out of any region must be equal such that charge does not build up. Since the electrons are lighter, and would tend to flow out faster in an unmagnetized plasma, an electric field must spring up to maintain the local flux balance. = + = AmbipolarΓ electric − field for = Γ − − = Γ Γ + − The common particle flux + = = + Γ − − The ambipolar diffusion coefficient for weakly ionized plasmas + T = + 1 + ~ T + Te e ≈ ≈ i 21/38 Introduction to Nuclear Engineering, Fall 2018 Steady-state plane-parallel solutions Diffusion equation volume source and sink = 2 − − For a plane-parallel geometry with no volume source or sink = 0 = + 2 B.C. = 0 = − 2 = Γ = /2 =Γ00 2 Γ0 − is determined For a uniform specified source of diffusing particles 0 B.C. symmetric at = 0 = 1 = 2 2 2 0 = ± /2 = 0 2 − 2 0 − The most common case is for a plasma8 consisting of positive ions and an equal number of electrons which are the source of ionization + = 0 where, = and is the ionization frequency 2 22/38 Introduction to Nuclear Engineering, Fall 2018 Boltzmann’s relation The density of electrons in thermal equilibrium can be obtained from the electron force balance in the absence of the inertial, magnetic, and frictional forces + = � − − − Setting = and assuming = −Φ = ln= 0 Integrating,Φ − we have Φ − = exp = exp T Boltzmann’s relation for electrons 0 Φ 0 Φ e ( ) For positive ions in thermal equilibrium at temperature T ( ) = exp i T 0 Φ However, positive ions are almost never in thermal equilibrium in low pressure− i discharges because the ion drift velocity is large, leading to inertial or frictional forces comparable to the electric field or pressure gradient forces. 23/38 Introduction to Nuclear Engineering, Fall 2018 Debye shielding (screening) Plasma Coulomb potential Poisson’s equation = = 1 / = 4 2 2 0 Φ 0 Φ Φ Φ − − ≈ Φ 2 Screened Coulomb0 potential Debye length0 0 / / = exp T 4 = 1 2 = 1 2 0 0 e Φ − 2 0 0 0 24/38 Introduction to Nuclear Engineering, Fall 2018 Debye length The electron Debye length is the characteristic length scale in a plasma. The Debye length is the distance scale over which significant charge densities can spontaneously exist. For example, low-voltage (undriven) sheaths are typically a few Debye lengths wide. 10 -3 Typical values for a processing plasma (ne = 10 cm , Te = 4 eV) T T eV 4 cm = = 743 = 743 0.14 mm 1⁄2 cm 10 0 e e −3 10 0 0 ≈ Quasi-neutrality? + It is on space scales larger than a Debye length that the - plasma will tend to remain neutral. - - + + The Debye length serves as a characteristic scale length + - - to shield the Coulomb potentials of individual charged λ + particles when they collide. D 25/38 Introduction to Nuclear Engineering, Fall 2018 Quasi-neutrality The potential variation across a plasma of length can be estimated from Poisson’s equation ≫ ~ ~ 2 Φ 2 Φ 0 − We generally expect that T = 2 Φ ≲ e Then, we obtain 0 1 2 − 2 Plasma approximation ≲ ≪ = The plasma approximation