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

= ( , ) + × ( , ) , = ( ) 𝑑𝑑𝒗𝒗 𝑑𝑑𝒓𝒓 𝑚𝑚 𝑞𝑞 𝑬𝑬 𝒓𝒓 𝑡𝑡 𝒗𝒗 𝑩𝑩 𝒓𝒓 𝑡𝑡 𝒗𝒗 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑  Motion in constant electric field

 For a constant electric field = with = 0,

𝑬𝑬 𝑬𝑬𝟎𝟎 𝑩𝑩 ( ) = + + 𝑞𝑞𝑬𝑬𝟎𝟎 2 𝒓𝒓 𝑡𝑡 𝒓𝒓𝟎𝟎 𝒗𝒗𝟎𝟎𝑡𝑡 𝑡𝑡  Electrons are easily accelerated𝟐𝟐𝑚𝑚 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

= × 𝑑𝑑𝒗𝒗 𝑚𝑚 𝑞𝑞𝒗𝒗 𝑩𝑩  For a constant𝑑𝑑𝑑𝑑 magnetic field = 0 with = 0,

= 𝑩𝑩 𝐵𝐵 𝒛𝒛 𝑬𝑬 𝑑𝑑𝑣𝑣𝑥𝑥 𝑚𝑚 𝑞𝑞𝐵𝐵0𝑣𝑣𝑦𝑦 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑣𝑣𝑦𝑦 𝑚𝑚 −𝑞𝑞𝐵𝐵0𝑣𝑣𝑥𝑥 𝑑𝑑𝑑𝑑 = 0 𝑑𝑑𝑣𝑣𝑧𝑧 𝑚𝑚  Cyclotron𝑑𝑑𝑑𝑑 (gyration) frequency

= = 2 𝑑𝑑 𝑣𝑣𝑥𝑥 2 𝑞𝑞 𝐵𝐵0 2 −𝜔𝜔𝑐𝑐 𝑣𝑣𝑥𝑥 𝜔𝜔𝑐𝑐 3/38 𝑑𝑑𝑑𝑑 Introduction to Nuclear𝑚𝑚 Engineering, Fall 2018 Motions in uniform magnetic field

 Particle velocity

= cos + = sin + 𝑥𝑥 ⊥ 𝑐𝑐 0 𝑣𝑣 = 0𝑣𝑣 𝜔𝜔 𝑡𝑡 𝜙𝜙 𝑣𝑣𝑦𝑦 −𝑣𝑣⊥ 𝜔𝜔𝑐𝑐𝑡𝑡 𝜙𝜙0 𝑧𝑧  Particle𝑣𝑣 position = sin + + sin = cos + + cos 𝑥𝑥 = 𝑟𝑟𝑐𝑐 + 𝜔𝜔𝑐𝑐𝑡𝑡 𝜙𝜙0 𝑥𝑥0 − 𝑟𝑟𝑐𝑐 𝜙𝜙0 𝑦𝑦 𝑟𝑟𝑐𝑐 𝜔𝜔𝑐𝑐𝑡𝑡 𝜙𝜙0 𝑦𝑦0 − 𝑟𝑟𝑐𝑐 𝜙𝜙0 0 𝑧𝑧�  Guiding𝑧𝑧 𝑧𝑧 center𝑣𝑣 𝑡𝑡 , , +

 Larmor𝑥𝑥0 𝑦𝑦0(gyration)𝑧𝑧0 𝑣𝑣𝑧𝑧�𝑡𝑡 radius

= = ⊥ ⊥ 𝑐𝑐 𝑣𝑣 𝑚𝑚𝑣𝑣 𝑟𝑟 𝑐𝑐 0 4/38 𝜔𝜔 𝑞𝑞 𝐵𝐵 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 𝑐𝑐𝑐𝑐 0 0 𝑓𝑓 3.37 𝐵𝐵 𝐵𝐵 = cm ( in volts)

𝑐𝑐𝑐𝑐 𝐸𝐸 𝑟𝑟 0 𝐸𝐸  For singly 𝐵𝐵charged ions

= 1.52 × 10 / Hz in gauss 3 𝑐𝑐𝑐𝑐 0 𝐴𝐴 0 𝑓𝑓 144 𝐵𝐵 𝑀𝑀 𝐵𝐵 = cm ( in volts, in amu) 𝐸𝐸𝑀𝑀𝐴𝐴 𝑟𝑟𝑐𝑐𝑐𝑐 𝐸𝐸 𝑀𝑀𝐴𝐴  Energy gain?𝐵𝐵0

5/38 Introduction to Nuclear Engineering, Fall 2018 Motions in uniform E and B fields

 Equation of motion

= + × 𝑑𝑑𝒗𝒗 𝑚𝑚 𝑞𝑞 𝑬𝑬 𝒗𝒗 𝑩𝑩 𝑑𝑑𝑑𝑑  Parallel motion: = 0 and = 0 ,

𝑩𝑩 𝐵𝐵 𝒛𝒛 𝑬𝑬 𝐸𝐸 𝒛𝒛 = 𝑑𝑑𝑣𝑣𝑧𝑧 𝑚𝑚 𝑞𝑞𝐸𝐸𝑧𝑧 𝑑𝑑𝑑𝑑 = + 𝑞𝑞𝐸𝐸𝑧𝑧 𝑣𝑣𝑧𝑧 𝑡𝑡 𝑣𝑣𝑧𝑧� 𝑚𝑚  Straightforward acceleration along B

6/38 Introduction to Nuclear Engineering, Fall 2018 × drift

𝑬𝑬 Transverse𝑩𝑩 motion: = 0 and = 0 ,

𝑩𝑩 𝐵𝐵 𝒛𝒛 𝑬𝑬 𝐸𝐸 𝒙𝒙 = + 𝑑𝑑𝑣𝑣𝑥𝑥 𝑚𝑚 = 𝑞𝑞𝐸𝐸0 𝑞𝑞𝐵𝐵0𝑣𝑣𝑦𝑦 𝑑𝑑𝑑𝑑 𝑑𝑑𝑣𝑣𝑦𝑦 𝑚𝑚 −𝑞𝑞𝐵𝐵0𝑣𝑣𝑥𝑥  Differentiating,𝑑𝑑𝑑𝑑

= 2 𝑥𝑥 2 𝑑𝑑 𝑣𝑣 𝑐𝑐 𝑥𝑥 2 = −𝜔𝜔 𝑣𝑣 + × 𝑑𝑑𝑑𝑑2 = 𝑑𝑑 𝑣𝑣𝑦𝑦 2 𝐸𝐸0 2 −𝜔𝜔𝑐𝑐 𝑣𝑣𝑦𝑦 𝑬𝑬 𝑩𝑩 𝑑𝑑𝑑𝑑 𝐵𝐵0 𝒗𝒗𝐸𝐸 𝟐𝟐  Particle velocity 𝐵𝐵

= cos + 𝑔𝑔𝑔𝑔 𝑣𝑣𝑥𝑥 = 𝑣𝑣⊥ sin𝜔𝜔𝑐𝑐𝑡𝑡 +𝜙𝜙0 𝑣𝑣 0 𝑦𝑦 ⊥ 𝑐𝑐 0 𝐸𝐸 𝑣𝑣 −𝑣𝑣 𝜔𝜔 𝑡𝑡 𝜙𝜙 − 0 7/38 Introduction𝐵𝐵 to Nuclear Engineering, Fall 2018 DC magnetron

 A magnetron which is widely used in the sputtering system uses the × drift motion for plasma confinement. 𝑬𝑬 𝑩𝑩

 What is the direction of × drift motion?

𝑬𝑬 𝑩𝑩 8/38 Introduction to Nuclear Engineering, Fall 2018 Motions in gravitational field

 Generally, the guiding center drift caused by general force

1 × 𝑭𝑭 = 𝑭𝑭 𝑩𝑩 𝒗𝒗𝑓𝑓 𝟐𝟐 𝑞𝑞 𝐵𝐵  If is the force of gravity ,

𝑭𝑭 × 𝑚𝑚𝒈𝒈 = 𝑚𝑚 𝒈𝒈 𝑩𝑩 𝒗𝒗𝑔𝑔 𝟐𝟐 𝑞𝑞 𝐵𝐵  What is the difference between and ?

𝒗𝒗𝐸𝐸 𝒗𝒗𝑔𝑔

9/38 Introduction to Nuclear Engineering, Fall 2018 ⊥ : Grad-B drift

𝜵𝜵𝐵𝐵The 𝑩𝑩gradient 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, perpendicular𝑩𝑩 to both and .

𝑩𝑩 𝜵𝜵𝐵𝐵

𝜵𝜵𝐵𝐵

 Guiding center motion

1 × = ± 2 𝑩𝑩 𝛁𝛁𝐵𝐵 𝒗𝒗𝛻𝛻𝐵𝐵 𝑣𝑣⊥𝑟𝑟𝑐𝑐 𝟐𝟐 𝐵𝐵 10/38 Introduction to Nuclear Engineering, Fall 2018 Curved B: Curvature drift

 The average centrifugal force

= = 2 𝑚𝑚𝑣𝑣∥ 2 𝑹𝑹𝑐𝑐 𝑭𝑭𝑐𝑐𝑐𝑐 𝒓𝒓� 𝑚𝑚𝑣𝑣∥ 2 𝑅𝑅𝑐𝑐 𝑅𝑅𝑐𝑐  Curvature drift

1 × × = = 2 1 𝑭𝑭𝑐𝑐𝑐𝑐 𝑩𝑩 𝑚𝑚𝑣𝑣∥ 𝑹𝑹𝑐𝑐 𝑩𝑩 𝒗𝒗𝑅𝑅 𝟐𝟐 𝟐𝟐 2 𝑞𝑞 𝐵𝐵 𝑞𝑞𝐵𝐵 𝑅𝑅𝑐𝑐 𝐵𝐵 ∝ 𝑅𝑅𝑐𝑐  Total drift in a curved vacuum field (curvature + grad-B)

× 1 + = + = 2 𝑐𝑐 𝑐𝑐 2 2 𝜵𝜵 𝐵𝐵 𝑹𝑹 𝑅𝑅 𝛻𝛻𝐵𝐵 𝑚𝑚 𝑹𝑹 𝑩𝑩 ∥ ⊥ 2 𝒗𝒗 𝒗𝒗 2 2 𝑣𝑣 𝑣𝑣 − 𝑐𝑐 𝑞𝑞 𝑅𝑅𝑐𝑐 𝐵𝐵 𝐵𝐵 𝑅𝑅 11/38 Introduction to Nuclear Engineering, Fall 2018 ∥ : Magnetic mirror

𝜵𝜵𝐵𝐵Adiabatic𝑩𝑩 invariant: Magnetic moment

1 2 = = 2 𝑚𝑚𝑣𝑣⊥ 𝜇𝜇 𝐼𝐼𝐼𝐼 𝐵𝐵  As the particle moves into regions of stronger or weaker , its Larmor radius changes, but remains invariant. 𝐵𝐵  Magnetic mirror 𝜇𝜇 1 1 2 2 2 = 2 𝑚𝑚𝑣𝑣⊥0 𝑚𝑚𝑣𝑣⊥𝑚𝑚 𝐵𝐵0 = +𝐵𝐵𝑚𝑚 2 2 2 2 𝑣𝑣⊥𝑚𝑚 𝑣𝑣⊥0 𝑣𝑣∥0 ≡ 𝑣𝑣0 = = = sin 2 2 𝐵𝐵0 𝑣𝑣⊥0 𝑣𝑣⊥0 2 2 2 𝑚𝑚 ⊥𝑚𝑚 0 𝜃𝜃 12/38 𝐵𝐵 𝑣𝑣 𝑣𝑣 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

+ = 𝜕𝜕𝑛𝑛 𝜵𝜵 � 𝑛𝑛𝒖𝒖 𝐺𝐺 − 𝐿𝐿 The net 𝜕𝜕number𝑡𝑡 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

𝐺𝐺 𝜈𝜈𝑖𝑖𝑖𝑖𝑛𝑛𝑒𝑒 𝐿𝐿 ≈ Hence, =

𝜵𝜵 � 𝑛𝑛𝒖𝒖 𝜈𝜈𝑖𝑖𝑖𝑖𝑛𝑛𝑒𝑒  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 𝑛𝑛 Introduction to Nuclear Engineering, Fall 2018 𝒖𝒖 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

+ = 𝜕𝜕𝒖𝒖 𝑚𝑚𝑚𝑚 𝒖𝒖 � 𝜵𝜵 𝒖𝒖 𝑞𝑞𝑞𝑞𝑬𝑬 − 𝜵𝜵𝑝𝑝 − 𝑚𝑚𝑚𝑚𝜈𝜈𝑚𝑚𝒖𝒖 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. = , , = ( , )

 I𝑝𝑝sothermal𝑝𝑝 𝑛𝑛 𝑇𝑇 EOS for𝜀𝜀 slow𝜀𝜀 𝑛𝑛 time𝑇𝑇 variations, where temperatures are allowed to equilibrate. In this case, the fluid can exchange energy with its surroundings. = , =  𝑝𝑝 The𝑛𝑛𝑛𝑛energy𝑛𝑛 conservation𝛻𝛻𝑝𝑝 equation𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 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) 𝛾𝛾 𝛻𝛻𝛻𝛻 𝛻𝛻𝛻𝛻 𝐶𝐶𝑝𝑝 𝑝𝑝 𝐶𝐶𝑛𝑛 𝛾𝛾 𝛾𝛾 𝑣𝑣  The energy conservation𝑝𝑝 equation𝑛𝑛 is not required.𝐶𝐶 2  Specific heat ratio vs degree of freedom ( ) = 1 +

𝑓𝑓 𝛾𝛾 17/38 Introduction to Nuclear Engineering, Fall 2018 𝑓𝑓 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𝜕𝜕𝜕𝜕 𝑐𝑐 velocity distribution

2 = 3 2 𝑚𝑚 2 𝑚𝑚𝑣𝑣 𝑓𝑓 𝑣𝑣 𝑣𝑣 𝑒𝑒𝑒𝑒𝑒𝑒 − 2𝜋𝜋 𝑘𝑘𝑘𝑘 / 2𝑘𝑘𝑘𝑘 = 1 2 𝜀𝜀 𝜀𝜀 𝑓𝑓 𝜀𝜀 𝑒𝑒𝑒𝑒𝑒𝑒 −  The mean speed𝜋𝜋 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘 /

= 1 2 8𝑘𝑘𝑘𝑘 𝑣𝑣̅ 18/38 𝜋𝜋𝑚𝑚 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

𝑧𝑧 / = 1 2 𝑘𝑘𝑘𝑘 𝑣𝑣𝑡𝑡푡 1 𝑚𝑚 = 4 Γ𝑧𝑧 𝑛𝑛𝑣𝑣̅  The average energy flux: the amount of energy per square meter per second in the +z direction 1 = = 2 2 𝑆𝑆𝑧𝑧 𝑛𝑛 𝑚𝑚𝑣𝑣 𝑣𝑣𝑧𝑧 2𝑘𝑘𝑘𝑘Γ𝑧𝑧  The average kinetic 𝒗𝒗energy per particle crossing = 0 in the positive direction

= 𝑊𝑊 𝑧𝑧

𝑊𝑊 2𝑘𝑘𝑘𝑘 19/38 Introduction to Nuclear Engineering, Fall 2018 Diffusion and mobility

 The fluid equation of motion including collisions

= + = 𝑑𝑑𝒖𝒖 𝜕𝜕𝒖𝒖 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚 𝒖𝒖 � 𝜵𝜵 𝒖𝒖 𝑞𝑞𝑞𝑞𝑬𝑬 − 𝜵𝜵𝑝𝑝 − 𝑚𝑚𝑚𝑚𝜈𝜈𝑚𝑚𝒖𝒖 𝑑𝑑𝑑𝑑 𝜕𝜕𝜕𝜕  In steady-state, for isothermal plasmas 1 1 = =

𝒖𝒖 𝑚𝑚 𝑞𝑞𝑞𝑞𝑬𝑬 − 𝜵𝜵𝑝𝑝 𝑚𝑚 𝑞𝑞𝑞𝑞𝑬𝑬 − 𝑘𝑘𝑘𝑘𝜵𝜵𝑛𝑛 = 𝑚𝑚𝑚𝑚𝜈𝜈 = ±𝑚𝑚𝑚𝑚𝜈𝜈 𝑞𝑞 𝑘𝑘𝑘𝑘 𝜵𝜵𝑛𝑛 𝜵𝜵𝑛𝑛 𝑬𝑬 − Drift𝜇𝜇𝑬𝑬 − Diffusion𝐷𝐷 𝑚𝑚𝜈𝜈𝑚𝑚 𝑚𝑚𝜈𝜈𝑚𝑚 𝑛𝑛 𝑛𝑛  In terms of particle flux = Einstein relation = = ± 𝑞𝑞 𝐷𝐷 𝜇𝜇 ∶ 𝑘𝑘𝑘𝑘 𝚪𝚪 𝑛𝑛𝒖𝒖 𝑛𝑛𝜇𝜇𝑬𝑬 − 𝐷𝐷𝜵𝜵𝑛𝑛 = Mobility = Diffusion coefficient 𝑞𝑞 𝑘𝑘𝑘𝑘 𝜇𝜇 ∶ 𝐷𝐷 ∶ 𝑚𝑚𝜈𝜈𝑚𝑚 𝑚𝑚𝜈𝜈𝑚𝑚 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.

= + =

 AmbipolarΓ𝑖𝑖 𝑛𝑛 electric𝜇𝜇𝑖𝑖𝐸𝐸 − 𝐷𝐷 field𝑖𝑖𝛻𝛻𝛻𝛻 for = Γ𝑒𝑒 −𝑛𝑛𝜇𝜇𝑒𝑒𝐸𝐸 − 𝐷𝐷𝑒𝑒𝛻𝛻𝛻𝛻

𝑖𝑖 𝑒𝑒 = Γ Γ + 𝐷𝐷𝑖𝑖 − 𝐷𝐷𝑒𝑒 𝛻𝛻𝛻𝛻 𝐸𝐸  The common𝜇𝜇𝑖𝑖 𝜇𝜇 particle𝑒𝑒 𝑛𝑛 flux + = = + 𝜇𝜇𝑒𝑒𝐷𝐷𝑖𝑖 𝜇𝜇𝑖𝑖𝐷𝐷𝑒𝑒 Γ − 𝛻𝛻𝛻𝛻 −𝐷𝐷𝑎𝑎𝛻𝛻𝛻𝛻  The ambipolar𝜇𝜇𝑖𝑖 diffusion𝜇𝜇𝑒𝑒 coefficient for weakly ionized plasmas + T = + 1 + ~ T 𝑒𝑒 𝑖𝑖 + 𝑖𝑖 𝑒𝑒 𝑖𝑖 Te 𝑎𝑎 𝜇𝜇 𝐷𝐷 𝜇𝜇 𝐷𝐷 𝑖𝑖 𝜇𝜇 𝑒𝑒 𝑖𝑖 𝑖𝑖 e 𝐷𝐷 𝑖𝑖 𝑒𝑒 ≈ 𝐷𝐷 𝑒𝑒 𝐷𝐷 ≈ 𝐷𝐷 i 𝜇𝜇 21/38 𝜇𝜇 𝜇𝜇 Introduction𝜇𝜇 to Nuclear Engineering, Fall 2018 Steady-state plane-parallel solutions

 Diffusion equation volume source and sink

= 𝜕𝜕𝜕𝜕 2 − 𝐷𝐷𝛻𝛻 𝑛𝑛 𝐺𝐺 − 𝐿𝐿  For 𝜕𝜕a𝜕𝜕 plane-parallel geometry with no volume source or sink = 0 = + 2 B.C. = 0 = 𝑑𝑑 𝑛𝑛 −𝐷𝐷 2 𝑛𝑛 = 𝐴𝐴𝐴𝐴 𝐵𝐵 Γ 𝑥𝑥 = /2 =Γ00 𝑑𝑑𝑥𝑥 2 Γ0 𝑙𝑙 𝑛𝑛 − 𝑥𝑥 is determined𝑛𝑛 𝑥𝑥 𝑙𝑙  For a uniform specified source of diffusing𝐷𝐷 particles 𝑛𝑛0 B.C. symmetric at = 0 = 1 = 2 2 2 0 = ± /2 = 0 𝑑𝑑 𝑛𝑛 𝐺𝐺 𝑙𝑙 2𝑥𝑥 𝑥𝑥 −𝐷𝐷 2 𝐺𝐺0 𝑛𝑛 −  The most𝑑𝑑𝑥𝑥 common case is for a plasma8𝐷𝐷 consisting𝑙𝑙 of positive ions𝑛𝑛 𝑥𝑥and an𝑙𝑙 equal number of electrons which are the source of ionization

+ = 0 where, = and is the ionization frequency 2 𝜈𝜈𝑖𝑖𝑖𝑖 𝛻𝛻 𝑛𝑛 𝑛𝑛 𝐷𝐷 𝐷𝐷𝑎𝑎 𝜈𝜈𝑖𝑖𝑖𝑖 22/38 𝐷𝐷 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

+ = 𝜕𝜕𝒖𝒖 𝑚𝑚𝑛𝑛𝑒𝑒 𝒖𝒖 � 𝜵𝜵 𝒖𝒖 −𝑒𝑒𝑛𝑛𝑒𝑒𝑬𝑬 − 𝜵𝜵𝑝𝑝𝑒𝑒 − 𝑚𝑚𝑛𝑛𝑒𝑒𝜈𝜈𝑚𝑚𝒖𝒖 Setting 𝜕𝜕=𝜕𝜕 and assuming =

𝑬𝑬 −𝜵𝜵Φ = 𝑝𝑝𝑒𝑒 ln𝑛𝑛𝑒𝑒𝑘𝑘𝑇𝑇𝑒𝑒= 0

Integrating,𝑒𝑒𝑛𝑛𝑒𝑒𝜵𝜵Φ − we𝑘𝑘𝑇𝑇 have𝑒𝑒𝜵𝜵𝑛𝑛𝑒𝑒 𝑛𝑛𝑒𝑒𝜵𝜵 𝑒𝑒Φ − 𝑘𝑘𝑇𝑇𝑒𝑒 𝑛𝑛𝑒𝑒

= exp = exp T Boltzmann’s relation for electrons 𝑒𝑒 0 𝑒𝑒Φ 𝒓𝒓 0 Φ 𝒓𝒓 𝑛𝑛 𝒓𝒓 𝑛𝑛 𝑒𝑒 𝑛𝑛 e ( )  For positive ions in thermal𝑘𝑘𝑇𝑇 equilibrium at temperature T ( ) = exp i T 𝑖𝑖 0 Φ 𝒓𝒓  However, positive ions are almost never in thermal equilibrium𝑛𝑛 𝒓𝒓 in low𝑛𝑛 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. 𝑢𝑢𝑖𝑖 23/38 Introduction to Nuclear Engineering, Fall 2018 Debye shielding (screening)

Plasma

 Coulomb potential  Poisson’s equation

= = 1 / = 4 2 𝑄𝑄 2 𝑒𝑒𝑛𝑛0 𝑒𝑒Φ 𝑘𝑘𝑇𝑇𝑒𝑒 𝑒𝑒 𝑛𝑛0 Φ Φ 𝑟𝑟 𝛻𝛻 Φ − − 𝑒𝑒 ≈ Φ 2  Screened𝜋𝜋 Coulomb𝜖𝜖0𝑟𝑟 potential  Debye length𝜖𝜖0 𝜖𝜖0𝑘𝑘𝑇𝑇𝑒𝑒 𝜆𝜆𝐷𝐷 / / = exp T 4 = 1 2 = 1 2 𝑄𝑄 𝑟𝑟 𝜖𝜖0𝑘𝑘𝑇𝑇𝑒𝑒 𝜖𝜖0 e Φ 𝑟𝑟 − 𝐷𝐷 2 𝜋𝜋𝜖𝜖0𝑟𝑟 𝜆𝜆𝐷𝐷 𝜆𝜆 0 0 24/38 Introduction to Nuclear Engineering,𝑒𝑒 𝑛𝑛Fall 2018 𝑒𝑒𝑛𝑛 Debye length

 The electron Debye length is the characteristic length scale in a plasma.

 The Debye length is the distance𝜆𝜆𝐷𝐷𝐷𝐷 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 𝜖𝜖0 e e 𝐷𝐷 −3 10 𝜆𝜆 0 0 ≈ 𝑒𝑒𝑛𝑛 𝑛𝑛 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 𝑙𝑙 ≫ 𝜆𝜆𝐷𝐷𝐷𝐷 ~ ~ 2 Φ 𝑒𝑒 2 𝑖𝑖 𝑒𝑒 𝛻𝛻 Φ 0 𝑍𝑍𝑛𝑛 − 𝑛𝑛 We generally expect that 𝑙𝑙 𝜖𝜖

T = 𝑒𝑒 2 Φ ≲ e 𝑛𝑛𝑒𝑒𝜆𝜆𝐷𝐷𝐷𝐷 Then, we obtain 𝜖𝜖0

1 2 𝑍𝑍𝑛𝑛𝑖𝑖 − 𝑛𝑛𝑒𝑒 𝜆𝜆𝐷𝐷𝐷𝐷 2 Plasma approximation 𝑒𝑒 ≲ ≪ 𝑛𝑛 = 𝑙𝑙

𝑖𝑖 𝑒𝑒  The plasma approximation is violated𝑍𝑍𝑛𝑛 𝑛𝑛 within a plasma sheath, in proximity to a material wall, either because the sheath thickness s , or because T .

26/38 Introduction to Nuclear Engineering, Fall≈ 2018𝜆𝜆𝐷𝐷𝐷𝐷 Φ ≫ e Plasma oscillations

 Electrons overshoot by inertia and oscillate around their equilibrium position with a characteristic frequency known as plasma frequency.

 Equation of motion (cold plasma) d22 ne me  eE  0  2 xe dt 0 d22 ne e 0  0 2 e dt m0  Electron plasma frequency / = 2 1 2 𝑛𝑛0𝑒𝑒 𝜔𝜔𝑝𝑝𝑝𝑝  If the assumption𝑚𝑚𝜖𝜖0 of infinite mass ions is not made, then the ions also move slightly and we obtain the natural frequency / = + where, = / / (ion plasma frequency) 2 2 1 2 2 1 2 𝑝𝑝 𝑝𝑝𝑝𝑝 𝑝𝑝𝑖𝑖 𝑝𝑝𝑝𝑝 0 0 27/38 𝜔𝜔 𝜔𝜔 𝜔𝜔 Introduction to Nuclear𝜔𝜔 Engineering,𝑛𝑛 𝑒𝑒 Fall𝑀𝑀 2018𝜖𝜖 Plasma frequency

 Plasma oscillation frequency for electrons and ions

= = 8980 Hz in cm 2 𝜔𝜔𝑝𝑝𝑝𝑝 −3 𝑓𝑓𝑝𝑝𝑒𝑒 𝑛𝑛0 𝑛𝑛0 = 𝜋𝜋 = 210 / Hz in cm , in amu 2 𝜔𝜔𝑝𝑝𝑖𝑖 −3 𝑓𝑓𝑝𝑝𝑖𝑖 𝑛𝑛0 𝑀𝑀𝐴𝐴 𝑛𝑛0 𝑀𝑀𝐴𝐴 𝜋𝜋  Typical values for a processing plasma (Ar)

= = 8980 10 Hz = 9 × 10 [Hz] 2 𝜔𝜔𝑝𝑝𝑝𝑝 10 8 𝑓𝑓𝑝𝑝𝑒𝑒 = 𝜋𝜋 = 210 10 /40 Hz = 3.3 × 10 [Hz] 2 𝜔𝜔𝑝𝑝𝑖𝑖 10 6 𝑓𝑓𝑝𝑝𝑖𝑖  Collective𝜋𝜋 behavior

> 1 Plasma frequency > collision frequency

𝜔𝜔𝑝𝑝𝑝𝑝𝜏𝜏 28/38 Introduction to Nuclear Engineering, Fall 2018 Criteria for plasmas

 The picture of Debye shielding is valid only if there are enough particles in the charge cloud. Clearly, if there are only one or two particles in the sheath region, Debye shielding would not be a statistically valid concept. We can compute the number of particles in a “Debye sphere”: 4 = = 4 /3 3 Wigner-Seitz radius 3 3 𝑁𝑁𝐷𝐷 𝑛𝑛 𝜋𝜋𝜆𝜆𝐷𝐷 𝑎𝑎 𝜋𝜋𝑛𝑛𝑒𝑒  Plasma parameter  Coupling parameter Coulomb energy /(4 ) = 4 = 3 = = ~ / Thermal energy 2 3 𝑞𝑞 𝜋𝜋𝜖𝜖0𝑎𝑎 −2 3 Λ 𝜋𝜋𝑛𝑛𝜆𝜆𝐷𝐷 𝑁𝑁𝐷𝐷 Γ Λ  Criteria for plasmas: 𝑘𝑘𝑇𝑇𝑒𝑒

𝐷𝐷 𝜆𝜆 ≪ 𝐿𝐿1

𝐷𝐷 𝑁𝑁 ≫> 1

𝑝𝑝𝑝𝑝 𝑐𝑐 29/38 𝜔𝜔 𝜏𝜏 Introduction to Nuclear Engineering, Fall 2018 Formation of plasma sheaths

 Plasma sheath: the non-neutral potential region between the plasma and the wall caused by the balanced flow of particles with different mobility such as electrons and ions.

30/38 Introduction to Nuclear Engineering, Fall 2018 Plasma-sheath structure

31/38 Introduction to Nuclear Engineering, Fall 2018 Collisionless sheath

 Ion energy & flux conservations (no collision) 1 1 ( ) + ( ) = = 2 2 2 2 𝑀𝑀𝑢𝑢 𝑥𝑥 𝑒𝑒Φ 𝑥𝑥 𝑀𝑀𝑢𝑢𝑠𝑠 𝑛𝑛𝑖𝑖 𝑥𝑥 𝑢𝑢 𝑥𝑥 𝑛𝑛𝑖𝑖𝑠𝑠𝑢𝑢𝑠𝑠  Ion density profile

( ) / = 1 −1 2 2𝑒𝑒Φ 𝑥𝑥 𝑛𝑛𝑖𝑖 𝑥𝑥 𝑛𝑛𝑖𝑖𝑠𝑠 − 2  Electron density profile𝑀𝑀𝑢𝑢𝑠𝑠 ( ) ( ) = exp T Φ 𝑥𝑥 𝑛𝑛𝑒𝑒 𝑥𝑥 𝑛𝑛𝑒𝑒𝑠𝑠  Setting = e

𝑛𝑛𝑒𝑒𝑒𝑒 𝑛𝑛𝑖𝑖𝑖𝑖 ≡ 𝑛𝑛𝑠𝑠 / = exp 1 2 T −1 2 𝑑𝑑 Φ 𝑒𝑒𝑛𝑛𝑠𝑠 Φ Φ 2 where, 0 e − − 𝑠𝑠 𝑑𝑑𝑑𝑑 𝜖𝜖 ℰ 1 2 32/38 Introduction to Nuclear Engineering, Fall𝑒𝑒 2018ℰ𝑠𝑠 ≡ 2 𝑀𝑀𝑢𝑢𝑠𝑠 Bohm sheath criterion

 Multiplying the sheath equation by / and integrating over

𝑑𝑑Φ 𝑑𝑑𝑑𝑑 / 𝑥𝑥 = exp 1 Φ Φ T −1 2 𝑑𝑑Φ 𝑑𝑑 𝑑𝑑Φ 𝑒𝑒𝑛𝑛𝑠𝑠 𝑑𝑑Φ Φ Φ �0 𝑑𝑑𝑑𝑑 0 �0 e − − 𝑠𝑠 𝑑𝑑𝑑𝑑  Cancelling𝑑𝑑𝑑𝑑 𝑑𝑑 𝑑𝑑 ’s𝑑𝑑𝑑𝑑and integrating𝜀𝜀 with𝑑𝑑𝑑𝑑 respect to ℰ

1 𝑑𝑑𝑑𝑑 Φ / = T exp T + 2 1 2 0 2 2 T 1 2 𝑑𝑑Φ 𝑒𝑒𝑛𝑛𝑠𝑠 Φ Φ e − e ℰ𝑠𝑠 − − ℰ𝑠𝑠 ≥ 𝑑𝑑𝑑𝑑 𝜀𝜀0 e ℰ𝑠𝑠 1 T 2 2 2 e ℰ𝑠𝑠 ≡ 𝑀𝑀𝑢𝑢𝑠𝑠 ≥  Bohm sheath criterion𝑒𝑒

T / / 1 2 1 2 (Bohm speed) 𝑒𝑒 e 𝑘𝑘𝑇𝑇𝑒𝑒 𝑢𝑢𝑠𝑠 ≥ ≡ ≡ 𝑢𝑢𝐵𝐵 𝑀𝑀 𝑀𝑀 33/38 Introduction to Nuclear Engineering, Fall 2018 Presheath

 To give the ions the directed velocity , there must be a finite electric field in the plasma over some region, typically much wider than the sheath, called the 𝐵𝐵 presheath. 𝑢𝑢  At the sheath–presheath interface there is a transition from subsonic ( < ) to supersonic ( > ) ion flow, where the condition of charge neutrality must 𝑖𝑖 𝐵𝐵 break down. 𝑢𝑢 𝑢𝑢 𝑢𝑢𝑖𝑖 𝑢𝑢𝐵𝐵  The potential drop across a collisionless presheath, which accelerates the ions to the Bohm velocity, is given by 1 T where, is the plasma potential with respect to the = = 2 2 potential at the sheath–presheath edge 2 e 𝑝𝑝 𝐵𝐵 𝑒𝑒 𝑝𝑝 Φ  The spatial𝑀𝑀𝑢𝑢 variation𝑒𝑒Φ of the potential ( ) in a collisional presheath (Riemann)

1 1 2 𝑝𝑝 exp = Φ 𝑥𝑥 2 2 T T Φ𝑝𝑝 Φ𝑝𝑝 𝑥𝑥  − − The ratio of the densitye ate the𝜆𝜆 𝑖𝑖sheath edge to that in the plasma = / 0.61 −Φ𝑝𝑝 Te 𝑠𝑠 𝑏𝑏 𝑏𝑏 34/38 𝑛𝑛 𝑛𝑛 𝑒𝑒 ≈Introduction𝑛𝑛 to Nuclear Engineering, Fall 2018 Sheath potential at a floating Wall

 Ion flux =

 ElectronΓ𝑖𝑖 𝑛𝑛 𝑠𝑠flux𝑢𝑢𝐵𝐵 1 1 a few = = exp 4 4 T 𝜆𝜆𝐷𝐷𝐷𝐷 Φ𝑤𝑤 Γ𝑒𝑒 𝑛𝑛𝑒𝑒𝑒𝑒𝑣𝑣𝑒𝑒̅ 𝑛𝑛𝑠𝑠𝑣𝑣𝑒𝑒̅  Ion flux = electron flux for a floatinge wall

T / 1 T / = exp 1 2 4 1 2 T 𝑒𝑒 e 8𝑒𝑒 e Φ𝑤𝑤 𝑛𝑛𝑠𝑠 𝑛𝑛𝑠𝑠  Wall potential𝑀𝑀 𝜋𝜋𝑚𝑚 e

T = ln 2.8T for hydrogen and 4.7T for argon 2e 2 𝑀𝑀 𝑤𝑤 e 𝑤𝑤 e Φ𝑤𝑤 − Φ ≈ − Φ ≈ − 𝜋𝜋𝑚𝑚 T T  Ion bombarding energy = + = 1 + ln 2 2 2 𝑒𝑒 e 𝑒𝑒 e 𝑀𝑀 ℰ𝑖𝑖𝑖𝑖𝑖𝑖 𝑒𝑒 Φ𝑤𝑤 35/38 Introduction to Nuclear Engineering, Fall 2018 𝜋𝜋𝑚𝑚 High-voltage sheath: matrix sheath

 The potential in high-voltage sheaths is highly negative with respect to the plasma–sheath edge; hence ~ / 0 and only ions are present in the sheath. Φ Φ Te 𝑛𝑛𝑒𝑒 𝑛𝑛𝑠𝑠𝑒𝑒 →  The simplest high-voltage sheath, with a uniform ion density, is known as a matrix sheath (not self-consistent in steady-state).  Poisson’s eq.

= = 2 22 𝑑𝑑 Φ 𝑒𝑒𝑛𝑛𝑠𝑠 𝑒𝑒𝑛𝑛𝑠𝑠 𝑥𝑥 2 − Φ −  Setting𝑑𝑑𝑥𝑥 = 𝜖𝜖0 at = , we obtain the𝜖𝜖 matrix0 sheath thickness / Φ2 −𝑉𝑉0 𝑥𝑥 𝑠𝑠 = 1 2 𝜖𝜖0𝑉𝑉0 𝑠𝑠  In terms of𝑒𝑒𝑛𝑛 the𝑠𝑠 electron Debye length at the sheath edge 2 / = where, = T / / T 1 2 0 1 2 𝐷𝐷𝐷𝐷 𝑉𝑉 𝐷𝐷𝐷𝐷 0 e 𝑠𝑠 𝑠𝑠 𝜆𝜆 e 𝜆𝜆 𝜖𝜖 𝑒𝑒𝑛𝑛 36/38 Introduction to Nuclear Engineering, Fall 2018 High-voltage sheath: space-charge-limited current

 In the limit that the initial ion energy is small compared to the potential, the ion energy and flux conservation equations reduce to 𝑠𝑠 1 ℰ ( ) = ( ) / 2 2 = 𝑀𝑀𝑢𝑢 𝑥𝑥 =−𝑒𝑒Φ 𝑥𝑥 −1 2 𝐽𝐽0 2𝑒𝑒Φ 𝑛𝑛 𝑥𝑥 −  Poisson’s𝑒𝑒𝑒𝑒 𝑥𝑥 𝑢𝑢 𝑥𝑥eq. 𝐽𝐽0 𝑒𝑒 𝑀𝑀 / = ( ) = 2 −1 2 𝑑𝑑 Φ 𝑒𝑒 𝐽𝐽0 2𝑒𝑒Φ 2 − 𝑛𝑛𝑖𝑖 − 𝑛𝑛𝑒𝑒 − −  Multiplying𝑑𝑑𝑥𝑥 by𝜖𝜖0 / and integrating𝜖𝜖0 𝑀𝑀 twice from 0 to / / 3 = 0 / = 𝑑𝑑Φ 𝑑𝑑𝑑𝑑 𝑥𝑥 B.C. 2 1 2 −1 4 𝑑𝑑Φ 3 4 𝐽𝐽0 2𝑒𝑒 � −Φ 𝑥𝑥 𝑑𝑑𝑑𝑑 𝑥𝑥=0  Letting = 𝜖𝜖0 at =𝑀𝑀 and solving for , we obtain 0 / / 0 4Φ −𝑉𝑉 𝑥𝑥 𝑠𝑠 Child law:𝐽𝐽 = 9 1 2 3 2 Space-charge-limited current in a plane diode 2𝑒𝑒 𝑉𝑉0 𝐽𝐽0 𝜖𝜖0 2 37/38 𝑀𝑀 𝑠𝑠Introduction to Nuclear Engineering, Fall 2018 High-voltage sheath: Child law sheath

 For a plasma = / / 4 𝐽𝐽0 𝑒𝑒𝑛𝑛𝑠𝑠𝑢𝑢𝐵𝐵 = 9 1 2 3 2 2𝑒𝑒 𝑉𝑉0 𝑒𝑒𝑛𝑛𝑠𝑠𝑢𝑢𝐵𝐵 𝜖𝜖0 2 𝑀𝑀 𝑠𝑠 2 T / 2 / 2 2 /  Child law sheath = 1 2 3 4 = 3 4 3 0 e T 0 3 T 0 𝜖𝜖 𝑉𝑉 𝐷𝐷𝐷𝐷 𝑉𝑉 𝑠𝑠 𝑠𝑠 e 𝜆𝜆 e  Potential, electric field and density𝑒𝑒𝑛𝑛 within the sheath / 4 / 4 / = = = 4 3 3 1 3 9 −2 3 𝑥𝑥 𝑉𝑉0 𝑥𝑥 𝜖𝜖0 𝑉𝑉0 𝑥𝑥  AssumingΦ −𝑉𝑉0 that an ion enters𝐸𝐸 the sheath with initial𝑛𝑛 velocity 2 0 = 0 𝑠𝑠 𝑠𝑠 𝑠𝑠 𝑒𝑒 𝑠𝑠 𝑠𝑠 / = where, is the characteristic ion velocity𝑢𝑢 in the sheath 2 3 / 𝑑𝑑𝑑𝑑 𝑥𝑥 0  𝑣𝑣0 𝑣𝑣 Ion𝑑𝑑𝑑𝑑 transit time𝑠𝑠 across the sheath = 1 2 2𝑒𝑒𝑉𝑉0 = = 𝑣𝑣0 3 𝑀𝑀 𝑥𝑥 𝑡𝑡 𝑣𝑣0𝑡𝑡 3𝑠𝑠 𝜏𝜏𝑖𝑖 38/38 𝑠𝑠 3𝑠𝑠 Introduction to Nuclear Engineering,𝑣𝑣0 Fall 2018