Gas Discharge Fundamentals

Fall, 2017

Kyoung-Jae Chung

Department of Nuclear Engineering Seoul National University Thermionic emission

 Theoretically, the current density, , of electrons emitted from a cathode at an absolute temperature, (K), is given by: 𝐽𝐽𝑒𝑒 = / 𝑇𝑇 Richardson-Dushman equation 2 −𝑒𝑒𝜑𝜑 𝑘𝑘𝑘𝑘 Work function (eV) 𝑒𝑒  Schottky𝐽𝐽 effect𝐴𝐴𝑇𝑇 𝑒𝑒(field enhanced thermionic emission): For a constant temperature, the current still slowly increases with the applied extraction potential by lowering the surface barrier.

= exp 4 2 𝑒𝑒 𝑒𝑒 𝑒𝑒𝑒𝑒 𝐽𝐽 𝐴𝐴𝑇𝑇 − 𝜑𝜑 − 0  Space charge limited current𝑘𝑘𝑘𝑘 : When𝜋𝜋 𝜀𝜀the external electric field is not sufficiently high, the emitted electron current density is limited by space charge. The space charge limited current is determined by the extraction voltage and independent of the cathode current.  Emission (source) limited current: When the external electric field is sufficiently high, the saturation emission electron current is determined by the cathode temperature.

2/29 Radiation Source Engineering, Fall 2017 Secondary electron emission (SEE)

 Electrons, ions or neutral particles cause secondary electron emission through different physical processes.  The secondary electron emission is described by the secondary emission coefficient, , which is defined as the number of secondaries produced per an incident primary for normal incidence. = 𝛾𝛾  Since depends on the state of the surface, e.g. depending on the 𝑛𝑛𝑒𝑒 𝛾𝛾 surface film, there is an extremely wide variation in the measured values. 𝑛𝑛𝑝𝑝 𝛾𝛾 SEE by ion impact ( )

𝑖𝑖 𝛾𝛾 SEE by electron impact ( )

𝛾𝛾𝑒𝑒

3/29 Radiation Source Engineering, Fall 2017 Secondary electron emission (SEE)

 The coefficient of the secondary emission for a proton incident on a gold-plated tungsten filament with an energy of 1.6-15 keV is given by

= (1.06 ± 0.01) / Ep is the proton energy in keV 1 2  The dependence𝛾𝛾𝑖𝑖 of SEE𝐸𝐸 on𝑝𝑝 the incidence angle is given approximately by a sec law.  SEE for an ion of a diatomic molecule is 𝜃𝜃 about double of that for an atomic ion with half the energy of the molecular ion.  The energy distribution of the secondary electrons is almost Maxwellian. The mean energy of secondary electrons is about 5 eV. The majority of electrons have energies less than 20 eV.  The angular distribution of the secondary electrons approximates a cosine law.

4/29 Radiation Source Engineering, Fall 2017 Surface ionization (thermal ionization)

 When an atom or molecule is adsorbed on a metal surface for a time long enough to reach thermodynamic equilibrium, the valence electron level of the adsorbed atom is sufficiently broadened and surface work function is modified, resulting in the valence electron moving between the adsorbed atom and the surface.  When the surface temperature is hot enough to desorb the particles, the evaporated particles can be in the state of ions (positive of negative) or neutrals. This ionization phenomenon is called the positive or negative surface ionization.  For a thermodynamic equilibrium process, the degree of positive surface ionization ( ) as a function of surface temperature is given by the Saha- Langmuir equation: 𝛼𝛼 ( ) = = exp = exp 𝑖𝑖 𝑛𝑛𝑖𝑖 𝑒𝑒 𝑈𝑈𝑖𝑖 − 𝜑𝜑 𝑊𝑊 − 𝐸𝐸  𝛼𝛼 𝐺𝐺 − 𝐺𝐺 Lowering of𝑛𝑛 𝐴𝐴ionization potential𝑘𝑘𝑘𝑘 by external electric𝑘𝑘𝑘𝑘 field similar to Schottky effect:

= exp 4 𝑒𝑒 𝑒𝑒𝑒𝑒 𝛼𝛼 𝐺𝐺 − 𝑈𝑈𝑖𝑖 − 𝜑𝜑 − 𝑘𝑘𝑘𝑘 𝜋𝜋𝜀𝜀0 5/29 Radiation Source Engineering, Fall 2017 Surface ionization

 The surface ionization efficiency ( ) is given by: 1 1 = = = 𝛽𝛽= + 1 + 1 ( ) 𝑛𝑛𝑖𝑖 𝑛𝑛𝑖𝑖 1 + exp 𝛽𝛽 −1 𝑛𝑛 𝑛𝑛𝑖𝑖 𝑛𝑛𝐴𝐴 𝛼𝛼 𝑒𝑒 𝑈𝑈𝑖𝑖 − 𝜑𝜑  For < and 1, 𝐺𝐺 𝑘𝑘𝑘𝑘 ( ) 𝜑𝜑 𝑈𝑈=𝑖𝑖 𝛼𝛼 ≪ = exp For 1 𝑒𝑒 𝑈𝑈𝑖𝑖 − 𝜑𝜑 𝑛𝑛𝑖𝑖 𝛽𝛽𝑛𝑛 ≈ 𝛼𝛼𝑛𝑛 𝑛𝑛𝐺𝐺 − 𝛼𝛼 ≪  For > and ( ) , the majority𝑘𝑘𝑘𝑘 of evaporated atoms will be ionized ( 1) 𝜑𝜑 𝑈𝑈𝑖𝑖 𝑒𝑒 𝜑𝜑 − 𝑈𝑈𝑖𝑖 ≫ 𝑘𝑘𝑘𝑘 1 𝛽𝛽 ≈ For 𝑛𝑛𝑖𝑖 ≈ 𝑛𝑛 𝛼𝛼 ≫

6/29 Radiation Source Engineering, Fall 2017 Collision parameters

 Let be the number of incident particles per unit volume at that undergo an “interaction” with the target particles within a differential distance , removing them𝑑𝑑 𝑑𝑑from the incident beam. Clearly, is proportional to , 𝑥𝑥 , and for infrequent collisions within . 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑛𝑛 𝑛𝑛𝑔𝑔 𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑 = 𝑔𝑔 𝑔𝑔  The collided𝑑𝑑𝑑𝑑 − flux:𝜎𝜎𝑛𝑛𝑛𝑛 𝑑𝑑𝑑𝑑 (x) = 1 𝑑𝑑/ 𝚪𝚪 −𝜎𝜎𝚪𝚪𝑛𝑛 𝑑𝑑𝑑𝑑 −𝑥𝑥 𝜆𝜆 Γ Γ10 − 𝑒𝑒  Mean free path: =

λ 𝑛𝑛𝑔𝑔𝜎𝜎  Mean collision time: = 𝜆𝜆 𝜏𝜏  Collision frequency: = 𝑣𝑣 =

𝜈𝜈 𝑛𝑛𝑔𝑔𝜎𝜎𝑣𝑣 𝑛𝑛𝑔𝑔𝐾𝐾  Rate constant: =

𝐾𝐾 𝜎𝜎𝑣𝑣 7/29 Radiation Source Engineering, Fall 2017 Elastic and inelastic collisions

 Collisions conserve momentum and energy: the total momentum and energy of the colliding particles after collision are equal to that before collision.  Electrons and fully stripped ions possess only kinetic energy. Atoms and partially stripped ions have internal energy level structures and can be excited, de- excited, or ionized, corresponding to changes in potential energy.  It is the total energy, which is the sum of the kinetic and potential energy, that is conserved in a collision.

 Elastic: the sum of kinetic energies of the collision partners are conserved.  Inelastic: the sum of kinetic energies are not conserved. ionization and excitation. the sum of kinetic energies after collision is less than that before collision.  Super-elastic: the sum of kinetic energies are increased after collision. de- excitation.

8/29 Radiation Source Engineering, Fall 2017 Elastic collision: energy transfer

, ′ 𝑚𝑚1 𝑣𝑣1

1 , , = 0 𝜃𝜃 𝜃𝜃2 , 𝑚𝑚1 𝑣𝑣1 𝑚𝑚2 𝑣𝑣2  The conservation of momentum and energy gives ′ 𝑚𝑚2 𝑣𝑣2 '' '' mv11 mv 11cos 1 mv 22 cos  2 0 mv11 sin  1 mv 22 sin 2 1 11 mv2  mv'2 mv'2 211 22 11 22

11'2 2 4mm12 2  Eliminate and , then mv22  mv11 2 cos 2 2 2(mm ) ′ 12 1 1 𝑣𝑣 𝜃𝜃 4mm  12 cos2  The fraction of energy lost by the projectile: L 2 2 ()mm12

9/29 Radiation Source Engineering, Fall 2017 Energy transfer by elastic collision

 Energy transfer rate (averaging over scattering angle)

4 2 = cos = + + 𝑚𝑚1𝑚𝑚2 2 𝑚𝑚1𝑚𝑚2 𝜁𝜁𝐿𝐿 2 𝜃𝜃2 2 𝑚𝑚1 𝑚𝑚2 𝑚𝑚1 𝑚𝑚2  Electron-electron collision ( = ) : effective energy transfer

2 1 1 2 = =𝑚𝑚 𝑚𝑚 : Quickly thermalized + 2 𝑚𝑚1𝑚𝑚2 𝐿𝐿 2 𝜁𝜁 1 2  Electron-ion or𝑚𝑚 electron𝑚𝑚 -neutral collision ( )

2 2 1 2 = 10 𝑚𝑚 ≪: Hard𝑚𝑚 to be thermalized + 𝑚𝑚1𝑚𝑚2 𝑚𝑚1 −4 𝐿𝐿 2 𝜁𝜁 1 2 ≈ 2 ≈  Therefore, in weakly𝑚𝑚 𝑚𝑚 ionized𝑚𝑚 plasma

: Non-equilibrium

𝑇𝑇𝑒𝑒 ≫ 𝑇𝑇𝑖𝑖 ≈ 𝑇𝑇𝑛𝑛 Table tennis ball (2.5 g) Bowling ball (10 lb.)

10/29 Radiation Source Engineering, Fall 2017 Actual velocity dependence of elastic e-n collision

 Probability of collision: the average number of collisions in 1 cm of path through a gas at 1 Torr at 273K

el  vp0 Pc

Hard-sphere-like

~1/v polarization scattering dominant ~1/v polarization scattering dominant

Ramsauer effect: quantum mechanical wave diffraction of the electron around the atom at low energy

11/29 Radiation Source Engineering, Fall 2017 Inelastic collision

 Inelastic collision: the sum of kinetic energies are not conserved. Where is the lost energy?  transferred to internal energy (ionization, excitation, dissociation) - Atomic gases : electronic transition - Molecules : excitation of rotational and vibrational states

, ′ 𝑚𝑚1 𝑣𝑣1

1 , , = 0 𝜃𝜃 2 𝜃𝜃 , , 𝑚𝑚1 𝑣𝑣1 𝑚𝑚2 𝑣𝑣2 ′ 𝑚𝑚2 𝑣𝑣2 ∆𝑈𝑈 • Find , Hint: = 0

𝑚𝑚𝑚𝑚𝑚𝑚 ∆𝑈𝑈 ∆𝑈𝑈 ′ 𝑎𝑎𝑎𝑎 ∆𝑈𝑈𝑚𝑚𝑚𝑚𝑚𝑚 𝑑𝑑𝑣𝑣2 12/29 Radiation Source Engineering, Fall 2017 Inelastic collision

 The maximum obtainable internal energy

= = cos 1 + 𝑚𝑚𝑚𝑚𝑚𝑚 2 2 𝐿𝐿 2∆𝑈𝑈 𝑚𝑚 2 𝜁𝜁 2 1 2 𝜃𝜃 𝑚𝑚1𝑣𝑣1 𝑚𝑚 𝑚𝑚  = 1 m1 << m2 + 𝑚𝑚2 𝜁𝜁 ≈  Almost all electron energy𝑚𝑚1 𝑚𝑚 may2 be transferred to the internal energy of the heavy particle

 = 0 m1 >> m2 + 𝑚𝑚2 𝜁𝜁 ≈ 𝑚𝑚1 𝑚𝑚2 1  m = m = 1 2 + 2 𝑚𝑚2 𝜁𝜁 ≈  When using H+ ions 𝑚𝑚to1 ionize𝑚𝑚2 hydrogen atoms, the minimum required H+ energy is double the ionization energy for atomic hydrogen.

13/29 Radiation Source Engineering, Fall 2017 Photon induced ionization

 Photo-ionization hν + A → e + A+ The physical process in which an incident photon ejects one or more electrons from an atom, ion or molecule. This is essentially the same process that occurs with the photoelectric effect with metals. To provide the ionization, the photo wavelength should be usually less than 1000 Å, which is ultraviolet radiation.

12,400 < ( ) 𝜆𝜆 Å . =𝐼𝐼 𝑒𝑒𝑒𝑒

𝐾𝐾 𝐸𝐸 ℎ𝜈𝜈 − 𝐼𝐼

14/29 Radiation Source Engineering, Fall 2017 Electron induced ionization

 Electron impact ionization e + A → e + e + A+ Electrons with sufficient energy (> 10 eV) can remove an electron from an atom and produce one extra electron and an ion. Thomson cross section:

15/29 Radiation Source Engineering, Fall 2017 Recombination of charged particles

 Radiative recombination e + A+ → A + hν

 Electron-ion recombination e + A+ + A → A* + A For electron-ion recombination, a third-body must be involved to conserve the energy and momentum conservation. Abundant neutral species or reactor walls are ideal third-bodies. This recombination process typically results in excited neutrals.

16/29 Radiation Source Engineering, Fall 2017 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 17/29 Radiation Source Engineering, Fall 2017 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.

= + =

 Ambipolar𝛤𝛤𝑖𝑖 𝑛𝑛 electric𝜇𝜇𝑖𝑖𝐸𝐸 − 𝐷𝐷 field𝑖𝑖𝛻𝛻𝛻𝛻 for = 𝛤𝛤𝑒𝑒 −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 18/29 𝜇𝜇 𝜇𝜇 Radiation𝜇𝜇 Source Engineering, Fall 2017 Diffusion across a magnetic field

 The perpendicular component of the force equation for either species

0 = ( + × )

 In terms𝑞𝑞 of𝑞𝑞 𝑬𝑬the rectangular𝒖𝒖⊥ 𝑩𝑩0 − components𝑘𝑘𝑘𝑘𝜵𝜵𝑛𝑛 − 𝑚𝑚𝑚𝑚𝜈𝜈𝑚𝑚𝒖𝒖⊥

= + = ± + 𝜕𝜕𝜕𝜕 𝐷𝐷 𝜕𝜕𝜕𝜕 𝜔𝜔𝑐𝑐 𝑚𝑚𝑚𝑚𝜈𝜈𝑚𝑚𝑢𝑢𝑥𝑥 𝑞𝑞𝑞𝑞𝐸𝐸𝑥𝑥 − 𝑘𝑘𝑘𝑘 𝑞𝑞𝑞𝑞𝑢𝑢𝑦𝑦𝐵𝐵0 𝑢𝑢𝑥𝑥 𝜇𝜇𝐸𝐸𝑥𝑥 − 𝑢𝑢𝑦𝑦 = 𝜕𝜕𝜕𝜕 = ± 𝑛𝑛 𝜕𝜕𝜕𝜕 𝜈𝜈𝑚𝑚 𝑐𝑐 𝑚𝑚 𝑦𝑦 𝑦𝑦 𝜕𝜕𝜕𝜕 𝑥𝑥 0 𝐷𝐷 𝜕𝜕𝜕𝜕 𝜔𝜔 𝑚𝑚𝑚𝑚𝜈𝜈 𝑢𝑢 𝑞𝑞𝑞𝑞𝐸𝐸 − 𝑘𝑘𝑘𝑘 − 𝑞𝑞𝑞𝑞𝑢𝑢 𝐵𝐵 𝑢𝑢𝑦𝑦 𝜇𝜇𝐸𝐸𝑦𝑦 − − 𝑢𝑢𝑥𝑥  In terms of the rectangular𝜕𝜕 components𝑦𝑦 𝑛𝑛 𝜕𝜕𝑦𝑦 𝜈𝜈𝑚𝑚 1 1 + = ± + 2 𝐷𝐷 𝜕𝜕𝜕𝜕 2 𝐸𝐸𝑦𝑦 2 𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝜔𝜔𝑐𝑐𝜏𝜏𝑚𝑚 𝑢𝑢𝑥𝑥 𝜇𝜇𝐸𝐸𝑥𝑥 − 𝜔𝜔𝑐𝑐𝜏𝜏𝑚𝑚 − 𝜔𝜔𝑐𝑐𝜏𝜏𝑚𝑚 1 1 + = ± 𝑛𝑛 𝜕𝜕𝜕𝜕 + 𝐵𝐵0 + 𝑞𝑞𝐵𝐵0 𝑛𝑛 𝜕𝜕𝑦𝑦 2 𝐷𝐷 𝜕𝜕𝜕𝜕 2 𝐸𝐸𝑥𝑥 2 𝑘𝑘𝑘𝑘 𝜕𝜕𝜕𝜕 𝜔𝜔𝑐𝑐𝜏𝜏𝑚𝑚 𝑢𝑢𝑦𝑦 𝜇𝜇𝐸𝐸𝑦𝑦 − 𝜔𝜔𝑐𝑐𝜏𝜏𝑚𝑚 𝜔𝜔𝑐𝑐𝜏𝜏𝑚𝑚 where, 1/ 𝑛𝑛 𝜕𝜕𝑦𝑦 𝐵𝐵0 𝑞𝑞𝐵𝐵0 𝑛𝑛 𝜕𝜕𝑥𝑥

𝜏𝜏𝑚𝑚 ≡ 𝜈𝜈𝑚𝑚 19/29 Radiation Source Engineering, Fall 2017 Perpendicular mobility and diffusion coefficient

 In vector form ExB drift Diamagnetic drift + × = ± + = 1 + ( ) 𝜵𝜵𝑛𝑛 𝒖𝒖𝐸𝐸 𝒖𝒖𝐷𝐷 𝑬𝑬 𝑩𝑩0 𝒖𝒖⊥ 𝜇𝜇⊥𝑬𝑬 − 𝐷𝐷⊥ −2 𝒖𝒖𝐸𝐸 2 where, 𝑛𝑛 𝜔𝜔𝑐𝑐𝜏𝜏𝑚𝑚 𝐵𝐵0 × = = = 𝑘𝑘𝑘𝑘 𝜵𝜵𝑛𝑛 𝑩𝑩0 1 + 1 + 𝒖𝒖𝐷𝐷 − 2 𝜇𝜇 𝐷𝐷 0 ⊥ 2 ⊥ 2 𝑞𝑞𝐵𝐵 𝑛𝑛 𝜇𝜇 𝑐𝑐 𝑚𝑚 𝐷𝐷  The mobility and𝜔𝜔 𝜏𝜏 diffusion fluxes perpendicular𝜔𝜔𝑐𝑐 𝜏𝜏to𝑚𝑚 the field exist only in the presence of collisions, and are slowed by the presence of the magnetic field. 1  Note that = 𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘𝜈𝜈𝑚𝑚 𝑘𝑘𝑘𝑘 ⊥ 2 2 ∥ 𝐷𝐷 ≈ 𝑚𝑚 𝑐𝑐 𝑚𝑚 𝑐𝑐 𝐷𝐷 ≈ 𝑚𝑚  Cross-field ambipolar diffusion𝑚𝑚𝜈𝜈 𝜔𝜔 coefficient𝜏𝜏 𝑚𝑚 𝜔𝜔(with assumption of𝑚𝑚 no𝜈𝜈 loss along B)

+ 1 + + 𝜇𝜇⊥𝑖𝑖𝐷𝐷⊥𝑒𝑒 𝜇𝜇⊥𝑒𝑒𝐷𝐷⊥𝑖𝑖 𝑇𝑇𝑖𝑖 𝐷𝐷⊥𝑎𝑎 ≈ ≈ 𝐷𝐷⊥𝑒𝑒 𝜇𝜇⊥𝑖𝑖 𝜇𝜇⊥𝑒𝑒 𝑇𝑇𝑒𝑒 20/29 Radiation Source Engineering, Fall 2017 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.

21/29 Radiation Source Engineering, Fall 2017 Plasma-sheath structure

22/29 Radiation Source Engineering, Fall 2017 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 23/29 Radiation Source Engineering, Fall 2017𝑒𝑒ℰ𝑠𝑠 ≡ 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 24/29 Radiation Source Engineering, Fall 2017 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 25/29 𝑛𝑛 𝑛𝑛 𝑒𝑒 ≈ Radiation𝑛𝑛 Source Engineering, Fall 2017 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 26/29 Radiation Source Engineering, Fall 2017 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 27/29 Radiation Source Engineering, Fall 2017 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 28/29 𝑀𝑀 𝑠𝑠 Radiation Source Engineering, Fall 2017 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 Ion𝑑𝑑 𝑑𝑑transit time𝑠𝑠 across the sheath = 1 2 2𝑒𝑒𝑉𝑉0 = = 𝑣𝑣0 3 𝑀𝑀 𝑥𝑥 𝑡𝑡 𝑣𝑣0𝑡𝑡 3𝑠𝑠 𝜏𝜏𝑖𝑖 29/29 𝑠𝑠 3𝑠𝑠 Radiation Source Engineering,𝑣𝑣0 Fall 2017