Basic Properties of Plasmas

Basic Properties of Plasmas

Fall, 2017

Kyoung-Jae Chung

Department of Nuclear Engineering

Characteristics of weakly-ionized plasmas

 A plasma is a collection of free charged particles moving in random directions that is, on the average, electrically neutral.

 Weakly-ionized plasmas have the following features:

(1) they are driven electrically;

(2) charged particle collisions with neutral gas molecules are important;

(3) there are boundaries at which surface losses are important;

(4) ionization of neutrals sustains the plasma in the steady state;

(5) the electrons are not in thermal equilibrium with the ions.

Classification of plasmas

Self-consistent character of plasmas

( )

mn dv qn E v B dt   

   

Equation of motion, etc

( , ) f r v  

dr v dt 

 

o E

     

B 0

   

o o o

B J E

    t

  



  

E B

t

  



 

Maxwell’s equations

(

) e Zn n

  

(

) J   e Zn u   n u  Density and velocity

, E B  

, , ,

i e i e

n n u u  

Maxwell’s equations

𝜖𝜖

𝜵𝜵 � 𝑬𝑬 = 𝜌𝜌

𝜵𝜵 � 𝑩𝑩 = 0

𝜵𝜵 × 𝑬𝑬 = − 𝜕𝜕𝑩𝑩

𝜕𝜕𝑡𝑡

𝜵𝜵 × 𝑩𝑩 = 𝜇𝜇

𝜖𝜖

𝜕𝜕𝑬𝑬

𝜕𝜕𝑡𝑡 + 𝜇𝜇

𝑱𝑱

 Gauss’s law

 Gauss’s law for magnetism

 Faraday’s law of induction

 Ampere’s law with Maxwell’s addition

𝜌𝜌 = 𝑒𝑒 𝑍𝑍𝑛𝑛

− 𝑛𝑛

𝑱𝑱 = 𝑒𝑒 𝑍𝑍𝑛𝑛

𝒖𝒖

− 𝑛𝑛

𝒖𝒖

𝜕𝜕𝜌𝜌

𝜕𝜕𝑡𝑡 + 𝜵𝜵 � 𝑱𝑱 = 0  Charge continuity equation

 Constitutive relation

Boltzmann’s equation & macroscopic quantities

 Particle density

 Particle flux

 Particle energy density

Particle conservation

𝜕𝜕𝑛𝑛

𝜕𝜕𝑡𝑡 + 𝜵𝜵 � 𝑛𝑛𝒖𝒖 = 𝐺𝐺 − 𝐿𝐿

The net number of particles per second generated within Ω either flows across the surface Γ or increases the number of particles within Ω.

For common low-pressure discharges in steady-state:

Hence,

𝐺𝐺 = 𝜈𝜈_{𝑖𝑖𝑖𝑖}𝑛𝑛_𝑒𝑒, 𝐿𝐿 ≈ 0

𝜵𝜵 � 𝑛𝑛𝒖𝒖 = 𝜈𝜈_{𝑖𝑖𝑖𝑖}𝑛𝑛_𝑒𝑒

 The continuity equation is clearly not sufficient to give the evolution of the density n, since it involves another quantity, the mean particle velocity u.

Momentum conservation

𝑚𝑚𝑛𝑛 𝜕𝜕𝒖𝒖

𝜕𝜕𝑡𝑡 + 𝒖𝒖 � 𝜵𝜵 𝒖𝒖 = 𝑞𝑞𝑛𝑛𝑬𝑬 − 𝜵𝜵𝑝𝑝 − 𝑚𝑚𝑛𝑛𝜈𝜈

𝒖𝒖

Convective derivative

 Pressure tensor

 isotropic for weakly ionized plasmas

 The time rate of momentum transfer per unit volume due to collisions with other species

Equation of state (EOS)

 An equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions.

𝑝𝑝 = 𝑝𝑝 𝑛𝑛, 𝑇𝑇 , 𝜀𝜀 = 𝜀𝜀(𝑛𝑛, 𝑇𝑇)

 Isothermal EOS for slow time variations, where temperatures are allowed to equilibrate. In this case, the fluid can exchange energy with its surroundings.

 The energy conservation equation needs to be solved to determine p and T.

𝑝𝑝 = 𝑛𝑛𝑛𝑛𝑇𝑇, 𝛻𝛻𝑝𝑝 = 𝑛𝑛𝑇𝑇𝛻𝛻𝑛𝑛

 Adiabatic EOS for fast time variations, such as in waves, when the fluid does not exchange energy with its surroundings

 The energy conservation equation is not required.

𝑝𝑝 = 𝐶𝐶𝑛𝑛

, 𝛻𝛻𝑝𝑝

𝑝𝑝 = 𝛾𝛾

𝛻𝛻𝑛𝑛

𝑛𝑛

𝐶𝐶_𝑝𝑝

𝐶𝐶_𝑣𝑣 (specific heat ratio)

 Specific heat ratio vs degree of freedom (𝑓𝑓) 𝛾𝛾 = 1 + 2 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

𝜕𝜕𝑓𝑓�

𝜕𝜕𝑡𝑡 _𝑐𝑐 = 0

 Then, we obtain the Maxwell-Boltzmann velocity distribution 𝑓𝑓 𝑣𝑣 = 2

𝜋𝜋 𝑚𝑚 𝑛𝑛𝑇𝑇

3 𝑣𝑣²𝑒𝑒𝑒𝑒𝑝𝑝 −𝑚𝑚𝑣𝑣² 2𝑛𝑛𝑇𝑇 𝑓𝑓 𝜀𝜀 = 2

𝜋𝜋 𝜀𝜀 𝑛𝑛𝑇𝑇

1/2𝑒𝑒𝑒𝑒𝑝𝑝 − 𝜀𝜀 𝑛𝑛𝑇𝑇

 The mean speed

̅𝑣𝑣 = 8𝑛𝑛𝑇𝑇 𝜋𝜋𝑚𝑚

1/2

Particle and energy flux

 The directed particle flux: the number of particles per square meter per second crossing the z = 0 surface in the positive direction

Γ

= 1 4 𝑛𝑛 ̅𝑣𝑣

 The average energy flux: the amount of energy per square meter per second in the +z direction

𝑆𝑆

= 𝑛𝑛 1

2 𝑚𝑚𝑣𝑣

𝑣𝑣

𝒗𝒗

= 2𝑛𝑛𝑇𝑇Γ

 The average kinetic energy 𝑊𝑊 per particle crossing z = 0 in the positive direction

𝑊𝑊 = 2𝑛𝑛𝑇𝑇

𝑣𝑣_𝑡𝑡𝑡 = 𝑛𝑛𝑇𝑇 𝑚𝑚

1/2

Boltzmann constant k

𝑃𝑃𝑃𝑃 = 𝑛𝑛𝑛𝑛𝑇𝑇 𝑃𝑃𝑃𝑃 = 𝑁𝑁𝑛𝑛𝑇𝑇

R (gas constant) = 8.31 JK^-1mol^-1

k (Boltzmann constant) = 1.38x10^-23 JK^-1

𝑆𝑆 = 𝑛𝑛 ln𝑊𝑊

Macroscopic state No. of microscopic states 𝑛𝑛 = 𝑛𝑛

𝑁𝑁_𝐴𝐴

 Bridge from macroscopic to microscopic physics

 Entropy

Temperature

 Temperature = Average kinetic energy at thermal equilibrium

 SI unit: Kelvin (K)

1

2 𝑚𝑚𝑣𝑣² = 3 2 𝑛𝑛𝑇𝑇

 The electron-volt (eV) unit is widely used for presenting the plasma temperature instead of Kelvin.

 Unit conversion between K and eV 𝐸𝐸 = 𝑛𝑛𝑇𝑇

 What is the difference between energy and temperature ?

1 eV = (1.6x10^-19 C) x (1 V) = (1.6x10^-19 J) / (1.38x10^-23 J/K) = 11,600 K

𝑛𝑛𝑇𝑇

= 𝑒𝑒T

Pressure

 Pressure = the force per unit area applied in a direction perpendicular to the surface of an object

 Units

• mmHg (millimeter of Hg, Torr) 1 atm = 760 mmHg = 760 Torr

• Pa (SI) = N/m²

1 atm = 101,325 Pa = 1,013.25 hPa

• Useful conversion relation 1 Pa = 7.5 mTorr

𝑝𝑝 = 𝑛𝑛𝑛𝑛𝑇𝑇

Vacuum and collision mean free path

𝜆𝜆[𝑐𝑐𝑚𝑚]~ 5 𝑝𝑝[𝑚𝑚𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚]

𝜈𝜈 = ̅𝑣𝑣

𝜆𝜆 = 𝑛𝑛^𝑔𝑔 < 𝜎𝜎𝑣𝑣 >

• Collision frequency

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 𝑬𝑬 = −𝜵𝜵Φ and assuming 𝑝𝑝_𝑒𝑒 = 𝑛𝑛_𝑒𝑒𝑛𝑛𝑇𝑇_𝑒𝑒

Integrating, we have 𝑚𝑚𝑛𝑛_𝑒𝑒 𝜕𝜕𝒖𝒖

𝜕𝜕𝑡𝑡 + 𝒖𝒖 � 𝜵𝜵 𝒖𝒖 = −𝑒𝑒𝑛𝑛^𝑒𝑒𝑬𝑬 − 𝜵𝜵𝑝𝑝_𝑒𝑒 − 𝑚𝑚𝑛𝑛_𝑒𝑒𝜈𝜈_𝑚𝑚𝒖𝒖

𝑒𝑒𝑛𝑛_𝑒𝑒𝜵𝜵Φ − 𝑛𝑛𝑇𝑇_𝑒𝑒𝜵𝜵𝑛𝑛_𝑒𝑒 = 𝑛𝑛_𝑒𝑒𝜵𝜵 𝑒𝑒Φ − 𝑛𝑛𝑇𝑇_𝑒𝑒ln𝑛𝑛_𝑒𝑒 = 0

𝑛𝑛_𝑒𝑒 𝒓𝒓 = 𝑛𝑛₀exp 𝑒𝑒Φ 𝒓𝒓

𝑛𝑛𝑇𝑇_𝑒𝑒 = 𝑛𝑛₀exp Φ 𝒓𝒓

T_e Boltzmann’s relation for electrons

 For positive ions in thermal equilibrium at temperature T_i 𝑛𝑛_𝑖𝑖(𝒓𝒓) = 𝑛𝑛₀exp −Φ(𝒓𝒓) T_i

 However, positive ions are almost never in thermal equilibrium in low pressure discharges because the ion drift velocity u_i is large, leading to inertial or frictional

Debye shielding (screening)

 Coulomb potential

 Screened Coulomb potential Φ 𝑚𝑚 = 𝑄𝑄

4𝜋𝜋𝜖𝜖₀𝑚𝑚

Φ 𝑚𝑚 = 𝑄𝑄

4𝜋𝜋𝜖𝜖₀𝑚𝑚exp − 𝑚𝑚 𝜆𝜆_𝐷𝐷

𝛻𝛻²Φ = −𝑒𝑒𝑛𝑛₀

𝜖𝜖₀ 1 − 𝑒𝑒^{𝑒𝑒Φ/𝑘𝑘𝑇𝑇𝑒𝑒} ≈ 𝑒𝑒²𝑛𝑛₀

𝜖𝜖₀𝑛𝑛𝑇𝑇_𝑒𝑒 Φ = Φ 𝜆𝜆_𝐷𝐷²

 Poisson’s equation

 Debye length 𝜆𝜆_𝐷𝐷 = 𝜖𝜖₀𝑛𝑛𝑇𝑇_𝑒𝑒

𝑒𝑒²𝑛𝑛₀

1/2

= 𝜖𝜖₀T_e 𝑒𝑒𝑛𝑛₀

1/2

Plasma

Debye length

𝜆𝜆_𝐷𝐷 cm = 𝜖𝜖₀T_e 𝑒𝑒𝑛𝑛₀

12

= 743 T_e eV

𝑛𝑛₀ cm⁻³ = 743 4

10¹⁰ ≈ 0.14 mm

 The electron Debye length 𝜆𝜆_{𝐷𝐷𝑒𝑒} 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.

+ -

- - - -

+ + +

+

λ_D Quasi-neutrality?

 Typical values for a processing plasma (n_e = 10¹⁰ cm^-3, T_e = 4 eV)

 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 particles when they collide.

Quasi-neutrality

 The potential variation across a plasma of length 𝑙𝑙 ≫ 𝜆𝜆_{𝐷𝐷𝑒𝑒} can be estimated from Poisson’s equation

We generally expect that

Then, we obtain

𝛻𝛻²Φ~Φ

𝑙𝑙² ~ 𝑒𝑒

𝜖𝜖₀ 𝑍𝑍𝑛𝑛_𝑖𝑖 − 𝑛𝑛_𝑒𝑒

Φ ≲ T_e = 𝑒𝑒

𝜖𝜖₀ 𝑛𝑛_𝑒𝑒𝜆𝜆_{𝐷𝐷𝑒𝑒}²

𝑍𝑍𝑛𝑛_𝑖𝑖 − 𝑛𝑛_𝑒𝑒

𝑛𝑛_𝑒𝑒 ≲ 𝜆𝜆_{𝐷𝐷𝑒𝑒}²

𝑙𝑙² ≪ 1 𝑍𝑍𝑛𝑛_𝑖𝑖 = 𝑛𝑛_𝑒𝑒

Plasma approximation

 The plasma approximation is violated within a plasma sheath, in proximity to a material wall, either because the sheath thickness s ≈ 𝜆𝜆_{𝐷𝐷𝑒𝑒}, or because Φ ≫ T_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)

2 2

0 2

0 e

x e

d n e

m eE

dt

 

    

2 2

0 2

0

0

e

e

d n e

dt m

 

  

 Electron plasma frequency

𝜔𝜔_{𝑝𝑝𝑒𝑒} = 𝑛𝑛₀𝑒𝑒² 𝑚𝑚𝜖𝜖₀

1/2

 If the assumption of infinite mass ions is not made, then the ions also move slightly and we obtain the natural frequency

𝜔𝜔 = 𝜔𝜔² + 𝜔𝜔^{2 1/2} where, 𝜔𝜔_{𝑝𝑝𝑖𝑖} = 𝑛𝑛₀𝑒𝑒²/𝑀𝑀𝜖𝜖₀ ^1/2 (ion plasma frequency)

Plasma frequency

 Plasma oscillation frequency for electrons and ions 𝑓𝑓_{𝑝𝑝𝑒𝑒} = 𝜔𝜔_{𝑝𝑝𝑒𝑒}

2𝜋𝜋 = 8980 𝑛𝑛⁰ Hz 𝑛𝑛₀ in cm⁻³ 𝑓𝑓_{𝑝𝑝𝑖𝑖} = 𝜔𝜔_{𝑝𝑝𝑖𝑖}

2𝜋𝜋 = 210 𝑛𝑛⁰/𝑀𝑀_𝐴𝐴 Hz 𝑛𝑛₀ in cm⁻³, 𝑀𝑀_𝐴𝐴 in amu

 Typical values for a processing plasma (Ar) 𝑓𝑓_{𝑝𝑝𝑒𝑒} = 𝜔𝜔_{𝑝𝑝𝑒𝑒}

2𝜋𝜋 = 8980 10¹⁰ Hz = 9 × 10⁸ [Hz]

𝑓𝑓_{𝑝𝑝𝑖𝑖} = 𝜔𝜔_{𝑝𝑝𝑖𝑖}

2𝜋𝜋 = 210 10¹⁰/40 Hz = 3.3 × 10⁶ [Hz]

 Collective behavior

𝜔𝜔_{𝑝𝑝𝑒𝑒}𝜏𝜏_𝑐𝑐 > 1 Plasma frequency > collision frequency

Criteria for plasmas

 Criteria for plasmas:

𝜆𝜆_𝐷𝐷 ≪ 𝐿𝐿

𝜔𝜔_{𝑝𝑝𝑒𝑒}𝜏𝜏_𝑐𝑐 > 1 𝑁𝑁_𝐷𝐷 ≫ 1 𝑁𝑁_𝐷𝐷 = 𝑛𝑛4

3 𝜋𝜋𝜆𝜆^𝐷𝐷3

 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”:

 Plasma parameter Λ = 4𝜋𝜋𝑛𝑛𝜆𝜆_𝐷𝐷³ = 3𝑁𝑁_𝐷𝐷

 Coupling parameter Γ = Coulomb energy

Thermal energy =

𝑞𝑞²/(4𝜋𝜋𝜖𝜖₀𝑎𝑎)

𝑛𝑛𝑇𝑇_𝑒𝑒 ~Λ^−2/3

Wigner-Seitz radius 𝑎𝑎 = ³ 4𝜋𝜋𝑛𝑛_𝑒𝑒/3