ECE507 - Plasma Physics and Applications

Lecture 1

Prof. Jorge Rocca and Dr. Fernando Tomasel Department of Electrical and Computer Engineering Introduction: What is a plasma?

• A quasi-neutral collection of charged (and neutral) particles which exhibits collective behavior.

ECE 507 - Lecture 1 2 Examples of naturally occurring plasmas:

• (99% of the visible universe is a plasma)

Gas Nebula Lightning Flames

Aurora Borealis Solar Corona

ECE 507 - Lecture 1 3 Examples of man-made plasmas:

Fluorescent Flat panel lamps (glow plasma discharge) display

Plasma etching reactor (plasmas play important role in the manufacturing of integrated circuits) Plasma torch

Z-pinch Laser-created plasmas

ECE 507 - Lecture 1 4 Particles found in plasma • Neutral and ionized atoms Densities: N(z) z = ion charge

• Free Electrons Ne • Photons ρ( )  if N(z = 0)  0 the plasma is partially ionized  if N(z = 0) = 0 the plasma is completely ionized (no neutral atoms)

All these particles interact with each other and with electric and magnetic fields making the plasma a very complex system Electrons Ions N e N(z)

ρ()

Photons ECE 507 - Lecture 1 5 Plasma Parameters

• Plasma Density Ne   N(z) z z: ion charge z • Electron temperature Te

• Ion temperature Ti • Mean ion charge Z These plasma parameters determine important plasma properties Examples 1 2 • Debye screening distance (distance beyond which individual  ε kT  λ   0 e  D  2  charges tend to be screened by other nearby charges)  e Ne 

2 1 2 π e m Where lnΛ is the • Electrical resistivity: η  e ln Λ (z 1) 2 3 2 Coulomb logarithm ≈10 4π ε0  kTe 

1 2  e2 N  • Plasma frequency (natural frequency at which electrons  e  ωp    tend to oscillate)  ε0me 

ECE 507 - Lecture 1 6 The Concept of Temperature

A gas has particles of all velocities

If a sufficiently large number of collisions occurred between these particles the most probable distribution of these velocities is known as the Maxwell Distribution For simplicity lets consider a gas in which the particles can move in only one direction (e.g. charged particles in a strong magnetic field). The one dimensional Maxwell Distribution is given by:

1 2 (1.1) f(vi )  A exp(  2 mvi /kT) 3 • f(vi)dvi is number of particles per m with velocity between vi and vi+dvi 2 • ½ mvi is the kinetic energy • k = 1.38 10-23J/K is Boltzmann’s constant  3 • The density of particles per m is N  f(vi )dvi (1.2)  - 1 2  m  • A is a normalization constant related to density A  N   (1.3) 2π kT ECE 507 - Lecture 1   7 The Concept of Temperature The width of the distribution is characterized by a parameter T we call the f(v ) Temperature i Gaussian functions, T > T T 2 1 1 푚 휎 = T2 푘푇

0 vi T is related to the average kinetic energy EAV  1 2 ( 2 mvi ) f(vi ) dvi E   - av  (1.4) f(vi ) dvi  - We will define the thermal (most probable) velocity as

1 2 2 mvTh  kT (1.5) 1 2  2kT  vTh     m  (1.6) Substituting (1.5) in (1.1)  v 2  f(v )  A exp i  (1.7) i  2   vTh  ECE 507 - Lecture 1 8 The Concept of Temperature

v Defining Υ  (1.8) vTh

f(v)  A exp(-Y) (1.9)

Substituting in 1.4 (and multiplying and dividing vi by vTh to form Y)

3 1  2 mAvTh 2 2 EAv  exp-Y  Y dY (1.10) N -

Integrating the numerator by parts:

   2 1 2 1 2 Yexp-Y  YdY   2 exp( Y )Y    2 exp-Y  dY  -   (1.11)  1 2 1  2 exp( Y ) dY  2  

ECE 507 - Lecture 1 9 The Concept of Temperature

Summarizing

3/ 2 1 m  2kT  2 mN    1 mA v3 1  2 kT  m  E  2 Th 2   1 kT av N N 2 (1.12)

1 Eav  2 kT Average kinetic energy (1.13) in one dimension

ECE 507 - Lecture 1 10 The Concept of Temperature Maxwell’s velocity distribution in three dimension can be written as 1 2 2 2 f(vx ,vy ,vz )  A3 exp  2 vx  vy  vz /kT  (1.14)

3  1 2   m  (1.15) A3  N     2π kT   The average kinetic energy is  1 2 2 2 1 2 2 2 A3 2 mvx  vy  vz  exp  2 mvx  vy  vz /kT  dvxdvydvz   Eav   (1.16) 1 2 2 2 A3 exp  2 mvx  vy  vz /kT  dvxdvydvz  

The expression is symmetric in vx, vy, vz since the Maxwell distribution is isotropic

1 2 1 2 1 2 2 3A 2 mv exp  2 mv /kT  dv exp  2 m v  v / kT  dv dv 3  x x x  y z y z 1.17) Eav  ( A exp  1 mv2 / kT dv exp  1 m v2  v2 / kT dv dv 3   2 x  x   2  y z   y z

Average kinetic energy E  3 kT av 2 in three dimensions (1.18)

ECE 507 - Lecture 1 11 The Concept of Temperature

Since T is so closely relate to Eav it is common in plasma physics to give the temperature in units of energy.

To avoid confusion in the number of dimensions involved it is not Eav but the energy corresponding to kT that is used to denote temperature.

1.6 10-19 J For kT 1 eV 1.6 1019 J  T  11,600 oK 1.38 10-23 J/oK

1 eV 11,600 K

By 2 eV usually we mean: kT = 2 eV → Eav = 3 eV in three dimensions

ECE 507 - Lecture 1 12 Temperature is an equilibrium concept Notice that to define the previous relations we assumed a Maxwellian distribution. If two groups of particles with different velocities are allowed to undergo a sufficient number of collisions, they will interchange energy and “thermalize” acquiring a Maxwellian distribution.

F1 F2 Mono-energetic distribution v v v1 v2 Collisions

FNM Non-Maxwellian distribution

More v collisions F Maxwellian distribution (thermalization has occurred) v

1 2 The Maxwellian distribution is defined by only   2 mv  f(v )  A exp  i  i   one parameter: the temperature T  kT  ECE 507 - Lecture 1 13 Electron, Ion and Atomic temperatures

Electron, ion and atoms in the same plasma can all have different temperatures • The interchange of energy in collisions between particles of equal mass is large (examples: collisions between electrons and electrons, ions and ions) • The e-e collision rate >> e-i collision rate Therefore electrons tend to be in “thermal equilibrium” with other electrons and ions with other ions, but often they are not in equilibrium with each other. F e This situation requires a different temperature to define each group Te= electron temperature

v 1 2 fe(vx )  Ae exp(  2 mvx /kTe )

Fi

1 2 fi(vx )  Ai exp(  2 mvx /kTi ) Ti= ion temperature with Te  Ti

v

ECE 507 - Lecture 1 14 Thermalization Electrons and Ions are often in Thermal Equilibrium with themselves but not with each other

Examples: Glow discharges (Neon sign, He-Ne laser discharge) Te > Ti Theta Pinch (Magnetically compressed plasma Ti > Te The electron-electron equilibration time is much Te e i Ti e i shorter than the electron-ion equilibration time e-e collisions i-i collisions Equilibration times in seconds (L. Spitzer – Physics fully ionized gases) Electron-electron equilibration time Electron-ion equilibration time 1.66 104 Te3 2 1.98 108 Te3 2 [Te] = eV, τeq(e  e)  s τeq(e  i)  s N = Z N N [N ] = cm-3 2 e mean i e e Ne Z 21 Examples: A carbon laser-created plasma: Te = 150 eV, Ne = 1 x 10 , Z = 6 3 1.66  104 150 2  (e - e)   3  10-14s  30 fs eq 1  1021 At t  10 ps

3 1.98  108 150 2  Te  Ti  (e -i)   1  10-11s  10 ps eq 1  1021  36 This motivates ‘two temperature plasma’ models ECE 507 - Lecture 1 15 Maxwell speed distribution The figure below shows the geometrical interpretation of the speed distribution function, and also serves to illustrate the conversion from velocity coordinates (vx, vy, vz) to that of speed, v.

1 2 1 2 2 2 2 mv  2 mvx  vy  vz   E

 - f v dv  f vx ,vy ,vz  dvxdvydvz  0   4πv2dv

Three-dimensional velocity space

f(v)

3 2 2  m   mv  2 f v  N   exp   4πv  2πkT   2kT  Maxwell speed distribution

v ECE 507 - Lecture 1 16 Maxwell speed distribution The zeroth moment of the speed distribution function (equal to the area under the function) is equal to the particle density:

 v0f v dv  N  0 The first moment of the velocity distribution is the arithmetic mean speed, mean thermal velocity, or average magnitude of the velocity:

3/ 2 2 1   m    mv  v  v f v dv    v exp 4v2dv   0  0   N  2πkT   2kT  3/ 2 2  m   2kT      4   u3 exp u 2 du  2πkT   m  0 = 1/2

1/ 2  8kT  v      m 

ECE 507 - Lecture 1 17 Maxwell speed distribution The second moment of the speed distribution function is related to the root mean square speed of the particles (related to the average energy):

3/ 2 2   2 1 2  m  2  mv  2 vrms  v f vdv    v exp 4v dv   0  0   N  2πkT   2kT  3/ 2 5/3  m   2kT      4   u 4 exp u 2 du  2πkT   m  0 3   8 3kT v2  rms m The most probable speed (some times called thermal velocity) is calculated by differentiating the distribution function once and setting it equal to zero:

1/ 2 d   mv2   2kT  v2 exp   0 v     th   dv   2kT   m 

ECE 507 - Lecture 1 18 Maxwell speed distribution

The speed distribution can be rewritten as a function of energy using the relation between speed and energy:

1  2E  2 v     m 

1 N 2  E  2  E  f(E)    exp   kT π  kT   kT  Maxwell energy distribution

Performing similar calculations to those in previous slides, you can easily show that the most probable energy and the mean energy are given by kT 3 E  E  kT m 2 2

ECE 507 - Lecture 1 19