ECE507 - Plasma Physics and ApplicationsThe first moment of the velocity distribution is the arithmetic mean speed, mean thermal velocity, or average magnitude of the velocity: f d

ECE507 - Plasma Physics and Applications

Lecture 1

Prof. Jorge Rocca and Dr. Fernando Tomasel

Department of Electrical and Computer Engineering

ECE 507 - Lecture 1 2

Introduction: What is a plasma?

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


Examples of naturally occurring plasmas:

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

Gas Nebula

Solar CoronaAurora Borealis

Lightning Flames


Z-pinch

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

Laser-created plasmas

Flat panel plasma display

Fluorescent lamps (glow

discharge)

Plasma torch

Examples of man-made plasmas:


• 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 magneticfields making the plasma a very complex system

Ne

Electrons

N(z)

Ions

ρ()

Photons

Particles found in plasma


Plasma Parameters

• Plasma Density

• Electron temperature

• Ion temperature

• Mean ion charge

These plasma parameters determine important plasma properties

Examples

• Debye screening distance (distance beyond which individual charges tend to be screened by other nearby charges)

• Electrical resistivity:

• Plasma frequency (natural frequency at which electrons tend to oscillate)

z

e zN(z)N

eT

21

2

0

e

eD

Ne

kTελ

1)( ln

4232

0

212

zΛkTπ ε

mπ eη

e

e

21

0

2

e

ep

mε

Neω

Z

Where lnΛ is the Coulomb logarithm ≈10

z: ion charge

iT


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 MaxwellDistribution

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:

(1.1)

• f(vi)dvi is number of particles per m3 with velocity between vi and vi+dvi

• ½ mvi2 is the kinetic energy

• k = 1.38 10-23J/K is Boltzmann’s constant

• The density of particles per m3 is (1.2)

• A is a normalization constant related to density (1.3)

/kT)mv(A )f(v ii2

21exp

-ii v)f(vN d

21

2

π kT

mN A

The Concept of Temperature


The width of the distribution is characterized by a parameter T we call theTemperature

T is related to the average kinetic energy EAV

(1.4)

We will define the thermal (most probable) velocity as

(1.5)

(1.6)Substituting (1.5) in (1.1)

(1.7)

f(vi)

vi0

T1

T2

T2> T1

-ii

-iii

av

v) f(v

v) ) f(vmv(E

d

d2

21

kTmvTh 2

21

212

m

kTvTh

2

2

expTh

ii

v

vA )f(v


Gaussian functions,

𝜎 =𝑚

𝑘𝑇


Defining (1.8)

(1.9)

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

(1.10)

Integrating the numerator by parts:

(1.11)

Thv

vΥ

(-Y)A f(v) exp

-

ThAv Y Y-Y

N

mAvE dexp 22

32

1

212

21

22

122

12

dexp

dexpexpdexp

Y) Y(

Y -Y)YY( Y Y-YY

-



Summarizing

(1.12)

(1.13)

kTN

m

kT

kT

mmN

N

mA vE Thav 2

1

2/3

21

213

21

2

2

Average kinetic energy in one dimension

kTEav 21



Maxwell’s velocity distribution in three dimension can be written as

(1.14)

(1.15)

The average kinetic energy is

(1.16)

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

(1.17)

(1.18)

/kTvvv A) ,v,vf(v zyxzyx 222213 exp 3

21

32

π kT

mNA

zyxzyx

zyxzyxzyx

av

vvv /kTvvvm A

vvv /kTvvvm vvvmAE

dddexp

dddexp

2222

13

2222

12222

13

vv / kTvvm v / kTmvA

vv / kTvvm v /kTmv mvAE

zyzyxx

zyzyxxx

avddexpdexp

ddexpdexp3

222

122

13

222

122

122

13

Average kinetic energy

in three dimensionskTE 23av



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

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

K 11,600KJ/10 1.38

J10 1.6 J10 1.6eV 1For

23-

-1919 o

o TkT

K 11,600eV 1


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


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 sufficientnumber of collisions, they will interchange energy and “thermalize” acquiring aMaxwellian distribution.

v1 v2

Mono-energetic distribution

Collisions

FNM

v

Non-Maxwellian distribution

More collisions

Maxwellian distribution (thermalization has occurred)

F2

v

F1

v

The Maxwellian distribution is defined by onlyone parameter: the temperature T

kT

mv A )f(v ii

2

21

exp

F

v

Temperature is an equilibrium concept


Electron, ion and atoms in the same plasma can all have different temperatures

• The interchange of energy in collisions between particles of equal mass islarge (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 electronsand ions with other ions, but often they are not in equilibrium with each

other.

Te= electron temperature

Ti= ion temperature

This situation requires a different temperature to define each group

)/kTmv( A)(vf exexe2

21exp

ie

ixixi

T T with

)/kTmv( A)(vf

2

21exp

Fi

v

Fe

v

Electron, Ion and Atomic temperatures


Electrons and Ions are often in Thermal Equilibrium with themselvesbut not with each other

Examples: Glow discharges (Neon sign, He-Ne laser discharge) Te > TiTheta Pinch (Magnetically compressed plasma Ti > Te

The electron-electron equilibration time is much shorter than the electron-ion equilibration time e-e collisions

Te e

ei-i collisions

Ti

i

i

Examples: A carbon laser-created plasma: Te = 150 eV, Ne = 1 x 1021, Z = 6

ps 10 s10 1

36 10 1

150 10 1.98 i)-e(

fs 30 s10 3 10 1

150 10 1.66 e)-e(

11-

21

8

eq

14-

21

4

eq

23

23

Ti Te

ps 10 At t

This motivates ‘two temperature plasma’ models

Equilibration times in seconds (L. Spitzer – Physics fully ionized gases)

sN

Te . e) (e τ

e

eq

23410661 s

ZN

Te . i) (e τ

e

eq 2

23810981

Thermalization

[Te] = eV, [Ne] = cm

-3

Electron-electron equilibration time Electron-ion equilibration time

Ne = Zmean Ni


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.

E vvvm mv zyx 22221221

zyx -

zyx

vvv ,v,vvf v vf dddd

0

vvπ d4 2

22

42

exp2

23

vπ kT

mv

kTπ

m N vf

f(v)

v

Three-dimensional velocity space

Maxwell speed distribution




The zeroth moment of the speed distribution function (equal to the area under the function) is equal to the particle density:

The first moment of the velocity distribution is the arithmetic mean speed, mean thermal velocity, or average magnitude of the velocity:

N d f

0

0

vvv

= 1/2

uuum

kT

πkT

m

vvkT

mvv

πkT

m v vfv

Nv

dexp2

42

d42

exp2

d1

2

0

3

22/3

0

222/3

0

2/1

8

m

kTv



The second moment of the speed distribution function is related to the root mean square speed of the particles (related to the average energy):

The most probable speed (some times called thermal velocity) is calculated by differentiating the distribution function once and setting it equal to zero:

uuum

kT

πkT

m

vvkT

mvv

πkT

m vvfv

Nv

rms

dexp2

42

d42

exp2

d1

2

0

4

3/52/3

0

22

2

2/3

0

22

m

kTvrms

32

8

3

2/122 20

2exp

d

d

m

kTv

kT

mvv

vth


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

Performing similar calculations to those in previous slides, you can easily show that the most probable energy and the mean energy are given by

21

2

m

E v

kT

E

kT

E

π

kT

N f(E) exp

2 21


Maxwell energy distribution

2

kT Em kT E

2

3

Documents

ECE507 - Plasma Physics and ApplicationsThe first moment of the velocity distribution is the arithmetic mean speed, mean thermal velocity, or average magnitude of the velocity: f d