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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.
ECE 507 - Lecture 1 3
Examples of naturally occurring plasmas:
• (99% of the visible universe is a plasma)
Gas Nebula
Solar CoronaAurora Borealis
Lightning Flames
ECE 507 - Lecture 1 4
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:
ECE 507 - Lecture 1 5
• 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
ECE 507 - Lecture 1 6
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
ECE 507 - Lecture 1 7
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
ECE 507 - Lecture 1 8
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
The Concept of Temperature
Gaussian functions,
𝜎 =𝑚
𝑘𝑇
ECE 507 - Lecture 1 9
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
-
The Concept of Temperature
ECE 507 - Lecture 1 10
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
The Concept of Temperature
ECE 507 - Lecture 1 11
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
The Concept of Temperature
ECE 507 - Lecture 1 12
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
The Concept of Temperature
By 2 eV usually we mean: kT = 2 eV → Eav = 3 eV in three dimensions
ECE 507 - Lecture 1 13
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
ECE 507 - Lecture 1 14
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
ECE 507 - Lecture 1 15
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
ECE 507 - Lecture 1 16
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
Maxwell speed distribution
ECE 507 - Lecture 1 17
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
ECE 507 - Lecture 1 18
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):
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
ECE 507 - Lecture 1 19
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 speed distribution
Maxwell energy distribution
2
kT Em kT E
2
3