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Transport propertiesq
( , )dm
j p r p' p'
v E
nej v ne E E
2meter /volt-second
j E
is the conductivity in mhos/meter or siemens/meter
Find the conductivity of a resistor
j E
0VI
area A
length L
conductivity
0VI
A L
0
IL
AV
Conductivity of a plasma
12 3
edv
m qEdt
cmax 0
e
qEv dt
m
ce
qE
m
average velocity ce
qE
2m c
e
qmobility
2m
e econductivity n q2
ec
e
n q
2m
Diffusion
R
C
I( z,t )
z
V( z,t )RI( z,t )
z
R
V( z,t )C
t
Diffusion2
2
V( z,t ) V( z,t )RC
z t
1D
RC
meter
ohms farads
2
meter
second
2
1
Dohms/meter farads/meter
2
z RClet
t
( ) ( , )U tV z t
2
2
( , ) ( , )V z t V z tRC
z t
2
2
1 ( , )V z t
RC z
22 2
2 2
1 ( , ) 1
2
V z t d U RC
RC z dRC t t
22
2
1 d U
zdRC t
2
z RC
t
( )( , )
UV z t
t
( , )V z t
t
3
2
1
2
U dU
d ttt
3 32 2
( , ) 1
2 2
V z t U dU z RC
dt tt t
2
z RC
t
( )( , )
UV z t
t
3 32 2
( , ) 1
2 2
V z t U dU z RC
dt tt t
32
( , ) 1
22
V z t U dU
d tt tt
( , )V z t
t
3
2
1
2
U dU
d ttt
22
2 32
1 1
22 2
d U RC U dU
d tdRC t t tt
2
2
1 ( , ) ( , )V z t V z t
RC z t
2
2
d Ud U
dd
2
2
d U dUU
dd
2
2
d Ud U
dd
constant - Ud U
Ud
dU
Ud
d
lncabin
cabincabin
2
ln( )2
U
2
2U e
2
2U e
2
z RC
t
( ) ( , )U tV z t
2
4( , )
RCz
toVV z t et
-5 0 50
0.2
0.4
0.6
0.8
1de
nsity
z
t = 1t = 2t = 3t = 4t = 5
2
0( , )4
RCzV z t V erf
t
2
0( , )4
RCzV z t V erfc
t
1erfc erf
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1de
nsity
z
t = 1t = 2t = 3t = 4t = 5
Diffusion
x
area A
2
2
n( x,t ) 1 n( x,t )
Dx t
Harmonic oscillatorhttp://www.kettering.edu/~drussell/Demos/SHO/mass.html
2202
d xx 0
dt
0 0x( t ) x cos t
dx( t )v( t )
dt 0 0 0x sin t
Maxwell Boltzmann distribution
2
Maxwell Boltzmann 3BB
1 pf exp
2m T2 m T
Boltzmann equationhttp://grus.berkeley.edu/~jrg/ay202/node32.html
Ludwig Eduard Boltzmann (1844-1906). The physicist whose greatest achievement was in the development of statistical mechanics, which explains and predicts how the properties of atoms determine the visible properties of matter such as viscosity, thermal conductivity, and diffusion. He is reputed to have smuggled wine into the Faculty Club during his 1904 visit to Berkeley--at that time Berkeley was a dry town
Boltzmann equation
is known at a time tf ( x,v,t ) dN( x,v,t )
f ( x x,v v,t t ) dN( x x,v v,t t ) f ( x v t ,v a t ,t t )
Boltzmann equation
f ( x,v,t )
?t
f f ( x,v,t )
xv
f
t
collisions
f f f dfFv mt x v dt
v Fm
f dv
v dt
f dx
x dtcollisions
Water bag distribution
2
Maxwell Boltzmann 3BB
1 pf exp
2m T2 m T
Boltzmann equation
collisions
f f f dfFv mt x v dt
f f
v 0t x
0f f ( x vt ) n fdv
Velocity modulationhttp://www2.slac.stanford.edu/vvc/accelerators/klystron.html
Velocity modulation
2e 0 1
1m v q V V sin t2
0 1e
2qv V V sin t
m
0 1
e 0
2qV Vv 1 sin t
m V
0 1
e 0
2qV Vv 1 sin t
m 2V
0V 1V
Velocity modulationhttp://www.google.com/search?hl=en&client=firefox-a&rls=org.mozilla%3Aen-
US%3Aofficial&hs=aBD&q=Applegate+diagram&btnG=Search
0 1
e 0
2qV Vv 1 sin t
m 2V
Phase space interpretation of velocity modulation
Velocity modulation animation
0 1
e 0
2qV Vv 1 sin t
m 2V
Velocity modulationhttp://www2.slac.stanford.edu/vvc/accelerators/klystron.html
“bump on a tail distribution”
2
Maxwell Boltzmann 3BB
1 pf exp
2m T2 m T
Summary of physics principles
• Waves and particles are related – photo electric effect
Summary of physics principles
• Schrödinger equation gives us empty states and quantum numbers
• Fermi function tells us if the state is filled• Boltzmann equation describes how the
distribution of states evolves in space and time
• Velocity modulation helped win World War II and demonstrates the Boltzmann equation
Energy bands• With more than one atom, one has to
inquire about possible interaction between individual particles.
• Think of two race cars (or two witches) – the one behind uses less energy if it is following very closely behind the first one.
• Splitting of individual energy levels yielding a band.
2 21 1m v v mv2 2
mv v
Energy bands• Probability density functions from two
adjacent atoms in close juxtaposition causes interaction and splitting of the lowest state yielding a band.
separated close together
band
Energy bands• Valence band – the top most energy band
containing electrons
• Conduction band – the energy band just above the valence band
• Electrons in the conduction band can move from one location to another
• Why did the chicken cross the road? • Aristotle: It is the nature of chickens to cross roads. • Issac Newton: Chickens at rest tend to stay at rest,
chickens in motion tend to cross roads. • Albert Einstein: Whether the chicken crossed the road or
the road moved beneath the chicken depends on your frame of reference.
• Werner Heisenberg: We are not sure which side of the road the chicken was on, but it was moving very fast.
• Wolfgang Pauli: There already was a chicken on this side of the road.
