Free Electron Modelatlas.physics.arizona.edu/~kjohns/downloads/phys242/... · 2006. 12. 4. · 1...
Transcript of Free Electron Modelatlas.physics.arizona.edu/~kjohns/downloads/phys242/... · 2006. 12. 4. · 1...
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Free Electron ModelThe free electron model (Fermi gas) can also be used to calculate a pressure called the degeneracy pressureAs the Fermi gas is compressed, the energy of the electrons increases and positive pressure must be applied to compress it
This degenerate motion is forced on the electrons by quantum mechanics and it cannot be stopped by cooling the solid even to absolute zero
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Free Electron ModelThe degeneracy pressure is given by
For a typical metal this is
This enormous pressure is counteracted by the Coulomb attraction of the electrons to the positive ions
3/523/14
deg
3/23/523/43/5
2356 Then
253
53 and
⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛=
==∂∂
−= −
VN
mP
VNm
NEUVUP F
h
h
π
π
( ) ( )Paatm
PaP
5
103/5282
31
34
deg
101 1 Recall
1051016109101
×=
×≈×××
≈ −
−
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Free Electron ModelThis degenerate pressure is important in stellar evolution
Stars burn by fusione.g.
For stars that are burning (fusion) the gas and radiation pressure supports the star against gravitational compressionWhen the star becomes primarily 12C, 16O or heavier the burning (fusion) stops and the star resumes gravitational collapse
The only barrier to this is the degeneracy pressure
MeVeHHH 44.1211 +++→+ ν
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Free Electron ModelThe gravitational potential energy is calculated as
( )
( ) 3/123/1
3
522
0
422
422
23
34
53
34 and using thisrewrite can we
1516
316
find wegintegratin and3
16
43
4
−⎟⎠⎞
⎜⎝⎛⎟⎠⎞
⎜⎝⎛−=
==
−=−=
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛
−=−=
∫
VmNGU
RVV
mN
RGdrrGU
drrGdU
r
drrr
Gr
mdmGdU
pnucg
pnuc
R
g
g
g
π
πρ
ρπρπ
ρπ
πρπρ
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Free Electron ModelThen the gravitation pressure is
Which can be compared to our previous result for the degeneracy pressure
( ) 3/423/1
34
51 −⎟
⎠⎞
⎜⎝⎛+=
∂∂
−= VmNGVU
P pnucg
gπ
2 where
22356 3/523/14
deg
nuce
nuc
e
NN
VN
mP
≈
⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛=
hπ
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Free Electron ModelOnce fusion ends, gravitational pressure causes the star to collapse until the gravitational pressure and degeneracy pressure balanceThis happens at
Stars that reach this state are called white dwarfs
Most small and medium size stars (like our sun) end up as white dwarfs
( )3/1233/12
23/12
1015.112881 −− ×≈⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛= nucnuc
pe
NkmNmGm
R hπ
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Free Electron ModelWhite dwarfs have initial masses of M < 3-4 Msun but radii on the order of Rearth!
We can calculate R using the results of our calculation
Or a teaspoon of white dwarf would weigh ~ 5 tons
5727
30
102.1107.1102
×=××
=≈ − kgkg
mMN
p
sun
( )
)2x (about 101.1 So
1015.112881 And
4
3/1233/12
23/12
earth
nucnucpe
RkmR
NkmNmGm
R
×≈
×≈⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛= −−hπ
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Free Electron ModelIf Mstar > ~1.4 Msun, there are larger numbers of electrons with larger energiesIf Ne is large enough, the electrons become relativistic and E≈pc instead of p2/2mThis changes the expression for the degeneracy pressure and in fact the gravitational pressure overcomes the degeneracy pressure and collapse continues
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Free Electron ModelThis further collapse causes
This means the star becomes a neutron star and now we calculate the degeneracy pressure using identical neutrons instead of identical electrons
For Mstar~2Msun, equilibrium between the gravitational and degeneracy pressure can be reached and R ~ 10 km!
If Mstar > several Msun, there will be more neutrons and their energy will become relativistic
At this point there is no mechanism to counterbalance the huge gravitational pressure and a black hole is formed
ν+→+− npe
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Band TheoryThe free electron model was successful in predicting a number of properties of metals
Conduction electrons had relatively long mean free paths l in order to gain agreement with experimental conductivity
Electron-nucleus (ion) and electron-electron interactions were ignoredThe model isn’t appropriate for semiconductors or insulators
ρσ 12
==Fmvlne
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Band TheoryThe free electron model was successful in predicting a number of properties of metals
Conduction electrons had to have relatively long mean free paths l in order to gain agreement with experimental conductivity
Electron-nucleus (ion) and electron-electron interactions were ignoredThe model isn’t appropriate for semiconductors or insulators
ρσ 12
==Fmvlne
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Band Theory
There are striking differences between conductors and semiconductors
Recall R=ρL/A
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Band Theory
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Band TheoryBand theory incorporates the effects of electrons interacting with the crystal lattice
The interaction with a periodic lattice results in energy bands (grouped energy levels) with allowed and forbidden energy regions
There are two standard approaches to finding the energy levels
Bring individual atoms together to form a solidApply the Schrodinger equation to electrons moving in a periodic potential
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Band Theory
Consider first two atoms (say Na) When the atoms are well separated there is little overlap between the electronic wave functions
The energy is two-fold degenerate
When the atoms are brought closer together, the electronic wave functions will overlap and we must use wavefunctionswith the proper symmetry
As the two atoms move closer together the degeneracy is broken
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Band Theory
The symmetric combination will be lower in energy because the ions exert a greater Coulomb attractive force
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Band Theory
Energy levels of two atoms as they are brought close together
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Band Theory
Consider next six atomsAgain, as the atoms are brought closer together there will be symmetric and antisymmetric wavefunctionsBut there will also be wavefunctions of mixed symmetry and the corresponding energies lie between the twoNote the energy increases as the separation R becomes smaller because of the Coulomb repulsion of the ions
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Band Theory
Energy levels of six atoms as they are brought close together
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Band Theory
Energy levels of N atoms as they are brought close together – band formation
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Band TheoryFor higher (or lower) energy levels an energy gap may or may not exist depending on the type of atom, type of bonding, lattice spacing, and lattice structure