1

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

2

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

×=

×≈×××

≈ −

−

3

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

4

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

5

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

6

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

7

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

8

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

9

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

10

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

11

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

12

Band Theory

There are striking differences between conductors and semiconductors

Recall R=ρL/A

13

Band Theory

14

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

15

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

16

Band Theory

The symmetric combination will be lower in energy because the ions exert a greater Coulomb attractive force

17

Band Theory

Energy levels of two atoms as they are brought close together

18

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

19

Band Theory

Energy levels of six atoms as they are brought close together

20

Band Theory

Energy levels of N atoms as they are brought close together – band formation

21

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