Fermi-Dirac and Bose-Einstein...

Phys 112 (S2005) 6 1 B.Sadoulet

Fermi-Dirac and Bose-EinsteinDistributions

Beg. chap. 6 and chap. 7 of K &KQuantum Gases

Fermions, BosonsPartition functions and distributionsDensity of states

Non relativisticRelativistic

Classical LimitFermi Dirac Distribution

Fermi Energy Electrons in solids

Nuclear matterWhite Dwarf

Bose Einstein DistributionBose-Einstein CondensationLiquid Helium


Quantum GasesBosons, Fermions

Integer spin= Boson = number of particles in a given state isarbitraryHalf Integer spin=Fermion= at most one on each orbital:

Pauli exclusion principlePartition Functions and mean occupation numbers

FermionAt most one

BosonSum on all integers

Z = 1+ expµ ! "#

$%&

'()

s !( ) =1

exp! " µ

#$ % & '

( ) +1

Z= exp sµ ! "#

$%&

'()

s=0

*

+ =1

1! exp µ ! "#

$%&

'()

s !( ) =1

exp! " µ

#$ % & '

( ) "1


Ideal quantum gases“Ideal gas” approximation

States are not modified by presence of other particles

“Density “ of statesmultiplicity x density in phase space change of variable to energy

Non relativistic

Ultra-Relativistic

gi !d3x d3p

h3

gi !p2dp d"

h3"# = D($ )d$ where D($) is the density of states

D !( )d! = 4"gip2dp

h3=

gi

4"22m

h2

#

$ %

&

' (

3

2 !d!

D !( )d! = 4"gip2dp

h3=

gi

2"2h3c3

!2d!


BehaviorSign is critical for (ε-µ)/τ small

Fermi-Dirac

Bose-EinsteinBose condensation

For (ε-µ)/τ large, classical limitOccupation number << 1

Same old resultsBoltzmann /Gibbs !

s !( ) =1

exp! " µ

#$ % & '

( ) + 1

=1

2 for ! = µ

= 1 for ! << µ

s !( ) =1

exp! " µ

#$ % & '

( ) "1

*+ for ! * µ

s !( ) " exp

µ # !$

%&'

()*= + exp #

!$

%&'

()*

independent of F.D. or B.E.

Prob !( ) =s !( )

N!!!!< N >= V s !( )" D !( )d! = V exp

µ

#$%&

'()exp *

!#

$%&

'()"d3p

h3

!!!< N >= V expµ

#$%&

'()nQ + µ = log

n

nQ

$

%&

'

()

<s(ε)>

µ ε

1


Thermo functions for ideal quantum gases

Number of Particles

µ(τ) set by requirement that N=total number of particles

Energy

- Bose- Einstein + Fermi-Dirac

Entropy

N =V s !( ) D !( )d!0

"# = V

D !( )d!

exp! $ µ

%& ' ( )

* + ±1

0

"#

U = V!D !( )d!

exp! " µ

#$ % & '

( ) ±1

0

*+

! "( ) =

# $ logZ( )

#$ !=! " µ

#s + logZ ! " log prob s =< s >( )$% &'


Fermi Gas Ground State (Non relativistic)

Fermi EnergyCalculation :

Energy/Free Energy

Pressure Repulsive!

s !( )

!

!F

! << µ

s !( ) =1

exp! " µ

#$ % & '

( ) + 1

*1 for ! < µ

! 0 for " > µ

N =V s !( ) D !( )d!0

"# = V D !( )d!

0

! F#

!F = µ " = 0( )

!"F =

h2

2m3#

2n( )

2

3 with n =N

V

U = F = V !D !( )d!0

!F

" =3

5N!

F

p = !"F

"V #

=2

5n$F %V

!5

3

Spin 1/2


Fermi Gas Ground State (Relativistic)Note that even at zero temperature very large kinetic energies: in

some case ultra-relativistic

Fermi Energy

Energy

Pressure

!F = µ " = 0( )

N =V D !( )d!0

! F" =

V

3#2h3c3!F3

!"F = 3#

2n( )1

3 hc

U = F = V !D !( )d!0

! F" =

3

4N!F

p = !"F

"V #

=1

4n$F %V

!4

3


Hole and Electron ExcitationsSymmetry of Fermi Dirac distribution

Basic symmetry (except for lower bound at δ =-µ ).Note:

Decomposition into hole-like and electron-likeexcitations

Number of excited electrons= number of excited holes

When referenced to the Fermi energy, the energy of holes are opposite tothat of the corresponding missing electrons and is positive

With !="-µ !!!!!!!! f ",#( ) =1

exp!#

$%&

'()+1

1* f ",#( ) =1

exp *!#

$%&

'()+1

N

V= D !( ) f !,"( )

0

#$ d! = D !( )

0

!F$ d!

D !( ) f !,"( )!F#$ d!

B = "electrons"

1 2444 3444= D !( ) 1% f !,"( )( )0

!F$ d!

A = "holes"

1 24444 34444

�

f ! ,"( )

�

!

u !( ) " u 0( ) = # " #F( )D #( ) f #,!( )#F$% d#

"electrons"

1 24444 34444+ #F " #( )D #( ) 1" f #,!( )( )0

#F% d#

"holes"

1 244444 344444

for ! << µ,!µ !( ) " µ 0( ) = #

F

f !,"( ) = s !( )"


Energy Band structurePeriodicity of the lattice (e.g., spacing a)

Resonant tunelling: free propagation of specific modes

gap (cf. Kittel, introduction to Solid States Physics Chap. 7))≈ 1 eV

Seen in projection on the energy axis: energy bands• Valence band• Conduction band

Metal: Fermi level = chemical potential in conduction band => conductioncan be described by free Fermi gas

Insulator: Fermi level in gap

Electrons in crystals: Quantum States

�

Discrete E k( )

E

k

εc

εv

conduction band

valence band

Gap


Electrons in metalsOrder of magnitude

τ=o : very good approximation at room temperature and below

Heat Capacity

use: • Conservation of number of particles• df /dτ only large close to εF• µ does not vary fast with τ• εF/τ large

!F " 5 eV # TF " 5 104

K >> Tlab

�

Cel = kBdU

d!= kB

d

d!V"D "( ) f ",!( )

0

#

$ d" = kBV "D "( )d

d!f " ,!( )

0

#

$ d"

Cel!"2

3kBVD #

F( )$

Ctot = Cel +C! = "T + AT3

�

!F

�

!�

f ! ,"( )


Insulators: Density of states cf. K&K chap 13

Often called intrinsic semiconductors (no role of impurities)

Statistical distributionStill good approximation to consider free electrons as quantum ideal gas=> occupation number

Density of states

We then get

f !( ) =1

exp ! " µ( ) /#( ) +1

Dh(!)d! =2

4" 22mh

*

h2

#

$ %

&

' (

3

2

(!v )! )d!

De(!)d! =2

4" 22me

*

h2

#

$ %

&

' (

3

2

(! ) !c)d!!c

!v

! !

Gap

D (!)

conduction band

valence band

2 spin states

Parabolicat gap edge

Electron state density below the gap

1

exp µ ! "( ) / #( ) +1Dh"( )d"

0

"v

$holes

1 244444 344444

=1

exp " ! µ( ) / #( ) +1De"( )d"

"c

%

$electrons

1 244444 344444

ne =2

4! 2

2me

*

h2

"#$

%&'

3/2

1

exp ( ') µ ) (c( )( ) / *( ) +1( 'd( '

0

+

, - nQe exp )(c ) µ*

"#$

%&'

with!nQe = 2me

**2!h

2

"#$

%&'

3

2

!!

for ! << "v = nh =2

4# 2

2mh

*

h2

$%&

'()

3/2

1

exp " '* "v * µ( )( ) / !( ) +1" 'd" '

0

+

, - nQh exp *µ * "v!

$%&

'()

with!nQh = 2mh

*!2#h

2

$%&

'()

3

2


Determining the chemical potentialNo impurities: intrinsic semiconductors

!v !

c

!µ

f

!v !

c

µµ

logne(µ)

lognhµ( )


SemiconductorsLarge role of impurities: localized states (Not band !) in gap

If they are shallow (≈ 40meV (Si) ≈10meV (Ge)) can be excited at roomtemperature. This modifies totally the behavior!Donors

Acceptors note: 2 A0 state because a bond is missing and the missing electron

can be spin up or down, A- bond established (pair of electrons of antiparallel spins) : 1state

⇒The number of free electrons(holes) is no moreconstant

Can be increased by donors and decreased by acceptorsBut we need to keep charge neutrality = method to compute the

Fermi level⇒For large enough impurities concentration, the Fermi level can

move close to the edge of the gap⇒(Thermally generated) conductivity either dominated by

• electron like excitation: negative carriers (n type)• hole like excitation: positive carriers (p type)

do! d

++ e

" nd = nd + + ndo

a!" a

o+ e

! na = na!+ n

ao

k

εc

εv

εDεA


Donors

negative carriers (n type)

Acceptors

positive carriers (p type)

Semiconductors cf. K&K fig 13.6

!v !

c

µ

µ

logne(µ)

lognhµ( )

d0 ! d

++ e

"

nd+ = nd

exp "#+

$%&'

()*

exp "#+

$%&'

()*+ 2exp "

#0 " µ

$%&'

()*

= nd

1

1+ 2expµ " #

d

$%&'

()*

with!!!#d+ #0 " #+ , #

c"0.04 !eV !Si

0.01!eV !Ge

lognd+ µ( )

!d

Electric neutrality

ne= n

d+ + nh ! nd+

!v !

c

µ

µ

logne(µ)

lognhµ( )

lognd+ µ( )

!a

Electric neutrality

ne+ n

a!= n

h" n

a!

a0+ e

! " d+

na! = na

exp !#! ! µ

$%&'

()*

2exp !#0$

%&'

()*+ exp !

#! ! µ

$%&'

()*

= nd

1

1+ 2exp#a! µ

$%&'

()*

with!!!#a+ #! ! #0 , #

v+0.04 !eV !Si

0.01!eV !Ge


Other examples of degenerate Fermigas3He

Spin 1/2Very different behavior from 4He: phase separation

Nuclear Matter

⇒Fermi momentumR ! 1.3 10

"13 A

1

3 cm

np ! nn ! 5 1037

cm-3"#F = 4 10

$12 J ! 30 MeV


White Dwarfs and Neutron StarsWhite dwarf stars (and core of massive stars)

=> Degenerate Fermi gasFermi pressure balances gravity => Condition for equilibriumNon relativistic minimum of total energy: stable!

Ultra RelativisticDegeneracy pressure cannot balance gravity if M too big!

Chandrasekhar limitNeutron stars

Same story for neutrons (uncertainty of equation of state)Similar Chandrasekhar limit if larger => black hole

! " 106 g/cm3 n " 1030 cm-3

!F " 0.5 10#13

J " 3 105

eV TF " 3 109

K >> Tstar

1.4 M!

3 M!

R

UUFD Non Relativistic

UT NR

UG

R

UFD Relativistic

UTRel UG


White Dwarf Explosions: SN Ia

time

Lum

inos

ity

Distance

Ωm=1ΩΛ=0

Fain

ter

An acceleratinguniverse?

Supernovae Type Ia at high redshift (2 groups) Ωm-ΩΛDistant supernovae appear dimmer

than expected in a flat universe

Potential problemsAre supernova properties

really constant?Dust?

The Uninvited Guest:Dark Energy

Large negative energy

aacceleration

{ =G

r2

1

c2

uenergy density

{ + 3 p

pressure

{

!

"#

$

%&

GR gravitational mass

1 24444 34444

V

Gravity becomes repulsive!


Bose-Einstein CondensationCalculation of chemical potential

Let us take the origin of energy at the ground state

Separating between the ground state s=1 and the other states

Forwe can solve the equation by having

Bose Einstein condensation. For large number of particles, not a verylow temperature phenomenon

�

1

exp!µ

"#

$ %

&

' ( !1

+1

exp)s! µ

"#

$ %

&

' ( !1s>1

* = N

�

exp!µ

"#

$ %

&

' ( ) 1

exp!s

"#$%

&'(>> 1+

1

N)" << !

sN

f 0,!( ) = s 0( ) =1

exp "µ

!#$%

&'("1

) "!µ) N ** µ ) "

!N

! << "sN

! Nexc = 2.612m"2#h

2

$

% &

'

( )

3 /2

V = 2.612nQV

�

!E

=2"h

2

m

N

2.612 V

#

$ %

&

' (

2/3

) Nexc

= N!!E

#

$ %

&

' (

3/2


Liquid Helium 4Properties of 4He

Loose coupling =>liquid (4.2K at 1 Atm) ≈ ideal gas 4He has spin 0 => boson ≠ 3He spin 1/2

=> expect condensation at ≈3.1KExperimentally “lambda point” 2.17K (Landau temp.)

Phase transition => peculiar propertiesMacroscopic quantum state

=> Quantization phenomenae.g. Vortex Equivalent of Josephson effect

=>Superfluidity

! = n1/2

ei" t,

r x ( )

!" = 2n# where n is integer

C 4He

T2.17K


Much cleaner system: Alcali VaporsBE condensation for atoms demonstrated in 1995=> 2001 Nobel Prize in Physics

awarded jointly to Eric A. Cornell of NIST / JILA; WolfgangKetterle of MIT; and Carl E. Wieman of CU / JILA.

Time sequence of imagesshowing one cycle of the ringingof a Bose-Einsteincondensate (BEC) in the JILATOP (time-averaged orbitingpotential) trap after being drivenby strong oscillationsof trap potential.


Pairing of FermionsSuperconductivity

Pairing of electrons s=0 (Cooper pairs) <= phonon interactionBut condensation theory bad approximation (not free)Similar effects

• Zero resistance• Quantization of flux : Vortices

3HeSpin 1/22 phases of pairing s=1

similar to superconductivity but magnetic properties

τcondensation = 0.95mK and 2.5mK


Energy Density of Ultra Relativistic GasesGeneralizationImportant for behavior of early universe (energy density =>expansion)

Suppose that particles are non degenerate (µ<<τ)

Density of energy

f !( ) "1

exp!#

$

% &

'

( ) ±1

u = gbosons +7

8gfermions

!

" #

$

% & 'aB

2T4

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