8. Ideal Fermi Systems

1. Thermodynamic Behavior of an Ideal Fermi Gas

2. Magnetic Behavior of an Ideal Fermi Gas

3. The Electron Gas in Metals

4. Ultracold Atomic Fermi Gas

5. Statistical Equilibrium of White Dwarf Stars

6. Statistical Model of the Atom

8.1. Thermodynamic Behavior of an Ideal Fermi Gas

ln ln 1F kT kT ze

Z

Ideal Fermi gas results from §6.1-2 : 1

1e

Z

N n

1

1n

e

z e

unrestricted

3

32V d k

k

2 2 2

2 2p km m

3/2

20

22

V mg d

3/2

20

2 ln 12

V mF gkT d ze

3/2 3/2

3/22 0

0

2 2 ln 12 3 1

V m zegkT ze dze

3/2 3/2

2 10

42 13

V mg dz e

( g = spin degeneracy )

Fermi Function

1

10

11x

xf z d xz e

Fermi-Dirac FunctionsLet

1

00

1 nx x

n

d x x ze ze

11 1

0 0

1 n n xn

n

z d x x e

11

0 0

1+ 1

nn y

n

z d y y en

1

0 1

nn

n

zn

1

1

nn

n

zf zn

2 3

2 3z zz

1

0

11

xx

xf z d x zeze

0

!n yd y y e n

1f zf z

z

Series diverges for z > 1, but f is finite.

Mathematica

F, P, N3/2 3/2

2 10

42 13

V mF g dz e

22mkT

3/2

130

413 x

V xF kT g dxz e

5/23

gF kTV f z

35 / 24

5/23

P g f zkT

3/2

2 10

22 1

V mN g dz e

130

21x

V xg dxz e

3/23

VN g f z

13 / 22

F U TS N PV

1

10

11x

xf z d xz e

5/2

3/2

f zPV N kT

f z F

U

,

ln

z V

U

Z,z V

F

3

5/2g V f z

5/2 3

32

kTg V f z

5/2

3/2

32

f zU N kT

f z

5/23

gF kTV f z

3/23

VN g f z

1/23/23

,

302 V N

f zV zg f zT z T

1f zf z

z

3/2

1/2,

32V N

f zz zT T f z

22mkT

,V

V N

UCT

5/2 5/2 1/2 3/22

3/2 1/23/2

3 312 2

f z f z f z f zT zN kf z z T f zf z

5/2

3/2

f zPV N kT

f z

23

PV U

CV , A, S

5/2 5/2 1/2 3/22

3/2 1/23/2

3 312 2V

f z f z f z f zT zC N kf z z T f zf z

5/2 3/2

3/2 1/2

15 94 4

V f z f zCN k f z f z

A F N

U AST

A U ST

5/2

3/2

32

f zU N kT

f z

5/2

3/2

f zPV N kT

f z PV N

5/2

3/2

lnf z

A N kT zf z

/ kTz e

5/2

3/2

5 ln2

f zS N k z

f z

3/23

VN g f z

Using f

0 1

E g d ae

1

10

11x

xf z d xz e

11

0 0 11 x

xd kT d xz ee

1

11kT f z

a c 2 2

2km

x

z e

111s

sg c kT s f z

3/2

23 / 2 2V mc

3/203/23 / 2N E gc kT f z

5/23

32

Vg kT f z

sa c

E.g., free particles :

1 23 3

P n pu EV

5/23

1g kT f z

3/23

13 / 2

VckT

3/23

Vg f z

5/23

5 / 23 / 2

Vg kT f z

5/2

5/25 / 2E g c kT f z

z << 1

5/2

3/2

f zF PV N kT

f z

5/2

3/2

32

f zU N kT

f z

5/2

3/2

lnf z

A N kT zf z

5/2 3/2

3/2 1/2

15 94 4

V f z f zCN k f z f z

5/2

3/2

5 ln2

f zS N k z

f z

F PV N kT

1 lnA N kT z

3 32 2

U N kT PV

5 ln2

S N k z

32

VCN k

1

1

nn

n

zf zn

0f z z

3/23

VN g f z

3

VN g z

3 3N nz

g V g

3

1 ln nN kTg

3

5ln2

gN kn

Virial Expansions ( z < 1, or )

5/2

3/2

f zF PV N kT

f z 1

1

nn

n

zf zn

5/2 3/2

3/2 1/2

15 94 4

V f z f zCN k f z f z

13

15/23

1

ll

ll

PV g nf z aN kT n g

3

1

l

ll

nz cg

al = Virial coefficients

3/23

gn f z

is inverted to give

0 1 2 3 41 1 1 = 0, = 1, = , = 9 4 3 , = 36 45 2 40 6 ,

36 2882 2c c c c c

1 2 3 41 1 2 11, , , 18 15 2 16 6 ,

8 1924 2 9 3a a a a

Mathematica

3

1ng

1nF Bn nc c

F Bn na a

CV

32

V

n

C TN k T

2 33 3 33 1 0.0883884 0.00660012 0.0003895122V

n n nC Nkg g g

13

15/23

1

ll

ll

PV g nf z aN kT n g

32

U PV 1

1

1

3 5 312 2

ll

ll

l nag

§ 7.1 :

Mathematica

13

1

3 5 32 2

lBV

ll

C laN k v

2ddT T

Degenerate Gas ( z >> 1, or )3

1ng

0

0

1101

ne

T 0 3n

g

22

mkT

0 0 FT 0

Fermi energy

0

N d a n

0

F

N d a

xkT

kT

Mathematica

E0

p

p k 2

2 32V d pa g p

d

3

2 36 FVN g p

F Fp

1/326

FNp

g V

Fermi momentum

0

F

N d a

2

2m

p 2/32 2 26

2 2F

Fp nm m g

Ground state / zero point energy :

00

F

E d a

22 3

02

FV d pg d pd

22

2 302 2

FpV pg d p pm

5

2 32 10FV pgm

2

310

FpNm

0 3

5 FEN

P0

0 35 F

EN

23

PV UIdeal gas : 00

23

EPV

5/3n

25 Fn

2/32 262F

nm g

2/32 25/3

06

5P n

m g

Zero point motion is a purely quantum effect ( vanishes for h = 0 )

due to Pauli’s exclusion principle.

n

7/2 15 /8

5/2 3/4

3/2 1/2

1/2 1

z > 1, or3

1ng

T 0 but is low :

1.Virial expansion not valid.

2.Only particles with |F | < O( k T ) active.

response functions ( e.g. CV ) much reduced than their classical counterparts.

Sommerfeld lemma :

1

odd

ln 1 111 2 11 2 ln jj

j

z jf z

j z

1

10

11x

xf z d xz e

25/2

5/2 28 5ln 1

15 8 lnf z z

z

23/2

3/2 24 ln 1

3 8 lnf z z

z

21/2

1/2 22 ln 1

24 lnf z z

z

n

2

26

Asymptotic approx.

3/23

gn f z

3/2 2

22

4 ln 123 8 lnmkTn g z

z

22mkT

2/32

02 3

4n

m g

2/32 26

2n

m g

F

2/32/32 2

22 3ln 1

4 8 lnkT z n

m g z

Lowest order :

2/32

218 lnF z

Next order :

2

20

112 lnF z

22

112F

F

kT

U, P, CV

5/2

3/2

32

f zU N kT

f z

25/2

2

23/2

2

8 5ln 115 8 ln3

2 4 ln 13 8 ln

zz

N kT

zz

23 ln5 2 ln

N kT zz

22

ln 112F

F

kTkT z

2 22 2 23 1 15 12 2 12

F

F F F

kT kT kTN kTkT

223 515 12F

F

U kTN

222 515 12F

F

kTP n

23

P V U

2

2V

F

C kTN k

CV ,

Adapting the Bose gas result ( § 7.1 ) gives

5/2 3/2

3/2 1/2

15 94 4V

f z f zC

f z f z

1

10

11x

xf z d xz e

2

2V

F

C kTN k

A, S22

112F

F

kT

222 515 12F

F

kTP n

A F N PV N 2 22 22 51 1

5 12 12F FF F

kT kTN N

223 515 12F

F

kTA N

U AST

2

2 F

S kTN k

223 515 12F

F

U kTN

Series Expansion of f (z)

1

0

11

x

x

x z ef z d xze

1 1

1 0

1 n n n x

n

z d x x e

1 1

1 0

1 nn x

n

z d x x en

1

1

nn

n

zf zn

0f z z

12

0

ln 1 1d e kT f z

1z f z f

2

2

ln 11 1

1 6 lnz

f zz

odd0 0

12 1 11 2

j

x j jj x

x dd x d x x je d x

Ref: 1. Pathria, App. E.2. A.Haug, “Theoretical Solid State Physics”, Vol.1, App. A.1.

8.2. Magnetic Behavior of an Ideal Fermi Gas

Boltzmannian treatment ( § 3.9 ) : Langevin paramagnetism

Saturation at low T. 1/T for high T

IFG :

1. Pauli paramagnetism ( spin ) :

No saturation at low T ; = (n) is indep of T.

2. Landau diamagnetism ( orbital ) : < 0

= (n) is indep of T at low T. 1/T for high T

8.2.A. Pauli Paramagnetism2

*

2m

p μ B = intrinsic magnetic moment* μ S

= gyromagnetic ratio

12

S 2 groups of particles :

2*

2B

m p

p

Highest filled level at T = 0 is F .*

FK B

3/2

2 2

26

FV mN g

3/2

*2 2

26 FV mN B

Net magnetic moment :

* ˆN N M B

* ˆˆ μ Bfor

Highest K.E.s are :

3/2

3/2 3/2* * *2 2

2 ˆ6 F FV m B B

B

0

3/2

3/2 3/2* * *2 2

2 ˆ6 F FV m B B

M B

00

1

B

MV B

3/2

* 20 2 2

1 22 F

m

3/2

2 2

26

FV mN g

* 2

032 F

n

3/2

2 2 3/2

1 23 F

m n

Langevin paramag for g = 2, J = ½, & low B or high T ( § 3.9 ) :

*2nkT

( T = 0, low B )

2 *2113

ng J JT k

0 3

2 F

kT

32 F

TT

Z ( N, T, B )

E n n n p p p p pp

* ˆˆ μ B2

*

2B

m p

p

2

*

2n n n n B

m

p p p pp

p

2

*

2E n n n N N B

m

p p pp

p

,

,

, E n

n n

Z N T e

p

p p

N E

0,1n p

n N pp

N N N

* 2 22

0

exp exp2 2

NN N B N N N

N n n

e n nm m

p p

p pp p

p p

2N N N N

Let

2

0 , exp2n

Z N T nm

p

pp

pN

*2

0 00

,N

N N B

N

Z N T e Z N Z N N

0 0lnA kT Z

*2

0 00

,N

N N B

N

Z N T e Z N Z N N

0 0lnA kT Z

*

0 02*

0

1 lnN B N A N A N N

N

N kT B eN

lnA kT Z

Method of most probable value : maxln lnf f

0 0*2 0N N N N

A N A N NB

N N

*0 02 0B N N N 0

0T

AN

chemical

potentialfor g = 1

Let maximizes N *0 02 B N A N A N N

e

* *0 0

1 2A N kT B B N A N A N NN

A ( T, N, B )

* *0 0

1 2A N kT B B N A N A N NN

*0 02A N N B A N A N N

,N T

AMB

*2N N *N N *r N

2 1N N NrN N

*0 02 0B N N N

where 1 12

N r N

*0 0

1 12 1 1 02 2

B r N r N

*0 02 0B N N N

( T )

*0 0

1 12 1 1 02 2

B r N r N

B = 0 : 0 01 11 12 2

r N r N r = 0

For r 0 00 0

1/2

1 112 2 2

x

xNrr N Nx

* 0

1/2

2 0x

xNB r

x

* 2

*

0

1/2

2

x

N BM r NxNx

* 2

0

1/2

2

x

nTxNx

valid T & low B

0 0 ,n T

,N T

MB

( T ), Low T * 2

0

1/2

2

x

nTxn

x

For T 0 :(g = 1)

2 2/30 2

0 62Fn n

m

2 2/32/320

1/21/2

62 xx

xN d xnx m d x

2 4/32/32 262 3

nm

2 2/323

2F nm

* 2

032 F

n

( g = 2 )

For low T :(g = 1)

220

0 0112F

F

kT

43 F

same as before

220

1/2

4 13 12F

Fx

xN kTx

22

0 112 F

kT

0 / 2F Fn n

( T ), High T

T :(g = 1)

3/23 3

V VN g f z g z

30 lnn kT n

* 2

0

1/2

2

x

nTxn

x

0

1/21/2

ln

xx

xN d xkTx d x

2kT * 2n

kT same as

before

For high T :(g = 1)

2

3 3/22V zN g z

32 1 1 2z n 3 3/2 31 2n n

3 3/2 30 ln 1 2n kT n n 3 3/2 3ln 2kT n n

0

1/21/2

ln

xx

xN d xkTx d x

3/2 32 2kT n

5/2 31 2 n

3/23 FTn

T

3/2Fn

In Terms of n

* ˆN N M B * * *ˆ n B n B

B

* *

3/2*

20

2 1 1ˆ2 1 1B B

V m de e

B

*3/2 3/23

ˆ V f z f z

B

*Bz e

3/2

*2

0

2 1 1ˆ1 12 1 1x x

V m k T dx xe e

z z

B

*z zB

1/23/2

f zf z

z

*3/2 3/23

1 f z f zB

* 2

1/2 1/23

1 f z f z

* 21/230

2B

f z

0z 31/2

12

f z z n * 2

0B

nk T

(g = 1; n/2)

8.2.B. Landau Diamagnetism

Free electron in B circular / helical motion about B with Be Bm c

Cyclotron frequency

Motion B like SHO See M.Alonso, “Quantum Mechanics”, § 4.12.

2 12 2

zB

p jm

0,1,2,j

Continuum of states with 2 21 1 3

2 2 2B x y Bj p p jm

are coalesced into the lower level

# of states in this level are ( degeneracy / multiplicity )

2

2j

B

pj

m2 2x y

B

L Lm

h

B x yeBg L Lhc

1

2 2

1 2j

j

px y

B x yp

L Lg dx dy dp dp d p p

h h

1 , , 02

B y x A

Z ( z << 1 ) 1 e

Z ln ln 1 e

Z

2 12 2

zB

p jm

B x yeBg L Lhc

0

ln ln 12

zz B

j

L d p g e

Z

For z << 1 :

0

ln2 z

j

eB V d p ehc

Z

z e

( Boltzmannian)

21

/ 2 2

02B

zj

p mz

j

eB V e d p e ehc

12

22 1

B

B

eB V e emkThc e

1

1ln 2sinh2 B

eB V zhc

Z 1z

22mkT

0

ln 12 z

j

eB V d p ehc

N, M ( z << 1 )1

1ln 2sinh2 B

eB V zhc

Z

, ,

ln

T V B

N kT

Z, ,

ln

T V B

zz

ZT T T T

z zz z

1

12sinh2 B

eB Vzhc

lnN Z

12 Bx Let

12

eBmc

eff B 2effemc

2 1

2sinhmkTx VzN

h x

3 sinhVz xN

x

, ,

ln

T z T

FM kTB B

Z

1z for

3 2

1 coshsinh sinheff

Vz x xMx x

effT

xB

1z

Bfor free e

3 sinhVz xN

x

3 2

1 coshsinh sinheff

Vz x xMx x

1 cotheffM N xx

effM N L x

1cothL x xx

Langevin function

Mathematica

L(x) 0 Diamagnetism

eff Bx

kT

2effemc

0 0M if

Landau diamagnetism is a quantum effect.

C.f. Bohr-van Leeuween theorem ( Prob. 3.43 ) :

No diamagnetism in classical physics.

z, x << 1

eff Bx

kT

3 sinhVz xN

x 1cothL x x

x effM N L x

3

3 45x xL x x <

1 :

2

1sinh 6

x xx

3

VzN

13

effeff

BM N

kT

Curie’s law for diamagnetism

213

effnkT

Net susceptibility = paramagnetism - diamagnetism

2 213B eff

nkT

Euler-Maclaurin Formula

( 1) ( 1)1

2 !

ppk kk

k

BB f n f m f n f m O fk

S I I

B1 = −1/2B2 = 1/6

n

j m

f j

S n

m

d x f xI

0kf n k ( 1)1

2 !

ppkk

k

BB f m f m O fk

S I I

Let0

12j

S f j

1/2

1 1 1 12 2 12 2

S dx f x f f O f

0kf k with

1/2

1/2 0 0

dx f x dx f x dx f x

1/2

0 0

1 1 1 12 2 12 2

dx f x dx f x f f O f

Z ( x << 1 )

0

ln ln 12 z

j

eB V d p ehc

Z

Weak B , all T with x << 1 :2B eff B

2 12 2

zB

p jm

Euler-Maclaurin formula :

ln 1f e 2

2z

Bpm

Let

1B ef

e

1 1

B

z e

0 0

1 1 02 24j

f j d f f

0f f

01

0

1ln ln 12 24 1

Bz

eB V d p d zehc z e

Z

01

0

1ln ln 12 24 1

Bz

eB V d p d zehc z e

Z

2

2z

Bpm

2B eff B

2 22 2

2 2

0 0

ln 1 ln 1z z

eff effp pB xm mB d ze d x ze

is indep of B

2

1/2

101 2

1 22 1

1z

z ypm

m yd p d yz e

z e

2

2zpym

2z

z zp yd y d p d pm m

1/22 m f z

1

10

11x

xf z d xz e

0 1/2ln ln 22 12

eff BeB V m f zhc

Z Z

2

0 1/2, , , 22 12

effeB VF T B F T m kT f zhc

2

0 1/2, , , 22 12

effeB VF T B F T m kT f zhc

1/2,

22 6

eff

T z

F eB VM m kT f zB hc

1/20

1 22 6

eff

B

M e V m kT f zV B h c

2effemc

3/2

21/22

1 224 eff

m kT f z

z << 1 : 31/2f z z n

213

eff nkT

3/2 3/22

22

1 2 224 eff

m kT nmkT

z >> 1 : 1/22 lnf z z

3/2

22

1 2 224

Feff

m kTkT

212

eff

F

n

2 2/326

2F nm