8.3.
The Electron Gas in Metals
History :
1. Drude (1900) - Lorentz (1904-5) : Free e, MB statistics.
a) Successes :
Qualitative features of transport.
Wiedemann-Franz law explained : ( K /  = const )
b) Failures :
Equipartition  C = Cvib + Cel .
Exp.: C  Cvib.
Paramagntism (  1/T ).
Exp.:  indep of T & too small.
2. Sommerfeld : Free e, FD statistics.
Sommerfeld Theory
g=2:
F 
2
2 m*
Cubic lattice :
Sodium (Na) :

 3 n 
2
n
2/3
m* = effective mass of e.
nA = # of atoms per unit cell
ne = # of electrons per atom
a = lattice constant
( length of unit cell edge )
ne n A
a3
ne  1, n A  2, a  4.29 A, m*  0.98me
n  2.53  1022 cm3
TF 
F
k
Mathematica
 3.72  104 K
Troom
 0.008
TF

 F  3.21 eV
Ce 
2
2
Highly degenerate Fermi system
Nk
kT
F
3

 102  N k 
2

~ few percents of Cclassical
Low T :
For Cu, 
CV  Ce  Cvib   T   T 3

D   343.8  0.5 K
c.f. 345K from elastic constants.
Intercept of CV / T vs T 2 with the vertical axis gives
   0.688  0.002 mJ mole1 K 2
c.f. 0.69 from DOS calculation.
Paramagnetism, Lorenz number are likewise improved .
Thermionic & Photoelectric Electron Emission
No spontaneous emission
 e in metal in potential well.
Simplest model : square well
Classically, only e’s with
 W
U 
 0
p2
 W such that u 
2m
inside metal
outside
2m W
can escape.
m
Low emission current  Remaining e’s in quasistatic equilibrium.
 Treatment analogous to gas effusion .
8.3.A. Thermionic Emission ( Richardson Effect )
From § 6.4 : # of e emitted ( in the z > 0 direction ) per unit area per unit time is
R


2
 2

 
d px
3



d py


2
 2 
3

3

d pz
2 mW

pz 2  m
d
p
 zm 
2 mW
e
 pz2

2
    p
 2m 
2m
 
e
e
 pz2

  
 2m  x
 
e 1
a 1e  x
1
d ln 1  a e   
d
x


dx
1  x
x
1 a e
a e 1
1  x

R
kT
2 2
 p2


   z    
 2m  

d
p
p
ln
1

e
 z z 

2 mW



3
e
   p    
1
1
1
dx
1
p2
 p 
2m
1

0
np 
pz
np
m
pz
d
p
 z m 2 0 d p p
2 mW

2
 2

p2
x
2m
R

kT
2 2
 p2


   z    
 2m  

d
p
p
ln
1

e
 z z 

2 mW



3
m kT
R 2 3
2
pz2
z 
2m

   z   


d

ln
1

e
 z 

W
 z     z   F  W   F   = Work function ~ eV ~ 104 K

mkT
R 2 3
2

 d
z
W
e
   z   
m  kT     W   

e
2 3
2
2
e m  kT     W  
J  eR 
e
2 2 3
2
Thermionic current density :
Boltzmannian
e m  kT     W  
J
e
2 2 3
2
J
Classical statistics ( z << 1 ) :
 mkT 

2
n 3z

2 

 2 
g

J classical  n e
n
3/2
g
3
f 3/2  z 
z
k T  W
e
2 m
 work function   W
FD statistics ( z >> 1 ) :
  F
class  n e
e m  kT     W  F 
e
2 3
2
2

J FD 
  W  F
 FD
k
2 m
e m k2
2
2


120.4
A
cm
K
2 2 3
J classical  class
T e  W 
J FD  FD T e
2
   W  F


ln
J classical
W
  class 
kT
T
Plot is straight line
with slope W / k
ln
J FD
W  F



FD
T2
kT
Plot is straight line with
slope (W  F ) / k
e beam impinging on metal :
outside 
h
poutside

h
2mE
inside 
 refractive index of metal for e’s is
h
pinside
n

h
2m  E  W 
outside

inside
 W can be estimated from e diffraction experiments.

W  13.5 eV for tungsten (W )
E W
E
ln
ln
J classical
W
  class 
kT
T
Tungsten :
 = 4.5 eV
J FD
W  F



FD
T2
kT
Dotted Line : r = 0
Solid line : r = ½
For tungsten :
W  13.5 eV,   4.5 eV, F  9 eV

FD statistics
For nickel :
W  17 eV,   5.0 eV, F  11.8 eV

Effects of reflection at surface :
J  (1r)J
FD statistics
Intercepts for most metal with clean surfaces range from 60 to 120 A cm2 K2 .
c.f.
 FD  120.4 A cm 2 K 2
Schottky Effect
Electric field  surface : E   F xˆ
( x = 0 at surface, x < 0 inside metal )
For an outside electron :
Force from its image is
f  x 
 e  e  xˆ
2
2x
Setting U(0) =0 , the corresponding potential energy is
x
2
e
U Image  x     f  x  d x  
4x
x
Potential energy due to E :
U E  x      e   F  d x   eFx
By definition, potential energy of an inside e is .
 Potential energy of an outside e w.r.t. an inside one is
e2
  x   eFx 

4x
e2
  x   eFx 

4x
e2
  x   eF  2  0
4x
e2
  x    3  0
2x
x  0
  xm   e eF  
 E lowers the barrier by e3/2 F1/2 .

J F  J0 e  e
eF
Cold emission :   xm   0

xm 
e
4F
 (xm) is a maximum
8.3.B. Photoelectric Emission ( Hallwachs Effect )
Condition for emission :
m kT
R 2 3
2

pz2
z 
2m
 z  h  W


Wh
d  z ln 1  e    z    
 z     z   F  W  h   F can be ~ k T for
x
where
 z  h  W
kT


m  kT 
R
2 2 3
h  W  
h    0 

kT
kT
2 
c.f. § 8.3A
h  W   F
 d x ln 1  e
x 
( Non-Boltzmannian )

0
0 
W 
W  F 


h
h
h
m  kT 
R
2 2 3
J  eR
J

e m  kT 
2 2 3
2 
x 
d
x
ln
1

e



0
2 
x  
d
x
ln
1

e



0

x  
x  
d
x
ln
1

e

x
ln
1

e





0

0

h    0 

kT

e  x  
x


f
e
 dx x

d
x


2
x 
x 

1 e
e
1
0
0
e m  kT 

J
f
e


2
2 2 3
2

f  e
2
2

e
m
kT



1 
 J
   1
2 2 3 2

f  e

1  e

2
e m  kT   1 
e m  kT 
J0 
1


2



J



0
2 2 3  2 
24 3
2
Threshold  = 0 :
2
e m  kT   e m  kT  h  0  / k T
J
e 
e
2 3
2 3
2
2
2


e m    0 
2
h    0 
kT
1
h    0 
kT
1
  2 
2
6
Thermionic
Pd (  = 4.97 eV )
8.4.
MOT :
Ultracold Atomic Fermi Gas
2
a   
3
2  0 
0  1 2 3 
1/3

N   d  a   n
FD

0
N

1
d   

e
2  
3
F
2  0 
3
 d 
2

0
 
3
1
 F  0  6 N 
1/3

3
 F  7.5 1011 eV
For 106 atoms in 100Hz trap :
Ground state energy : U 0 
 F3
1
2  0 
2
0
0
1
( same as Bose gas )
F
1
2  0 
3
 d 
0
3

TF  870 nK
 F4
1
2  0 
3
4

U0 
3
NF
4
U

1
d   

e
2  
3
0

4U 0

 kT 
4
F
3
 
0
1

4
x3
0 d x z 1e x  1
U0 
x  
T 
U
 24   f 4  z 
U0
 TF 
N

1
d   

e
2  
3
0

3N
 F3
2
 
0
 kT 
1

3
x2
0 d x z 1e x  1
40K
3

T 
1  6   f3  z 
 TF 
3
NF
4
 F  0  6 N 
1/3
 0 
4


x 1
f  z  
d x 1 x
   0
z e 1
1
See §11.9 for
BCS condensation
3

 F3
6N

 F4
8 U0
8.5.
Statistical Equilibrium of White Dwarf Stars
White dwarf stars ~ abnormally faint white stars.
Reason: emitted light due NOT to fusion, but to gravitational contraction.
Model : Star = Ball of He of
M  1033 g ,   107 gcm 3 , Tcenter  107 K
  kB Tcenter  900 eV
k B  8.617  105 eV K 1
Mathematica
Ionization energy of He :
24.6 eV, 54.4 eV
me  9.109 1028 g,
 Ball = N e’s + ½ N (He nuclei) :
mp  1.673 1024 g
1


M  N  me    4m p    2 N m p
2


n

N M / 2m p

 3  1030 cm3

2m p
V
M /
 dynamics of He ions negligible
Model
n  3  1030 cm 3
 6 n 
pF  

g



2/3
 6 2 n 
5

8

10
eV
F 


2me  g 
2
me c 2  0.511  106 eV
Tcenter  107 K
1/3
2

TF 
 5  1017 g cm s 1
F
k
 9  109 K
 e dynamics relativistic
Tcenter
 103
TF
 e gas completely degenerate
Simplistic Model : Star = uniform, relativistic e gas
Caution : n must varies
for structural stability
Ground State Properties
N
2V
 2 
4
3
pF
dp p
  mc 2

0
Relativistic particle :
with rest mass m

2

2 3
pF3
3


p   , p
c 
2
u



p
m
pF 


 p 
1 

 mc 
 = hamiltonian 
V
 3 2n 
1/3
p  p   mc 
2

K    mc  mc 

2
2
 
(g=2)
2
    p2
c
2

 p 
1 
 1 
 mc 

p
1   p / mc 
2
m  me
From § 6.4 :
( gas inside V )
1
P0  n p u
3
0
1
V
 n
3 N 2
pF
3
dp p
0
p2
2
m
1   p / mc 
2
N
V

2 3
3
F
p
3
p
sinh  
mc
p
u
1   p / mc 
m

uc
2
P0 
sinh 
1  sinh 2 
3
pF
1
3

2 3
dp p
0
where
3
1/3
pF
2

x  sinh  F 
 3 n 
mc
mc
dp
cosh  d 
mc

m 4c5
P0 
3 2 3
F
4
d

sinh


0
m
u  c tanh 
V  mc 
V  mc  3
3
N
sinh


F

 x
2 
2 
3 
3 



p2
2
1   p / mc 
2
P0
m 4c5
P0 
3 2 3
F
 d  sinh
0
4

m 4c5

24  2
3
A x
x  sinh  F 
pF
mc
F
where
A  x   8  d  sinh 4   x  2 x 2  3 1  x 2  3sinh 1 x
0

A x  0  
8 5 4 7 1 9 5 11 35 13
x  x  x 
x 
x 
5
7
3
22
208
5
35
 7

A  x     2 x 4  2 x 2     3ln 2 x  


2
4
4
4
x
64
x


Mathematica
Equilibrium Configuration
Adiabatic change of spherical volume :
dE0   P0  n  dV   P0  R  4 R2 d R
Accompanying gravitational energy change :
d Eg 
 Eg
R
dR
GM 2

dR
2
R
 ~ 100 depends on the spatial variation of n.
Equilibrium :
d E0  d E g  0


GM 2
 P0  R  4 R  
0
2
R
GM 2
P0  R   
4 R 4
2
m 4c5
P0 
24  2
3
A x
x
 3 n 
mc
2
1/3
A  x   x  2 x 2  3 1  x 2  3sinh 1 x
m  me
GM 2
P0  R   
4 R 4
Equilibrium condition :
M  2 Nm p
1/3


 2 N 
x
 3 4

mc 
3
R 
3


4 5
mc
24  2

3

A
m

 Equilibrium condition :
( Mass-radius relationship )
mc
 1011 cm
1/3

9

1

N
R


mc  4

9M

c  8 m p

A
m

9M


m c  8 m p
1/3

1
 R

1/3


GM 2
1
 R   
4
4

R



9M

c  8 m p


1
 R   6



1/3
= Compton wavelength of electron
3

  GM 2 
 mc R   mc 2 R 

 

QM+SR+Gr

A
m

9M

c  8 m p
1/3


1
 R   6



9M
x

m c  8 m p
3

  GM 2 
 mc R   mc 2 R 

 

1/3

1
  5  108 cm  R 1
 R

A x 
1. R >> 108 cm ( x << 1 ) :
1/3

9  3 2 
R


40  M 
9M
R

m c  8 m p

M  M0
Mathematica
for M = 1033 g
8 5
x
5
2
m m 5/3
p
 M 1/3
A x   2 x4  2 x2
2. R << 108 cm ( x >>1 ) :

A  x   x  2 x 2  3 1  x 2  3sinh 1 x
1/3



M 
1 

M
 0
as observed
2/3
M0 
9 3
64
 c 



G


Chandrasekhar limit :
e 
M
2
N mp
3/2
m p2  1033 g
M0 
5.75
e2
M  M0
 Star collapse into neutron star or black hole
Influx from companion binary star  Type Ia supernovae (see Chap.9)
8.6.
Statistical Model of the Atom
Thomas-Fermi model :
1/2
For a completely degenerate e gas
 6 2n 
pF  

 2 
Let the gas be under a Coulomb potential (r).
For slowly varying n :
pF  r   3 2n  r  
Energy of e at top of Fermi sea :
1/3
pF2  r 
 r  
 e  r 
2m
At boundary of system, pF = 0 & we can set  = 0 so that  = 0.

pF2  r 
 r  
 e  r   0
2m
r
pF  r   3 n  r  
2

1  p r  
n r   2  F

3 

Poisson eq. :

3
1  2m

 2  2 e  r 
3 

pF2  r 
 e  r 
2m
1/3
3/2
2 r   4  r   4 e n r 
4 e  2m

  r  
e

r



3  2
3/2
2
 4 e  2m
1 d  2 d

r

r

e

r




Spherical symmetry : 2
 3  2

r d r  d r


3/2
ThomasFermi eq.
 4 e  2m
1 d  2 d

r

r

e

r




 3  2

r 2 d r  d r


  r  
Let


Set
3/2
 r
Ze / r
 d d
d  2 d 
d 2 

r
r
  Ze  

dr  dr 
d r2 
 dr dr
  1 d 
d
 Ze  2 

dr
r dr 
 r
1 d  2 d  Ze d 2 
r

2
r d r  d r  r d r2
d 2   3/2
 1/2
2
dx
x

2
3 Ze d 

x d x2

4 e  2m Ze 2





3  2 x

4 e  2m 2 
 Ze 
 2 Ze  
3 

3/2
x  r
3/2
3
Dimensionless T-F eq.

 4 
  
 3 
2/3
2m e 2
2
1/3
Z
Z 1/3

0.88534 aB
aB 
2
me2
Bohr radius
  r  
 r
d 2   3/2
 1/2
d x2
x
Ze / r
Z 1/3
x
r
0.88534 aB
B.C. :
Ze
r
 r 
For x , r  0 :
For x , r  r0 (boundary) :

 r  0
  0  1
( neutral atom )
   x0   0
For r0   :
  x  0   1  1.5886 x 
  x 3  
  x  10   1  


  144  
 x    
144
x3
4 3/2
x 
3
1 / 

73  7
 0.257
6
Sommerfeld,
Z.Physik 78, 283 (32)
Complete solution tabulated by
Bush & Caldwell, PR 38,1898 (31)
1  2m

n r   2  2 e  r 
3 


n r  
3/2
  r  



r



4 Z  e
1
3/2
Z  4  dr r n  r   Z 
3/2
 dr r
x  r




r


 r

1/2

3/2
 r 
0
0
 4 
  
 3 
Ze / r


2
Z

4
 r
3/2

Z
1/2
3/2
d
x
x

 x

0


1/2
3/2
d
x
x

 x  1

0
Z 1/3

0.88534 aB

1
1
Ze
 r 
  r   Z 3  Z 4/3
r
Z
n r 
4




r


 r

2/3
3/2
Z
1 13
1   
 3 32
 Z2
2m e 2
2
Z1/3
Radial Distribution Function
D  r   4 r 2 n  r 

Hg atom
Z   dr D  r 
0
Actual
T-F
a
2
me 2
pF  r   3 2n  r  
Binding Energy
Mean ground state K.E.:
1 2V
 
4
N  2 3
pF2  r 
 e  r 
2m
pF
2
2 p
0 dp p 2m

pF5
10m
1
n2
3
Total ground
state P.E.:
3
3
5
 r   e  r 


Total ground
state K.E.:
1  pF   F
3


 F r 


n2 
5
5

K0  4  dr r 2 n  r    r 
0
3
 e  4
5

2
dr
r
n r  r 

0

Ze  
 Ze 1 
U 0  4  dr r 2 n  r  e  
   r    
r 
 r 2
0
nuclei
other e’s
cancels double counting
Total ground
state E:

E0  K0  U 0
1
1 Z e2 
 4  dr r n  r   e   r  

10
2
r


0
2
1/3

1
1 Z e2 
2
E0  4  dr r n  r   e   r  

10
2
r


0
1  2m

n  r   2  2 e   r 
3 

3/2
E.Milne, Proc.Camb.Phil.Soc. 23,794 (1927) :
1.5886  e2  7/3  1 
E0 

 Z   1
0.88534  2aB 
7 
E B   E0  1.538 Z
 Binding energy :
7/3
e2

 13.6 eV
2a B

EB of H atom
E0 Z 7/3

 Z 4/3
N
Z
E0  Z 7/3
 4 
  
 3 
2/3
2m e 2
2
Classical regime :
1/2
 Z 2/3
 Linear size of e cloud l   1  Z1/3
Z1/3

h
E
    0 
p N
l

Z 2/3
Z 1/3

Z
1