1. 8.3. The Electron Gas in Metals
History :
Drude (1900) - Lorentz (1904-5) : Free e, MB statistics.
Successes :
Qualitative features of transport.
Wiedemann-Franz law explained : ( K /  = const )
Failures :
Equipartition  C = Cvib + Cel Exp.: C  Cvib.
Paramagntism (  1/T ). Exp.:  indep of T & too small.
Sommerfeld : Free e, FD statistics.

2. Sommerfeld Theory g = 2 : m* = effective mass of e.
nA = # of atoms per unit cell
ne = # of electrons per atom
a = lattice constant
( length of unit cell edge )
Cubic lattice :
Sodium (Na) : 
Mathematica
Highly degenerate Fermi system 
~ few percents of Cclassical

3. Low T :
For Cu,  
c.f. 345K from elastic constants.
Intercept of CV / T vs T 2 with the vertical axis gives
c.f from DOS calculation.
Paramagnetism, Lorenz number are likewise improved .

4. Thermionic & Photoelectric Electron Emission
No spontaneous emission  e in metal in potential well.
Simplest model : square well
Classically, only e’s with such that can escape.
Low emission current  Remaining e’s in quasistatic equilibrium.
 Treatment analogous to gas effusion .

5. 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 

6. 
= Work function ~ eV ~ 104 K
Boltzmannian 
Thermionic current density :

7. J Classical statistics ( z << 1 ) :   work function
FD statistics ( z >> 1 ) : 

8. Plot is straight line with slope W / k
Plot is straight line with slope (W  F ) / k 
e beam impinging on metal :
 refractive index of metal for e’s is
 W can be estimated from e diffraction experiments.
 W  13.5 eV for tungsten (W )

9. Effects of reflection at surface : J  ( 1  r ) J
Tungsten :
 = 4.5 eV
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  eV
 FD statistics
Effects of reflection at surface :
J  ( 1  r ) J
Intercepts for most metal with clean surfaces range from 60 to 120 A cm2 K2 .
c.f.

10. Schottky Effect Electric field  surface :
( x = 0 at surface, x < 0 inside metal )
For an outside electron :
Force from its image is
Setting U(0) =0 , the corresponding potential energy is
Potential energy due to E :
By definition, potential energy of an inside e is .
 Potential energy of an outside e w.r.t. an inside one is

11. 
 (xm) is a maximum
 E lowers the barrier by e3/2 F1/2 . 
Cold emission :

12. 8.3.B. Photoelectric Emission ( Hallwachs Effect )
Condition for emission : 
c.f. § 8.3A
can be ~ k T for
( Non-Boltzmannian ) 
where

13.    
Threshold  = 0 : 

14. Thermionic
Pd (  = 4.97 eV )

15. 8.4. Ultracold Atomic Fermi Gas
MOT :
( same as Bose gas ) 
For 106 atoms in 100Hz trap :
Ground state energy : 

16. 
40K 
See §11.9 for
BCS condensation

17. 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
Mathematica
Ionization energy of He : eV, eV
 Ball = N e’s + ½ N (He nuclei) :
 dynamics of He ions negligible

18. Model   e dynamics relativistic   e gas completely degenerate
Simplistic Model : Star = uniform, relativistic e gas
Caution : n must varies for structural stability

19. Ground State Properties
( g = 2 )
Relativistic particle :
with rest mass m  
 = hamiltonian 
From § 6.4 :
( gas inside V )

20.   
where 

21. P0
where
Mathematica 

22. Equilibrium Configuration
Adiabatic change of spherical volume :
Accompanying gravitational energy change :
 ~ 100 depends on the spatial variation of n.
Equilibrium :  

23. Equilibrium condition :
Equilibrium condition :
( Mass-radius relationship )
QM+SR+Gr
= Compton wavelength of electron

24. 1. R >> 108 cm ( x << 1 ) :
for M = 1033 g
Mathematica
1. R >> 108 cm ( x << 1 ) : 
2. R << 108 cm ( x >>1 ) : 
Chandrasekhar limit : 
as observed

25.  Star collapse into neutron star or black hole
Influx from companion binary star  Type Ia supernovae (see Chap.9)

26. 8.6. Statistical Model of the Atom
Thomas-Fermi model :
For a completely degenerate e gas
Let the gas be under a Coulomb potential (r).
For slowly varying n :
Energy of e at top of Fermi sea :
At boundary of system, pF = 0 & we can set  = 0 so that  = 0. 

27. 
Poisson eq. : 
Thomas-Fermi eq.
Spherical symmetry :

28. Let  
Set 
Dimensionless T-F eq. 
Bohr radius

29. Complete solution tabulated by Bush & Caldwell, PR 38,1898 (31)
B.C. :
For x , r  0 : 
For x , r  r0 (boundary) :
( neutral atom ) 
For r0   :
Sommerfeld, Z.Physik 78, 283 (32)
Complete solution tabulated by Bush & Caldwell, PR 38,1898 (31)

30.   

31. Radial Distribution Function
Hg atom
Actual
T-F

32. Binding Energy Mean ground state K.E.:  Total ground state K.E.:
Total ground state P.E.:
nuclei
other e’s
cancels double counting
Total ground state E:

33. E.Milne, Proc.Camb.Phil.Soc. 23,794 (1927) :
 Binding energy :
EB of H atom
 Linear size of e cloud l   1  Z1/3
Classical regime :  