1. 8. Ideal Fermi Systems Thermodynamic Behavior of an Ideal Fermi Gas
Magnetic Behavior of an Ideal Fermi Gas
The Electron Gas in Metals
Ultracold Atomic Fermi Gas
Statistical Equilibrium of White Dwarf Stars
Statistical Model of the Atom

2. 8.1. Thermodynamic Behavior of an Ideal Fermi Gas
Ideal Fermi gas results from §6.1-2 :
 unrestricted 
( g = spin degeneracy )

3. Fermi Function Let Fermi-Dirac Functions   
Series diverges for z > 1, but f is finite.
Mathematica  

4. F, P, N    

5. U    

6. CV , A, S    

7. Using f 
E.g., free particles :  

8. z << 1  

9. Virial Expansions ( z < 1, or )
is inverted to give
Mathematica 
al = Virial coefficients 

10. CV 
§ 7.1 :
Mathematica

11. Degenerate Gas ( z >> 1, or )
Fermi energy 
Mathematica

12. E0   
Fermi momentum 
Ground state / zero point energy : 

13. P0
Ideal gas :  
Zero point motion is a purely quantum effect ( vanishes for h = 0 )
due to Pauli’s exclusion principle.

14. z > 1, or T  0 but is low : Virial expansion not valid.
Only particles with |F   | < O( k T ) active.
 response functions ( e.g. CV ) much reduced than their classical counterparts.
Sommerfeld lemma :
Asymptotic approx. n
7/2
15 /8
5/2
3/4
3/2
1/2 1 

15.   
Lowest order : 
Next order :

16. U, P, CV   

17. CV ,
Adapting the Bose gas result ( § 7.1 ) gives

18. A, S  

19. Series Expansion of f (z)
Ref:
1. Pathria, App. E.
2. A.Haug, “Theoretical Solid State Physics”, Vol.1, App. A.1.

20. 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 :
Pauli paramagnetism ( spin ) :
No saturation at low T ;  = (n) is indep of T.
Landau diamagnetism ( orbital ) :  < 0
 = (n) is indep of T at low T.   1/T for high T

21. 8.2.A. Pauli Paramagnetism = intrinsic magnetic moment
 = gyromagnetic ratio
 2 groups of particles :
for
Highest filled level at T = 0 is F .
 Highest K.E.s are : 
Net magnetic moment :

22. 0 
( T = 0, low B )  
Langevin paramag for g = 2, J = ½, & low B or high T ( § 3.9 ) : 

23. Z ( N, T, B ) 
Let 

24. Method of most probable value :
Method of most probable value :
Let maximizes 
chemical potential
for g = 1

25. A ( T, N, B ) 
where  

26.  ( T )
B = 0 :
 r = 0
For r  0   
valid  T & low B

27.  ( T ), Low T For T  0 : (g = 1)   same as before ( g = 2 )
For low T :
(g = 1)  

28.  ( T ), High T T   : (g = 1)  same as before   For high T :

29. In Terms of  n 
(g = 1; n/2)  

30. 8.2.B. Landau Diamagnetism Cyclotron frequency
Free electron in B  circular / helical motion about B with
Motion  B like SHO 
See M.Alonso, “Quantum Mechanics”, § 4.12.
Continuum of states with
are coalesced into the lower level
# of states in this level are ( degeneracy / multiplicity ) 

31. Z ( z << 1 ) 
For z << 1 :
( Boltzmannian) 

32. N, M ( z << 1 ) 
for
Let
for free e   

33.  Landau diamagnetism is a quantum effect.
L(x)  0
 Diamagnetism 
Langevin function 
Mathematica
 Landau diamagnetism is a quantum effect.
C.f. Bohr-van Leeuween theorem ( Prob ) :
No diamagnetism in classical physics.

34. z, x << 1 x < 1 :  Curie’s law for diamagnetism 
Net susceptibility = paramagnetism - diamagnetism 

35. Euler-Maclaurin Formula
B1 = −1/2
B2 = 1/6  
Let
with 

36. Z ( x << 1 ) Weak B , all T with x << 1 : Let  
Euler-Maclaurin formula : 

37. is indep of B   

38.  
z << 1 : 
z >> 1 : 