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

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

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

F, P, N    

U    

CV , A, S    

Using f  E.g., free particles :  

z << 1  

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

CV  § 7.1 : Mathematica

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

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

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

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 

   Lowest order :  Next order :

U, P, CV   

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

A, S  

Series Expansion of f (z)  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 : 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

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 :

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

Z ( N, T, B )  Let 

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

A ( T, N, B )  where  

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

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

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

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

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 ) 

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

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

 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. 3.43 ) : No diamagnetism in classical physics.

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

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

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

is indep of B   

  z << 1 :  z >> 1 : 