8 Ideal Fermi Systems 1 Thermodynamic Behavior of

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 Ideal Fermi gas results from § 6. 1 -2 : unrestricted ( g = spin degeneracy )

Fermi Function Let Fermi-Dirac Functions

F, P, N

U

CV , A, S

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

CV see § 7. 1 Mathematica

Degenerate Gas ( z >> 1, or ) T 0 Fermi energy Mathematica

E 0 Fermi momentum Ground state / zero point energy :

P 0 Ideal gas : Zero point motion is a purely quantum effect due to Pauli’s exclusion principle.

z > 1, or 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 : 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

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 Paramagnetism = intrinsic magnetic moment = gyromagnetic ratio 2 groups of particles : Highest filled level at T = 0 is F. Net magnetic moment : for Highest K. E. s are :

0 ( T = 0, low B ) Langevin paramag for g = 2, J = ½, & high T ( § 3. 9 ) :

Z ( N, T, B ) Let

Method of most probable value : Let maximizes chemical potential

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 ), T : (g = 1) High T same as before For high T : (g = 1)

8. 2. B. Landau Diamagnetism Free electron in B circular / helical motion about B with Larmor frequency Quantization of L Continuum of states with # of states in this level are ( degeneracy / multiplicity ) are coalesced into the lower level

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

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

L(x) 0 Diamagnetism Langevin function Landau diamagnetism is a quantum effect. C. f. Bohr-van Leeuween theorem ( Prob. 3. 43 ) : No diamagnetism in classical physics. Mathematica

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

Euler-Maclaurin Formula B 1 = − 1/2 B 2 = 1/6 Let with

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

is indep of B

8. 3. The Electron Gas in Metals

8. 4. Ultracold Atomic Fermi Gas

8. 5. Statistical Equilibrium of White Dwarf Stars

8. 6. Statistical Model of the Atom