Introduction to Thermodynamics and Statistical Physics phần 8 ppt

Chapter 3. Bosonic and Fermionic Systems

√ε

2

µ β

α2

¶3/2

dε = n2dn .

Thus, by introducing the density of states

D (ε) = ( V

2π2

¡ 2m

~2

¢3/2

ε1/2 ε ≥ 0

0 ε < 0 , (3.97)

one has

log Zgc = 1

2

X

l

Z∞

−∞

dε D (ε) log (1 + λ exp (−β (ε + El))) . (3.98)

3.3.3 Energy and Number of Particles

Using Eqs. (1.80) and (1.94) for the energy U and the number of particles

N, namely using

U = −

µ∂ log Zgc

∂β ¶

η

, (3.99)

N = λ∂ log Zgc

∂λ , (3.100)

one finds that

U = 1

2

X

l

Z∞

−∞

dε D (ε) (ε + El) fFD (ε + El) , (3.101)

N = 1

2

X

l

Z∞

−∞

dε D (ε) fFD (ε + El) , (3.102)

where fFD is the Fermi-Dirac distribution function [see Eq. (2.35)]

fFD () = 1

exp [β ( − µ)] + 1 . (3.103)

3.3.4 Example: Electrons in Metal

Electrons are Fermions having spin 1/2. The spin degree of freedom gives rise

to two orthogonal eigenstates having energies E+ and E− respectively. In the

absent of any external magnetic field these states are degenerate, namely

E+ = E−. For simplicity we take E+ = E− = 0. Thus, Eqs. (3.101) and

(3.102) become

Eyal Buks Thermodynamics and Statistical Physics 112