Chapter 19: Fermi-Dirac gases

19.1 The Fermi energy

• Fermi-Dirac statistics governs the behavior of indistinguishable particles (fermions).

• Fermions have a half-integer spin.

• Fermions obey the Pauli exclusion principle, which prohibits the occupancy of an available quantum state by more than one particle.

Fermi function• Considering an ideal gas comprising N non inter

acting fermions, each of mass m

• For ε=μ, f(ε) has the value ½ at any temperature.• μ is the chemical potential and its value at T = 0

, μ(0) ,is called the Fermi energy.

1

1)(

kTu

j

jj j

eg

Nf

1

1)( )(

kTu

eg

Nf

• Consider the Fermi function at a temperature of absolute zero;

• This tells us that at T = 0 all states with energy ε<μ(0) are occupied.

• All N particles will be crowded into the N lowest energy levels.

)0()0({/))0((

if

ifkT

)0(1)0(0{)(

ififf

At T = 0• There is only one possible configuration

(microstate), i.e. the thermodynamic probability w is 1.

• According to the Boltzmann relationship, S = klnW = 0.

how does the Fermi energy μ(0) depend on m, N, and V?

• Obviously it doesn’t depend on the temperature since T = 0. We need g(ε) , the density of quantum states. For particles of spin 1/2 , the spin factor is 2,

18.38

19.2 The calculation of μ(T)

• To obtain these curves, we must determine μ(T) . The calculation is considerably more complicated than it was for T = 0. We have

• the number of particles at T = 0 is the same as at T ≠ 0,

• At high temperatures the fermion gas approximates the classical ideal gas. In the classical limit,

• The spin degeneracy factor is 2 for fermions.

• For T >> TF ,μ/kT takes on a large negative value and exp(-μ/kT) >>1.

• As an example, consider a kilomole of 3He gas atoms (which are fermions) at STP. The Fermi temperature is 0.069 K, so that T /TF = 3900. Using eqns (9.16) and (9.17), we find thatμ/kT = -12.7 and exp(-μ/kT) = 3.3x105. The average occupancy of single particle states is indeed very small, as in the case of an ideal dilute gas obeying the M-B distribution.

19.3 Free electrons in a metal

• Electrons are spin 1/2 fermions.

• Statistical thermodynamics provides profound insights into the behavior of conduction electrons in metals at moderate temperatures.

• Each atom in the crystal lattice of the metal is assumed to part with some number of its outer valence electrons, which can then move freely about in the metal.

• There is an electric field due to the positive ions that varies widely from point to point. However, the effect of the field is canceled out except at the surface of the metal where there is a strong potential barrier, called the work function that draws back into the metal any electron that happens to make a small excursion outside.

• The free electrons are therefore confined to the interior of the metal as gas molecules are confined to the interior of a container.

• In this model, the free electrons move in a potential box or well whose walls coincide with the boundaries of the specimen. They occupy energy states up to the so-called Fermi level, which is the chemical potential μ(T) . The work function φ is the energy required to remove an electron at the Fermi level from the metal surface. The depth of the potential well is equal toμ(T) + φ

A more realistic picture of the potential well is given in the following Figure, which shows how the potential varies in the vicinity of the positive ions in the crystal lattice. The periodicity leads to a band structure in the density of quantum states, which is the foundation of semiconductor physics.

The Fermi level of the free electrons in most metals at room temperature is only fractionally less than the Fermi energy εF. It is often assumed that the two are equal, and this leads to confusion. The Fermi level, strictly speaking, is μ(T), which is an approximation to the Fermi energy valid forT << TF.

19.2) consider a kilomole of 3He gas atoms under STP conditions

(a) What is the Fermi temperature of the

gas?

KTeTemperaturFermi

KK

v

N

mk

hT

F

F

07.0

07.0069.0

4.22504.1

10023.6

1038.11098.42

10626.6

504.12

32

26

2327

234

322

(b) Calculate

kTandkT exp

5

26

23

234

2327

23

2

1028.3

7.12

?exp

7.12

10023.6

4.22

10626.6

2731038.11098.422ln

22ln

?

kTe

kT

kT

kT

N

V

h

mkT

kT

thatknowwekT

(c) Find the average occupancy of a single particle state that has energy of

f kT23

7

7

6

7.1223

108.6

108.6

10469.1

11

11

1

?:

7.12:23:

fOccupancyAverage

f

e

ef

thatknowWe

fOccupancyAverage

kTPotentialChemical

kTEnergy

kTkT

KT

19.3) For a system of noninteracting electrons, show that the probability of finding an electron in a state with energy ∆ above the chemical potential μ is the same as the probability of finding an electron absent from the with energy ∆ below μ at any given temperature T