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How does the ideal Fermi gas behave when the occupancy N n is relatively high, especially when the occupancy N n approaches its maximum value of 1?

We are particularly concerned to know if the relatively high occupancy regime is equivalent to the low temperature regime and, if so, whether the ideal Fermi gas observes the third law.

While we cannot solve ( 6.15 ), ( 6.17 ), and ( 6.20 ) analytically for the func-tion S T

( )

, we can use these equations to parameterize S and T in terms of the occupancy N n for a given particle number N and volume V . Then we can use these parametric equations to plot S T

( )

^.

In order to make our plot as meaningful as possible, we search for a nor-malization of T that casts Eqs. ( 6.17 ) and ( 6.20 ) into simplest form. In doing

NkT

And since the entropy of an ideal Fermi gas is a function of its temperature through ( 6.15 ), and ( 6.25 ), we i nd, after some algebra, that

6.3 Ideal Fermi gas 117 ( 6.27 ) of an ideal Fermi gas given the relation ( 6.25 ) that parameterizes the normalized temperature NkT EE in terms of the occupancy N n . ₀

Ground state

As the normalized temperature NkT EE passes from values high compared ₀ to 1 to values low compared to 1, the ideal Fermi gas passes from its clas-sical regime to its quantum regime. Because, given ( 6.23 ), the normalized temperature is

low temperature, high particle density, and small particle mass all contribute to placing the ideal Fermi gas in the quantum regime.

0.5 between normalized temperature NkT EE and occupancy N n . The thin lines ₀ represent classical limits. The volume V and particle number N are constant.

In the extreme quantum limit NkT EE₀2 the ground state pressure P1 PP of 0 adjective degeneracy simply means “relatively high occupancy” or “in the quantum limit.” The phrase ground state pressure is more descriptive. A non-vanishing ground state energy EE , a non-vanishing ground state pressure P₀ PP , ₀ and the vanishing of the heat capacity in the T→ 0 limit are outstanding quan-tum features of the ideal Fermi gas.

6.4 Average energy approximation

The reader who consults texts listed in Appendix VI will i nd that the usual way of describing ideal Fermi gas is via the occupation number method accord-ing to which the entropy is maximized over occupation number macrostates while constrained to conserve the system energy E and particle number N . This approach has become standard and produces an expression for the proba-bility p_j ⎡=

⎣⎡⎡ ⎤

⎦⎤⎤

e−^{εj kT} Z₁ that a particle of ideal gas will occupy a single-particle microstate whose energy is ε_j . Consequently, procedures for determining the single-particle partition function Z₁ and the quantized energies ε_j are required.

Then the entropy of the gas is determined from the Gibbs entropy formula S kNkk p_j p

j j

∑

^{ln .}

In contrast, we have, in Section 6.3 , employed and will, in Sections 7.4 and 7.5 , employ the average energy approximation according to which all the particles of an ideal gas have the same energy: the average energy of a particle E N . Then the system entropy S

[

⁼

]

is determined through its multi-plicity Ω by the number of ideal gas particles N , the number of single-particle microstates n those particles may occupy, and the kind of indistinguishable par-ticles, fermions or bosons, that compose the gas. The number of single-particle

6.4 Average energy approximation 119

microstates n proceeds from quantizing phase space into Plank’s constant-sized chunks and depends, according to ( 6.13 ), on the system volume V and average energy per particle E N . In this way an entropy function S E V N

(

, ,V

)

^is

produced that encapsulates all that can be known of the thermodynamics of a quantum ideal gas.

These two methods, one based on maximizing the entropy over occupation number macrostates and the other based on the average energy approximation, produce many of the same results. Both produce the Sackur–Tetrode entropy in the high temperature, low density, classical limit and both produce important quantum features: third law compliant entropy and heat capacity and the non-vanishing ground state energy and pressure of an ideal Fermi gas. But there are differences. In particular, the ground state energy ( 6.23 ) and pressure ( 6.30 ) of an ideal Fermi gas predicted by the average energy approximation are a factor of 1.2 higher than those produced by standard methods. Quantities determined by the derivatives of the entropy, like the heat capacity C_V , differ even in their functionality.

There is no doubt that the standard approach to the ideal quantum gases incorporates more physics and is more accurate than the approach of the average energy approximation. However, we have employed the average energy approx-imation because it generates important results with a minimum of mathematics and directly exploits the properties of the entropy function S E V N

(

, ,V

)

^.

Example 6.1 conduction electrons, nuclei, and white dwarfs The conduction electrons within a typical metal such as copper form an ideal Fermi gas that is essentially ground state at room temperature. Since each cop-per atom contributes approximately one electron to its gas of conducting elec-trons, the density of copper atoms, N V ≈ 8 48 148 1× 0× 0²⁸²⁸atoms/m³ , is the density of its conduction electrons. Given T = 300 K for room temperature, the nor-malized temperature ( 6.29 ) becomes a number, 1 15 10. ⁻¹² , that is well below 1. Thus the conduction electrons in a metal contribute per electron much less to the heat capacity than that classically expected, 3 2k

[

⁼

]

^.

The ground state pressure PPP can be quite large for the conduction electrons ₀ in metals – 46.2 billion N/m ² for copper at standard density – but is effectively neutralized by the electrostatic attraction between the metallic ions and the conduction electrons of the ideal Fermi gas.

The nucleons (protons and neutrons) within a nucleus also form an ideal Fermi gas. The ground state pressure within a nucleus opposes the strong-force attraction between nucleons and, in this way, provides for the stability of the nucleus and determines its size. A similar balancing occurs between the

outward pressure of the electrons of white dwarf stars, stars that at the end of their nuclear fuel cycle have collapsed into objects a million times more dense than the Sun, and the inward force of gravity.

Problems

6.1 Extensivity

Show that the entropy of an ideal Fermi gas is an extensive function of its l uid variables, that is, show that the function S E V N

(

, ,V

)

implied by ( 6.15 ) and ( 6.17 ) has the property λλλS

(

, ,V

) (

S λλE,λλλλλV,λλN

)

^.

6.2 De Broglie wavelength

Another way to indicate when an ideal gas enters the quantum regime is to indicate when the de Broglie wavelength λ

[

⁼

]

of particles with average energy E N becomes larger than the average spacing V N

( )

^{1 3} between parti-cle centers. (a) Derive an inequality that expresses this condition in terms of the average particle energy E N , the particle density N V , and universal con-stants. (b) What values of the occupancy N n does this condition imply?

6.3 Heat capacity

(a) Derive the heat capacity of an ideal Fermi gas ( 6.27 ), that is, derive

C

Nk

N N n

N n

V =

( ) (

⁻

)

( ) { ⁽

^{N n}

^{) (}

^{−N n}

⁾ }

⎡⎣⎡⎡ ⎤⎦⎤⎤

1 n N

3

( ) (

1 3+ ^{−N n}

) (

l

from C_V =T

(

∂S ∂T

)

_V . (b) Show that this heat capacity recovers the result expected for a classical, ideal, monatomic gas when N n2 1 .

6.4 Laser fusion

One method of creating the temperatures and densities necessary for nuclear fusion is to shine laser beams on a pellet of the isotopes of hydrogen: deu-terium and tritium. The result sought is a temperature of at least 1 keV and an electron density of about N V = 10³³ m³ in the imploded core of the fuel pellet. Determine the size of the normalized temperature NkT EE that cor-₀ responds to these parameters. Is the imploded core of the fuel pellet in the quantum regime?

Problems 121

6.5 White dwarfs

When a star consumes all its nuclear fuel it no longer generates enough energy to maintain its size and the star will collapse into a compact object, either a white dwarf, a neutron star, or a black hole, depending upon its mass. The ground state or degeneracy pressure of its electron gas is the mechanism that keeps white dwarf stars with masses up to 1.4 times the mass of the sun from further gravitational collapse. Consider that a spherical, homogeneous white dwarf of mass M and radius R will have a negative gravitational potential energy of U_g 3GGMGG ² 5R , and that the inward gravitational pressure exerted on the surface of the white dwarf isP_g

(

^{∂UUgg ∂V}

)

. A white dwarf is hot enough to be a fully ionized plasma and dense enough to be in the fully quan-tum regime. Therefore, its electron pressure is PPP₀

(

N V

)

^{5 3}

(

^{h2 2 em}

)

^{, and}

the condition P_g PPPP₀ =0 establishes the equilibrium of a white dwarf. Assume that a white dwarf is composed of equal numbers of electrons with mass m and protons with mass m_p . Determine how the radius R of a white dwarf depends on its mass M . (Hint: You will need to use the relationship between the volume V and the radius R of a sphere.)

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