Degenerate Fermi Gas: Equation of State & White Dwarf Stability

Welcome back to our study of ideal Fermi gases. In our previous lesson, we established the general integral expressions for the particle number, internal energy, and pressure of a Fermi gas at any temperature $T$ . These integrals form the exact starting point for analyzing the thermodynamics of fermionic systems.

Today, we will focus on the important and illuminating limit of absolute zero temperature, $T = 0$ . In this "completely degenerate" regime, the quantum nature of fermions becomes starkly apparent. Our primary goal is to derive the T=0 equation of state for a degenerate Fermi gas and discuss its application to the stability of white dwarf stars, culminating in the derivation of the Chandrasekhar limit. This analysis will not only solidify our understanding of quantum pressure but also reveal a profound connection between fundamental constants and the maximum mass of a star.

1. The Equation of State of a Degenerate Fermi Gas

At absolute zero, the Fermi-Dirac distribution, $⟨ n (E)⟩ = (e^{β (E - μ)} + 1)^{- 1}$ , simplifies dramatically. As $T \to 0$ ( $β \to \infty$ ), the distribution becomes a Heaviside step function:

⟨ n (E)⟩ = {10 if E < μ if E > μ

This means all single-particle energy states are completely filled up to the chemical potential, and all states above it are empty. The chemical potential at T=0 is given a special name: the Fermi energy, $E_{F} \equiv μ (T = 0)$ .

With this, we can now evaluate the integrals for the particle number $N$ and internal energy $U$ from our last lesson. The integrals become sharp cutoffs at $E_{F}$ .

The MIT lecture notes by Prof. Robert Jaffe provide a clear, thermodynamically-minded derivation of the key quantities. We will follow this approach, which treats all intensive properties as functions of a single variable, the Fermi momentum $p_{F} = 2 m E_{F}$ .

Let's summarize the key results for a non-relativistic gas ( $E = p^{2} /2 m$ ).
By integrating the density of states $g (E)$ up to $E_{F}$ , we find the relationship between the particle density $n = N / V$ and the Fermi energy:

E_{F} = \frac{ℏ ^{2}}{2 m} (\frac{6 π ^{2} n}{g _{s}})^{2/3}

where $g_{s}$ is the spin degeneracy ( $g_{s} = 2$ for electrons). This shows that the maximum energy of the occupied states scales with density as $E_{F} \propto n^{2/3}$ .

The total internal energy is found by integrating $E \cdot g (E)$ up to $E_{F}$ , which yields:

U = \frac{3}{5} N E_{F}

This non-zero energy at absolute zero is a purely quantum mechanical effect called zero-point energy. It arises because the Pauli exclusion principle forces fermions to occupy higher energy states even in the ground state of the many-body system.

Finally, the pressure exerted by the gas, known as degeneracy pressure, is obtained from the general relation $P V = \frac{2}{3} U$ that we found in the last lesson.

P = \frac{2}{3} \frac{U}{V} = \frac{2}{5} n E_{F}

Substituting the expression for $E_{F}$ , we arrive at the T=0 equation of state for a non-relativistic degenerate Fermi gas:

P = \frac{ℏ ^{2}}{5 m} (\frac{6 π ^{2}}{g _{s}})^{2/3} n^{5/3}

The pressure is non-zero even at T=0 and scales with density as $P \propto n^{5/3}$ . This quantum pressure is what prevents certain stars from collapsing under their own gravity.

2. Relativistic Limit and White Dwarf Stability

As stars like our sun exhaust their nuclear fuel, they collapse under gravity, increasing their density dramatically. Stars with masses up to about 8 solar masses end their lives as white dwarfs, incredibly dense objects where a sea of degenerate electrons provides the pressure to halt gravitational collapse.

As the density of the electron gas increases, the Fermi energy $E_{F}$ can become comparable to the electron's rest mass energy, $m_{e} c^{2}$ . In this regime, the non-relativistic approximation $E = p^{2} /2 m$ breaks down, and we must use the full relativistic expression $E = p^{2} c^{2} + m_{e}^{2} c^{4}$ . In the extreme or ultra-relativistic limit where $E_{F} ≫ m_{e} c^{2}$ , this simplifies to $E \approx p c$ .

This change in the energy-momentum relation fundamentally alters the equation of state. Following a similar derivation as before, but with $E = p c$ , one finds:

The Fermi energy scales as $E_{F} \propto n^{1/3}$ .
The internal energy scales as $U \propto N E_{F} \propto n^{4/3} V$ .
The pressure scales as $P \propto U / V \propto n^{4/3}$ .

So, we have two different equations of state depending on the density:

Non-Relativistic (low density): $P \propto n^{5/3}$
Ultra-Relativistic (high density): $P \propto n^{4/3}$

The video from the Physics Almanac channel provides a detailed walkthrough of these derivations.

3. The Chandrasekhar Limit

The fate of a white dwarf is decided by a battle between the outward push of electron degeneracy pressure and the inward pull of gravity. A particularly elegant way to analyze this, which you will often encounter in theoretical physics, is to find the configuration that minimizes the total energy of the star.

The total energy $E_{t o t a l}$ is the sum of the electron degeneracy energy ( $U$ ) and the gravitational potential energy ( $E_{g r a v}$ ).

E_{t o t a l} = U + E_{g r a v}

For a spherical star of mass $M$ and radius $R$ , the gravitational potential energy is approximately $E_{g r a v} \approx - \frac{3}{5} \frac{G M ^{2}}{R}$ .

The following video from Physics Explained provides a masterful, step-by-step derivation of the stability condition using this energy minimization approach. It clearly explains the physics without getting bogged down in the most complex mathematics, making it an ideal guide for our purposes.

What is the Chandrasekhar limit for White Dwarf Stars?

This video brilliantly explains the entire concept of white dwarf stability and the Chandrasekhar limit from first principles. It uses the energy minimization argument we are following.

We will work through the core argument of this video. First, familiarize yourself with the strategy of minimizing the total energy. Next, watch the derivation of the non-relativistic case. The key result is that the total energy has a stable minimum, leading to a stable radius R where E_{total} is minimized. Notice the final relation: R \propto M^{-1/3}, meaning more massive white dwarfs are smaller! Now, the crucial part. Watch the segment explaining the ultra-relativistic limit. This section shows how the degeneracy energy changes when electrons move near the speed of light. Finally, watch the derivation of the Chandrasekhar limit. This is the climax of the argument, where the relativistic degeneracy energy is balanced against gravity.

Let's formalize the argument from the video.

Case 1: Non-Relativistic Electrons

The degeneracy energy is $U \propto n^{5/3} V$ . Since the electron density $n \propto M / R^{3}$ and $V \propto R^{3}$ , we get:
$U \propto (\frac{M}{R ^{3}})^{5/3} R^{3} = \frac{M ^{5/3}}{R ^{2}}$
So, the total energy is:
$E_{t o t a l} (R) = A \frac{M ^{5/3}}{R ^{2}} - B \frac{M ^{2}}{R}$
where A and B are positive constants. This function has a stable minimum at a specific radius $R_{e q}$ , which can be found by setting $\frac{d E _{t o t a l}}{d R} = 0$ . As long as the electrons remain non-relativistic, gravity can always be balanced by degeneracy pressure at some finite radius.

Case 2: Ultra-Relativistic Electrons

Here, the degeneracy energy is $U \propto n^{4/3} V$ . Following the same scaling:
$U \propto (\frac{M}{R ^{3}})^{4/3} R^{3} = \frac{M ^{4/3}}{R}$
The total energy now has a strikingly different form:
$E_{t o t a l} (R) = A^{'} \frac{M ^{4/3}}{R} - B \frac{M ^{2}}{R} = \frac{1}{R} (A^{'} M^{4/3} - B M^{2})$
The radial dependence is identical for both terms! The stability of the star now depends entirely on the sign of the term in the parenthesis.

If $A^{'} M^{4/3} > B M^{2}$ , then $E_{t o t a l} > 0$ . The energy is minimized as $R \to \infty$ . The star will expand until the electrons become non-relativistic and it finds the stable equilibrium of Case 1.
If $A^{'} M^{4/3} < B M^{2}$ , then $E_{t o t a l} < 0$ . The energy is minimized as $R \to 0$ . There is no stable equilibrium. The star is doomed to catastrophic collapse.

The critical condition occurs when the two terms are equal. This defines the Chandrasekhar Mass Limit, $M_{C h}$ :
$A^{'} M_{C h}^{4/3} = B M_{C h}^{2} ⟹ M_{C h}^{2/3} = \frac{A ^{'}}{B}$
Plugging in the full expressions for the constants A' and B reveals a remarkable result:
$M_{C h} \approx (\frac{ℏ c}{G})^{3/2} \frac{1}{m _{p}^{2}}$
(Here $m_{p}$ is the proton mass, which dominates the star's mass). The maximum mass a white dwarf can have is determined solely by fundamental constants of nature! A more careful calculation, accounting for composition and non-uniform density, yields $M_{C h} \approx 1.44 M_{⊙}$ , where $M_{⊙}$ is the mass of the sun. Any star that ends its life with a core more massive than this cannot become a stable white dwarf and must collapse further, potentially forming a neutron star or a black hole.

The MIT notes provide a more formal treatment using the Lane-Emden equation for hydrostatic equilibrium, but the physical conclusion is identical and can be seen with this simpler energy argument. You can find this powerful, physical argument summarized in the notes.

Conclusion

In this lesson, we have taken the T=0 limit of our general Fermi gas formalism to derive the equation of state for a completely degenerate system. We discovered that quantum mechanics, via the Pauli exclusion principle, gives rise to a powerful "degeneracy pressure" that can support massive objects against gravity.

Key Takeaways:

At T=0, a Fermi gas fills all energy states up to the Fermi energy $E_{F}$ , resulting in significant zero-point energy and pressure.
The equation of state depends on whether the fermions are non-relativistic ( $P \propto n^{5/3}$ ) or ultra-relativistic ( $P \propto n^{4/3}$ ).
In a white dwarf, the outward degeneracy pressure of electrons balances the inward pull of gravity.
The stability analysis shows that for non-relativistic electrons, a stable equilibrium always exists, with more massive stars being smaller ( $R \propto M^{- 1/3}$ ).
For ultra-relativistic electrons, the balance becomes precarious. There exists a maximum mass, the Chandrasekhar limit ( $\approx 1.44 M_{⊙}$ ), beyond which degeneracy pressure fails and the star must collapse. This limit is determined by a combination of fundamental constants: $ℏ$ , $c$ , $G$ , and $m_{p}$ .

So far, we have focused on the idealized T=0 case. While this is a very good approximation for dense systems like white dwarfs and electrons in metals, real systems exist at finite temperatures. In our next lesson, we will develop the Sommerfeld expansion, a powerful mathematical tool for calculating thermodynamic properties of a Fermi gas at low but non-zero temperatures ( $T ≪ T_{F}$ ). This will allow us to derive one of the most important results in condensed matter physics: the linear temperature dependence of the electronic heat capacity of metals.