Fermi–Dirac Statistics

Fermi–Dirac (FD) statistics is a fundamental concept in quantum mechanics that describes the behavior of a system of identical fermions. Fermions are particles that obey the Pauli exclusion principle, meaning no two identical fermions can occupy the same quantum state simultaneously. Examples of fermions include electrons, protons, neutrons, and quarks. This statistical framework is crucial for understanding the properties of materials like metals, semiconductors, and white dwarf stars.

The Fermi–Dirac Distribution Function

The Fermi–Dirac distribution function, denoted by $f(E)$, gives the probability that a given energy state $E$ is occupied by a fermion at a given temperature $T$. It is derived using the principles of statistical mechanics and quantum mechanics. The function is expressed as:

$f(E) = \frac{1}{e^{(E - \mu) / k_B T} + 1}$

Where:

• $E$ is the energy of the quantum state.
• $\mu$ is the chemical potential, also known as the Fermi level.
• $k_B$ is the Boltzmann constant ($1.381 \times 10^{-23} \, J/K$).
• $T$ is the absolute temperature in Kelvin.

Let's analyze the behavior of this function under different conditions:

Case 1: At Absolute Zero ($T = 0$ K)

At absolute zero, the distribution function simplifies significantly.

If $E < \mu$: The exponent $(E - \mu) / k_B T$ approaches $-\infty$. Thus, $e^{-\infty} = 0$. $f(E) = \frac{1}{0 + 1} = 1$. This means all energy states below the Fermi level are completely occupied.

If $E > \mu$: The exponent $(E - \mu) / k_B T$ approaches $+\infty$. Thus, $e^{+\infty} = \infty$. $f(E) = \frac{1}{\infty + 1} = 0$. This means all energy states above the Fermi level are completely empty.

If $E = \mu$: The expression is undefined at $T=0$ in this form, but it represents the sharp boundary. The Fermi level at $T=0$ is called the Fermi energy, $E_F$.

Case 2: At Temperatures Above Absolute Zero ($T > 0$ K)

As temperature increases, the sharp distinction between occupied and unoccupied states blurs.

If $E \ll \mu$ and $k_B T \ll \mu$: The exponent $(E - \mu) / k_B T$ is large and negative. $e^{(E - \mu) / k_B T}$ is close to 0. $f(E) \approx \frac{1}{0 + 1} = 1$. States well below the Fermi level are still almost fully occupied.

If $E \gg \mu$ and $k_B T \ll \mu$: The exponent $(E - \mu) / k_B T$ is large and positive. $e^{(E - \mu) / k_B T}$ is large. $f(E) \approx \frac{1}{e^{(E - \mu) / k_B T}} \approx 0$. States well above the Fermi level are still almost completely unoccupied.

If $E \approx \mu$ and $k_B T$ is comparable to $(E - \mu)$: The exponential term is not negligible. The probability $f(E)$ drops from near 1 to near 0 over an energy range of a few $k_B T$ around the Fermi level. The value of $f(E)$ at $E = \mu$ becomes: $f(\mu) = \frac{1}{e^{( \mu - \mu) / k_B T} + 1} = \frac{1}{e^0 + 1} = \frac{1}{1 + 1} = \frac{1}{2}$. So, at the Fermi level, the probability of occupation is 0.5.

Electron Gas

An electron gas is a model used to describe the behavior of electrons in a metal. In this model, the valence electrons of the metal atoms are treated as being free to move throughout the entire volume of the solid, forming a "gas" of electrons. These electrons are assumed to interact only weakly with each other and with the positive ion cores of the metal lattice.

The quantum mechanical treatment of an electron gas is crucial because electrons are fermions. According to FD statistics, even at absolute zero, these electrons do not all occupy the lowest energy state. Instead, they fill up energy levels from the bottom up to the Fermi energy ($E_F$). This implies that even at $0$ K, the electrons possess significant kinetic energy.

The Fermi energy ($E_F$) for a free electron gas in three dimensions can be calculated. For $N$ electrons in a volume $V$, the number density is $n = N/V$. The Fermi energy is given by:

$E_F = \frac{\hbar^2}{2m} \left( \frac{3\pi^2 N}{V} \right)^{2/3} = \frac{\hbar^2}{2m} (3\pi^2 n)^{2/3}$

Where:

• $\hbar$ is the reduced Planck constant ($\hbar = h / 2\pi$).
• $m$ is the mass of the electron.
• $n$ is the number density of electrons.

The Fermi velocity ($v_F$) is the velocity of an electron with energy $E_F$:

$v_F = \sqrt{\frac{2E_F}{m}}$

The Fermi temperature ($T_F$) is the temperature at which the thermal energy $k_B T_F$ is comparable to the Fermi energy $E_F$:

$T_F = \frac{E_F}{k_B}$

For most metals, the Fermi temperature is very high (e.g., for copper, $T_F \approx 8 \times 10^4$ K). This means that at room temperature (around 300 K), the electrons in a metal behave as if they are at very low temperatures, and FD statistics are essential for describing their behavior. The thermal energy $k_B T$ is much smaller than $E_F$, so the distribution function is close to the $T=0$ case for most states.

Pauli Paramagnetism

Paramagnetism is a property of materials that are weakly attracted by an external magnetic field. In typical paramagnets, the magnetic moments arise from unpaired electron spins in atoms. However, in metals, even though most electrons are paired, the unpaired spins of the conduction electrons contribute to paramagnetism. This contribution, arising from the spin degeneracy of energy levels and the Pauli exclusion principle, is known as Pauli paramagnetism.

Consider electrons in a metal subjected to an external magnetic field $B$. The magnetic field splits the energy levels for electrons with spin up ($\uparrow$) and spin down ($\downarrow$). According to FD statistics, electrons will fill these split energy levels.

In the absence of a magnetic field, the density of states for spin up and spin down electrons is the same. At $T=0$, all states up to $E_F$ are filled.

When a magnetic field $B$ is applied, the energy of a spin $\uparrow$ electron decreases by $\mu_B B$, and the energy of a spin $\downarrow$ electron increases by $\mu_B B$, where $\mu_B$ is the Bohr magneton ($\mu_B = \frac{e\hbar}{2m_e}$).

Because of the Pauli exclusion principle, electrons cannot simply flip their spins to occupy lower energy states if those states are already filled. The applied magnetic field causes a slight redistribution of electrons between spin-up and spin-down states. A small number of electrons with spin down just below the Fermi level will flip their spins to become spin up, occupying empty states just above the Fermi level. This redistribution leads to a net excess of spin-up electrons.

This excess of spin-up electrons creates a net magnetic moment in the direction of the applied field, resulting in paramagnetism.

The key characteristics of Pauli paramagnetism are:

• It is independent of temperature at low temperatures because only electrons near the Fermi level are involved in the spin-flipping process, and the density of these electrons does not change significantly with temperature.
• The susceptibility is proportional to the density of states at the Fermi level.
• It is generally a weak form of paramagnetism compared to the paramagnetism arising from localized magnetic moments.

Thermionic Emission

Thermionic emission is the process by which a material emits electrons when its surface is heated. This phenomenon is important in devices like vacuum tubes, cathode ray tubes, and some types of light bulbs.

In a metal, electrons are in constant motion. At room temperature, most of these electrons are bound within the metal by the work function ($\phi$), which is the minimum energy required to remove an electron from the surface of the metal.

When the metal is heated to a sufficiently high temperature, the electrons gain kinetic energy. According to Fermi-Dirac statistics, even at high temperatures, most electrons are below the Fermi level. However, a small fraction of electrons near the top of the Fermi distribution gain enough thermal energy to overcome the work function and escape from the metal surface.

The rate of thermionic emission is described by the Richardson-Dushman equation. This equation relates the current density ($J$) of emitted electrons to the temperature ($T$) and the work function ($\phi$) of the material:

$J = A T^2 e^{-\phi / k_B T}$

Where:

• $A$ is the Richardson constant, which depends on the material and fundamental constants ($A = \frac{4\pi m_e k_B^2 e}{h^3}$).
• $T$ is the absolute temperature of the emitter.
• $\phi$ is the work function of the material.
• $k_B$ is the Boltzmann constant.
• $e$ is the elementary charge.

The exponential term $e^{-\phi / k_B T}$ shows that the emission current is highly sensitive to temperature. A small increase in temperature can lead to a significant increase in the number of electrons with energy greater than the work function.

The Fermi-Dirac distribution function helps understand why only a fraction of electrons escape. Only those electrons whose energy $E$ is greater than the work function ($\phi$) and whose initial velocity component perpendicular to the surface is sufficient to overcome the surface potential barrier will be emitted.

Elementary Ideas of Phase Transition

A phase transition is a physical process where a thermodynamic system undergoes a change from one phase (e.g., solid, liquid, gas) to another. These transitions occur at specific temperatures and pressures and involve changes in the macroscopic properties of the system. Examples include melting, boiling, and condensation.

In the context of statistical mechanics and quantum statistics, phase transitions can also occur in systems of interacting particles, such as electrons in solids or quantum fluids. These transitions are often driven by changes in temperature, pressure, or external fields, and they are characterized by abrupt changes in the system's order parameter or symmetry.

Phase transitions are broadly classified into two types:

First-Order Phase Transitions

These transitions are characterized by a discontinuity in the first derivative of the thermodynamic potential (e.g., Gibbs free energy) with respect to temperature or pressure. They involve latent heat, meaning energy must be absorbed or released for the transition to occur.

• Examples: Melting of ice into water, boiling of water into steam.
• Thermodynamic Property Change: Entropy changes discontinuously.

Second-Order Phase Transitions (and Continuous Transitions)

These transitions are characterized by a discontinuity in the second derivative of the thermodynamic potential. They do not involve latent heat, and the change occurs continuously. They are often associated with changes in symmetry.

• Examples: Ferromagnetic to paramagnetic transition (Curie point), superconducting transition, superfluid transition in Helium-4.
• Thermodynamic Property Change: Specific heat, magnetic susceptibility, or compressibility often changes discontinuously or diverges.

In systems governed by Fermi-Dirac or Bose-Einstein statistics, phase transitions can occur due to collective behavior of particles. For instance, superconductivity is a phase transition where electrons near the Fermi surface form Cooper pairs and condense into a coherent quantum state, exhibiting zero electrical resistance. This transition is a second-order phase transition.

Properties of Liquid Helium

Liquid helium is a fascinating substance that exhibits unique quantum mechanical properties at very low temperatures, primarily due to its light atomic mass and weak interatomic forces. It exists in two distinct liquid phases: Helium-I and Helium-II.

Helium-I (He-I)

Above the lambda point ($T_\lambda \approx 2.17$ K at saturated vapor pressure), liquid helium exists as Helium-I.

• He-I behaves like a normal liquid, although it has a very low viscosity and a high thermal conductivity compared to other liquids.
• It exhibits properties consistent with classical fluids, though quantum effects are still present.
• As temperature increases towards $T_\lambda$, its viscosity decreases and thermal conductivity increases.

Helium-II (He-II)

Below the lambda point ($T < T_\lambda$), liquid helium enters a state known as Helium-II, which is a superfluid. This transition at $T_\lambda$ is a classic example of a second-order phase transition.

• Superfluidity: He-II exhibits superfluidity, meaning it can flow without any viscosity. It can flow through extremely narrow capillaries or pores without resistance.
• Zero Viscosity: A consequence of superfluidity is zero viscosity ($\eta = 0$).
• Second Sound: In He-II, it is possible to have a wave of temperature fluctuations propagating through the liquid, known as second sound. This is distinct from the usual first sound (pressure/density wave) found in normal fluids.
• Fountain Effect: When He-II is heated in a narrow tube, it can produce a fountain. This is because the superfluid component is driven towards regions of lower temperature (or higher entropy), and if heated locally, it creates a pressure gradient that can eject the fluid.
• Quantized Vortices: When He-II is stirred or rotated, the vorticity does not form a continuous spectrum but is quantized into elementary vortices. The circulation around each vortex is quantized in units of $h/m$, where $h$ is Planck's constant and $m$ is the mass of a helium atom.
• Two-Fluid Model: The behavior of He-II is often described by the two-fluid model, proposed by Tisza and Landau. This model suggests that He-II can be considered as a mixture of two interpenetrating fluids: a normal fluid component (with viscosity and entropy) and a superfluid component (with zero viscosity and zero entropy). The proportion of each component depends on temperature.

The superfluidity of Helium-II is a macroscopic manifestation of quantum mechanics, arising from the Bose-Einstein condensation of Helium-4 atoms (which are bosons). While Helium-3 atoms are fermions and do not undergo Bose-Einstein condensation, liquid Helium-3 also exhibits superfluidity at even lower temperatures due to the formation of Cooper pairs, analogous to superconductivity in electrons.