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--- un:fermi-dirac-statistics [2025/10/26 13:00] – created asad
+++ un:fermi-dirac-statistics [2025/10/27 12:10] (current) – [Fermi–Dirac statistics] asad
@@ Line 1: / Line 1: @@
 ====== Fermi–Dirac statistics ======
-The particles of the [[uv:standard model]] of particle physics carry an intrinsic angular momentum known as **spin**, represented by the spin quantum number \(S\).
+The particles of the [[standard model]] of particle physics carry an intrinsic angular momentum known as **spin**, represented by the spin quantum number \(S\).
 The magnitude of the angular momentum vector is
@@ Line 8: / Line 8: @@
 $$
-where \(\hbar = h/(2\pi)\) and \(S\) can be either zero or a half-integer.
+where \(\hbar = h/(2\pi)\) is the reduced Planck constant.
-Particles with **half-integer spin** are called **fermions**, and those with **integer spin** are called **bosons**.
+Particles with **half-integer spin** (\(S = 1/2, 3/2, \dots\)) are called **fermions**, while those with **integer spin** (\(S = 0, 1, 2, \dots\)) are called **bosons**.
   - **Fermions:** electrons, protons, neutrons, neutrinos, and any nucleus with an odd number of nucleons
-  - **Bosons:** photons, gluons, and nuclei such as \(^{4}\)He and \(^{12}\)C
+  - **Bosons:** photons, gluons, and nuclei such as \(^{4}\mathrm{He}\) and \(^{12}\mathrm{C}\)
-{{:courses:ast301:pauli.png?nolink&500|Pauli exclusion principle illustrated in phase space.}}
 ===== Pauli exclusion principle =====
+{{:courses:ast301:pauli.png?nolink&500|Pauli exclusion principle illustrated in one-dimensional phase space.}}
 The **Pauli exclusion principle** states that no two identical fermions can occupy the same quantum state simultaneously.
@@ Line 29: / Line 29: @@
 $$
-and each state can hold only two fermions of opposite spins.
+and each quantum state can hold only two fermions of opposite spins.
+This exclusion is purely quantum mechanical and has no classical analogue.
+It is the fundamental reason why matter resists compression at very high densities — the basis of **degeneracy pressure**.
 ===== From Maxwell–Boltzmann to Fermi–Dirac =====
-At high temperature or low density, the number of available states greatly exceeds the number of particles, and the exclusion principle has negligible effect.
+At high temperature or low density, the number of available quantum states greatly exceeds the number of particles, and the exclusion principle has negligible effect.
-The distribution of particles then follows **Maxwell–Boltzmann statistics (MBS)**.
+The particle distribution then follows **Maxwell–Boltzmann statistics (MBS)**.
-At very low temperatures or high densities, however, nearly all the low-energy states become filled.
+At very low temperatures or high densities, however, nearly all low-energy states become filled.
 This regime is described by **Fermi–Dirac statistics (FDS)**, and the gas is said to be **degenerate**.
   - In MBS: most particles occupy low-energy states, with a long tail at high energies.
-  - In FDS: all states below a certain energy are filled, and very few particles occupy higher levels.
+  - In FDS: all states below a certain limiting energy are filled, and very few particles occupy higher ones.
-{{:courses:ast301:degeneracy.webp?nolink&700|Filling of quantum states in degeneracy.}}
+{{:courses:ast301:degeneracy.webp?nolink&700|Filling of quantum states under degeneracy: in the degenerate case, all states below a limiting energy are occupied.}}
-This creates an effective pressure even at zero temperature, called **degeneracy pressure**.
+This complete filling of low-energy quantum states produces an effective **pressure** even at zero temperature — **degeneracy pressure**.
-It arises not from thermal motion, but from the Pauli principle itself: the particles cannot all occupy the lowest state.
+It does not arise from thermal motion, but from the Pauli exclusion principle itself: compressing the gas forces fermions into higher momentum states, increasing the mean momentum and thus the pressure.
+This pressure supports compact stellar objects such as **white dwarfs** and plays a major role in the structure of **brown dwarfs** and **planetary interiors**.
 ===== The Fermi distribution =====
-The Fermi–Dirac occupation probability for a particle of energy \(E\) is
+The **Fermi–Dirac occupation probability** for a particle of energy \(E\) at temperature \(T\) is
 $$
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 $$
-where \(E_F\) is the **Fermi energy** — the highest occupied energy level at \(T=0\).
+where \(E_F\) is the **Fermi energy**, the highest occupied energy level at absolute zero (\(T = 0\)).
-At absolute zero, all states with \(E < E_F\) are filled and those with \(E > E_F\) are empty, giving a nearly rectangular distribution.
+At \(T = 0\), all states with \(E < E_F\) are filled (\(F = 1\)) and those with \(E > E_F\) are empty (\(F = 0\)), producing a sharp step in the distribution.
-The general form of the **phase-space distribution function** is
+{{:courses:ast301:fd-probability.webp?nolink&700|Fermi–Dirac occupation probability versus energy for various temperatures.}}
+The figure above shows how \(F(E)\) changes with temperature.
+At \(T = 0\), the probability falls abruptly from 1 to 0 at \(E = E_F\).
+As the temperature increases, this transition becomes smoother — some particles gain energy and occupy states with \(E > E_F\), while some lower-energy states are vacated.
+Even for \(kT \ll E_F\), the curve remains steep near \(E_F\), meaning that most fermions remain below the Fermi energy.
+The general **phase-space distribution function** is
 $$
-f_{FD} = \frac{2}{h^3}\,F(E)
+f_{FD} = \frac{2}{h^3}\,F(E),
 $$
-where the prefactor \(2/h^3\) accounts for spin degeneracy (two electrons per quantum state).
+where the factor \(2/h^3\) accounts for spin degeneracy (two spin orientations per quantum state).
-In the non-degenerate limit (\(E_F \ll kT\)), the exponential term dominates and FDS reduces to MBS.
+In the non-degenerate limit (\(E_F \ll kT\)), the exponential term dominates and Fermi–Dirac statistics reduce to Maxwell–Boltzmann statistics.
 ===== Degeneracy and Fermi momentum =====
-In three dimensions, the occupied region in momentum space at \(T=0\) is a sphere of radius \(p_F\), called the **Fermi sphere**.
+In three dimensions, all occupied momentum states at \(T = 0\) form a sphere in momentum space of radius \(p_F\), called the **Fermi momentum**.
 The total number of electrons within this sphere is
@@ Line 77: / Line 88: @@
 where \(V_x\) is the physical volume.
-The **number density** of electrons is therefore
+The **number density** of electrons is then
 $$
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 $$
-Hence the **Fermi momentum** depends only on the electron density:
+Rearranging gives the **Fermi momentum** in terms of the electron density:
 $$
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 $$
-At zero temperature, all states up to \(p_F\) are filled. The Fermi energy follows as
+The corresponding **Fermi energy** is
 $$
-E_F = \frac{p_F^2}{2m_e} \quad \text{(nonrelativistic)}
+E_F = \frac{p_F^2}{2m_e} \quad \text{(nonrelativistic)}, \qquad
+E_F = \sqrt{p_F^2 c^2 + m_e^2 c^4} - m_e c^2 \quad \text{(relativistic)}.
 $$
-or, for relativistic electrons,
+{{:courses:ast301:fds.png?nolink&700|Fermi sphere in momentum space showing all occupied states up to the Fermi momentum \(p_F\). The surface of this sphere represents the Fermi energy boundary at \(T=0\).}}
-$$
-E_F = \sqrt{p_F^2 c^2 + m_e^2 c^4} - m_e c^2.
-$$
-===== Physical meaning =====
-In a degenerate gas:
-  - Pressure arises from the quantum mechanical restriction on fermions.
-  - The pressure remains nonzero even at \(T=0\).
-  - It depends only on the **density**, not on the temperature.
-  - Electrons dominate the pressure, while protons and ions dominate the mass.
-Degeneracy pressure supports compact objects such as **white dwarfs**, and partially degenerate matter occurs in brown dwarfs and planetary cores.
+The **Fermi sphere** above visualizes all occupied quantum states in momentum space.
+Each point inside the sphere corresponds to a filled state with momentum \(\mathbf{p}\).
+At \(T = 0\), all states with \(|\mathbf{p}| < p_F\) are filled, and those outside remain empty.
+The radius \(p_F\) depends only on the density \(n_e\), and thus determines the Fermi energy and the magnitude of degeneracy pressure.
+Since degeneracy pressure depends only on \(n_e\), not on temperature, it persists even when \(T = 0\).
 ===== Insights =====
-  - Fermi–Dirac statistics governs particles that obey the Pauli exclusion principle.
+  - Fermi–Dirac statistics describe particles that obey the Pauli exclusion principle.
-  - The Fermi energy defines the boundary between filled and empty states at \(T=0\).
+  - The Fermi energy \(E_F\) marks the transition between filled and empty states at \(T = 0\).
-  - Degeneracy pressure is independent of temperature.
+  - Degeneracy pressure arises from the exclusion principle and increases with density, not temperature.
-  - FDS reduces to MBS in the high-temperature or low-density limit.
+  - The occupation probability \(F(E)\) softens with temperature but remains sharply defined near \(E_F\).
+  - The Fermi sphere in momentum space contains all filled states up to \(p_F\); its radius determines \(E_F\) and the strength of degeneracy pressure.
+  - FDS converges to MBS in the high-temperature or low-density limit.
 ===== Inquiries =====
   - Derive the expression for \(p_F\) from the number density \(n_e\).
-  - What happens to the Fermi distribution as \(T \to 0\)?
+  - Sketch \(F(E)\) at \(T = 0\) and at a finite temperature, labeling \(E_F\).
-  - Why does degeneracy pressure persist even when temperature is zero?
+  - Why does degeneracy pressure persist even when \(T = 0\)?
-  - How does Fermi–Dirac statistics differ from Maxwell–Boltzmann statistics?
+  - How does the Fermi–Dirac distribution differ from the Maxwell–Boltzmann distribution?
+  - Explain the physical significance of the Fermi sphere and its radius \(p_F\).