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un:fermi-dirac-statistics [2025/10/26 23:21] – asadun:fermi-dirac-statistics [2025/10/27 12:10] (current) – [Fermi–Dirac statistics] asad
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 ====== Fermi–Dirac statistics ====== ====== Fermi–Dirac statistics ======
  
-The particles of the [[un: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 The magnitude of the angular momentum vector is
  
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 $$ $$
  
-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     - **Fermions:** electrons, protons, neutrons, neutrinos, and any nucleus with an odd number of nucleons  
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 corresponds to one quantum state. Because electrons can have two spin orientations (up and down), at most **two electrons** can occupy a single phase-space cell. corresponds to one quantum state. Because electrons can have two spin orientations (up and down), at most **two electrons** can occupy a single phase-space cell.
  
-In the full six-dimensional phase space,+In full six-dimensional phase space,
  
 $$ $$
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 and each quantum 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 ===== ===== 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**. 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 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 under degeneracy: in the degenerate case, all states below a limiting energy are occupied.}} {{: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 filling of quantum states 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 exclusion itself: the particles cannot all occupy the same low-energy 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 distribution =====
  
-The **Fermi–Dirac occupation probability** for a particle of energy \(E\) at temperature \(T\) is given by+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 absolute zero (\(T=0\)).   +where \(E_F\) is the **Fermi energy**, the highest occupied energy level at absolute zero (\(T = 0\)).   
-At \(T=0\), all states with \(E < E_F\) are completely filled and those with \(E > E_F\) are empty, resulting in a step-like (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.
  
 {{:courses:ast301:fd-probability.webp?nolink&700|Fermi–Dirac occupation probability versus energy for various temperatures.}} {{: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.   The figure above shows how \(F(E)\) changes with temperature.  
-At absolute zero, the probability drops sharply from 1 to 0 at \(E = E_F\).   +At \(T = 0\), the probability falls abruptly from 1 to 0 at \(E = E_F\).   
-As the temperature increases, the sharp step softens — a few particles gain enough energy to occupy states with \(E > E_F\), while some lower-energy states become partially empty.   +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.   
-However, even for moderate temperatures (\(kT \ll E_F\)), the curve remains steep around \(E_F\), meaning that most fermions stay below the Fermi energy.+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 The general **phase-space distribution function** is
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 $$ $$
  
-where the factor \(2/h^3\) accounts for spin degeneracy — two allowed spin orientations 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 Fermi–Dirac statistics reduce to Maxwell–Boltzmann statistics. 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 ===== ===== Degeneracy and Fermi momentum =====
  
-In three dimensions, all occupied states at \(T=0\) form a **Fermi sphere** in momentum space of radius \(p_F\), called the **Fermi momentum**.   +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 inside this sphere is+The total number of electrons within this sphere is
  
 $$ $$
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 where \(V_x\) is the physical volume.   where \(V_x\) is the physical volume.  
-Hence the **number density** of electrons is+The **number density** of electrons is then
  
 $$ $$
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 $$ $$
  
-Rearranging, the Fermi momentum is expressed in terms of the electron density as+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 occupied. The corresponding **Fermi energy** is +The corresponding **Fermi energy** is
- +
-$$ +
-E_F = \frac{p_F^2}{2m_e} \quad \text{(nonrelativistic)}, +
-$$ +
- +
-and for relativistic electrons, +
- +
-$$ +
-E_F = \sqrt{p_F^2 c^2 + m_e^2 c^4} - m_e c^2. +
-$$ +
- +
-===== Fermi energy and degeneracy pressure ===== +
- +
-At zero temperature, electrons occupy all momentum states up to the **Fermi momentum** \(p_F\): +
- +
-$$ +
-n_e = \frac{8\pi p_F^3}{3h^3} \Rightarrow +
-p_F = h\left(\frac{3n_e}{8\pi}\right)^{1/3}. +
-$$ +
- +
-The corresponding Fermi energy is+
  
 $$ $$
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 {{: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 \(T=0\).}} {{: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 \(T=0\).}}
  
-The diagram above visualizes the Fermi sphere in momentum space.   +The **Fermi sphere** above visualizes all occupied quantum states in momentum space.   
-Each point inside the sphere represents an occupied quantum state of momentum \(\mathbf{p}\).   +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, while those outside remain empty.   +At \(T = 0\), all states with \(|\mathbf{p}| < p_F\) are filled, and those outside remain empty.   
-The radius of this sphere, \(p_F\), depends only on the density of electrons and determines the Fermi energy, which separates occupied and unoccupied states even in the absence of thermal motion. +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 \(T = 0\).
-===== Physical meaning ===== +
- +
-In a degenerate fermion gas: +
- +
-  - Pressure arises from the quantum mechanical restriction on identical particles.   +
-  - The pressure remains nonzero even at \(T=0\).   +
-  - It depends only on the **density**, not on the **temperature**.   +
-  - Electrons dominate the pressure, while ions provide most of the mass. +
- +
-Degeneracy pressure is what supports compact objects such as **white dwarfs**, while partially degenerate matter occurs in **brown dwarfs** and **planetary cores**.+
  
 ===== Insights ===== ===== Insights =====
-  - Fermi–Dirac statistics describe systems where the Pauli exclusion principle is active.   +  - 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 \(T = 0\).   
-  - The occupation probability \(F(E)\) becomes smoother with temperature but remains steep near \(E_F\).   +  - Degeneracy pressure arises from the exclusion principle and increases with density, not temperature.   
-  - Degeneracy pressure originates from the exclusion principle, not from thermal motion.  +  - 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.   - FDS converges to MBS in the high-temperature or low-density limit.
  
 ===== Inquiries ===== ===== Inquiries =====
   - Derive the expression for \(p_F\) from the number density \(n_e\).     - Derive the expression for \(p_F\) from the number density \(n_e\).  
-  - Sketch \(F(E)\) at \(T=0\) and at a finite temperature, labeling \(E_F\).  +  - Sketch \(F(E)\) at \(T = 0\) and at a finite temperature, labeling \(E_F\).  
   - Why does degeneracy pressure persist even when \(T = 0\)?     - Why does degeneracy pressure persist even when \(T = 0\)?  
   - How does the Fermi–Dirac distribution differ from the Maxwell–Boltzmann distribution?     - How does the Fermi–Dirac distribution differ from the Maxwell–Boltzmann distribution?  
   - Explain the physical significance of the Fermi sphere and its radius \(p_F\).   - Explain the physical significance of the Fermi sphere and its radius \(p_F\).
  
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