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Fermi–Dirac statistics is a type of quantum statistics that applies to the physics of a system consisting of many non-interacting, identical particles that obey the Pauli exclusion principle. A result is the Fermi–Dirac distribution of particles over energy states.
In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-odd-integer spin (spin 1/2, spin 3/2, etc.) and obey the Pauli exclusion principle.
Electrons are an example of a type of particle called a fermion. Other fermions include protons and neutrons. In addition to their charge and mass, electrons have another fundamental property called spin. A particle with spin behaves as though it has some intrinsic angular momentum. This causes each electron to have a small magnetic dipole.
Consider a gas of N non-interacting fermions, e.g., electrons, whose one-particle wave-functions ϕr( r) are plane-waves. In this case, a complete set of quantum numbers r is given, for instance, by the three cartesian components of the wave vector k and the z spin projection ms of an electron: r ≡ (kx,ky,kz,ms) .
1. What are the basic steps used to derive the Fermi-Dirac distribution? 2. Where did the Fermionic properties of the electrons enter in the derivation? 3. What is the physical significance of the Fermi energy and Fermi k-vector? 4. How does the position of Fermi level with respect to band structure determine the materials electron transport ...
• Denote the energy of a particle in state r by ε r. • Denote the number of particles in state r by n r. • Label the possible quantum states of the whole gas by R. Since the particles in the gas are not interacting or are interacting weakly, we can describe the state R of the system as having n 1 particles in state r = 1, n 2 particles ...
Its partition function \(\Xi\) is given by \[\Xi=\operatorname{Tr}\left[e^{\mu \beta \hat{n}-\beta H}\right],\tag{8.43}\] where \(H\) is the many-body Hamiltonian, \(\beta=1 / k_{B} T\) and \(\mu\) is the chemical potential. The average number of particles with single particle energy \(\boldsymbol{E}_{j}\) is then given by
