Fermi gas
Fermions are elementary or composite particles with half-integer spin, thus follow Fermi–Dirac statistics.
The equivalent model for integer spin particles is called the Bose gas (an ensemble of non-interacting bosons).
[4] The total energy of the Fermi gas at absolute zero is larger than the sum of the single-particle ground states because the Pauli principle implies a sort of interaction or pressure that keeps fermions separated and moving.
Only when a star is sufficiently massive to overcome the degeneracy pressure can it collapse into a singularity.
It is possible to define a Fermi temperature below which the gas can be considered degenerate (its pressure derives almost exclusively from the Pauli principle).
This temperature depends on the mass of the fermions and the density of energy states.
Since interactions are neglected due to screening effect, the problem of treating the equilibrium properties and dynamics of an ideal Fermi gas reduces to the study of the behaviour of single independent particles.
In these systems the Fermi temperature is generally many thousands of kelvins, so in human applications the electron gas can be considered degenerate.
It is a standard model-system in quantum mechanics for which the solution for a single particle is well known.
Since the potential inside the box is uniform, this model is referred to as 1D uniform gas,[5] even though the actual number density profile of the gas can have nodes and anti-nodes when the total number of particles is small.
Without loss of generality, the zero-point energy is chosen to be zero, with the following result:
A three-dimensional infinite square well, (i.e. a cubical box that has a side length L) has the potential energy
The states are now labelled by three quantum numbers nx, ny, and nz.
When the box contains N non-interacting fermions of spin-1/2, it is interesting to calculate the energy in the thermodynamic limit, where N is so large that the quantum numbers nx, ny, nz can be treated as continuous variables.
For the 3D uniform Fermi gas, with fermions of spin-1/2, the number of particles as a function of the energy
In this sense, systems composed of fermions are also referred as degenerate matter.
Standard stars avoid collapse by balancing thermal pressure (plasma and radiation) against gravitational forces.
Using the Fermi gas as a model, it is possible to calculate the Chandrasekhar limit, i.e. the maximum mass any star may acquire (without significant thermally generated pressure) before collapsing into a black hole or a neutron star.
For the case of metals, the electron degeneracy pressure contributes to the compressibility or bulk modulus of the material.
Stars known as white dwarfs have mass comparable to the Sun, but have about a hundredth of its radius.
This density must be divided by two, because the Fermi energy only applies to fermions of the same type.
The presence of neutrons does not affect the Fermi energy of the protons in the nucleus, and vice versa.
The density of states (or more accurately, the degree of degeneracy) for a given spin species is:
Related to the Fermi energy, a few useful quantities also occur often in modern literature.
The grand potential is related to the number of particles at finite temperature in the following way
Many systems of interest have a total density of states with the power-law form:
The results of preceding sections generalize to d dimensions, giving a power law with: For such a power-law density of states, the grand potential integral evaluates exactly to:[12]
From this grand potential and its derivatives, all thermodynamic quantities of interest can be recovered.
For particles with energies close to their respective rest mass, the equations of special relativity are applicable.
The relativistic Fermi gas model is also used for the description of massive white dwarfs which are close to the Chandrasekhar limit.
