Fermi gas

As a Fermi gas (after Enrico Fermi, who first presented it in 1926 ) is known in quantum physics, a system of identical particles of type fermion present in such great numbers that you have to be limited to the System Description on statistical statements. In contrast to the gas in classical physics applies the quantum theoretical exclusion here.

The ideal Fermi gas is a model concept for this purpose, in which one the mutual interaction of the particles completely neglected, analogous to the ideal gas. This represents a strong simplification efforts, however, allowed in many practically important cases, physically correct predictions, eg for

• The electron gas, which provides in metallic solids and semiconductors for the electrical conductivity
• Protons and neutrons in the nucleus
• Neutrons in neutron stars
• Liquid helium-3.

• 2.1 Simplified derivation

Ground state ( vanishing temperature)

As can occupy because of the exclusion principle only a few particles, the ( single-particle ) level with the lowest energy ( as a set) must occupy most of the particles have higher levels in the lowest energy state of the whole gas. The energy of the highest occupied level is called the Fermi energy. It depends on the particle density (number per volume ):

It is

• The Planck ( divided by ) the quantum of action
• The particle mass.

The formula is valid for particles with spin, such as electrons and is due to the quantum statistics.

In a spatial density of particles per cm3 (about as conduction electrons in the metal ) gives the Fermi energy to a few electron volts. This is of the same order of magnitude as the energy of atomic excitations and significantly affects the macroscopic behavior of the gas out. One then speaks of a degenerate Fermi gas. The Fermi energy makes its salient characteristic that has far-reaching consequences for the physical properties of the ( condensed ) matter.

Only in extremely dilute Fermi gas the Fermi energy is negligible. It then behaves " non-degenerate ", ie as a normal (classic ) diluted gas.

Simplified derivation

When a gas of particles in a spatial volume ( with zero potential energy ) is in the ground state, then much of the bottom of states with different kinetic energy reception, until all of the particles are located. The highest energy is thus achieved, which is called the Fermi momentum. In the three-dimensional momentum space then come all particle momenta between and in front, in all directions. They form a ball ( Fermi sphere ) with radius and volume. Had the particle mass points, they would fill the volume in her six - dimensional phase space. For particles with spin is multiplied by the spin multiplicity. Since each ( linearly independent ) state of a cell the size claimed in the phase space, different states that can accommodate each one of the particles result ( occupation number 1):

By converting to and inserting follows the above formula.

Excited state (finite temperature)

If an ideal Fermi gas in which in reality is not accessible, ie hypothetical temperature T = 0 K (→ Third Law of Thermodynamics) supplied energy, particles must pass above from levels below the Fermi energy in levels. In thermal equilibrium, a profile of the occupation numbers of the levels is emerging, which drops steadily from one to zero. This profile which is of great importance in different physical areas, is the Fermi -Dirac distribution, or the Fermi distribution. The mean occupation number of a state with the energy is:

This is

• The Fermi level or chemical potential
• Temperature and
• Is the Boltzmann constant.

The Fermi distribution can be derived in the framework of statistical physics with the help of large canonical ensemble.

Simplified derivation

A simple derivation of having recourse to the classical Boltzmann statistics, the principle of detailed balance and the exclusion principle follows here:

Consider the equilibrium state of a temperature T in the Fermi gas in thermal contact with a classical gas. A fermion with energy can then by a classical system of particles absorb energy and move to a state with energy. Because of the conservation of energy, the classical particles changes to its state in the opposite direction from where. The occupation numbers are respectively for the two states of the fermion, respectively, for the two states of the classical particle. Thus, these processes do not change the equilibrium distribution, they have forward and reverse occur with equal frequency in total. The frequency (or entire transition rate ) is determined by the product of the transition probability as it applies to individual particles when no other particles would be there, with statistical factors that take into account the presence of other particles:

In words: The total number of transitions of a fermion by (left-hand side of the equation ) to be proportional to the number of fermions in state 1, the number of Reaktionspartnerteilchen in state 2 ', and - so that the exclusion principle is taken into account - the number of free places for in state 2 the fermion analog for the reverse reaction (right hand side of the equation ). Since according to the principle of detailed balance for return jump has the same value ( ), the statistical factors for themselves are the same. Now applies to the classical particles the Boltzmann factor

By inserting this relation, and using the above equation, the following:

This size has therefore for both states of the fermion same value. Since the choice of these states was free, this equality holds for all possible states, thus represents a constant for all single-particle states around the Fermi size we parameterize with:

Solving for n follows:

The derivation of these parameters thus proves to be the Fermi level.