Ideal Fermigas

As a Fermi gas (after Enrico Fermi , which it introduced in 1926 for the first time) is called in quantum physics a system of identical particles of the type fermion present in such great numbers that you look for system description statistical statements must be limited. In contrast to gas in classical physics , the quantum-theoretical exclusion principle applies here .

The ideal Fermigas is a model for this, in which one completely neglects the mutual interaction of the particles, analogous to the ideal gas in classical physics . This represents a great simplification, but simplifies the formulas in such a way that physically correct predictions can be made in many practically important cases, e.g. B. for

• the electron gas , which in metallic solids and semi-conductors for the electrical conductivity ensures
• Protons and neutrons in the atomic nucleus
• Neutrons in neutron stars
• liquid helium-3 .

Ground state (vanishing temperature)

Since only a few particles can occupy the (single-particle) level with the lowest possible energy (than set) because of the exclusion principle, most of the particles must occupy higher levels in the lowest possible energy state of the entire gas. The energy of the highest occupied level is called the Fermi energy . It depends on the particle density (number per volume): ${\displaystyle E=0}$ ${\displaystyle E_{\mathrm {F} }}$ ${\displaystyle \rho }$

${\displaystyle E_{\mathrm {F} }={\frac {\hbar ^{2}}{2m}}\;(3\pi ^{2}\rho )^{\frac {2}{3}}\approx {\frac {\rho ^{\frac {2}{3}}}{m}}h^{2}\cdot 0{,}121215345}$

In it is

• ${\displaystyle \hbar }$the Planck quantum of action ( divided by) ${\displaystyle 2\pi }$
• ${\displaystyle m}$ the particle mass.

The formula applies to particles with spin such as B. electrons and is justified in quantum statistics . ${\displaystyle {\tfrac {1}{2}}}$

With a spatial density of 10 ²² particles per cm ³ (roughly like conduction electrons in metal), the Fermi energy results in a few electron volts . This is in the same order of magnitude as the energy of atomic excitations and has a clear effect on the macroscopic behavior of the gas. One then speaks of a degenerate Fermigas . The Fermi energy is its most prominent characteristic, which has far-reaching consequences for the physical properties of ( condensed ) matter.

The Fermi energy can only be neglected in extremely dilute Fermi gas. It then behaves “not degenerate”, i.e. H. like a normal (classic) diluted gas.

Simplified derivation

If a gas made of particles in a spatial volume (with zero potential energy ) adopts the ground state, so many states with different kinetic energies are occupied from below until all particles are accommodated. The highest energy achieved in this way is what is called the Fermi impulse . In the three-dimensional momentum space then all particle momenta are between and in front, in all directions. They form a sphere with a radius and volume or a Fermi sphere with a radius and volume . If the particles were mass points , they would fill the volume in their 6-dimensional phase space . For particles with spin , multiply by the spin multiplicity. Since each (linearly independent) state in phase space demands a cell of the same size , there are different states that can each accommodate one of the particles ( occupation number 1 ): ${\displaystyle \,N}$ ${\displaystyle \,V}$ ${\displaystyle \,E_{\mathrm {kin} }{\mathord {=}}{\tfrac {p^{2}}{2m}}\geq 0}$ ${\displaystyle \,E_{\mathrm {F} }{\mathord {=}}{\tfrac {{p_{\mathrm {F} }}^{2}}{2m}}}$ ${\displaystyle \,p_{\mathrm {F} }}$ ${\displaystyle \,p{\mathord {=}}0}$ ${\displaystyle \,p{\mathord {=}}p_{\mathrm {F} }}$ ${\displaystyle \,p_{\mathrm {F} }}$ ${\displaystyle V_{p}{\mathord {=}}{\tfrac {4\pi }{3}}{p_{\mathrm {F} }}^{3}}$ ${\displaystyle \,k_{\mathrm {F} }={\frac {p_{\mathrm {F} }}{\hbar }}}$ ${\displaystyle V_{\mathrm {F} }{\mathord {=}}{\tfrac {4\pi }{3}}{k_{\mathrm {F} }}^{3}}$ ${\displaystyle \,\Omega =V\cdot V_{p}}$ ${\displaystyle \,s}$ ${\displaystyle {\mathord {(}}2s+1)}$ ${\displaystyle (2\pi \,\hbar )^{3}}$ ${\displaystyle \,(2s+1)\Omega /(2\pi \hbar )^{3}}$ ${\displaystyle \,N}$

${\displaystyle N={\frac {(2s+1)\Omega }{(2\pi \hbar )^{3}}}={\frac {(2s+1)V\cdot V_{p}}{(2\pi \hbar )^{3}}}={\frac {(2s+1)V}{(2\pi \hbar )^{3}}}{\frac {4\pi }{3}}{p_{\mathrm {F} }}^{3}}$

By converting to and substituting for , the above formula follows. ${\displaystyle \,E_{\mathrm {F} }{\mathord {=}}{\tfrac {{p_{\mathrm {F} }}^{2}}{2m}}}$ ${\displaystyle \,s={\tfrac {1}{2}}}$

Excited state (finite temperature)

If an ideal Fermi gas is supplied with energy at the hypothetical temperature T = 0 K (→ Third Law of Thermodynamics ), which in reality cannot be achieved, particles must move from levels below the Fermi energy to levels above. In thermal equilibrium , the occupation numbers develop for the levels , which continuously decreases from one to zero. This course, which is of great importance in various physical fields, is called the Fermi distribution or Fermi-Dirac distribution. The mean occupation number of a state with the energy is: ${\displaystyle \langle n_{i}\rangle }$ ${\displaystyle |i\rangle }$ ${\displaystyle E_{i}}$

${\displaystyle \langle n_{i}\rangle ={\frac {1}{e^{\frac {E_{i}-\mu }{k_{\mathrm {B} }T}}+1}}}$

Here is

• ${\displaystyle \mu }$the Fermi level or chemical potential
• ${\displaystyle T}$ the temperature and
• ${\displaystyle k_{\mathrm {B} }}$the Boltzmann constant .

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

Simplified derivation

A simple derivation using the classic Boltzmann statistics , the principle of detailed equilibrium and the principle of exclusion follows:

Let us consider the equilibrium state of a Fermi gas at temperature T in thermal contact with a classical gas. A fermion with energy can then take up energy from a particle of the classical system and change into a state with energy . Because of the conservation of energy, the classical particle changes its state in the opposite sense from to , where . The occupation numbers are or for the two states of the fermion, or for the two states of the classical particle. So that these processes do not change the equilibrium distribution, they must occur forwards and backwards with the same frequency overall. The frequency (or total transition rate) is determined from the product of the transition probability as it applies to individual particles if there were no other particles, with statistical factors that take into account the presence of the other particles: ${\displaystyle E_{1}}$ ${\displaystyle E_{2}}$ ${\displaystyle E_{2}'}$ ${\displaystyle E_{1}'}$ ${\displaystyle E_{2}'-E_{1}'=E_{2}-E_{1}}$ ${\displaystyle n_{1}}$ ${\displaystyle n_{2}}$ ${\displaystyle n_{1}'}$ ${\displaystyle n_{2}'}$ ${\displaystyle W}$

${\displaystyle n_{1}\cdot n_{2}'\cdot (1-n_{2})\cdot W_{1\rightarrow 2}=n_{2}\cdot n_{1}'\cdot (1-n_{1})\cdot W_{2\rightarrow 1}}$

In words: The total number of transitions of a fermion from to (left side of the equation) is proportional to the number of fermions in state 1, to the number of reaction partner particles in state 2 ', and - so that the principle of exclusion is taken into account - to the number of free places for the fermion in state 2. Analogous for the reverse reaction (right side of the equation). Since, according to the principle of detailed equilibrium, return and return have the same value ( ), the statistical factors are also the same. Now the Boltzmann factor applies to the classical particles ${\displaystyle E_{1}}$ ${\displaystyle E_{2}}$ ${\displaystyle W}$ ${\displaystyle W_{2\rightarrow 1}{\mathord {=}}W_{1\rightarrow 2}}$

${\displaystyle {\frac {n_{2}'}{n_{1}'}}=e^{-{\tfrac {E_{2}'-E_{1}'}{k_{\mathrm {B} }T}}}.}$

Substituting this relationship and using the above equation, we get: ${\displaystyle E_{2}'-E_{1}'=E_{2}-E_{1}}$

${\displaystyle {\frac {n_{1}}{1-n_{1}}}\;e^{\tfrac {E_{1}}{k_{\mathrm {B} }T}}={\frac {n_{2}}{1-n_{2}}}\;e^{\tfrac {E_{2}}{k_{\mathrm {B} }T}}.}$

This quantity has the same value for both states of the fermion. Since the choice of these states was free, this equality applies to all possible states, i.e. it represents a constant value for all single-particle states in the whole of Fermigas, which we parameterize with : ${\displaystyle e^{\tfrac {\mu }{k_{\mathrm {B} }T}}}$

${\displaystyle {\frac {n}{1-n}}\;e^{\tfrac {E}{k_{\mathrm {B} }T}}=e^{\tfrac {\mu }{k_{\mathrm {B} }T}}.}$

Solved for n it follows:

${\displaystyle n={\frac {1}{e^{\tfrac {E-\mu }{k_{\mathrm {B} }T}}+1}}.}$

The parameter of this derivation thus turns out to be the Fermi level. ${\displaystyle \mu }$

See also

• Fermion condensate
• Ideal Bosegas

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1. ↑ Enrico Fermi: On the quantization of the monatomic ideal gas , Zeitschrift für Physik Vol. 36, 1926, pp. 902–912 DOI: 10.1007 / BF01400221 .
2. ↑ Robert Eisberg; Robert Resnick: Quantum physics of atoms, molecules, solids, nuclei and particles , Verlag Wiley, 1974 (NY), ISBN 0-471-23464-8 .