### FERMI-DIRAC STATISTICS

Fermions obey Fermi-Dirac Statistics whereby no two can occupy the same quantum numbers, being Pauli's exclusion principle. This means that no two fermions in close proximity can have exactly the same energy levels, they must have slightly differing energy levels. When a great many fermions are forced into close proximity, as are the electrons within white dwarf stars, then the electrons are forced into all having slightly differing and ever higher energy levels, leading to a continuum of energy levels, called an energy band. With each electron added, it is forced to adopt an infintesimally higher energy level than the rest. The electrons are said to be 'degenerate'. This gives rise to degeneracy pressure. It is the degeneracy pressure of the electrons that stops the white dwarf collapsing to form a neutron star (where the electrons are forced into the protons, turning them into neutrons by a process of forced neutron capture, which is similar to the reverse of beta decay). When the energy (equivalent to velocity) of any of the electrons needs to exceed that of the speed of light in order to accomodate the next added electron, the degeneracy pressure can no longer support the weight of the white dwarf, and it collapses, becoming a neutron star. See Supernova

Helium-3, with an odd number of nucleons, must have half-integral spin, and hence is a Fermi gas, obeying Fermi-Dirac statistics, where the Pauli exclusion principle applies and no two atoms can occupy the same energy level. The boiling point of helium-3 is 3.2 Kelvin at atmospheric pressure. Below the boiling point, helium-3 like helium-4, remains liquid right down to absolute zero temperatures at normal pressures because the binding energy between atoms is so weak that it is easily overcome by zero point motions. [The zero point energy is the energy that a system has at zero temperature, it's ground state energy. The energy is forbidden from being zero by Heisenbergs' Uncertainty Principle and takes on the value of a half a one-half quantum, hv/2]. Only under a pressure greater than 29 atmospheres and a temperature below 0.5 Kelvin can helium-3 be solidified. Strangely, however, solid helium-3 is more disordered than liquid helium-3 because in the solid state, its' spins are totally disordered. Thus by cooling liquid helium-3 to 0.3 Kelvin and then pressurising it into a solid, latent heat is consumed which reduces the temperature to as low as 2 milliKelvin, and this provides a good method of cryogenic refrigeration.

[Helium-3 can also exist in a superfluid form; but this is a Bose Einstein Condensate rather than a Fermi liquid].

Heavy fermion compounds are compounds that contain such rare earth elements as Cerium or Ytterbium, or actinide elements such as Uranium. Examples are CeRu2Si2, CeCu6, YbCuAl and URu2Si2, UPt3. The f-electrons are nearly localised in atomic-like configurations. The motion of these itinerant electrons is highly correlated due to the strong electrostatic interaction between electrons in the same ionic shell. This results in large effective quasi-particle masses which can be up to 100 times greater than for normal rare earth and actinide compounds. At low temperatures, some of the materials are magnetically ordered, others strongly paramagnetic, and some display superconductivity at high pressures (URu2Si2, UPt3).