Degenerate Gas

a gas whose properties differ considerably from those of a classical ideal gas as a consequence of the quantum mechanical effect of identical particles on each other. This mutual effect is due to the indistinguishability (identity) of identical particles in quantum mechanics rather than to force interactions, which are absent in an ideal gas. As a result of such an effect, the filling of possible energy levels by particles, even in an ideal gas, depends on the presence of other particles at a given level. Consequently the heat capacity and pressure of such a gas depend differently on temperature from the case of a classical ideal gas; entropy, free energy, and so on are expressed differently.

Degeneration of a gas sets in when its temperature is lowered to a certain value, called the degeneration temperature. Complete degeneration corresponds to a temperature of absolute zero.

The effect of particle identity becomes more pronounced as the average distance r between particles compared with the de Broglie wavelength of the particles, λ = h/mv, decreases (m is particle mass, v is particle velocity, and h is Planck’s constant). This is explained by the fact that classical mechanics is applicable to the motion of gas particles only when r » λ. Since the velocity of the gas particles is related to temperature (the greater the velocity, the higher the temperature), the degeneration temperature (which determines the limit of validity of classical theory) is higher the smaller the mass of the gas particles and the greater their density (that is, the smaller the average distance between particles). Consequently, the degeneration temperature is especially high (on the order of 10,000° K) for an electron gas in metals: the electron’s mass is very small (~10^-27 g), and the density of electrons in the metal is very high (10²² electrons per cu cm). In metals, an electron gas is degenerate at all temperatures for which the metal remains in the solid state.

For ordinary atomic and molecular gases, the degeneration temperature is near absolute zero; therefore such a gas virtually always behaves classically. (At such low temperatures all substances—except helium, which is a quantum liquid no matter how close the temperature to absolute zero—are solids.)

Since the nature of the nonforce effect of identical particles on each other differs for particles with integer spin (bosons) and half-integer spin (fermions), the behavior of a gas of fermions (Fermi gas) and of bosons (Bose gas) will also differ during degeneration.

In a Fermi gas (such as an electron gas in a metal) upon complete degeneration (at T = 0° K), all the lower energy levels down to some maximum level, called the Fermi level, are filled, and all the subsequent levels remain empty. A temperature increase only slightly changes the energy distribution of the electrons in a metal: a small fraction of the electrons at levels near the Fermi level pass into empty levels with higher energy, thus freeing the levels below the Fermi level from which the transition was performed.

Upon degeneration of a boson gas of particles with non-zero mass (atoms and molecules can be such bosons), a certain fraction of the system’s particles must pass into a state with zero momentum; this is called Bose-Einstein condensation. The nearer the temperature to absolute zero, the more particles must be in this state. However, at very low temperatures, systems of such particles pass into the solid or liquid (in the case of helium) states, in which the force interactions between particles are considerable and to which the ideal-gas approximation is not applicable. Bose-Einstein condensation in liquid helium, which can be considered a nonideal gas of so-called quasi-particles, leads to the appearance of superfluidity.

For gases of zero-mass bosons, among them photons (with spin 1), the degeneration temperature is infinity; consequently, a photon gas is always degenerate, and classical statistics does not apply to it under any conditions. A photon gas is the only degenerate, ideal Bose gas of stable particles. However, Bose-Einstein condensation does not occur in it, since there are no photons of zero momentum (photons always move with the velocity of light). A photon gas ceases to exist at absolute zero.