Question

Consider a dense electron gas in a given metal. What is the characteristic temperature? What does this mean physically?

Answer 1

In metallic atoms electrons do not belong to any particular atom or molecule, instead they are delocalized and extend throughout the system as free electrons, which means that their Coulombic interaction can be neglected. So we regard these electrons as a gas but since their density is high, their de Broglie wavelength is larger than their mean separation. This means that the classical approximation is no longer valid we need quantum statistics.
The characteristic temperature or the Fermi temperature is given by-

where N is the number of electrons and V is the volume of the metal and the other symbols have their usual meanings.

Usually the Fermi temperature is of the order of 10,000K.

The Fermi temperature can be considered as the temperature at which the thermal effects are comparble or related to the quantum effects in Fermi statistics.

The Fermi temperature leads to fermi energy-

This energy denotes the highest energy level occupied by electrons in a metal at 0K.

I hope this helps. Have a nice day.