= Solution
In the <free electron model>, <Conduction electron>[conduction electrons] occupy independent <Plane wave>[plane-wave] states with <energy> $\varepsilon_{\mathbf k}=\hbar^2k^2/(2m)$ in a uniform mean background. <Periodic potential>[Crystal-periodic] band structure and explicit <electron>–<electron> correlations are omitted. <Electron>[Electrons] still obey the <Pauli exclusion principle> and <Fermi-Dirac statistics>[Fermi–Dirac statistics]; free does not mean distinguishable classical particles. A scattering time can be added phenomenologically to describe transport.
The <Fermi energy> is the highest occupied single-particle <energy> at zero <temperature>, measured here from the bottom of the <Parabolic dispersion>[parabolic band]. The <Fermi temperature> is the corresponding scale $T_F=E_F/k_B$, not the actual <Temperature>[thermal temperature] of the sample. Metallic <electron> densities are large, so filling all distinct states up to the required <momentum> gives <Fermi energy>[Fermi energies] of several electronvolts, equivalent to tens of thousands of kelvin or more. At ordinary <Temperature>[temperatures], only a narrow region around the <Fermi surface> has appreciably changing occupations.
A room-temperature metal and its lattice are nevertheless at room <temperature>, not at $T_F$. High occupied <Kinetic energy>[kinetic energies] are present even in the zero-temperature ground state and cannot all be liberated as <heat> without violating the allowed occupancy structure. A metal below skin <temperature> often feels cold because it transfers <heat> rapidly out of the skin; its <thermal conductivity> and <thermal effusivity> matter, not the value of its <Fermi temperature>. At the same actual <temperature>, a poorly conducting material can therefore feel warmer while neither material is thermodynamically hotter.
Back to article page
