ε_{k} = ℏ^{2} k^{2} / (2 m)

in a uniform mean background. Crystal-periodic band structure and explicit electron–electron correlations are omitted. Electrons still obey the Pauli exclusion principle and 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 band. The Fermi temperature is the corresponding scale

T_{F} = E_{F} / k_{B}

, not the actual thermal temperature of the sample. Metallic electron densities are large, so filling all distinct states up to the required momentum gives Fermi energies of several electronvolts, equivalent to tens of thousands of kelvin or more. At ordinary 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 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.

b

Solution

Use periodic boundaries in a box of side

L

and include two spin states for each wavevector. One orbital state occupies

(2 π / L)^{d}

v_{d}

d

, namely

2, π, 4 π /3

. Filling the ball of radius

k_{F}

gives

N = 2 (\frac{L}{2 π})^{d} v_{d} k_{F}^{d}, k_{F}^{d} = \frac{( 2 π ) ^{d}}{2 v _{d}} n .

Since

E_{F} = ℏ^{2} k_{F}^{2} / (2 m)

a_{d} = \frac{( 2 π ) ^{d}}{2 v _{d}}, a_{1} = \frac{π}{2}, a_{2} = 2 π, a_{3} = 3 π^{2} .

n

means number per length, area or volume as appropriate; it is not assigned three-dimensional units in every case.

For the same spin-summed states, the cumulative number per

d

-dimensional volume below energy

E

n (E) = 2 v_{d} (2 m E / ℏ^{2})^{d /2} / (2 π)^{d}

g (E) = (d /2) n (E) / E

. Evaluating at the filling energy,

g (E_{F}) = b_{d} \frac{n}{E _{F}}, b_{1} = \frac{1}{2}, b_{2} = 1, b_{3} = \frac{3}{2} .

In one, two and three dimensions the density of states varies respectively as

E^{- 1/2}

, a constant and

E^{1/2}

. The normalization counts both signs of momentum and both spin states exactly once. A single-spin density of states or extra valley degeneracy would require a different counting convention, not an extra unexplained factor in these constants.

n = \frac{12}{( 0.405 \times 1 0 ^{- 9} m ) ^{3}} = 1.8064 \times 1 0^{29} m^{- 3}, k_{F} = (3 π^{2} n)^{1/3} = 1.7488 \times 1 0^{10} m^{- 1} .

E_{F} = \frac{ℏ ^{2} k _{F}^{2}}{2 m _{e}} = 11.65 eV, T_{F} = 1.352 \times 1 0^{5} K .

The Drude transport relation is

ρ = m_{e} / (n e^{2} τ)

τ = \frac{m _{e}}{n e ^{2} ρ} = 7.02 \times 1 0^{- 15} s .

v_{F} = ℏ k_{F} / m_{e} = 2.025 \times 1 0^{6}

m/s:

ℓ_{F} = v_{F} τ = 1.42 \times 1 0^{- 8} m = 14.2 nm .

This is a transport relaxation time and associated transport mean free path inferred from the specified resistivity. It need not equal a single-particle quantum lifetime when scattering is strongly angle dependent. The free-electron valence count also neglects the detailed band structure of aluminium, as the stipulated model requires. A classical thermal speed is not the appropriate velocity scale for this strongly degenerate metal.