Fermi Energy of a 1D Electron Gas

Calculate the Fermi energy for a one-dimensional metal with one free electron per atom and an atomic spacing of 2.5 A at T = 0.

SOLUTION

For a one-dimensional gas the number of quantum states in the interval dp is

where g = 2s + 1 = 2 and L is the “length” of the metal. The total number of electrons N (which in this case is equal to the number of atoms) is

(1)

Therefore,

where d is the atomic spacing. The Fermi energy

(2)

where m is the electron mass.

(3)