A filled Fermi sphere has total kinetic energy (the check-yourself result of Section 8.5)

PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Free electrons, energy bands and electronic structure

Condensed Matter Physics, Ch The free-electron gas: what Drude gets right and what only Fermi fixes

Problem

A filled Fermi sphere has total kinetic energy

U=\tfrac35NE_F

(the check-yourself result of Section 8.5), with

E_F=(\hbar^2/2m)(3\pi^2N/V)^{2/3}

depending on the volume

V

through

N/V=n

. The gas exerts an outward pressure

P=-(\partial U/\partial V)_N

, exactly as a classical gas does. (a) Differentiate to show that

P=\tfrac25nE_F

. (b) Evaluate this degeneracy pressure for copper, using

n=8.47\times10^{28}\,\mathrm{m^{-3}}

and

E_F=7.03\,\mathrm{eV}

, in gigapascals. (c) Differentiate once more to find the gas's own bulk modulus,

B=-V\,\mathrm{d}P/\mathrm{d}V

, in terms of

n

and

E_F

, evaluate it for copper and compare it with copper's measured bulk modulus of

140\,\mathrm{GPa}

(Chapter 7). Separately, what must balance the outward pressure

P

for copper to sit in equilibrium with no pressure applied from outside?

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