According to a prediction made by Wigner, at low temperatures and sufficiently low densities, an electron gas should undergo a phase transition to a crystalline structure (bcc). For a qualitative analysis, consider the energy of a lattice of electrons embedded in a homogeneous, positively charged background. Assume that the potential in which each electron moves can be approximated by the potential of a uniformly charged sphere of radius r0=rsa0 surrounding each electron.
Here, r0 is the mean particle separation in the Wigner crystal with electron density n, i.e., 34πr03=1/n. This leads to a model of independent electrons (Einstein approximation) in an oscillator potential
H=2mp2+2r03e2r2−2r03e2.
Determine the zero-point energy E0 of this three-dimensional harmonic oscillator and compare this with the result found in the literature:
E0=2a0e2{−rs1.792+rs3/22.638}.
By minimizing the zero-point energy, determine the mean separation of the electrons.