 Solution

A Wigner crystal is an ordered electron arrangement favored when repulsion overwhelms the kinetic cost of localization. In a three-dimensional electron gas with a neutralizing background, define a mean spacing $a=(3/4\pi n{)}^{1/3}$ and effective Bohr radius $a_B^{\ast }=4\pi \epsilon {\hslash }^2/(m^{\ast }{e}^2)$. The Coulomb-to-kinetic energy ratio of an electron gas scales asThus low density favors the crystal in the degenerate quantum model. Translational symmetry is broken, and excitations include collective lattice vibrations. The melting criterion depends on dimension, screening, disorder and quantum fluctuations, so this scaling argument does not supply a universal numerical transition density.

The electron crystallization on liquid helium experiment provides an important related observation: a dilute two-dimensional electron sheet above liquid helium crystallized on cooling. It was a classical electron system, with thermal rather than Fermi motion competing with Coulomb energy; its coupling parameter is $\Gamma =E_C/(k_{B}T)$. It therefore illustrates Coulomb ordering without being a direct measurement of the zero-temperature three-dimensional quantum transition.^[ref]