is given by with the Planck constant, the electron mass and ω the characteristic frequency of the oscillations. The latter can be estimated by considering the electrostatic potential energy for an electron displaced by r from its lattice point. Say that the Wigner–Seitz cell associated to the lattice point is approximately a sphere of radius a/2. The uniform, neutralizing background then gives rise to a smeared positive charge of density with the electron charge. The electric potential felt by the displaced electron as a result of this is given by with ε0 the vacuum permittivity. Comparing to the energy of a harmonic oscillator, one can read off or, combining this with the result from the quantum harmonic oscillator for the root-mean-square displacement The Lindemann criterion than gives us the estimate that rs > 40 is required to give a stable Wigner crystal. Quantum Monte Carlo simulations indicate that the uniform electron gas actually crystallizes at rs = 106 in 3D and rs = 31 in 2D. For classical systems at elevated temperatures one uses the average interparticle interaction in units of the temperature: .. The Wigner transition occurs at G = 170 in 3D and G = 125 in 2D. It is believed that ions, such as those of iron, form a Wigner crystal in the interiors of white dwarf stars. Experimental realisation In practice, it is difficult to experimentally realize a Wigner crystal because quantum mechanical fluctuations overpower the Coulomb repulsion and quickly cause disorder. Low electron density is needed. One notable example occurs in quantum dots with low electron densities or high magnetic fields where electrons will spontaneously localize in some situations, forming a so-called rotating "Wigner molecule", a crystalline-like state adapted to the finite size of the quantum dot. Wigner crystallization in a two-dimensional electron gas under high magnetic fields was predicted (and was observed experimentally) to occur for small filling factors (less than ) of the lowest Landau level. For larger fractional fillings, the Wigner crystal was thought to be unstable relative to the fractional quantum Hall effect (FQHE) liquid states. A Wigner crystal was observed in the immediate neighborhood of the large fractional filling , and led to a new understanding (based on the pinning of a rotating Wigner molecule) for the interplay between quantum-liquid and pinned-solid phases in the lowest Landau level. Another experimental realisation of the Wigner crystal occurred in single-electron transistors with