Why do the potential energy dominant the kinitic energy at low densiti

[hokhani]

In explaining the Wigner crystal, It is always said that " the potential energy dominates the kinetic energy at low densities". Why?

 

[csopi]

Let's take in one dimension for simplicity. If you have density n, the average distance between two electrons is 1/n, so the potential energy (=Coulomb potential) scales roughly as e^2/epsilon * n.

As for the kinetic energy: if you take free electrons, you have that the energy is hbar^2*k^2/(2*m). Because the density of states is proportional to 1/sqrt(energy) in 1D, you have, that the Fermi energy (that can be thought as kinetic energy in a free electron gas) scales as constant * n^2 (this can be seen if you express the number of electrons in terms of the Fermi energy).

Thus at low densities the kinetic energy becomes much smaller than the potential energy. The situation is quite the same in higher dimensions, you might just have different scaling exponents, but I haven't calculated them.