Abstract
Wigner showed that a sufficiently thin electron gas will condense into a crystal of localized electrons. Here, we show, using a model based on cubic charge distributions that gives exact results, that the Coulomb repulsion energy of localized charge distributions is lower than that of delocalized distributions in spite of the fact that the total overall charge distribution is the same. Assuming a simple cubic geometry, we obtain an explicit result for the energy reduction. This reduction results from the exclusion of self-interactions of the electrons. The corresponding results for electron pairs are also discussed.
Highlights
Energy for Localized versusThe Coulomb interaction may appear mathematically trivial, especially for point particles, but as anyone who studies ionic crystals quickly learns, this is not necessarily the case
The Coulomb interaction energy of two charged particles with charges e1 and e2 is given by e1 e2
We study the implications of this interaction energy for conduction electrons in a metal by means of cubic charge distributions
Summary
The Coulomb interaction may appear mathematically trivial, especially for point particles, but as anyone who studies ionic crystals quickly learns, this is not necessarily the case. Calculation of the Coulomb binding energy per ion in an ionic crystal, the Madelung energy [1,2], is quite intricate When it comes to electrons, it is intuitively obvious that since they repel each other they should attempt to be as far apart as possible, as long as they can compensate any positive charge density present. The energy is significantly lower for the localized case Our model compares the Coulomb energy of N 3 electrons delocalized in a cube of side-length L to the energy when these electrons are localized with one in each of the N 3 subcubes of side-length L/N. In our model the Coulomb repulsion energies can be calculated exactly, after some preliminary results. A fixed electron density is assumed when comparing results for single electrons to electron pair results
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