Wigner Solid Research Articles

Electrostatic structural transitions in a Yukawa‐Wigner solid

A derivation is supplied for a functional relation between the Fuchs energy e and the Madelung energy S for a Yukawa‐Wigner solid (YWS) in which the usual uniform background of a Wigner solid (WS) is replaced by a periodic array of Yukawa charge distributions with variable ‘‘ripple’’ parameter λ allowing the WS and the empty lattice in the limits of small λ and large λ, respectively. It is the zeros of Δe, and not of ΔS, that are relevant for structural transitions between two lattices. It is knwon that 2eWS=SWS, and Medeiros and Mokross incorrectly assumed 2e=S for the YWS. Here it is first shown by elementary means that the relation between e and S varies with λ, and then the functional relation is supplied for all λ. When applied to the bcc‐fcc system, it is found that Δe has two zeros whereas ΔS has one not equal to either of those of Δe. Starting with small λ, the sequence of lowest energy structures is bcc, fcc, and bcc if these are the only two allowed to compete. The equations for the sc case have...

Response to "Comment on the average potential of a Wigner solid"

I give a counterexample to Kleinman's claim concerning certain two-dimensional of charge, I categorize according to equivalence six definitions (three from the literature) of average potential $〈\ensuremath{\Phi}〉$ and illustrate the results with the spherical (SWS) and regular (WS) Wigner solids, and I treat finite clusters and slabs of three-dimensional cells. Results for the SWS approach those for the WS in the limit of infinite SWS ball radius provided $〈\ensuremath{\Phi}〉$ is treated consistently in the two.

Electrostatic structural transitions in a Gaussian Wigner solid

A derivation is supplied for a functional relation between the Fuchs energy ε and the Madelung energy S for a Gaussian Wigner solid (GWS) in which the usual uniform background of a Wigner solid (WS) is replaced by a periodic array of Gaussians with variable ’’ripple’’ parameter p allowing the WS and the empty lattice in the limits of small p and large p, respectively. It is the zeros of Δε, and not of ΔS, that are relevant for structural transitions between two lattices. Much can be determined about the transitions with minimal computations by utilizing a modest amount of information about order relations on certain theta functions. With increasing p, the sequence of lowest-energy structures restricted to the cubics is bcc, fcc, and sc. A later report will treat the Yukawa WS (YWS) in which the Gaussians are replaced with Yukawa distributions. The functional relation is again derivable, and it is not given by Medeiros and Mokross’s recent assumption that 2εYWS = SYWS.

Comments on the electrostatic energy of a Wigner solid

A recent contention by Hall that Fuchs's calculation of the electrostatic energy of an infinite Wigner solid contains a fundamental error is not true. Fuchs calculated the average electrostatic energy per electron ${\ensuremath{\Phi}}_{i}$, while Hall derived this quantity from the energy of the interaction of one electron with the rest of the lattice ${\ensuremath{\Phi}}_{\mathrm{el}}$ by use of the relation ${\ensuremath{\Phi}}_{i}=(\frac{1}{2}){\ensuremath{\Phi}}_{\mathrm{el}}$; however, this relation is not valid in the case studied by Hall. Preliminary results for finite Wigner solids are discussed.