Functional relations between Fuchs and Madelung energies of generalized Wigner solids

Abstract

Functional relations existing between the Fuchs en«gy e and the Madelung energy S for Wigner solids with the usual uniform background replaced by periodic arrays of either Gaussian or Yukawa charge distributions are utilized in a study of electrostatic structural transitions between the cubic lattices. The Wigner' solid (WS) model is composed of a lattice of point charges Q (of either sign) with a neutralizing uniform background. Changing the background of a WS to a displaced lattice of point charges -Q yields the classical Coulomb lattice (CCL). Here

Antiferromagnetic phase of a 2‐D Wigner crystal

“…For the Wigner crystal, it is reasonable to have a nonuniform distribution of positive charge, which does not have to be uniform 19–21. Here, the nonuniform positive background is represented by a periodic Gaussian‐type array with variable parameter α′ and a Yukawa‐type distribution with variable ripple parameter λ.…”

“…The Gaussian‐type distribution is represented by 19, 20 With this background, the expression for V c ( r 1 ) is obtained as 22 where The Yukawa‐type charge distribution is given by 19, 20 where λ is the variational parameter. If we can define then the expression for V c ( r 1 ) is 23 With this H eff , we compute the Hartree–Fock ground‐state energy by extremizing the functional For the 2‐D electron system, Jonson and Srinivasan 25 interpolate the result of Singwi et al 26 at r s = 0.5 and a low‐density Einstein solid result of −1.103/ r s + O ( r )Ry to get Using Eq.…”

Structural stability analysis of Wigner crystal with Gaussian and Yukawa‐type positive background

“…In the Gaussian–Wigner electron crystal, the background is formed by centering about each lattice site, a charge distribution 5 of the Gaussian type, With this background, the expression for V c ( r 1 ) turns out to be and the expression for the electrostatic energy is derived as where In the present calculation, we need the matrix element of V c ( r 1 ) with respect to the trial Wannier functions.…”

“…In the Yukawa–Wigner electron crystal, the background is formed by centering about each lattice site, a charge distribution 5 of the Yukawa‐type: where λis the variational parameter.…”

“…Hall 3 suggested that the positive background may be represented by a periodic array of Gaussian distribution with variable ripple parameter α or Yukawa distribution with variable ripple parameter λ. Hall 5 also found that the limits of zero and infinity for α (or λ) yield the electron crystal or Wigner solid and the empty lattice, respectively.…”