Ground state of a confined Yukawa plasma including correlation effects

The ground state of an externally confined one-component Yukawa plasma is derived analytically using the local density approximation (LDA). In particular, the radial density profile is computed. The results are compared with the recently obtained mean-field (MF) density profile [Henning et al., Phys. Rev. E 74, 056403 (2006)]. While the MF results are more accurate for weak screening, the LDA with correlations included yields the proper description for large screening. By comparison with first-principles simulations for three-dimensional spherical Yukawa crystals, we demonstrate that the two approximations complement each other. Together they accurately describe the density profile in the full range of screening parameters.

Figure 1

Radial density profile for a parabolic confinement potential $\Phi \left(r\right)=(\alpha ∕2)r^2$, a constant screening parameter $\kappa d_c=1$, and three different particle numbers $N=100$, 700, 2000. The result for $\kappa d_c=0.4$, $N=2000$ is also shown by the dashed line. For comparison, the nonlocal MF results for $\kappa d_c=0.4,1.0$, $N=2000$ are given by the dotted lines.

Figure 2

(Color online) Radial MF density profile (solid lines) compared to the LDA-MF (dashed lines) for three different density parameters $x=\left(\kappa d_c{)}^3\right(N-1)$. The abscissa is normalized with the MF radius $R^{\mathrm{MF}}$, while the ordinate is normalized with the corresponding MF density $n^{\mathrm{MF}}\left(0\right)$ at the trap center.

Figure 3

Radial density profiles of a spherical plasma of $N=1000$ and $\kappa d_c=0.3$ calculated by the LDA-MF with (fs corrected) and without finite-size effects included. For comparison, the exact MF result is also given (solid line). The difference between the finite-size correction and the partial finite-size correction is described in the text.

Figure 4

Radial density profile of a confined spherical plasma of $N=2000$ particles calculated with the LDA including correlation contributions (solid lines) compared to the LDA-MF (dashed lines) for three different screening parameters.

Figure 5

(Color online) Smoothed density of a simulated spherical Yukawa crystal with $N=1000$ and $\kappa d_c=1$ with application of different sphere sizes for substitution of the particles: $0.3d_c$ (top curve), $0.4d_c$ and $0.5d_c$. The symbols show the averaged densities of the shells placed on the centers of mass of the shells.

Figure 6

Radial density profiles of a three-dimensional plasma of $N=1000$ particles calculated with the exact mean-field model (solid lines) and with the LDA including correlation contributions (dashed lines) for four different screening parameters: $\kappa d_c=1$, 2, 3, and 5. Averaged shell densities of molecular dynamics results of a plasma crystal for the same parameters are shown by the symbols. For details, see the discussion in Sec. 4.

Figure 7

Radial density profiles of a three-dimensional plasma with $N=\mathrm{10\hspace{0.17em}000}$ and two different screening parameters ($\kappa d_c=1$, $\kappa d_c=3$). The solid (dashed) lines show MF (LDA) calculation results. Symbols denote molecular dynamics results of a plasma crystal for the same parameters where the average density at the positions of the shells is shown.

Figure 8

Comparison of the MF method and the different LDA methods for calculating the energy density of interaction $u^{\mathrm{MF}}\left(\mathbf{r}\right)$ in the case of finite screening: (a) MF, (b) LDA (infinite reference system), (c) LDA (fs corrected), and (d) LDA (partly fs corrected). Within the graphics the system is represented by the dashed line. The hatched region shows the integration area used within the method, whereas the solid gray region shows the effective integration area due to finite screening. The color gradient within (a) represents the nonconstant density of the system, which is taken into account within the MF method in contrast to the LDA methods, which take the density at point $\mathbf{r}$ for the whole integration area.