Physical Review A

Charged Bose gas at T=0

Álvaro A. Caparica, Oscar Hipólito

Published Nov 1, 1982• 26 citations

Abstract

The ground state of a charged Bose gas is studied throughout a large range of densities in a self-consistent-field approximation formalism which includes the short-range correlations between bosons through a local-field correction. Numerical self-consistent calculations have been carried out in order to determine the static structure factor $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$. Hence, from $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$ we have obtained the pair-correlation functions, the ground-state energy which essentially is the correlation energy, the pressure of the gas, and, from Feynman's relation, the energy spectrum of elementary excitations. The response of the system to a static impurity charge has also been investigated. In the high-density limit our calculations show the same results as those given by Bogoliubov's perturbation theory. In the intermediate-density region, corresponding to the strongly coupled system, our results are in very good agreement with recent calculations based on the hypernetted-chain integral equation as well as Monte Carlo computations.

DOI: 10.1103/physreva.26.2832

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Physical Review A

Charged Bose gas at T=0

Álvaro A. Caparica, Oscar Hipólito

Published Nov 1, 1982• 26 citations

Publisher

Abstract

S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})

. Hence, from

S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})

we have obtained the pair-correlation functions, the ground-state energy which essentially is the correlation energy, the pressure of the gas, and, from Feynman's relation, the energy spectrum of elementary excitations. The response of the system to a static impurity charge has also been investigated. In the high-density limit our calculations show the same results as those given by Bogoliubov's perturbation theory. In the intermediate-density region, corresponding to the strongly coupled system, our results are in very good agreement with recent calculations based on the hypernetted-chain integral equation as well as Monte Carlo computations.

DOI: 10.1103/physreva.26.2832

Related papers

Ground state of the strongly coupled, charged Bose gas

Jean-Pierre Hansen, Redha Mazighi

Using a Jastrow trial wave function and the hypernetted-chain integral equation, as well as Monte Carlo computations, to calculate energy expectation values, we have obtained a variational upper bound to the ground-state energy of the charged Bose gas over an extensive range of densities. In the high-density region our data are in good agreement with previous weak-coupling results. In the low-density (strong-coupling) limit, comparison of our fluid energies with the self-consistent phonon calculations of Glyde and Keech in the solid allow us to locate the Wigner transition for charged bosons at ${r}_{s}=139$ , where ${r}_{s}$ is the usual density parameter. We propose a simple and accurate equation of state based on our numerical data. Furthermore, we find that the Bose condensate fraction drops from a value of about 80% at ${r}_{s}=2$ , to a small but finite value (\ensuremath{\sim}1%) at crystallization. Finally we use our results for the charged Bose gas to derive a heuristic equation of state of the electron gas in the strong-coupling limit (${r}_{s}\ensuremath{\gg}1$).Using a Jastrow trial wave function and the hypernetted-chain integral equation, as well as Monte Carlo computations, to calculate energy expectation values, we have obtained a variational upper bound to the ground-state energy of the charged Bose gas over an extensive range of densities. In the high-density region our data are in good agreement with previous weak-coupling results. In the low-density (strong-coupling) limit, comparison of our fluid energies with the self-consistent phonon calculations of Glyde and Keech in the solid allow us to locate the Wigner transition for charged bosons at ${r}_{s}=139$, where ${r}_{s}$ is the usual density parameter. We propose a simple and accurate equation of state based on our numerical data. Furthermore, we find that the Bose condensate fraction drops from a value of about 80% at ${r}_{s}=2$, to a small but finite value (\ensuremath{\sim}1%) at crystallization. Finally we use our results for the charged Bose gas to derive a heuristic equation of state of the electron gas in the strong-coupling limit (${r}_{s}\ensuremath{\gg}1$).

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Ground State of the Charged Bose Gas

Marvin D. Girardeau

The ground-state energy of the charged Bose gas is calculated by the pair-correlation variational method of Girardeau and Arnowitt. The method is exact in the high-density limit (${r}_{s}\ensuremath{\ll}1$) and provides a variational extrapolation to intermediate densities. The leading terms of the high-density expansion, obtained by iteration of the variational integral equation, are ${u}_{0}=\ensuremath{-}0.804{{r}_{s}}^{\ensuremath{-}\frac{3}{4}}\ensuremath{-}(\frac{1}{8})\mathrm{ln}{r}_{s}+O(1)$, where ${u}_{0}$ is the ground-state energy per particle in Rydbergs and ${r}_{s}$ is the ratio of the mean interparticle spacing to the Bohr radius. The first term was obtained previously by Foldy, but the logarithmic term is new; it is related to screening of the long-range correlations at a distance ${r}_{0}\ensuremath{\sim}{{r}_{s}}^{\ensuremath{-}\frac{1}{2}}{\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}}$, in analogy with the logarithmic term in the correlation energy of the electron gas. Results of numerical solutions for the intermediate-density region are presented, ranging up to ${r}_{s}=10$ . On the basis of a comparison with the energy calculated from the known low-density expansion, it is estimated that the transition into Wigner's electron crystal (here a boson crystal) should occur at ${r}_{s}\ensuremath{\sim}5$.The ground-state energy of the charged Bose gas is calculated by the pair-correlation variational method of Girardeau and Arnowitt. The method is exact in the high-density limit (${r}_{s}\ensuremath{\ll}1$) and provides a variational extrapolation to intermediate densities. The leading terms of the high-density expansion, obtained by iteration of the variational integral equation, are ${u}_{0}=\ensuremath{-}0.804{{r}_{s}}^{\ensuremath{-}\frac{3}{4}}\ensuremath{-}(\frac{1}{8})\mathrm{ln}{r}_{s}+O(1)$, where ${u}_{0}$ is the ground-state energy per particle in Rydbergs and ${r}_{s}$ is the ratio of the mean interparticle spacing to the Bohr radius. The first term was obtained previously by Foldy, but the logarithmic term is new; it is related to screening of the long-range correlations at a distance ${r}_{0}\ensuremath{\sim}{{r}_{s}}^{\ensuremath{-}\frac{1}{2}}{\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}}$, in analogy with the logarithmic term in the correlation energy of the electron gas. Results of numerical solutions for the intermediate-density region are presented, ranging up to ${r}_{s}=10$. On the basis of a comparison with the energy calculated from the known low-density expansion, it is estimated that the transition into Wigner's electron crystal (here a boson crystal) should occur at ${r}_{s}\ensuremath{\sim}5$.

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Chung‐In Um, W. H. Kahng, Eui-Soon Yim +1 more

The ground state of a two-dimensional charged-boson system is investigated over the range of densities 1\ensuremath{\lesssim} ${R}_{s}$ \ensuremath{\lesssim}10 in the self-consistent-field approximation; ${R}_{s}$ =( ${a}_{0}^{2}$ \ensuremath{\pi}\ensuremath{\rho}${)}^{\mathrm{\ensuremath{-}}1/2}$, where ${a}_{0}$ is the Bohr radius and \ensuremath{\rho} is the number density. Starting with numerical self-consistent calculations of the static structure factor, the elementary excitation, pair-correlation functions, pressure, and ground-state energy are evaluated. These results are compared with those of the two-dimensional and three-dimensional systems obtained from other methods. The ground-state energy is given as ${\mathit{E}}_{0=\mathrm{\ensuremath{-}}1.2918{\mathit{R}}_{\mathit{s}}}^{\mathrm{\ensuremath{-}}2/3}$+0.03, which improves that result from the ring-diagram approximation.The ground state of a two-dimensional charged-boson system is investigated over the range of densities 1\ensuremath{\lesssim}${R}_{s}$\ensuremath{\lesssim}10 in the self-consistent-field approximation; ${R}_{s}$=(${a}_{0}^{2}$\ensuremath{\pi}\ensuremath{\rho}${)}^{\mathrm{\ensuremath{-}}1/2}$, where ${a}_{0}$ is the Bohr radius and \ensuremath{\rho} is the number density. Starting with numerical self-consistent calculations of the static structure factor, the elementary excitation, pair-correlation functions, pressure, and ground-state energy are evaluated. These results are compared with those of the two-dimensional and three-dimensional systems obtained from other methods. The ground-state energy is given as ${\mathit{E}}_{0=\mathrm{\ensuremath{-}}1.2918{\mathit{R}}_{\mathit{s}}}^{\mathrm{\ensuremath{-}}2/3}$+0.03, which improves that result from the ring-diagram approximation.

Physical review. B, Condensed matterJan 1, 199016 citations

S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})

. Hence, from

S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})

{r}_{s}=139

, where

{r}_{s}

is the usual density parameter. We propose a simple and accurate equation of state based on our numerical data. Furthermore, we find that the Bose condensate fraction drops from a value of about 80% at

{r}_{s}=2

, to a small but finite value (\ensuremath{\sim}1%) at crystallization. Finally we use our results for the charged Bose gas to derive a heuristic equation of state of the electron gas in the strong-coupling limit (${r}_{s}\ensuremath{\gg}1$).

{r}_{s}.

We find that the static and dynamical properties are described rather well in this approach in comparison to previous theories. Our results are in good agreement with the available quantum Monte Carlo simulations.

{u}_{0}

is the ground-state energy per particle in Rydbergs and

{r}_{s}

is the ratio of the mean interparticle spacing to the Bohr radius. The first term was obtained previously by Foldy, but the logarithmic term is new; it is related to screening of the long-range correlations at a distance ${r}_{0}\ensuremath{\sim}{{r}_{s}}^{\ensuremath{-}\frac{1}{2}}{\ensuremath{\rho}}^{\ensuremath{-}\frac{1}{3}}$, in analogy with the logarithmic term in the correlation energy of the electron gas. Results of numerical solutions for the intermediate-density region are presented, ranging up to

{r}_{s}=10

. On the basis of a comparison with the energy calculated from the known low-density expansion, it is estimated that the transition into Wigner's electron crystal (here a boson crystal) should occur at ${r}_{s}\ensuremath{\sim}5$.

The ground state of a two-dimensional charged-boson system is investigated over the range of densities 1\ensuremath{\lesssim}

{R}_{s}

\ensuremath{\lesssim}10 in the self-consistent-field approximation;

{R}_{s}

{a}_{0}^{2}

\ensuremath{\pi}\ensuremath{\rho}${)}^{\mathrm{\ensuremath{-}}1/2}$, where

{a}_{0}

is the Bohr radius and \ensuremath{\rho} is the number density. Starting with numerical self-consistent calculations of the static structure factor, the elementary excitation, pair-correlation functions, pressure, and ground-state energy are evaluated. These results are compared with those of the two-dimensional and three-dimensional systems obtained from other methods. The ground-state energy is given as ${\mathit{E}}_{0=\mathrm{\ensuremath{-}}1.2918{\mathit{R}}_{\mathit{s}}}^{\mathrm{\ensuremath{-}}2/3}$+0.03, which improves that result from the ring-diagram approximation.