Physical Review

Charged Boson Gas at High Density

K. A. Brueckner

Published Apr 5, 1967• 67 citations

Abstract

The ground-state energy of a charged boson gas at high density is evaluated exactly to second order in the density expansion, making use of complete sums of one- and two-ring diagrams. The result agrees numerically with Feenberg's variational calculation of the energy, although the methods appear to be substantially different. The presence of logarithmic terms in the energy, which are absent in the exact result, is shown to be a result of truncation of the Hamiltonian in the pair approximation or of inadequacies of variational wave functions which omit interpair correlations.

DOI: 10.1103/physrev.156.204

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Physical Review

Charged Boson Gas at High Density

K. A. Brueckner

Published Apr 5, 1967• 67 citations

Publisher

Abstract

DOI: 10.1103/physrev.156.204

Related papers

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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Charged Bose Gas

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Green-function techniques are applied to the charged Bose gas at the absolute zero of temperature. The procedure permits some general remarks to be made concerning the expansions of the ground-state energy and depletion parameter in terms of ${\mathcal{r}}_{S}$ , the ratio of the interparticle spacing to the Bohr radius. It is found that these series contain not only all integer powers of ${{\mathcal{r}}_{S}}^{\frac{3}{4}}$ as had been conjectured, but additional functions of ${\mathcal{r}}_{S}$ as well. The first-two terms in the expansion of the ground-state energy are calculated exactly as are the first-four terms in the expansion of the depletion parameter. The former expansion agrees with the results of Lee and Feenberg. The latter, containing a noninteger power of ${{\mathcal{r}}_{S}}^{\frac{3}{4}}$ , is new. It is shown that the ground-state energy diagrams which must be summed in the Bose gas are characterized by equal numbers of variables of integration. It is emphasized that these are not the most divergent diagrams as in the electron-gas calculation of Gell-Mann and Brueckner. This difference leads us to reformulate the diagrammatic procedure so as to emphasize its self-consistency. The underlying similarity of the charged Fermi and Bose-gas calculations is then made apparent.Green-function techniques are applied to the charged Bose gas at the absolute zero of temperature. The procedure permits some general remarks to be made concerning the expansions of the ground-state energy and depletion parameter in terms of ${\mathcal{r}}_{S}$, the ratio of the interparticle spacing to the Bohr radius. It is found that these series contain not only all integer powers of ${{\mathcal{r}}_{S}}^{\frac{3}{4}}$ as had been conjectured, but additional functions of ${\mathcal{r}}_{S}$ as well. The first-two terms in the expansion of the ground-state energy are calculated exactly as are the first-four terms in the expansion of the depletion parameter. The former expansion agrees with the results of Lee and Feenberg. The latter, containing a noninteger power of ${{\mathcal{r}}_{S}}^{\frac{3}{4}}$, is new. It is shown that the ground-state energy diagrams which must be summed in the Bose gas are characterized by equal numbers of variables of integration. It is emphasized that these are not the most divergent diagrams as in the electron-gas calculation of Gell-Mann and Brueckner. This difference leads us to reformulate the diagrammatic procedure so as to emphasize its self-consistency. The underlying similarity of the charged Fermi and Bose-gas calculations is then made apparent.

Physical ReviewJan 5, 196918 citations

{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 ${r}_{s}\ensuremath{\sim}5$.

{{r}_{s}}^{\frac{3}{4}}

. The ground-state energy reduces to that calculated correctly by Brueckner, and earlier by Lee and Feenberg in a variational treatment. The excitation spectrum agrees with that calculated in our preceding paper using an entirely different method.

{\mathcal{r}}_{S}

, the ratio of the interparticle spacing to the Bohr radius. It is found that these series contain not only all integer powers of

{{\mathcal{r}}_{S}}^{\frac{3}{4}}

as had been conjectured, but additional functions of

{\mathcal{r}}_{S}

as well. The first-two terms in the expansion of the ground-state energy are calculated exactly as are the first-four terms in the expansion of the depletion parameter. The former expansion agrees with the results of Lee and Feenberg. The latter, containing a noninteger power of

{{\mathcal{r}}_{S}}^{\frac{3}{4}}

, is new. It is shown that the ground-state energy diagrams which must be summed in the Bose gas are characterized by equal numbers of variables of integration. It is emphasized that these are not the most divergent diagrams as in the electron-gas calculation of Gell-Mann and Brueckner. This difference leads us to reformulate the diagrammatic procedure so as to emphasize its self-consistency. The underlying similarity of the charged Fermi and Bose-gas calculations is then made apparent.