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Physical Review

Analysis of a Model Exhibiting Condensation with Application to He3

C. M. Andersen, S. Gartenhaus, G. Stranahan

Published Jun 3, 1966• 7 citations
Publisher

Abstract

A study is made of a certain model Hamiltonian which describes a system of fermions. Making use of the variational results of Girardeau one can obtain, in the limit of infinite volume, an explicit formula for the free energy in terms of the solution of a nonlinear integral equation. It is established that for certain values of the coupling constant the model exhibits condensation, and that qualitatively at least, the isotherms for the model reproduce those for actual physical systems. The model is applied to ${\mathrm{He}}^{3}$ and it is shown that by a suitable choice for the parameters of the model, reasonable agreement with the low-temperature properties of ${\mathrm{He}}^{3}$ is obtained. The second virial coefficient for ${\mathrm{He}}^{4}$ is also reproduced with the same parameter values which give the correct virial coefficient for ${\mathrm{He}}^{3}$.

DOI
10.1103/physrev.146.101

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ФИЗИКА ЭЛЕМЕНТАРНЫХ ЧАСТИЦ И АТОМНОГО ЯДРА•May 3, 2024•0 citations
A study is made of a certain model Hamiltonian which describes a system of fermions. Making use of the variational results of Girardeau one can obtain, in the limit of infinite volume, an explicit formula for the free energy in terms of the solution of a nonlinear integral equation. It is established that for certain values of the coupling constant the model exhibits condensation, and that qualitatively at least, the isotherms for the model reproduce those for actual physical systems. The model is applied to He3{\mathrm{He}}^{3}He3 and it is shown that by a suitable choice for the parameters of the model, reasonable agreement with the low-temperature properties of He3{\mathrm{He}}^{3}He3 is obtained. The second virial coefficient for He4{\mathrm{He}}^{4}He4 is also reproduced with the same parameter values which give the correct virial coefficient for He3{\mathrm{He}}^{3}He3.
Girardeau has shown that a particular class of variational approximations for the free energy of a Fermi-Dirac system may be expressed in terms of the exact free energy of a certain soluble "model Hamiltonian." This exact formula for the partition function associated with the model system involves the solution of a certain nonlinear integral equation, and in the present work it is shown in detail that there is a correspondence between phase transitions in the system and the existence of multiple solutions of this equation. The connection between the existence of phase transitions and the properties of the interaction is studied, and it is shown that these phase transitions may occur for a very wide class of interactions. It is established that there is an upper bound to the temperature, above which no phase transitions will occur at any density. The extension of these results to a classical and a Bose-Einstein system is indicated, and the lattice-gas model previously studied by Mermin is re-examined in the light of the present results.
The virial expansion method is applied within a harmonic approximation to an interacting N-body system of identical fermions. We compute the canonical partition functions for two and three particles to get the two lowest orders in the expansion. The energy spectrum is carefully interpolated to reproduce ground-state properties at low temperature and the noninteracting high-temperature limit of constant virial coefficients. This resembles the smearing of shell effects in finite systems with increasing temperature. Numerical results are discussed for the second and third virial coefficients as functions of dimension, temperature, interaction, and transition temperature between low- and high-energy limits.
The quantum theory of the third virial coefficient CCC is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of CBE{C}_{\mathrm{BE}}CBE​ are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for He4{\mathrm{He}}^{4}He4 and He3{\mathrm{He}}^{3}He3, by means of a comparison of the corresponding expansion for the second virial coefficient BBB with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to BBB. Then the leading term of CCC is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for CCC is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.
The authors study a one dimensional model system of fermions which proves to be exactly soluble in the thermodynamic limit. It is shown that the thermodynamic functions for the model system are the same as for a system of noninteracting fermions in a harmonic oscillator well. As a special case of the model system, with a particular choice for the coupling constant, it reduces to a particular two-species system of fermions interacting via pairwise harmonic forces. For this case of the two-species system the free energy per particle, the pressure the specific heat at constant volume, and the isothermal compressibility are calculated in the canonical ensemble in closed form. It is shown that this system does not undergo a phase transition.
Within the conserving Φ-derivable and virial approaches to equilibrium dilute strongly-interacting systems we study a system of non-relativistic fermions of one kind (e.g., neutron matter) interacting via a pair potential in the limit nλ3≪1n{{\lambda }^{3}} \ll 1nλ3≪1 , where n is the fermion number density and λ is the thermal wavelength. The variety of potentials of the quantum and classical origin is used for the quantitative comparison of results obtained within both approaches. In all cases relevant to the nucleon-nucleon interaction, the attraction volume of the interaction is larger than the repulsion one. General properties of the Φ derivable approach are studied. Expressions for the second and third virial coefficients are derived and analyzed for the Φ functional including the tadpole and sandwich diagrams. The classical high-temperature limit of the Φ functional is reproduced by the tadpole diagram. Next, classical, semiclassical (up to order ℏ3{{\hbar }^{3}}ℏ3 ), and purely quantum (Beth-Uhlenbeck) virial expansions of the equation of state are studied. Various extrapolations of the virial equation of state are considered including the van der Waals form and the excluded volume models. We derive the expression for the second virial coefficient within the effective range approximation for the scattering amplitude. Comparison with purely quantum Beth–Uhlenbeck result employing experimental phase shifts shows approximate agreement for low temperatures, T≲20T \lesssim 20T≲20 MeV. Then, we study the problem of the anomalously large empirical value of the nucleon-nucleon scattering length, appearing due to existence of the quasi-bound state in nucleon-nucleon scattering. The latter is destroyed already in a dilute matter due to the Pauli blocking. Subtracting the quasi-bound state term, we show that the second virial coefficient has the same low temperature dependence as one obtained within the Φ derivable approach in the Born approximation. Also, we discuss origins of differences arising beyond the region of the common validity of the Φ derivable and virial approaches.
Within the conserving Φ-derivable and virial approaches to equilibrium dilute strongly-interacting systems, we study a system of nonrelativistic fermions of one kind (e.g., neutron matter) interacting via a pair potential in the limit nλ3 ≪\ll≪ 1, where n is the fermion number density and λ is the thermal wavelength. The variety of potentials of the quantum and classical origin is used for the quantitative comparison of results obtained within both approaches. In all cases relevant to the nucleon–nucleon interaction, the attraction volume of the interaction is larger than the repulsion one. General properties of the Φ-derivable approach are studied. Expressions for the second and third virial coefficients are derived and analyzed for the Φ functional, including the tadpole and sandwich diagrams. The classical hightemperature limit of the Φ functional is reproduced by the tadpole diagram. Next, classical, semiclassical (up to order ¯ ħ3) and purely quantum (Beth–Uhlenbeck) virial expansions of the equation of state are studied. Various extrapolations of the virial equation of state are considered, including the van der Waals form and the excluded volume models. We derive the expression for the second virial coefficient within the effective range approximation for the scattering amplitude. Comparison with purely quantum Beth–Uhlenbeck result employing experimental phase shifts shows approximate agreement for low temperatures, T≲ 20 MeV. Then, we study the problem of the anomalously large empirical value of the nucleon–nucleon scattering length, appearing due to existence of the quasi-bound state in nucleon–nucleon scattering. The latter is destroyed already in a dilute matter due to the Pauli blocking. Subtracting the quasi-bound state term, we show that the second virial coefficient has the same low-temperature dependence as the one obtained within the Φ-derivable approach in the Born approximation. Also, we discuss origins of differences arising beyond the region of the common validity of the Φ-derivable and virial approaches.