Lowest-order Constrained Variational Method Research Articles

Neutron and nuclear matter properties with chiral three-nucleon forces

The equation of state of neutron and symmetric matter is calculated within the lowest order constrained variational (LOCV) method using the Argonne V18 (AV18) nucleon–nucleon interaction supplemented by the local, coordinate space versions of the chiral three-nucleon force (TNF) at next-to-next-to-leading order (N2LOL). We present the first results of studying the nuclear equation of state in the LOCV framework by employing a chiral TNF. It is shown that although some calculated values of saturation properties lie well in the empirically determined ranges, all of the saturation quantities can not be predicted simultaneously. We also study other possible parametrizations of the N2LOL TNF by fitting the low-energy constants (LECs) to nuclear saturation point. It is found that by employing the proposed sets of LECs, LOCV method is able to successfully produce the equilibrium properties of symmetric nuclear matter.

A microscopic study of spin-polarized asymmetric hot nuclear matter within the static fluctuation approximation (SFA)

In this work, the thermodynamic properties of spin-polarized asymmetric hot nuclear matter are studied within the Static Fluctuation Approximation (SFA). Specifically, the energy per unit volume, pressure, entropy per unit volume, and specific heat capacity are determined as functions of density up to 0.6 fm−3, and temperature up to 80 MeV. Also, the effects of the asymmetry and spin-polarization parameters on these quantities are explored. The main input in the formalism is the Urbana V14 nucleon–nucleon potential. It is found that these parameters have significant effects on the thermodynamic properties at high densities or low temperatures where quantum features become pronounced. As expected, the entropy results show that the system is more ordered when polarized or for large values of the asymmetry parameter. The quadratic dependence of the energy on the asymmetry and spin-polarization parameters is confirmed at different densities and temperatures. The results are compared to those obtained with different approaches, such as the lowest-order constrained variational method, the conventional variational method, and the relativistic mean-field approximation. It is found that the present results are consistent with those.

Appearance of hyperons in neutron stars within LOCV method

Chemical potentials of nucleons in asymmetric nuclear matter are calculated in a completely self-consistent manner, using the lowest order constrained variational (LOCV) method. Employing different nucleon–nucleon potentials, threshold density of free hyperons formation and the maximum mass of hypernuclear star are determined. Furthermore, the particles fraction in beta stable and charge neutral matter is calculated as well. The effects of three-body force (TBF) on the onset of hyperons, particle composition and maximum mass of the star are discussed. The results show that, for all different nucleon–nucleon interactions considered, onset of hyperons formation sets in at about 2–3 times of the nuclear saturation density. In particular, it is shown that by considering the TBF, the density of appearance of hyperons shifts to a lower value. The maximum masses of hypernuclear star are obtained about 1.3068M⊙ and 1.1032M⊙ with and without TBF respectively. Although these results are much below the ones required by observations of astrophysics, they are in agreement with the previous works based on Brueckner–Hartree–Fock (BHF) approach as well as other non-relativistic potential models.

Role of nuclear correlations and kinematic effects on neutrino emission from the modified Urca processes

The neutrino emissivity from the proton branch of the modified Urca process is calculated. Angular integrations in momentum space are performed numerically without recourse to widely used, but unjustified in this specific case, approximations. To deal with nuclear correlation effects, the independent-pair approximation for nucleon quasiparticles is assumed. The realistic nuclear correlation effects are extracted from a modified extended version of the lowest-order constrained variational method in asymmetric nuclear matter in $\ensuremath{\beta}$ equilibrium with electrons and muons that allows for three-body nucleon interactions and fits all semiempirical saturation parameters of nuclear matter quite well. Two-body nucleon interaction is modeled by a realistic Argonne AV18 potential and the three-body potential used is a phenomenological Urbana UIX, generalizing parametrization of its component strengths in a manner suitable for application in asymmetric nuclear matter. Limiting to the two-body forces only, we find that, at fixed temperature, neutrino emissivity is a (weakly) decreasing function of density. This is also the case for the neutron branch of the modified Urca process analyzed by us in a recent paper. Inclusion of the modified three-body force effects changes this behavior. Quantitatively the value of the emissivity at each density increases and qualitatively the overall effect of the modified three-body force addition is such that dependence on density becomes weak in a density range from ${n}_{0}$ to $3.5{n}_{0}$. An updated version of neutron branch results obtained using our modified three-body force model is also presented. This is done using exact treatment of the angular integrations of nuclear transition rate in momentum space.

Studies of a single component Fermi gas near a p-wave resonance with the lowest order constrained variational method

We study a single component Fermi gas near a p-wave resonance with the lowest order constrained variational (LOCV) method. We obtain the energy per particle for the ground state of the single component Fermi gas near a p-wave resonance with the LOCV method. We also calculate the compressibility of the single component Fermi gas near a p-wave resonance and it shows that in the strongly interacting BCS side, the system would lose its stability and collapse. The width of the unstable region is proportional to the Fermi energy which reflects the universality of the dilute Fermi gas. The two p-wave contacts are also obtained and their variation tendencies with the interaction strength are qualitatively in agreement with recent experimental results.

Erratum: Proto-neutron star structure within an extended lowest-order constrained variational method at finite temperature [Phys. Rev. C 92 , 035806 (2015)

Received 18 December 2017DOI:https://doi.org/10.1103/PhysRevC.97.049904©2018 American Physical Society

Nuclear correlations and neutrino emissivity from the neutron branch of the modified Urca process

The neutrino emissivity from the neutron branch of the modified Urca process is calculated. The nuclear correlation effects are taken into account by employing the correlation functions extracted from the lowest-order constrained variational (LOCV) method applied to asymmetric nuclear matter. Two-body nucleon interaction is modeled by a realistic Argonne AV18 potential. In order to get consistency with semiempirical saturation parameters of nuclear matter and the existence of $2{M}_{\ensuremath{\bigodot}}$ pulsars, we add a phenomenological Urbana UIX three-body potential to the nucleon Hamiltonian and apply a newly formulated version of the LOCV method that allows for three-body nucleon interactions. We find that at fixed temperature neutrino emissivity is a (weakly) decreasing function of density, due to quenching of the contribution from tensor correlations with increasing density. This is in variance with all previous works. We also find that three-body forces allow for the opening of the direct Urca process at nucleon density $0.3\phantom{\rule{0.28em}{0ex}}{\mathrm{fm}}^{\ensuremath{-}3}$.

The effect of gap in n(k, $ \rho$ ) on the single-particle properties of nucleons and the ground-state binding energy of closed-shell nuclei

The present work evaluates the effect of gap in the density-dependent one-body momentum distribution, \( n(k,\rho)\), at the Fermi surface on the calculation of the single-particle properties of nucleons, i.e., the momentum- and density-dependent single-particle potential and the nucleon effective mass, and also on the calculation of the ground-state binding energy of the selected closed-shell nuclei, i.e., 16O, 40Ca, and 56Ni. In order to do this, \( n(k,\rho)\) is constructed by use of the calculations of the lowest-order constrained variational method for the symmetric nuclear matter with the \( Av_{18}\) potential up to \( J_{max}=2\) and 5. It is shown that the gap in \( n(k,\rho)\) at the Fermi surface has no significant effect on the calculation of single-particle properties in the case of \( J_{max}=5\). In the relevant evaluation of the ground-state binding energy of selected nuclei, it is seen that the binding energy of 16O, improved by including \( n(k,\rho)\), is closer to the experimental data, contrary to 40Ca and 56Ni.

The Calculation of Single-Nucleon Energies of Nuclei by Considering Two-Body Effective Interaction, n(k,ρ), and a Hartree-Fock Inspired Scheme

The nucleon single-particle energies (SPEs) of the selected nuclei, that is, O16, Ca40, and Ni56, are obtained by using the diagonal matrix elements of two-body effective interaction, which generated through the lowest-order constrained variational (LOCV) calculations for the symmetric nuclear matter with the Aυ18 phenomenological nucleon-nucleon potential. The SPEs at the major levels of nuclei are calculated by employing a Hartree-Fock inspired scheme in the spherical harmonic oscillator basis. In the scheme, the correlation influences are taken into account by imposing the nucleon effective mass factor on the radial wave functions of the major levels. Replacing the density-dependent one-body momentum distribution functions of nucleons, n(k,ρ), with the Heaviside functions, the role of n(k,ρ) in the nucleon SPEs at the major levels of the selected closed shell nuclei is investigated. The best fit of spin-orbit splitting is taken into account when correcting the major levels of the nuclei by using the parameterized Wood-Saxon potential and the Aυ18 density-dependent mean field potential which is constructed by the LOCV method. Considering the point-like protons in the spherical Coulomb potential well, the single-proton energies are corrected. The results show the importance of including n(k,ρ), instead of the Heaviside functions, in the calculation of nucleon SPEs at the different levels, particularly the valence levels, of the closed shell nuclei.

Proto-neutron star structure within an extended lowest-order constrained variational method at finite temperature

Proto-neutron star structure within an extended lowest-order constrained variational method at finite temperature

The ground state binding energy of the closed shell nuclei with the density dependent Av18 effective interaction in LOCV method

This work presents the calculation of the ground state binding energy of some light closed shell nuclei such as 4He, 12C, 16O, 28Si, 32S, 40Ca and 56Ni. We use our channel-dependent effective two-body interactions data that are produced through the lowest order constrained variational (LOCV) method for asymmetric nuclear matter with the charge-dependent Av18 bare NN potential up to Jmax = 2 and Jmax = 5. The local density approximation approach in the harmonic oscillator basis and Brody-Moshinsky coefficients are used to produce the relative and center of mass dependent effective two-body potentials. We show that the ground state energies of the closed shell nuclei with Av18 (Jmax = 2) gives more binding with respect to Reid68 bare NN potential as well as Δ – Reid68. However, there is not much difference between the Av18 (Jmax = 5) and Reid68Day which has been defined up to Jmax=5. We conclude that the contributions of higher partial waves (J > 2) are not very important and two-body kinetic energy in J=1 channel is twice as that of J=0 which is not the case for the two-body potential energy. Finally, our work results are compared with other theoretical approaches and experimental data.

Ferromagnetic and antiferromagnetic spin-ordering stabilities of asymmetric nuclear matter: Lowest-order constrained variational method

In this paper we study the possibility of spontaneous ferromagnetic and antiferromagnetic phase transitions of asymmetrical nuclear matter using the lowest-order constrained variational technique with ${\mathit{AV}}_{18}$ potential and employing a microscopic point of view. Our results show that the spontaneous transition to ferromagnetic and antiferromagnetic phases cannot occur for asymmetric nuclear matter.

LOCV calculation of nuclear matter with phenomenological two-nucleon interaction operators

The lowest-order constrained variational (LOCV) method is developed for the wide range of phenomenological two-nucleon interaction operators such as , and U potentials. The calculation is performed for both nuclear and neutron matter with the state-dependent correlation operators. The validity of our lowest-order approximation is tested by calculating the three-body cluster energy with the state-averaged correlation functions. It is shown that while the three-body cluster energy improves the nuclear matter saturation density, the LOCV method still overbinds nuclear matter with the above potentials. Finally, we find that our LOCV results are similar to those calculations which have been performed by using more sophisticated many-body techniques.

Neutron matter equation of state and thermal energy of nuclear matter

The equation of state of neutron matter is calculated by using the lowest-order constrained variational method and compared with the results obtained by other methods. The modified Reid potential which contains N-N and N- Delta interactions and fits the scattering data is used in the neutron matter Hamiltonian. The explicit dependence of correlation functions on spin, isospin, angular momentum, density and tensor operators for all of the NN and N Delta channels as well as temperature is taken into the account in our calculation. Free energy, effective mass, etc. of neutron matter plus thermal and compression energies of nuclear matter are calculated. These results are compared with the recent calculations which have been obtained by using other methods.

LOCV calculations of pressure in nuclear matter at finite temperature

The lowest-order constrained variational method is used to calculate the equation of state of hot nuclear matter. The calculations are performed for a wide range of densities of interest in heavy-ion collisions and astrophysics. A realistic nuclear Hamiltonian that contains N-N and N- Delta interactions and fits N-N scattering as well as nuclear matter saturation data is used. Besides spin, isospin, total angular momentum and density dependence, the correlation functions are also arranged to depend on the temperature of the system. The liquid-vapour phase equilibrium, as well as the behaviour of the effective mass in nuclear matter, is discussed.

Spin-polarized hydrogen, deuterium, and tritium: II. Pressure and compressibility calculated by a lowest order constrained-variation method

The pressure and the compressibility of spin-polarized H↓, D↓, and T↓ are obtained from ground-state energies calculated by means of a modified variational lowest order constrained-variation method. The pressure and the compressibility are calculated or estimated from the dependence of the ground-state energy on density or molar volume, generally in a density region from 0 to 1.5σ−3 corresponding to a molar volume of more than 20 cm3/mol, where σ = 3.69 Å (1Å = 10−10 m); the calculations are done for five different two-body potentials. Theoretical results for the pressure are 54.1–57.9 atm for spin-polarized H↓ 18.4–23.4 atm for spin-plolarized D↓, and 5.6–12.9 atm for spin-polarized T↓ at a particle density of 0.50σ−3 or a molar volume of 60 cm3/mol (1 atm = 101 kPa). Theoretical results for the compressibility are 51 × 10−4 −54 × 10−4 atm−1 for spin-polarized H↓, 108 × 10−4 −120 × 10−4 atm−1 for spin-polarized D↓, and 162 × 10−4 −224 × 10−4 atm−1 for spin-polarized T↓ at a particle density of 0.50σ−3 for a molar volume of 60 cm3/mol. The relative agreement between results for different potentials is somewhat better for higher densities.

Solid hydrogen and deuterium. II. Pressure and compressibility calculated by a lowest-order constrained-variation method

The pressure and the compressibility of solid H2 and D2 are obtained from ground-state energies calculated by means of a modified variational lowest-order constrained-variation (LOCV) method. Both fcc and hcp structures are considered, but results are given for the fcc structure only. The pressure and the compressibility are calculated or estimated from the dependence of the ground-state energy on density or molar volume, generally in a density region of 0.65Σ−3 to 1.3Σ−3, corresponding to a molar volume of 12–24 cm3/mole, where Σ = 2.958 a, and the calculations are done for five different two-body potentials. Theoretical results for the pressure are 340–460 atm for solid H2 at a particle density of 0.82Σ−3 or a molar volume of 19 cm3/mole, and 370–490 atm for solid 4He at a particle density of 0.92Σ−3 or a molar volume of 17 cm3/mole. The corresponding experimental results are 650 and 700 atm, respectively. Theoretical results for the compressibility are 210 × 10−6 to 260 × 10−6 atm−1 for solid H2 at a particle density of 0.82Σ−3 or a molar volume of 19 cm3/mole, and 150 × 10−6 to 180 × 10−6 atm−1 for solid D2 at a particle density of 0.92Σ−3 or a molar volume of 17 cm3/mole. The corresponding experimental results are 180 × 10−6 and 140 × 10−6 atm−1, respectively. The agreement with experimental results is better for higher densities.

Solid hydrogen and deuterium. I. Ground-state energy calculated by a lowest order constrained-variation method

The ground-state energy of solid hydrogen and deuterium is calculated by means of a modified variational lowest order constrained-variation (LOCV) method. Both fcc and hcp H2 and D2 are considered, and the calculations are done for five different two-body potentials. For solid H2 we obtain theoretical results for the ground-state binding energy per particle from −74.9 K at an equilibrium particle density of 0.700 σ−3 or a molar volume of 22.3 cm3/mole to −91.3 K at a particle density of 0.725 σ−3 or a molar volume of 21.5 cm3/mole, where σ=2.958 A. The corresponding experimental result is −92.3 K at a particle density of 0.688 σ−3 or a molar volume of 22.7 cm3/mole. For solid D2 we obtain theoretical results for the ground-state binding energy per particle from −125.7 K at an equilibrium particle density of 0.830 σ−3 or a molar volume of 18.8 cm3/mole to −140.1 K at a particle density of 0.843 σ−3 or a molar volume of 18.5 cm3/mole. The corresponding experimental result is −137.9 K at a particle density of 0.797 σ−3 or a molar volume of 19.6 cm3/mole.

Solid helium. II. Pressure and compressibility calculated by a lowest-order constrained-variation method.

The pressure and the compressibility of solid $^{3}\mathrm{He}$ and $^{4}\mathrm{He}$ are obtained from ground-state energies calculated by means of a modified variational lowest-order constrained-variation (LOCV) method. The structures bcc, fcc, and hcp are considered, but results are given for the bcc structure only. The pressure and the compressibility are calculated or estimated from the dependence of the ground-state energy on density or molar volume, generally in a density region of 0.4${\ensuremath{\sigma}}^{\mathrm{\ensuremath{-}}3}$ to 1.0${\ensuremath{\sigma}}^{\mathrm{\ensuremath{-}}3}$ corresponding to a molar volume of 10 ${\mathrm{cm}}^{3}$/mole to 25 ${\mathrm{cm}}^{3}$/mole, where \ensuremath{\sigma}=2.556 A\r{}, and the calculations are done for five different two-body potentials. Theoretical results for the pressure are 72 to 94 atm for solid $^{3}\mathrm{He}$ at a particle density of 0.50${\ensuremath{\sigma}}^{\mathrm{\ensuremath{-}}3}$ or a molar volume of 20 ${\mathrm{cm}}^{3}$/mole, and 56 to 87 atm for solid $^{4}\mathrm{He}$ at a particle density of 0.55${\ensuremath{\sigma}}^{\mathrm{\ensuremath{-}}3}$ or a molar volume of 18.3 ${\mathrm{cm}}^{3}$/mole. The corresponding experimental results are 97 and 84 atm, respectively. Theoretical results for the compressibility are 121\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}5}$ to 166\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}5}$ ${\mathrm{atm}}^{\mathrm{\ensuremath{-}}1}$ for solid $^{3}\mathrm{He}$ and 172\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}5}$ to 246\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}5}$ ${\mathrm{atm}}^{\mathrm{\ensuremath{-}}1}$ for solid $^{4}\mathrm{He}$ at a particle density of 0.55${\ensuremath{\sigma}}^{\mathrm{\ensuremath{-}}3}$ or a molar volume of 18.3 ${\mathrm{cm}}^{3}$/mole. The corresponding experimental results are 108\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}5}$ ${\mathrm{atm}}^{\mathrm{\ensuremath{-}}1}$ and 165\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}5}$ ${\mathrm{atm}}^{\mathrm{\ensuremath{-}}1}$, respectively. The agreement with experimental results is reasonably good since higher-order cluster terms are not included in our LOCV approximation.

Fermi-Hypernetted-Chain Calculations for Liquid 3He

The Fermi hypernetted chain (FHNC) method for quantum many-body calculations is investigated in some detail by means of calculations of pair correlation functions, structure functions and binding energy for liquid 3He. The results are compared with experimental results and other theoretical results obtained by the lowest-order constrained variation (LOCV) method and variational Monte Carlo (MC) calculations. The FHNC results agree quite well with MC results, but are very different from results obtained by the LOCV method. The LOCV binding energies are generally reduced by approximately 2K in the FHNC calculations.