Lowest Order Constrained Variational Research Articles

The effect of Fermi momentum cutoff on the binding energy of closed shell nuclei in the LOCV framework

The ground state binding energies of the light symmetric closed-shell nuclei, i.e., 4He, 12C, 16O and 40Ca and the heavy asymmetric ones, i.e., 48Ca, 90Zr and 120Sn are calculated in the harmonic oscillator (HOS) basis, by imposing the relative Fermi momentum cutoff of two point-like interacting nucleons on the density dependent average effective interactions (DDAEI). The DDAEI are generated through the lowest order constrained variational (LOCV) method calculations for the asymmetric nuclear matter with the operator and the channel dependent type bare nucleon-nucleon potentials, such as the Argonne \(Av_{18}^{j_{\max } = 2}\) and the Reid soft core, Reid68, interactions. In the framework of harmonic oscillator shell model, the cutoff is imposed by defining the maximum value of the relative quantum numbers (RQNmax) in two ways: (1) The RQNmax of the last shell and (2) the RQNmax of each shell, in the ground state of the nucleus. It is shown that present results on the binding energies and the root means square radius are closer to the corresponding experimental data than, our previous works with the same DDAEI potentials, but without the cutoff constraint. However, for the light symmetric nuclei, the second scheme gives less binding energy and larger root mean square radius compare to the first one. While the situation is reversed for the heavier nuclei.

Imposing Fermi momentum cut-off on the channel- and density-dependent effective interaction and the ground-state properties of closed shell nuclei

The ground-state properties of light-moderate symmetric closed shell nuclei, i.e. 4He, 12C, 16O, 28Si, and 40Ca, and moderate-heavy asymmetric ones, i.e. 48Ca, 90Zr, and 120Sn, are calculated using the channel- and density-dependent effective interaction with the imposed Fermi momentum cut-off (CDDEI+FMC). CDDEI+FMC are generated through the lowest-order constrained variational (LOCV) nuclear matter calculations with the $ Av_{18}(j_{max}=2)$ nucleon-nucleon bare potential. FMC is considered by defining the maximum value of the relative quantum numbers according to the last shell in the ground state of a nucleus. The Hamiltonian matrix of the truncated effective interactions is diagonalized in the harmonic oscillator basis, regarding the nucleon configurations. The FMC effect on the different interaction channels is considered by comparing the contributions of each channel in the binding energies of the sample nuclei. CDDEI+FMC are replaced by the density-dependent average effective interactions with the imposed FMC (DDAEI+FMC) for the channels with $ j > j_{max}$ . For the light nuclei, the results show that the nuclei are more binding relative to the CDDEI-FMC case. The 4He nucleus is about 1.4MeV more binding per nucleon, and the root mean square of its radius equals the experimental data exactly. For the moderate-heavy closed shell nuclei, there are no significant differences between the results of the two cases, both with FMC and without it.

Thermal properties of nuclear matter in a variational framework with relativistic corrections

The properties of hot symmetric nuclear matter for a wide range of densities and temperatures are investigated by employing the $A{V}_{14}$ potential within the lowest order constrained variational (LOCV) method with the inclusion of a phenomenological three-body force as well as relativistic corrections. The relativistic corrections of many-body kinetic energies as well as the boot interaction corrections are presented for a wide range of densities and temperatures. The free energy, pressure, incompressibility, and other thermodynamic quantities of symmetric nuclear matter are obtained and discussed. The critical temperature is found, and the liquid-gas phase transition is analyzed both with and without the inclusion of three-body forces and relativistic corrections in the LOCV approach. It is shown that the critical temperature is strongly affected by the three-body forces but does not depend on the relativistic corrections. Finally, the results obtained in the present study are compared with other many-body calculations and experimental predictions.

Equation of State for Neutron Star Matter with the Lowest Order Constrained Variational Method and Kaon Condensation

We investigate the equation of state of kaon condensed neutron star matter using the lowest order constrained variational (LOCV) method. In this paper, we employ the AV18 and AV14 potentials for nucleon-nucleon interaction and use the chiral model for kaon-nucleon interaction. According to our results, the kaon condensation occurs at some density values and softens the equation of state as other techniques have shown.

The transport properties of neutron matter in the LOCV channel dependent effective two-body interactions framework

Abstract The channel dependent effective two-body interactions ( CDEI ) which are generated through the lowest order constrained variational ( LOCV ) method for pure neutron matter ( PNM ) are used to calculate the in-medium neutron–neutron ( nn ) cross-sections and transport properties of PNM within the Landau–Abrikosov–Khalatnikov ( LAK ) formalism. Similar to our previous work, the Reid 68, Δ - Reid 68 , and Av 18 interactions as well as the V 6 ′ potential are considered as the input bare nucleon–nucleon phenomenological potentials. In present work, we improve our previous results by calculating the shear viscosity and thermal conductivity of PNM in the framework of LOCV CDEI and the comparisons are made with those coming from the channel independent calculations, i.e. the averaged effective interactions ( AEI ) results. It is shown that our new results are in a good agreement with the other approaches. It is also demonstrated that, the choice of in-medium effective interaction has a crucial role in the resulting nn cross-section as well as the transport properties of PNM .

LOWEST-ORDER CONSTRAINED VARIATIONAL CALCULATIONS FOR TWO-DIMENSIONAL LIQUID 3He

We have used the lowest-order constrained variational (LOCV) method based on the cluster expansion of the energy functional to calculate some ground state properties of two-dimensional liquid 3 He at zero temperature employing the Lennard-Jones and Aziz pair potentials. We have seen that the total energy increases by increasing density. It is shown that the two-dimensional liquid 3 He system has no self-bound state.

The nucleonic matter LOCV calculations in a periodic box versus the FHNC method

Abstract The lowest order constrained variational ( LOCV ) method is reformulated for a fixed number of nucleons in a period box ( PBLOCV ) with the inter-nucleon interactions such as the Bethe homework potential, v B , and the scalar ( v 4 ) part of the Argonne interaction ( A v 8 ′ ) c ( 4 ) . The two-body cluster approximation is used in the ( PB ) LOCV formalism to obtain the energy per particle subjected to the normalization constraint for an infinite (a finite) number of nucleons. It is shown that our ( PB ) LOCV results are reasonably consistent with those of Fantoni and Schmidt (Periodic-Box) Fermi HyperNetted Chain ( ( PB ) FHNC ) calculations. In addition, unlike the thermodynamic limit, it is found that the normalization constraint in the periodic box is not exactly satisfied. However, it is realized that, by increasing the number of particles, the value of normalization constraint in the periodic box improves. Therefore, the two-body cluster approximation in the PBLOCV method becomes more reliable in the larger periodic boxes than the smaller ones. On the other hand, the PBLOCV energy per particle approaches the corresponding LOCV prediction for the large number of particles. The anisotropic behavior of the PB two-body distribution function, which is one of the main point of this report, is discussed. Finally, it is found that the PBLOCV results, i.e. the total (two-body cluster) energy and the normalization constraint, similar to the PBFHNC calculations depend on the number of particles in the periodic box and this dependence principally becomes negligible for the large number of particles.

The in-medium nn cross section and the transport properties of neutron matter using the LOCV effective two-body interaction

Abstract In order to calculate the in-medium neutron–neutron (nn) differential cross section and the transport properties of the pure neutron matter, the lowest order constrained variational (LOCV) method is applied to obtain the effective nucleon–nucleon (NN) interactions from the various bare realistic phenomenological NN interaction. The wide range of potentials, such as the Reid 68 , the Δ - Reid 68 , the V 8 ′ and the Av 18 are used as the input phenomenological interactions. The approach, based on the effective NN interaction, allows for a consistent description of the neutron matter and the nucleon–nucleon scattering in the nuclear medium. In this work, by using the above LOCV effective NN interactions, beside the nn differential cross section, the related quantities such as the shear viscosity and the thermal conductivity of the pure neutron matter, which are the important gradient for the stellar structure, are calculated within the Landau–Abrikosov–Khalatnikov (LAK) formalism. It is shown that, while the density dependence of effective mass are not important, the choice of in-medium effective interaction has crucial rule on the resulting nn differential cross section as well as the shear viscosity and the thermal conductivity of pure neutron matter. Finally, our above LOCV calculations are compared with those predicted by the other theoretical methods.

Effect of nondiagonal lowest order constrained variational effective two-body matrix elements on the binding energy of closed shell nuclei

The binding energies of the light, the moderate and the heavy closed shell nuclei, that is, ${}^{4}\mathrm{He}$, ${}^{12}\mathrm{C}$, ${}^{16}\mathrm{O}$, ${}^{28}\mathrm{Si}$, ${}^{32}\mathrm{S}$, ${}^{40}\mathrm{Ca}$, ${}^{56}\mathrm{Ni}$, ${}^{48}\mathrm{Ca}$, ${}^{90}\mathrm{Zr}$, ${}^{120}\mathrm{Sn}$, and ${}^{208}\mathrm{Pb}$ are calculated, using all of the channels-dependent effective two-body interactions (CDEI) matrix elements, which are generated through our lowest order constrained variational (LOCV) nuclear matter calculations with the $A{v}_{18}$ $({J}_{\mathrm{max}}=2$ and $5)$ phenomenological nucleon-nucleon potential. The $\mathcal{J}$-coupling scheme is applied to construct the interaction Hamiltonian matrix elements in the spherical harmonics oscillator shell model basis. In the channels with $Jg{J}_{\mathrm{max}}$, the CDEI are replaced by the average effective two-body interactions. It is shown that the nondiagonal matrix elements with the $A{v}_{18,{J}_{\mathrm{max}}=2}$ interaction increase the binding energies of nuclei; that is, the maximum magnitude of them is about 1.49 MeV for the ${}^{56}\mathrm{Ni}$ nucleus. However, for the similar calculations with the $A{v}_{18,{J}_{\mathrm{max}}=5}$ potential, they increase the binding energies of the symmetric nuclei more than the asymmetric ones; that is, the maximum magnitude of them is about 2.74 MeV for the ${}^{56}\mathrm{Ni}$ nucleus. Owing to the huge computational time that is obviously needed for the calculation of the matrix elements of the heavy closed shell nuclei, their binding energies are evaluated only at their saturation points, which are available from our previous works.

Correlations in nuclear matter

We analyze the nuclear matter correlation properties in terms of the pair correlation function. To this aim we systematically compare the results for the variational method in the Lowest Order Constrained Variational (LOCV) approximation and for the Bruekner-Hartree-Fock (BHF) scheme. A formal link between the Jastrow correlation factor of LOCV and the Defect Function (DF) of BHF is established and it is shown under which conditions and approximations the two approaches are equivalent. From the numerical comparison it turns out that the two correlation functions are quite close, which indicates in particular that the DF is approximately local and momentum independent. The Equations of State (EOS) of Nuclear Matter in the two approaches are also compared. It is found that once the three-body forces (TBF) are introduced the two EOS are fairly close, while the agreement between the correlation functions holds with or without TBF.

Two-nucleon spectral function of the16O nucleus using the lowest-order constrained variational state-dependent correlation functions of the Reid and Av18interactions

In this work, the two-nucleon spectral functions (TNSFs) are defined in terms of the state- and the density-dependent correlation functions in the framework of the lowest-order constrained variational (LOCV) method to calculate the TNSF of the ${}^{16}\text{O}$ nucleus in the ${}^{16}\text{O}$$(e,{e}^{\ensuremath{'}}NN)$${}^{14}\text{C}$ reaction. The Reid soft-core (Reid68) and the ${\text{Av}}_{18}$ potentials are used as the internucleon interactions. Since, the short-range correlation effects are imposed on the wave functions for the individual channels (e.g., the ${}^{1}{S}_{0}$ and ${}^{3}{P}_{J}$ channels); therefore, the defect wave functions are obtained for various channels such that the high relative momenta ($p>4\phantom{\rule{0.28em}{0ex}}{\text{fm}}^{\ensuremath{-}1}$) are ignored. The resulting TNSFs for the ${}^{16}\text{O}$ nucleus are compared with those of the dressed random phase approximation (DRPA) calculations of Geurts et al. and the experimental predictions, especially those of Onderwater et al., (NIKHEF group), where reasonable agreement is found. It is shown that the optimized state-dependent defect wave functions have substantial effects on the TNSF and it is not justified to use the simplified parametrized two-body correlation functions in all of the channels. In agreement with the experimental data of Onderwater et al., the knockout of a ${}^{1}{S}_{0}$ pair proton dominates the above reaction cross section. Finally, it is demonstrated that the ${0}^{+}$ and ${2}^{+}$ peaks, which are expected to be observed in the above reaction cross section, are moved to the lower momenta of out-going protons when the state-dependent correlation functions are imposed.

Lowest-order constrained variational calculation of asymmetric nuclear matter in a strong magnetic field

In the present work we study asymmetric nuclear matter in the presence of strong external magnetic fields using the lowest-order constrained variational (LOCV) method with the AV${}_{18}$ potential and employing a microscopic point of view. According to our results, the induced phase transition to antiferromagnetic states occurs for asymmetric nuclear matter in a strong magnetic field.

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.

The single-particle potential of nuclear matter in the LOCV framework

Abstract The density and momentum dependence of single-particle potential ( SPP ) and effective mass of symmetric nuclear matter are studied in the framework of lowest order constrained variational ( LOCV ) method. The R e i d 68 , the R e i d 68 - Δ and the A v 18 interactions are considered as the input nucleon–nucleon potentials. It is shown that the SPP of nuclear matter, at fixed density, is an increasing function of nucleon momentum, and it has different behavior for the Reid type potentials with respect to A v 18 interaction. We find good agreements between our LOCV SPP and those coming from others many-body techniques such as the ( Dirac –) Brueckner–Hartree–Foch ( ( D ) B H F ), the fermion hypernetted chain ( FHNC ), mean field ( MF ), etc. On the other hand SPP dramatically depends on the density at low and high nucleon momentums. While the effective mass of nuclear matter increases as we increase the nucleon momentum, it decreases at the Fermi surface. Again, good agreements are observed between our calculated effective mass and those coming from the methods mentioned above.

LOCV CALCULATIONS FOR POLARIZED LIQUID 3He: THE EFFECT OF THREE-BODY CLUSTER ENERGY

We have used the lowest order constrained variational (LOCV) method to calculate some of the ground state properties of the polarized liquid 3 He by inclusion of the three-body cluster energy contribution. It is shown that the contribution of the three-body cluster term for the ground state energy of this system is substantial, especially at high densities. It is also shown that the inclusion of the three-body energy contribution improves our result for the binding energy of the liquid 3 He . It is seen that the magnitude of the three-body cluster energy decreases by increasing the polarization. The magnetic susceptibility of the liquid 3 He has also been computed, showing no spontaneous ferromagnetic phase transition.

Finite temperature calculations for the bulk properties of a strange star using a many-body approach

We have considered a hot strange star matter, just after the collapse of a supernova, as a composition of strange, up and down quarks to calculate the bulk properties of this system at finite temperature with the density dependent bag constant. To parameterize the density dependent bag constant, we use our results for the lowest order constrained variational (LOCV) calculations of asymmetric nuclear matter. Our calculations for the structure properties of the strange star at different temperatures indicate that its maximum mass decreases by increasing the temperature. We have also compared our results with those of a fixed value of the bag constant. It can be seen that the density dependent bag constant leads to higher values of the maximum mass and radius for the strange star.

The effect of density dependent Av18 effective interaction on the ground state properties of heavy closed shell nuclei

Abstract The channel-dependent Argonne Av18 effective two-body interactions (CDEI) which are generated through the lowest order constrained variational (LOCV) calculation for asymmetric nuclear matter with the charge-dependent Av18 bare nucleon–nucleon potential are used to calculate the ground state properties of heavy closed shell nuclei such as 48Ca, 90Zr, 120Sn and 208Pb. The harmonic oscillator basis, and the local density approximation (LDA) are applied to create the relative and the center of mass dependent effective two-body potential. We get more binding with respect to the similar calculation with the Reid types potentials. It is tried to omit the LDA and perform full calculation with the Av18 CDEI for light nuclei. The results indicate that the LDA works quite well. It is also shown that in case of heavy closed shell nuclei and unlike our previous report with Reid68Day interaction, the contributions of higher partial waves ( J > 2 ) are very important for the calculations with Av18 potential and we get reasonable agreement between our calculated binding energies and RMS radii, with those predicated by the others methods, and the experimental data. Finally, the various aspects of channel and density dependent two-body effective interactions are discussed.

Investigation of the field-induced ferromagnetic phase transition in spin-polarized neutron matter: A lowest order constrained variational approach

In this paper, the lowest order constrained variational (LOCV) method has been used to investigate the magnetic properties of spin polarized neutron matter in the presence of strong magnetic field at zero temperature employing $AV_{18}$ potential. Our results indicate that a ferromagnetic phase transition is induced by a strong magnetic field with strength greater than $10^{18}\ G$, leading to a partial spin polarization of the neutron matter. It is also shown that the equation of state of neutron matter in the presence of magnetic field is stiffer than the case in absence of magnetic field.

THE DENSITY-DEPENDENT Av18 EFFECTIVE INTERACTION AND GROUND STATE OF CLOSED SHELL NUCLEI

The ground state properties of light closed shell nuclei, i.e. 4He, 12C, 16O, 28Si, 32S, 40Ca and 56Ni are studied by using the channel-dependent effective two-body interactions (CDEI's). The CDEI's are generated through the lowest-order constrained variational (LOCV) calculation for asymmetric nuclear matter with the charge-dependent Av18 bare nucleon–nucleon potential. The work is performed on the harmonic oscillator basis, and the local density approximation is applied to create the relative and the center of mass dependent effective two-body potential. Unlike nuclear matter, and similar to our previous calculations with the Reid68 interaction, while the Av18 potential under binds above nuclei up J max = 2 channel, it gives ground state binding energies closer to the experimental data with respect to the Δ- Reid68 and the Reid68 potentials. There are not much difference between the results of Av18 interaction with J max = 5, and those of Reid68Day potential which has been define up to J max = 5. The different CDEI's up to J max = 5 are discussed and the results of our calculations are compared with the other theoretical approaches and experimental data. Finally, it is shown that the contributions of higher partial waves (J>2) are not very important and the two-body kinetic energy in J = 1 channel is roughly twice as that of J = 0 which is not the case for the two-body potential energy.

The Density Dependence of Homogenous Normal Liquid Helium 3 One-Body Momentum Distribution in the LOCV and Ristig–Clark Formalisms

The lowest order constrained variational (LOCV) and the extended LOCV (ELOCV) method in the frame work of the Ristig–Clark formalism is used to calculate the density dependence of the normal liquid helium 3 one-body momentum distribution, n(k), at zero and finite temperatures. We impose the familiar 6–12 Lennard–Jones potential as the inter-atomic interaction. It is shown that the normal liquid helium 3 one-body momentum distribution, n(k), decreases at low momentums by increasing the density, but its tail becomes longer. The discontinuity of n(k) also decreases as we increase the density of the normal liquid helium 3. The discontinuity disappears at finite temperature but the same density dependent is observed as that of frozen case.