A new table of the nuclear equation of state (EOS) based on realistic nuclear potentials is constructed for core-collapse supernova numerical simulations. Adopting the EOS of uniform nuclear matter constructed by two of the present authors with the cluster variational method starting from the Argonne v18 and Urbana IX nuclear potentials, the Thomas Fermi calculation is performed to obtain the minimized free energy of a Wigner-Seitz cell in non-uniform nuclear matter. As a preparation for the Thomas Fermi calculation, the EOS of uniform nuclear matter is modified so as to remove the effects of deuteron cluster formation in uniform matter at low densities. Mixing of alpha particles is also taken into account following the procedure used by Shen et al. (1998, 2011). The critical densities with respect to the phase transition from non-uniform to uniform phase with the present EOS are slightly higher than those with the Shen EOS at small proton fractions. The critical temperature with respect to the liquid gas phase transition decreases with the proton fraction in a more gradual manner than in the Shen EOS. Furthermore, the mass and proton numbers of nuclides appearing in non-uniform nuclear matter with small proton fractions are larger than those of the Shen EOS. These results are consequences of the fact that the density derivative coefficient of the symmetry energy of our EOS is smaller than that of the Shen EOS. (C) 2017 Elsevier B.V. All rights reserved.

We investigate the effects of the odd-state part of bare Lambda Lambda interactions on the structure of neutron stars (NSs) by constructing equations of state (EOSs) for uniform nuclear matter containing Lambda and Sigma(-) hyperons with use of the cluster variational method. The isoscalar part of the Argonne v18 two-nucleon potential and the Urbana IX three-nucleon potential are employed as the interactions between nucleons, whereas, as the bare Lambda N and even-state Lambda Lambda interactions, two-body central potentials that are determined so as to reproduce the experimental data on single-and double-Lambda hypernuclei are adopted. In addition, the Sigma(-) N interaction is constructed so as to reproduce the empirical single-particle potential of Sigma(-) in symmetric nuclear matter. Since the odd-state part of the Lambda Lambda interaction is not known owing to lack of experimental data, we construct four EOSs of hyperonic nuclear matter, each with a different odd-state part of the Lambda Lambda interaction. The EOS obtained for NS matter becomes stiffer as the odd-state Lambda Lambda interaction becomes more repulsive, and correspondingly the maximum mass of NSs increases. It is interesting that the onset density of Sigma(-) depends strongly on the repulsion of the odd-state Lambda Lambda interaction. Furthermore, we take into account the three-baryon repulsive force to obtain results that are consistent with observational data on heavy NSs.

© 2016 American Physical Society. We investigate the effects of the odd-state part of bare ΛΛ interactions on the structure of neutron stars (NSs) by constructing equations of state (EOSs) for uniform nuclear matter containing Λ and Σ- hyperons with use of the cluster variational method. The isoscalar part of the Argonne v18 two-nucleon potential and the Urbana IX three-nucleon potential are employed as the interactions between nucleons, whereas, as the bare ΛN and even-state ΛΛ interactions, two-body central potentials that are determined so as to reproduce the experimental data on single- and double-Λ hypernuclei are adopted. In addition, the Σ-N interaction is constructed so as to reproduce the empirical single-particle potential of Σ- in symmetric nuclear matter. Since the odd-state part of the ΛΛ interaction is not known owing to lack of experimental data, we construct four EOSs of hyperonic nuclear matter, each with a different odd-state part of the ΛΛ interaction. The EOS obtained for NS matter becomes stiffer as the odd-state ΛΛ interaction becomes more repulsive, and correspondingly the maximum mass of NSs increases. It is interesting that the onset density of Σ- depends strongly on the repulsion of the odd-state ΛΛ interaction. Furthermore, we take into account the three-baryon repulsive force to obtain results that are consistent with observational data on heavy NSs.

The variational method with explicit energy functionals (EEFs) is applied to infinite neutron matter. Starting from the Argonne v8' two-body potential and the repulsive part of the Urbana IX three-body potential, the energy per neutron is expressed explicitly with the spin-dependent central, tensor, and spin-orbit distribution functions. This EEF is constructed as an extension of the previously proposed one for the Argonne v6' potential, including central and tensor forces. The Euler-Lagrange equations derived from this EEF are solved numerically. The obtained fully minimized energy with this EEF is in good agreement with that obtained from the auxiliary field diffusion Monte Carlo calculation.

The equation of state (EOS) for hot asymmetric nuclear matter that is constructed with the variational method starting from the Argonne v18 and Urbana IX nuclear forces is applied to spherically symmetric core-collapse supernovae (SNe). We first investigate the EOS of isentropic beta-stable SN matter, and find that matter with the variational EOS is more neutron-rich than that with the Shen EOS. Using the variational EOS for uniform matter supplemented by the Shen EOS for non-uniform matter at low densities, we perform general-relativistic spherically symmetric simulations of core-collapse SNe with and without neutrino transfer, starting from a presupernova model of 15 M-circle dot. In the adiabatic simulation without neutrino transfer, the explosion is successful, and the explosion energy with the variational EOS is larger than that with the Shen EOS. In the case of the simulation with neutrino transfer, the shock wave stalls and then the explosion fails, as in other spherically symmetric simulations. The inner core with the variational EOS is more compact than that with the Shen EOS, due to the relative softness of the variational EOS. This implies that the variational EOS is more advantageous for SN explosions than the Shen EOS.

We construct a new nuclear equation of state (EOS) for core-collapse supernova (SN) simulations using the variational many-body theory. For uniform nuclear matter, the EOS is constructed with the cluster variational method starting from the realistic nuclear Hamiltonian composed of the Argonne v18 two-body potential and the Urbana IX three-body potential. The masses and radii of neutron stars calculated with the obtained EOS at zero temperature are consistent with recent observational data. For non-uniform nuclear matter, we construct the EOS in the Thomas-Fermi approximation. In this approximation, we assume a functional form of the density distributions of protons, neutrons, and alpha-particles, and minimize the free energy density in a Wigner-Seitz cell with respect to the parameters included in the assumed density distribution functions. The phase diagram of hot nuclear matter at a typical temperature is reasonable as compared with that of the Shen EOS.

We calculate nuclear composition in supernova (SN) matter explicitly taking into account the temperature dependence of nuclear shell effects. The abundance of nuclei in SN matter is important in the dynamics of core-collapse supernovae and, in recently constructed equations of state (EOS) for SN matter, the composition of nuclei are calculated assuming nuclear statistical equilibrium wherein the nuclear internal free energies govern the composition. However, in these EOS, thermal effects on the shell energy are not explicitly taken into account. To address this shortfall, we calculate herein the shell energies of hot nuclei and examine their influence on the composition of SN matter. Following a simplified macroscopic-microscopic approach, we first calculate single-particle (SP) energies by using a spherical Woods-Saxon potential. Then we extract shell energies at finite temperatures using Strutinsky method with the Fermi distribution as the average occupation probability of the SP levels. The results show that at relatively low temperatures, shell effects are still important and magic nuclei are abundant. However, at temperatures above approximately 2 MeV, shell effects are almost negligible, and the mass fractions with shell energies including the thermal effect are close to those obtained from a simple liquid drop model at finite temperatures.

We have applied the variational method using explicit energy functionals (EEFs) to energy calculations of infinite nuclear matter. In EEFs, the energy per nucleon is explicitly expressed with spin-isospin-dependent two-body distribution functions, which are regarded as variational functions, and fully minimized energies are conveniently calculated with the EEF. A remarkable feature of this approach is that EEFs guarantee non-negativeness of structure functions. In this study, we extend the EEF variational method so as to consider stateindependent three-body forces for neutron matter at finite temperatures following the procedure proposed by Schmidt and Pandharipande. For neutron matter, the free energies obtained with the Argonne v4' two-body potential and the repulsive part of the Urbana IX (UIX) threebody potential are quite reasonable. Furthermore, we improve the EEF of nuclear matter using the two-body central and tensor forces by considering the main three-body cluster terms and guaranteeing non-negativeness of tensor structure functions. In addition, healing distances are introduced for two-body distribution functions so that Mayer's condition is satisfied. The obtained energies per neutron of neutron matter with the Argonne v6' two-body potential and the repulsive part of the UIX potential are in good agreement with those obtained by auxiliary field diffusion Monte Carlo calculations.

We report on an equation of state (EOS) of hot asymmetric nuclear matter constructed using the variational method and its application to hydrodynamic simulations of core-collapse supernovae. This nuclear EOS is based on the AV18 two-body potential and UIX three-body potential, and the energy per nucleon at zero temperature is constructed with the cluster variational method. At finite temperatures, the free energies per nucleon are calculated with an extension of the variational method devised by Schmidt and Pandharipande. This EOS is in good agreement with that by the Fermi hypernetted chain variational calculations at zero and finite temperatures, and the structure of neutron stars calculated with this EOS is consistent with recent observational data. Using this nuclear EOS, we perform a spherically symmetric general-relativistic adiabatic simulation of the SN explosion. The explosion energy calculated with our EOS in the present simulation is larger than that obtained with the Shen EOS, implying that the variational EOS is softer than the Shen EOS.

We report the current status of our project to construct a new nuclear equation of state (EOS) with the variational method for core-collapse supernova (SN) simulations. Starting from the realistic nuclear Hamiltonian, the EOS for uniform nuclear matter is constructed with the cluster variational method: For non-uniform nuclear matter, the EOS is calculated with the Thomas-Fermi method. The obtained thermodynamic quantities of uniform matter are in good agreement with those with more sophisticated Fermi Hypernetted Chain variational calculations, and phase diagrams constructed so far are close to those of the Shen-EOS. The structure of neutron stars calculated with this EOS at zero temperature is consistent with recent observational data, and the maximum mass of the neutron star is slightly larger than that with the Shen-EOS. Using the present EOS of uniform nuclear matter, we also perform the 1D simulation of the core-collapse supernovae by a simplified prescription of adiabatic hydrodynamics. The stellar core with the present EOS is more compact than that with the Shen-EOS, and correspondingly, the explosion energy in this simulation with the present EOS is larger than that with the Shen-EOS.

An equation of state (EOS) for uniform asymmetric nuclear matter (ANM) is constructed at zero and finite temperatures by the variational method starting from the nuclear Hamiltonian that is composed of the Argonne v18 and Urbana IX potentials. At zero temperature, the two-body energy is calculated with the Jastrow wave function in the two-body cluster approximation which is supplemented by Mayer's condition and the healing-distance condition so as to reproduce the result by Akmal, Pandharipande and Ravenhall. The energy caused by the three-body force is treated somewhat phenomenologically so that the total energy reproduces the empirical saturation conditions. The masses and radii of neutron stars obtained with the EOS are consistent with recent observational data. At finite temperatures, thermodynamic quantities such as free energy, internal energy, entropy, pressure and chemical potentials are calculated with an extension of the method by Schmidt and Pandharipande. The validity of the frozen-correlation approximation employed in this work is confirmed as compared with the result of the fully minimized calculation. The quadratic proton-fraction-dependence of the energy of ANM is confirmed at zero temperature, whereas the free energy of ANM deviates from the quadratic proton-fraction-dependence markedly at finite temperatures. The obtained EOS of ANM will be an important ingredient of a new nuclear EOS for supernova numerical simulations. (C) 2013 Elsevier B.V. All rights reserved.

We report the current status of our project to construct a new nuclear equation of state (EOS), which may be used for supernova numerical simulations, based on the cluster variational method starting from the realistic nuclear Hamiltonian. We also take into account a higher-order correction to the energy of the nuclear three-body force (TBF). The nuclear EOSs with and without the higher-order TBF correction at zero temperature are very close to each other, when parameters are readjusted so as to reproduce the empirical saturation data.

エネルギー汎関数を用いた変分法により、一様核物質の一核子当たりのエネルギーを求める研究の現状を報告する。本変分法では一核子当たりのエネルギーを2体分布関数の汎関数として陽に表すことにより、十分に最小化されたエネルギーを得る。現状で特に2体核力の非中心力成分を考慮すると、本変分法で計算されたエネルギーが低くなる傾向があるため、拡張したMayer条件を考慮するなど、本変分法の改良を進めている。

We calculate the equation of state (EOS) of asymmetric nuclear matter at finite temperatures with the cluster variational method based on the realistic nuclear Hamiltonian composed of the AV18 and UIX nuclear potentials. The free energy is calculated with an extension of the variational method proposed by Schmidt and Pandharipande. The obtained thermodynamic quantities such as entropy, internal energy, pressure and chemical potential derived from the free energy are reasonable. It is also found that the present variational calculation is self-consistent. These thermodynamic quantities are essential ingredients in our project for constructing a new nuclear EOS applicable to supernova simulations.

We calculated the energies of asymmetric nuclear matter at zero and finite temperatures with the cluster variational method. At zero temperature, the expectation value of the two-body Hamiltonian composed of the kinetic energies and the AV18 two-body forces is calculated with the Jastrow wave function in the two-body cluster approximation. The obtained two-body energy is in good agreement with the result with the Fermi Hypernetted Chain (FHNC) calculation by Akmal et al. The energy caused by the UIX three-body forces is treated somewhat phenomenologically so that the total energy reproduces the empirical saturation point. Furthermore, the parameters included in the three-body energy are readjusted so that the Thomas-Fermi (TF) calculations with use of the obtained energy of nuclear matter reproduce the gross feature of the experimental data on atomic nuclei. The nuclear species in the neutron star crust obtained by the TF calculation are reasonable. The free energies of asymmetric nuclear matter at finite temperatures are calculated with an extension of the method proposed by Schmidt and Pandharipande. The obtained free energies are in good agreement with those with the FHNC method, and it is also found that the present variational method is self-consistent. It is remarkable that the symmetry free energy is not proportional to (1 - 2x)(2), where x is the proton fraction. With use of the obtained thermodynamic quantities, we are constructing a new nuclear equation of state for supernova simulations.

The equation of state (EOS) for neutron star (NS) crusts is studied in the Thomas-Fermi (TF) approximation using the EOS for uniform nuclear matter obtained by, the variational method with the realistic nuclear Hamiltonian. The parameters associated with the nuclear three-body force, which are introduced to describe the saturation properties, are finely adjusted so that the TF calculations for isolated atomic nuclei reproduce the experimental data on masses and charge distributions satisfactorily. The resulting root-mean-square deviation of the masses from the experimental data for mass-measured nuclei is about 3 MeV. With the use of the nuclear EOS thus obtained, the nuclear species in the NS crust at zero temperature are determined. The predicted proton numbers of the nuclei in the NS crust are close to the gross behavior of the results obtained by Negele and Vautherin, while they are larger than those for the EOS obtained by Shen et al. owing to the difference in the symmetry energy. The density profile of NS is calculated with the constructed EOS.

The equation of state (EOS) is calculated for uniform nuclear matter at zero and finite temperatures with the variational method. Making use of uncertainty of the three-body nuclear force, adjustable parameters in the nuclear EOS are tuned so that the Thomas-Fermi calculations for beta-stable nuclei with the EOS reproduce the empirical data. The calculated nuclear properties imply that larger symmetry energy of the EOS is preferable to reproduce the empirical beta-stability line. The expectation value of the nuclear Hamiltonian caused by the 2 pi-exchange three-body nuclear force is uncertain and related to the symmetry energy.

The approximate energy expression for neutron matter is refined with respect to the tensor forces. The refined energy expression includes the main part of the three-body-cluster kinetic-energy terms caused by tensor correlations appropriately. Furthermore, the new energy expression guarantees necessary conditions on tensor structure functions. Significant improvement is seen in the numerical results of the variational calculation with the refined energy expression for neutron matter.

We construct an equation of state (EOS) for uniform nuclear matter at zero and finite temperatures with the variational method starting from the realistic nuclear Hamiltonian composed of the AV18 two-body potential and the UIX three-body interaction. Using the obtained EOS, we perform the Thomas-Fermi calculation of atomic nuclei, and tune adjustable parameters in the EOS for uniform matter to reproduce empirical data of beta-stable nuclei, toward a nuclear EOS table for supernova simulations.

We refine the approximate energy expression for neutron matter by taking into account three-body cluster terms with tensor correlations. The energy expression is an explicit functional of two-body distribution functions and used conveniently in the variational method. The energy expression proposed previously does not include kinetic energy terms caused by noncentral correlations sufficiently, and the obtained energy is too low. In this study, we refine the energy expression by introducing the main part of the missing three-body cluster terms that are related to the tensor correlations. The refined energy expression automatically guarantees necessary conditions on tensor structure functions. The obtained energy per neutron with the v6'potential is considerably higher than that of the old energy expression.

An equation of state (EOS) for uniform nuclear matter is constructed at zero and finite temperatures with the variational method starting from the realistic nuclear Hamiltonian composed of the Argonne V18 and UIX potentials. The energy is evaluated in the two-body cluster approximation with the three-body-force contribution treated phenomenologically so as to reproduce the empirical saturation conditions. The obtained energies for symmetric nuclear matter and neutron matter at zero temperature are in fair agreement with those by Akmal, Pandharipande and Ravenhall, and the maximum mass of the neutron star is 2.2M(circle dot). At finite temperatures, a variational method by Schmidt and Pandharipande is employed to evaluate the free energy, which is used to derive various thermodynamic quantities of nuclear matter necessary for supernova simulations. The result of this variational method at finite temperatures is found to be self-consistent. (c) 2007 Elsevier B.V. All rights reserved.

An equation of state (EOS) for uniform nuclear matter is constructed at zero and finite temperatures with the variational method starting from a realistic nuclear Hamiltonian composed of the AV18 two-body potential and the UIX three-nucleon interaction (TNI). The two-body energy is calculated with a Jastrow wave function, and the TNI energy with the Fermi-gas wave function. In this calculation, the Fujita-Miyazawa (FM) 2 pi-exchange term of the TNI energy is positive for symmetric nuclear matter. By taking into account the correlation between nucleons perturbatively, the expectation value of the FM term is modified to be negative in the low density region.

Asymmetric nuclear matter at zero temperature is studied using a variational method which is an extension of the methods used by the present authors previously for simpler systems. An approximate expression for the energy per nucleon in asymmetric nuclear matter is derived through a combination of two procedures, one used for symmetric nuclear matter and the other for spin-polarized liquid He-3 with spin polarization replaced by isospin polarization. The approximate expression for the energy is obtained as a functional of various spin-isospin-dependent radial distribution functions, tenser distribution functions, and spin-orbit distribution functions. The Euler-Lagrange equations are derived to minimize this approximate expression for the energy; they consist of 16 coupled integro differential equations for various distribution functions. These equations were solved numerically for several values of the nucleon number density rho and for many degrees of asymmetry zeta[zeta = (rho(n) - rho(p)/rho, where rho(n)(rho(p)) is the neutron (proton) number density]. Unexpectedly, we find that the energies at a fixed density cannot be represented by a power series in zeta(2). A new energy term, epsilon(1)(zeta(2) + zeta 0(2))(1/2) where zeta(0) is a small number and epsilon(1) is a positive coefficient, is proposed. It is shown that if the power series is supplemented with this new term, it reproduces the energies obtained by variational calculations very accurately. This new term is studied in relation to cluster formation in nuclear matter, and some mention is made of a possible similar term in the mass formula for finite nuclei.

A new scenario for neutron-star cooling is suggested by the correspondence between pion condensation, induced by critical spin-isospin fluctuations, and the metal-insulator phase transition in a 2D electron gas. Above the threshold density for pion condensation, the neutron single-particle spectrum acquires an insulating gap that quenches neutron contributions to neutrino production. In the liquid phase just below the transition, the fluctuations play dual roles by (i) creating a multisheeted neutron Fermi surface that extends to low momenta and activates the normally forbidden direct Urca cooling mechanism, and (ii) amplifying the nodeless P-wave neutron superfluid gap while suppressing S-wave pairing. Lighter stars without a pion-condensed core undergo slow cooling, whereas enhanced cooling occurs in heavier stars via direct Urca emission from a thin shell of the interior.

Two variational methods are used to calculate energies per nucleon of asymmetric nuclear matter. One is the explicit energy functional method, in which approximate energy expressions are proposed as explicit functionals of the variational functions. The other is the two-body, cluster variational method. The numerical results show that, in both cases, the saturation densities decrease markedly as the proton fraction decreases.

An approximate energy expression is proposed for arbitrarily spin-polarized Fermi liquids with central two-body forces. It is explicitly expressed as a functional of spin-dependent radial distribution functions and can be used conveniently in the variational method. It includes the potential energies completely and the kinetic energies up to main parts of the three-body cluster terms. This approximation is similar to that used previously for spin-unpolarized and fully polarized matter. A notable feature of this expression is that it guarantees the necessary conditions on arbitrarily spin-polarized structure functions automatically. The Euler-Lagrange equations are derived from this energy expression and are numerically solved for arbitrarily spin-polarized liquid 3 He. The results for liquid He-3 with the HFDHE2 potential are consistent with the nearly ferromagnetic property.

The energy per nucleon for infinite nuclear matter with the parametrized Paris potential is calculated using the variational method recently proposed by Takano and Yamada. Approximate energy expressions playing an important role in this variational method are refined for the Paris potential, and in particular for the quadratic momentum parts in it. Since this variational method without any constraints gives extreme overbinding, an effective theory also proposed in conjunction with the above-mentioned variational method is applied, in a modified manner, to the calculations for symmetric nuclear matter and neutron matter with the Paris potential, as well as the Hamada-Johnston and AV14 potentials. The obtained equation of state (EOS) for the Paris potential is, unlike the unrealistically soft EOS for the AV14 potential, capable of describing neutron stars with a reasonable maximum mass of 1.65 M-..

Approximate energy expressions are proposed for infinite zero-temperature neutron matter and symmetric nuclear matter by taking into account noncentral forces. They are explicitly expressed as functionals of spin-isospin-dependent radial distribution functions, tensor distribution functions and spin-orbit distribution functions: and can be used conveniently in the variational method. The two-body noncentral cluster terms are fully included in these expressions, while the degree of inclusion of the three-body cluster terms related to noncentral forces is less than that of the purely central three-body terms. The Euler-Lagrange equations are derived from these energy expressions and numerically solved for neutron matter and symmetric nuclear matter. The Hamada-Johnston and AV14 potentials are used as the two-body nuclear force. The results show that the noncentral forces cause the total energy of symmetric nuclear matter to be decreased too much with a saturation density which is too high. Discussion is given as to the reason for this disagreement with experiment, and the long tails of the noncentral distribution functions obtained in the numerical calculations are suspected to be the main reason. Then, an effective theory is proposed by introducing a density-dependent modification of the noncentral part of the energy expression to suppress the long tails of the noncentral distribution functions. With a suitable choice of the value of a parameter included in the modification, the saturation point (both the energy and the density) of symmetric nuclear matter can be reproduced with the Hamada-Johnston potential. Neutron stars are studied by use of this effective theory, and reasonable results are obtained.

A refinement is made on the approximate energy expression proposed previously by Takano and Yamada for infinite zero-temperature systems of spin-1/2 fermions interacting through two-body central forces which may depend on the two-body spin states. This refinement is accomplished through incorporation of a certain part of the three-body-cluster exchange terms into the energy expression. The Euler-Lagrange equation is derived from the refined energy expression and solved numerically for liquid He-3 and for neutron matter. The energy per particle for liquid He-3 is in better agreement with experimental data, while the effect of this refinement is relatively small for neutron matter. The three-body cluster terms not included in the refined energy expression are evaluated perturbatively and shows to be negligibly small for liquid He-3 as well as for neutron matter with central forces.

An approximate energy expression is proposed for infinite zero-temperature symmetric nuclear matter in which nucleons interact through two-body central forces depending on the two-body spin-isospin states. It is a functional of spin-isospin-dependent radial distribution functions, and includes the potential energies completely and the kinetic energies up to the main part of the three-body cluster terms. A notable feature of this expression is that it automatically guarantees the necessary conditions on the spin-isospin-dependent structure functions. This expression is conveniently used in the variational method; the Euler-Lagrange equations are derived and numerically solved for four typical potentials. The obtained energies per nucleon are shown together with the spin-isospin-dependent radial distribution functions and structure functions.

Approximate energy expressions are proposed for infinite zero-temperature systems of spin-1/2 fermions interacting through two-body central forces which may depend on the two-body spin states. They are explicitly expressed as functionals of spin-dependent radial distribution functions, and can be used conveniently in the variational method. They include the potential energies completely and the kinetic energies up to main parts of the three-body cluster terms. A notable feature of these expressions is that they guarantee the necessary conditions on the spin-dependent structure functions automatically. The Euler-Lagrange equations are derived from these energy expressions and numerically solved for liquid He-3 and hypothetical neutron matter interacting only through central forces. The parts of the three-body cluster terms which are not included in the above expressions are treated perturbatively. The results for liquid He-3 are in fair agreement with experiment. The obtained spin-dependent radial distribution functions and structure functions are discussed in relation with pairing and cluster formation.

Spin-isospin-dependent liquid structure functions are defined in terms of the projection operators onto particular spin-isospin pair states. Then, necessary conditions on these structure functions are derived in the form of inequalities. The systems studied are liquid He-3, neutron matter, symmetric nuclear matter, asymmetric nuclear matter, and partially spin-polarized asymmetric nuclear matter. More than one inequalities is obtained for each of these systems, and the number of the inequalities increases with increase of the spin-isospin degree of freedom. It is argued that these conditions can be used to find approximate kinetic-energy expressions of fermion systems.