Hugenholtz-Van Hove Theorem Research Articles

Equation of state of neutron-rich matter in [formula omitted]-dimensions

Nuclear systems under constraints, with high degrees of symmetries and/or collectivities may be considered as moving effectively in spaces with reduced spatial dimensions. We first derive analytical expressions for the nucleon specific energy E0(ρ), pressure P0(ρ), incompressibility coefficient K0(ρ) and skewness coefficient J0(ρ) of symmetric nucleonic matter (SNM), the quadratic symmetry energy Esym(ρ), its slope parameter L(ρ) and curvature coefficient Ksym(ρ) as well as the fourth-order symmetry energy Esym,4(ρ) of neutron-rich matter in general d spatial dimensions (abbreviated as “dD”) in terms of the isoscalar and isovector parts of the isospin-dependent single-nucleon potential according to the generalized Hugenholtz–Van Hove (HVH) theorem. The equation of state (EOS) of nuclear matter in dD can be linked to that in the conventional 3-dimensional (3D) space by the ϵ-expansion which is a perturbative approach successfully used previously in treating second-order phase transitions and related critical phenomena in solid state physics and more recently in studying the EOS of cold atoms. The ϵ-expansion of nuclear EOS in dD based on a reference dimension df=d−ϵ is shown to be effective with −1≲ϵ≲1 starting from 1≲df≲3 in comparison with the exact expressions derived using the HVH theorem. Moreover, the EOS of SNM (with/without considering its potential part) is found to be reduced (enhanced) in lower (higher) dimensions, indicating in particular that the many-nucleon system tends to be more bounded but saturate at higher densities in spaces with lower dimensions. The symmetry energy perturbed from its counterpart in 3D is found to strongly depend on the momentum-dependence of the nucleon isovector potential. Moreover, the specific structure of the fourth-order symmetry energy in dD is also analyzed generally, and it is found to be naturally small, confirming the parabolic approximation for the EOS of neutron-rich matter from an even wider viewpoint. The links between the EOSs in 3D and dD spaces from the ϵ-expansion provide new perspectives to the EOS of neutron-rich matter. Further studies and potential applications of these links in nuclear physics and/or astrophysics are discussed.

The relationship of the neutron skin thickness to the symmetry energy and its slope

Abstract The neutron skin thickness of asymmetric semi-infinite nuclear matter (δRM) is shown to be a function of the Coulomb energy (Vc), the asymmetry energy coefficient (J), the slope (L) of the asymmetry energy, and the incompressibility coefficient (K), in addition to the Fermi momentum (kF) and the asymmetry parameter (I = (N − Z)/A). The relational formula is derived on the basis of the Hugenholtz–Van Hove theorem in the mean-field (MF) approximation for nuclear matter. Using the formula as a guide, the neutron skin thickness (δR) in 208Pb is examined in the MF models. The apparently spurious linear correlation between L and δR appears through the model-dependent correlation of L with J that is included in the main components of the formula.

The mean-square radius of the neutron distribution in the relativistic and non-relativistic mean-field models

Abstract It is investigated why the root-mean-square radius of the point neutron distribution is smaller by about 0.1 fm in non-relativistic mean-field models than in relativistic ones. The difference is shown to stem from the different values of the product of the effective mass and the strength of the one-body potential in the two frameworks. The values of those quantities are constrained by the Hugenholtz–Van Hove theorem. The neutron skin is not a simple function of the symmetry potential, but depends on the nucleon effective mass.

Basic quantities of the equation of state in isospin asymmetric nuclear matter

Based on the Hugenholtz–Van Hove theorem, six basic quantities of the EoS in isospin asymmetric nuclear matter are expressed in terms of the nucleon kinetic energy t(k), the isospin symmetric and asymmetric parts of the single-nucleon potentials \(U_0(\rho ,k)\) and \(U_{\text {sym},i}(\rho ,k)\). The six basic quantities include the quadratic symmetry energy \(E_{\text {sym,2}}(\rho )\), the quartic symmetry energy \(E_{\text {sym,4}}(\rho )\), their corresponding density slopes \(L_2(\rho )\) and \(L_4(\rho )\), and the incompressibility coefficients \(K_2(\rho )\) and \(K_4(\rho )\). By using four types of well-known effective nucleon–nucleon interaction models, namely the BGBD, MDI, Skyrme, and Gogny forces, the density- and isospin-dependent properties of these basic quantities are systematically calculated and their values at the saturation density \(\rho _0\) are explicitly given. The contributions to these quantities from t(k), \(U_0(\rho ,k)\), and \(U_{\text {sym},i}(\rho ,k)\) are also analyzed at the normal nuclear density \(\rho _0\). It is clearly shown that the first-order asymmetric term \(U_{\text {sym,1}}(\rho ,k)\) (also known as the symmetry potential in the Lane potential) plays a vital role in determining the density dependence of the quadratic symmetry energy \(E_{\text {sym,2}}(\rho )\). It is also shown that the contributions from the high-order asymmetric parts of the single-nucleon potentials (\(U_{\text {sym},i}(\rho ,k)\) with \(i>1\)) cannot be neglected in the calculations of the other five basic quantities. Moreover, by analyzing the properties of asymmetric nuclear matter at the exact saturation density \(\rho _{\text {sat}}(\delta )\), the corresponding quadratic incompressibility coefficient is found to have a simple empirical relation \(K_{\text {sat,2}}=K_{2}(\rho _0)-4.14 L_2(\rho _0)\).

Decomposition of nuclear symmetry energy based on Lorentz-covariant nucleon self-energies in relativistic Hartree-Fock approximation

Using the Lorentz-covariant decomposition of nucleon self-energies based on the Hugenholtz–Van Hove theorem, we study the effect of Fock terms on the nuclear symmetry energy and its slope parameter within relativistic Hartree-Fock approximation. It is found that the exchange contribution suppresses the nuclear symmetry energy and prevents the slope parameter from increasing monotonically at high densities. Furthermore, not only the isovector-vector (ρ) meson but also the isoscalar mesons (σ,ω) and pion give significant influence on the potential term of nuclear symmetry energy through the exchange diagrams.

Rearrangement term in the folding model of the nucleon optical potential

Based on the mean-field determination of the single-particle energy in nuclear matter that contains naturally a rearrangement term (RT) implied by the Hugenholtz–van Hove theorem, the folding model of the nucleon optical potential (OP) is extended to take into account the RT using the effective, density dependent CDM3Yn interaction. With the exchange part of the nucleon folded OP treated exactly in the Hartree–Fock manner, a compact nonlocal version of the folding model is suggested in the present work to determine explicitly the isospin-dependent, nonlocal central term of the nucleon OP. To solve the optical model (OM) equation with a complex nonlocal OP, the calculable R-matrix method is used to analyse the elastic neutron and proton scattering on 40,48Ca, 90Zr, and 208Pb targets at low energies. The inclusion of the RT into the folding model calculation of the nonlocal nucleon OP was shown to be essential for the overall good OM description of elastic nucleon scattering. To validate the nonlocal version of the folding model, the OM results given by the nonlocal folded nucleon OP are also compared with those given by the global parametrization of the nonlocal OP using the analytical nonlocal form factor suggested by Perey and Buck.

Mean-field description of heavy-ion scattering at low energies and fusion

The nuclear mean-field potential built up during the $${}^{12}\hbox {C}+{}^{12}\hbox {C}$$ and $${}^{16}\hbox {O}+{}^{16}\hbox {O}$$ collisions at low energies relevant for the carbon- and oxygen-burning processes is constructed within the double-folding model (DFM) using the realistic ground-state densities of $$^{12}\hbox {C}$$ and $$^{16}$$ O, and CDM3Yn density-dependent nucleon–nucleon (NN) interaction. The rearrangement term, indicated by the Hugenholtz–van Hove theorem for the single-particle energy in nuclear matter, is properly considered in the DFM calculation. To validate the use of the density-dependent NN interaction at low energies, an adiabatic approximation was suggested for the dinuclear overlap density. The reliability of the nucleus–nucleus potential predicted through this low-energy version of the DFM was tested in the optical model (OM) analysis of the elastic $${}^{12}\hbox {C}+{}^{12}\hbox {C}$$ and $${}^{16}\hbox {O}+{}^{16}\hbox {O}$$ scattering data at energies below 10 MeV/nucleon. These OM results provide a consistently good description of the elastic angular distributions and 90 $$^\circ$$ excitation function. The dinuclear mean-field potential predicted by the DFM is further used to determine the astrophysical S factor of the $${}^{12}\hbox {C}+{}^{12}\hbox {C}$$ and $${}^{16}\hbox {O}+{}^{16}\hbox {O}$$ fusions in the barrier penetration model. Without any adjustment of the potential strength, our results reproduce the non-resonant behavior of the S factor of the $${}^{12}\hbox {C}+{}^{12}\hbox {C}$$ and $${}^{16}\hbox {O}+{}^{16}\hbox {O}$$ fusions very well over a wide range of energies.

Nuclear mean field and double-folding model of the nucleus-nucleus optical potential

Realistic density dependent CDM3Yn versions of the M3Y interaction have been used in an extended Hartree-Fock (HF) calculation of nuclear matter (NM), with the nucleon single-particle potential determined from the total NM energy based on the Hugenholtz-van Hove theorem that gives rise naturally to a rearrangement term (RT). Using the RT of the single-nucleon potential obtained exactly at different NM densities, the density- and energy dependence of the CDM3Yn interactions was modified to account properly for both the RT and observed energy dependence of the nucleon optical potential. Based on a local density approximation, the double-folding model of the nucleus-nucleus optical potential has been extended to take into account consistently the rearrangement effect and energy dependence of the nuclear mean-field potential, using the modified CDM3Yn interactions. The extended double-folding model was applied to study the elastic $^{12}$C+$^{12}$C and $^{16}$O+$^{12}$C scattering at the refractive energies, where the Airy structure of the nuclear rainbow has been well established. The RT was found to affect significantly the real nucleus-nucleus optical potential at small internuclear distances, giving the potential strength close to that implied by the realistic optical model description of the Airy oscillation.

Quantifying correlations between isovector observables and the density dependence of the nuclear symmetry energy away from saturation density

According to the Hugenholtz-Van Hove theorem, the nuclear symmetry energy $S(\rho)$ and its slope $L(\rho)$ at arbitrary densities can be decomposed in terms of the density and momentum dependence of the single-nucleon potentials in isospin-asymmetric nuclear matter which are potentially accessible to experiment. We quantify the correlations between several well-known isovector observables and $L(\rho)$ to locate the density range in which each isovector observable is most sensitive to the density dependence of the $S(\rho)$. We then study the correlation coefficients between those isovector observables and all the components of the $L(\rho)$. The neutron skin thickness of $^{208}$Pb is found to be strongly correlated with the $L(\rho)$ at a subsaturation density of $\rho = 0.59 \rho_0$ through the density dependence of the first-order symmetry potential. Neutron star radii are found to be strongly correlated with the $L(\rho)$ over a wide range of supra-saturation densities mainly through both the density and momentum dependence of the first-order symmetry potential. Finally, we find that although the crust-core transition pressure has a complex correlation with the $L(\rho)$, it is strongly correlated with the momentum derivative of the first-order symmetry potential, and the density dependence of the second-order symmetry potential.

Constraining the neutron–proton effective mass splitting using empirical constraints on the density dependence of nuclear symmetry energy around normal density

According to the Hugenholtz–Van Hove theorem, nuclear symmetry energy Esym(ρ) and its slope L(ρ) at an arbitrary density ρ are determined by the nucleon isovector (symmetry) potential Usym(ρ,k) and its momentum dependence ∂Usym∂k. The latter determines uniquely the neutron–proton effective k-mass splitting mn−p⁎(ρ,δ)≡(mn⁎−mp⁎)/m in neutron-rich nucleonic matter of isospin asymmetry δ. Using currently available constraints on the Esym(ρ0) and L(ρ0) at normal density ρ0 of nuclear matter from 28 recent analyses of various terrestrial nuclear laboratory experiments and astrophysical observations, we try to infer the corresponding neutron–proton effective k-mass splitting mn−p⁎(ρ0,δ). While the mean values of the mn−p⁎(ρ0,δ) obtained from most of the studies are remarkably consistent with each other and scatter very closely around an empirical value of mn−p⁎(ρ0,δ)=0.27⋅δ, it is currently not possible to scientifically state surely that the mn−p⁎(ρ0,δ) is positive within the present knowledge of the uncertainties. Quantifying, better understanding and then further reducing the uncertainties using modern statistical and computational techniques in extracting the Esym(ρ0) and L(ρ0) from analyzing the experimental data are much needed.

Extracting the nuclear symmetry potential and energy from neutron–nucleus scattering data

According to the Hugenholtz–Van Hove theorem, the symmetry energy and its density slope can be decomposed uniquely in terms of the single-nucleon potential in asymmetric nuclear matter which, at normal density, can be constrained by the nucleon optical model potential extracted from analyzing the nucleon–nucleus scattering data. To more accurately extract information on the symmetry energy and its density slope at normal density from neutron–nucleus scattering data, going beyond the well-known Lane potential, we include consistently the second-order terms in isospin asymmetry in both the optical model potential and the symmetry energy decomposition. We find that the strength of the second-order symmetry potential Usym,2 in asymmetric nuclear matter is significant compared to the first-order one Usym,1, especially at high nucleon momentum. While the Usym,1 at normal density decreases with nucleon momentum, the Usym,2 is found to have the opposite behavior. Moreover, we discuss effects of the Usym,1 and Usym,2 on determining the density dependence of the symmetry energy, and we find that the available neutron–nucleus scattering data favor a softer density dependence of the symmetry energy at normal density with a negligible contribution from the Usym,2.

Lorentz covariant nucleon self-energy decomposition of the nuclear symmetry energy

Using the Hugenholtz–Van Hove theorem, we derive analytical expressions for the nuclear symmetry energy Esym(ρ) and its density slope L(ρ) in terms of the Lorentz covariant nucleon self-energies in isospin asymmetric nuclear matter. These general expressions are useful for determining the density dependence of the symmetry energy and understanding the Lorentz structure and the microscopic origin of the symmetry energy in relativistic covariant formulism. As an example, we analyze the Lorentz covariant nucleon self-energy decomposition of Esym(ρ) and L(ρ) and derive the corresponding analytical expressions within the nonlinear σ–ω–ρ–δ relativistic mean field model.

MEASUREMENT OF PHOTOFISSION CROSS-SECTION OF238UUSING MICROTRON FACILITY

Photofission cross-section of238U was measured using bremsstrahlung radiation energy 7.4–9.0 MeV with energy step of 0.4 MeV by employing Lexan polycarbonate film as solid state nuclear track detector (SSNTD). The photon intensity from the Microtron accelerator at a distance of 15 cm from the bremsstrahlung converter (tantalum target) facility was estimated to be 1010photons/sec using the code EGS-4. In this paper, details of the fission fragment angular distribution measurements of238U target using Lexan polycarbonate have been discussed. The photofission cross-section was calculated using the angular distribution of fission fragments and the results were compared with those obtained using the code EMPIRE-II and various barrier parameters of the RIPL-1, RIPL-2 libraries and with the new analytical fission barrier formula based on the Hugenholtz–Van Hove theorem. The present experimental measurements were in good agreement with the results obtained from the Empire-II code predictions for potential parameter taken from RIPL-1 and a newly developed analytical fission barrier formula.

GENERALIZED HUGENHOLTZ–VAN HOVE THEOREM FOR MULTI-COMPONENT FERMI SYSTEMS WITH MULTI-BODY FORCES

We show here that the Hugenholtz–Van Hove theorem for an infinite Fermi system remains valid even with all possible multi-body interaction terms taken into account either separately or in any combination under Hartree–Fock approximation. We further extend the theorem to obtain its most generalized form that would hold good for any multi-component Fermi system.

Analytical relations between nuclear symmetry energy and single-nucleon potentials in isospin asymmetric nuclear matter

Using the Hugenholtz-Van Hove theorem, we derive general expressions for the quadratic and quartic symmetry energies in terms of single-nucleon potentials in isospin asymmetric nuclear matter. These analytical relations are useful for gaining deeper insights into the microscopic origins of the uncertainties in our knowledge on nuclear symmetry energies especially at supra-saturation densities. As examples, the formalism is applied to two model single-nucleon potentials that are widely used in transport model simulations of heavy-ion reactions.

Measurement of photofission excitation of 237NP

Abstract The angular distribution of 237Np fission fragments was measured over the energy range 7.4 MeV–9.2 MeV in steps of 0.4 MeV of bremsstrahlung radiation. Lexan polycarbonate films (SSNTD) were used to detect the fission fragments. The photon intensity from the Microtron accelerator at a distance of 15 cm from the bremsstrahlung converter (tantalum target) facility was estimated to be 1010 photons s−1 using the code EGS-4. The photofission cross-section was calculated using the angular distribution of fission fragments and the results were compared with those obtained using the code EMPIRE-II and various barrier parameters of the RIPL-1, RIPL-2 libraries and with the new analytical fission barrier formula based on the Hugenholtz–Van Hove theorem. The results are in good agreement with the prediction of the RIPL-1 barrier parameter and analytical fission barrier formula.

Relativistic chiral Hartree-Fock description of nuclear matter with constraints from nucleon structure and confinement

We present a relativistic chiral effective theory for symmetric and asymmetric nuclear matter taken in the Hartree-Fock scheme. The nuclear binding is ensured by a background chiral invariant scalar field associated with the radial fluctuations of the chiral quark condensate. Nuclear matter saturation is obtained once the scalar response of the nucleon-generating three-body repulsive forces is incorporated. For those parameters related to the scalar sector and quark confinement mechanism inside the nucleon we make use of an analysis of lattice results on nucleon mass evolution with quark mass. The other parameters are constrained as much as possible by standard hadron and nuclear phenomenology. Special attention is paid to the treatment of the propagation of the scalar fluctuations. The rearrangement terms associated with in-medium modified mass and coupling constants are explicitly included to satisfy the Hugenholtz-Van Hove theorem. We point out the important role of the tensor piece of the rho exchange Fock term to reproduce the asymmetry energy of nuclear matter. We also discuss the isospin dependence of the Landau nucleon effective mass.

Properties of nuclear and neutron matter in a nonlinear σ– ω– ρ mean-field approximation with self- and mixed-interactions

Properties of nuclear and neutron matter are discussed in a nonlinear σ– ω– ρ mean-field approximation with self- and mixed-interactions of mesons and baryons. The nonlinear interactions of mesons are naturally renormalized as effective masses of mesons by thermodynamic consistency required by the theory of conserving approximations. The nonlinear mean-field approximation results in a thermodynamically consistent approximation that maintains Hugenholtz–Van Hove theorem and Landau's requirement of quasiparticles, and it is generally proved that the nonlinear approximation is equivalent to the Hartree approximation with effective masses of baryons and mesons. The approximation is applied to nuclear and neutron matter, which suggests that the lower bound of nuclear compressibility, K ∼ 150 MeV , be required to be consistent with the binding energy ( − 15.75 MeV at k F = 1.30 fm −1 ) and the maximum mass of neutron stars ( M max ⩾ 2.00 M ⊙ ). Since the contributions of nonlinear interactions are strictly confined as the effective masses of mesons and baryons, properties of nuclear and neutron matter will enable us to study the validity of nonlinear interactions of mesons. The implication of thermodynamic consistency to the hypothesis of naturalness is discussed. The density-dependence of the compressibility, symmetry energy together with effective masses of hadrons will be important constraints for nonlinear mean-field approximations, and the accumulating data together with accurate measurements of observables in high density and energy region will supply significant information in order to testify theoretical consistency of nuclear models.

The Theory of Conserving Approximations and the Density Functional Theory in Approximations for Nuclear Matter

Self-consistent relativistic approximations for nuclear matter are discussed in terms of the theory of conserving approximations and a self-consistent Green’s function approach. Moreover, the nuclear density functional theory is explained in terms of the theory of conserving approximations. The analysis is applied to a relativistic field-theoretical model for nuclear matter, in order to construct self-consistent Dirac-Hartree-Fock and Dirac-Ring approximations that maintain the Hugenholtz-Van Hove theorem and Landau’s hypothesis for quasiparticles. A self-consistent set of functional equations for self-energies is obtained by the stationary condition required by the theory of conserving approximations. The exact self-energy solutions are derived, and retardation corrections that maintain the HugenholtzVan Hove theorem and Landau’s hypothesis for quasiparticles are generated explicitly in the self-energies. This approach is a direct method in order to construct self-consistent approximations for nuclear matter, and the results are applied to Fermi-liquid properties of nuclear matter and properties of neutron stars.

Erratum: Return of the EMC effect [Phys. Rev. C65, 015211 (2002)

The relationship between the properties of nuclear matter and structure functions measured in lepton-nucleus deep inelastic scattering is investigated using light front dynamics. We find that relativistic mean field models such as the Walecka, Zimanyi-Moszkowski (and point-coupling versions of the same) and Rusnak-Furnstahl models contain essentially no binding effect, in accord with an earlier calculation by Birse. These models are found to obey the Hugenholtz-van Hove theorem, which is applicable if nucleons are the only degrees of freedom. Any model in which the entire Fock space wave function can be represented in terms of free nucleons must obey this theorem, which implies that all of the plus momentum is carried by nucleons, and therefore that there will be essentially no binding effect. The explicit presence of nuclear mesons allows one to obtain a modified form of the Hugenholtz-van Hove theorem, which is equivalent to the often-used momentum sum rule. These results argue in favor of a conclusion that the depletion of the deep inelastic structure function observed in the valence quark regime is due to some interesting effect involving dynamics beyond the conventional nucleon-meson treatment of nuclear physics.