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.

Born–Jordan pseudodifferential operators and the Dirac correspondence: Beyond the Groenewold–van Hove theorem

Quantization procedures play an essential role in operator theory, time–frequency analysis and, of course, in quantum mechanics. Roughly speaking the basic idea, due to Dirac, is to associate to any symbol, or observable, a(x,ξ) an operator Op(a), according to some axioms dictated by physical considerations. This led to the introduction of a variety of quantizations. They all agree when the symbol a(x,ξ)=f(x) depends only on x or a(x,ξ)=g(ξ) depends only on ξ:Op(f⊗1)u=fu,Op(1⊗g)u=F−1(gFu) where F stands for the Fourier transform. Now, Dirac aimed at finding a quantization satisfying, in addition, the key correspondence[Op(a),Op(b)]=iOp({a,b}) where [,] stands for the commutator and {,} for the Poisson brackets, which would represent a tight link between classical and quantum mechanics. Unfortunately, the famous Groenewold–van Hove theorem states that such a quantization does not exist, and indeed most quantization rules satisfy this property only approximately.In this work we show that the above commutator rule in fact holds for the Born–Jordan quantization, at least for symbols of the type f(x)+g(ξ). Moreover we will prove that, remarkably, this property completely characterizes this quantization rule, making it the quantization which best fits the Dirac dream.

Extended Hartree-Fock study of the single-particle potential: The nuclear symmetry energy, nucleon effective mass, and folding model of the nucleon optical potential

The nucleon mean-field potential has been thoroughly investigated in an extended Hartree-Fock (HF) calculation of nuclear matter (NM) using the CDM3Y3 and CDM3Y6 density dependent versions of the M3Y interaction. The single-particle (SP) energies of nucleons in NM are determined according to the Hugenholtz--Van Hove theorem, which gives rise naturally to a rearrangement term (RT) of the SP potential at the Fermi momentum. Using the RT obtained exactly at the different NM densities and neutron-proton asymmetries, a consistent method is suggested to take into account effectively the momentum dependence of the RT of the SP potential within the standard HF scheme. To obtain a realistic momentum dependence of the nucleon optical potential (OP), the high-momentum part of the SP potential was accurately readjusted to reproduce the observed energy dependence of the nucleon OP over a wide range of energies. The impact of the RT and momentum dependence of the SP potential on the density dependence of the nuclear symmetry energy and nucleon effective mass has been studied in detail. The high-momentum tail of the SP potential was found to have a sizable effect on the slope of the symmetry energy and the neutron-proton effective mass splitting at supranuclear densities of the NM. Based on a local density approximation, the folding model of the nucleon OP of finite nuclei has been extended to take into account consistently the RT and momentum dependence of the nucleon OP in the same mean-field manner, and successfully applied to study the elastic neutron scattering on the lead target at the energies around the Fermi energy.

Probing the existence of phase transitions in one-dimensional fluids of penetrable particles.

Phase transitions in one-dimensional classical fluids are usually ruled out by using van Hove's theorem. A way to circumvent the conclusions of the theorem is to consider an interparticle potential that is everywhere bounded. Such is the case of, e.g., the generalized exponential model of index 4 (GEM-4 potential), which in three dimensions gives a reasonable description of the effective repulsion between flexible dendrimers in a solution. An extensive Monte Carlo simulation of the one-dimensional GEM-4 model [S. Prestipino, Phys. Rev. E 90, 042306 (2014)] has recently provided evidence of an infinite sequence of low-temperature cluster phases, however, also suggesting that upon pushing the simulation forward what seemed a true transition may eventually prove to be only a sharp crossover. We hereby investigate this problem theoretically by use of three different and increasingly sophisticated approaches (i.e., a mean-field theory, the transfer matrix of a lattice model of clusters, and the exact treatment of a system of point clusters in the continuum) to conclude that the alleged transitions of the one-dimensional GEM-4 system are likely just crossovers.

Cluster phases of penetrable rods on a line.

Phase transitions are uncommon among homogeneous one-dimensional fluids of classical particles owing to a general nonexistence result due to van Hove. A way to circumvent van Hove's theorem is to consider an interparticle potential that is finite everywhere. Of this type is the generalized exponential model of index 4 (GEM4 potential), a model interaction which in three dimensions provides an accurate description of the effective pair repulsion between dissolved soft macromolecules (e.g., flexible dendrimers). Using specialized free-energy methods, I reconstruct the equilibrium phase diagram of the one-dimensional GEM4 system, showing that, apart from the usual fluid phase at low densities, it consists of an endless sequence of cluster fluid phases of increasing pressure, having a sharp crystal appearance for low temperatures. The coexistence line between successive phases in the sequence invariably terminates at a critical point. Focussing on the first of such transitions, I show that the growth of the two-cluster phase from the metastable ordinary fluid is extremely slow, even for large supersaturations. Finally, I clarify the apparent paradox of the observation of an activation barrier to nucleation in a system where, due to the dimensionality of the hosting space, the critical radius is expected to vanish.

A differential equation for the transition probability B(E2)↑ and the resulting recursion relations connecting even–even nuclei

We obtain here a new relation for the reduced electric quadrupole transition probability B(E2)↑ of a given nucleus in terms of its derivatives with respect to neutron and proton numbers based on a similar local energy relation in the Infinite Nuclear Matter (INM) model of atomic nuclei, which is essentially built on the foundation of the Hugenholtz–Van Hove (HVH) theorem of many-body theory. Obviously, such a relation in the form of a differential equation is expected to be more powerful than the usual algebraic difference equations. Although the relation for B(E2)↑ has been perceived simply on the basis of a corresponding differential equation for the local energy in the INM model, its theoretical foundation otherwise has been clearly demonstrated. We further exploit the differential equation in using the very definitions of the derivatives to obtain two different recursion relations for B(E2)↑, connecting in each case three neighboring even–even nuclei from lower to higher mass numbers and vice versa. We demonstrate their numerical validity using available data throughout the nuclear chart and also explore their possible utility in predicting B(E2)↑ values.

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.

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.

Symmetry energy, its density slope, and neutron-proton effective mass splitting at normal density extracted from global nucleon optical potentials

Based on the Hugenholtz--Van Hove theorem, it is shown that both the symmetry energy $E$${}_{\mathrm{sym}}(\ensuremath{\rho})$ and its density slope $L(\ensuremath{\rho})$ at normal density ${\ensuremath{\rho}}_{0}$ are completely determined by the nucleon global optical potentials. The latter can be extracted directly from nucleon-nucleus scatterings, ($p,n$) charge-exchange reactions, and single-particle energy levels of bound states. Averaging all phenomenological isovector nucleon potentials constrained by world data available in the literature since 1969, the best estimates of ${E}_{\mathrm{sym}}({\ensuremath{\rho}}_{0})=31.3$ MeV and $L({\ensuremath{\rho}}_{0})=52.7$ MeV are simultaneously obtained. Moreover, the corresponding neutron-proton effective mass splitting in neutron-rich matter of isospin asymmetry $\ensuremath{\delta}$ is estimated to be $({m}_{n}^{*}\ensuremath{-}{m}_{p}^{*})/m=0.32\ensuremath{\delta}$.

A numerical test of a high-penetrability approximation for the one-dimensional penetrable-square-well model

The one-dimensional penetrable-square-well fluid is studied using both analytical tools and specialized Monte Carlo simulations. The model consists of a penetrable core characterized by a finite repulsive energy combined with a short-range attractive well. This is a many-body one-dimensional problem, lacking an exact analytical solution, for which the usual van Hove theorem on the absence of phase transition does not apply. We determine a high-penetrability approximation complementing a similar low-penetrability approximation presented in previous work. This is shown to be equivalent to the usual Debye-Hückel theory for simple charged fluids for which the virial and energy routes are identical. The internal thermodynamic consistency with the compressibility route and the validity of the approximation in describing the radial distribution function is assessed by a comparison against numerical simulations. The Fisher-Widom line separating the oscillatory and monotonic large-distance behaviors of the radial distribution function is computed within the high-penetrability approximation and compared with the opposite regime, thus providing a strong indication of the location of the line in all possible regimes. The high-penetrability approximation predicts the existence of a critical point and a spinodal line, but this occurs outside the applicability domain of the theory. We investigate the possibility of a fluid-fluid transition by the Gibbs ensemble Monte Carlo techniques, not finding any evidence of such a transition. Additional analytical arguments are given to support this claim. Finally, we find a clustering transition when Ruelle's stability criterion is not fulfilled. The consequences of these findings on the three-dimensional phase diagrams are also discussed.