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Nuclear Matter Properties, Phenomenological Theory of Clustering at the Nuclear Surface, and Symmetry Energy Q. N. Usmania*, No

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Nuclear Matter Properties, Phenomenological Theory of Clustering at the Nuclear Surface, and Symmetry Energy Q. N. Usmania*, No Nuclear matter properties, phenomenological theory of clustering at the nuclear surface, and symmetry energy Q. N. Usmani a*, Nooraihan Abdullah a, K. Anwar a and Zaliman Sauli b aInstitute of Engineering Mathematics, University Malaysia Perlis, Malaysia bSchool of Microelectronics Engineering, University Malaysia Perlis, Malaysia Abstract We present a phenomenological theory of nuclei that incorporates clustering at the nuclear surface in a general form. The theory explains the recently extracted large symmetry energy by Natowitz et al . at low densities of nuclear matter and is fully consistent with the static properties of nuclei. In phenomenological way clusters of all sizes, shapes along with medium modifications are included. Symmetric nuclear matter properties are discussed in detail. Arguments are given that lead to an equation of state of nuclear matter consistent with clustering in the low density region. We also discuss properties of asymmetric nuclear matter. Because of clustering, an interesting interpretation of the equation of state of asymmetric nuclear matter emerges. As a framework, an extended version of Thomas Fermi theory is adopted for nuclei which also contain phenomenological pairing and Wigner contributions. This theory connects the nuclear matter equation of state, which incorporate clustering at low densities, with clustering in nuclei at the nuclear surface. Calculations are performed for various equation of state of nuclear matter. We consider measured binding energies of 2149 nuclei for N, Z ≥ 8. The importance of quartic term in symmetry energy is demonstrated at and below the saturation density of nuclear matter. It is shown that it is largely related to the use of, ab initio , realistic equation of state of neutron matter, particularly the contribution arising from the three neutron interaction and somewhat to clustering. Reasons for these are discussed. Because of clustering the neutron skin thickness in nuclei is found to reduce significantly. Theory predicts new situations and regimes to be explored both theoretically and experimentally. * [email protected] 1. INTRODUCTION There is mounting evidence [1], both theoretically and experimentally, that clustering ( 2 H, 3 H, 3 He , α−particles and possibly heavier nuclei) occurs at the nuclear surface; a phenomenon which normally can not be described in Mean Field Theories. This clustering is evident from the studies of Natowitz et al . who have reported large values of symmetry energy at nuclear matter (NM) densities ρ < 0.01 fm -3 at low temperatures [2,3]. The clustering arises because extra binding energies are gained because of cluster formation of various shapes and sizes in the equation of state (EOS), E (ρ), of symmetric nuclear matter 1 (SNM) at subsaturation densities [4]. It finds explanation in Quantum Statistical (QS) [2, 4] approach which includes specific cluster correlations and then interpolates between the low density limit and the relativistic mean field (RMF) approaches near the saturation density. In QS approach only clusters with A ≤ 4 has been included. In the present study, we adopt a different approach. We introduce a thermodynamically consistent phenomenology in which we include the possibility of formation of clusters of all shapes and sizes along with medium modifications. We begin with the assertion that the binding energy of SNM per nucleon in the limit of zero density -3 must approach uv(≈16 MeV) at the saturation density ρ0(≈0.16 fm ), where uv is the volume term in the Weizsäcker mass formula. The foregoing assertion is based upon the principle that nuclear matter in its ground state (T=0 MeV) at a given density will attain the lowest possible energy. Our assertion can be proven in the following way. It was demonstrated in Ref. [5] that an idealized α-matter picture of SNM in the neighborhood of zero density gives energy per nucleon, E/A ≈ –7.3 MeV, which is the energy of α- 1 By SNM we understand nuclear matter with equal numbers of neutrons and protons with Coulomb interaction turned off. particle per nucleon with Coulomb interaction switched off. The above conclusion is at variance with the mean field theories results where E(ρ) →0 as the density ρ→0, for example, in the Skyrme-Hartree-Fock calculations. But why α-particles, why not heavier nuclei which have lower energies per nucleon compared to α-particle? For example, an ideal 40 Ca -matter will have lower E/A compared to an ideal α-matter. This line of argument can be extended further, since heavier the nuclei lower are the E/A as the Coulomb interaction is switched off. We may thus consider SNM as an ideal gas of chunks of NM or very heavy nuclei in which we can ignore the surface effects. We consider the densities low enough so that interactions between clusters are negligibly small and the concept of ideal cluster-matter is valid. Clearly in such a situation, we obtain the exact result E(ρ) → – uv as ρ→0. We therefore obtain the relation ρ → → ρ =ρ = − E( )0 E( 0) uv (1) Relation (1) is counter intuitive and, to the best of our knowledge, has never been exploited in any nuclear physics calculations. We shall make extensive use of relation (1) in our formulation. At finite but low densities, the SNM is not an ideal cluster matter. The ideal cluster matter exists only in the limit of zero density. Thus nuclear matter at low densities will be a highly correlated system with complicated structure. Nonetheless, based on somewhat general considerations, we demonstrate in section II that EOS of SNM will have one maximum between 0 and ρ0. This conclusion along with the identity (1) then includes the possibility of presence of clusters at low densities and leads to a compelling interpretation of the symmetry energy data of Natowitz et al . We postulate an EOS of SNM which contains the identity (1), a maximum between 0 and ρ0, and explicitly other empirically found quantities, namely the compression modulus K, the saturation density and the volume term uv. To determine the parameters of EOS we make use of the binding energy and charge rms radii data of nuclei; for which we use an approximate but well tested theory. One of the parameters of EOS is found sensitive to the symmetry energy data of Natowitz et al . [2]. This way we determine a semi-empirical EOS in the entire region between 0 and ρ0 with the materialization of four new empirical quantities. These are the spinodal density ρsp at which the homogeneous nuclear matter changes its character, ρmin the density at which symmetry energy shows a minimum, and the density ρmax at which the SNM attains a maximum value Emax . Another novel feature of the present study is the use of neutron matter EOS obtained recently through ab initio calculations with realistic interactions [6-8]. It is demonstrated that these EOS give rise to a quartic isospin term in the symmetry energy for a consistent description of nuclei. Though it has been suggested [7] that these EOS can be incorporated, or more appropriately indicate, an adjustment of the coefficient of the isovector gradient term in Skyrme density functional. But, the isovector gradient term contributes mainly in the surface region whereas the quartic isospin term affects the entire region. It is shown that the quartic term is much more efficient; its inclusion considerably decreases or almost entirely eliminates the importance of the isovector gradient term. Relation (1) should also hold for other quantum liquids, namely the two isotopes of helium 3He and 4He at zero temperature. They have some relevance to the present study, particularly 4He , since exact Green’s Function Monte Carlo (GFMC) and Diffusion Monte Carlo (DMC) calculations exist for this system. Experimental data on these liquids also exist. Incorporation of (1) into a theory is far from trivial. Clearly Hartree-Fock methods cannot take into account clustering at the nuclear surface since they do not satisfy (1); though mean field theories do contain some aspect of clustering but they are not related to surface properties. There are cluster models of nuclei and hypernuclei in terms of specific clusters, mostly α-particle clusters, but these are confined to light nuclei [9]. On the other hand we have ab initio theories which give accurate or nearly exact results, but they are also confined to light nuclei [10]. In future, it may become possible to extend ab initio theories to heavy nuclei with realistic Hamiltonians, but it would be difficult to entangle cluster properties particularly its universal character, if any, at the nuclear surface. Recently, there has been a spurt of activity in developing microscopically based nuclear density functional theories [11-13] for medium and heavy nuclei with reference to realistic two- and three-nucleon or Skyrme interactions. The realistic interactions used are either the Argonne AV18 (two-body) [14] and Urbana IX (three- body) [15] or they are derived from chiral effective field theory. These studies have started with the motivation to search for a Universal Nuclear Energy Density Functional – a new model for nuclear theory [16], which in principle is supposed to have all the many body correlations. But none of these theories, because of constrains or choice of density functional, conform to identity (1); on the contrary they give E(ρ)→0 as ρ→0. They do not include the clustering aspect in nuclei to the full extent. Besides, on the basis of fits to energy alone it will be difficult to see the impact of (1) unless one probes those properties of nuclei which are sensitive to surface region such as the neutron skin thickness, the symmetry energy at low densities, and the one neutron separation energies for those pair of nuclei where separation energies are small.
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