Hadron-quark Pasta Phase in Massive Neutron Stars

The structured hadron-quark mixed phase, known as the pasta phase, is expected to appear in the core of massive neutron stars. Motivated by the recent advances in astrophysical observations, we explore the possibility of the appearance of quarks inside neutron stars and check its compatibility with current constraints. We investigate the properties of the hadron-quark pasta phases and their influences on the equation of state (EOS) for neutron stars. In this work, we extend the energy minimization (EM) method to describe the hadron-quark pasta phase, where the surface and Coulomb contributions are included in the minimization procedure. By allowing different electron densities in the hadronic and quark matter phases, the total electron chemical potential with the electric potential remains constant, and local β equilibrium is achieved inside the Wigner–Seitz cell. The mixed phase described in the EM method shows the features lying between the Gibbs and Maxwell constructions, which is helpful for understanding the transition from the Gibbs construction to the Maxwell construction with increasing surface tension. We employ the relativistic mean-field model to describe the hadronic matter, while the quark matter is described by the MIT bag model with vector interactions. It is found that the vector interactions among quarks can significantly stiffen the EOS at high densities and help enhance the maximum mass of neutron stars. Other parameters like the bag constant can also affect the deconfinement phase transition in neutron stars. Our results show that hadron-quark pasta phases may appear in the core of massive neutron stars that can be compatible with current observational constraints.

Pairing effects in nuclear pasta phase within the relativistic Thomas-Fermi formalism

Pairing effects in non-uniform nuclear matter, surrounded by electrons, are studied in the protoneutron star early stage and in other conditions. The so-called nuclear pasta phases at sub saturation densities are solved in a Wigner-Seitz cell, within the Thomas-Fermi approximation. The solution of this problem is important for the understanding of the physics of a newly born neutron star after a supernova explosion. It is shown that the pasta phase is more stable than uniform nuclear matter on some conditions and the pairing force relevance is studied in the determination of these stable phases.

Proximity effect of pair correlation in the inner crust of neutron stars

We study proximity effect of pair correlation in the inner crust of neutron stars by means of the Skyrme-Hartree-Fock-Bogoliubov theory formulated in the coordinate space. We describe a system composed of a nuclear cluster immersed in neutron superuid, which is confined in a spherical box. Using a density-dependent effective pairing interaction which reproduces both the pair gap of neutron matter obtained in ab initio calculations and that of finite nuclei, we analyze how the pair condensate in neutron superuid is affected by the presence of the nuclear cluster. It is found that the proximity effect is characterized by the coherence length of neutron superuid measured from the edge position of the nuclear cluster. The calculation predicts that the proximity effect has a strong density dependence. In the middle layers of the inner crust with baryon density 5 × 10-4 fm-3 ≲ ρb ≲ 2 × 10-2 fm-3, the proximity effect is well limited in the vicinity of the nuclear cluster, i.e. in a sufficiently smaller area than the Wigner-Seitz cell. On the contrary, the proximity effect is predicted to extend to the whole volume of the Wigner-Seitz cell in shallow layers of the inner crust with ρb ≲ 2 × 10-4 fm-3, and in deep layers with ρb ≲ 5 × 10-2 fm-3.

Waves and particles in the crystal

This chapter provides the groundwork necessary to mathematically describe the crystalline solid that is to be the semiconductor used to explore the variety of interactions and cause and chance behaviors that physics builds insights into. The crystalline environment can be portrayed as a space-filling periodic arrangement consisting of a lattice with an atomic basis. The periodic arrangement leads to real space and reciprocal space descriptions with unit cells—the Wigner-Seitz cell and the Brillouin and Jones zones—where a variety of characteristics can be represented. Bloch’s theorem with its modulation function of the plane wave of a quantized wavevector, momentum versus crystal momentum, together with the consequences of symmetries and periodic perturbation in the appearance of bandgaps, is discussed for electron states. Phonons as particles for periodic oscillations of atoms, their modes and various branches, and consequences of ionicity leading to frequency-dependent permittivity are discussed.

Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice An

The orthogonal projections of the Voronoi and Delone cells of root lattice An onto the Coxeter plane display various rhombic and triangular prototiles including thick and thin rhombi of Penrose, Amman–Beenker tiles, Robinson triangles, and Danzer triangles to name a few. We point out that the symmetries representing the dihedral subgroup of order 2h involving the Coxeter element of order h=n+1 of the Coxeter–Weyl group an play a crucial role for h-fold symmetric tilings of the Coxeter plane. After setting the general scheme we give samples of patches with 4-, 5-, 6-, 7-, 8-, and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice A3 , whose Wigner–Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an h=4-fold symmetry.

The Study of Symmetry Energy in Pasta of Neutron Star From Compressible Liquid Drop Model Approximation

The properties of pasta which is located at the bottom of inner crust from neutron star has been studied by using compressibl e liquid drop model. Compressible liquid drop model is a modified liquid drop model as a density function. Liquid drop model based on assumption that the magnitude of nucleus bonding energy is contribution of surface, Coulomb, volume, symmetry, and proton -neutron pair effect. Pasta of neutron star behaves like liquid crystals (mesomhorpic phase). The top layer of pasta filled by free neutron gas, while in the lowest layer of the pasta is filled by proton-neutron gas. The properties of pasta are observed at temperatures close to zero Kelvin with the assumption that neutron star is on ground state and non accretion. The study of pasta emphasizes on symmetry energy’s influence. Symmetry energy reduces the magnitude of bonding energy of nucleon in the nucleus and it causes nucleon to be more easily released from nucleus. After that, symmetry energy influence the properties of pasta, such as the shape of nucleus that is non spherical (some like plates, rods, and bubbles), the fluctuative values of Wigner-Seitz cell, and uneven distribution of protons and neutrons in the pasta region of neutron star.

Neutronic Methodological Benchmarks With Simplified Geometries for the Gas Cooled Reactor Group Constant Generating Tools

In the ESNII+ EU FP7 project, a reactor physics benchmark aiming at the whole core calculation with the reflectors and detailed description of the structural elements was specified. This benchmark is based on the 2009 CEA concept of the ALLEGRO core. Fixed nominal technological data at nominal reactor state (geometry, composition) were prescribed which had to be modified in specified calculation branches according to different types of the thermal expansion and control rod positions. The parameters of the point kinetic model to be applied in a system thermal hydraulic code had to be determined this way. Static mechanical models of the expansion processes were specified by the benchmark. The goal of the calculation exercise was to verify the reactor physics codes, namely to get information about the modelling uncertainties and — after — their influence on the calculated results of the safety analyses. The obtained deviations between the participants are characterizing the user effects, the modelling uncertainties and the influence of the nuclear data differences all, without the possibility of their separation because of the complexity of the benchmark problem. A conclusion could be drawn that a step by step procedure starting from simple problems (homogenous material, Wigner-Seitz cell or subassembly in asymptotic approach) is necessary if we wish to identify the reasons of the deviations. For the Doppler effect, a decision was made in this direction already in the ESNII+ project where an infinite regular lattice problem without any leakage had to be solved. This approach of the simplicity is followed by the present benchmarks (one rod and one assembly), but extending the simple benchmarks with burnup calculations and taking into account leakage in asymptotic approximation by neglecting the complicated processes necessary in the reflector regions.