Symmetry energy

State Symmetry Energy (eV) Calc. Exp. Oscillator Strength... 

Abstract The hadronic equation of state for a neutron star is discussed with a particular emphasis on the symmetry energy. The results of several microscopic approaches are compared and also a new calculation in terms of the self-consistent Green function method is presented. In addition possible constraints on the symmetry energy coming from empirical information on the neutron skin of finite nuclei are considered. 

The aim of this contribution is to review the present understanding of the hadronic EoS, and the nuclear symmetry energy (SE) in particular. In the past the latter quantity has been computed mostly in terms of the Brueckner-Hartree-Fock (BHF) approach. In order to get an idea about the accuracy of the BHF result we present a recent calculation [1] in terms of the self-consistent... 

Although the BHF approach has several shortcomings it provides a numerically simple and convenient scheme to provide insight in some aspects of the symmetry energy. 

As noted above at higher densities the EoS is sensitive to 3NF contributions. Whereas the 3NF for low densities seems now well understood its contribution to nuclear matter densities remains unsettled. In practice in calculations of the symmetry energy in the BHF approach two types of 3NF have been used in calculations in ref.[4] the microscopic 3NF based upon meson exchange by Grange et al. was used, and in ref. [15] as well in most VCS calculations the Urbana interaction. The latter has in addition to an attractive microscopic two-pion exchange part a repulsive phenomenological part constructed in such a way that the empirical saturation point for SNM is reproduced. Also in practice in the BHF approach to simplify the computational efforts the 3NF is reduced to a density dependent two-body force by averaging over the position of the third particle. 

Recently the density dependence of the symmetry energy has been computed in chiral perturbation effective field theory, described by pions plus one cutoff parameter, A, to simulate the short distance behavior [23]. The nuclear matter calculations have been performed up to three-loop order the density dependence comes from the replacement of the free nucleon propagator by the in-medium one, specified by the Fermi momentum ItF... 

In view of the existing uncertainties in the calculation of the SE one may ask whether from finite nuclei one can obtain experimental constraints on the symmetry energy as a function of density. In this section some recent activities pertaining to this issue are reviewed. 

Here, the mean field potential includes the phenomenological isoscalar part Uq x) along with the isovector U (x) and the Coulomb Uc(x) parts calculated consistently in the Hartree approximation Uo(r) and Uso(x) = Uso r)a l are the central and spin-orbit parts of the isoscalar mean field, respectively, and, SPot(r) is the potential part of the symmetry energy. 

In the right panel of Fig. 4 we display the symmetry energy as a function of the nucleon density p for different choices of the TBF. We observe results in agreement with the characteristics of the EOS shown in the left panel. Namely, the stiffest equation of state, i.e., the one calculated with the microscopic TBF,... 

Molecular orbital models are valuable aids in understanding the reactivity, regioselectivity, and stereospecificity phenomena exhibited by cycloaddition reactions and in predicting reactivity and product identities for addend pairs. Symmetry-energy correlation diagrams indicate that the 1,3-dipolar cyclo-... 

Fig. 3 Simplified molecular orbital (MO) diagram of the fragment [Tcv=N]2+ in C4v symmetry (energy scale is arbitrary). The xy highest occupied MO (HOMO) and the xz,yz pair of lowest unoccupied MOs (LUMOs) are the frontier orbitals regulating the reactivity of the fragment...

Table 3.5 Energy Levels of d Orbitals in Crystal Fields of Different Symmetries. Energies are in Units of Dq where lODq = A. (Reproduced with permission from Huheey et al.7)...

Semi-quantitative molecular orbital scheme for VOX3 under C3, symmetry energy scale in electron-volts (eV). The LUMOs, at around — lO.SeV, are represented by essentially empty V(3d) levels. Main HOMOs contributing to AE are in bold. Indications underneath the energy level bars are percentages of V(3d) contributions to these levels. The nonbonding n(X) do not contribute. Redrawn from ref. 2. 

When all LF parameters are evaluated, they can be introduced into a favoured LF program [37-39] to yield all multiple energies and expectation values of all operators for comparison with experiment. In the case of d Fe04 with tetrahedral symmetry, energy matrices can be written explicitly (Table 19-2) the role of single excitation (for the T2 and Tj terms) and double excitations (for Aj, E, T2 and Tj) is important - we return to this point when we look at applications later in the article. According to this procedure, both dynamical correlation (via the DFT... 

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