Finite nucleus

Finally, we mention that Filatov [10, 11] recently in an interesting new approach discussed the effect of finite nuclei in detail. He suggested calculating the isomer shift from the variation to the total electron energy in dependence of the nuclear charge extension. 

A possible application for the formation of a-like condensates are selfconjugate 4n nuclei such as 8Be, 12C, 160,20Ne, 24Mg, and others. Of course, results obtained for infinite nuclear matter cannot immediately be applied to finite nuclei. However, they are of relevance, e.g., in the local density approximation. We know from the pairing case that the wave function for finite systems can more or less reflect properties of quantum condensates. 

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. 

Green function method, that can be considered as a generalization of the BHF approach. The results are compared with those from various many-body approaches, such as variational and relativistic mean field approaches. In view of the large spread in the theoretical predictions we also examine possible constraints on the nuclear SE that may be obtained from information from finite nuclei (such the neutron skin). 

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. 

The interpretation of the nuclear matter results in part depends on the question whether there is a surface contribution to the SE in finite nuclei. In ref. [24] it was found that for heavy nuclei the latter is of minor importance, which has also been confirmed in ref. [20],... 

Phenomenological TBF. A second class of TBF that are widely used in the literature, in particular for variational calculations of finite nuclei and nuclear matter [5], are the phenomenological Urbana TBF [19]. We remind that the Urbana IX TBF model contains a two-pion exchange potential supplemented by a phenomenological repulsive term VRk,... 

The limit c —can now be taken provided that (1) V is everywhere non-singular, which is true for finite nuclei [42] but not point nuclei, and that (2) E < c which is true for the (shifted) positive-energy solutions only. With this procedure all relativistic effects are eliminated and one obtains the four-component non-relativistic Levy-Leblond equation [34,43]... 

There has been some success in developing effective interactions for finite nuclei using realistic two-body interacti n3. Much of this work has been done on the lighter nuclei. Kuo and Brown [KU066] developed a g-matrix interaction for the sd and fp shells. They used the Hamada-Johnson nucleon-nucleon interaction in their calculations with a core polarization correction. This work was extended to the region near Ca and [KU068] and... 

Fig. 4.3. Relativistic correction factor for the LDA exchange potential. The values of the densities of Kr and Hg at the origin (r = 0) and the r-expectation values of the Is-orbitals (r = < r >, 5) from RLDA-calculations using finite nuclei are also indicated...

It is well known that for heavy atoms the effect of the finite nucleus charge distribution has to be taken into account (among other effects) in order to describe the electronic structure of the system correctly (see e.g. (36,37)). As a preliminary step in the search for the effect of the finite nuclei on the properties of molecules the potential energy curve of the Th 73+ has been calculated for point-like and finite nuclei models (Table 5). For finite nuclei the Fermi charge distribution with the standard value of the skin thickness parameter was adopted (t = 2.30 fm) (38,39). 

In a many-electron molecular system the effect of finite nuclei should result in increasing its total energy. Whether the binding energy and the bond length are also changed should be further investigated. 

The relativistic mean meson field (R.MF) theory formulated by Teller and others [8, 9, 10] and by Walecka [11] is quite successful in both infinite nuclear matter and finite nuclei[12, 13, 14]. In the RMF model, only positive-energy baryonic states are considered to study the properties of ordinary nuclei. This is the so-called no-sea-approximation . However, an interesting feature of the RMF theory is the existence of bound negative-energy baryonic states. This happens because the interaction with the vector field generated by the baryon-... 

Tho critical baryon density corresponds to pmax = 0. This leads to the ( (juations Uv + Us = G or Uv — Us — 2Mg = 0 or both. The first condition Uv + Us = 0 is fulfilled earlier than the second one. When the first condition is reached the nucleus becomes unbound, i.e., unstable with respect to emission of nucleons. So it is impossible to compress the nucleus more than the critical density in a self-consistent manner such densities should occur only as shortlived intermediate stages in a heavy-ion collision. Wc performed a constraint calculation with the monopole moment [23, 24, 25], which produced self-consistent solutions up to w 3po for the case of Pb. A chart for the critical densities of nuclear matter with different parameter sets is given in ref. [26]. We found that the critical densities are much larger for nuclear matter c ompared to finite nuclei in all available parameter sets. The TMl parameter set was chosen for our calculations, because it gives a larger critical density of about 3po ... 

J. Kobus, H. M. Quiney, S. Wilson, A comparison of finite difference and finite basis set Hartree-Fock calculations for the N2 molecule with finite nuclei, J. Phys. B 34 (2001) 2045-2056. 

Unlike the electron gas and nuclear matter, where a major role is played by ring and ladder diagrams, respectively, the atomic and molecular correlation problem—or, in fact, the finite nuclei problem—requires a proper account of both types of diagrams, as well as of their combinations. This was certainly the main motivation for the development of CC methodology, which we shall discuss in the next section. 

This is similar to the closed form of the Bethe-Goldstone equation for finite nuclei. The latter is, however, for particles in the Brueckner sea" instead of the H.F. sea represented by the Vf of Eq. (103). A Bethe-Goldstone type equation in a form which would be useful for actual calculations on finite nuclei, the properties of its solutions and its variational principle are given in the Appendix of Reference 9b. 

Gaussian basis sets and finite nuclei. - We have already noted in Section... 

Differences between point and finite nuclei finite nuclear models ... 

Presently it is widely accepted that the relativistic mean-field (RMF) model [40] gives a good description of nuclear matter and finite nuclei [41]. Within this approach the nucleons are supposed to obey the Dirac equation coupled to mean meson fields. Large scalar and vector potentials, of the order of 300 MeV, are necessary to explain the strong spin-orbit splitting in nuclei. The most debated... 

The admixture of antibaryons to finite nuclei provides an almost unique doorway to the study of cold compressed nuclear and/or quark matter in the laboratory. This region of the phase diagram of strongly interacting matter, see Figure 8.31, is not accessible by collisions of heavy ions, which produce matter which is simultaneously hot and dense. 

Furthermore, we have investigated the influence that differences between different NN potentials have on nuclear structure predictions. It turns out that for potentials that fit the NN data reasonably well, on-shell differences have only a negligible effect. However, potentials that are essentially identical on-shell, may differ substantially off-shell. Such off-shell differences may lead to large differences in nuclear structure predictions. Relativistic, meson-theory based potentials (which are non-local) are in general weaker off-shell than their local counterparts. In particular, the weaker (off-shell) tensor force component (as quantified by a small deuteron D-state probability, Pd) leads to more binding in finite nuclei. For several examples shown, these predictions compare favourably with experiment. 

In this work we discuss various approaches to the effective interaction appropriate for finite nuclei. The methods we review are the folded-diagram method of Kuo and co-workers and the summation of folded diagrams as advocated by Lee and Suzuki. Examples of applications to sd-shell nuclei from previous works are discussed together with hitherto unpublished results for nuclei in the pf-shell. Since we find the method of Lee and Suzuki to yield the best converged results, we apply this method to calculate the effective interaction for nuclei in the pf-shell. To calculate this effective interaction we have used three recent versions of the Bonn meson-exchange potential model. These versions are fitted to the same set of data and differ only in the strength of the tensor force. The importance of the latter for finite nuclei is also discussed. 

To evaluate the OBE amplitudes we use in this work the Bonn potential as it is defined by the meson parameters in Table A.2 of Ref. [7]. For further discussion of this topic, see e.g. Machleidt s contribution in these proceedings. There are three sets of meson parameters which then define three potentials, referred to as the Bonn A, B and C potentials. These potentials differ in the strength of the tensor force, which is reflected in the probability of the D-state of the deuteron. The significance of the tensor force for both nuclear matter and finite nuclei will be discussed in Section 2.3. The coupling constants, cutoffs and masses of the various mesons of Table A.2 of [7] are redisplayed in Table 1. These meson parameters are obtained through a solution of the scattering equation for... 

Vq is the central part of the NN interaction while Vj is the tensor force. Thus, if the tensor force is weak (strong), a stronger (weaker) central force is needed to arrive at the same on-shell. -matrix. A similar mechanism is present when we evaluate the G-matrix for either nuclear matter or finite nuclei as well, though, anticipating the discussion in the next subsection, in these cases we must also account for medium effects such as the modification of the energy denominator i and the inclusion of the Pauli principle. 

In this subsection we briefly discuss how to construct the G-matrix for both nuclear matter and finite nuclei. Several methods to solve the Bethe-Goldstone equation can be found in the literature, for a review see Ref [20]. The basic difficulty resides in the fact that the Pauli operator Q is diagonal in the laboratory representation of the two-particle states, while it is nondiagonal in the... 

For finite nuclei we will not use the angle-average option, but rather a formally exact technique for handling Q, originally presented by Tsai and Kuo [21] and discussed in Ref. [22]. Tsai and Kuo employed the matrix identity... 

We see then that the G-matrix for finite nuclei is expressed as the sum of two terms the first term is the free G-matrix with no Pauli corrections included, while the second term accounts for medium... 

The bulk of the Gnm-matrix (and the G-matrix for finite nuclei) behaves similarly to the scattering matrix. -matrix in Eq. (3), i.e. 

For finite nuclei the situation is not as transparent as in nuclear matter. The density dependence of the Pauli operator is couched by the fact that one integrates over all relative momenta k and k to obtain a G-matrix in a HO basis. Moreover, the summation over partial waves in Eq. (9) may include several channels representing contributions from central, spin-orbit and tensor contributions. However, the quenching of the tensor force due to the energy denominator can easily be studied. 

In this work we have calculated the G-matrix for finite nuclei in the pf-shell with the above-mentioned three potentials. The scheme described in the previous subsection and in Ref. [22] has been used to calculate G. The Pauli operator for the pf-shell Q is defined so as to prevent scattering into intermediate states with one nucleon in states up to the IsOd-shell or two nucleons in the IpOf-or 2sld0g-shells. The oscillator energy for Ca was set to lOMeV. 

Finite nuclei Just a droplet of nuclear matter ... 

---