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Symmetry Projected Density Functional Theory and Neutron Halo’s E.C. Lopes1 and P. Ring1 Physik Department, Technische Universit¨ at M¨ unchen, 85748 Garching, Germany we show neutron density distributions of Ne-isotopes as a function of the radius r on a logarithmic scale. Number projected results (full lines) are compared with those without number projection (dashed line). In both cases we find halo phenomena and there is only a small difference in the density distributions. Going from 30 Ne to 40 Ne we find a phase transition from zero pairing in 30 Ne to rather strong pairing in 34 Ne and all the heavier isotopes. Because of vanishing pairing there is no difference between projected and unprojected calculations in 30 Ne. 32 Ne is a transitional nucleus with week pairing. In this case we find the largest differences between the projected and the unprojected calculations. For heavier nuclei with strong pairing, simple Hartree-Bogoliubov theory without projection provides already a rather good approximation and therefore number projection is not necessary and the conclusions found in the earlier calculations are valid.
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The appearance of halo phenomena near the drip line nuclei has challenged our traditional understanding of the nuclei as an incompressible charged liquid drop and extended nuclear physics to low density and inhomogeneous system, where the coupling to the continuum has to be treated in a consistent way. Recently Relativistic Hartree Bogoliubov (RHB) theory in the continuum has been applied successfully to the description of halo phenomena in light and medium heavy nuclei [1, 2, 3]. This theory provides a self-consistent treatment of pairing correlation in the presence of the continuum and allows a microscopic description of halo phenomena in the framework of density functional theory. Essential conditions for the formation of a neutron halo have been found: (a) the Fermi surface of the neutrons has to be close to the continuum limit, (b) there has to be a large level density in the vicinity of the Fermi surface, such that pairing correlations can develop and (c) there have to be single particle states with zero or small angular momentum barrier in close neighborhood. This is the case in 11 Li [1], where the 2s1/2 level is shifted downwards and where a neutron halo as been observed, but also in Ne-nuclei [2] close to the neutron drip line, where 2p3/2 and 2p1/2 reach the continuum limit and where a neutron halo has been predicted. The same theory also predicts further halo phenomena in the Zr-region [3] with multi-particle halos containing up to 6 neutrons. One of the essential shortcoming of mean field theories is the fact, that they violate symmetries. In HartreeBogoliubov (HFB) approximation gauge symmetry of particle number conservation is broken and the wave functions are linear combinations of wave functions with different particles. In principle one has to project onto the eigenspace of good particle number and this projection has to be carried out before the variation. In heavy nuclei, where many particles contribute to the pairing correlations one has found that an explicit projection is not necessary and the BCS-approach, where the particle number is only conserved on the average presents already a very reasonable approximation. However, in light nuclei it is a priori not clear, whether such arguments are not applicable and whether the simple Hartree-Bogolubiov theory is applicable. Recently a new method has been developed to solve the number projected Hartree-Fock-Bogoliubov problem [4]. So far this method has only been applied to simple non-relativistic models. We have now implemented this method for realistic RHB theory. For this purpose a new code has been developed to solve the number projected relativistic Hartree-Bogoliubov equations in by a variation after projection. This method has been applied in several areas of the periodic system [5]. Here we show as an example calculations for the chain of Ne-isotopes, where in Ref. [2] extended halo phenomena have been found. In Fig. 1
ρ(r) [fm ]
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Figure 1: The neutron density distributions as a unction of radius r for 30 Ne, 32 Ne, 34 Ne 36 Ne, 38 Ne, and 40 Ne in Projected RHB theory (solid line) compared with that in the case without projection (dashed line).
Concluding we have implemented full number projection before variation in relativistic Hartree-Bogoliubov theory. This will allow many other applications in transitional nuclei in the future.
References [1] J. Meng and P. Ring, Phys. Rev. Lett. 77, 3963(1996). [2] W. P¨ oschl, D. Vretenar, G.A. Lalazissis, and P. Ring, Phys. Rev. Lett. 79, 3841 (1997). [3] J. Meng and P. Ring, Phys. Rev. Lett. 80, 460(1998). [4] J.A. Sheikh and P. Ring, Nucl. Phys. A665, 71 (2000). [5] E. C. Lopes, PhD thesis, Technische Universit¨ at M¨ unchen (2002)
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Renormalization group approach to neutron matter Achim Schwenka , Bengt Frimanb and Gerald E. Brownc Department of Physics, The Ohio State University, Columbus; b GSI, Darmstadt; Astronomy, SUNY Stony Brook
The main idea of the renormalization group (RG) approach to Fermi liquids [1] is to “adiabatically” include the in-medium corrections to the effective interaction by solving the RG equations in the relevant channels. In this exploratory calculation [2], we include the particle-hole (ph) channels, which play a special role in Fermi liquid theory. The main effect of scattering in the particle-particle channel, the taming of the short-range repulsion, is taken care of by using Vlow k [3] as the starting point of the RG flow. The low-lying excitations in this channel, which are responsible, e.g., for superfluidity, are not included. These are then treated explicitly when we compute the superfluid gap by employing BCS theory for the fully reducible scattering amplitude. We derive the one-loop RG equations for the quasiparticle interaction and the scattering amplitude at zero temperature. The evolution of the effective mass is included in the RG flow, as well as a simplified treatment of the renormalization of the quasiparticle strength. As we decimate down to the Fermi surface, by taking the cutoff Λ → 0, we obtain not only the forward scattering amplitude for low-lying quasiparticle-quasihole excitations, but also the amplitude for general (non-forward) scattering processes of quasiparticles on the Fermi surface. The one-loop RG equations in the ph channels are solved at zero temperature. The corresponding diagrams, which renormalize the four-point vertex, are shown in Fig. 1. Here the momenta pi of the intermediate ph pair lie in a shell Λ − dΛ ≤ |pi − kF | ≤ Λ. When only one channel is considered, the one-loop RG equation is exact in the sense that it is equivalent to the corresponding scattering equation, and when both the direct and exchange ph channels are included, the scattering amplitude remains antisymmetric under the RG flow. The RG approach includes the induced interaction. We work in the approximation that both ph momentum transfers are small compared to the Fermi momentum. The physical system considered is neutron matter, where complications of the tensor force do not enter in the S-wave.
Figure 1: The one-loop contributions to the RG equation. γ denotes the running effective four-point vertex.
∆ [MeV]
a
c
Department of Physics and
3.5 3
direct
2.5 2 1.5 1
full
0.5 0 0
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Figure 2: The 1 S0 superfluid gap versus the Fermi momentum kF . The dots denote the pairing obtained from Vlow k only, whereas the squares and the triangles are computed from the full RG solution using different approximations for the quasiparticle strength [2]. In comparison, the dashed line is obtained by solving the BCS equation with the bare NN interaction [5], while the solid line includes the ph polarization effects [4]. Using the resulting Fermi liquid parameters, we compute the 1 S0 superfluid pairing gap in weak coupling BCS theory, Fig. 2. This application probes the angular dependence of the scattering amplitude. We generally find good agreement with the results obtained in the polarization potential model by Wambach et al. [4]. The RG method is a promising tool for studying a wide range of nuclear many-body problems. For a similar analysis of symmetric or asymmetric nuclear matter, it is necessary to extend the flow equations to incorporate tensor interactions. Fairly large renormalization effects are expected and spin non-conserving interactions are generated in the medium [6]. In particular, polarization effects on the 3 P2 –3 F2 pairing of neutrons, where the tensor and spin-orbit forces play a crucial role, may have important effects on neutron star properties.
References [1] R. Shankar, Rev. Mod. Phys. 66 (1994) 129. [2] A. Schwenk, B. Friman and G.E. Brown, Nucl. Phys. A713 (2003) 191. [3] S.K. Bogner, T.T.S. Kuo and L. Coraggio, Nucl. Phys. A684 (2001) 432c; S.K. Bogner, et al., nuclth/0108041. [4] J. Wambach, T.L. Ainsworth and D. Pines, Nucl. Phys. A555 (1993) 128. [5] U. Lombardo and H.-J. Schulze, astro-ph/0012209. [6] A. Schwenk and B. Friman, to be published.
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Effective NN-Interaction Including Radial and Tensor Correlations T. Neff and H. Feldmeier
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