Structure of exotic nuclei from relativistic Hartree Bogoliubov model (II) Shan-Gui Zhou Email: sgzhou@itp.ac.cn; URL: http://www.itp.ac.cn/~sgzhou Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou HISS-NTAA 2007 Dubna, Aug. 7-17

Magic numbers in super heavy nuclei Zhang et al. NPA753(2005)106 2018/11/19

Contents Introduction to Relativistic mean field model Basics: formalism and advantages Pseudospin and spin symmetries in atomic nuclei Pairing correlations in exotic nuclei Contribution of the continuum BCS and Bogoliubov transformation Spherical relativistic Hartree Bogoliubov theory Formalism and results Summary I Deformed relativistic Hartree Bogoliubov theory in a Woods-Saxon basis Why Woods-Saxon basis Formalism, results and discussions Single particle resonances Analytical continuation in coupling constant approach Real stabilization method Summary II 2018/11/19

Deformed Halo? Deformed core? Decoupling of the core and valence nucleons？ Misu, Nazarewicz, Aberg, NPA614(97)44 11,14Be Ne isotopes … Bennaceur et al., PLB296(00)154 Hamamoto & Mottelson, PRC68(03)034312 Hamamoto & Mottelson, PRC69(04)064302 Poschl et al., PRL79(97)3841 Nunes, NPA757(05)349 Pei, Xu & Stevenson, NPA765(06)29 2018/11/19

Hartree-Fock Bogoliubov theory Deformed non-relativistic HFB in r space Deformed relativistic Hartree-Bogoliubov or Hartree-Fock-Bogoliubov theory in harmonic oscillator basis Terasaki, Flocard, Heenen & Bonche, NPA 621, 706 (1996) Stoitsov, Dobaczewski, Ring & Pittel, PRC61, 034311 (2000) Terán, Oberacker & Umar, PRC67, 064314 (2003) Vretenar, Lalazissis & Ring, PRL82, 4595 (1999) No deformed relativistic Hartree-Bogoliubov or Hartree-Fock-Bogoliubov theory in r space available yet 2018/11/19

Harmonic oscillator basis and r-space Average potential in atomic nucleus Woods-Saxon potential: no analytic solution harmonic oscillator potential: a good approx. for stable nuclei; matrix diagonalization Drip line nuclei: large space distribution, contribution of continuum HO basis: localization r-space: complicated and time-consuming (deformation and pairing) Woods-Saxon basis: a reconciler of r-space & HO basis? Basic idea Numerical solutions for spherical WS potential in r space Large-box boundary condition to discretize the continuum WS wave functions used as a complete basis  matrix diagonalization problem We know that there is an Woods-Saxon-like average field in an atomic nucleus. Due to the fact that there is no analytical solution for the WS potential, as a good approximation for stable nuclei, one often works in the Harmonic oscillator basis in mean field calculations. For drip line nuclei, which are characterized by many exotic properties, one meets difficulties to solve the mean field equations in the HO basis, due to the localization property of this potential. A proper representation for solving the mean field equations, e.g., the HFB theory, for drip line nuclei is the coordinate space, the r space. Nevertheless for deformed nuclei, working in r space becomes much more difficult. where wave functions are approximated on spatial lattice and the continuum is discretized in suitable large-box boundary conditions The HFB method solved in r space can take fully into account all the mean-field effects of coupling to the continuum. 2018/11/19

Schroedinger Woods-Saxon basis VWS(r) Rmax Shooting Method First the SRHSWS, the SRH in Shrodinger equation generated WS basis. For the spherical WS potential, in order to discritize the continuum, we apply a box-boundary condition. By solving this radial Schroedinger equation with the shooting method, we get the WS basis. For simplicity, I only include the radial part of the wave functions here. n is the radial quantum number and l the angular momentum. 2018/11/19

Spherical RMF in Schroedinger WS basis The upper and lower components of the Dirac spinor are expanded in terms of this WS basis independently. Solving the radial Dirac equation is easily transformed to diagonalize this matrix. 2018/11/19

Dirac Woods-Saxon basis Now let me move to the SRH theory in the Dirac equation generated WS basis. The WS basis is obtained by solving the Dirac equation for nucleons with initial WS-like vector and scalar potentials. The point is we must include not only positive states in the Fermi sea but also negative states in the Dirac sea. Then the Dirac spinors for nucleons are expanded in terms of this basis. To solve the Dirac equation is also transferred to to diagonalize a matrix. I should mention that for spherical nuclei, we can solve the Dirac equation directly with the method we used to generate the basis. 2018/11/19

Dirac-WS: negative energy states V0 [MeV] E/A [MeV] Rrms [fm] 54 8.013 | 8.547 2.568 | 2.385 72 8.015 | 8.117 2.567 | 2.531 90 8.012 | 8.427 2.567 | 2.610 Completeness of the basis (no contradiction with no-sea) Underbound without inclusion of n.e. states Results independent of basis parameters I already stressed that we must include negative levels in the Dirac sea when we expand the Dirac spinor in terms of the WS basis generated by solving the Dirac equation. How important are the negative states? From this table you see, if we do not include negative states in the expansion, the final results depend on the WS basis potential very much. I’ve shown the figures in black. The figures in red are the results when we don’t include the negative states. If we do not include them, for instance, when the potential depth changes from 54 to 72 MeV, the average binding energy changes about 5 % and rms radius changes by almost 10 %. The next question is, how many negative states should be included. 2018/11/19

Basis: Dirac-WS versus Schroedinger-WS Smaller Basis! Schroedinger WS Dirac WS n-max < n+max nFmax = nGmax + 1 2018/11/19

Neutron density distribution: 48Ca 2018/11/19

Spherical Rela. Hartree calc.: 72Ca SGZ, Meng & Ring, PRC68,034323(03) Woods-Saxon basis reproduces r space 2018/11/19

RMF in a Woods-Saxon basis: progress Shape Model Schrödinger W-S basis Dirac Spherical Rela. Hartree SRH SWS SRH DWS √ Axially deformed Rela. Hartree + BCS DRH DWS Rela. Hartree-Bogoliubov DRHB DWS Triaxially deformed TRHB DWS SGZ, Meng & Ring,PRC68,034323(03) SGZ, Meng & Ring, AIP Conf. Proc. 865, 90 (06) SGZ, Meng & Ring, in preparation Woods-Saxon basis might be a reconciler between the HO basis and r space 2018/11/19

Deformed RHB in a Woods-Saxon basis Axially deformed nuclei 2018/11/19

DRHB matrix elements , even , 0 , even or odd , 0 or 1 2018/11/19

Pairing interaction Phenomenological pairing interaction with parameters: V0, 0, and  ( = 1) Soft cutoff Bonche et al., NPA443,39 (1985) Smooth cutoff 2018/11/19

RHB in Woods-Saxon basis for axially deformed nuclei (-force in pp channel) 2018/11/19

How to fix the pairing strength and the pairing window Zero pairing energy for the neutron 2018/11/19

Convergence with E+cut and compared to spherical RCHB results E+cut: 100 MeV ~16 main shells dE ~ 0.1 MeV dr ~ 0.002 fm 2018/11/19

Routines checks: comparison with available programs Compare with spherical RCHB model Spherical, Bogoliubov Compare with deformed RMF in a WS basis Deformed, no pairing Compare with deformed RMF+BCS in a WS basis Deformed, BCS for pairing 2018/11/19

Compare with spherical RCHB model 2018/11/19

Properties of 44Mg 2018/11/19

Density distributions in 44Mg 2018/11/19

Density distributions in 44Mg 2018/11/19

Density distributions in 44Mg 2018/11/19

Pairing tensor in 44Mg 2018/11/19

Canonical single neutron states in 44Mg 2018/11/19

Contents Introduction to Relativistic mean field model Basics: formalism and advantages Pseudospin and spin symmetries in atomic nuclei Pairing correlations in exotic nuclei Contribution of the continuum BCS and Bogoliubov transformation Spherical relativistic Hartree Bogoliubov theory Formalism and results Summary I Deformed relativistic Hartree Bogoliubov theory in a Woods-Saxon basis Why Woods-Saxon basis Formalism, results and discussions Single particle resonances Analytical continuation in coupling constant approach Real stabilization method Summary II 2018/11/19

Analytical continuation in coupling constant Kukulin et al., 1989 Padé approximant 2018/11/19

Analytical continuation in coupling constant Zhang, Meng, SGZ, & Hillhouse, PRC70 (2004) 034308 2018/11/19

Analytical continuation in coupling constant Zhang, Meng, SGZ, & Hillhouse, PRC70 (2004) 034308 2018/11/19

Real stabilization method Hazi & Taylor, PRA1(1970)1109 Box boundary condition Stable against changing of box size: resonance Stable behavior: width 2018/11/19

Real stabilization method Zhang, SGZ, Meng, & Zhao, 2007 RMF (PK1) 2018/11/19

Real stabilization method Zhang, SGZ, Meng, & Zhao, 2007 RMF (PK1) 2018/11/19

Comparisons ACCC: analytical continuation in coupling constant S: scattering phase shift RSM: real stabilization method RMF (NL3) Zhang, SGZ, Meng, & Zhao, 2007 2018/11/19

Summary II Deformed exotic nuclei, particularly halo Weakly bound and large spatial extension Continuum contributing Deformed relativistic Hartree Bogoliubov model in a Woods-Saxon basis for exotic nuclei W-S basis as a reconciler of the r space and the oscillator basis Preliminary results for 44Mg Halo in deformed nucleus tends to be spherical Single particle resonances: bound state like methods Analytical continuation in the coupling constant approach Real stabilization method 2018/11/19