Session KK: Nuclear Theory III

 Thursday, September 22, 2005 2:00PM - 2:15PM KK.00001: Critical Point Symmetry in A Fermion Monopole and Quadrupole Pairing Model Joseph N. Ginocchio Recent interest in symmetries at a critical point of phase transitions in nuclei prompts a revisit to the fermion monopole and quadrupole pairing model [1]. This model has an exactly solvable symmetry limit that is transitional between spherical nuclei and gamma unstable deformed nuclei. The eigenenergies, eigenfunctions, pairing strength and quadrupole transition rates in this limit are derived. Comparison with empirical quadrupole transition rates suggests that the Xenon isotopes may have this symmetry [2]. 1. Joseph N. Ginocchio, Ann. Phys. 126, 234 (1980). 2. Joseph N. Ginocchio, Phys. Rev. C (2005). Thursday, September 22, 2005 2:15PM - 2:30PM KK.00002: All-order core polarization for shell-model effective interactions Jason D. Holt , Jeremy Holt , T.T.S. Kuo , G.E. Brown , Scott Bogner Although core polarization, calculated to second-order in perturbation theory, has been successful in describing a range of nuclear observables, the effect of high-order diagrams has been a long-standing issue. In this talk we present an all-order summation of a large class of core polarization diagrams using the low-momentum NN interaction $V_{low-k}$. Our calculation, based on the elegant formalism of Kirson and Babu-Brown, involves solving a set of coupled non-linear equations in which the vertex functions are generated self-consistently. By using $V_{low-k}$, which is energy independent, and true Green functions in the particle-particle and particle-hole channels, we can simplify the solution and include a class of diagrams whose calculation has been previously intractable. We apply this procedure to the $sd$-shell effective interactions and find that the all-order calculation serves to mildly suppress the second order results, typically by less than 10\%. Thursday, September 22, 2005 2:30PM - 2:45PM KK.00003: Very-low Momentum Nucleon-Nucleon Interaction Based upon Chiral Perturbation Theory Ruprecht Machleidt , Luigi Coraggio , David Entem Recently, several groups have constructed low-momentuum nucleon-nucleon (NN) interactions that have become known as $V_{\rm low-k}$. One starts from a conventional high-momentum NN potential and applies renormalization group techniques that preserve the (half)-on-shell T-matrix to obtain a new potential that is charcterized by a low-momentum cutoff, typically around 2 fm$^{-1}$. The general justification for this proceedure comes from low-energy effective field theory (EFT). This fact suggests that there may be a more efficient way to construct a $V_{\rm low-k}$. Namely, instead of taking the detour through a high-momentum NN potential, one may as well construct a low-momentum potential from scratch---and this is what our contribution is about. We use chiral perturbation theory at next-to-next-to-next-to-leading order (N3LO) and apply a sharp cutoff at 2.1 fm$^{-1}$. This potential reproduces the NN phase shifts up to about 300 MeV lab energy and the deuteron properties. While the $V_{\rm low-k}$ constructed in the past allow only for a rather cumbersome numerical representation, our low-momentum potential is given in analytic form. Moreover, the low-energy constants are explicitly known such that the chiral three-nucleon forces consistent with our NN potential can be properly defined. Thursday, September 22, 2005 2:45PM - 3:00PM KK.00004: Fermi liquid theory and Kuo-Brown effective interactions Jeremy W. Holt , G.E. Brown , J.D. Holt , T.T.S. Kuo , S.K. Bogner We study the properties of nuclear matter using the low-momentum nucleon-nucleon interaction $V_{\rm low-k}$ and Landau's theory of normal Fermi liquids. The Landau $f$-function, which describes the quasiparticle-quasihole interaction at the Fermi surface, can be expanded in Legendre polynomials whose coefficients are directly related to the effective mass, symmetry energy, and compression modulus of nuclear matter. It is found that in the single-bubble approximation to the induced interaction of Babu and Brown, the compression modulus is much too repulsive compared with experiment. This is remedied by solving the Babu-Brown equation self-consistently using $V_{\rm low-k}$ as the driving term. The result is a reasonable agreement with experiment, both for the compression modulus and the remaining Fermi liquid parameters. In addition, we discuss the effect of high-order direct and exchange terms in the quasiparticle scattering amplitude. Thursday, September 22, 2005 3:00PM - 3:15PM KK.00005: Shell Model Analysis of the $^{56}$Ni Spectrum in the Full $pf$ Model Space Mihai Horoi , B. Alex Brown , T. Otsuka , M. Honma , T. Mizusaki We present a full $pf$-shell spectroscopy of the low-lying states of $^{56}$Ni using the GXPF1A interaction.[M. Honma et al., Proceedings of ENAM, 2004] Both, the ground state band and the first deformed band, as well as the transition probabilities compares favorably with the experimental data.[D. Rudolpf et al., Phys. Rev. Lett. {\bf 82}, 5763 (1999)] We analyze the significance of the $np-nh$ contributions to the full model calculations, similar to the analysis done for $^{28}$Si in the $sd$-shell some twenty years ago.[Brown and Wildenthal, Ann. Rev. Nucl. Part. Sci. {\bf 38}, 29 (1988)] Thursday, September 22, 2005 3:15PM - 3:30PM KK.00006: Exactly Solvable Nuclear Models and Richardson-Gaudin Algebraic Structures V.G. Gueorguiev , J. Dukelsky , P. Van Isacker , S.S. Dimitrova We discuss the exact solution of the isovector pairing (T=1) in nuclei within the SO(5) Richardson-Gaudin model. $^{64}$Ge is used to illustrate the parameter space and shed light on the solutions of the Richardson-Gaudin models. Basic properties of the integrable Richardson-Gaudin models are summarized and their possible applications to variety of nuclear physics models are emphasized. This work was partially performed under the auspices of the U. S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. Thursday, September 22, 2005 3:30PM - 3:45PM KK.00007: The Redmond Formula with Seniority Larry Zamick , Alberto Escuderos As we get to heavier nuclei, we find more states with different seniorities and several states of a given seniority. There is a recursion formula by Redmond that relates an $n \to (n+1)$ coefficient of fractional parentage (cfp) to that of $(n-1) \to n$. However, this involves an {\it overcomplete} set of principal parent (pp) cfp's. For example, for a 3-particle system, we can form basis states $[ [12]^{J_0} 3]^J$, where $J_0$ is the pp; we then antisymmetrize and normalize $\Psi [J_0]=N[J_0] (1-P_{12}-P_{13}) \left[ [12]^{J_0} 3\right]^J$, and form a ppcfp expansion $\Psi[J_0]=\sum_{J_1} [j^2 (J_1) j |\} j^ 3 [J_0] J] \left[ [12]^{J_1} 3\right]^J$. But for, say, $J=j=9/2$, there are five $\Psi[J_0]$'s, but only two independent wave functions, one with seniority 1 and one with seniority 3. We note that $[j^2 (J_0) j |\} j^3 [J_0] J]=1/(3 N[J_0])$. We are able then to obtain the following relation between overcomplete ppcfp's and complete orthonormal cfp's: $A=B=C$, where $$A=(n+1)[j^n(J_0 v_0) j |\} j^{n+1} [J_0 v_0] J] \; \; [j^n (J_1 v_1) j |\} j^{n+1} [J_0 v_0] J],$$ $$B=(n+1) \sum_v [j^n (J_0 v_0) j |\} j^{n+1} J v] \; \; [j^n (J_1 v_1) j |\} j^{n+1} J v],$$ $$C=\delta_{J_0 J_1} \delta_{v_0 v_1} + n (-1)^{J_0+J_1} \sqrt{(2J_0+1)(2J_1+1)} \sum_{v_2 J_2} \begin{Bmatrix} J_2 & j & J_1 \\ J & j & J_0 \end{Bmatrix} \times$$ $$\times [j^{n-1} (J_2 v_2) j |\} j^n J_0 v_0] \; \; [ j^{n-1} (J_2 v_2) j |\} j^n J_1 v_1].$$ Thursday, September 22, 2005 3:45PM - 4:00PM KK.00008: Applications of RPA in the nuclear shell model Calvin Johnson , Ionel Stetcu In recent work we have described a computational implementation of the random phase approximation (RPA) in the interacting shell model. Such an implementation is computationally much cheaper than full scale diagonalization, and provides a reasonable approximation, to binding energies and transitions, including charge-changing transitions. Here we discuss the latest applications of our computer code, SHERPA, with an emphasis on astrophysics.