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Coexisting shapes and shape-phase transition in shell models

This study discusses the coexistence of different shapes and shape-phase transitions in shell models, focusing on specific isotopes and their properties. The research explores the impact of isospin symmetry breaking and many-body effects on nuclear shapes. Experimental data and theoretical calculations are utilized to understand the behavior of nuclei with different deformations. The study also addresses the concept of softness in well-deformed nuclei and the challenges in describing transitional nuclei.

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Coexisting shapes and shape-phase transition in shell models

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  1. Coexisting shapes and shape-phase transition in shell models Yang Sun Shanghai Jiao Tong University, China SDANCA15, Oct. 8-11, 2015

  2. To be discussed ... • Along the N=Z line, sudden phase transition from less-deformed to well-deform shape at N=Z=36. • 72Kr marks the boundary, shows oblate to prolate shape transition near the ground state. • In Sm, Gd, Dy isotopes, spherical to deformed shape transition at N=90. • At very high spins of deformed nuclei, states with broken pairs tend to rotate uniformly with a same frequency.

  3. Shapes on the N=Z line 104 slow b decay (waiting point) 80 76 72 68 64 abundance Mass number

  4. Isospin symmetry • Nuclear two-body interaction is approximately charge symmetric and charge-independent. • Charge symmetry: vnn = vpp • Invariance under a rotation by 180o about an axis in isospin space perpendicular to z-direction • Charge independence: vnp = (vnn + vpp) / 2 • Invariance of any rotation in isospin space • Scattering data show that both symmetries are broken • R. Machleidt & H. Muether, Phys. Rev. C 63 (2001) 034005

  5. Isospin-symmetry breaking in nuclei • Reasons for isospin-symmetry breaking: • Coulomb interaction between protons (Nolen–Schiffer anomaly) Annu. Rev. Nucl. Sci. 19 (1969) 471 • Isospin-nonconserving (INC) nuclear interactions • Many-body effects • Wigner et al. (1957): Assuming the two-body nature for any charge-dependent effects and the Coulomb force between the nucleons as a perturbation, they noted that mass excess of the 2T+1members of an isobaric multiplet T are related by the isobaric multiplet mass equation (IMME).

  6. Isospin-symmetry breaking • Isobaric Multiplet Mass Equation (IMME): which depends on Tz up to the quadratic term. • Higher orders of Tz (dTz3, eTz4,…) in IMME are possible, due to • Higher order perturbation • Effective three-body forces • Other many-body effects such as: shape changes Connecting the problem to nuclear many-body effects ! W. Benenson, E. Kashy, Rev. Mod. Phys. 51 (1979) 521

  7. Isospin-symmetry breaking IMME has been tested to be very accurate up to about A~40 M.A. Bentley, S.M. Lenzi, Prog. Part. Nucl. Phys. 59 (2007) 497

  8. Advanced facility: Storage rings ESR at GSI, Germany HIRFL-CSR at Lanzhou, China

  9. Experiment at IMP, China Tz=-1/2, (78Kr Beam) Tz= -1, -3/2, (58Ni Beam)

  10. Deviations from the quadratic IMME • New data suggested non-zero Tz3 term: Y.-H. Zhang et al, Phys. Rev. Lett. 109 (2012) 102501 + d(A,T)Tz3 ME=mass excess Large deviation for A=53, T=3/2 quartet. A non-zero d term is found. Intruder structure influence beyond N=Z=20

  11. Isospin-symmetry breaking seen in excited nuclear states • Isospin is to classify different nuclear states having same quantum numbers (e.g. same J and p). 51Fe: N=25, Z=26,Tz=-1/2 51Mn: N=26, Z=25,Tz=1/2 • States of same T are clearly different. • A shape effect? D. Warner et al., Nature Phys. 2 (2006) 311

  12. Shape coexistence in waiting-point nuclei 68Se and 72Kr Bouchez et al, PRL (2003)

  13. Shape phase transition along the N=Z line • Shape changes suddenly at N=Z=36 Hasegawa, Kaneko, Mizusaki, Sun, Phys. Lett. B656 (2007) 51

  14. Oblate-prolate shape transition in 72Kr Blue: oblate band Pink: prolate band • large-scale shell-model calculation in pf5/2g9/2d5/2 space • Monopole interactions derived from the tensor force: repulsive V(f5/2,f5/2) and attractive V(f5/2,g9/2), push the gd orbits down toward the fp shell with tensor without tensor Calculated by K. Kaneko Model: Kaneko, Mizusaki, Sun, Tazaki, Phys. Rev. C 89, 011302(R) (2014)

  15. Heavy nuclei: Projected Shell Model • Take a set of quasiparticle states at a fixed deformation (e.g. solutions of HF, HFB or HF + BCS) • Select configurations (qp vacuum + multi-qp states near the Fermi level) • Project them onto good angular momentum (if necessary, also parity, particle number) to form a basis in laboratory frame • Diagonalize a two-body Hamiltonian in the projected basis • This model has worked well for spectrum description for nuclei with stable deformation (and super-deformed or superheavy nuclei) • K. Hara, Y. Sun, Int. J. Mod. Phys. E 4 (1995) 637

  16. Example of a good axially-deformed rotor • Angular-momentum-projected energy calculation shows a deep prolate minimum • A very good rotor with axially-deformed shape • Quasi-particle excitations based on the same deformed potential

  17. Example of softness – no definite shapes Mean-field calculation shows a spherical shape. Projected calculation shows shallow minima separated by a low energy barrier. Definite shapes are developed with rotation.

  18. g-softness in well-deformed nuclei Angular-momentum-projected energy surfaces as functions of e and g Soft in g-direction in nuclei that are believed to be well-deformed axial rotor

  19. Description of a system with soft potential surfaces • A spherical nucleus described by spherical shell model. • A deformed nucleus described by deformed shell model. • Transitional ones are difficult. A better wavefunction is a superposition of many states of deformation parameter b. Spherical Transitional Deformed Schematic energy potential for spherical (red), transitional (dashed), and deformed (blue) nuclei.

  20. Generate Coordinate Method (GCM) • GCM starts with a general ansatz for a trail wave function • with being generate coordinates • is a weight function, determined by solving the Hill-Wheeler Equation • with the overlap functions • Ring and Schuck, Nuclear Many-Body Problem, 1980

  21. Projected Generate Coordinate Method (PGCM) • Choosing generate coordinate as e2, an improved wave function • Hamiltonian • with a fixed set of parameters (fixed c, GM, and GQ) is diagonalized for a chain of isotopes. • F.-Q. Chen, Y. Sun, P. Ring, Phys. Rev. C88 (2013) 014315

  22. Energy levels • Comparison of energy levels of 21+, 41+, and 61+ of ground band and excited 02+ state • Exp data (filled squares) • Calculations (open circles) • for isotopes from N=90 (transitional) to N=98 (well-deformed) nuclei N=90 N=98

  23. Spherical-deformed shape phase transition Rotor Critical point Vibrator

  24. Spherical-deformed shape phase transition • Drastic changes in electric quadrupole transition B(E2, 2+ 0+) from vibrator 152Gd (N=88), to critical point 154Gd (N=90), to rotor 156-160Gd (N>90). • Black squares show if use only one fixed deformation e2 in the calculation, transitional feature cannot be reproduced.

  25. Distribution function • The Hill-Wheeller Equation diagonalizes the Hamiltonian in a non-orthogonal basis, and therefore, f(e2) is not a proper quantity to analyze the GSM wave function. • Transformation of f(e2) to an orthogonal basis gives • whichcan be used to present the distribution of the GCM wave functions. • g2(e2) represent the probability function.

  26. Distribution function of deformation Calculated distribution function of deformation for the first three 0+ states in 154Gd and 160Gd

  27. Probability function of deformation Calculated probability function of deformation for ground state 01+ and excited 02+ state in 154Gd and 160Gd.

  28. Probability function of deformation • Peak of the Gaussian defines deformation • 160Gd being more deformed than 154Gd • The distribution is wider for 154Gd • reflecting the softness of this nucleus • The distribution for 02+ is much more fragmented • reflecting a vibrational nature of these states • For 01+ , system stays mainly at system’s deformation with the largest probability • For 02+ , system shows two peaks having different heights lying separately at both sides of the equilibrium • indicating an anharmonic oscillation • prefering to have a larger probability in the site of larger deformation

  29. Multi-quasiparticle computation using the Pfaffian algorithm • Calculation of projected matrix elements usually uses the generalized Wick theorem • A matrix element having n (n’) qp creation or annihilation operators respectively on the left- (right-) sides of the rotation operator contains (n + n − 1)!! terms in the expression – a problem of combinatorial complexity • Use of the Pfaffian algorithm: • L.M. Robledo, Phys. Rev. C 79 (2009) 021302(R). • L.M. Robledo, Phys. Rev. C 84 (2011) 014307. • T. Mizusaki, M. Oi, Phys. Lett. B 715 (2012) 219. • M. Oi, T. Mizusaki, Phys. Lett. B 707 (2012) 305. • T. Mizusaki, M. Oi, F.-Q. Chen, Y. Sun, Phys. Lett. B 725 (2013) 175 • Q.-L. Hu, Z.-C. Gao, Y. S. Chen, Phys. Lett. B 734 (2014) 162

  30. Go to very high-spin states Record extension of qp-basis: including up to10-qp states At very high spins, all states with different configurations rotate similarly – indication of a new collectivity?

  31. Collaborators Fang-Qi Chen Long-Jun Wang (Shanghai Jiao Tong University, China) K. Kaneko (Fukuoka, Japan) T. Mizusaki (Tokyo, Japan) M. Oi (Tokyo, Japan) P. Ring (Munich, Germany)

  32. Summary • Examples of shape phase transition • Along the N=Z line, 72Kr shows oblate to prolate shape transition near ground state (large-scale spherical shell model with inclusion of monopole interactions derived from the tensor force) • In Sm, Gd, Dy isotopes, spherical to deformed shape transition at N=90 (extended PSM with Generate Coordinate Method) • At very high spins, states with broken pairs tend to rotate uniformly with a same frequency (Phaffian algorithm) • New development in the Projected Shell Model: • Improved PSM wave function by superimposing (angular-momentum and particle-number) projected states with different deformation e2 • High-order multi-quasiparticle states with angualr momentum projection by using the Phaffian algorithm

  33. CGS16 Shanghai, 2017The 16th International Symposium on Capture Gamma-Ray Spectroscopy and Related TopicsEarly September, 2017 Shanghai Jiaotong University, Shanghai, China

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