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Critical Point Symmetries

V( β ). Approximate potential at phase transition with infinite square well. β. Solve Bohr Hamiltonian with square well potential. Result is analytic solution in terms of zeros of special Bessel functions. Predictions for energies and electromagnetic transition probabilities. γ -soft. E(5).

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Critical Point Symmetries

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  1. V(β) Approximate potential at phase transition with infinite square well β Solve Bohr Hamiltonian with square well potential Result is analytic solution in terms of zeros of special Bessel functions Predictions for energies and electromagnetic transition probabilities γ-soft E(5) Symmetric Rotor Spherical Vibrator X(5) Critical Point Symmetries Two solutions depending on γ degree of freedom F. Iachello, Phys. Rev. Lett. 85, 3580 (2000); 87, 052502 (2001).

  2. τ = 1 Key Signatures τ = 0 E(41)/E(21) = 2.91 ξ = 2 E(02)/E(21) = 5.67 R4/2 = 2.20 E(02)/E(21) = 3.03 E(03)/E(21) = 3.59 ξ = 1 X(5) and E(5)

  3. P= NpNn Np+Nn Searching for X(5)-like Nuclei β-decay studies at Yale 152Sm R.F. Casten and N.V. Zamfir, Phys. Rev. Lett. 87, 052503 (2001). N.V. Zamfir et al., Phys. Rev. C 60, 054312 (1999).. 156Dy M.A. Caprio et al., Phys. Rev. C 66, 054310 (2002). 162Yb E.A.M. et al., Phys. Rev. C 69, 024308 (2004). 166Hf Good starting point: R4/2 or P factor E.A.M. et al., Phys. Rev. C- submitted. Other Yale studies: 150Nd - R.Krücken et al., Phys. Rev. Lett. 88, 232501 (2002).

  4. Searching for E(5)-like Nuclei Ce 58 3.06 2.93 2.80 2.69 2.56 2.38 2.32 Ba 134Ba 56 2.96 2.89 2.83 2.78 2.69 2.52 2.43 2.32 2.28 Xe 54 2.33 2.40 2.47 2.50 2.48 2.42 2.33 2.24 2.16 2.04 R.F. Casten and N.V. Zamfir, Phys. Rev. Lett. 85, 3584 (2000). Te 52 2.09 2.00 1.99 2.07 2.09 2.07 2.04 2.01 1.94 1.72 Sn 50 1.54 1.67 1.75 1.81 1.79 1.68 1.84 1.85 1.87 1.88 1.86 1.80 1.71 1.63 102Pd Cd 48 2.33 1.79 2.11 2.27 2.36 2.38 2.33 2.29 2.30 2.38 2.39 2.38 46 Pd 1.79 2.12 2.29 2.38 2.40 2.42 2.46 2.53 2.56 2.58 N.V. Zamfir et al., Phys. Rev. C 65, 044325 (2002). Ru 44 1.82 2.14 2.27 2.32 2.48 2.65 2.75 2.76 2.73 42 Mo 1.81 2.09 1.92 2.12 2.51 2.92 3.05 2.92 130Xe P~2.5 Zr 40 1.60 1.60 1.63 1.51 2.65 3.15 3.23 Sr 38 1.99 2.05 3.01 3.23 Z/N 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 Good starting point: R4/2 or P factor

  5. Symmetries and phases transitions in the IBM • Challenges for neutron-rich: • New collective modes in three fluid systems (n-skin). • New regions of phase transition • New examples of critical point nuclei? • Rigid triaxiality? D.D. Warner, Nature 420 (2002) 614

  6. 126 82 proton number Z N=Z 50 82 28 20 50 8 28 2 20 neutron number N 8 2 Symmetry Near the Drip Lines • N-p exchange symmetry (Isospin) • proton-neutron pairing: T=1 versus T=0 • Neutron excess =>Neutron skin • N-p (boson) exchange symmetry (Fspin) • New collective modes

  7. Dynamical Symmetries with Neutrons and Protons

  8. p n j n p Only new structure is U+(6) • Describes the symmetry under exchange of neutron and proton bosons. • Corresponds to the “orbital symmetry” of n’s and p’s. • Describes whether n,p contributions to collective motion couple in-phase (symmetric) or out-of-phase (non-symmetric).

  9. Algebra of U+(6) New quantum number --- F-Spin

  10. Properties of F-Spin

  11. Example:Vibrational U(5) case

  12. Experimental Evidence

  13. THE NEUTRON SKIN n Core n+p p t r(r) t n Collective n-p modes p n j n p “Soft modes” core n

  14. Up(6) x Un(6) x Us(6)  Upn(6) x Us(6)  Upns(6) SUpns(3) Upns(5) Opns(6) THE ALGEBRAIC APPROACH • Treat as a system with 3 constituents • Assume weaker coupling of the neutron skin

  15. Consider lowest irreps. Upn (6) x Us(6)  Upns(6) [Nc-f , f] [Ns] [N-f1-f2 , f1 , f2] Gr. State - skin: [N, 0, 0] [Nc, 0] [Ns] Mixed sym. - skin: [N-1 , 1, 0] [Nc-1 , 1] [Ns] [N-1, 1, 0] Now two non-symmetric modes: Onep versus n in core: Onep + n in core versus skin

  16. Consider SU(3) case: Soft Scissors mode E = l1 C2[Upn(6)] + l2 C2[Upns(6)] “Normal” 1+ “Soft” ~ 2 l1 Nc + 2 l2 N 1+ ~ 2 l2 N 0+ Similarly in U(5) and O(6) limits

  17. Properties of Soft Scissors Mode

  18. Relativistic Coulomb Excitation Coulex probability: With strength function: Hence:

  19. Shapes and Excitation Modes of Halos & Skins E2 isovector deformation E1 “pygmy” resonance Scissors mode core skin

  20. The Algebraic Approach • There are two isospin invariant versions of the IBM:- IBM-3:S=0, T=1 IBM-4: T=1, S=0 AND T=0, S=1 p n p p n n n p • IBM-4 incorporates Wigners’s spin-isospin SU(4) symmetry. • [H,s] = [H,t] = [H,s.t] = 0 N-P Pairing

  21. Even-even NO (,d) T=0 T=0 (d,) (d,) Odd-odd (T=1) T=1 and 0 T=0 T=1 and 0 (p,3He) (3He,p) (p,3He) Z NO (,d) T=0 (3He,p) N Deuteron Transfer on the N=Z Line

  22. T=0 vs. T=1 pairing • P. Van Isacker and D. D. Warner, PRL 78, 3266 (1997) • N=Z Masses in 28-50 shell • E.Baldini-Nero et al, Phys. Rev. C65, 064303, (2002) Estimate from IBM • Schematic Hamiltonian: • H = a C2[SUTS(4)] + b C1[US(3)} • L = 0 IBM-4 • C1[US(3)} determines relative energy of T=0 and T=1 bosons (pairs) b/a IBM-4 SU(4) IBM-3 No SU(4)

  23. T=0 or 1 transfer Nb Nb + 1 b/a > 1 T = 1 g.s. in odd-odd b/a < 1 T = 0 g.s. in odd-odd b/a =5 from fit to masses b/a >> 1 C02/C12 ~ 3/(Nb+3) for EE to OO Conclusion: T=1 transfer likely to dominate (Van Isacker, Frank and Warner, PRL 94, 162502, 2005) Deuteron Transfer

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