1. Partial Dynamical Symmetry in Odd-Mass Nuclei A. Leviatan Racah Institute of Physics The Hebrew University, Jerusalem, Israel P. Van Isacker, J. Jolie, T. Thomas, A. Leviatan, Phys. Rev. C 92, 011301(R) (2015) International Workshop ``Shapes and Dynamics of Atomic Nuclei: Contemporary Aspects” (SDANCA-15) Sofia, Bulgaria, October 8-10, 2015

2. Dynamical Symmetry U(6)  U(5)  SO(5)  SO(3)  [N] n d  n  L  Spherical vibrator U(6)  SU(3)  SO(3)  [N] (,  ) K L  Axial rotor U(6)  SO(6)  SO(5)  SO(3)  [N]   n  L   -unstable rotor Complete solvability Good quantum numbers for all states

3. Dynamical Symmetry U(6)  U(5)  SO(5)  SO(3)  [N] n d  n  L  Spherical vibrator U(6)  SU(3)  SO(3)  [N] (,  ) K L  Axial rotor U(6)  SO(6)  SO(5)  SO(3)  [N]   n  L   -unstable rotor Complete solvability Good quantum numbers for all states Partial dynamical symmetry (PDS) Leviatan, Prog. Part. Nucl. Phys. 66, 93 (2011) Exact DS provides considerable insight, however, it is broken in most nuclei Remaining degeneracies e.g., SU(3)-DS degenerate  -  bands Monotonic in-band splitting e.g., L(L+1) splitting in SO(5)  -multiplets More often some states fulfill the DS, other do not e.g., mixed dynamics, QPT

4. Bosonic PDS G dyn = U B (  B ) SU(3)-PDS Leviatan, PRL 77, 818 (1996) Casten, Cakirli, Blaum,Couture, PRL 113, 112501 (2014) Couture, Casten, Cakirli, PRC 91, 014312 (2015) SO(6)-PDS Leviatan, Garcia Ramos, Van Isacker, PRC 87, 021302(R) (2013) Leviatan, Isacker, PRL 89, 222501 (2002) Garcia-Ramos, Leviatan, Van Isacker, PRL 102, 112502 (2009) Kremer et al., PRC 89, 041302(R) (2014) Fermionic PDS G dyn = Sp(6), U F (  F ) SU(3)-PDS Escher, Leviatan, PRL 84, 1866 (2000); PRC 65, 054309 (2002) Rowe, Rosensteel, PRL 87, 172501 (2001) Van Isacker, Heinze, PRL 100, 052501 (2008); Ann. Phys. 349, 73 (2014) Bose-Fermi PDS g,  g,  Symplectic Shell Model Seniority g G dyn = U B (  B )  U F (  F ) This talk Sp(2j+1)-PDS

5.  N    |  N   0   = 0 n-particle annihilation operator for all possible  contained in the irrep  0  of G Condition is satisfied if  0    N-n  DS is broken but solvability of states with  =  0  Is preserved n-body |  N   0   = 0  Lowest weight state  Equivalently: Garcia-Ramos, Leviatan, Van Isacker, PRL 102, 112502 (2009 ) Alhassid, Leviatan, J. Phys. A 25, L1265 (1992 ) Constructing Hamiltonians with PDS

6. U B (6)  U F (12) [N] [1 M ] = 1/2= 0, 2 j =  N bosons M=1 fermion Odd-Mass in the IBFM 195 Pt N=6, M=1, 3p 1/2, 3p 3/2, 2f 5/2

7. ground band 1 st excited band 2 nd excited band SO BF (6) Dynamical Symmetry

8. 2 0 [2,0]  0,0  (0,0) 0 0 1 1 [2,0]  0,0  (0,0) 0 1/2 1 1 [1,1]  1,1  (1,1) 1 1/2 1 1 [1,1]  1,1  (0,1) 1 3/2 1 1 [1,1]  1,1  (1,1) 3 5/2 1 1 [1,1]  1,1  (1,1) 3 7/2 1 1 [1,1]  1,1  (1,1) 2 3/2 1 1 [1,1]  1,1  (1,1) 2 5/2 =  1/2 N M [N 1,N 2 ]  1,  2  (  1,  2 ) Two-particle tensor operators

9. [N-1]  [1,1] = [N,1]  [N-1,1,1]  [N+1]  N+1,0  0,0  =  N+1,0   [N-2], [N-1]   N,1  0,0  =  N,1   [N-2], [N-1] or

10. a, b, c, d, d’ U BF (6), SO BF (6), SO BF (5), SO BF (3), Spin BF (3) a 0, a 2, a 3 (L=S=0, ), a’ 1 ( ) Parameters SO BF (6)-PDS Hamiltonian or

11. Ground band [7,0]  7,0  remains solvable with E DS = E PDS c, d, d’ [6,1]  6,1  levels: improved description a, b, a’ 1, a 2, a 3 1/2 - - 3/2 - inversion reproduced without changing the order of other doublets [6,1]  5,0  levels: position corrected by the a 0 term with marginal effect on lower bands Rms deviation 23 keV

12. Ground band [7,0]  7,0  remains solvable with E DS = E PDS c, d, d’ [6,1]  6,1  levels: improved description a, b, a’ 1, a 2, a 3 1/2 - - 3/2 - inversion reproduced without changing the order of other doublets [6,1]  5,0  levels: position corrected by the a 0 term with marginal effect on lower bands Rms deviation 23 keV

13. E While the states [7,0]  7,0  of the ground band are pure, other eigenstates of in excited bands can have substantial SO BF (6) mixing

14. E2 transition rates 195 Pt Solvable  solvable B(E2) DS = B(E2) PDS Solvable  Mixed Mixed  Solvable slight differences Mixed  mixed biggest changes PDS vs DS

15. PDS and Intrinsic States Ground band[N+1] Excited band [N,1] U BF (6)

16. PDS and Intrinsic States Ground band[N+1] Excited band [N,1], become lowest weight states for the SO BF (6) irreps  N+1 ,  N,1  from which the  (  1,  2 )LJM J  states are obtained by SO BF (5) projection The above operators coincide with those previously discussed For  =1: U BF (6)

17. Summary PDS notion extended to Bose-Fermi systems and exemplified in the odd-mass nucleus 195 Pt Analysis highlights the ability of a PDS to select, in a controlled fashion, required symmetry-breaking terms, yet retain a good DS for a segment of the spectrum Extensions: Additional PDSs in odd mass-nuclei (e.g., SU BF (3)-PDS for  = ) Partial supersymmetry With P. Van Isacker (GANIL) J. Jolie, T. Thomas (Cologne)

18. Thank you