Symmetry Principles and Nuclear Structure J. Jolie, Institut für Kernphysik, Universität zu Köln

1. Symmetry Principles and Nuclear Structure J. Jolie, Institut für Kernphysik, Universität zu Köln 1. Introduction 2. Dynamical symmetries 3. The interacting boson approximation 4. More advanced applications 4.1 Shape phases and critical point symmetries. 4.2 The N= 52 isotopes and mixed symmetry states. 4.3 Studies of stable even-even Cd isotopes: multiphonon states and intruder analog states. 4.4. The light unstable even-even Cd isotopes. Towards a microscopic foundation of collectivity and symmetries. 5. Conclusion

2. 1. Introduction Space-time symmetries for closed systems: Conservation of momentum P space Conservation of energy E time Conservation of angular momentum L

3. Are there more constants of motion? Niels Bohr description of the Hydrogen atom (1913) n=4 L=0,1,2,3 0 n=3 L=0,1,2 n=2 L=0,1 E n=1 L=0 • n is also a conserved quantity (constant of motion) independent of • the Rydberg constant. • 2) wavefunctions do not depend on Rydberg but on n, L and M • 3) n gives values ofLwhich in turn determinesM V. Fock (1934): The equations of the Hydrogen atom are invariant under orthogonal rotations in four dimensions O(4). They exhibit a dynamical symmetry.

4. 2. Dynamicalsymmetries Hamiltonian Lie AlgebraG [Gi,Gj]=SkcijkGk. Operators {Gi} Casimir Operator [Cn[G] ,Gk]=0Gk H = a Cn[G] {Gm} {Gl}  {Gk}  {Gi} EY = a f(a) Y => EYk = a f(a) Yk with Yk {GkY} GmGlGkGi H = Si aiCn[Gi] E(abg) = Si aifabg(n,Gi)

5. Example: Angular momentum algebra +J L 0 -J M Hamiltonian Lie AlgebraG O(3) Operators {Gi} Casimir Operator H = a L2 EY = a f(L) Y => EYk = a L(L+1)Yk with Yk {L+Y, L-Y LzY} O(2)O(3) H = = a L2 + bLz E(LM) = a L(L+1) + b M

6. Experiment Theory Light atomic nuclei L=0,2 Nuclei far from closed shells Pairing property between nucleons Interacting boson model Collective motion Quantum liquid drop model Two nucleon systems Strong force Few nucleon systems Shell model with single particle orbits Magic numbers Nuclei near closed shells

7. 2N fermions cj nucleon pairs M valence nucleons IBA s,d 1974-1979 A. Arima, F. Iachello, T. Otsuka, I Talmi 3. The Interacting Boson Approximation even-even nuclei A nucleons N bosons L = 0 and 2 pairs

8. Schrödinger equation in second quantisation N=cte N s,d boson system with

9. Dynamical symmetries of a N s,d boson system  U(5) O(5)  SO(3)  SO(2) {nd} (v) L M U(6)  O(6)  O(5)  SO(3)  SO(2) [N] <S> (t) L M  SU(3)  SO(3)  SO(2) (l,m) L M 110Cd U(5) SU(3) 156Gd 196Pt O(6) U(5): Vibrational nuclei O(6): g-unstable nuclei SU(3): Rotational nuclei (prolate)

10. A dynamical symmetry leads to very strict selection rules that can be used to test it. Example: 196Pt and its E2 properties. If an operator is a generator of a subalgebra G then due to the property: [Gi,Gj]=SkcijkGk. and EY = a f(a) Y =>EYk = a f(a) Yk with Yk {GkY} it cannot connect states having a different quantum number with respect to G. is an O(6) generator E2 transitions between different O(6) representations are forbidden.

11. 195Pt(n,g)196Pt + GRID method H.G. Borner, J. Jolie, S.J. Robinson, R.F. Casten, J.A. Cizewski, Phys. Rev. C42(1990) R2271 ? B(E2)= 0 Also quadrupole moments are equal to zero because of O(5) (seniority) and the d-boson number changing E2 operator. Experimentally: Q(2+1) = +0.66(12) eb

12. 4. More advanced applications 4.1 Shape phases and critical point symmetries. generates deformed shape generates spherical shape c h Most nuclei are very well described by a very simple IBA hamiltonian of Ising form: with with two structural parameters h and c and a scaling factor a.

13. U(5) limit U(6)  U(5) O(5)  SO(3) O(6) limit U(6)  O(6)  O(5)  SO(3) SU(3) limit U(6)  SU(3)  SO(3) SU(3) limit U(6)  SU(3)  SO(3) SU(3) O(6) c = 0 SU(3) U(5) The simple hamiltonian has four dynamical symmetries The rich structure of this simple hamiltonian are illustrated by the extended Casten triangle h = 1 h = 0

14. From a detailed analysis the following shapes phase transitions are obtained: oblate deformed dynamical symmetry First order phase transition Second order phase transition (isolated) spherical prolate deformed J. Jolie, R.F. Casten, P. von Brentano, V. Werner, Phys.Rev.Lett. 87(2001)162501 J. Jolie, P. Cejnar, R.F. Casten, S. Heinze, A. Linnemann, V. Werner, Phys.Rev.Lett. 89(2002)182502

15. 104 106 108 110 112 114 116 118 120 122 124 126 Pb Hg Pt Os W Hf Yb 200Hg 198Hg 194Pt 196Pt 190Os 188Os 192Os 184W 186W 182W R4/2 B(E2;2+2 ->2+1)[W.u] Q(2+1)[eb] 180Hf J.Jolie, A. Linnemann Phys. Rev. C 68 (2003) 031301. Experimental examples for the prolate-oblate phase transition

16. X(5) 152Sm R.F. Casten, V. Zamfir, Phys. Rev. Lett. 85 (2000)3584 Following a collective model approach F. Iachello introduced new symmetries that describe certain nuclei at the phase transition: Critical point symmytries, i.e. X(5) and E(5) F. Iachello, Phys. Rev. Lett.85 (2000) 3580 and 87 (2001) 052502.

17. Further examples of X(5) were identified: P 4.4 4.8 5.1 5.3 E.A.McCutchan et al., Phys. Rev. C69 (2004) 024308 A RISING experiment will look to test this 150Nd: R. Kruecken et al., Phys. Rev. Lett. 88, 232501 (2002) 152Sm: R.F. Casten et al., Phys. Rev. Lett. 85, 3584 (2000) 154Gd: D. Tonev et al., Phys. Rev. C. 69, 034334 (2004) 156Dy: M.A Caprio et al., Phys. Rev. C. 66, 054310 (2002) O. Moeller et al., Phys. Rev, C 74, 024313 (2006)

18. 4.2 The N= 52 isotopes and mixed symmetry states. Introduction of the neutron-proton degree of freedom F-spin Nnneutron bosons: Npproton bosons: [H,Fz] =0 always fulfilled [H,F2] = 0 good F-spin F= Fmax -1 F=Fmax P. Van Isacker, K. Heyde, J. Jolie, A. Sevrin, Ann. of Phys. 171 (1986) 253

19. Identification of MS states via typical F-spin vector M1 transitions.

20. Mixed symmetry states near N=52 C. Fransen et al. in prep. 98 N. Pietralla et al. (2001) Kr

21. b-decay and (a,2n) allowed to establish multiphonon states in several isotopes 4.3 Studies of stable even-even Cd isotopes: multiphonon states and Intruder Analog States. M. Bertschy, S. Drissi, P.E. Garrett, J. Jolie, J. Kern, S.J. Manannal, J.P. Vorlet, N. Warr, J. Suhonen, Phys. Rev C 51 (1995) 103 The cadmium isotopes are unique in several respects. -) protons nearly fill the Z=50 (Sn) shell; -) neutrons are near mid-shell (N=66) -) there are 8 stable isotopes of Cd. This is clearly the mass region where we can learn about nuclear structure.

22. E[N,nd,v,L] = e nd + and(nd +4) + bv(v+3) +g L(L+1) But, 0.7ps No lifetimes known for states with nd > 2 and L 2nd in 1995. < 2.8ps < 2.1ps 0.73ps and 0.68ps Additional states exist and build a second collective structure. 5.39ps The additional states are intruder states presenting 2 particle- 2 hole excitations across Z=50. They lead to shape coexistence.

23. Intruder states Normal states 82 50 110 Cd This can be described in the IBM by a N (normal) plus N+2 (intruder) system which might mix. : D = E(2p-2h) - Dpair + DQnp + ... with and K. Heyde, et al. Nucl. Phys. A466 (1987) 189.

24. U(6) U U(5) O(6) U O(5) U O(3) Symmetries can play a dominant role in shape coexistence. is a O(5) scalar. Wavefunctions with O(5) symmetry have fixed seniority of d-bosons. normal states Intruder states D 112 Cd J. Jolie and H. Lehmann, Phys. Lett B342 (1995)

25. Inelastic Neutron Scattering (INS) experiment at the Van de Graaff Accelerator of the University of Kentucky (Prof S.W. Yates, Lexington USA). (n,n’ g) Elevel Jpand placements of Eg from excitation function varying En d,t from angular distributions n t = 1.25ps1.200.42 t = 1.20ps0.830.35 t = 1.16ps0.490.27 t = 0.42ps0.100.07 q t = 0.67ps0.210.13 t = 0.51ps0.170.10

26. Absolute B(E2) values for the decay of three phononstates in 110Cd Harmonic vibrator (collective model) + finite N effects (IBM in the U(5)-limit) + intruder states within a U(5)-O(6) model + neutron-proton degree of freedom and symmetry breaking

27. Confirmation of the U(5)-O(6) picture. F. Corminboeuf, T.B. Brown, L. Genilloud, C.D. Hannant, J. Jolie, J. Kern, N. Warr and S.W. Yates, Phys. Rev. C 63 (2001) 014305.

28. Also new kinds of symmetries are possible: Intruder or I-spin U(6) IBM Up(6) Iz = + 1/2 EIBM Uh(6) Iz = - 1/2 [H,Iz] =0 always fulfilled [H,I2] = 0 good intruder Spin [H,I+] = [H,I-] = 0 intruder-analog state K. Heyde, C. De Coster, J. Jolie, J.L. Wood, Phys. Rev. C 46 (1992), 541

29. Intruder analog states I=2 I=3/2 I=1 I=1/2 I=0 -2 -3/2 -1 -1/2 0 +1/2 +1 +3/2 +2 Iz H. Lehmann, J. Jolie, C. De Coster, B. Decroix, K. Heyde, J.L. Wood, Nucl. Phys. A 621 (1997) 767

30. Neutron Cd Intruder Ru Ba ® ® ® Number B(E2;2 0 ) B(E2;2 0 ) B(E2;2 0 ) 3 A 1 1 1 1 + 27 - - - - 62 23 - 18 ± ± - - 64 56 17 58 5 ± 154 14 ± ± 66 61 8 70 5 + 24 ± ± 68 86 74 7 116 6 - 30 ± - - - - 98 16 70 B(E2) Values in Six Valence Proton Configurations Fribourg-Kentucky-Köln Data It works well only in a given shell. M. Kadi, N. Warr, P.E. Garrett, J. Jolie, S.W. Yates, Phys. Rev. C68 (2003) 031306(R).

31. 4.4. The light unstable Cd isotopes: towards a microscopic foundation of collectivity and symmetries. Recent theoretical and experimental work on 102Cd and 104Cd paves the way to a microscopic description of symmetries describing collective motion. Clue: Large scale shell model calculations, i.e. ANTOINE, can be performed today. Model space: core 88Sr and full open shell Realistic interaction with modified monopole part (see talk of E. Caurier). Today the large scale shell model calculation offer the way to go from the basics (NN potentials) to the collective phenomena (see also contributions of A. Covello, A. Faessler, J. Wambach and W. Nazarewicz)

32. Result: N. Boelaert, N. Smirnova, K.L.G. Heyde, J. Jolie, subm to PRC (2006)

33. Can the shell model describe the onset of collectivity? But, also other collective states like mixed- symmetry states (in 106Cd) are well described. O(6) U(5) R(4/2) ratio Experimental data from A. Linnemann (PhD Köln 2006)

34. Is If Further experimental tests used the Recoil Distance Doppler Method at the Tandem Accelerator in Cologne. High sensitivity High beam current compared to radioactive beams Long experiments can be made. Cologne Plunger Device T=0 Detector T = d/vR Stopper Target v Energie t T Ge Detectors dE/E = 2. 10-3 v/c=0.02-0.15 with c = 300 mm/ps

35. Cologne set-up: 102Cd N. Boelaert, A. Dewald, Ch. Fransen, J. Jolie, A. Linnemann, B. Melon et al. to be publ.

36. 5. Conclusions Symmetries, whether they are dynamical or fundamental, are an essential tool to guide us through the complex behavior of atomic nuclei: They provide benchmarks and robust predictions which experiments can directly verify or falsify. This is why experimenters love them. They lay the basis for perturbative treatments refining our understanding of nuclei. They provide understanding in complex situations. Nobody can read a wavefunction with 1011 components within his/her lifetime. Let´s stand make sense of it. They urge experiments to adopt new approaches in order to get the unreachable but crucial, while robust, number. They should be provided a solid microscopic foundation by comparing their predictions with those of more microscopically funded models. New symmetries like supersymmetry, not treated here, might offer even new ways to unify our understanding of nuclei. This work was done in colaboration with many people (see quoted references) and with substantial support for JJ from DFG, BMBF, and the Swiss National Science Foundation.