fermions c j N bosons A nucleons valence nucleonsN nucleon pairs L = 0 and 2 pairs s,d  even-even nuclei 2.2 The Interacting Boson Approximation A. Arima, F. Iachello, T. Otsuka, I Talmi 

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

The hamiltonian is written in terms of the 36 generators of U(6):. Construction of a dynamical symmetry as an example U(5) limit: Now, consider the 10 odd k generators only and then the angular momentum ones.

U(5) SU(3) SO(6) 196 Pt 156 Gd 110 Cd U(5): Vibrational nuclei SO(6):  -unstable nuclei SU(3): Rotational nuclei (prolate) Dynamical symmetries of a N s,d boson system  U(5)  SO(5)  SO(3)  SO(2) {n d } (  ) L M U(6)  SO(6)  SO(5)  SO(3)  SO(2) [N] (  ) L M  SU(3)  SO(3)  SO(2) (  ) L M

Analytic solutions are associated with each dynamical symmetry: U(5): H =  1 C 1 [U(5)] +  ’ 1 C 2 [U(5)] +  4 C 2 [O(5) ] +  5 C 2 [ SO(3)] E(n d ,L) =  1 n d +  ’ 1  n d ( n d + 4) +  4  (  +3) +  5 L(L+1) H =  1 C 1 [U(5)] +  ’ 1 C 2 [U(5)] +  2 C 2 [SU(3) ] +  3 C 2 [SO(6) ] +  4 C 2 [SO(5) ] +  5 C 2 [ SO(3)] Again using first and second order Casimir operators a six parameter hamiltonian results:: which needs to be solved numerically. SU(3): H =  2 C 2 [SU(3)] +  5 C 2 [ SO(3)] E( ,L) =  2      +  5 L(L+1) SO(6): H =  3 C 2 [SO(6)] +  4 C 2 [O(5) ] +  5 C 2 [ SO(3)] E( ,L) =  3  (  +4) +  4  (  +3) +  5 L(L+1)

A dynamical symmetry leads to very strict selection rules that can be used to test it. If an operator is a generator of a subalgebra G then due to the property: [G i,G j ]=  k c ijk G k. and E  = a f(  )  =>E  k = a f(  )  k with  k  {G k  } it cannot connect states having a different quantum number with respect to G. Example: 196 Pt and its E2 properties. is an SO(6) generator E2 transitions between different SO(6) representations are forbidden.

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

Nuclear shapes associated with the four dynamical symmetries The shapes can be studied using the coherent state formalism. using the intrinsic state (Bohr) variables: Then the energy functional: can be evaluated for each value of  and 

U(5) limit: irrelevant: spherical vibrator SO(6) limit: flat:  -unstable rotor SU(3) limit: prolate rotor SU(3) limit: oblate rotor   SU(3)  E U(5) SO(6) 0° 60°

Shape phases and critical point solutions. Most nuclei are very well described by a very simple IBA hamiltonian of Ising form: with two structural parameters  and  and a scaling factor a  with generates spherical shape generates deformed shape  

U(5)  The rich structure of this simple hamiltonian are illustrated by the extended Casten triangle  SU(3) U(5) limit U(6)  U(5)  SO(5)  SO(3) SO(6) limit U(6)  SO(6)  SO(5)  SO(3) SU(3) limit U(6)  SU(3)  SO(3) The simple hamiltonian has four dynamical symmetries SO(6)  SU(3) SU(3) limit U(6)  SU(3)  SO(3)

Energy functional in coherent state formalism

Shape phase transitions in the atomic nucleus. When studying the changes of the nuclear shape one might observe shape phase transitions of the groundstate configuration. They are analogue to phase transitions in crystals

Thermodynamic potential: External parameters Order parameter Energy functional: L. Landau Landau theory of continuous phase transitions (1937) describes these shape phase transitions. J.Jolie, P. Cejnar, R.F. Casten, S. Heinze, A. Linnemann, V. Werner, Phys. Rev. Lett. 89 (2002) P. Cejnar, S. Heinze, J.Jolie, Phys. Rev. C 68 (2003) P T

T cc  c  c   min T cc  P 0,T,  min  T cc  P 0,T,  First order phase transition with P = P 0 = const cc  c  c   min cc  P 0,T,  min  T cc  P 0,T,  Second order phase transition

should be continuous everywhere. if discontinuous at  0 : first order phase transition. if discontinuous at  0 : second order phase transition. with

Extremum are at: Our case: always: and

Both minima become degenerated at: or at  cc  To fullfill this equation and the one for  0

B -A 00 Solution: First order phase transitions at: Second order at:

So we can absorb it by allowing negative  values ! and Energy functional in coherent state formalism

One obtains then: when we fix N: prolate-oblate The first order phase transitions should occur when spherical-deformed The isolated second order transition at:

spherical  = 0 prolate deformed  > 0 oblate deformed  < 0 Triple point of nuclear deformation : first order transition : isolated second order transition I III II P T J.Jolie, P. Cejnar, R.F. Casten, S. Heinze, A. Linnemann, V. Werner, Phys. Rev. Lett. 89 (2002) Landau theory and nuclear shapes. Thermodynamic potential: External parameters Order parameter Energy functional:  (P,T;  ) E(N,  )

The shape phase transitions can be seen by the groundstate energies.  E  SU(3) O(6) SU(3) U(5) (N=40)

The quadrupole moment corresponds to the control parameter  0 : N=10N=40 N=10N=40 A sensitive signature is in particular the B(E2; > )

J. Jolie, P. Cejnar, R.F. Casten, S. Heinze, A. Linnemann, V. Werner, Phys.Rev.Lett. 89(2002) P. Cejnar and J. Jolie, Rep. on Progress in Part. and Nucl. Phys. 62 (2009) 210. dynamical symmetry First order phase transition Second order phase transition (isolated) spherical prolate deformed oblate deformed Conclusion: the following shapes phase transitions are obtained: J. Jolie, R.F. Casten, P. von Brentano, V. Werner, Phys.Rev.Lett. 87(2001)162501

Examples for the prolate-oblate and sherical-prolate phase transition Pb Hg Pt Os W Hf Yb Hg 198 Hg 196 Pt 194 Pt 192 Os 190 Os 188 Os 186 W 184 W 182 W 180 Hf Samarium isotopes

N=10 N=20

50 82 Normal states Intruder states 110 Cd  2p-2h  pair +  Q  +... K. Heyde, et al. Nucl. Phys. A466 (1987) 189. This can be described in the IBM by a N (normal) plus N+2 (intruder) system which might mix. : with and Core excitations

  I z = + 1/2 I z = - 1/2 U p (6) U h (6) U(6) Also new kinds of symmetries are possible: Intruder or I-spin always fulfilled [H,I z ] =0 [H,I 2 ] = 0 [H,I + ] = [H,I - ] = 0 intruder-analog state good intruder Spin K. Heyde, C. De Coster, J. Jolie, J.L. Wood, Phys. Rev. C 46 (1992), 541

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

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. 2.3 A case study: 112Cd M. Délèze, S. Drissi, J. Jolie, J. Kern, J.P. Vorlet, Nucl. Phys, A554 (1993) 1  Pd ( ,2n)  Cd reaction allowed the establishment of multiphonon states up to nd=6.

5.39ps 0.68ps 0.73ps But, above 1.2 MeV additional states exist and build a second collective structure. < 2.1ps < 2.8ps 0.7ps and The additional states are intruder states presenting 2 particle- 2 hole excitations across Z=50. They lead to shape coexistence and can be described using the SO(6) limit. E[N,n d, ,L] =    n d +    (  +3) +    L(L+1)  intr  L ] =    (  +3)  +    (  +3) +    L(L+1)

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

Cannot connect intruder with normal states  =N+2  max=N

Moreover one can rewrite:

Inelastic Neutron Scattering (INS) experiment at the Van de Graaff Accelerator of the University of Kentucky (Prof S.W. Yates, Lexington USA). (n,n’  ) E level J  and placements of E  from excitation function varying E n  from angular distributions  n  = 1.25ps  = 1.16ps  = 0.67ps  = 1.20ps  = 0.42ps  = 0.51ps

(n,n´) with 3.4 and 4 MeV neutrons for lifetimes and coincidences allowed the extension to higher low-spin states (P.E. Garrett et al. Phys. Rev. C75 (2007) There it becomes difficult to describe the details.

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 Absolute B(E2) values for the decay of three phonon states in 110 Cd

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)

Fribourg-Kentucky-Köln Data B(E2) Values in Six Valence Proton Configurations NeutronCdIntruderRuBa NumberB(E2;2 3  0 A ) 1  0 1 ) B(E2;2 1  0 1 )            It works well only in a given shell. M. Kadi, N. Warr, P.E. Garrett, J. Jolie, S.W. Yates, Phys. Rev. C68 (2003) (R).