even-even nuclei odd-even nuclei odd-odd nuclei 3.1 The interacting boson-fermion model

s,d,a j N bosons 1 fermion e-o nucleus s,d N+1 bosons  IBFA e-e nucleus  fermions c j M valence nucleons A nucleons L = 0 and 2 pairs nucleon pairs A. Arima, F. Iachello, T. Otsuka, O. Scholten, I Talmi Odd-A nuclei: the interacting boson-fermion approximation 

and

H B the IBM-1 Hamiltonian the single particle Hamiltonian with the energies  j. the boson-fermion interaction The most general hamiltonian contains much too many parameter and is replaced by a simpler one based on shell model considerations and BCS.

It has three terms

BCS calculation gives: quasiparticle energies E j and occupation numbers u j and v j as a function of  j and . O. Scholten PhD + ODDA code mit the excitation energy of the first 2+ state in the corresponding semimagical nucleus we now have n single particle energies, the gap  and three parameters + six for the boson part.

Example: odd Rhodium isotopes (J.Jolie et al. Nucl.Phys. A438 (1985)15 H B from fit of Pd isotopes by Van Isacker et al.  =+  =-  =1.5 MeV

N s,d bosons: U(6) symmetry for model space 36 generators N s,d bosons+ j fermion: U B (6)xU F (2j+1) Bose-Fermi symmetry 36 boson generators + (2j+1) 2 fermion generators, which both couple to integer total spin and fullfil the standard Lie algebra conditions. 3.2 Bose-Fermi symmetries

Bose-Fermi symmetries Two types of Bose-Fermi symmetries: spinor and pseudo spin types Spinor type: uses isomorphism between bosonic and fermionic groups Spin(3): SO B (3) ~ SU F (2) Spin(5): SO B (5) ~ Sp F (4) Spin(6): SO B (6) ~ SU F (4) Exemple: SO(6) core and j=3/2 fermion Balantekin, Bars, Iachello, Nucl. Phys. A370 (1981) 284. U B (6)xU F (4)  SO B (6)xSU F (4)  Spin(6)  Spin(5)  Spin(3) ¦ ¦ ¦ ¦ ¦ ¦ ¦ [N] [1] (  1,  2 ) J H= A’ C 2 [SO B (6)] + A C 2 [Spin(6)] + B C 2 [Spin(5)]+ C C 2 [Spin(3)] E = A’ (  (  +4)) + A(  1 (  1 +4) +  2 (  2 +2) +  3 2 ) + B(  1 (  1 +3) +  2 (  2 +1)) + CJ(J+1)

Pseudo Spin type: uses pseudo-spins to couple bosonic and fermionic groups Example: j= 1/2, 3/2, 5/2 for a fermion: P. Van Isacker, A. Frank, H.Z. Sun, Ann. Phys. A370 (1981) 284. U B (6)xU F (12)  U B (6)xU F (6)xU F (2)  U B+F (5)xU F (2)...  U B+F (6)xU F (2)  SU B+F (3)xU F (2)...  SO B+F (3)xU F (2)  Spin (3)  SO B+F (6)xU F (2)... H= B 0 + A 1 C 2 [U B+F (6)] + A C 1 [U B+F (5)] + A´ C 2 [U B+F (5)] + B C 2 [SO B+F (6)] + C C 2 [SO B+F (5)] + D C 2 [SU B+F (3)] + E C 2 [SO B+F (3)] + F Spin(3) L=2 L=0 1/2 3/2 5/2 L=2 L=0 x x L=2 x S= 1/2 U B (6) x U F (12)  U B (6) x U F (6) x U F (2) This hamiltonian has analytic solutions, but also describes transitional situations.

Example: the SO(6) limit H= B 0 + A 1 C 2 [U B+F (6)] + B C 2 [SO B+F (6)] + C C 2 [SO B+F (5)] + EC 2 [SO B+F (3)] + FSpin(3) E= A(N 1 (N 1 +5)+ N 2 (N 2 +3)) + B(  1 (  1 +4)+  2 (  2 +2)) + C(  1 (  1 +3)+  2 (  2 +1)) +EL(L+1) + F J(J+1)

Can we connect atomic nuclei using supersymmetry?  s,d,a j fermions c j s,d N+1 bosons N bosons 1 fermion M valence nucleons A nucleons  IBFA L = 0 und 2 pairs Nucleon pairs e-e nucleus odd-A nucleus SUSY a j b l F. Iachello, Phys. Rev. Lett. 44 (1980) 672

(6+ 2j+1) 2 generators of bosonic or fermionic type Supersymmetrie: U(6/2j+1) symmetry Note: graded Lie algebras U(6/m) are no Lie algebras. Their generators fullfil a mixture of commutation and anticommutation relations! By removing the mixed generators one finds that the Bose-Fermi symmetry is always a subalgebra of the graded Lie algebra:

[N}[N} [N][N] [N-1]x[1] x[1] <N><N> <N><N> (0) (1) (2) (1/2,1/2) (3/2,1/2) (5/2,1/2) 3/2 + 1/2 + 5/2 + 7/2 + U(6/4)  U B (6)xU F (4)  SO B (6)xSU F (4)  Spin(6)  Spin(5)  Spin(3) ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ [ N } [N] [1 m ] (  1,  2 ) J In the case of a dynamical supersymmetry the same parameter set describes states in both nuclei.

E = A’ (  (  +4)) + A(  1 (  1 +4) +  2 (  2 +2) +  3 2 )+ B(  1 (  1 +3) +  2 (  2 +1)) + CJ(J+1) d 3/2 s 1/2 h 13/2 d 5/2 g 7/2 191 Ir 190 Os A’=-18.3 keV, A= keV, B= 32.3 keV, C= 9.5 keV 

H = B 0 + A 1 C 2 [U B+F (6)] + B C 2 [SO B+F (6)] + C C 2 [SO B+F (5)] + D C 2 [SO B+F (3)] + E C 2 [Spin(3)] E = B 0 + A (  1 (  1 +6) +  2 (  2 +4)) + B (  1 (  1 +4) +  2 (  2 +2)) + C (  1 (  1 +3) +  2 (  2 +1)) + D L(L+1) + E J(J+1) f 5/2 p 1/2 i 13/2 h 9/2 f 7/2 p 3/2 SO(6) limit and j = 1/2, 3/2, 5/2 Example: SO(6) limit of U(6/12)

E o-e = A (  1 (  1 +6) +  2 (  2 +4)) + B (  1 (  1 +4) +  2 (  2 +2)) + C (  1 (  1 +3) +  2 (  2 +1)) + D L(L+1) + E J(J+1) E e-e = A  (  +6) + B (  1 (  1 +4) +  2 (  2 +2)) + C (  1 (  1 +3) +  2 (  2 +1)) + (D+E) L(L+1) 3.5 A case study: 195 Pt and the SO(6) Limit of U(6/12) A. Mauthofer et al., Phys. Rev. C 34 (1986) 1958.

Electromagnetic transition rates B(E2) values B(M1) values

One particle transfer reactions (pick-up): New results for 195 Pt Angular distribitions: Spectroscopic strenghts:

Q3D Spectrometer at accelerator laboratory (TUM-LMU Garching ) /.. Q3D Spectrometer Particle detector

196 Pt (p,d) Fribourg/Bonn/Munich Detailed studies of 195 Pt and 196 Au were performed in parallel The angular distributions reveal the parity and orbital angular momentum of the transferred neutron. Model space relevant information. The spin cannot be uniquely determined. p: 1/2 or 3/2 f: 5/2 or 7/2

Pt (d,t) 196 Pt Unique spin assignments can be obtained from polarised transfer. Then the cross sections become sensitive to the orientation of the spin of the transferred particle. lj 195 Pt jj with  =(-1) l

New result for 195 Pt A =46.7, B+B´= C= 52.3, D = 5.6 E = 3.4 (keV) A. Metz, Y. Eisermann, A. Gollwitzer, R. Hertenberger, B.D. Valnion, G. Graw,J. Jolie, Phys. Rev. C61 (2000)

Comparison of the transfer strenghts with theory Microscopic transfer operator: J. Barea, C.E. Alonso, J.M. Arias, J. Jolie Phys. Rev. C71 (2005)

3.5 Supersymmetry without dynamical symmetry U(6/12)  U B (6)xU F (12)  U B (6)xU F (6)xU F (2)  U B+F (6)xU F (2)  U B+F (5)xU F (2)...  SU B+F (3)xU F (2)...  SO B+F (3)xU F (2)  Spin (3)  SO B+F (6)xU F (2)... H= B 0 + A 1 C 2 [U B+F (6)] + A C 1 [U B+F (5)] + A ´C 2 [U B+F (5)] + B C 2 [SO B+F (6)] + C C 2 [SO B+F (5)] + D C 2 [SU B+F (3)] + E C 2 [SO B+F (3)] + F Spin(3) This hamiltonian has analytic solutions, but also describes transitional situations, in even-even and odd-A nuclei. Example: The Ru-Rh isotopes A. Frank, P. Van Isacker, D.D. Warner Phys. Lett. B197(1987)474 H= (7N-42) C 2 [U B+F (6)] + (841-54N) C 1 [U B+F (5)] C 2 [SO B+F (6)] C 2 [SO B+F (5)] -9.5 C 2 [SO B+F (3)] + 15 Spin(3) (all in keV)

Even-even Ru Odd proton Rh

Proton pick-up reactions on Palladium isotopes: 1/2 3/2 5/2

Phase transitions in odd-A nuclei: changing single particle orbits, AND finding a simple hamiltonian Partial solution: use the U(6/12) supersymmetry U(6/12): U(5), O(6) and SU(3) limits + j=1/2,3/2,5/2 An extension of the Casten triangle for odd-A nuclei was proposed: D.D. Warner, P. Van Isacker, J. Jolie, A.M. Bruce, Phys. Rev. Lett. 54 (1985)1365. Here we apply the very simple Hamiltonian with the quadrupole operator of U B+F (6). P. Van Isacker, A.Frank, H.Z. Sun, Ann. of Phys. 157 (1984) 183.

SU(3)-SU(3) with 10 bosons and one fermion  (J= 1/2 states) (J= 0 + states)  A phase transition at as expected. Groundstate energies in SU(3) to SU(3) transitions.

0 +- states 1/2 - states  Similar for the U(5)-SU(3) first order phase transition

The extended Casten triangle for odd-A nuclei becomes: Fig 1 J. Jolie, S. Heinze, P. Van Isacker, R.F. Casten, Phys. Rev. C 70 (2004) (R).

But is everything so normal and expected?   No crossings except at symmetries. Additional crossings occur! 

L 1/2 J Conserved quantities allow real crossings  

But there is even more to the story !  

Applications to real nuclei: there are no symmetry related constraints needed are dominant j = 1/2,3/2, 5/2 orbits D.D. Warner, P. Van Isacker, J. Jolie, A.M. Bruce, Phys. Rev. Lett. 54 (1985)1365. W,Pt Ru,Rh A. Frank, P. Van Isacker, D.D. Warner, Phys. Lett. B197 (1987)474. Se,As A. Algora et al.Z. f. Phys. A352 (1995) 25

Odd- neutron nuclei in the W-Pt region J. Jolie, S. Heinze, P. Van Isacker, R.F. Casten, Phys. Rev. C 70 (2004) (R). a= -47 keV A= 52 keV B=3.4 keV