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even-even nuclei

3.1 The interacting boson-fermion model. even-even nuclei. odd-odd nuclei. odd-even nuclei. Odd-A nuclei: the interacting boson-fermion approximation. N bosons 1 fermion. N+1 bosons. s,d. s,d,a j. IBA. IBFA. e-e nucleus. e-o nucleus. fermions c j. Nukleonen. A nucleons.

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even-even nuclei

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  1. 3.1 The interacting boson-fermion model even-even nuclei odd-odd nuclei odd-even nuclei

  2. Odd-A nuclei: the interacting boson-fermion approximation N bosons 1 fermion N+1 bosons s,d s,d,aj IBA IBFA e-e nucleus e-o nucleus fermions cj Nukleonen A nucleons M valence nucleons nucleon pairs L = 0 and 2 pairs 1974-1979 A. Arima, F. Iachello, T. Otsuka, O. Scholten, I Talmi

  3. and and

  4. HB the IBM-1 Hamiltonian the single particle Hamiltonian with the energies ej. 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.

  5. It has three terms

  6. BCS calculation gives: quasiparticle energies Ej and occupation numbers uj and vj as a function of ej and D. mit the excitation energy of the first 2+ state in the corresponding semimagical nucleus we now have n single particle energies, the gap D and three parameters + six for the boson part. O. Scholten PhD + ODDA code

  7. Example: odd Rhodium isotopes (J.Jolie et al. Nucl.Phys. A438 (1985)15 HBfrom fit of Pd isotopes by Van Isacker et al. D=1.5 MeV p =- p =+

  8. 3.2 Bose-Fermi symmetries N s,d bosons: U(6) symmetry for model space 36 generators N s,d bosons+ j fermion: UB(6)xUF(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.

  9. 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): SOB(3) ~ SUF(2) Spin(5): SOB(5) ~ SpF(4) Spin(6): SOB(6) ~ SUF(4) Exemple: SO(6) core and j=3/2 fermion Balantekin, Bars, Iachello, Nucl. Phys. A370 (1981) 284. UB(6)xUF(4)  SOB(6)xSUF(4)  Spin(6)  Spin(5)  Spin(3) ¦ ¦ ¦ ¦ ¦ ¦ ¦ [N] [1] <s ><1/2,1/2,1/2> <s1,s2 ,s3 > (t1 ,t2) J H= A’ C2[SOB(6)] + A C2[Spin(6)] + B C2[Spin(5)]+ C C2[Spin(3)] E = A’ (s(s+4)) + A(s1(s1+4) + s2(s2+2) + s32) + B(t1(t1+3) +t2(t2+1)) + CJ(J+1)

  10. 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. 5/2 L=2 L=2 L=2 3/2 x x x S= 1/2 L=0 1/2 L=0 L=0 UB(6) x UF(12)  UB(6) x UF(6) x UF(2) UB(6)xUF(12)  UB(6)xUF(6)xUF(2) UB+F(5)xUF(2)... UB+F(6)xUF(2) SUB+F(3)xUF(2)... SOB+F(3)xUF(2)  Spin (3)  SOB+F(6)xUF(2)... H= B0 + A1 C2[UB+F(6)] + A C1[UB+F(5)] + A´ C2[UB+F(5)] + B C2[SOB+F(6)] + C C2[SOB+F(5)] + D C2[SUB+F(3)] + E C2[SOB+F(3)] + F Spin(3) This hamiltonian has analytic solutions, but also describes transitional situations.

  11. Example: the SO(6) limit H= B0 + A1 C2[UB+F(6)] + B C2[SOB+F(6)] + C C2[SOB+F(5)] + EC2[SOB+F(3)] + FSpin(3) E= A(N1(N1 +5)+ N2(N2 +3)) + B(s1(s1 +4)+ s2(s2 +2)) + C(t1(t1 +3)+ t2(t2 +1)) +EL(L+1) + F J(J+1)

  12. Can we connect atomic nuclei using supersymmetry? ajbl SUSY F. Iachello, Phys. Rev. Lett. 44 (1980) 672 fermions cj Nukleonen A nucleons N bosons 1 fermion N+1 bosons M valence nucleons Nucleon pairs s,d s,d,aj L = 0 und 2 pairs IBA IBFA e-e nucleus odd-A nucleus

  13. Supersymmetrie: U(6/2j+1) symmetry (6+ 2j+1)2 generators of bosonic or fermionic type 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:

  14. U(6/4)  UB(6)xUF(4)  SOB(6)xSUF(4)  Spin(6)  Spin(5)  Spin(3) ¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦ [N} [N] [1m] <S ><1/2,1/2,1/2> <s1,s2 ,s3 > (t1 ,t2) J <N-4> <N -4> 4+ 2+ (2) 2+ <N -2> <N -2> (1) 0+ (0) [N] <N> <N> <N-5>x[1] 7/2+ (5/2,1/2) <N-1/2,1/2,1/2> 5/2+ <N-3>x[1] 1/2+ <N-1/2,3/2,1/2> (3/2,1/2) 3/2+ <N-1>x[1] [N} [N-1]x[1] <N+1/2,1/2,1/2> (1/2,1/2) In the case of a dynamical supersymmetry the same parameter set describes states in both nuclei.

  15. s1/2 191Ir 190Os d3/2 h13/2 d5/2 g7/2 E = A’ (s(s+4)) + A(s1(s1+4) + s2(s2+2) + s32)+ B(t1(t1+3) +t2(t2+1)) + CJ(J+1) t s A’=-18.3 keV, A= -27.3 keV, B= 32.3 keV, C= 9.5 keV

  16. Example: SO(6) limit of U(6/12) f5/2 p3/2 f7/2 i13/2 h9/2 p1/2 SO(6) limit and j = 1/2, 3/2, 5/2 H = B0 + A1 C2[UB+F(6)] + B C2[SOB+F(6)] + C C2[SOB+F(5)] + D C2[SOB+F(3)] + E C2[Spin(3)] E = B0 + A (N1(N1+6) + N2(N2+4)) + B (s1(s1+4) + s2(s2+2)) + C (t1(t1+3) +t2(t2+1)) + D L(L+1) + E J(J+1)

  17. 3.5 A case study: 195Pt and the SO(6) Limit of U(6/12) Eo-e = A (N1(N1+6) + N2(N2+4)) + B (s1(s1+4) + s2(s2+2)) + C (t1(t1+3) +t2(t2+1)) + D L(L+1) + E J(J+1) Ee-e = A N(N+6)+ B (s1(s1+4) + s2(s2+2)) + C (t1(t1+3) +t2(t2+1)) + (D+E) L(L+1) A. Mauthofer et al., Phys. Rev. C 34 (1986) 1958.

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

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

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

  21. Detailed studies of 195Pt and 196Au were performed in parallel Fribourg/Bonn/Munich 196Pt (p,d) 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

  22. 196Pt (d,t) 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 195Pt 196Pt with p =(-1)l jp 0+

  23. New result for 195Pt A =46.7, B+B´= -42.2 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) 064313

  24. Comparison of the transfer strenghts with theory Microscopic transfer operator: J. Barea, C.E. Alonso, J.M. Arias, J. Jolie Phys. Rev. C71 (2005) 014314

  25. 3.5 Supersymmetry without dynamical symmetry U(6/12) UB(6)xUF(12)  UB(6)xUF(6)xUF(2) UB+F(6)xUF(2) UB+F(5)xUF(2)... SUB+F(3)xUF(2)... SOB+F(3)xUF(2)  Spin (3)  SOB+F(6)xUF(2)... H= B0 + A1 C2[UB+F(6)] + A C1[UB+F(5)] + A ´C2[UB+F(5)] + B C2[SOB+F(6)] + C C2[SOB+F(5)] + D C2[SUB+F(3)] + E C2[SOB+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)C2[UB+F(6)] + (841-54N) C1[UB+F(5)] -23.3 C2[SOB+F(6)] + 30.8 C2[SOB+F(5)] -9.5 C2[SOB+F(3)] + 15 Spin(3) (all in keV)

  26. Even-even Ru Odd proton Rh

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

  28. 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 P. Van Isacker, A.Frank, H.Z. Sun, Ann. of Phys. 157 (1984) 183. 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 UB+F(6).

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

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

  31. 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) 011305(R).

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

  33. 0.3 0.4 0.2 c c Conserved quantities allow real crossings L 1/2 J 0.4 0.0 0.2 c c

  34. 0.3 0.4 0.2 c c But there is even more to the story ! 0.3 0.4 0.2 c c

  35. 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 Se,As Ru,Rh A. Algora et al.Z. f. Phys. A352 (1995) 25 A. Frank, P. Van Isacker, D.D. Warner, Phys. Lett. B197 (1987)474.

  36. a= -47 keV A= 52 keV B=3.4 keV Odd- neutron nuclei in the W-Pt region J. Jolie, S. Heinze, P. Van Isacker, R.F. Casten, Phys. Rev. C 70 (2004) 011305(R).

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