Partial dynamical symmetries in Bose-Fermi systems* Jan Jolie, Institute for Nuclear Physics, University of Cologne What are dynamical symmetries? Illustration.

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Presentation transcript:

Partial dynamical symmetries in Bose-Fermi systems* Jan Jolie, Institute for Nuclear Physics, University of Cologne What are dynamical symmetries? Illustration with the interacting boson model. What are partial dynamical symmetries? Illustration with the interacting boson model. What are dynamical Bose-Fermi symmetries? Illustration with the interacting boson fermion model. How to extend partial dynamical symmetries to Bose-Fermi systems. Application to 195 Pt. Work done with Piet Van Isacker, Tim Thomas and Ami Leviatan, *: work supported by BriX during a one month sabbatical stay at GANIL in June 2013

Dynamical symmetries Hamiltonian Operators {Gi} Lie Algebra G [gi,gj]=  k c ij k gk. {g m }  {g l }  {g k }  {g i } Casimir Operator [Cn[ G ],gk]=   gk H =  i a i C n [ G i ] E  = a f(  )  => E  k = a f(  )  k with  k  {g k  } E(  =  i a i f  (n, G i ) Gm  Gl  Gk  Gi H = a Cn[ G ]

Example: Angular momentum algebra Hamiltonian Operators {Gi} Lie Algebra G Casimir Operator H = a L 2 + bL z E  = a f(L)  => E  k = a L(L+1)  k with  k  {L +  L -  L z  } E(LM  = a L(L+1) + b M O(2)  O(3) H = a L 2 O(3) L -J 0 +J M

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

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

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

Partial dynamical symmetries Dynamical symmetries lead to: 1.Solvability of the complete spectrum. 2.Existence of exact quantum numbers for all states. 3.Pre-determined structure of the wavefunctions independent of the used parameters. H = (1-  )  1 C 1 [U(5)]+  3 C 2 [SO(6) ] +  4 C 2 [SO(5) ] +  5 C 2 [ SO(3)] Example of case 2: SO(5)  SO(3)  SO(2) (  ) L M All states still have: We want to relax these conditions such that: 1.Only some states keep all quantum numbers. 2.All eigenstates keep some quantum numbers. 3.Some eigenstates keep some quantum numbers. System exhibiting one of the three conditions have a partial dynamical symmetry.

Projection of a O(6) and SU(3) wavefunction on a U(5) basis Due to SO(5) only even or odd d-bosons in a given state with v is even (odd) occur

Example of case 1: Partial dynamical symmetry within the SO(6) limit SO(6) limit for 6 bosons }  =n=6  =4  =2 Step 1: construct operators having a definite tensorial character under all groups Ex: n=2

Step 2 chose an interaction V pds such that : for certain states because the by the irrep coupling allowed final states do not exist. Example: withthe boson pairing operator but since anddoes not exist. Note:and this is true for all other states with  <n

H = aP + n s P - +  3 C 2 [SO(6) ] +  4 C 2 [SO(5) ] +  5 C 2 [ SO(3)]  3 = keV  4 = 45.0 keV  5 = 25.0 keV a= 0 keV  3 = keV  4 = 45.0 keV  5 = 25.0 keV a= 34.9 keV J.E. Garcia-Ramos, P. van Isacker, A. Leviatan, Phys. Rev. Lett. 102 (2009)

odd-odd nuclei 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 Even-even nuclei: the interacting boson approximation Odd-A nuclei: the interacting boson-fermion approximation

N s,d bosons+ single j fermion: U B (6)xU F (2j+1) 36 boson generators + (2j+1) 2 fermion generators, which both couple to integer total spin and fullfil the standard Lie algebra conditions. 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)

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. L=2 L=0 1/2 3/2 5/2 L=2 L=0 x x L´=0 L´=2 x S´= 1/2 U B (6) x U F (12)  U B (6) x U F (6) x U F (2)  U B+F (5)xU F (2)... U B (6)xU F (12)  U B (6)xU F (6)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= AC 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.

Example: the SO(6) limit of U B (6)xU F (12) H= A 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 Spin(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)) +D L(L+1) + E J(J+1)

Result for 195 Pt A =46.7, 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)

wavefunction We consider now a system with 1 boson and 1 fermion. They can form [N 1,N 2 ] states with [2,0] and [1,1]. The [2,0] and [1,1] states are: Using them we can now construct operators which have a tensorial character under all groups in the group chain

We now construct two body interactions: which have the property that: Because in the N-1 boson the highest representation is [N-1] and [N-1]x[1,1] = [N,1]+[N-1,1,1]. Note that:

P. Van Isacker, J. Jolie, T. Thomas, A. Leviatan, subm to Phys. Rev. Lett. H= A C 2 [U B+F (6)] + B C 2 [SO B+F (6)] + C C 2 [SO B+F (5)] + DC 2 [SO B+F (3)] + ESpin(3) A=37.7, B=-41.5, C =49.1, D= 1.7, E= 5.6, a0=306, a 11 =10., a 22 =-97., a 33 = 112

Conclusion For the first time the concept of partial dynamical symmetries was applied to a mixed system of bosons and fermions. Using the partial dynamical symmetries part of the states keep the original symmetry, while other loose it. The description of 195 Pt could be improved. Thanks for your attention.