Collective properties of stable even-even Cd isotopes ochniak 1 , Ph. - PowerPoint PPT Presentation

Collective properties of stable even-even Cd isotopes ochniak 1 , Ph. Quentin 2 & M. Imadalou 3 L. Pr´ 1 Maria Curie-Skłodowska University, Lublin 2 CEN Bordeaux-Gradignan 3 Ecole Normale Sup´ erieure, Kouba, Alger 18th NPW, Kazimierz, Sept 2011 1 /15

Motivation 1. Vibrations, rotations or something more general? On the robustness of surface vibrational modes: case studies in the Cd region P .E. Garrett and J. L. Wood, J.Phys.G: Nucl.Part.Phys. 37 064028, 2010 18th NPW, Kazimierz, Sept 2011 2 /15

Motivation cont. 2. Treatment of pairing and particle number conservation, going beyond BCS? Mass parameters for large amplitude collective motion: A perturbative microscopic approach E.Kh. Yuldashbaeva, J. Libert, P . Quentin, M. Girod, Phys. Lett. B 461 1 (1999). 18th NPW, Kazimierz, Sept 2011 3 /15

The Bohr Hamiltonian Hamiltonian H GBH ( β, γ, Ω ) = T vib + T rot + V � r � r � 1 1 � � � � � � β 4 β 3 T vib ( β, γ ) = − ∂ β wB γγ ∂ β − ∂ β wB βγ ∂ γ + 2 √ wr β 4 � � r � � r �� 1 ∂ β + 1 � � � − ∂ γ w sin3 γ B βγ β∂ γ w sin3 γ B ββ ∂ γ + β sin3 γ 3 1 � J k ( β, γ ) = 4 B k ( β, γ ) β 2 sin 2 ( γ − 2 π k / 3) I 2 T rot ( β, γ, Ω ) = k ( Ω ) / J k ; 2 k = 1 w = B ββ B γγ − B 2 βγ ; r = B x B y B z Collective variables β , γ , Euler angles Ω q 0 = � Q 0 � = � � A i 3 z 2 i − r 2 β cos γ = Dq 0 , i � √ q 2 = � Q 2 � = � � A i x 2 i − y 2 β sin γ = 3 Dq 2 , i � D = √ π/ 5 / Ar 2 , r 2 = 3 5 ( r 0 A 1 / 3 ) 2 , r 0 = 1 . 2 fm 18th NPW, Kazimierz, Sept 2011 4 /15

Microscopic formulas from the ATDHFB theory Simplest (most approximate) version Vibrational mass parameters for constrained HFB calculations B kj = � 2 � � M − 1 (1) M (3) M − 1 (1) 2 kj � µ | Q k | ν �� ν | Q j | µ � � ( u µ v ν + u ν v µ ) 2 M ( n ) , kj = ( E µ + E ν ) n µ,ν B kj −→ B ββ , B βγ , B γγ Moments of inertia |� µ | j k | ν �| 2 J k = 2 � 2 � ( u µ v ν − u ν v µ ) 2 E µ + E ν µ,ν Thouless-Valatin correction, factor 1.3 18th NPW, Kazimierz, Sept 2011 5 /15

Microscopic calculations Even-even 106 − 116 Cd isotopes The Skyrme interaction, SIII parameters Seniority (constant G ) pairing interaction Two variants of the pairing strength ( g i = G i / (11 + N i ) , i = n , p ) G n G p old 17.1 16.5 new 16.1 17.5 � ( e i − λ ) 2 + ∆ 2 E qp = min Experimental gap from the 5-point mass formula 18th NPW, Kazimierz, Sept 2011 6 /15

7 /15 30 25 20 15 10 5 0 30 25 20 15 10 5 0 ] ] V V e e M M [ 0˚ 0˚ 0˚ [ 0˚ 0˚ 0˚ V 10˚ 10˚ 10˚ V 10˚ 10˚ 10˚ γ 20˚ 20˚ 20˚ γ 20˚ 20˚ 20˚ 0 0 0 . . . 7 7 7 0 0 0 . . . 7 7 7 ˚ ˚ ˚ ˚ ˚ ˚ 0 0 0 0 0 0 3 3 3 3 3 3 40˚ 40˚ 40˚ 40˚ 40˚ 40˚ 0 0 0 . . . 6 6 6 0 0 0 . . . 6 6 6 50˚ 50˚ 50˚ 50˚ 50˚ 50˚ n 0 0 0 . . . 5 5 5 n 0 0 0 . . . 5 5 5 e e s s ˚ ˚ ˚ ˚ ˚ ˚ 0 0 0 0 0 0 . . . 4 4 4 0 0 0 0 0 0 . . . 4 4 4 6 6 6 6 6 6 , , I β I β I I I 0 0 0 . . . 3 3 3 I 0 0 0 . . . 3 3 3 S S , 0 0 0 . . . 2 2 2 , 0 0 0 . . . 2 2 2 d d C C 0 0 0 . . . 1 1 1 0 0 0 . . . 1 1 1 0 6 1 1 1 1 0 0 0 0 0 0 18th NPW, Kazimierz, Sept 2011 30 25 20 15 10 5 0 30 25 20 15 10 5 0 ] ] V V e e M M [ [ 0˚ 0˚ 0˚ 0˚ 0˚ 0˚ V 10˚ 10˚ 10˚ V 10˚ 10˚ 10˚ γ 20˚ 20˚ 20˚ γ 20˚ 20˚ 20˚ 0 0 0 . . . 7 7 7 0 0 0 . . . 7 7 7 ˚ ˚ ˚ ˚ ˚ ˚ 0 0 0 0 0 0 3 3 3 3 3 3 40˚ 40˚ 40˚ 40˚ 40˚ 40˚ 0 0 0 . . . 6 6 6 0 0 0 . . . 6 6 6 50˚ 50˚ 50˚ 50˚ 50˚ 50˚ n 0 0 0 . . . 5 5 5 n 0 0 0 . . . 5 5 5 e e s s ˚ ˚ ˚ 0 0 0 . . . 4 4 4 ˚ ˚ ˚ 0 0 0 . . . 4 4 4 0 0 0 0 0 0 , 6 6 6 , 6 6 6 β β I I I I . . . 3 3 3 . . . 3 3 3 β = 0 . 7 → q 0 ≈ 19 . 3 b ( A = 110 ) I 0 0 0 I 0 0 0 S S , , 0 0 0 . . . 2 2 2 0 0 0 . . . 2 2 2 d d C C 0 0 0 . . . 1 1 1 0 0 0 . . . 1 1 1 8 4 0 1 1 1 0 0 0 0 0 0 106 − 116 Cd, Potential energy 30 25 20 15 10 5 0 30 25 20 15 10 5 0 ] ] V V e e M M [ 0˚ 0˚ 0˚ [ 0˚ 0˚ 0˚ V V 10˚ 10˚ 10˚ 10˚ 10˚ 10˚ γ 20˚ 20˚ 20˚ γ 20˚ 20˚ 20˚ 0 0 0 . . . 7 7 7 0 0 0 . . . 7 7 7 0 0 0 ˚ ˚ ˚ 0 0 0 ˚ ˚ ˚ 3 3 3 3 3 3 40˚ 40˚ 40˚ 0 0 0 . . . 6 6 6 40˚ 40˚ 40˚ 0 0 0 . . . 6 6 6 50˚ 50˚ 50˚ 50˚ 50˚ 50˚ n 0 0 0 . . . 5 5 5 n 0 0 0 . . . 5 5 5 e e s ˚ ˚ ˚ s ˚ ˚ ˚ 0 0 0 . . . 4 4 4 0 0 0 . . . 4 4 4 0 0 0 0 0 0 , 6 6 6 , 6 6 6 β β I I I I 0 0 0 . . . 3 3 3 0 0 0 . . . 3 3 3 I I S S , 0 0 0 . . . 2 2 2 , 0 0 0 . . . 2 2 2 d d C C 0 0 0 . . . 1 1 1 0 0 0 . . . 1 1 1 6 2 0 1 1 1 0 0 0 0 0 0

106 − 116 Cd, energy levels 18th NPW, Kazimierz, Sept 2011 8 /15

106 − 116 Cd, energy levels, cont Two variants of pairing 18th NPW, Kazimierz, Sept 2011 9 /15

106 − 116 Cd, B(E2) transition probabilities . . → → . . . . . . . . . . . . . . → → . . . . . . . . . → . . . . . . 18th NPW, Kazimierz, Sept 2011 10 /15

11 /15 B ββ = B γγ = B k = B , . . . . . . SKE -0.514 0.259 -0.591 B βγ = 0 . . . SKE . . . micr -0.480 0.213 -0.220 18th NPW, Kazimierz, Sept 2011 . . . Diagonal matrix elements of the Q el operator 2 1 2 2 2 3 10˚ 10˚ 0˚ 0˚ 0 0 ˚ ˚ 2 2 0 0 . . 7 7 . . . ˚ ˚ 0 0 3 3 40˚ 40˚ 0 0 . . 6 6 80 50˚ 50˚ 0 0 . . 5 5 40 80 ˚ ˚ 0 0 0 0 . . 4 4 6 6 40 80 0 0 . . 3 3 80 0 0 . . 2 2 0 0 . . 1 1 γ γ B 0 0 10˚ 10˚ 0˚ 0˚ ˚ ˚ 2 2 0 0 0 0 . . 7 7 ˚ ˚ 0 0 3 3 40˚ 40˚ 0 0 . . 6 6 50˚ 50˚ 20 0 0 . . 5 5 � i || Q el || i � [eb] for 2 1 , 2 , 3 levels ˚ ˚ 0 0 0 0 . . 4 4 20 6 6 20 0 0 . . 3 3 0 0 . . 2 2 0 0 . . 1 1 γ β 0˚ 0˚ 0˚ B 10˚ 10˚ 10˚ 0 0 20˚ 20˚ 20˚ 0 0 0 . . . 7 7 7 10˚ 10˚ 0˚ 0˚ ˚ ˚ ˚ 0 0 0 ˚ ˚ 3 3 3 2 2 0 0 0 0 . . 7 7 0 0 ˚ ˚ 40˚ 40˚ 40˚ 0 0 0 . . . 6 6 6 3 3 40˚ 40˚ 0 0 . . 6 6 50˚ 50˚ 50˚ 0 0 0 . . . 5 5 5 80 50˚ 50˚ 0 0 . . 5 5 120 ˚ ˚ ˚ 0 0 0 0 0 0 . . . 4 4 4 6 6 6 ˚ ˚ 0 0 . . 4 4 0 0 6 6 160 0 0 0 . . . 3 3 3 80 Test case 0 0 . . 3 3 ] V 0 0 0 . . . 2 2 2 micr 0 0 . . 2 2 e M 0 0 0 . . . 1 1 1 0 0 . . 1 1 β [ β B V 0 0 0 0 0

Higher Tamm-Dancoff Approximation 1. Mean-field (HF+BCS) calculations −→ one particle basis 2. Many-particle basis built from m particle-hole states (for protons and neutrons) 3. Diagonalization of the δ residual interaction in the many-particle basis States with a good particle number Density matrix ρ ii = � Φ | a + i a i | Φ � −→ v 2 i Sum � i u i v i for the HF+BCS solution 1 1 0 S I I I , s e n C d , 1 1 0 I , s e n o t C d , S I I Σ u v , p r i i 6 6 6 6 Σ u v , p r o t 0 0 0 0 ˚ ˚ ˚ ˚ i i 50˚ 50˚ 10 50˚ 50˚ 10 40˚ 40˚ 40˚ 40˚ γ γ 3 3 3 3 0 0 0 0 ˚ ˚ ˚ ˚ 20˚ 20˚ 5 20˚ 20˚ 5 10˚ 10˚ 10˚ 10˚ 0˚ 0˚ 0 0˚ 0˚ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . . . . . . . . . . . . 1 1 2 2 3 3 4 4 5 5 6 6 . . . . . . . . . . . . . . 1 1 2 2 3 3 4 4 5 5 6 6 7 7 β β 18th NPW, Kazimierz, Sept 2011 12 /15