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

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

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

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SLIDE 1

Collective properties of stable even-even Cd isotopes

  • L. Pr´
  • chniak1, Ph. Quentin2 & M. Imadalou3

1Maria Curie-Skłodowska University, Lublin 2 CEN Bordeaux-Gradignan 3 Ecole Normale Sup´

erieure, Kouba, Alger

18th NPW, Kazimierz, Sept 2011 1 /15

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SLIDE 2

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

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SLIDE 3

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

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SLIDE 4

The Bohr Hamiltonian

Hamiltonian HGBH(β, γ, Ω) = Tvib + Trot + V Tvib(β, γ) = − 1 2 √wr 1 β4

  • ∂β
  • β4

r wBγγ

  • ∂β − ∂β
  • β3

r wBβγ

  • ∂γ
  • +

+ 1 β sin3γ

  • − ∂γ

r w sin3γBβγ

  • ∂β + 1

β∂γ r w sin3γBββ

  • ∂γ
  • Trot(β, γ, Ω) =

1 2

3

  • k=1

I2

k(Ω)/Jk;

Jk(β, γ) = 4Bk(β, γ)β2 sin2(γ − 2πk/3) w = BββBγγ − B2

βγ; r = BxByBz

Collective variables β, γ, Euler angles Ω β cos γ = Dq0, q0 = Q0 = A

i 3z2 i − r2 i

β sin γ = √ 3Dq2, q2 = Q2 = A

i x2 i − y2 i

D = √π/5/Ar2, r2 = 3

5(r0A1/3)2, r0 = 1.2 fm

18th NPW, Kazimierz, Sept 2011 4 /15

slide-5
SLIDE 5

Microscopic formulas from the ATDHFB theory Simplest (most approximate) version Vibrational mass parameters for constrained HFB calculations Bkj = 2 2

  • M−1

(1)M(3)M−1 (1)

  • kj

M(n),kj =

  • µ,ν

µ|Qk|νν|Qj|µ (Eµ + Eν)n (uµvν + uνvµ)2 Bkj −→ Bββ, Bβγ, Bγγ Moments of inertia Jk = 22

µ,ν

|µ|jk|ν|2 Eµ + Eν (uµvν − uνvµ)2 Thouless-Valatin correction, factor 1.3

18th NPW, Kazimierz, Sept 2011 5 /15

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SLIDE 6

Microscopic calculations Even-even 106−116Cd isotopes The Skyrme interaction, SIII parameters Seniority (constant G) pairing interaction

Two variants of the pairing strength (gi = Gi/(11 + Ni), i = n, p) Gn Gp

  • ld

17.1 16.5 new 16.1 17.5 Eqp = min

  • (ei − λ)2 + ∆2

Experimental gap from the 5-point mass formula

18th NPW, Kazimierz, Sept 2011 6 /15

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SLIDE 7

106−116Cd, Potential energy

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

V [ M e V ]

1 6

C d , S I I I , s e n β γ

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 5 10 15 20 25 30 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

V [ M e V ]

1 8

C d , S I I I , s e n β γ

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 5 10 15 20 25 30 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

V [ M e V ]

1 1

C d , S I I I , s e n β γ

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 5 10 15 20 25 30 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

V [ M e V ]

1 1 2

C d , S I I I , s e n β γ

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 5 10 15 20 25 30 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

V [ M e V ]

1 1 4

C d , S I I I , s e n β γ

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 5 10 15 20 25 30 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

V [ M e V ]

1 1 6

C d , S I I I , s e n β γ

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 5 10 15 20 25 30

β = 0.7 → q0 ≈ 19.3 b (A = 110)

18th NPW, Kazimierz, Sept 2011 7 /15

slide-8
SLIDE 8

106−116Cd, energy levels

18th NPW, Kazimierz, Sept 2011 8 /15

slide-9
SLIDE 9

106−116Cd, energy levels, cont

Two variants of pairing

18th NPW, Kazimierz, Sept 2011 9 /15

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SLIDE 10

106−116Cd, B(E2) transition probabilities

. . . . . . .

→

. . . . . . .

→

. . . . . . .

→

. . .

→

. . . . . . .

→

18th NPW, Kazimierz, Sept 2011 10 /15

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SLIDE 11

Diagonal matrix elements of the Qel operator i||Qel||i [eb] for 21,2,3 levels

. . . . . . . . . . . . . . . . . .

Test case micr

80 80 120 160

0˚ 10˚ 2 ˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

B

β β 0˚ 10˚ 2 ˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 20 20 20 0˚ 10˚ 2 ˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

B

β γ 0˚ 10˚ 2 ˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 40 40 80 80 80 80 0˚ 10˚ 2 ˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

B

γ γ 0˚ 10˚ 2 ˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

SKE Bββ = Bγγ = Bk = B, Bβγ = 0

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

V [ M e V ]

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

micr SKE 21

  • 0.480
  • 0.514

22 0.213 0.259 23

  • 0.220
  • 0.591

18th NPW, Kazimierz, Sept 2011 11 /15

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SLIDE 12

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 ρii = Φ|a+

i ai|Φ −→ v2 i

Sum

i uivi for the HF+BCS solution

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6

Σ u

i

v

i

, p r

  • t

1 1

C d , S I I I , s e n β γ

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 5 10 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

Σ u

i

v

i

, p r

  • t

1 1

C d , S I I I , s e n β γ

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 5 10

18th NPW, Kazimierz, Sept 2011 12 /15

slide-13
SLIDE 13

HTDA, potential energy

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

V [ M e V ]

1 8

C d , H T D A β γ

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

  • 5

5 10 15 20 25 30 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

V [ M e V ]

1 1

C d , H T D A β γ

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 5 10 15 20 25 30 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

V [ M e V ]

1 1 2

C d , H T D A β γ

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 5 10 15 20 25 30 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

V [ M e V ]

1 8

C d , S I I I , s e n β γ

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 5 10 15 20 25 30 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

V [ M e V ]

1 1

C d , S I I I , s e n β γ

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 5 10 15 20 25 30 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7

V [ M e V ]

1 1 2

C d , S I I I , s e n β γ

0˚ 10˚ 20˚ 3 ˚ 40˚ 50˚ 6 ˚ . 1 . 2 . 3 . 4 . 5 . 6 . 7 5 10 15 20 25 30

18th NPW, Kazimierz, Sept 2011 13 /15

slide-14
SLIDE 14

HTDA, mass parameters How to get mass parameters in the HTDA?

  • 1. More fundamental theory (TD HTDA)
  • 2. Analogy with ATDHFB, density matrix ρii → v2

i , what about quasiparticle

energies ukvk = ∆k/2Ek ∆k = −

j ujvjk¯

k|Vres|j¯ j

  • 3. Schematic estimations

18th NPW, Kazimierz, Sept 2011 14 /15

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SLIDE 15

Conclusions

◮ The Bohr Hamiltonian is flexible enough to describe quadrupole dynamics

even in the region near to closed shell nuclei

◮ The Cd region is an interesting field for testing extensions of the present

approach (to explain e.g. low lying 0+ levels, improper A dependence of some energy levels)

18th NPW, Kazimierz, Sept 2011 15 /15