A Quadrupole-octupole Collective Approach A. Dobrowolski, A. G o - PowerPoint PPT Presentation

A Quadrupole-octupole Collective Approach A. Dobrowolski, A. G´ o´ zd´ z September 16, 2015 1 Construction of quadrupole+octupole collective model (outlook) 2 Symmetrization of collective solutions 3 Some preliminary results () September 16, 2015 1 / 17

Collective quadrupole+octupole approach in intrinsic frame We construct our deformed collective model already in the intrinsic frame–contrarly to the usual procedure which starts from the spherical Hamiltonian expressed in laboratory coordinates BUT the resulting rotational symmetry in the intrinsic frame is conserved! The set of collective variables in the intrinsic coordinate system: α 20 , α 22 , { α 3 ν } , Ω Nuclear surface in the intrinsic coordinate system: R ( θ, ϕ ) = R 0 [1 + + α 20 Y 20 ( θ, ϕ ) + α 22 ( Y 22 ( θ, ϕ ) + Y 2 , − 2 ( θ, ϕ )) 3 � + 2 α 3 ν Re ( Y 3 ν ( θ, ϕ )) ] , ν =0 () September 16, 2015 2 / 17

Center-of-mass problem It is believed that the dipole parameters { α 1 µ } , µ = {− 1 , 0 , 1 } are responsible for the center of mass motion. Let us expand the c.m. vector up to the first order in α 1 µ . � r CM = � r CM ( α 1 µ , α 20 , α 22 , { α 3 ν } ) . (1) This equation can be solved with respect to the variables α 1 µ with the condition � r CM = 0 , (2) α 1 µ = α 1 µ ( � r CM = 0 , α 20 , α 22 , { α 3 ν } ) . (3) Obtained in this way α 1 µ ’s ensure the nuclear surface to be defined in the center of mass frame. The above consideration indicates that the quadrupole and octupole deformations are the only independent collective variables. () September 16, 2015 3 / 17

Intrinsic rotation group G Action of the rotation intrinsic group ¯ g ∈ SO(3) . Transformations of coordinates: λµ ) ′ = ¯ ( α lab g α lab λµ = α lab λµ ( α λµ ) ′ = ¯ � µ ′ µ ( g − 1 ) α λµ ′ D λ g α λµ = µ ′ Ω ′ = ¯ g Ω = Ω g . Action in the space of functions of intrinsic variables: g − 1 Ω) ¯ g ψ ( α λµ , Ω) = ψ (¯ g α λµ , ¯ () September 16, 2015 4 / 17

Relation between intrinsic and laboratory frame The relation between collective laboratory and intrinsic shape variables λ � D λ ∗ α lab λµ ( α λν ) = µν (Ω) α λν ν = − λ with, at least, additional 3 conditions: f k ( α λµ , Ω) = 0 , { k = 1 , 2 , 3 } , which determine the orientation of both intrinsic vs laboratory frame. () September 16, 2015 5 / 17

Intrinsic frame The transformation from the laboratory to intrinsic coordinate system is, in general, non-reversible. It means that, for one given set of laboratory variables { α lab λν } usualy may correspond several sets of intrinsic variables { α λµ , Ω } , (well known problem e.g. for the Bohr Hamiltonian) ′ ′ ) α lab λν ( α λν , Ω) = α lab λν ( α λν , Ω ′ ′ ) where ( α λν , Ω) � = ( α λν , Ω How to omit this disadvantage? () September 16, 2015 6 / 17

Symmetrization group and uniqueness of eigensolutions It is possible to find the intrinsic transformation group of the intrinsic variables which does not change the transformation relation between intrinsic and laboratory variables α lab g ( α λν , Ω)) = α lab λν (¯ λν ( α λν , Ω) The set of all transformations ¯ g forms the so called symmetrization group G s . REMARK: generally while working in the intrinsic frame, for most of square integrable functions Ψ( α λµ , Ω) � = Ψ(( α λµ ) ′ , Ω ′ ). The symmetrization condition for states. For all ¯ g ∈ G s : g Ψ( α λµ , Ω) = +1 · Ψ( α λµ , Ω) ¯ () September 16, 2015 7 / 17

Construction of the collective basis Initial (before projection) H.O. one-phonon basis functions √ Ψ ( ± ) ( α 2 , α 3 , Ω) = u 0 ( η 2 , α 20 − ˚ α 20 ) u 0 ( 2 η 2 , α 22 − ˚ α 22 ) u n 0 ( η 3 , ± α 30 − ˚ α 30 ) k √ √ u n 1 ( 2 η 3 , ± α 31 − ˚ α 31 ) u n 2 ( 2 η 3 , ± α 32 − ˚ α 32 ) √ α 33 ) R J u n 3 ( 2 η 3 , ± α 33 − ˚ MK (Ω) √ with R J 2 J + 1 D J MK (Ω) = MK ∗ (Ω) and 3 � n k = 1 k =0 Basis functions of good (positive or negative) parity Ψ k ( α 2 , α 3 , Ω; π = +1) = 1 2(Ψ (+) ( a α 2 , α 3 , Ω) + Ψ ( − ) ( α 2 , α 3 , Ω)) k k Ψ k ( a α 2 , α 3 , Ω; π = − 1) = 1 2(Ψ (+) ( α 2 , a α 3 , Ω) − Ψ ( − ) ( α 2 , α 3 , Ω)) k k () September 16, 2015 8 / 17

Applying the projection operator onto the scalar irreducible representation of the symmetrization group one obtains the basis function card ( G s ) √ 1 Ψ ( A 1) ≡ ˆ P ( A 1) (¯ � 8 u 0 ( η 2 , ˆ 2 η 2 , ˆ g )Ψ k = g l α 20 − ˚ ¯ α 20 ) u 0 ( g l α 22 − ˚ ¯ α 22 ) × k l =1 √ u n 0 ( η 3 , ˆ 2 η 3 , ˆ g l α 30 − ˚ ¯ α 30 ) u n 1 ( g l α 31 − ˚ ¯ α 31 ) × √ √ 2 η 3 , ˆ 2 η 3 , ˆ α 33 ) R J u n 2 ( g l α 32 − ˚ ¯ α 32 ) u n 3 ( g l α 33 − ˚ ¯ MK (¯ g Ω) where J √ � R J MK (Ω g − 1 ) = D J KK ′ ( g ) D J ∗ 2 J + 1 MK ′ (Ω) K ′ = − J but after projection it may happen that � Ψ ( A 1) | Ψ ( A 1) � � = δ kk ′ . k k ′ How to orthogonalize them efficiently? 1 1 Standard Gram-Shmidtt procedure, 2 2 Solving the eigenproblem of the overlap operator (as in the general GCM method). () September 16, 2015 9 / 17

Collective Hamiltonian A realistic collective Hamiltonian with variable mass tensor H coll ( α 2 , α 3 , Ω) = 3 − � 2 ∂ B ( α 2 µ , α 3 µ ′ ) − 1 � ij ∂ � � � | g | + ∂α i ∂α j � 2 | g | µ { i , j } =2 µ ′ H rot (Ω) + ˆ ˆ V ( α 2 , α 3 ) where g is the metric tensor corresponding to α λµ manifold. The collective 6D potential ˆ V ( α 2 , α 3 ) is obtained through the macroscopic-microscopic Strutinsky-like method. () September 16, 2015 10 / 17

Rotor Hamiltonian The generalized rotor Hamiltonian ˆ H rot of given symmetry G ( g ∈ G ) and rank n can be constructed out of the angular momentum operators in the following way: λ ˆ � c λµ ( n ) ˆ H rot (Ω) ≡ T λµ ( n , Ω) , µ = − λ where � � ˆ (((ˆ I ⊗ ˆ I ) λ 2 ⊗ ˆ I ) λ 3 ⊗ ... ⊗ ˆ T λµ ( n ; λ 2 , λ 3 , ..., λ n − 1 , Ω) ≡ I ) λ n − 1 λµ I I and (ˆ I ⊗ ˆ � � (1 , µ ; 1 , µ ′ | λ 2 µ 2 )ˆ I 1 µ ˆ I ) λ 2 µ 2 = I 1 µ ′ ; λ 2 = { 0 , 1 , 2 } µ =1 µ ′ =1 and I 1+1 = − 1 I 1 − 1 = + 1 ˆ I 10 = ˆ ˆ (ˆ I x − i ˆ ˆ (ˆ I x + i ˆ √ I y ) , √ I y ) . I z , 2 2 () September 16, 2015 11 / 17

Preliminary estimates of intraband B ( E 2) transitions Experiment, ILL Grenoble on 156 Gd (PRL 104, 222502, 2010), ” Ultrahigh-Resolution-Ray Spectroscopy of 156 Gd: A Test of Tetrahedral Symmetry ”, M. Jentschel et al. Transition, No. of state B(E2) Dominating I π i → I π Nucleus (W.u.) excitation j 2 + → 0 + 156 Gd 211 (exp.187(5)) 4 + → 2 + 183 (exp.263(5)) 5 − → 3 − 168 (exp.293 +61 (1 → 1) − 134 ) α 30 → α 30 5 − → 3 − (2 → 2) 170 α 32 → α 32 5 − → 3 − (6 → 4) 179 α 31 → α 31 5 − → 3 − (10 → 7) 175 α 33 → α 33 E γ (2 + → 0 + ) ≈ 200 keV ( exp . 88 keV ) E γ (4 + → 2 + ) ≈ 350 keV ( exp . 199 keV ) E γ (5 − → 3 − ) ≈ 0 . 23 − 0 . 28 MeV ( exp . 0 . 13 MeV ) E (3 − 1 ) ≈ 1 . 07 MeV ( exp . 1 . 27 MeV ) () September 16, 2015 12 / 17

Collective quadrupole electric transition operators The intrinsic frame is chosen to fix quadrupoles in the principal axes frame . The quadrupole intrinsic operator: = 3 ZR 2 � ˆ Q intr 0 α 20 + 20 4 π 1 � 10 7 α 20 α 20 − 20 7 α 2 − 2 α 22 + 4 3 α 30 α 30 − 2 α 3 − 1 α 31 + 10 �� √ 3 α 3 − 3 α 33 5 π = ˆ Q quadr (1 st ) + ˆ Q quadr (2 nd ) + ˆ Q oct 20 (2 nd ) 20 20 () September 16, 2015 13 / 17

Four possible types of solutions of the ATDHF 2-D quadr.+oct. Hamiltonian ( IJMP E, vol.20, 2011, p. 500-505 ) | Ψ vib | 2 as function of α 20 and α ” 32 for 156 Dy. | ! | 2 | ! | 2 0.2 0.2 1.00 1.00 0.95 0.95 0.90 0.90 0.85 0.85 Deformation " 32 Deformation " 32 0.80 0.80 0.1 0.1 0.75 0.75 0.70 0.70 0.65 0.65 0.60 0.60 0.55 0.55 0.0 0.0 0.50 0.50 0.45 0.45 0.40 0.40 0.35 0.35 0.30 0.30 -0.1 0.25 -0.1 0.25 0.20 0.20 0.15 0.15 0.10 0.10 0.05 0.05 -0.2 0.00 -0.2 0.00 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 Deformation " 20 Deformation " 20 156 156 Dy 90 Dy Dy 90 Dy 66 | ! | 2 66 | ! | 2 0.2 0.2 1.00 1.00 0.95 0.95 0.90 0.90 0.85 0.85 Deformation " 32 0.80 Deformation " 32 0.80 0.1 0.1 0.75 0.75 0.70 0.70 0.65 0.65 0.60 0.60 0.55 0.55 0.0 0.50 0.0 0.50 0.45 0.45 0.40 0.40 0.35 0.35 0.30 0.30 -0.1 0.25 -0.1 0.25 0.20 0.20 0.15 0.15 0.10 0.10 0.05 0.05 -0.2 0.00 -0.2 0.00 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 Deformation " 20 Deformation " 20 156 156 Dy 90 Dy Dy 90 Dy 66 66 () September 16, 2015 14 / 17 Figure: Probability density distributions for the ground-state and the first excited