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

A Quadrupole-octupole Collective Approach

• A. Dobrowolski, A. G´
• ´

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: α20, α22, {α3ν}, Ω Nuclear surface in the intrinsic coordinate system: R(θ, ϕ) = R0[1 + +α20Y20(θ, ϕ) + α22(Y22(θ, ϕ) + Y2,−2(θ, ϕ)) +

3

• ν=0

2 α3νRe (Y3ν(θ, ϕ)) ],

() September 16, 2015 2 / 17

Center-of-mass problem

It is believed that the dipole parameters {α1µ}, 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 α1µ.

• rCM =

rCM(α1µ, α20, α22, {α3ν}). (1) This equation can be solved with respect to the variables α1µ with the condition

• rCM = 0,

(2) α1µ = α1µ( rCM = 0, α20, α22, {α3ν}). (3) Obtained in this way α1µ’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:

(αlab

λµ)′ = ¯

gαlab

λµ = αlab λµ

(αλµ)′ = ¯ gαλµ =

• µ′

Dλ

µ′µ(g−1)αλµ′

Ω′ = ¯ gΩ = Ωg.

Action in the space of functions of intrinsic variables:

¯ gψ(αλµ, Ω) = ψ(¯ gαλµ, ¯ g−1Ω)

() September 16, 2015 4 / 17

Relation between intrinsic and laboratory frame

The relation between collective laboratory and intrinsic shape variables αlab

λµ(αλν) = λ

• ν=−λ

Dλ∗

µν (Ω) αλν

with, at least, additional 3 conditions: fk(αλµ, 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 {αlab

λν } usualy may

correspond several sets of intrinsic variables {αλµ, Ω}, (well known problem e.g. for the Bohr Hamiltonian) αlab

λν (αλν, Ω) = αlab λν (α

′

λν, Ω

′)

where (αλν, 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 αlab

λν (¯

g(αλν, Ω)) = αlab

λν (αλν, Ω)

The set of all transformations ¯ g forms the so called symmetrization group Gs. REMARK: generally while working in the intrinsic frame, for most of square integrable functions Ψ(αλµ, Ω) = Ψ((αλµ)′, Ω′). The symmetrization condition for states. For all ¯ g ∈ Gs: ¯ gΨ(αλµ, Ω) = +1 · Ψ(αλµ, Ω)

() September 16, 2015 7 / 17

Construction of the collective basis

Initial (before projection) H.O. one-phonon basis functions Ψ(±)

k

(α2, α3, Ω) = u0(η2, α20 − ˚ α20)u0( √ 2η2, α22 − ˚ α22)un0(η3, ±α30 − ˚ α30) un1( √ 2η3, ±α31 − ˚ α31)un2( √ 2η3, ±α32 − ˚ α32) un3( √ 2η3, ±α33 − ˚ α33)RJ

MK(Ω)

with RJ

MK(Ω) =

√ 2J + 1 DJ

MK ∗ (Ω) and 3

• k=0

nk = 1 Basis functions of good (positive or negative) parity Ψk(α2, α3, Ω; π = +1) = 1 2(Ψ(+)

k

(aα2, α3, Ω) + Ψ(−)

k

(α2, α3, Ω)) Ψk(aα2, α3, Ω; π = −1) = 1 2(Ψ(+)

k

(α2, aα3, Ω) − Ψ(−)

k

(α2, α3, Ω))

() September 16, 2015 8 / 17

Applying the projection operator onto the scalar irreducible representation

• f the symmetrization group one obtains the basis function

Ψ(A1)

k

≡ ˆ P(A1)(¯ g)Ψk =

card(Gs)

• l=1

1 8 u0(η2, ˆ ¯ glα20 − ˚ α20)u0( √ 2η2, ˆ ¯ glα22 − ˚ α22) × un0(η3, ˆ ¯ glα30 − ˚ α30)un1( √ 2η3, ˆ ¯ glα31 − ˚ α31) × un2( √ 2η3, ˆ ¯ glα32 − ˚ α32)un3( √ 2η3, ˆ ¯ glα33 − ˚ α33)RJ

MK(¯

gΩ) where RJ

MK(Ωg−1) =

√ 2J + 1

J

• K ′=−J

DJ

KK ′(g)DJ∗ MK ′(Ω)

but after projection it may happen that Ψ(A1)

k

|Ψ(A1)

k′

= δkk′. 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 Hcoll(α2, α3, Ω) = −2 2

• |g|

3

• {i,j}=2

∂ ∂αi

µ

• |g|
• B(α2µ, α3µ′)−1ij ∂

∂αj

µ′

+ ˆ Hrot(Ω) + ˆ V (α2, α3) where g is the metric tensor corresponding to αλµ manifold. The collective 6D potential ˆ V (α2, α3) is obtained through the macroscopic-microscopic Strutinsky-like method.

() September 16, 2015 10 / 17

Rotor Hamiltonian

The generalized rotor Hamiltonian ˆ Hrot of given symmetry G (g ∈ G) and rank n can be constructed out of the angular momentum operators in the following way: ˆ Hrot(Ω) ≡

λ

• µ=−λ

cλµ(n) ˆ Tλµ(n, Ω), where ˆ Tλµ(n; λ2, λ3, ..., λn−1, Ω) ≡

• (((ˆ

I ⊗ ˆ I)λ2 ⊗ ˆ I)λ3 ⊗ ... ⊗ ˆ I)

λn−1

• λµ

and (ˆ I ⊗ ˆ I)λ2µ2 =

I

• µ=1

I

• µ′=1

(1, µ; 1, µ′|λ2µ2)ˆ I1µ ˆ I1µ′; λ2 = {0, 1, 2} and ˆ I10 = ˆ Iz, ˆ I1+1 = − 1 √ 2 (ˆ Ix − iˆ Iy), ˆ I1−1 = + 1 √ 2 (ˆ Ix + iˆ Iy).

() September 16, 2015 11 / 17

Preliminary estimates of intraband B(E2) transitions

Experiment, ILL Grenoble on 156Gd (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 Nucleus I π

i → I π j

(W.u.) excitation

156Gd

2+ → 0+ 211 (exp.187(5)) 4+ → 2+ 183 (exp.263(5)) 5− → 3− (1 → 1) 168 (exp.293+61

−134)

α30 → α30 5− → 3− (2 → 2) 170 α32 → α32 5− → 3− (6 → 4) 179 α31 → α31 5− → 3− (10 → 7) 175 α33 → α33 Eγ(2+ → 0+) ≈ 200 keV (exp.88 keV ) Eγ(4+ → 2+) ≈ 350 keV (exp.199 keV ) Eγ(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: ˆ Qintr

20

= 3ZR2 4π

• α20 +

1 √ 5π 10 7 α20α20 − 20 7 α2−2α22 + 4 3α30α30 − 2α3−1α31 + 10 3 α3−3α33

• = ˆ

Qquadr

20

(1st) + ˆ Qquadr

20

(2nd) + ˆ Qoct

20 (2nd)

() 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)

|Ψvib|2 as function of α20 and α”32 for 156Dy.

• 0.2
• 0.1

0.0 0.1 0.2 0.3 0.4

• 0.2
• 0.1

0.0 0.1 0.2

Dy90 Dy

156 66

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

|!|2

Deformation "20 Deformation "32

• 0.2
• 0.1

0.0 0.1 0.2 0.3 0.4

• 0.2
• 0.1

0.0 0.1 0.2

Dy90 Dy

156 66

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

|!|2

Deformation "20 Deformation "32

• 0.2
• 0.1

0.0 0.1 0.2 0.3 0.4

• 0.2
• 0.1

0.0 0.1 0.2

Dy90 Dy

156 66

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

|!|2

Deformation "20 Deformation "32

• 0.2
• 0.1

0.0 0.1 0.2 0.3 0.4

• 0.2
• 0.1

0.0 0.1 0.2

Dy90 Dy

156 66

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

|!|2

Deformation "20 Deformation "32

Figure: Probability density distributions for the ground-state and the first excited

() September 16, 2015 14 / 17

SUMMARY: We have constructed a realistic collective model able to give the collective eigenfunctions and reduced transition probabilities B(Eλ) This model treats all quadrupole and octupole degrees of freedom in the same footing. The basis in which the Hamiltonian is diagonalized contains zero- and

• ne–phonon H.O. shifted 6D solutions.

Since only real parts of complex αλµ collective variables are considered the symmetrization group is no longer the octahedral group ¯ O as for purely quadrupole vibrations but its subgroup ¯ D4. The intraband B(E2, 4+ → 2+) transition is too small compared to the experimental one. We have generated the fragment of the lowest odd-spin negative-parity band in 156Gd nucleus. Considering the structure of this band and the only measured intraband transition B(E2, 5− → 3−) we are not able to judge which kind of 1-ph excitation is dominating in both those states

() September 16, 2015 15 / 17

FUTURE: The collective potential may be additionally minimized with respect to the higher multipole degrees of freedom which do not break the D4 symmetry. The basis set may be extended to account for the 2-ph, 3-ph,...collective exitations Testing the credibility of the model using other nuclei.

() September 16, 2015 16 / 17

COLLABORATION

Andrzej G´

• ´

zd´ z, IF UMCS, Lublin, Poland Jerzy Dudek IPHC, Strasbourg, France Aleksandra P¸ edrak, IF UMCS, Lublin, Poland Agnieszka Szulerecka, IF UMCS, Lublin, Poland Katarzyna Mazurek IFJ, Krak´

• w, Poland
• S. Vinitsky, A. Gusev

JINR, Dubna, Russia

() September 16, 2015 17 / 17