Development of deformation and smart valence spaces Silvia M. Lenzi - - PowerPoint PPT Presentation

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Development of deformation and smart valence spaces Silvia M. Lenzi - - PowerPoint PPT Presentation

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Development of deformation and smart valence spaces Silvia M. Lenzi University of Padova and INFN Lecture 2 Joint ICTP-IAEA Workshop on Nuclear Structure and Decay Data Shell model and deformation Can the shell model describe deformed

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SLIDE 1

Development of deformation and “smart” valence spaces

Silvia M. Lenzi University of Padova and INFN Lecture 2 Joint ICTP-IAEA Workshop on Nuclear Structure and Decay Data

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SLIDE 2

Shell model and deformation

Can the shell model describe deformed structures?

In the last two decades the improvement in computing power together with the development of powerful shell model methods and codes has allowed to describe well deformed nuclear states, provided the number of degrees of freedom (number of valence particles and the model space are not too large. For this purpose it is essential to identify the smallest valence space that includes the relevant degrees of freedom

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SLIDE 3

The effective interaction

M m eff

V V V  

A multipole expansion

monopole Multipole

m

V

  • represents a spherical mean field extracted

from the interacting shell model

  • determines the single particle energies

and the shell evolution

  • correlations
  • energy gains

Deformation

M

V

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SLIDE 4

The multipole interaction

The multipole interaction is responsible

  • f the collective behaviour

The main components are: Pairing and Quadrupole Pairing dominates in semi-magic nuclei  superfluidity When quadrupole correlations dominate  deformation

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SLIDE 5

Interplay: Monopole and Multipole

5

The interplay of the monopole with the multipole terms, like pairing and quadrupole, determines the different phenomena we observe. In particular, far from stability new magic numbers appear and new regions of deformation develop giving rise to new phenomena such as:

  • islands of inversion
  • shape phase transitions
  • shape coexistence
  • haloes, etc.
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SLIDE 6

Quadrupole correlations:

Shapes and symmetries

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SLIDE 7

The usual model spaces

One is used to think that light and medium nuclei can be described in a single major HO shell For heavier nuclei, the spin-orbit (SO)

  • n top of the HO takes over

and new boundaries appear

s1/2 d5/2 d3/2 p3/2 s1/2 p1/2

20 8 2

g9/2 f5/2 d5/2 p3/2 f7/2 p1/2

82 50 28

…

126 … sd p s fpg

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SLIDE 8

Quadrupole deformation: a simple model

The spherical nuclear field is close to the harmonic oscillator potential. In the limit of degeneracy of the single-particle energies of a major harmonic oscillator shell, and in the presence of an attractive Q.Q proton-neutron interaction, the ground state of the many-body nuclear system is maximally deformed Elliott SU(3) in the sd shell So, at low energy, nuclear states tend to maximize the intrinsic quadrupole moment where the principal quantum number 𝑂 = 𝑜𝑦 + 𝑜y + 𝑜𝑨

q0 = (2𝑜z − 𝑜x − 𝑜y)

The single-particle quadrupole moment is:

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SLIDE 9

Example in the sd shell

The “intrinsic orbits” in SU3

  • start filling from below  prolate deformation
  • start filling from above  oblate deformation

Intrinsic states are the Slater “determinants” obtained by filling these fourfold (2p + 2n) degenerate “orbits” along the N=Z line In the sd shell N = 2 𝑂 = 𝑜𝑦 + 𝑜y + 𝑜𝑨 there are 6 possibilities: (2,0,0) (0,2,0) (0,0,2) (1,1,0)(1,0,1)(0,1,1)

𝑟0 = 2𝑜z − 𝑜x − 𝑜y 𝑟0 = 4, 1, −2

20Ne

Elliott’s SU3 works well in the sd shell but fails for upper shells where the SO interaction introduces large energy shifts

𝑟0 = 2𝑜z − 𝑜x − 𝑜y

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SLIDE 10

SU3 approximate symmetries

Quasi SU3

applies to the lowest Δj = 2, Δℓ = 2

  • rbits in a major HO shell

40

d3/2 g7/2 d5/2 g9/2 s1/2

quasi

N=4

Pseudo SU3

applies to a HO space where the largest j orbit has been removed. p3/2 p1/2

28

f5/2 f7/2

N=3

pseudo

Two variants of SU3 apply in specific spaces

A.P. Zuker et al., PRC 52, R1741 (1995). Zuker, Poves, Nowacki, Lenzi, PRC 92, 024320 (2015)

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SLIDE 11
  • 4
  • 2

2 4 6

q0 (in units of b2)

Quadrupole moments in Pseudo SU3

p3/2 p1/2

28

f5/2 f7/2

N=3

pseudo

We obtain Q0 by summing those

  • f the single particles/holes in each “orbit”

Q0= 20 Q0= 26 Q0= 30 Q0= -30 triaxial

60Zn 64Ge 68Se

shape coexistence

A.P. Zuker et al., PRC 92, 024320 (2015)

Prolate Oblate

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SLIDE 12

Quadrupole moments in Quasi SU3

We obtain Q0 by summing those

  • f the single particles in each “orbit”

gds

2d5/2 1g9/2 3s1/2

quasi

q0/b2

  • A.P. Zuker et al., PRC 92, 024320 (2015)
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SLIDE 13

Maximizing quadrupole correlations

  • 2

2 4 6

Q0/b2

pseudo SU3 for fp space

s1/2 d5/2 g9/2

40 quasi SU3 pseudo SU3

f5/2 p

pseudo quasi

Particle-hole excitations in the pseudo + quasi space maximize the quadrupole moment. The quadrupole correlation energy results much larger than the energy cost to promote the particles

  • 7.5
  • 4.5
  • 1.5

1.5 4.5 7.5 72Kr

K=1/2 K=3/2 K=5/2 K=7/2 K=9/2 K=11/2

68Se

28

A.P. Zuker et al., PRC 92, 024320 (2015)

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SLIDE 14

Quadrupole moments in N=Z nuclei

Quadrupole moments can be obtained from this simple schemes for different np-nh configurations between pseudo and quasi SU3 spaces. Non-degenerate single-particle energies erode slightly the quadrupole collectivity.

A.P. Zuker et al, PRC 92, 024320 (2015)

B(E2) values can be deduced and compared to experiment. B(E2: 2+→0+) = Q0

2/50.3

B(E2: 4+→2+) = Q0

2/35.17

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SLIDE 15

Shape coexistence in 80Zr

T.R.Rodríguez & J.L. Egido, PLB 705, 255 (2011) 2

  • 2
  • 4
  • 6
  • Q0/b2

pseudo SU3 for fp space

pseudo quasi

Our scheme predicts a gamma band due to the two platforms available in the quasi SU3 space

A.P. Zuker et al., PRC 92, 024320 (2015)

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SLIDE 16

Islands of inversion and symmetries

Islands of Inversion at the magic numbers can be understood in terms of dynamical symmetries

32Mg20

sd

p3/2 f7/2 20 quasi SU3

N=2 N=3

pseudo SU3 quasi SU3

12Be8

p

d5/2 s1/2 8

N=1 N=2

pseudo SU3

64Cr40

pf

s1/2 d5/2 g9/2 40 quasi SU3

N=3 N=4

pseudo SU3

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SLIDE 17

The region south of 68Ni

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SLIDE 18

Deformation and SM in the fpgd space

18

LNPS interaction: renormalized realistic interaction

+ monopole corrections

48Ca core

protons: full pf shell neutrons: p3/2,f5/2, p1/2, g9/2, d5/2

40 28 28 f7/2 p3/2 p1/2 f5/2

g9/2 d5/2

48Ca

Lenzi, Nowacki, Poves, Sieja (LNPS), PRC 82, 054301 (2010)

π ν

N=40

fp-gds gap

quasi SU3 pseudo SU3

s1/2 Other effective interactions: Vlow k: L. Coraggio et al., PRC 89, 024319 (2014). A3DA: Tsunoda et al., PRC 89, 031301 (2014).

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SLIDE 19

The N=40 isotones

19

B(E2;2+0+) A change of structure is

  • bserved along the isotonic

chain in good agreement with the available data Occupation of intruder orbitals and percentage of p-h in g.s. configurations

LNPS, PRC 82, 054301 (2010)

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SLIDE 20

Measurement of deformation with radioactive beams

  • H. L. Crawford et al., PRL 110, 242701 (2013)
  • T. Baugher et al., PRC 86, 011305(R) (2012)

Intermediate-energy Coulomb excitation measurements at NSCL-MSU

These data constitute a stringent test for the effective interaction and give direct information on the collectivity and deformation at N=40

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SLIDE 21

Spectroscopy of Mn isotopes

J.J. Valiente-Dobon et al., PRC 78, 024302 (2008) First level schemes from multi-nucleon transfer reactions using CLARA + PRISMA at LNL Calculations without the quasi-SU3 partners in the gds space were unable to reproduce the data for the neutron-rich isotopes

2 2 9 2 2 5 / / g

f

4 2 9 2 2 5 / / g

f

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SLIDE 22

Spectroscopy with radioactive beams

  • T. Baugher et al., PRC 93, 014313 (2016)

258 975 1294

Exp LNPSm

11/2- 9/2- 7/2- 5/2-

Inelastic scattering following fragmentation with SEGA @ MSU

Excitation energy and lifetimes in agreement with data.

More data on heavier Mn isotopes coming soon from RIKEN 63Mn

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SLIDE 23

Shape coexistence in 67Co and 68Ni

  • F. Recchia et al., PRC 85, 064305 (2012)

491 680 (3/2-,5/2-)

1613 2273

67Co

(7/2-) (11/2-) (9/2-) (1/2-)

  • D. Pauwels et al., PRC 78, 041307 (2008)

and PRC 79, 044309 (2009)

The LNPS interaction is able to reproduce these structures

The deformation driven by the neutrons induces a reduction of the Z=28 gap and gives rise to a deformed low-lying 1/2- state

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SLIDE 24

Shape coexistence in 67Co

68Ni

0+ 2034

] ) ( ) [( ] ) ( [

4 4 2 3 2 / 7

gd pf fp f

 

 

Ni 2

68 1 2 / 7   

f 

] ) ( ) [( ] ) ( [

4 4 1 2 2 / 7

gd pf fp f

 

 

the largest B(E2) in the region

  • F. Recchia et al., PRC 85, 064305 (2012)

Up to 11p-11h excitations across the N=40, Z=28 gap 2+

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SLIDE 25

Triple shape coexistence in 68Ni

In first approximation, 68Ni has a doubly closed shell structure in the g.s.

See also: Y. Tsunoda et al., Phys.Rev.C 89, 024313(R) (2014), for 70Ni, 70Co: A.I. Morales et al., PLB 765 (2017) 328 and for 72Ni, A.I. Morales et al., PRC 93, 034328 (2016)

The first three 0+ states are predicted to have different shapes Shell model calculations reproduce well all these structures

  • F. Nowacki, LNPS calculations

More data on heavier Ni isotopes coming soon from RIKEN