N SHELL MODEL C : A Ca - GitHub Pagesnuclearphysicsworkshops.github.io/ICNTatMichiganStateUniversity/... · outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure

outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure at N=90 Other applications Conclusions

NEW ADVANCES IN SHELL MODEL

CALCULATIONS: APPLICATIONS AROUND

DOUBLY MAGIC NUCLEI40Ca AND

132Sn

Houda NAÏDJA

IPHC Strasbourg, FranceGSI Darmstadt, Germany.

LPMS, Université Constantine 1, Algeria.

ICNT Program: Theory for open-shell near the limits of stability,

May 11-29, 2015

H. Naïdja IPHC

ICNT


Outline

PART 1 : AROUND40Ca

1. Ca isotopes shift

2. Isomer shift in 38K

3. Effect of pairing correlations

PART 2 : AROUND132Sn

1. Choose Valence space/Core

2. develop an effective interaction

3. Calculation of the energies, B(E2)

and masses in 134,136,138Sn,◮ Effect of core excitations◮ Closure or no of the sub-shell at

N=90

4. Other applications to the open n-p

systems : Te, Xe, Ba, Ce and Nd.

2/44


PART I : ISOMER SHIFT IN38K

3/44


Ca isotopes shift

REMINDER ON Ca ISOTOPES SHIFT

νP

O16

d5/2 d5/2

s1/2

d3/2

p3/2

f7/2

s1/2

d3/2

f7/2

p3/2��

��

��

��

��

��

��

��

4/44


Ca isotopes shift

REMINDER ON Ca ISOTOPES SHIFT

ZBM2 interaction :◮ based on realistic TBME

◮ monopole corrections to ensure 40Ca and48Ca gaps

◮ full space calculations

◮ almost free of center of masscontamination

◮ provides very good spectroscopy at sd-pfinterface

5/44


Ca isotopes shift

✞

✝

☎

✆δ r2

c = 1Z nπ

fp(A)b2

{

nπfp

π occupation probability in fpshell

b = 1.974fm oscillateur parameter

39 40 41 42 43 44 45 46 47 48 49A

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

r p

2 (A)−

r p

2 (40)

(fm

2 )

Isotope shifts in Ca chain

exp sm

Phys.Lett B 522, 240 (2001)

The parabolic dependence on A is

related to the partial breaking of the

Z = 20 shell closure due to the πνcorrelations

correlation energies

40Ca 42Ca 44Ca 46Ca 48Ca

HMulti -13.1 -17.5 -18.1 -13.6 -6.2

HP -9.7 -12.5 -11.9 -8.7 -4.0

HP1 -8.9 -11.8 -11.3 -8.4 -3.9

Hpn

P1 -2.95 -2.2 -1.4 -0.8 -0.2

6/44


38K

ISOMER SHIFT IN38K

δ r2c (0

+)− δ r2c (3

+) = 0.1fm2

3+GS

0+m

HMulti -7.9 -13.2

HP -4.2 -9.7

HP1 -4.2 -9.2

Hpn

P1 -1.4 -6.1

The strong neutron-proton correlations in 0+

compared to 3+

⇓proton occupancy of fp shell is reduced in 3+

compared to 0+ 7/44


38K

SM RESULTS : ZBM2

✗ bad level scheme ✓ good variation of rc

T=1 matrix elements of the ZBM2 are too strong with respect to theT=0, producing an inversion of the 0+ and 3+

⇓a correction of the ZBM2 is necessary preserving the description ofthe Ca isotopes shift

8/44


38K

SM RESULTS : MODIFIED ZBM2Hm = ∑εi ni +∑aijni ·nj +bij(Ti ·Tj −

3

4nδij)

modified ZBM2 : monopole correction of bij to (d3/2)2 with ∆aij = 0

✓ good level scheme ✓ good variation of rc

The modified version of ZBM2 give us a correct order of the states,with conserving a similar composition of the wave functions.

9/44


PART II : SPECTROSCOPIC PROPERTIES OF THE

NUCLEI AROUND132Sn

10/44


Introduction

Introduction

133Sn 134Sn 135Sn

BE/A keV 8310.091 (18) 8275.160 (24) 8230.687 (23)

Taken from NNDC

11/44


Interest field : Experimental

Experimental interest☞New results of 136,138Sn obtained at RIKEN Nishina center.

136Sn

138Sn

12/44


Interest field : Theoretical

Theoretical interest

G,132Sn core,π(gdsh)⊗ν(hfpi)

SMPN,132Sn core,π(gdsh)⊗ν(hfpi)

Vlow−k , 132Sn core, π(gdsh)⊗ν(hfpi)

13/44


Interest field : Astrophysical

Astrophysical interest

☞ responsible of the synthesis of the heavy elements by r-process,and their nuclear model properties predictions give the inputs for

r-process simulations.

100

150

200

250

10

−3

10

−2

10

−110

010

1

Solar r abundances

r−process waiting point (ETFSI−Q)

Known mass

Known half−life

N=184

N=126

N=82

N=50

r−path

28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 5860

6264

666870

727476

7880

8284

86 88 9092 94

9698

100102104

106108

110112114

116118120

122124

126

128130132

134136138

140142

144146

148150

152154

156158

160162

164166

168170

172174

176178

180182184

186188190

26283032

3436

3840

424446

48

50 5254

5658

6062

6466

6870

7274

7678

8082

8486

8890

9294

9698

100

Adapted from K.-L.Kratz

14/44


Core and valence space


15/44




☞ 1h11/2 and 1g9/2 closed ≡132 Sn core

☞ 1h11/2 and 1g9/2 opened ≡110 Zr core

✓ Opening the 132Sn core constitutes anumerical chalenge in the

diagonalisation of the matrix.

⇓Diagonalization in Antoine ∗ andNathan† codes using Lanczos procedure,

Exemple : 140Sm : D=10 1010

(*)E.Caurier et al, Rev.Mod.Phys 77

(2007)427, and Antoine website

(†)no public version

16/44


effective interaction

EFFECTIVE INTERACTION

REALISTIC

1. derived from realistic interaction :ArgonneV18, CD-Bonn,N3LO,...

2. renormalised by Vlow−k or G matrix approach to exclude the

repulsive part at short range.

3. adapted to the model space by many body perturbation theory,

usingP̂ and Q̂ projection operators into model space andexcluded space respectively

P = ∑di=1 |Ψi ><Ψi |, Q = ∑∞

i=d |Ψi ><Ψi |, P+Q=1

Veff = V +VQ

E −H0V

︸︷︷︸

second order

+V QE−H0

V QE−H0

V

V → Vlow−k

M. Hjorth-Jensen et al. Phys.Rep 261 (1995)125

boundary of

lowkV

VNN

0

∞

0∞

Λ

Λ

17/44


spe

Single particle energies

The re-adjustments of the monopole part of Vlow−k interaction to

obtain the experimental level scheme of 133Sn and 133Sb and theirmasses relative to 132Sn.

NNS110 interaction.18/44


be2

B(E2,6+ → 4+)

✗ NNS110 : predicts well the B(E2)

in 134,138Sn, but it underestimatesit for 136Sn.

0

10

20

30

40

50

133 134 135 136 137 138 139 140 141 142

BE

2(6+

-> 4+)

[e2 .fm4 ]

A

NNS110Exp

eeff (ν) = 0.64e

19/44


be2

B(E2,6+ → 4+)

✗ NNS110 : predicts well the B(E2)

in 134,138Sn, but it underestimatesit for 136Sn.

✗ open core : still underestimated

of 136Sn. 0

10

20

30

40

50

133 134 135 136 137 138 139 140 141 142

BE

2(6+

-> 4+)

[e2 .fm4 ]

A

NNS110Exp

NNS110+(1p1h)

eeff (ν) = 0.5e

eeff (π) = 1.5e

20/44


be2

B(E2,6+ → 4+)

✗ NNS110 : predicts well the B(E2)in 134,138Sn, but it underestimates

it for 136Sn.

✗ open core : still underestimated

of 136Sn.

✓ reducing the pairing(NNS110P) : gives us a goodagreement with the Exp.

0

10

20

30

40

50

133 134 135 136 137 138 139 140 141 142

BE

2(6+

-> 4+)

[e2 .fm4 ]

A

NNS110Exp

NNS110+(1p1h)NNS110P

H.Naïdja et al . J.Phys.Conf.series, 580 (2015)012030

21/44


seniority

seniority mixing

υ=0 98%

υ=2 95%

υ=2 82%

υ=2 96%

υ=4 84%

0

10

20

30

40

50

133 134 135 136 137 138 139 140 141 142B

E2(

6+->

4+)

[e2 .fm4 ]

A

NNS110Exp

NNS110+(1p1h)

B(E2,υf = υi ,J → J −2) ∝ (1− 2n2j+1 )

2

◮ 6+ and 4+1 are dominately

seniority υ = 2 ⇒ vanishing

B(E2)

◮ 4+2 (υ = 4) is above the 6+

22/44


seniority

seniority mixing

υ=0 98%

υ=2 93%

υ=2 52%, υ=4 48%

υ=2 45%, υ=4 55%

υ=2 95%

0

10

20

30

40

50

133 134 135 136 137 138 139 140 141 142B

E2(

6+->

4+)

[e2 .fm4 ]

A

NNS110Exp

NNS110+(1p1h)NNS110P

seniority-mixing

B(E2,υf = υi ,J → υi ,J −2) ∝ (1− 2n2j+1 )

2

◮ 4+2 is below the 6+

◮ mixed seniority υ = 2 and υ = 4 in

4+1 and 4+

2 states ⇒ allowed E2transitions.

23/44


seniority

seniority mixing

6+2 (υ = 4) is above the 8+ 6+

2 (υ = 2 and 4) is below the 8+

Pushing down of the 6+2 (υ = 4) state opens up a new channel for the

fast E2 decay of the 8+ state.

24/44


tins : core

Tins : NNS110 and opening the core

H.Naïdja et al . J.Phys.Conf.series,580 (2015)012030

Small effect of the core excitations (1p1h) to the level scheme of tin isotopes.

The robust nature of 132Sn core 25/44


tins :NNS

Tins : NNS110P interaction

ν(f7/2)2 ≈80%-96% ν(f7/2)

4 ≈60%-80% ν(f7/2)6 ≈50%-70%


Reducing the pairing strength (NNS110P interaction), improves

clearly the agreement with the data. 26/44


masses

Masses◮ Binding energies relative to 132Sn

-30

-25

-20

-15

-10

-5

0

5

132 133 134 135 136 137 138 139 140

BE

rel

ativ

e to

13

2 Sn

[MeV

]

A

ExpNNS110P

NNS110Vlowk

◮ one neutron separation energy

0

500

1000

1500

2000

2500

3000

3500

4000

4500

133 134 135 136 137S

n[ke

V]

A

ExpNNS110

NNS110PVlowk


✓ Our masses are consistent with the data

✓ The 2 body monopole corrections are believed to come from 3

body interaction not included in Vlow−k27/44


sub-shell closure at N=90 ?

28/44


bref

sub-shell closure at N=90 ?

Taken

from

O.Sorlin

notes

:

◮ Analogy between 22O, 48Ca, and 140Sn in theclosure of the (sub-)shell at N=14,28, and 90 ?

A.P.Zuker, PRL 90, 042502 (2003)T.Otsuka et al. PRL 105,032501 (2010) 29/44


Sn140

Sub-shell closure at N=90 ?ν(f7/2)

6(p3/2)2

ν(f7/2)7(p3/2)

1

ν(f7/2)6(p3/2)

2

ν(f7/2)8

0

10

20

30

40

50

133 134 135 136 137 138 139 140 141 142B

E2(

6+->

4+)

[e2 .fm4 ]

A

NNS110Exp

NNS110P

H.Naïdja and al . J.Phys.Conf series, 580 (2015)012030

◮ the excited states are characterized by mixed configurations.

our predictions to 140Sn structure using NNS110P

30/44


Sn140

Sub-shell closure at N=90 ?

0

0.5

1

1.5

2

2.5

3

133 134 135 136 137 138 139 140 141

E [M

eV]

N

2+

4+

6+

8+


◮ the spacing 0+−2+ remainsnearly constant at around 700

keV, except for a small increase at140Sn owing to the filling of the(f7/2).

S.Sarkar and M.S.Sarkar Phys.Rev.C 81, 064328(2010)

◮ a sudden increase for N=90,

indicating a closed-shell structurefor 140Sn.

2+ energy in 140Sn in not higher than in the neighboring nuclei. 31/44


ESPE

Evolution of neutron ESPE

-0.5

0

0.5

1

1.5

133 134 135 136 137 138 139 140 141

espe

(MeV

)

A

f7/2

p3/2


◮ The gap between νf7/2 and νp3/2

remains constant

⇓No sub-shell closure at N=90


◮ The gap νf7/2 and νp3/2

increases to 2.246 MeV⇓

140Sn is doubly magic nucleus

2.2 MeV880 keV

750 keV

32/44


ESPE

Evolution of neutron ESPE

-1

-0.5

0

0.5

1

1.5

2

133 134 135 136 137 138 139 140 141

espe

(MeV

)

A

f7/2

p3/2

◮ Increasing the gap by changing

the monopole part


◮ The gap νf7/2 and νp3/2

increases to 2.246 MeV

⇓140Sn is a doubly magic nucleus

2.2 MeV2 MeV

33/44


gap

Increasing the gap

0

10

20

30

40

50

133 134 135 136 137 138 139 140 141 142

BE

2(6+

-> 4+)

[e2 .fm4 ]

A

NNS110PExp

increasing gap

Increasing the gap by changing the monopole part

⇓◮ Transitions are slightly inconsistent with the experience◮ sudden decrease of B(E2) in 140Sn

1. E(2) favor the transitions from j → j +2(p3/2 → f7/2) compared as

j → j +1(f5/2 → f7/2)

ν(f7/2)7(f5/2)

1

ν(f7/2)7(p3/2)

1

ν(f7/2)7(p3/2)

1

ν(f7/2)8

34/44


gap

Tins : increasing the gap

increasing f7/2 −p3/2 gap, has an apparent effect to 138Sn structure.

⇓Losing the spectroscopic properties.

No sub-shell closure at N=9035/44


gap

OTHER APPLICATIONS TO THE OPEN N-P SYSTEMS

36/44


Te

Tellerium

37/44


xe

Xenon

38/44


Ba

Barium

39/44


Ce

Cerium

40/44


Nd

Neodymium

41/44


Be

Collective propertiesB(E2,2+ → 0+)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

134Te136

Te138

Te136

Xe138

Xe140

Xe138

Ba140

Ba142

Ba140

Ce142

Ce144

Ce142

Nd144

Nd146

Nd

BE

2(2+

-> 0+)

[e2 .fm4 ]

A

NNS110PExp

✓ B(E2) are reporduced with no explicit adjustement of the eeff .

✓ BE2 in 136Te is close to that 134Te, so anomaly in 136Te

eeff (ν) = 0.5e

eeff (π) = 1.5e

Tobe

published

42/44


Triaxiality

Triaxiality◮ Q(2+

γ ) =−Q(2+yrast ).

◮ the presence of B(E2,2+γ → 2+

1 ) transition

◮ Q(3+) = 0.

◮ strong B(E2,3+ → 2+2 ) transition

work under progress with Bounseng Bounthong using the deformed

HF with constraints

138Te 140Xe 142Ba 144Ce 146Nd

Q(2+1)e.fm2 -45.67 -62.64 -70.74 -75.24 -61.76

Q(2+2 )e.fm2 40.08 63.84 69.18 75.49 -60.92

Q(3+)e.fm2 -0.34 -1.31 -0.62 -0.79 -0.10

B(2+2 → 2+

1 )e2.fm4 43 155 180 210 294

B(3+ → 2+2)e2.fm4 745 1911 2054 2569 2153

Qi(Q0) 157 220 248 262 248Qi(B(E2)) 160 225 252 268 255β 0.1 0.14 0.15 0.15 0.14

43/44


CONCLUSIONS-PERSPECTIVES

✓ Using NNS110P interaction, the agreement between the experience andcalculated energy levels is improved.

✓ The pairing force must be reduced to reproduce the experimental transition ratesin 136Sn, leading to mixing seniority.

✓ The core excitations seem to have a negligible effect to the tin isotopes energies,confirming the strong magicity of 132Sn

✓ 140Sn doesn’t exhibit the features of a doubly magic nucleus.

✓ The applications to other nuclei allowed us to test our interaction to differentssystems.

PROJECTS UNDER PROGRESS

◮ Triaxiality in 140Sm : in collaboration with Andreas Görgen, University of Oslo.

◮ Isomer in 140Sb : in collaboration with Radomira Lozeva ; IPHC Strasbourg

◮ High spin states in 138Te and 140Xe : in collaboration with W.Urban,

◮ The deformation in the nuclei beyond 132Sn with B.Bounthong.

◮ ...

44/44

Documents

N SHELL MODEL C : A Ca - GitHub Pagesnuclearphysicsworkshops.github.io/ICNTatMichiganStateUniversity/... · outline isomer shift in 38K Beyond 132Sn Tins results sub-shell closure