advantages and limitations of the shell...

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Advantages and LimitationsOf

The Shell Model

F. Nowacki1

Atelier de Structure Nucleaire TheoriqueApril 7th/11th-2008

1Strasbourg-Madrid-GSI Shell-Model collaborationF. Nowacki Advantages and Limitations Of The Shell Model

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What do we mean by SHELL MODEL nowadays

An approximation to the exact solution of the nuclearA-body problem

Using effective interactions in restricted valence spaces (orregularized interactions in the No Core Shell Modeldescription of very light nuclei)

Where the Monopole hamiltonian is the (spherical) meanfield

and the Multipole hamiltonian is the “correlator”

F. Nowacki Advantages and Limitations Of The Shell Model

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The three pillars of the shell model

The Effective Interaction

Valence Spaces

Algorithms and Codes

E. Caurier, G. Martınez-Pinedo, F. Nowacki, A. Poves and A. P.Zuker. “The Shell Model as a Unified View of NuclearStructure”, Reviews of Modern Physics, 77 (2005) 427-488


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Shell Model Problem

CORE

Define a valence space


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Shell Model Problem

CORE

Define a valence spaceDerive an effective interaction

HΨ = EΨ → HeffΨeff = EΨeff


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Shell Model Problem

CORE



Build and diagonalize theHamiltonian matrix.


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Shell Model Problem

CORE



Build and diagonalize theHamiltonian matrix.

In principle, all the spectroscopic properties are describedsimultaneously (Rotational band AND β decay half-life).


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Basis

Choice of the basis:

m-scheme : |Φα〉 =∏

i=nljmτ

a†i |0〉 = a†

i1...a†iA|0〉

Simple HIJ but Maximal size: D ∼(dπ

p

)

.(dν

n

)

coupled-scheme (J or JT):

[

[

|(j1)n1v1γ1x1〉 |(j2)

n2v2γ2x2〉]Γ2 ... |(jk )nk vkγkxk 〉

]Γk

Reduced dimensions BUT complicated and much more nonzero terms


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Diagonalization

Lanczos Method: iterative process

HΨi = Ei−1 iΨi−1 + EiiΨi + Ei i+1Ψi+1

E11 E12 0 0 0 0E12 E22 E23 0 0 0

0 E32 E33 E34 0 00 0 E43 E44 E45 0

storage of “many” vectors to build the eigenvectors solvedby increase of disk capacity

Extension RAM memory (CPU time/ELAPSED time) 1

regular increase of CPU power


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Giant Matrices

Giant matrices: HIJ recalculated in the iterative process.Basic idea : Factorization |I〉 ≡ |iα〉|i〉: proton state, |α〉: neutron state

dim(i), dim(α) ≪ dim(I)

Precalculations in each separate space

m scheme (code Antoine):HIJ = V(K)two body ME inthe decoupled basis

coupled scheme (code Nathan): HIJ = hij .hαβ .W(K)I=i+α, J=j+β, K=r+µ

Recents improvements:generalized factorization i,αex: semi-magic nuclei N=126i≡ 1i 13

2, 1h 9

2α ≡ 2f 7

2, 2f 5

2, 3p 3

2, 3p 1

2


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Actual Limitations

• m schemeHuge dimensions of the matrices (109-1010)

storage of Lanczos vectors on diskAMD Opteron 64bits 2.2 GHz / 8 Gb RAM

56Ni: D=109 1h30/it.

Very large cases:splitting of the initial and final vectors

Ψi,f =⋃

mΨm

i,f

Ψ(m)f =

∑

nH(m,n)Ψ

(n)i

• Coupled schemeSmall dimensions of the matrices (107-108)Parallelization : each processor has the ini-tial and a final vector

Ψ(k)f = H(k)Ψi

final vectors are added

Ψf =∑

kΨ

(k)f

ImaBIO Cluster 24 nodes Xeon 2.8 GHzNuc Theo Cluster 10 nodes Xeon 2.7 GHz

128Xe GS: D=70 106 (DM=0=1010)

400 Gb precalculation storage

1014 non zero terms !!! 8h/it.(34 procs)

2+ out of reach


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The Effective Interaction(s): Key aspects

The evolution of the Spherical mean field in the valencespaces. What is missing in the monople hamiltonian derivedfrom the realistic NN interactions, be it through a G-matrix,Vlow−k or other options?

The multipole hamiltonian does not seem to demand majorchanges with respect to the one derived from the realisticnucleon-nucleon potentials

Do we really need three body forces? Would they be reducibleto simple monopole forms?

Much more to come with Andres Zuker’s talk


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Ab initio calculations

Valence space : all states with excitation energy in the H.O.basis until N~ω

N=10 for A≤8N=8 for A≤16

Specific problem : number of |nljm〉 states

276 shells for protons or neutrons4600 states in the 22 ~ω space (20 in fp shell)

Lee-Suzuki unitary transformation to derive effectivehamiltonian in the model spaceuse of modern realistic interactions CD-Bonn, Argonne,N3LO, Idaho-A ...


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NSM: 4He

4He CD-Bonn 2000

-36-34-32-30-28-26-24-22-20-18-16-14-12-10

-8-6-4-202468

10

0 2 4 6 8 10 12 14 16 18 20 22 24

N max

E [M

eV] 40; 0+0 exc

28; 0+0 exc19; 0+0 exc40; 0+0 gs28; 0+0 gs19; 0+0 gs


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p shell: Halo nuclei

8Li and 8B quadrupole moments

Quadrupole moment (e.fm2)N=0 N=2 N=4 N=6 N=8 N=10 N=12 exp. QFMC

8B 4.27 4.51 4.76 5.03 5.29 5.55 5.76 6.83(21) 8.2(3)8Li 1.79 2.24 2.43 2.60 2.72 2.83 2.92 3.19(7) 3.0(2)

Two first 32−

states in 11Li

N=2 N=4 N=6 N=8E( 3

2 )−1 - E( 32 )−2 22.5 20.4 15.2 12.8

n components Ψ1 Ψ2 Ψ1 Ψ2 Ψ1 Ψ2 Ψ1 Ψ2n=0 0.72 0.0 0.67 0.0 0.62 0.0 0.59 0.0n=2 0.28 1.0 0.16 0.77 0.16 0.67 0.14 0.59n=4 0.17 0.23 0.13 0.17 0.14 0.19n=6 0.08 0.15 0.06 0.13n=8 0.06 0.08


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Spectroscopy in 10B


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3N: status

0

4

8

NN+NNN Exp NN

3 + 3

+1

+

1 +

0 +; 1

0 +; 1

1 +

1 +

2 +

2 +

3 +

3 +

2 +; 1

2 +; 1

2 +

2 +

4 +

4 +

2 +; 1

2 +; 1

hΩ=14

0

4

8

12

16

NN+NNN Exp NN

3/2-

3/2-

1/2-

1/2-

5/2-

5/2-

3/2-

3/2-

7/2-

7/2-

5/2-

5/2-

5/2-

5/2-

1/2-; 3/2

1/2-; 3/2

0

4

8

12

16

NN+NNN Exp NN

0 +

0 +

2 +

2 +

1 +

1 +

4 +

4 +

1 +; 1

1 +; 1

2 +; 1

2 +; 1

0 +; 1

0 +; 1

0

4

8

12

16

NN+NNN Exp NN

1/2- 1/2

-

3/2-

3/2-

5/2-

5/2-

1/2-

1/2-

3/2-

3/2-

7/2-

7/2-

3/2-; 3/2

3/2-; 3/2

10B

11B

12C 13

C

P. Navratil, V. G. Gueorgiev, J. P. Vary, W. E. Ormand, and A. Nogga

Phys. Rev. Lett. 99, 042501 (2007)


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3N: statusNucleus/property Expt. NN+NNN NN

6Li : |E(1+1 0)| [MeV] 31.995 32.63 28.98

Q(1+1 0) [e fm2] -0.082(2) -0.124 -0.052

µ(1+1 0) [µN ] +0.822 +0.836 +0.845

Ex (3+1 0) [MeV] 2.186 2.471 2.874

B(E2;3+1 0 → 1+

1 0) 10.69(84) 3.685 4.512

B(E2;2+1 0 → 1+

1 0) 4.40(2.27) 3.847 4.624

B(M1;0+1 1 → 1+

1 0) 15.43(32) 15.038 15.089

B(M1;2+1 1 → 1+

1 0) 0.149(27) 0.075 0.03110B : |E(3+

1 0)| [MeV] 64.751 64.78 56.11rp [fm] 2.30(12) 2.197 2.256

Q(3+1 0) [e fm2] +8.472(56) +6.327 +6.803

µ(3+1 0) [µN ] +1.801 +1.837 +1.853

rms(Exp − Th) [MeV] - 0.823 1.482B(E2;1+

1 0 → 3+1 0) 4.13(6) 3.047 4.380

B(E2;1+2 0 → 3+

1 0) 1.71(0.26) 0.504 0.082

B(GT;3+1 0 → 2+

1 1) 0.083(3) 0.070 0.102

B(GT;3+1 0 → 2+

2 1) 0.95(13) 1.222 1.48712C : |E(0+

1 0)| [MeV] 92.162 95.57 84.76rp [fm] 2.35(2) 2.172 2.229

Q(2+1 0) [e fm2] +6(3) +4.318 +4.931

rms(Exp − Th) [MeV] - 1.058 1.318B(E2;2+0 → 0+0) 7.59(42) 4.252 5.483B(M1;1+0 → 0+0) 0.0145(21) 0.006 0.003B(M1;1+1 → 0+0) 0.951(20) 0.913 0.353B(E2;2+1 → 0+0) 0.65(13) 0.451 0.301

P. Navratil, V. G. Gueorgiev, J. P. Vary, W. E. Ormand, and A. NoggaPhys. Rev. Lett. 99, 042501 (2007)


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3N: statusNucleus/property Expt. NN+NNN NN

11B : |E( 32−1

12 )| [MeV] 76.205 77.52 67.29

rp( 32−1

12 ) [fm] 2.24(12) 2.127 2.196

Q( 32−1

12 ) [e fm2] +4.065(26) +3.065 +2.989

µ( 32−1

12 ) [µN ] +2.689 +2.063 +2.597

rms(Exp − Th) [MeV] - 1.067 1.765

B(E2; 32−1

12 → 1

2−1

12 ) 2.6(4) 1.476 0.750

B(GT; 32−1

12 → 3

2−1

12 ) 0.345(8) 0.235 0.663

B(GT; 32−1

12 → 1

2−1

12 ) 0.440(22) 0.461 0.841

B(GT; 32−1

12 → 5

2−1

12 ) 0.526(27) 0.526 0.394

B(GT; 32−1

12 → 3

2−2

12 ) 0.525(27) 0.762 0.574

B(GT; 32−1

12 → 5

2−2

12 ) 0.461(23) 0.829 0.236

13C : |E( 12−1

12 )| [MeV] 97.108 103.23 90.31

rp( 12−1

12 ) [fm] 2.29(3) 2.135 2.195

µ( 12−1

12 ) [µN ] +0.702 +0.394 +0.862

rms(Exp − Th) [MeV] - 2.144 2.089

B(E2; 32−1

12 → 1

2−1

12 ) 6.4(15) 2.659 4.584

B(M1; 32−1

12 → 1

2−1

12 ) 0.70(7) 0.702 1.148

B(GT; 12−1

12 → 1

2−1

12 ) 0.20(2) 0.095 0.328

B(GT; 12−1

12 → 3

2−1

12 ) 1.06(8) 1.503 2.155

B(GT; 12−1

12 → 1

2−2

12 ) 0.16(1) 0.733 0.263

B(GT; 12−1

12 → 3

2−2

12 ) 0.39(3) 1.050 0.221

B(GT; 12−1

12 → 3

2−1

32 ) 0.19(2) 0.400 0.151

Total energy rms [MeV] - 1.314 1.671

P. Navratil, V. G. Gueorgiev, J. P. Vary, W. E. Ormand, and A. NoggaPhys. Rev. Lett. 99, 042501 (2007)


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Valence space

The choice of the valence space:

In light nuclei the harmonic oscillator closures determinethe natural valence spaces:

4He −→ 16O −→ 40Ca −→ 80Zrp shell sd shell pf shell ↑Cohen/ Brown/ DeformedKurath Wildenthal


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Valence space




In heavier nuclei:−→ jj closures due to the spin-orbit term show up

N=28, 50, 82, 126


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Valence space




In heavier nuclei:−→ jj closures due to the spin-orbit term show up

N=28, 50, 82, 126

the transition HO −→ jj : occurs between 40Ca and 100Snwhere the protagonism shifts from the 1f7/2 to the 1g9/2


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Valence space

A valence space can be adequate to describe someproperties and completely wrong for others

48Cr (f 72)8 (f 7

2p 3

2)8 (fp)8

Q(2+) (e.fm2) 0.0 -23.3 -23.8E(2+) (MeV ) 0.63 0.44 0.80E(4+)/E(2+) 1.94 2.52 2.26

BE2(2+ → 0+) (e2.fm4) 77 150 216

B(GT) 0.90 0.95 3.88


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Valence space

For the quadrupole properties f 72p 3

2is a good space

(Quasi-SU3 orbitals)whereas for magnetic and Gamow-Teller processes thepresence of the spin orbit partners is compulsory.

In the tin isotopesthe natural valence space consists in a 100Sn core andvalence orbits:

(d 52g 7

2s 1

2d 3

2h 11

2)ν

However, numerous E1 transitions have been measuredthat are forbidden in this space because:(a†

11/2.aj)λ6=1,

it is then necessary to incorporate the 1g9/2 orbit.


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Physics Goals

Precision Spectroscopy towards larger masses

Description of the nuclear correlations in the laboratory frame

Changing Magic Numbers far from Stability: The competingroles of spherical mean field and correlations

Double β decay, the key to the nature of the neutrinos, theabsolute scale of their masses and their hierarchy

No core shell model for light nuclei. Ab initio description of thelow-lying intruder states and of the origin of the Gamow-Tellerquenching

Nuclear Structure and Nuclear Astrophysics


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The most popular “flaws” of the standard SMdescription

Not all the regions of the nuclear chart are amenable to aSM description yet

Quadrupole effective charges are needed (But their valueis universal and rather well understood)

Spin operators are quenched by another universal factorwhich relates to the regularization of the interaction (alsoknown as short range correlation). Indeed, BMFapproaches share this shortcoming


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Accessible Regions

Ν→

Ζ↑

*

* *

*

*

*

* *

*

*

* *

* ** **

* * *

**

*

* **

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n 0 H 1

He 2 Li 3

Be 4 B 5 C 6

N 7 O 8

F 9 Ne 10

Na 11 Mg 12 Al 13 Si 14

P 15 S 16

Cl 17 Ar 18

K 19 Ca 20

Sc 21 Ti 22

V 23 Cr 24

Mn 25 Fe 26

Co 27 Ni 28

Cu 29 Zn 30

Ga 31 Ge 32

As 33 Se 34

Br 35 Kr 36

Rb 37 Sr 38

Y 39 Zr 40

Nb 41 Mo 42

Tc 43 Ru 44

Rh 45 Pd 46

Ag 47 Cd 48

In 49 Sn 50

Sb 51 Te 52

I 53 Xe 54

Cs 55 Ba 56

La 57 Ce 58

Pr 59 Nd 60

Pm 61 Sm 62

Eu 63 Gd 64

Tb 65 Dy 66

2 4 6

8 10

12 14 16

18 20

22 24

26

28 30 32

34

36 38

40 42

44 46 48 50

52

54

56 58

60 62 64

66

68 70

72

74 76 78 80 82

84

86 88

90 92 94 96

98

100

*** **

**

** **

*

****

* ** *** * *

*

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*******

***

** *** **

***********

*

*** ********************

Ho 67 Er 68

Tm 69 Yb 70

Lu 71 Hf 72

Ta 73 W 74

Re 75 Os 76

Ir 77 Pt 78

Au 79 Hg 80

Tl 81 Pb 82

Bi 83 Po 84

At 85 Rn 86

Fr 87 Ra 88

Ac 89 Th 90

Pa 91 U 92

Np 93 Pu 94

Am 95 Cm 96 Bk 97

Cf 98 Es 99

Fm 100Md 101

No 102Lr 103Rf 104

Ha 105Sg 106Ns 107

Hs 108Mt 109

10 11011 111

80 82 84 86 88 90 92 94 96 98 100 102 104106

108110

112

114 116

118 120

122124

126128

130 132

134136 138

140142 144

146

148 150

152

154 156

158

160


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Quadrupole excitations on CS nuclei

Ca40

(sd)

(fp)

(gds)

N=2

N=3

N=4

100 Sn

N=5

N=6

N=4(gds)

(hfp)

(igds)


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SD band in 36Ar

C. E. Svensson, et al.,Phys. Rev. Lett. 85, 2693 (2000)C. E. Svensson, et al.,Phys. Rev. C63, 061301 (2001)

good description in terms of phexcitations


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SD band in 36Ar



Decay of the SD bands shows themixing between npnh configurations


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SD band in 36Ar




Mixing should not destroy theagreement of 4p4h calculations


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SD band in 36Ar




Mixing should not destroy theagreement of 4p4h calculations

Complex mecanism and theoreticalchallenge


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Decay out of the SD band in 36Ar

0 1000 2000 3000 4000 5000

Eγ (keV)

0

2

4

6

8

10

12

14

16

18

20

J

exp.4p-4h SDPF

0 1000 2000 3000 4000 5000

Eγ (keV)

0

2

4

6

8

10

12

14

16

18

20

J

exp.sdpf

improvment upon 4p-4h calculations

backbending reproduced and correct moment of inertia

E. Caurier, F. Nowacki, and A. PovesPhys. Rev . Lett. 95, 042502 (2005)


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0 2 4 6 8 10 12 14 16 18J

0

100

200

300

400

500

B(E

2) in

e2

fm4

exp.4p-4h sdpf

good overall agreement between experiment and calculations


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0 2 4 6 8 10 12 14 16 18J

0

100

200

300

400

500

B(E

2) in

e2

fm4

exp.4p-4h sdpf


reconstruction of 4p-4h results due to mixing with 6p-6h states (as deformed as 4p-4h ones)


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0 2 4 6 8 10 12 14 16 18J

0

100

200

300

400

500

B(E

2) in

e2

fm4

exp.4p-4h sdpf


reconstruction of 4p-4h results due to mixing with 6p-6h states (as deformed as 4p-4h ones)

E. Caurier, F. Nowacki, and A. PovesPhys. Rev. Lett. 95, 042502 (2005)


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Out-band transitions in the SD band of 36Ar(B(E2)’s in e2fm4 and energies in keV)

Eγ B(E2)experiment theory experiment theory

2+SD → 0+

1 4950 4846 4.6(23) 4.04+

SD → 2+1 4166 3917 2.5(4) 1.2

4+SD → 2+

2 1697 946 19.2(30) 18.46+

SD → 4+1 3552 2787 5.3(8) 0.25

10+1 → 8+

SD 1975 1192 43.6(74) 13.112+

SD → 10+1 3448 3655 15.0(30) 1.5

E. Caurier, F. Nowacki, and A. Poves

Phys. Rev. Lett. 95, 042502 (2005)


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The Superdeformed band of 40Ca

0 1000 2000 3000 4000Eγ (in keV)

0

2

4

6

8

10

12

14

16

18

20

J

expsdpf-full

exp: E. Ideguchi et al., Phys. Rev. Lett. 87, 222501 (2001)F. Nowacki Advantages and Limitations Of The Shell Model

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Transition Quadrupole Moments

0 2 4 6 8 10 12 14 16 18J

100

150

200

250T

rans

ition

qua

drup

ole

mom

ent (

e fm

2 )


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B(E2)’s in Tin isotopes

0

100

200

300

400

500

600

50 60 70 80

B(E

2)

(e2fm

4)

Neutron number

expt=0t=2

t=4 g9g7d5t=4 gds

A. Banu et al.,

Phys. Rev. C72, 061305(R) (2005)


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Renormalization of the spin operator in the pf -shell

Nucleus Uncorrelated Correlated Expt.

Unquenched Q = 0.7451V 5.15 2.42 1.33 1.2 ± 0.154Fe 10.19 5.98 3.27 3.3 ± 0.555Mn 7.96 3.64 1.99 1.7 ± 0.256Fe 9.44 4.38 2.40 2.8 ± 0.358Ni 11.9 7.24 3.97 3.8 ± 0.459Co 8.52 3.98 2.18 1.9 ± 0.162Ni 7.83 3.65 2.00 2.5 ± 0.1


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Quenching of GT strength in the pf -shell


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48Ca(p,n)48Sc Strength Function


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Quenching of GT operator in the pf -shell|i〉 = α|0~ω〉 +

∑

n 6=0

βn|n~ω〉,

|f 〉 = α′|0~ω〉 +∑

n 6=0

β′n|n~ω〉

then

〈f ‖ T ‖ i〉2 =

αα′ T0 +∑

n 6=0

βnβ′n Tn

2

,

n 6= 0 contributions negligible

α ≈ α′

projection of the physical wavefunction in the0~ω space is Q ≈ α2

transition quenched by Q2


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Quenching of M1 operator in the pf -shell

0

0.04

0.08

6 8 10 12 140.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5

6 8 10 12 14Excitation Energy (MeV)

0.0

0.5

1.0

Orbital

Spin

Shell−ModelTotal

Expt.

B(M

1) (

µ N2)

52Cr

K. Langanke, G. Martinez-Pinedo,

P. Von Neumann-Cosel, and A. Richter

Phys. Rev. Lett. 93, (2004) 202501

KB3G interaction


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Quenching of M1 operator in the pf -shell

KB3 interaction

Neumann-Cosel et al.

Phys. Lett. B433 1 (1998)


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Correlations in nuclei

V. R. Pandharipande, I. Sick and P. K. A. deWitt

Huberts, Rev. mod. Phys. 69 (1997) 981


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50 ≤ Z , N ≤ 82 region


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β decay systematics

Nucleus 128Sn 130Sn 132Sb 132Te 133Te

Transition 0+ → 1+ 0+ → 1+ 4+ → 3, 4, 5+ 0+ → 1+ 32

+→ 1

2 , 32 , 5

2+

T1/2exp. 59.07m 3.72m 2.79m 3.2d 12.5mT1/2calc. (0.74) 32.21m 2.47m 1.56m 1.73d 6.42m

Renorm. 0.54 0.6 0.55 0.54 0.53

134Te 135Xe 136Cs

0+ → 1+ 32

+→ 1

2 , 32 , 5

2+

5+ → 4, 5, 6+

41.8m 9.14h 13.16d29.19m 7.07h 8.1d

0.62 0.63 0.57

Our valence space leads to a renormalization of the στ

operator of a factor 0.57


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Integrated strength

Integrated B(GT) values for 13078 Te52 (left)

and 12876 Te52 (right) calculated within the

(g 72

d 52

d 32

s 12

h 112

) space

(experimental data are from (p,n) reactions)

0 2 4 6 8 10 12 14Excitation energy (MeV)

0

10

20

30

B(G

T)

cum

ul

exp.théo. (espace 50-82)

0 2 4 6 8 10 12 14Excitation energy (MeV)

0

10

20

30

B(G

T)

cum

ul

exp.théo. (espace 50-82)


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A=136

136Cs half-life

Gamow-Teller strength function from the 5+

ground state of 136Cs :

T1/2 th. 13 dT1/2 exp. 13.16 d


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Shell Model at the limits of the stability

At the very neutron rich or very proton rich edges, the T=0 andT=1 channels of the effective nuclear interaction weight verydifferently than they do at the stability line. Therefore theeffective single particle structure may suffer important changes,leading in some cases to the vanishing of established shellclosures or to the appearance of new ones.


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Neutron rich nuclei: N=20

-20

-10

0

10

2016148

ES

PE

(M

eV

)

Proton number

d5/2

s1/2

d3/2

f7/2

p3/2

p1/2

f5/2


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Dipole response in Neon chain

Neons bound from stability to n-rich

Shell evolution Island of inversion

Shape evolutionSU3 Deformation (N=Z)Sphericity (N=14)Intruder Deformation (N=20)

Full sd diagonalization + full 1~ω excitations

Exact removal of Center of Mass spurious components


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Neons Dipole Strength

20Ne


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22Ne


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24Ne


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26Ne


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28Ne


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30Ne


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Neutron rich nuclei: N=20

1 2 3 4(1) 028 (2) Si34 (3) S36 (4) Ca40

-5

0

5

10

15E

nerg

ies

in M

eV

f5

p1

f7p3d3

s1

d5


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26Ne Pygmy Resonance

recently observed at RIKEN:Eexp= 9 MeVESM = 8.5 MeV


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B(E1)exp= 0.60 ± 0.06 e2.fm2, 5.9 ± 1.0 % STRKB(E1)SM = 0.42 e2.fm2, 4.0 % STRK


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Main contributions :ν(2s−1

1/2 2p3/2)

ν(2s−11/2 2p1/2)


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Summary

Thanks to: E. Caurier,K. Langanke,G. Martinez-Pinedo,J. Menedez,A. Poves,K. Sieja,O. Sorlin,A. Zuker


advantages and limitations of the shell...

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