Realistic shell-model calculations for neutron-rich ...vietri2010.na.infn.it/talks/Coraggio_Vietri10.pdfRealistic shell-model calculations for neutron-rich calcium isotopes Luigi Coraggio

Realistic shell-model calculations forneutron-rich calcium isotopes

Luigi Coraggio

Istituto Nazionale di Fisica Nucleare - Sezione di Napoli

May, 21st 2010

Luigi Coraggio INFN, Napoli

10th International Spring Seminar on Nuclear Physics “New Quests in Nuclear Structure”

Why to perform realistic shell-model calculations forcalcium isotopes?

I To ascertain if modern realistic shell-model potentials areable to overcome the deficiences of the previous ones

I If the answer is in the affirmative, to investigate theshell-structure of these nuclei lying far from the stabilityvalley

L. Coraggio, A. Covello, A. Gargano, and N. Itaco, Phys. Rev. C 80, 044311 (2009)



Why to perform realistic shell-model calculations forcalcium isotopes?

I To ascertain if modern realistic shell-model potentials areable to overcome the deficiences of the previous ones

I If the answer is in the affirmative, to investigate theshell-structure of these nuclei lying far from the stabilityvalley

L. Coraggio, A. Covello, A. Gargano, and N. Itaco, Phys. Rev. C 80, 044311 (2009)



40 44 48 52 56

A

–10.0

–9.5

–9.0

–8.5

–8.0

–7.5

–7.0

g.s.

Ene

rgy/

Nva

l [M

eV]

Expt.

Kuo–Brown interaction



40 44 48 52 56

Nucleus [A]

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

4.0E

xcita

tion

ener

gy [M

eV]

Jπ=2

+ 1st excited states

Expt.




There has been a significant progress about the derivation ofrealistic effective shell-model hamiltonians:

I Renormalization of modern NN potentials by way of theVlow−k approach

I A sound theory to derive the effective shell-modelhamiltonian: Q-box plus folded-diagram method



There has been a significant progress about the derivation ofrealistic effective shell-model hamiltonians:

I Renormalization of modern NN potentials by way of theVlow−k approach

I A sound theory to derive the effective shell-modelhamiltonian: Q-box plus folded-diagram method



Low-momentum nucleon-nucleon potentials: the Vlow−k

Inspiration to renormalize VNN :

I Effective field theory (EFT)I Renormalization group (RG)

from EFT: we restrict the configurations of VNN(k , k ′) to thosewith k , k ′ < kcutoff = Λ

S. Bogner, T. T. S. Kuo, L. Coraggio, A. Covello, and N. Itaco, Phys. Rev. C 65,

051301(R) (2002)








051301(R) (2002)








051301(R) (2002)



The two-nucleon hamiltonian:In the full space:∫ ∞

0[H0(k , k ′) + VNN(k , k ′)]〈k |Ψν〉k2dk = Eν〈k ′|Ψν〉

In a reduced model space P =

∫ Λ

0|k〉〈k |k2dk∫ Λ

0[H0(k , k ′) + Vlow−k(k , k ′)]〈k |Φµ〉k2dk = Eµ〈k ′|Φµ〉

Fundamental constraint: Eµ ∈ {Eν}

How to construct 〈k |Heff|k ′〉?

⇓Unitary transformation: Andreozzi-Lee-Suzuki method






∫ Λ











∫ Λ








A numerical test: phase shifts in the 1S0 channel (in degrees)

Elab (MeV) CD-Bonn Vlow−k Expt.1 62.1 62.1 62.110 60.0 60.0 60.025 50.9 50.9 50.950 40.5 40.5 40.5100 26.4 26.4 26.8150 16.3 16.3 16.9200 8.3 8.3 8.9250 1.6 1.6 2.0300 -4.3 -4.3 -4.5



The shell-model effective hamiltonian

A very useful way to derive Heff is the time-dependentperturbative approach as developed by Kuo and is co-workersin the 1970s (see T. T. S. Kuo and E. Osnes, Lecture Notes inPhysics vol. 364 (1990))In this approach the effective hamiltonian Heff is expressed as

Heff = Q − Q′∫

Q + Q′∫

Q∫

Q − Q′∫

Q∫

Q∫

Q · · · ,

I The so-called Q-box is a collection of irreduciblevalence-linked diagrams

I The integral sign represents a generalized foldingoperation






Q + Q′∫

Q∫

Q − Q′∫

Q∫

Q∫

Q · · · ,








Q + Q′∫

Q∫

Q − Q′∫

Q∫

Q∫

Q · · · ,





M. Hjorth-Jensen, E. Osnes, and T. T. S. Kuo, Phys. Rep. 261 (1995)





Shell-model hamiltonian for calcium isotopes

I Input VNN : Vlow−k derived from the high-precision NNCD-Bonn potential.

I Heff obtained including diagrams up to the 3rd order inVlow−k .

I Effective electric quadrupole operator calculated at 3rdorder in perturbation theory, consistently with Heff.

I Single-particle spacings fixed to reproduce the observedenergies of single-particle states in 47Ca and 49Ca;absolute energies fixed so to reproduce experimentalbinding energy of 49Ca with respect 48Ca.
























Single-particle energies

Orbital SP energies (in MeV)

0f7/2 0.0 (-8.2)1p3/2 2.71p1/2 5.50f5/2 8.5



Odd-mass neutron-rich isotopes

0

1

2

3

4

5

E (

MeV

)

3/2-

1/2-

5/2-

49Ca

Expt Th

3/2-

1/2-

5/2-

51Ca

Expt Th

(3/2-)

(1/2-)

(5/2-)

(3/2-)

(7/2-)

(5/2-)

5/2-3/2-

1/2-

3/2-

1/2-1/2

-

5/2-

3/2-

5/2-

3/2-

11/2-

9/2-

7/2-

7/2-

(9/2-)

3/2-

1/2-

5/2-

3/2-

7/2-

5/2-

9/2-

1/2-7/2-



Even-mass neutron-rich isotopes

0

1

2

3

4

5

E (

MeV

)

0+

2+

(2+)

50Ca

Expt Th

0+

2+

2+

52Ca

Expt Th

0+

2+

(1+)

0+

2+

1+

(1+)

(2+)

(2+)

1+

4+

(0+)

2+

4+

4+

3+

0+

2+

5+1+

0+

2+3+

4+

In 50Ca: B(E2; 2+1 → 0+

1 ) = 10.9 e2fm4 (7.5± 0.2 e2fm4)



Saturation properties: ground-state energies per valence neutron

40 44 48 52 56

A

–10.0

–9.5

–9.0

–8.5

–8.0

–7.5

–7.0

g.s.

Ene

rgy/

Nva

l [M

eV]

Expt.

Present calculations




Shell-closure properties: excitation energies of Jπ = 2+1

40 44 48 52 56

Nucleus [A]

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

4.0E

xcita

tion

ener

gy [M

eV]

Jπ=2

+ 1st excited states

Expt.


GXPF1A




Effective single-particle energies

28 30 32 34 36 38

Neutron number

0.0

1.0

2.0

3.0

4.0

5.0

6.0E

SP

E [M

eV]


GXPF1A

f5/2

p1/2

f5/2p3/2=-0.09 MeV (present calculations)

f5/2p3/2=0.12 MeV (GXPF1A)



Concluding remarks

I The agreement of our reults with the experimental datatestifies the reliability of our Veff

I No need of T = 1 monopole correctionsI No evidence of the magic number N = 34I Looking forward for more experimental data for

neutron-rich calcium isotopes



Concluding remarks






Concluding remarks






Concluding remarks






Calculations for valence proton-neutron nuclei: Ti isotopes

40 42 44 46 48 50 52 54 56 58

Nucleus [A]

40

80

120

160

200

B(E

2 ; 2

+ →

0+)

[e2 fm

4 ]

Expt.

Calc.

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2 1 +

exci

tatio

n en

ergy

[MeV

]

Expt.

Calc.



Documents

Realistic shell-model calculations for neutron-rich ...vietri2010.na.infn.it/talks/Coraggio_Vietri10.pdfRealistic shell-model calculations for neutron-rich calcium isotopes Luigi Coraggio