1 Gamow-Teller strength in deformed QRPA with np-pairing Eun Ja Ha (Soongsil University) in...

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Gamow-Teller strength in deformed QRPA with np-pairing

Eun Ja Ha (Soongsil University)

in collaboration with Myung-Ki Cheoun (Soongsil University) F. Simkovic (Comenius University,

Slovakia)

ECT*-APCTP Joint Workshop, Sep. 15, 2015

• Motivation - Deformation & Neutron-proton(np) pairing correlation

• Formalism- Deformed Woods-Saxon (MF)

- Deformed Bardeen Cooper Schrieffer (DBCS) ; without (with) np-pairing - Deformed quasi-particle random phase approximation (DQRPA)

; pn-QRPA (without np-pairing)

; pp+nn+pn QRPA (with np-pairing)

• Results - Gamow-Teller (GT) strength with deformation

- np-pairing effect in DBCS approach

• Summary

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ContentsContents


site 1 : Supernovae Type II

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Motivations Results Summary Formalism


Known nuclides : 2,500Stable nuclides : 270Unstable nuclides : 6,000~8,000

site 1 : Supernovae Type II

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In the core collapsing supernovae(SNe), medium and heavy elements are believed to be produced by r-process and s-process.

Since most of these nuclei are thought to be more or less deformed, we need to explicitly take into account of the deformation in the nuclear structure.

Why do we consider the deformation in the nuclear structure?


Since the high density (≈104g/cm3)and low temperature on the neutron star crust make electrons degenerated. The degenerated electrons block the beta decay and induce electric captures. Therefore, the valley of stability is shifted toward neutron-rich nuclei.Ordinary nuclei become highly unstable, and RI become the normal stable nuclei at the neutron-star crusts !!

site 2 : Neutron star crusts

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The rapid proton process(rp-process) is thought to be occurred on the binary star system composed of a massive compact star and a companion star.

Deformation could be of practical importance on the understanding of the nucleosynthesis.

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Are the np-pairing correlations restricted only to the vicinity of the N = Z ?


proton-drip line

PRL 106, 252502(2011)

The neutron-proton (np) pairing correlations are important in nuclear structure and decay for proton-rich nuclei with N ≈ Z : protons and neutrons occupy identicalorbitals and have maximal spatial overlap.

S=1,T=0,J=1

S=0,T=1,J=0

T=0,1

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Are the np-pairing correlations restricted only to the vicinity of the N = Z ?


PRL 106, 252502(2011)

(±20, 0) S=0 (T=1) (0, ±22) S=1 (T=0) (±11, ±22) S=0,1 (T=0,1)

(a) 13260Nd72 (b)

13266Dy66 (c) 132

64Gd68

Neodymium Dysprosium Gadolinium

The nuclear structure of the N ≠ Z nuclei may also be affected by np pairing correlations.Does the np-pairing depends on the nuclear deformation ?

)2.1()2(

34 3/1

0

2/1

220

2 AReEB

ZR

8

ℓ is a distance function of a given point r to the nuclear surface as

In experimental side, β2 can be extracted from E2 transition probability.

prolate,02

oblate,02

spherical,02

How to include the deformation?



Deformed Woods-Saxon potential (cylindrical WS, Damgaard et al 1969)

Cv,Cuz,)v,u(S/)v,u(CS),;v,u(

)p)((Vgrad)mc2/(V,a/exp1

V)(V

v,u42

2so

0

distance function surface function

In our calculation, β2 value is the input parameter.

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Shell evolution change according to deformation.


0f7/2 : 3301/20d3/2 : 2001/2

The breaking of magic number comes from the burrowing of f7/2 state below d3/2 state by the deformation.

E. Ha and MK Cheoun, Phys. Rev. C88(2013)


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To exploit G-matrix elements, which is calculated on the spherical basis, deformed bases are expanded in terms of the spherical bases.

Deformed single particle state (SPS)

ECT*-APCTP Joint Workshop, Sep. 15, 2015As the deformation increase, expansion term is also increasing.

Deformed BCS

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J=0T=1 j m j -m Ω -Ω

J=0,1, 2, 3 ∙∙∙T=0 J=1, 3, 5, ∙∙∙T=1 J=0, 2, 4, ∙∙∙

BCS deformed BCS

Ω= ½j ≥ Ω

2

1

2

72

1

2

52

1

2

32

1

2

1

j

K=0


K

JLaboratory frame

Intrinsic frame

Since the deformed SPS are expanded in terms of the spherical SP bases the different total angular momenta of the SP basis states would be mixed.

Realistic two body interaction was taken by Brueckner G-matrix, which is asolution of the Bethe-Goldstone Eq., derived from the Bonn-CD one-bosonexchange potential.

DQRPA eq (nn+pp+pn DQRPA).

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without np-pairing (pn-DQRPA)

Motivations Summary Formalism Results

13ECT*-APCTP Joint Workshop, Sep.

15, 2015


15, 2015

The particle model space 5 ħω is not enough to reproduce the empirical pairing gap. Therefore, the particle model space can be used beyond 6 ħω in G−matrix. In this calculation we use Nmax=10 ħω in G−matrix. (5 ħω in deformed basis)


Particle model space Nmax : pairing strength gpair

15ECT*-APCTP Joint Workshop, Sep. 15, 2015


GT strength in deformed basis & expanded basis

ISR = 98.4 %

ISR = 98.5 %

Since we will apply our DQRPA to other transition, such as electric or magnetic transition, the calculation expanded in spherical basis is more proper than in deformed basis to calculate Wigner Ecart theorem.


Particle-hole strength gph

; determined from the GTGR


Particle-particle strength gpp

; tuned from the double beta decay

The position of the GTGR energy is roughly reproduced.

All GT peaks get shifted to smaller energies as gpp increase.


The high-lying GT excited states beyond one nucleon threshold were already measured at the charge exchange reaction experiments. It is consistent with our calculation.

β2= 0.157 by RMF 0.262 from B(E2)

GT(-) strength for 76Ge with different β2 value



Running sum of GT(-) strength for 76Ge

ISRexp = 55% (up to 12MeV)There may be a possibility of the high-lying GT state above 12.0 MeV.ISRDQRPA ≈ 98 %

Results by DQRPA reproduce well experimental data without quenching factor.




15, 2015

GT(+) strength for 76Se with different β2 value

β2= -0.244 by RMF, 0.309 from B(E2)

(2008) (1997)


Running sum of GT(+) strength for 76Se

Results by DQRPA reproduce well experimental data.



GT(-) strength for 82Se with different β2 value

β 2= 0.133 by RMF 0.193 from B(E2)



Running sum of GT(-) strength for 82Se

Results by DQRPA reproduce well experimental data.


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Motivations Summary ResultsFormalism


Empirical neutron-proton pairing gaps

N=Z

N

Z

The values of np pairing gaps are not negligible even for large neutron excess isotopes

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Motivations Summary

The values of np pairing gaps δpnemp are not negligible even for large neutron

excess isotopes. The attractive short-range interaction between one unpaired proton and neutron with energies close to the Fermi surface is considered to be the origin of the np paring interaction. The np-pairing interaction can be associated with the deformation effect, which is changing the distribution of proton and neutron SP levels.

ResultsFormalism

Empirical pairing gaps for 64Ge ~76Ge


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β2=0.217 (RMF) , In (a), below some critical value (~0.97) : only pp & nn-pairing modes. Above this value : the system prefers to form only np-pair.In (a) pp, nn, and np-pairs coexist in the narrow region.In (b) the coexistence region is more wide and the phase transition becomes less sharp.

)b(MeV5.1),a(MeV25.0gn,p


simple schematic phenomenological force

Pairing gap for 64Ge (Z=N)

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Pairing gap for 70Ge (ZǂN)


β2= - 0.261 (RMF) , The realistic interaction by the G-matrix makes the phase transitions more slowly than the schematic force. np-pairing mode does exist only in coexistence with pp & nn-pairing mode.

)b(MeV5.1),a(MeV229.0gn,p


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Gnp =0.275 in (a), gnp=1.75 in (b) The np-pairing gap decreases with neutron excess. It heavily depends on the deformation parameter β2 for 64Ge. If we use Gnp ≥ 0.275(gnp ≥ 1.75), np-pairing gap is not zero for N-Z =8~10 nuclei.


np-pairing gap with a fixed pairing strength for 64Ge ~76Ge

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Does the δpn depend on the deformation parameter β2 ?


The np pairing gap is sensitive to the β2.

Since the spherical SPS are split by the deformation, the energy gaps of protons and neutrons are also scattered. Therefore the overlap of wave functions becomes smaller and energy gaps becomes larger. The np pairing gaps decrease with the deformation.

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ISR(Ikeda sum rule) = 3( N - Z ) =0 for 64Ge : spherical nucleus

::: deformed nucleus

ISRde/3(N-Z) =100% for64Ge with pn-pairing at BCS process. The modified smearing of Fermi surface were found with np-pairing for N = Z.


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How about N ≠ Z nucleus ?


The modified smearing of Fermi surface were found with np-pairing for N ≠ Z but not as much as 64Ge.

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GT(-) strength for 24Mg (nn+pp+pn DQRPA)

Motivations Summary Formalism Results preliminar

y


0 5 10 15 20 250

1

2

0 5 10 15 20 250

1

2

0 5 10 15 20 250

1

2

0 5 10 15 20 25

10

15

(c) 2= 0.6

with np-pairing

(d) 2= 0.482

with correction

Eex

[MeV]

B(G

T - )

partl.

num

.B

(GT

- )

B(G

T - )

24Mg(3He,t ) (a) Exp.

Sp = 1.871 MeV

Sn = 14.894 MeV

Q = 13.878 MeV

(b) proton neutron

There is a particle number fluctuation in QRPA formalism in (b).

We need a correction of the excitation energy in (d).

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Excitation Energy correction



Correction term

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Motivations Results Formalism Summary

1. We used the deformed WS potential and then performed the deformed BCS and deformed QRPA with realistic two-body interaction calculated by Brueckner G- matrix based on Bonn potential.2. Results of the Gamow-Teller strength, B(GT±), for 76Ge and 76,82Se show that the deformation effect leads to a fragmentation of the GT strength into high-lying GT excited states.3. We examined isovector(T=1) and isoscalar(T=0) np-pairing correlations for the ground state of even-even Ge isotopes, A=64–76, within the deformed BCS approach.4. For N=Z 64Ge a sharp phase transition from the pp(nn)-pairing mode to the np -pairing mode is observed.5.The T=0,1 np-pairing correlations should be considered also for medium- heavy nuclei with large neutron excess since the np-pairing effect is not negligible.6.The change of Fermi level and the modified smearing of Fermi surface were found for N ≠ Z as well as N = Z nuclei. These variations may affect many important nuclear electro-magnetic and weak transitions in nuclear physics.7.The GT strength for 24Mg in DQRPA with np-pairing reproduce well experimental data.

SummarySummary


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Mean Field (Deformed WS )

Deformed BCS w/o np-pairing (O) (T=1) with np-pairing (O) Generalized DBCS(X)

(T=0,1 ) (T=0) 64~76Ge

Deformed QRPA w/o np-pairing (O): pn-DQRPA with np-pairing (O): nn+pp+pn DQRPA for only light nuclei Medium and heavy nuclei (X)

Calculation: Gamow-Teller(GT) transition (pn-DQRPA)(O) : all nucleus GT transition with np (nn+pp+DQRPA) (O) : 24,26Mg M1 spin (X)

Status The gray colored letters are next work !!

Thanks for your attention !!Thanks for your attention !!


15, 2015

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