1. 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)
2nd HaPhy meeting, July

2. Contents Motivation - Deformation & Neutron-proton(np) pairing
Formalism
- Deformed Woods-Saxon (MF)
- Deformed Bardeen Cooper Schrieffer (DBCS)
- Deformed quasi-particle random phase approximation (DQRPA)
Results
- Gamow-Teller strength : 76Ge, 82Se ,90, 92Zr
- np-pairing effect : Ge isotopes
Summary
The contents are Motivation, Farmalism, Results, and Summary.
2nd HaPhy meeting, July

3. site 1 : Supernovae Type II
Motivations
Formalism
Results
Summary
Black : stable nuclei
Yellow: unstable (known)
Green: unstable(unknown)
지구상에는 100여종의 원소가 존재하며 지구상에 존재하는 원자핵은 안정핵으로
시간이 지나도 절대 변하지 않으며 약 270종류가 있다. 그러나 생성된 후 자연적으로 파괴되어 다른 원자핵으로 변하는 불안정핵이 6,000~8,000개에 이른다.
이것을 불안정핵 또는 radio isotope (RI)라고 부른다. Black circle denote stable nuclides and unstable nuclides lie in the shaded regions.
최근까지 원자핵물리의 연구는 안정핵과 안정핵부근의 원자핵으로 한정되었기 때문에 불안정핵에 관한 연구는 거의 이루어지지 않았다.
최근에 불안정핵의 고속입자를 사용하여 산란이나 반응을 일으키는 새로운 방법인 RI빔법을 이용해 불안정핵에 대한 정보가 증가, 지금까지 미처 생각하지 못한 새로운 구조가 발견. RI빔법이 개발되기전까지 연구는 “ 마치 골짜기만 관찰하고 산맥전체를 이해하려는 것처럼 억지스러운 부분이 있었다
site 1 : Supernovae Type II
Known nuclides : 2,500
Stable nuclides : 270
Unstable nuclides : 6,000~8,000
2nd HaPhy meeting, July

4. site 1 : Supernovae Type II
Motivations
Formalism
Results
Summary
Why do we consider the deformation in the nuclear structure?
site 1 : Supernovae Type II
In the core collapsing supernovae(SNe), medium and heavy elements are believed to be produced by r-process and s-process.
Since most of the nuclei produced in these processes are thought to be more or less deformed, we need to explicitly take into account of the deformation in the nuclear structure.
2nd HaPhy meeting, July

5. site 2 : Neutron star crusts
Motivations
Formalism
Results
Summary
site 2 : Neutron star crusts
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.
We have another interesting process associated with deformed nuclei.
This process is r-p process.Because of strong gravity on the massive star surface, one expect hydrogen rich mass flow from the companion star.
The outermost layers of a neutron stars are composed of iron. At densities above ten thousand grams per cubic centimeters, the atoms are fully ionized due to the pressure of the upper layers. The free electrons are degenerate
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 !!
2nd HaPhy meeting, July

6. Motivations
Formalism
Results
Summary
Are the pn-pairing correlations restricted only to the vicinity of the N = Z ?
PRL 106, (2011)
S=0, T=1, J=0
S=1, T=0, J=1
T=0,1
proton-drip line
There are many publications in pn-pairing correlations.
The proton-neutron (pn) pairing correlations are important in nuclear structure and decay for proton-rich nuclei with N ≈ Z : proton and neutrons occupy identical
orbitals and have maximal spatial overlap.
2nd HaPhy meeting, July

7. the np-pairing may exist in isoscalar(T=0) and isovector(T=1) pairing.
Motivations
Formalism
Results
Summary
In contrast to the proton-proton(pp) and the neutron-neutron(nn) pairing,
the np-pairing may exist in isoscalar(T=0) and isovector(T=1) pairing.
Chen and Goswami, Nucl. Phys. 11, 263(1979)
Both single- and double-beta decay transitions are affected by the np-pairing.
M-K. Cheoun, Nucl. Phys. A561,74(1993), A564,329(1993), Phys.Rev.C53,
695(1996)
Does the pn-pairing strength depend on the deformation parameter β2 ?
We examine isovector (T=1 ) and isocalar (T=0 ) np-correlations for the ground
state of even-even 64~76Ge isotopes within the deformed BCS approach with a
realistic two-body interaction, given by the Brueckner G-matrix based on the CD
Bonn potential.
There are many publications in pn-pairing correlations.
2nd HaPhy meeting, July

8. Single particle state (SPS) in spherical basis in deformed basis
Motivations
Formalism
Results
Summary
Single particle state (SPS)
in spherical basis in deformed basis
[ n ℓ j]
[N nzɅ Ω]
This is the dependence of the expansion coef. on beta_2 value in 202 5/2 of 76Ge.
Ω=0 + ½
2nd HaPhy meeting, July

9. E. Ha and MK Cheoun, Phys. Rev. C88(2013)
Motivations
Formalism
Results
Summary
Shell evolution in the neutron-rich nuclei.
E. Ha and MK Cheoun, Phys. Rev. C88(2013)
Here is an example for N=20 isotones. This is the spse of neutron of N=20 isotones as a function of proton number.
We utilize two kind of deformation paramers beta_2. In this work beta_2 is input.
One is from RMF for the upper panel and the other is from E2 transition probability for the lower panel.
We can see magic number. But in this case, magic number disappeared.
0d3/2 : 2001/2
0f7/2 : 3301/2
The breaking of magic number comes from the burrowing of f7/2 state below d3/2 state by the deformation.
2nd HaPhy meeting, July

10. How to include the deformation?
Motivations
Formalism
Results
Summary
How to include the deformation?
Deformed Woods-Saxon potential
(cylindrical WS, Damgaard et al 1969)
distance function
surface function
As you can see, the nuclear shape in the deformed WS potential depends on epsilon,
which is related to as the deformation parameter β2.
Furthermore, in experimental side, β2 can be extracted from E2 transition probability.
If beta_2 value is larger than 0, it is called as the prolate deformation,
And it is less than 0, the shape of nucleus is oblate.
In experimental side,
β2 can be extracted from E2 transition probability.
In our calculation, β2 value is the input parameter.
2nd HaPhy meeting, July

11. Here, alpha prime denotes a proton or neutron. And alpha is ~
Motivations
Formalism
Results
Summary
In many-body system, the Hamiltonian can be written as
: deformed axially symmetric
WS potential
where, = single particle state
= sign of the angular momentum projection of state
= projection of the total angular momentum J on the nuclear
symmetry axis
Spherical symmetry is broken.
J is a not good quantum number any more in deformed basis. `
This is a Hamiltonian.
Here, alpha prime denotes a proton or neutron. And alpha is ~
As I mentioned before, omega is a ~
2nd HaPhy meeting, July

12. Deformed single particle state (SPS)
Motivations
Formalism
Results
Summary
Deformed single particle state (SPS)
To exploit G-matrix elements, which is calculated on the spherical basis, deformed bases are expanded in terms of the spherical bases.
This is the dependence of the expansion coef. on beta_2 value in 7/2 of 76Ge.
Expanded terms increase with large beta_2 value
2nd HaPhy meeting, July

13. J K Ω= ½ j ≥ Ω Deformed BCS BCS deformed BCS J=0 T=1 K=0 j m j -m Ω -Ω
Motivations
Formalism
Results
Summary
Deformed BCS
BCS deformed BCS
Ω= ½
j ≥ Ω
J=0
T=1
K=0
j m j -m
Ω -Ω
J=0,1, 2, 3 ∙∙∙
T=0 J=1, 3, 5, ∙∙∙
T=1 J=0, 2, 4, ∙∙∙
Laboratory frame J K
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.
2nd HaPhy meeting, July

14. We obtain the following Deformed HFB (DHFB) equation :
Motivations
Formalism
Results
Summary
We obtain the following Deformed HFB (DHFB) equation :
The pairing potentials ∆ are calculated in the deformed basis by using the
G-matrix calculated from the realistic Bonn CD potential for the N-N interaction.
In order to renormalize the G-matrix, strength parameters, g is multiplied to the G-matrix by adjusting the pairing potentials to the empirical pairing potentials.
In contrast, pn-pairing potential has two component. One is T=1 the other is T=0.
In order to renormalize the G-matrix, strength parameters, g is multiplied to the G-matrix by adjusting the pairing potentials to the empirical pairing potentials
J= even
J= even
J= odd
2nd HaPhy meeting, July

15. Empirical pairing gaps
Motivations
Formalism
Results
Summary
Empirical pairing gaps
The empirical pairing potentials of proton and neutron are evaluated by the following symmetric five term formula for neighboring nuclei.
In macroscopic model, there are one unpaired proton and one unpaired neutron near Fermi level.
p-n pairing
interaction energy δ
Z ≈ N
with pn-pairing without pn-pairing
An attractive short-range residual interaction between one unpaired proton and neutron is considered to be the origin of the proton-neutron interaction energy.
2nd HaPhy meeting, July

16. Motivations
Formalism
Results
Summary
DQRPA eq.
without np-pairing
Residual interaction contains two parts.
A particle-hole force are defined with positive sign, which is repulsive, and particle-particle force is negative sign which is attractive.
Realistic two body interaction was taken by Brueckner G-matrix, which is a
solution of the Bethe-Goldstone Eq., derived from the Bonn-CD one-boson
exchange potential.
2nd HaPhy meeting, July

17. Gamow-Teller(GT) Transition
Motivations
Formalism
Results
Summary
Gamow-Teller(GT) Transition
∆T=1, ∆S=1, ∆L=0, ∆ J=1+ (K=0,1,-1)
Forward ampl. X Backward ampl. Y
We apply our formalism to GT and beta-decay.
p n
p n
2nd HaPhy meeting, July

18. (a) constant value (b) G-matrix as the pairing interaction.
Motivations
Formalism
Results
Summary
(a) constant value (b) G-matrix as the pairing interaction.
2nd HaPhy meeting, July

19. Particle model space Nmax : pairing strength gpair
Motivations
Formalism
Results
Summary
Particle model space Nmax : pairing strength gpair
The particle model space 5hω is not enough to reproduce the empirical pairing gap. Therefore, the particle model space can be used beyond 6hω in G−matrix.
In this calculation we use Nmax=10hω in G−matrix. (5hω in deformed basis)
N is the maxim major oscillator quantum number.
2nd HaPhy meeting, July

20. Gamow-Teller strength in deformed basis & expanded basis
Motivations
Formalism
Results
Summary
Gamow-Teller strength in deformed basis & expanded basis
ISR = 98.4 %
There is different at low-lying GT state.
The difference of ISR between two case is about 0.5%.
ISR = 98.5 %
2nd HaPhy meeting, July

21. Particle-hole strength gph
Motivations
Formalism
Results
Summary
Particle-hole strength gph
The energy of the GTGR
is roughly reproduced.
This is the GT strength distributions for different particle hole strengths. Gpp is fixed by 0.99.
2nd HaPhy meeting, July

22. Particle-particle strength gpp
All GT peaks get shifted to smaller energies as gpp increase.
This is the GT strength distributions for different particle hole strengths. Gpp is fixed by 0.99.
2nd HaPhy meeting, July

23. It is consistent with our calculation.
GT(-) strength for 76Ge with different β2 value
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.
For a reference,
β2= by RMF
0.262 from B(E2)
2nd HaPhy meeting, July

24. Running sum of GT(-) strength for 76Ge
Results by DQRPA reproduce well experimental data without quenching factor.
ISRexp = 55% (up to 12MeV)
There may be a possibility of the high-lying GT state above 12.0 MeV.
ISRDQRPA ≈ 98 %
2nd HaPhy meeting, July

25. GT(+) strength for 76Se with different β2 value
(2008)
(1997)
Grewe[greve]
β2= by RMF, from B(E2)
2nd HaPhy meeting, July

26. Running sum of GT(+) strength for 76Se
Results by DQRPA reproduce well experimental data without quenching factor.
2nd HaPhy meeting, July

27. GT(-) strength for 82Se with different β2 value
β 2= by RMF
0.193 from B(E2)
2nd HaPhy meeting, July

28. Running sum of GT(-) strength for 82Se
Results by DQRPA reproduce well experimental data without quenching factor.
2nd HaPhy meeting, July

29. GT(-,+) strength for 90Zr with different β2 value
β 2= by RMF
0.089 from B(E2)
Is DQRPA proper at β2=0 ???
Two uppermost panel
2nd HaPhy meeting, July

30. Does DQRPA go back to pn-QRPA at β2=0 ??
Deformed WS goes back to the spherical WS at spherical limit.
2nd HaPhy meeting, July

31. Does DQRPA go back to pn-QRPA at β2=0 ??
Deformed SPS | Ω =½ > are linear combination of the deformed basis even if we take the β2 = 0 limit.
Deformed basis I000 ½ > can be composed of different j values in the β2 = 0.3.
One may notice | Ωj = ½> > state has other components |n s½>
although main component is |0 s½>at spherical limit.
The minor states 1s½, 2s ½, ∙∙∙ should be correction terms to the pnQRPA.
2nd HaPhy meeting, July

32. Running sum of GT(-) strength for 90Zr
The running sum of the experiment show monotonous increase along with the Eex, while our theoretical results show a quantum leap around 5MeV and 12MeV.
2nd HaPhy meeting, July

33. Running sum of GT(+) strength for 90Zr
2nd HaPhy meeting, July

34. GT(-) strength for 92Zr with different β2 value
The experimental B(GT−) values are extracted from 92Zr(p,n)92Nb reaction at 26MeV.
Since the projectile energy was too low to expect high-lying excited states, most GT excitation are observed below 9 MeV.
Our theoretical calculations address a possibility of another peak around 14 MeV.
β 2= by RMF
0.103 from B(E2)
2nd HaPhy meeting, July

35. Motivations
Formalism
Results
Summary
submitted in PRC
(a) constant value (b) G-matrix as the pairing interaction.
2nd HaPhy meeting, July

36. Empirical pairing gaps for 64Ge ~76Ge
Motivations
Formalism
Results
Summary
Empirical pairing gaps for 64Ge ~76Ge
The values of p-n interaction energies δpnemp are not negligible even for large neutron excess isotopes.
The pn-pairing interaction is expecting to play a significant role in construction of the quasiparticle mean field for these nuclei.
It is supposed that the origin of this phenomenon is associated with the deformation effect, which is changing the distribution of proton and neutron SP levels.
2nd HaPhy meeting, July

37. Pairing gap for 64Ge (Z=N)
Motivations
Formalism
Results
Summary
Pairing gap for 64Ge (Z=N)
(a) constant value (b) G-matrix as the pairing interaction.
β2=0.217 (RMF) ,
In (a), below some critical value (~0.97), there are only pp & nn-pairing modes, which seems to be a result of a simple monopole pair(K=0) Hamiltonian.
Above this value, the system prefers to form only pn-pair.
In (a) pp, nn, and pn-pairs coexist in the narrow region(blue square).
In (b) the coexistence region is more wide and the phase transition becomes less sharp.
2nd HaPhy meeting, July

38. Pairing gap for 70Ge (ZǂN)
Motivations
Formalism
Results
Summary
Pairing gap for 70Ge (ZǂN)
For nucleus with Z is not equal to N,there is a different situation.
β2= (RMF) ,
There is a less sharp phase transition to the pn-pairing in comparison with 64Ge.
pn-pairing mode does exist only in coexistence with pp & nn-pairing mode.
2nd HaPhy meeting, July

39. Pairing strengths for 64Ge ~76Ge
Motivations
Formalism
Results
Summary
Pairing strengths for 64Ge ~76Ge
N=Z 64Ge seems to be special. g_pp is stable with N-Z difference.
G_nn decreases with increasing N-Z. g_pn is growing.
The T=0, Gpn (gpn) are larger in comparison with T=1, Gpp and Gnn (gpp and gnn ).
The largest differences among gpp(T=1), gnn(T=1), and gpn(T=0) forces are visible for maximal value of N−Z=12 (76Ge), which undergoes double β-decay.
Double-beta decay transitions might be affected by the pn-pairing.
2nd HaPhy meeting, July

40. Does the gpn depend on the deformation parameter β2 ?
Motivations
Formalism
Results
Summary
Does the gpn depend on the deformation parameter β2 ?
It is an open issue whether the value of pairing strength Gpn T=0 depends on the deformation of the considered isotope.
Gpn and gpn are sensitive to the change of the deformation parameter β2.
These phenomena were found irrespective of nuclear shapes and species.
2nd HaPhy meeting, July

41. ISR(Ikeda sum rule) = 3( N - Z ) =0 for 64Ge : spherical nucleus
Motivations
Formalism
Results
Summary
ISR(Ikeda sum rule) = 3( N - Z ) =0 for 64Ge : spherical nucleus
::: deformed nucleus
After consideration of pn-pairing, proton occupation probabilities become almost same as neutron occupation probabilities.
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.
2nd HaPhy meeting, July

42. Motivations
Formalism
Results
Summary
How about N ≠ Z nucleus ?
It is an open issue whether the value of pairing strength Gpn T=0 depends on the deformation of the considered isotope.
The modified smearing of Fermi surface were found with np-pairing for N ≠ Z.
2nd HaPhy meeting, July

43. 2nd HaPhy meeting, July 18 2015 Motivations Formalism Results Summary
Two upper panels are calculated without pn-pairing.
And two bottom panels are ~
After consideration of pn-pairing
2nd HaPhy meeting, July

44. preliminary preliminary GT(-) strength for 24Mg
Motivations
Formalism
Results
Summary
GT(-) strength for 24Mg
preliminary
preliminary
2nd HaPhy meeting, July

45. Motivations
Formalism
Results
Summary
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, 76,82Se, and 90,92Zr 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) pn-pairing correlations for the ground state of even-even Ge isotopes, A=64–76, within the deformed BCS approach.
4.For N=Z 64Ge pure T=0 pairing mode is found and 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.
Work on spin-M1 transition is now in the progress.
2nd HaPhy meeting, July

46. 2nd HaPhy meeting, July

47. Thanks for your attention !!
2nd HaPhy meeting, July

48. G-matrix They used the constant and as a pairing strength.
In principle, these strength have to be evaluated from the scattering of two particles in deformed mean field.
Therefore, we will exploit the Brueckner reaction matrix G, which is obtained by solving the Bethe-Goldstone equation
Black : stable nuclei
Yellow: unstable (known)
Green: unstable(unknown)
지구상에는 100여종의 원소가 존재하며 지구상에 존재하는 원자핵은 안정핵으로
시간이 지나도 절대 변하지 않으며 약 270종류가 있다. 그러나 생성된 후 자연적으로 파괴되어 다른 원자핵으로 변하는 불안정핵이 6,000~8,000개에 이른다.
이것을 불안정핵 또는 radio isotope (RI)라고 부른다. Black circle denote stable nuclides and unstable nuclides lie in the shaded regions.
최근까지 원자핵물리의 연구는 안정핵과 안정핵부근의 원자핵으로 한정되었기 때문에 불안정핵에 관한 연구는 거의 이루어지지 않았다.
최근에 불안정핵의 고속입자를 사용하여 산란이나 반응을 일으키는 새로운 방법인 RI빔법을 이용해 불안정핵에 대한 정보가 증가, 지금까지 미처 생각하지 못한 새로운 구조가 발견. RI빔법이 개발되기전까지 연구는 “ 마치 골짜기만 관찰하고 산맥전체를 이해하려는 것처럼 억지스러운 부분이 있었다
a,b,c,d : single nucleon basis states(oscillator wave functions with
s.p.e from a Woods-Saxon potential)
w : starting energy
Vab,cd : one boson exchange potential of the Bonn group
Qp : Pauli operator
H0 : harmonic oscillator Hamiltonian
2nd HaPhy meeting, July