Study of deformed two-body weakly bound nuclei in the strong-coupling and application to reactions

Study of deformed two-body weakly bound nuclei in the strong-coupling

and application to reactions

Pedro Punta, José Antonio Lay and Antonio M. Moro

Departamento de Física Atómica Molecular y Nuclear Universidad de Sevilla

Junta de Andalucía

Introduction

• Light exotic nuclei

• Two-body model

• Nilsson Model

• THO diagonalization

• Transfer reactions

2

Nilsson Model

• Deformed potential

• Single particle Hamiltonian

• and are not conserved

• and are constant of the motion

⃗ l ⃗ j

π Ω

3

1

2 3

⃗

r′

θ′ ^neutron

core

H

= − ℏ

2 μ ∇

+ V

( r ) + V

( r )( l ⃗ ⋅ ⃗ s ) − rβ dV

( r )

dr Y

( θ ′ ) V ( r ′ ⃗ ) ≈ V

( r ) − rβ dV

( r )

dr Y

( θ ′ )

1 2

3

⃗j Ω

Nilsson Diagram for 17 C

4

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

-8 -7 -6 -5 -4 -3 -2 -1 0

(MeV)

Niveles monoparticulares

[2s_1/2, =1/2] [1d_5/2, =1/2] [1d_5/2, =3/2] [1d_5/2, =5/2] [1p_1/2, =1/2]

[8]

IKUKO HAMAMOTO PHYSICAL REVIEW C 76, 054319 (2007)

some particular nuclei, both bound and resonant one-neutron levels are calculated as a function of quadrupole deformation.

The change of nuclear shell structure for neutrons is seen in both negative and positive one-particle energies of the Nilsson diagrams. The change comes from the unique behavior of neutron orbits with small ! values, in particular ! = 0 and 1.

The modified shell structure has direct relevance to the ground and low-lying states of neutron-drip-line nuclei, in which weakly bound neutrons are present. Considering the possible absence of many-body pair-field in light nuclei, the study of the present type of Nilsson diagrams can definitely help us to understand the origin of possible deformation and the related spectroscopic properties of light neutron-drip-line nuclei.

In Sec. II some points of our model are summarized.

Numerical results are presented in Sec. III. Conclusions and discussions are given in Sec. IV.

II. MODEL

The occupancy of weakly bound one-particle levels has a contribution especially to the tail of the self-consistent potentials. However, even for light nuclei presently considered the number of weakly bound neutron(s) is much smaller than that of well-bound core nucleons. In other words, the major part of the nuclear potential is provided by well-bound nucleons.

Thus, for simplicity, the parameters of Woods-Saxon potentials are taken from the standard ones [11] for stable nuclei except for the depth, V_WS. Namely, the diffuseness, the strength of spin-orbit potentials, and the radius parameter are taken from those on p. 239 of Ref. [11]. The depth is adjusted so that a particular one-neutron level obtains a given desirable binding-energy in respective examples.

The way in which bound one-particle levels are calculated is described in Ref. [6], while the eigenphase formalism that is used to estimate one-particle resonant levels for a deformed potential is given in Refs. [7,8]. The essential point is that the coupled equations obtained from the Schr¨odinger equation are solved in coordinate space with the correct asymptotic behavior of wave functions forr → ∞. The solution obtained in this way is totally independent of the upper limit of radial integration, R_max, if both the potential and the coupling term are already negligible at r = R_max. One-particle resonant energy for β #= 0 is defined as the energy at which one of the eigenphases increases through π/2 as energy increases. In the limit ofβ → 0 this definition in the eigenphase formalism is in agreement with the definition of one-particle resonance in spherical potentials described in textbooks [12]; the phase shift increases through π/2 as energy increases.

One-particle resonance is not obtained if none of the calculated eigenphases do not increase through π/2 as energy increases. For example, we have no corresponding resonance in the case where a calculated eigenphase starts to decrease before reaching π/2 as energy increases. Even if one fails to obtain one-particle resonance defined in terms of eigenphase, for a certain small region of energy just after the disappearance of resonance the concentration of the wave functions inside the potential may still be found. However, the concentration will easily disappear after a short time if a resonance is no longer obtained in the eigenphase formalism. This situation is

analogous to the case of the spherical potential, in which the phase shift starts to decrease before reaching π/2 as energy increases [12].

Compared with the Nilsson diagram based on modified oscillator potentials, the striking difference of the level scheme obtained in the present work comes from the behavior of levels with low ! values (in particular, ! = 0 and 1) for β = 0 and those with small $ values (mainly $^π = 1/2⁺,1/2⁻, and 3/2⁻) for β #= 0, in both the weakly bound and positive- energy regions. Note that the minimum ! value of possible components of $^π = 1/2⁺,1/2⁻, and 3/2⁻ levels is equal to 0, 1 and 1, respectively. The absence of the centrifugal barrier for the ! = 0 channel produces the unique behavior of weakly bound and positive-energy $^π = 1/2⁺ orbits. However, we find that some $^π = 1/2⁺ resonant levels survive in a higher- energy region (see, for example, the [200 1/2] level in Fig. 1) if the relative probability of the s_1/2 component inside the potential is smaller than a certain critical value [8]. Because the height of the centrifugal barrier becomes lower for a larger nuclear radius, the unique behavior of ! = 1 components will be more easily seen in nuclei with larger mass.

III. NUMERICAL RESULTS A. Neutron-rich C isotopes

Taking V_WS = −40.0 MeV and the radius parameter for A = 17, at β = 0 in Fig. 1 we obtain ε(1d_5/2) = −560 keV

8 16

Neutron one-particle levels in Woods-Saxon potential

VWS = − 40.0 MeV R = 3.266 fm (A = 17) a = 0.67 fm

[211 1/2]

[202 5/2]

[211 3/2]

[220 1/2]

[101 1/2]

[200 1/2]

[330 1/2]

[202 3/2]

1/2 3/2

1/2 5/2

-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 -8

-6 -4 -2 0 2 4 6 8 10

quadrupole-deformation parameter β εΩ (MeV)

FIG. 1. Neutron one-particle levels as a function of quadrupole deformation. Parameters of the Woods-Saxon potential are designed approximately for the nucleus ¹⁷C. One-particle levels are denoted by the asymptotic quantum numbers [N n_z&$]. The $ values are denoted for four positive-parity levels for β < 0, because it may be difficult to see the connection to the levels for β > 0. One-particle levels appearing at β = 0 are 1p_1/2,1d_5/2,2s_1/2, and 1d_3/2 levels at

−6.77, −0.56, −0.42, and +5.60 MeV, respectively. One-particle levels in the positive-energy region, of which the phase shift (one of the eigenphases) for β = 0 (β #= 0) does not increase through π/2 as energy increases, are not plotted. The neutron numbers 8 and 16, which are obtained by filling in all lower-lying levels, are indicated with circles.

054319-2

I. Hamamoto, PRC76 (2007) 054319

• Transformed Harmonic Oscillator (THO)

• Local Scale Transformation (m=4)

Full Hamiltonian and Basis

• Collective Hamiltonian

5

ϕ

( r ) = ds

dr ϕ

[ s ( r )]

s ( r ) = [ r

+ ( γ r )

m

]

−_m¹ 1

2

3

x

y z

M

−Ω

= ⃗ +

J j⃗ I⃗

⃗I

⃗j

H = H

+ ℏ

2𝒥 I

⃗

Full Hamiltonian Basis

S. Karataglidis et al. PRC71 (2005) 064601

6

Experimental data:

J. H. Kelley et al., NPA880 (2012) 88

-0.8 -0.6 -0.4 -0.2 0 0.2

E (MeV)

3/2⁺ 1/2⁺ 5/2⁺

Experimental

3/2⁺ 1/2⁺ 5/2⁺

Nilsson

3/2⁺ 1/2⁺ 5/2⁺

PAMD

16C+n

PAMD:

J. A. Lay et al. PRC89 (2014) 014333

Level Schemes

Experimental data:

Z. Elekes et al., PLB614 (2005) 1743

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5

E (MeV)

1/2⁺ 1/2^- 5/2⁺ 3/2⁺

Experimental

1/2⁺ 1/2^- 5/2⁺ 3/2⁺

Nilsson-2V

1/2⁺ 1/2^- 5/2⁺ 3/2⁺

Nilsson-1V

1/2⁺ 1/2^- 5/2⁺ 3/2⁺

PAMD

10Be+n

17 C 11 Be

Ground State of 17 C

7

0 5 10 15 20 25

r(fm)

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

u(r) (fm-1/2 )

s_1/2

d_3/2, =1/2 d_5/2, =1/2 d_3/2, =3/2 d_5/2, =3/2

0 5 10 15 20 25

r(fm)

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3

u(r) (fm-1/2 )

Nilsson PAMD d_3/2, 0⁺ s_1/2, 2⁺ d_3/2, 2⁺ d_5/2, 2⁺

E = − 0,73 MeV u

( r ) = 2 ∑

Ω

(− 1)

⟨ I 0 | j − Ω J Ω⟩ u

( r )

Excited States of 17 C

8

0 5 10 15 20 25

r(fm)

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

u(r) (fm-1/2 )

s_1/2 d_3/2 d_5/2

0 5 10 15 20 25

r(fm)

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

u(r) (fm-1/2 )

s_1/2

d_3/2, =1/2 d_5/2, =1/2 d_3/2, =3/2 d_5/2, =3/2 d_5/2, =5/2

17

C

(1/2

), E = − 0,46 MeV

C

(5/2

), E = − 0,44 MeV

Bound States of 11 Be

9

11

Be

(1/2

), E = − 0,50 MeV

Be

(1/2

), E = − 0,18 MeV

0 5 10 15 20 25

r(fm)

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

u(r) (fm-1/2 )

Nilsson-2V Nilsson-1V PAMD s_1/2, 0⁺ d_3/2, 2⁺ d_5/2, 2⁺

0 5 10 15 20 25

r(fm)

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

u(r) (fm-1/2 )

Nilsson-2V Nilsson-1V PAMD p_1/2, 0⁺ p_3/2, 2⁺ f_5/2, 2⁺

Neutron Transfer Reactions

10

( dσ

d Ω ) βα = μ α μ β

(2 π ℏ 2 ) 2 ∫ χ p (−)* ( k p ⃗ , R ′ ⃗ ) ψ *

C ( r ′ ⃗ )Δ V β χ d (+) ( k d ⃗ , R ) ⃗ ψ d ( r ⃗ ) d rd ⃗ R ⃗

2 d-

C Adiabatic Potential

p-

C Optical potential

Our Model p-n potential

n

p

16C

17C

⃗

r′

⃗R′

n p

16C d

⃗

r R⃗

16 C(d,p) 17 C

Post-Form ADWA

Neutron Transfer Reactions

11

( dσ

d Ω ) βα = μ α μ β

(2 π ℏ 2 ) 2 ∫ χ d (−)* ( k d ⃗ , R ′ ⃗ ) ψ d * ( r ′ ⃗ )Δ V α χ p (+) ( k p ⃗ , R ) ⃗ ψ

Be ( r ⃗ ) d rd ⃗ R ⃗

2

d-

Be Adiabatic Potential p-

Be Optical potential

Our Model

p-n potential

11 Be(p,d) 10 Be

p n

10Be

11Be

⃗

r

⃗R n

10Be d p ^r′⃗

⃗R′

Prior-Form ADWA

Experimental Data

12

11 Be(p,d) 10 Be

16 C(d,p) 17 C

• X. Pereira-López et al., PLB811 (2020) 135939

• GANIL (France)

• 17,2 MeV/nucleon beam

• Y. Jiang et al., Chin. Phys. Lett.

35 (2018) 082501

• RCNP (Osaka)

• 26,9 MeV/nucleon beam

13

16 C(d,p) 17 C

0 5 10 15 20 25 30 35 40

CM (deg.)

10^-2 10^-1 10⁰ 10¹

d/d (mb/sr)

16C(d,p)¹⁷C(1/2⁺)

Nilsson PAMD

Exp. (GANIL) [9]

E = 17.2 MeV/u E = 17.2 MeV/u

0 5 10 15 20 25 30 35 40

CM (deg.)

10^-1 10⁰ 10¹ 10²

d/d (mb/sr)

16C(d,p)¹⁷C(5/2⁺)

Nilsson PAMD

Exp. (GANIL) [9]

14

16 C(d,p) 17 C

E = 17.2 MeV/u

Sum for 3 bound states

0 5 10 15 20 25 30 35 40

CM (deg.)

10⁰ 10¹

d/d (mb/sr)

16C(d,p)¹⁷C

Nilsson PAMD

Exp. (GANIL) [9]

15

11 Be(p,d) 10 Be

E = 26.9MeV/u E = 26.9 MeV/u

0 5 10 15 20 25 30

CM (deg.)

10^-2 10^-1 10⁰ 10¹

d/d (mb/sr)

11Be(p,d)¹⁰Be(gs)

Nilsson-2V Nilsson-1V PAMD

Exp. (RCNP) [13]

0 5 10 15 20 25 30

CM (deg.)

10⁰ 10¹

d/d (mb/sr)

11Be(p,d)¹⁰Be(2⁺)

Nilsson-2V Nilsson-1V PAMD

Exp. (RCNP) [13]

Conclusions and Perspectives

• We have explored the suitability of the Nilsson model to study the structure of deformed weakly bound nuclei.

• Preliminary calculations have been performed for

C and

Be, using a phenomenological Hamiltonian with adjustable

parameters.

• Calculated overlap functions have been applied to describe (d,p) reactions, showing encouraging agreement with existing data.

• Extensions to other nuclei in the island of inversion (e.g.

Ne) and observables (e.g. breakup) in progress.

16

Thanks for your

attention