alpha-6he scattering using mcas and the quest of ... · of microscopic guidance for orthogonalizing...

Alpha-6He scattering using MCAS and the questof microscopic guidance for orthogonalizing

pseudopotentials

L. Canton

Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Italia

L. Canton Alpha-6He scattering using MCAS and the quest of microscopic guidance for orthogonalizing pseudopotentials

COLLABORATION

Yuliya A. Lashko BITP, Kyiv, Ukraine

MCAS Collaboration withKen Amos, Dirk van der Knijff, University of MelbournePaul R. Fraser, Curtin UniversitySteven Karataglidis University of JohannesburgJuris P. Svenne University of Manitoba

14th Varenna NRM Int Conf 2015 and the quest ...

INTRODUCTION

The existing radioactive ion beam (RIB) facilities, and theconstruction of new ones, allow the formation and investigation ofnew species, particularly at or beyond the drip lines (continuumregime).

The goal of the MCAS collaboration is to investigate exotic andnon-exotic systems via a coupled-channel method, in thelow-energy domain (scattering, bound states, resonances).


Treating scattering on light-medium nuclei

COUPLED-CHANNEL dynamics: including the collectivelow-energy excitations of the core.

C13 (n-C12), N13 (p-C12), C15 (n-C14), F15 (p-O14), He7, B7,Be7, Li7, Be9, B9, C17 (n-C16), Na-17 (p-Ne16) C-19 (n-C18),9

ΛBe,13

ΛC,current ... Ne23 (n-Ne22), Mn23, Na23 (p-Ne22), Al23, O17(n-O16), (p-F17), O19 ... O16 (alpha-C12), Be10 (alpha-He6)


Model of nuclear interaction

Current description: nucleon-nucleus scattering (light-mediumnuclei with 0+ g.s.) including first core excitations of collectivenature (quadrupole, octupole, etc).

Vcc ′(r) =∑

n=C ,LS ,LL,SI

Vn < (`s)jI ; Jπ|Onfn(r ,R, θr,R)|(`′s)j ′I ′; Jπ >

For all operators, the functional forms are expanded to secondorder in the core-deformation parameter (R = R0(1 + β2P2(θ))

fn(r ,R, θ) = f(0)n (r)− β2R0P2(θ)

d

drf

(0)n (r)

+β2

2R20

2√π

(P0 −

2√

5

7P2(θ) +

2

7P4(θ)

)d2

dr2f

(0)n (r)


MCAS: A low-energy scattering tool

Sturmians are used as basis for Finite-Rank expansion of realisticCC interaction

Vcc ′(p, q) ∼ VNcc ′(p, q) =

N∑n=1

χcn(p) η−1n χc ′n(q) .

Optimal representations of χcn(p) in terms of Sturmians |Φcn〉∑c ′

G 0c Vcc ′ |Φc ′n〉 = −ηn|Φcn〉

with χcn(p) defined as |χcn〉 = Vcc ′ |Φc ′n〉.The S-matrix can be rewritten also as

Scc ′(E ) = δcc ′ − iπk∑i

χci (E(+); k)

1

1− ηi (E (+))χc ′i (E

(+); k)

(3)


First application n − C12/p − C12 aborted: Why?

Bound states13C four observed → 12 computed

13N one observed → 8 computed

The deep forbidden states contaminate the physical solution due toCoupled-Channel dynamics. Problems in CC formalisms (but notonly)...


The solution: Elimination of these forbidden states in the definitionof the Hamiltonian.

The OPP technique highly nonlocal. ( Kukulin et al. ’74)

(OCM) constrained hamiltonian. (Saito ’69)

(susyQM ) mainly local potentials. (Witten ’81/ Baye ’87)

C.V.Sukumar, D.M. Brink (2004) examined connections withinverse scattering approaches (J. Phys. A 37).

The OPP approach (Kukulin, Pomerantsev et al.) eliminates thedeep bound states introducing a new term in the nuclear potential.


The full nuclear potential Vcc(r) is not the local potential n−C12:The “complete” potential is (in partial-wave decomposition)

Vcc ′(r , r ′) = Vcc ′(r)δ(r − r ′)

+δcc ′λcAc(r)Ac(r ′)(δc=s 1

2

+) + δcc ′λcAc(r)Ac(r ′)(δc=p 3

2

−)

Ac(r) are the Pauli-forbidden deep (CC-uncoupled) bound states.

A state in the OPP approach is:forbidden in the limit λ→ +∞allowed when λ→ 0


12C

-5

0

5

n + 12

C

13C

exp

0+

2+

0+

1-

1+

3-

5+

5+

7+3

+3+

5-

9+

1+

5+

13C

th

Ex (

MeV

)

0 5


BUT THERE IS MORE, in many applications we had to loosen theOPP constraints

[Eric.Schmid, 1978] “The states can be Pauli-forbidden,Pauli-allowed, or Pauli-suppressed” Trieste, IAEA Few-BodyConference.

[Langanke-Friedrich, 1986] “The role of partially redundantstates” Adv in Nucl. Phys.

RGM theory:

Forbidden - Eigenvalue in the Norm Kernel e = 0Allowed - Eigenvalue in the Norm Kernel e = 1Quasi-Rendundant - Eigenvalue in the Norm Kernel 0 < e < 1

OPP Approach:

Forbidden - Strength of λ =∞Allowed - Strength of λ = 0Hindered - Strength of 0 < λ <∞


BUT THERE IS MORE, in many applications we had to loosen theOPP constraints

[Eric.Schmid, 1978] “The states can be Pauli-forbidden,Pauli-allowed, or Pauli-suppressed” Trieste, IAEA Few-BodyConference.

[Langanke-Friedrich, 1986] “The role of partially redundantstates” Adv in Nucl. Phys.

RGM theory:

Forbidden - Eigenvalue in the Norm Kernel e = 0Allowed - Eigenvalue in the Norm Kernel e = 1Quasi-Rendundant - Eigenvalue in the Norm Kernel 0 < e < 1Super-allowed - Eigenvalue in the Norm Kernel e > 1

OPP Approach:

Forbidden - Strength of λ =∞Allowed - Strength of λ = 0Hindered - Strength of 0 < λ <∞- Super-allowed - Strength of λ < 0


α-He6 scattering/cross-sections

Table : The states of 6He used in the coupled-channel evaluations Allenergies are in units of MeV.

state Centroid Width

0+g.s. 0.000 0.002+

1 1.797 0.1132+

2 5.60 10.0

The CC α-6He interaction

Vcc′(r) =

[V0f (r) + V``f (r)[` · `] + VII f (r)[I · I] + V`Ig(r)[` · I]

]cc′

+ Vmonoδc′cδI0+g.sf (r),


α-6He parameters

Table : The potential parameters used for the interactions in the α+6Hesystem. All strengths are in MeV and lengths are in fermi.

Pot. strengths Negative Positive GeometryV0 -41.4 -41.4 R0 = Rc 2.58V`` 1.5 1.0 a0 = ac 0.7V`I -0.3 -1.5 β2 0.7VII 1.7 1.6

Vmono -4.5

Table : The states of 6He used in the coupled-channel evaluations. TheOPP bocking (ie, λ’s) was tentatively assigned for s and p waves. Allenergies are in units of MeV.

state/OPP (λ1s) (λ2s) (λ1p)0+g.s. 106 0.0 10.22+

1 106 0.0 10.02+

2 106 0.0 8.0


α-6He parameters

Table : The potential parameters used for the interactions in the α+6Hesystem. All strengths are in MeV and lengths are in fermi.

Pot. strengths Negative Positive GeometryV0 -41.4 -41.4 R0 = Rc 2.58V`` 1.5 1.0 a0 = ac 0.7V`I -0.3 -1.5 β2 0.7VII 1.7 1.6

Vmono 0.0

Table : The states of 6He used in the coupled-channel evaluations. TheOPP bocking (ie, λ’s) was tentatively assigned for s and p waves. Allenergies are in units of MeV.

state/OPP (λ1s) (λ2s) (λ1p)0+g.s. 106 -1.0 10.22+

1 106 0.0 10.02+

2 106 0.0 8.0


α-He6 (10Be) spectrum

0

2

4

6

8

10

Ex(M

eV)

0+

2+

2+ 1

_0+ 2

_

3_2

+

(4_)

2+

3_

exp. MCAS

0+

2+

0+

2+

1+

2+

3+

4+

1_

2_

1_

A

B

(α+6He)

1_

2_

3_

1_

2_

2+

exp.

(α+6He)

MCAS

3_

Figure : The low-excitation states in 10Be. Positive and negative paritystates are shown separately. Arrows ‘A’ and ‘B’ indicate the n+9Be andthe α+6He thresholds respectively.



40 60 80 100 120 140θ

c.m.(deg)

100

101

102

dσ

/dΩ

(m

b/s

r)

FullC+0+1C+0+1+2C+0+1+2+3

Coulomb

C+0

Figure : Cross section measured at 3.8 MeV (c.m.) as partial waves areadded to the evaluations.



100

101

102

E = 2.7 MeV E = 3.3 MeV

100

101

102

dσ

/dΩ

(m

b/s

r)

E = 3.8 MeV E = 4.2 MeV

40 60 80 100 120 140

100

101

102

E = 4.7 MeV

60 80 100 120 140

θc.m.

(deg)

E = 5.1 MeV

Figure : Exp data D. Suzuki et al., Phys. Rev. C 87, 054301 (2013). The solid curves are the results when amonopole interaction is used, the dashed ones when 2s-orbit enhancement in OPP is applied


α-Be8 (12C ) spectrum

Figure : The low-excitation states in 12C. PRELIMINARY


α-C12 (16O) spectrum

Figure : The low-excitation states in 16O. PRELIMINARY


α-O16 (20Ne) spectrum

Figure : The low-excitation states in 20Ne. PRELIMINARY


Microscopic guidance...

Within a microscopic two-cluster model, the Schrodinger equation in thediscrete representation is reduced to a set of linear equations forexpansion coefficients of wave functions of discrete states with the energyEκ = −κ2/2 < 0, and of continuum states with the energy E > 0 :

Ψκ (E)(r) =∑n

Cκ (E)n Ψn(r),

∑n

< n|H|n >√ΛnΛn

Cn − ECn = 0.

To understand the results of action of the antisymmetrization operator,first, let us discuss a set of the algebraic equations where only theoperator of the kinetic energy of the relative motion of clusters (in thec.o.m. frame) is retained:

∑n

< n|T |n >√ΛnΛn

Cn − ECn = 0.


in the simplest case, the collision of two- (0s)shell clusters in the statewith angular momentum `, the equation for T can be written in the formof the finite-difference equations

− 1

2

(1 +

Λn−2

Λn

)(n +

3

2− (2l + 1)2

8n

)+ 1− Λn−2

Λn

×

× 1

4(Cn+2 − 2Cn + Cn−2)− 1

4(Cn+2 − Cn−2)×

× 1

2

1 +

Λn−2

Λn+

(1− Λn−2

Λn

)(n +

3

2− (2l + 1)2

8n

)+

+

(1 +

Λn−2

Λn

)(2l + 1)2

32n+

1

4

(1− Λn−2

Λn

)(n +

1

2

)Cn =

=mr2

0

~2ECn.

The term in red is the m.e. of operator effective cluster-clusterinteraction originated by Pauli exclusion in the kinetic energy.(Attractive if Λn − Λn−2 < 0, repulsive in other case)see Lashko-Filippov Phys Part Nucl. 2005, Lashko-Filippov-LC Ukr. J. Phys. 2015


Table : Eigenvalues Λ(λ,µ) of the norm kernel of 6He+α

States with n = 2k States with n = 2k + 1

k (n + 2, 0) (n, 1) (n − 2, 2) (n + 2, 0) (n, 1) (n − 2, 2)

0 0 0 0 0 0 01 0 0 0 0 0 02 0 0 1.2056 0 1.0549 0.45213 0.9419 0.2721 1.1587 0.1831 1.2192 0.76504 1.2922 0.5698 1.0834 0.4045 1.1795 0.90115 1.3264 0.7645 1.0408 0.5983 1.1160 0.9581. . . . . . .10 1.0367 0.9921 1.0009 0.9718 1.0051 0.9994

Two SU(3)-branches of positive parity and one SU(3)-branch of negativeparity show eigenvalues which exceed unity. Hence, the states belongingto (2k+2,0), (2k-2,2) and (2k+1,1) SU(3) branches can be consideredsuperallowed states, while states (2k,1), (2k+3,0) and (2k-1,2) are partlyforbidden. Effective attraction in the former case and effective repulsionin the latter case.


α-He6 estimate of effective interaction generated by Pauliprinciple in the Kinetic Operator

Figure : Rough estimate of the Pauli potential by analyzing the Paulieffective term (in red)


α-He6 phase shifts for elastic scattering in Lπ=0+

,Phase shifts generated only by forbidden states (left), Phase shiftsgenerated by all states, including quasiforbidden and superallowed(right). Evidence of overall attraction in Lπ=0+ channel.


CONCLUSIONS

MCAS has been extended to calculate α scattering andα-nucleus interaction from Coupled-Channel dynamics.

Handling of Pauli principle via the OPP in CC dynamics is afundamental requisite.

Our results in microscopic AVRGM could provide guidance forchoosing sign and value of the strength of orthogonalizingpseudopotential to simulate the effect of partly forbidden andsuperallowed states on scattering of light and medium-lightnuclei.


alpha-6he scattering using mcas and the quest of ... · of microscopic guidance for orthogonalizing...

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