Consistent analysis of nuclear level structures

and nucleon interaction data of Sn iso-topes

J.Y. Lee1*, E. Sh. Soukhovitskii2,Y. D. Kim1, R. Capote3, S. Chiba4, and J. M. Quesada5

1Dep. of Physics, Sejong University, Korea

2Joint Institute for Energy and Nuclear Research, Belarus3Nuclear Data Section, IAEA, Austria

4Advanced Science Research Center, JAEA, Japan5Universidad de Sevilla, Spain

* yeon@sejong.ac.kr

Why Study Tin Isotopes ?

A main component of nuclear reactor material. A candidate material for superconducting

magnets in fusion reactors. Energy splittings of yrast 0+, 2+, 4+ and 6+

levels are irregular. ⇒ may suggest non-harmonic vibrational

states? Sn isotopes are single-closed-shell nuclei of

Z=50.determine whether the calculations using a

self-consistent CC optical model may produce different nuclear deformations for different ex-ternal probes (protons, neutrons) for Sn iso-topes.

Present soft-rotator model

- Lee et al., PRC 79, 064612 (‘09) - Soukhovitskii et al., PRC 72, 024604 (‘05) - Capote et al., PRC 72, 064610 (‘05)  - Soukhovitskii et al., J. Physics G 30, 905(‘04)

Non-axial quadrupole, octupole, hexade-capole deformations

γ-vibrations Soft-octupole and rigid hexadecapole de-

formations Identify positive and negative parity bands,

associated with octupole surface vibrations

Calculations

i) Nuclear Hamiltonian parameters to reproduce experimental collective levels (determined by fitting the calculated levels to

the evaluated nuclear structure data )ii) Contruct wave functions from these parame-

ters.iii) CC optical Model calculations ⇒ “ Self-consistent ! ”

Present soft-rotator model⇒ Quite successful in explaining

– Nuclear collective level structures, – Nucleon interaction cross sections,– Proton non-elastic scattering cross sec-

tions,– γ -transition probabilities,

for 12C, 28Si, 56Fe, 58Ni, & 238U.

- Lee et al.,, PRC 79, 064612 (‘09) - Soukhovitskii et al, PRC 72, 024604 (‘05) - Capote et al, PRC 72, 064610 (‘05)  - Soukhovitskii et al., J. Physics G 30, 905(‘04) - Soukhovitskii et al., J. Nucl. Sci. Tech. 40, 69 (‘03), - Soukhovitskii et al., PRC 62, 044605(‘00), NPA 624, 305

(‘98).

Goals

Consistent description of collective nuclear level structures & nucleon scattering proper-

ties for 116,118,120Sn using the soft-rotator model.

50

Description of soft-rotator model

ASSUME : An excited state of even-even non-spherical nucleus can be described as a combination of rotation, β-quadrupole and octupole vibrations, & γ-quadrupole vibra-

tion.

Multipole-deformed instant nuclear shape

Deformations

4,3,2 ,)','(1)','(

θYRθR

:

Hamiltonian of the soft-rotator model

where,

)()()(ˆ2

ˆ2

ˆ1ˆ2

ˆ322

2

420

3

22

222

2

32

VVVTB

TTTB

H r

.ˆˆ

ˆ,1ˆ

,3sin3sin

1ˆ,1ˆ

3

1)4()3()2(

23

1

2

3

33

333

2

42

242

3

2

i iii

i

i i

ir JJJ

I

J

ITT

TT

inertia of moments Principal:ˆ

axisth in operator momentumangular theof Projection:

energy rotationalnuclear deformed ofOperator :ˆ

i

i

J

i-I

T

Description of soft-rotator model

ASSUME : An excited state of even-even non-spherical nucleus can be described as a combination of rotation, β-quadrupole and octupole vibrations, & γ-quadrupole vibra-

tion.

Multipole-deformed instant nuclear shape

Deformations

4,3,2 ,)','(1)','(

θYRθR

:

(Review)

Deformed nuclear potential :

max

1

00 )','(

!

),(),())','(,(

0t

tt

RR

t

t

Yt

RrR

R

VRrVRrV

)','( YASSUME : is small.

))','(,()ˆˆ)(,(

))','(,()()()(

))','(,()()()(

))','(,()()(

CoulCoulSO

SWSSCoulSS

VWSVCoulVV

RWSCoulRHF

RrVlErV

RrgEiWEVEV

RrfEiWEVEV

RrfEVEV

(Non-spherical) Dispersive Optical Potential

(“Lane consistent dispersive CC OMP”)deal with (p,n) charge exchange reactions [to the elastic Isobaric Analogue States(IAS)]

)(4

)()(

)(4

)()()(

)(4

)()()(

1

10

10

EVA

ZNEV

EVA

ZNEVEV

EVA

ZNEVEV

pn

FFnnn

FCFCppp

Isospin-dependent dispersive CC OMP

Lane equations Soukhovitskii et al, PRC 72, 024604 (‘05) Capote et al, PRC 72, 064610 (‘05) 

Applications to Sn isotopes

For 120Sn(32.59%), 118Sn (24.22%), 116Sn (14.54%),

Collective nuclear level structures Total neutron & proton reaction cross sections Nucleon elastic & inelastic scattering cross

sections [(n,n), (n,n’), (p,p), (p,p’)] Quasi-elastic (p,n) reactions.

Collective level structures of 120Sn & 118Sn ⇒ All the levels are involved in CC calcula-tions.

EXP.

CALCULATIONS

CALCULATIONS

EXP.

(i) K≈0,nβ=nγ=0 (g.s. rotational band) (ii) K≈0,nβ=1,nγ=0

(iii) K≈2,nβ=0,nγ=0 (positive parity band) (iv) K≈0,nβ=0,nγ=0 (negative parity band) (v)

K≈0,nβ=0,nγ=1

120Sn total neutron & proton reaction cross sec-tions

Neutron elastic scattering cross sec-tions

116Sn(n,n) 118Sn(n,n) 120Sn(n,n)

Neutron inelastic scattering cross sec-tions

116Sn(n,n’)2+ 118Sn(n,n‘)2+ 120Sn(n,n’)2+

Neutron inelastic scattering cross sec-tions

116Sn(n,n’)3- 118Sn(n,n‘)3- 120Sn(n,n’)3-

Proton elastic scattering cross sections

116Sn(p,p) 118Sn(p,p) 120Sn(p,p)

Proton inelastic scattering cross sections

116Sn(p,p’)2+ 118Sn(p,p‘)2+ 120Sn(p,p’)2+

Proton inelastic scattering off 3- state

116Sn(p,p’)3- 118Sn(p,p‘)3- 120Sn(p,p’)3- •

Quasi-elastic (p,n) reactions

116Sn(p,n) 118Sn(p,n) 120Sn(p,n)

Deformation Parameters

Isotopeβ20 β30 β40

n p n p

116Sn 0.025 0.027 0.050 0.053 -0.080

118Sn 0.029 0.035 0.047 0.049 -0.060

120Sn 0.034 0.037 0.043 0.045 -0.094

Summary

For 116Sn, 118Sn, 120Sn, Collective level structures Total neutron cross sections Nucleon elastic/inelastic scattering cross sec-

tions Quasi-elastic (p,n) reactions.

⇒ well described within the soft-rotator model self-consistently. [ χ2 : 6.882(116Sn), 8.369(118Sn), 6.74(120Sn) ]