Myaing Thi Win, K.Hagino and T.Koike

Tohoku University

Shape of hypernuclei in() deformation plane

University of Aizu-JUSTIPEN-EFES Symposium,“Cutting-Edge Physics of Unstable Nuclei

13th November 2010

Background Motivation Formalism and Method Results Summary & Conclusion

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Outline of this talk

1

Shrinkage Effect in 7Li

Theoretically 　 by 3-body cluster model

Hiyama et al. PRC 59 (1999) 2351

Experimentally by B(E2) valueK.Tanida,H.Tamura, D.Abe et al ucl.PhysA,684,560(2001) 　　

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Impurity effect of particle

Λ

Smaller size or

smaller deformation

• No Pauli principle between nucleons and lambda particle 　　　　　　　　　　　　　“Glue-Like　Role　of　” an impurity to explore the various changes of the nuclear

structure

NNHe5

Self-consistent Mean Field studies on hypernucleiCore polarization effect , such as

Change of total energy and radius of the core– Non-relativistic HF method c.f , D.E.Lanskoy PRC58,3351(98)

c.f, M.Rayet, Annals of Phys102,226(76)

– Relativistic Mean Field c.f, J.Mares and J.Zofka, Z.Phys.A333,209(89) c.f Y. Sugahara and H.Toki , Prog.Theo.Phys.92,803(94)

Extension of Neutron drip line – Relativistic HFB c.f, D.Vretenar et al, PRC57,R1060(98)

– Non-relativistic MF c.f, X.R.Zhou et al, PRC78, 054306(08)the presence of

strange -particle

makes the nucleus more stable extends the drip lines　makes the nucleus shrink

My interest

within Mean-field approach

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Deformation?

Theoretical and experimental interestChange　of　properties　of　nuclei　induced　by　(size,　shape　and　cluster　structure,　changes　of　collective　motion,　etc...)

Nuclear deformation• Shape of nuclei important role in determining nuclear

properties, such as quadrupole moment and radius.• Evidence for nuclear deformation

– Rotational BandsSchematic level schemes of deformed nuclei

Triaxial deformation

L.M.Robledo,R.Rodriguez-Guzman and P.SarrigurenJ.Phys.G.Nucl.Part.Phys.36(2009)

6

Previous Studies of Deformation of hypernuclei

Deformed Skyrme-Hartree Fock approach No change in deformation due to addition of lambda

X.R.Zhou et.al, PRC76,034312

Relativistic Mean Field Theory Disappearance of nuclear ｄ eformation induced by a

M.T.Win and K.Hagino, PRC78, 054311(08)6

SHF and RMF H.J.Schulze, M.Thi Win, K.Hagino,H.Sagawa Prog.Theo.Phys123,3(10)

Physical origin of deformation changes in terms of lambda and nucleon mean field potentials

compare the results of SHF and RMF•RMF stronger interaction of lambda with Nuclear core•Disappearance of deformation only in the case of

)SHF(MeV1|)0(E)(EE 0

Motivation• The previous studies of the deformation of Lambda

hypernuclei Axial symmetryDisapperance or change of deformation minimum

Extend the previous studies by taking

into account triaxial degree of freedom Skyrme Hartree Fock +BCS approachTo study the deformation of

hypernuclei in the full () plane arXiv:1010.5561v1

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Triaxial deformation?7

Skyrme HF Formalism for Hypernuclei M.Rayet, Nucl.Phys.A367,381(81)

Total Hypernuclear energy

22

pnN

i

2NiN

i

2NiN

,

,

0rderdeE 232i

3A

1iiH

,i

n,pq,e)r(U)r(V)r(m2

. qqqNq

q*

2

e)r(U

)r(m2. N

2

HF equations for hypernuclear system

)()( 33 rHrdrHrdE NH

N2141

2

*

2

)tt(m2m2

)tt()tt(

)tt(m2m2

2141

q1281

N2141

q

2

*q

2

)J.J.(W21

)2(4t).(

4tt3

)(4

)tt()x1(tm2

H

NN0

pn2N

3N

21

NN21

N021

0

2

03210 W,t,t,t,x

Skyrme parameters for -N interaction

)i.(WJ.W)2(t

)tt3()tt()x1(tU

N021

N021

pn2N34

1

N2

2141

N2141

N021

0

Cqqq VWUV

)i.(WJ.W))(22(t

)tt3()tt()x1(tU

021

021

qNN341

2214

1214

102

10N

n,pq,e)r(U)r(V)r(m2

. qqqNq

q*

2

e)r(U

)r(m2. N

2

HF equations for hypernuclear system

arising from N interaction

03210 W,t,t,t,x

Skyrme parameters for -N interaction

Parameter sets of Skyrme type N Interaction • five parameters : t0

, x0, t1

,t2, t3

,

a1

= t1+t2/4• Fitted to reproduce

• Small spin-orbit part is not considered.

)H(Hex

)Pb(B),O(Bt

),He(Bt

440

209171

50

Y.Yamamoto et.al, Prog.Theo.Phys.80,5(88)

Using these parameter sets B and level spacings （ sp, sd ) as a function of A-

2/3 are calculated •the results are within experimental error bar

)kx).(rr(kiWk)rr(kt

)k)rr()rr(k(t21

)rr()Px1(t)rr(V

N0N2

2NN

21

N00NN

)rr()rr(t)r,r,r(V 21321NN

Calculation methodExtend ‘EV8 Code’ for hypernuclei HF+BCS equations solved by discretizing the single-particle wave function on 3-D

Cartesian mesh Imaginary Time step method

Pairing BCS approximation with Density dependent pairing interaction

)rr(4.1))r(1(V)r,r(V 21

0021

0 = 0.16 fm-3

V0 = -1000MeV fm3

)1i(k

)i()i(k

~)ht1(~

P.Bonche et.al. Comp.Phys.comu.171(05)

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Application to sd-shell hypernuclei

Skyrme Interaction SGII

• Deformation energy surface of sd shell hypernuclei in () plane

Si29 Mg25

Mg27Si27

Results: 28Si , 28Si+

142814Si

Oblate magic number

Results: 24Mg, 24Mg+

122412Mg

prolate

Results: 26Mg, 26Mg+

Eprolate – Eoblate = 0.39MeV

142612Mg

prolate obate

Results: 26Si , 26Si+

122614Si

oblate prolate

Eprolate – Eoblate = 0.12MeV

),(E),(EMgMg 2626

),(E),(ESiSi 2828

),(E),(EMgMg 2424

),(E),(ESiSi 2626

Gain in Binding energy of in spherical configuration

prefers spherical configuration even if the core is deformed.

Discussion on softer energy surface for 26+Mg and 24+Mg

Due to larger overlap between and nucleon densities at the prolate side

Due to the effect of smaller value of at oblate side

Overlap between nucleon and lambda densities

• particle density

• Nucleon density

22 b/r3 e)b(

1)r(

]a/)),(Rrexp[(1),,r( 0

N

2

0N

1

1

2

0

2 ),,r()r(ddrrdrrOverlap

))],(Y),(Y(sin21

)(Ycos).[,(R),(R

2222

200

Overlapl argest for prolate configuration

• vibrational energy• 24+Mg, 26+Mg

0+

2+

4+

22+

24Mg

32+

24Mg = 4.23MeV(expt)

22+

0+2+

4+

24+Mg

+

Calculated value

MeV10.4Mg24

24MgK=0 g.s rotational band and K=2 rotational band built upon the -vibrational state

Mg

Mg

Mg

Mg

Mg24

22000

24

24

24

24

24

E

E

,DD

D21),(E),(E

MgMg2424 97.0

• Similarly, vibration energy for 26+Mg

Summary Triaxial calculations by Skyrme Hartree-Fock method PES in () deformation plane

– sd shell hypernuclei28 Si+p=14, n=14), 24Mg +p=12, n=12)26Mg +p=12, n=14), 26Si +p=14, n=12)

- similar between core and hypernuclei - Slightly softer along the triaxial degree of freedom Qualitatively similar results with SGII and SIII parameter sets Estimation of vibration energy

– vibration energy is lowered by about 0.15 MeV with the addition of hyperon