The The ii13/2 13/2 Proton and Proton and jj15/215/2 Neutron Orbital and the Neutron Orbital and the

SD Band in A~190 SD Band in A~190 RegionRegion

Xiao-tao He Xiao-tao He En-guang ZhaoEn-guang Zhao

Institute of Theoretical Physics, CAS, Beijing, Institute of Theoretical Physics, CAS, Beijing, ChinaChina

Summer school • Peking University •2004 Summer school • Peking University •2004

Outline• Introduction• Particle-number conserving method• Numerical result• Summary

Introduction• In experiment, more than 80 SD bands have

been observed in A~190 mass region.

• For even-even nuclei, the dynamical moment of inertia (J(2)) exhibits a gradual increase with the increasing rotational frequency ћω.

A. V. Afanasjev, P. Ring, J. König, Nucl. Phys A 676

(2000) 196.. Y. Sun, J. Y. Zhang, and M. Guidry, Phys. Rev. Lett. 78

(1997) 2321; Phys. Rev. C 63 (2001)

047306.

For the odd-odd nuclei, quite a part of the moments of inertia for SD bands keep constant.

X. T. He, S. X. Liu, S. Y. Yu, E. G. Zhao HEP & NP 27 (2003) 124

X. T. He, S. Y. Yu, S. X. Liu, Y. X. liu, E. G. Zhao Chin. Phys. Lett. 21 (2004) 813

X. T. He, S. X. Liu, S. Y. Yu, J. Y. Zeng, E. G. Zhao J. Phys. G: Nucl. Part. Phys. (submited)

Some different methods:

HF method B. -Q. Chen, et al., Phys. Rev. C 46 (1992) R1582.

J. Terasaki, et al,. Phys. Rev. C 55 (1997) 1231.

RMF J. D. Walecka, Ann. Phys. 83 491 (1974)

PRM X.Q.Chen and Z Xing J.Phys.G:Nucl.Part.Phys.19(1993)1869

PSM Y. Sun, J. Y. Zhang, and M. Guidry, Phys. Rev. Lett. 78 (1997) 2321

IBM (IBFM) Y. X. Liu, et al., Phys. Rev. C 63, (2001) 054314 Phys. Rev. C 59 (1999)

2511

CSM D. R. Inglis, Phys. Rev. 95, (1954) 1059

Particle-number conserving (PNC)

methodadvantage：• Particle number is conserved • A cranked many-particles configuration

(CMPC) truncation is adopted instead of the single-particle level (SPL) truncation.

• The blocking effects is taken into account strictly and consistently.

disadvantage：• Angular momentum is not conserved

FormulaCSM Hamiltonian ： H0 ： single particle part Hp ： pairing interaction part

ppxSPCSM HHHJHH 0

ii hH ))(( 00 xNil jhh )(0

)2()0( ppp HHH

1.   Nilsson level single-particle Nilsson hamiltonian ： good quantum number ： (parity),

[N,nz,L] : is the good quantum number of H0 ,Hz ,lz and jz under the extreme large deformation

degeneracy: （ ± ）。

'22

2VV

Mh oscNil

0 anN z 0 anN z

2. Cranked Nilsson levelCranked Nilsson hamiltonian

: signatureThe eignstate of Rx() and j2z ：

diagonalize h0() in space|, we get | ：

xNil jhh )(0

xjix eR )(

)(12

1x

i Re 0 b

0C

3. Many-body systemn particles system ， the configuration |i ：

According to configuration |i, the system

possess certain energy Ei 、 parity Pi 、 signaturei and particle number N:

02121

niiiniiii

)(occupied

ii

iE

)(occupied

ii

iP

2mod)()(

occupied

ii

i

４ .paring interactionIn Nilsson single particle space ：

In | space ：

In | space ： Similar to HP(0), we can get HP(2)

0,

)0(

aaaaGH p

,)()()2( 222,1,0

2

aaaaqqGHP

bbbbGHP0,

'''

'''

ffGHP

b

aa N

21

C

bC )(

In | space ： Diagonalize HCSM in the CMPC

space, we get the solution of CSM Hamiltonian ：

Di： interger

pCSM HHH 0

i

i iD

５ . moment of inertia

The angular momentum alignment in | ：

Dynamic ：Kinematic ：

ji

xjii

xixx jJiDDiJiDJJ 22

xJJ )1(

dJd

J x)2(

Result The cranked Nilsson orbital

in A~190 mass region

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.66.5

6.6

6.7

6.8

6.9

7.0

7.1

7.2

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.65.7

5.8

5.9

6.0

6.1

6.2

6.3

6.4

6.5

114

[651]1/2

[761]1/2

[633]5/2

[512]3/2

[510]1/2

[631]3/2

[752]5/2

[514]7/2

[624]9/2

[512]5/2[505]11/2

[761]3/2

[642]3/2

[640]1/2

[770]1/2

cran

ked

neut

ron

Nils

son

leve

l (0)

/Mev

80

[651]1/2[633]7/2

[521]3/2

[402]5/2

[523]5/2

[404]7/2

[514]9/2

[642]5/2

[530]1/2

[651]3/2

[532]3/2

[411]1/2

[660]1/2

cr

anke

d pr

oton

Nils

son

leve

l (0)

/Mev

Parameters• The Nilsson parameters (κ) are taken from Lund

systematics. T.Bengtsson and I. Ragnarsson, Nucl. Phys. A436, 14

(1985)

• The deformation parameters 2=0.46 , 4=0.03. M. A. Riley et al., Nucl. Phys. A512

178 (1990) • The effective pairing interaction strengths ( G0

for monopole pairing and G2 for quadrupole pairing ) in unite of MeV are given as follow:

G0p=0.3, G2p=0.01, G0n=0.2, G2n=0.013

Experimental and calculated J(1) and J(2) of the yrast SD band in even-even

nuclei 192Hg

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.4580

90

100

110

120

130

140

150

80

90

100

110

120

130

140

150

J(2)

J(1)

(MeV)

192Hg(1)

mom

ents

of i

nerti

a (

2 MeV

-1)

Experimental and calculated J(1) and J(2) of the SD bands in odd-A

nuclei 193Tl and 195Tl

90

100

110

120

130

140

150

90

100

110

120

130

140

150

0.0 0.1 0.2 0.3 0.480

90

100

110

120

130

140

150

0.0 0.1 0.2 0.3 0.480

90

100

110

120

130

140

150

J(2)

J(1)

193Tl(1) =-1/2([642]5/2,a=-1/2)

mom

ent o

f ine

rtia

(2 M

eV-1)

J(2)

J(1)

193Tl(2) =+1/2([642]5/2,a=+1/2)

J(1)

J(2)

195Tl(2) =+1/2([642]5/2,a=+1/2)

(MeV)

J(1)

J(2)

195Tl(1) =-1/2([642]5/2,a=-1/2)

Occupation probability of each proton cranked orbital near the

Fermi surface in 193Tl

0

1

2

0

1

2

0.0 0.1 0.2 0.3 0.40

1

2

0.0 0.1 0.2 0.3 0.40

1

20

1

20

1

2193Tl(1) proton N=6

[651]1/2

[642]5/2 =-1/2

[651]3/2

proton N=5

[514]9/2

[532]3/2[530]1/2

prot

on o

ccup

atio

n pr

obab

lity

proton N=4

[404]7/2

[411]1/2proton N=4

[404]7/2

[411]1/2

(MeV)

proton N=5

[514]9/2

[530]1/2

193Tl(2) proton N=6

[633]7/2

[642]5/2 =+1/2

[651]3/2

The separate contributions to J(2) from each cranked proton orbitals

near the Fermi surface

-20

0

20

40

60

0.0 0.1 0.2 0.3 0.4

-20

0

20

40

60

-20

0

20

40

60

0.0 0.1 0.2 0.3 0.4

-20

0

20

40

60[633] 7/2 [642] 5/2

193Tl(1) Proton

[651] 1/2 [642] 5/2[651] 1/2

[642] 5/2

[651] 3/2

[660] 1/2

Sep

arat

e co

ntrib

utio

ns to

J(2

) (2 M

eV-1)

[532] 3/2

[523] 7/2

[541] 1/2

[532] 5/2

[541] 3/2

[550] 1/2

[651] 1/2 [651] 3/2[642] 5/2 [651] 3/2

[633] 7/2 [642] 5/2

193Tl(2) Proton

[642] 5/2

[651] 3/2

[660] 1/2

(MeV)

[532] 3/2

[523] 7/2

[541] 1/2

[532] 5/2

[541] 3/2

[550] 1/2

Experimental and calculated J(1) and J(2) of the six SD bands in odd-

odd nuclei 194Tl

90

100

110

120

130

140

150

90

100

110

120

130

140

150

0.0 0.1 0.2 0.3 0.490

100

110

120

130

140

150

0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.490

100

110

120

130

140

150

J(2)

J(1)

194Tl(1a) =0([642]5/2+ [512]5/2-)

J(2)

J(1)

194Tl(2a) =0([642]5/2- [624]9/2+)

J(2)

J(1)

194Tl(3a) =0([642]5/2- [512]5/2=+)

J(2)

J(1)

194Tl(1b) =1([642]5/2+ [512]5/2+)

mom

ent o

f ine

rtia

(2 M

eV-1)

J(2)

J(1)

194Tl(2b) =1([642]5/2- [624]9/2-)

(MeV)

J(2)

J(1)

194Tl(3b) =1([642]5/2- [512]5/2=-)

Occupation probability of each neutron cranked orbital near the

Fermi surface in 194Tl

0.0 0.1 0.2 0.3 0.40

1

2

0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.40

1

20

1

2

0

1

20

1

2

0

1

2

N=5

(MeV)

[514]7/2

[512]5/2

[505]11/2

N=5

[514]7/2

[512]5/2

[505]11/2

N=5

[514]7/2

[512]5/2

[505]11/2

N=6

[642]3/2

[624]9/2

occu

patio

n pa

obab

ility

n

N=6

[624]9/2

[642]3/2

N=6

[642]3/2

[624]9/2

N=7

[752]5/2

[761]3/2

N=7

[752]5/2

[761]3/2

N=7

[752]5/2

[761]3/2

Experimental and calculated J(1) and J(2) of the four SD bands in

odd-odd nuclei 192Tl

90

100

110

120

130

140

150

90

100

110

120

130

140

150

0.0 0.1 0.2 0.3 0.490

100

110

120

130

140

150

0.0 0.1 0.2 0.3 0.490

100

110

120

130

140

150

J(1)

J(2)

192Tl(a) [642]-5/2[761]-3/2

J(1)

J(2)

192Tl(b) [642]+5/2[761]-3/2

J(1)

J(2)

192Tl(c) [642]5/2[512]5/2

mom

ent o

f ine

rtia

(2 M

eV-1)

J(1)

J(2)

192Tl(d) [642]5/2[512]-5/2

/MeV

Occupation probability of each neutron cranked orbital near the

Fermi surface in 192Tl

0.0 0.1 0.2 0.3 0.40

1

2

0

1

20

1

2

0

1

20

1

2

0

1

2

[512] 5/2 [514] 7/2[512] 5/2

[505]11/2

/MeV

[640] 1/2[642] 3/2

[624] 9/2

occu

patio

n pa

obab

ility

n

[770] 1/2[761] 3/2

[752] 5/2

Summary• The PNC method in the frame of CSM is

one of the useful method to deal with the SD nuclei.

• The i13/2 Proton and j15/2 Neutron intruder Orbital plays a very important role in the variation in both kinematic and dynamical moments of inertia (MOI) with rotational frequency in A~190 Region.

Thank you !