Theoretical Study on Rotational Bands and Shape Coexistence of $^{183,185,187}${Tl} in the

a r X i v :n u c l -t h /0602037v 1 11 F e

b 2006

Theoretical Study on Rotational Bands and Shape

Coexistence of 183,185,187Tl in the Particle

Triaxial-Rotor Model

Guo-Jie Chen 1,?,Yu-xin Liu 2,3,4,5,?,Hui-chao Song 6,and Hui Cao 1

1School of Science,Foshan University,Foshan 528000,China

2

Department of Physics,Peking University,Beijing 100871,China

3

The Key Laboratory of Heavy Ion Physics,Ministry of Education,Beijing 100871,China

4

Institute of Theoretical Physics,Academia Sinica,Beijing 100080,China

5

Center of Theoretical Nuclear Physics,National Laboratory of Heavy Ion Accelerator,

Lanzhou 730000,China

6

Department of Physics,Ohio State University,Columbus,OH43210,USA

February 9,2008

Abstract

By taking the particle triaxial-rotor model with variable moment of inertia,we investigate the energy spectra,the deformations and the single particle con?gura-tions of the nuclei 183,185,187Tl

systemically.The calculated energy spectra agree with experimental data quite well.The obtained results indicate that the aligned

bands observed in

183,185,187Tl

originate from the [530]1

2

?,[660]1

2

?isomeric states in

183,185,187Tl

are formed by a

proton with the [505]9

2

+isomeric state in 187Tl is generated

by a proton with con?guration [606]13

?

Email:chengj126@https://www.sodocs.net/doc/dd13015388.html, ?

Corresponding author,Email address:liuyx@https://www.sodocs.net/doc/dd13015388.html,

1Introduction

It has been known that nuclei in the Z=82region are rich in shape coexistence.In particular,the important deformation driving orbitals has been assigned as the h9/2and i13/2proton shells[1,2].In odd-mass Tl isotopes(with Z=81),one-particle–two-hole(1p-2h)intruder states and shape coexistence have been discovered through the observation of low-lying9/2?isomeric states[3].The structure of these isomeric states was con?rmed to be decided by the odd proton occupying the h9/2intruder orbital[4,5].Later,the rotational bands associated with both oblate(πh9/2,πi13/2)and prolate(πh9/2,πi13/2,πf7/2)structures have been observed in lighter isotopes185,187Tl[6].The band-head of the 1p-2h oblateπh9/2intruder band has been observed to lie lowest in energy near N=108. In contrast,the band-head of the prolate intruder band based on the i13/2structure has been predicted to decrease continuously in excitation energy as the neutron number decreases beyond the neutron mid-shell.This prolate structure is presumably formed by coupling the odd i13/2proton to the prolate Hg core with4p-6h structure[6].Recently,a rotational-like yrast cascade was established in183Tl and assigned to associate with the prolate i13/2structure[7].Furthermore the band-head energy of its yrast band was later determined[8].

Besides the coexistence of prolate and oblate shapes mentioned above,the signature splitting observed in the[505]9/2?band in187Tl which is signi?cantly larger than that observed in its heavier odd-mass isotopes with A≥191suggests that there may exist triaxial deformation[1,6]and the discrepancy between the calculated equilibrium energy and the experimental data of the band-head energy of the[606]13

2?(h

9/2

)state in185,187Tl can be distinguished from the[530]1

triaxial-rotor model with variable moment of inertia of the core to analyze the structure and deformation of the energy bands in183,185,187Tl systemically and to identify their microscopic con?guration.

The paper is organized as follows.After this introduction,we describe brie?y the formalism of the particle triaxial-rotor model in Section II.In Section III,we describe our calculation and obtained results.In Section IV,we give a summary and brief remark.

2Particle Triaxial-Rotor Model

In the particle rotor model,the Hamiltonian of an odd-A nucleus is usually written as [13,14,15]

?H=?H

core

+?H s.p.+?H pair.(1) In the case of triaxial deformation,the Hamiltonian of the even-even core is given as

?H core =

3

i=1ˉh2R2i2?i,(2)

where R,I and j are the angular momentum of the core,the nucleus and the single par-ticle,respectively.The three rotational moments of inertia are assumed to be connected by a relation of hydrodynamical type

?κ=43κ),(3) with

?0(I)=?0

2m?2+1

√

?H

is the Hamiltonian to represent the pairing correlation which can be treated in pair

the Bardeen-Cooper-Schrie?er(BCS)formalism.

The single-particle wavefunction can be expressed as

|ν = Nlj?C(ν)Nlj?|Nlj? ,(6)

whereνis the sequence number of the single-particle orbitals,|Nlj? represents the corresponding Nilsson state,C(ν)Nlj?is the coe?cient to identify the con?guration mixing. Diagonalizing the single-particle Hamiltonian in the basis|Nlj? ,we can obtain the C(ν)Nlj?and the single-particle eigenvalueεν.The corresponding quasi-particle energy can then be determined by Eν=

2I+1

A.To improve the agreement between calculated results and experimental data,we introduce a Coriolis attenuation factorξand take value as that giving the best agreement between the calculated and experimental energy spectra.We found that,when ξ=0.95,the calculated results agree best with the experimental data of183,185,187Tl.In general principle,in order to describe the nuclear property more accurately and to make better agreement between calculated and experimental data,it is necessary to involve su?cient single-particle orbitals near the Fermi surface in the calculation.Then we take13 orbitals near the Fermi surface to couple with the core for183Tl,185Tl,187Tl,respectively. Practical calculation shows that the Fermi levels of the bands6(we denote the band labels here as the same as those for nucleus187Tl in Ref.[6],so that the similar bands can be compared)of the nuclei lie between the20th and the21st single particle orbitals,and

the others lie between the19th and the20th.For the deformation parametersβandγof 185,187Tl,we take those given in Ref.[6]as the trial initial values to?t.For the deformation parameters of183Tl,since there does not exist any report to discuss them,we take the values of its neighbor nucleus185Tl[6]as the trial initial ones.Then we accomplished a series diagonalization of the total Hamiltonian with various values ofβandγto make the calculation errorχ2=1

2 ,23.0%|5p3/212 con?gurations.Since the largest component is|5f7/21

2?(πf

7/2

).Meanwhile,from Table4,

we can see that the band2consists of mixing of about93%20th and7%19th orbitals. Seen from Table3,the20th orbital contains84.3%of|5h9/23

2

?con?guration.Similarly,combining Table4with Table3,we can recognize that the band5,6originates mainly from the 23rd,21st single particle orbital,respectively,the21st orbital contains87.4%of|6i13/21

Table 1:The deformation parameters and the main components of the single-particle levels |ν near the Fermi surface in terms of the Nilsson levels of the bands in 183

Tl (the

initial values of the deformation parameters are taken as those of

185

Tl in Ref.[6].)

β

γband

?tted

?tted

value

value

|19 0.856|5h 11/21

2

?0.182|5f 7/21

|20 0.986|5h 9/292 band 3?0.168

15?

2 +0.601|5f 7/2

72 ([505]9

|22 0.819|5h 9/25

2

?0.210|5f 5/2

5

|23 0.733|5h 9/23

2 +0.378|5h 9/2

5

2

2

+0.217|4g 7/25

2

band 6

0.270

2 +0.545|6g 9/2

1

2

+)

2 +0.511|6g 9/23

|23 0.902|4d 3/2

3

2 +0.216|5d 5/2

3

initial

initial

ν wave function in terms of |Nlj ?

value

value

2

2

?0.420|5f 5/21

2 band 20.2470

2 +0.335|5f 5/2

3

2

([532]3

|21 0.997|5h 11/211|22 0.764|5f 7/21

2 +0.321|5h 9/21

2 +0.449|5f 5/21

2

2 +0.121|5h 9/2

5

?0.1620|21

0.770|5h 9/292 ?0.182|5h 9/2

3

2

?)

2 +0.394|5h 9/21

2

2 ?0.392|5h 9/21

2

|19

0.998|4g 9/27

|20 0.935|4d 5/252 +0.151|4g 9/25

0.267

|21 0.941|6i 13/212 ([660]1

|22

0.952|6i 13/232

2 +0.238|4s 1/21

2

Table 3:The deformation parameters and the main components of the single-particle levels |ν near the Fermi surface in terms of the Nilsson levels of the bands in 187

Tl (the

initial values of the deformation parameters are taken from Ref.[6].)

β

γband

?tted

?tted

value

value

|18 0.997|5h 11/29|19 0.849|5h 9/212

?0.169|5f 7/21

0.250

|20 0.909|5h 9/23

2

+0.191|5f 7/23

2

?)

2

2

+0.480|5p 3/21

2

|18 0.997|5h 11/29|19 0.868|5h 9/212

?0.162|5f 7/21

0.234

|20

0.916|5h 9/23

2

+0.189|5f 7/23

2

?)

2

2

+0.469|5p 3/21

2

|19 0.695|5h 11/212 ?

0.360|5f 7/21

|20

0.986|5h 9/29

2 band 3?0.162

15?

2 +0.348|5f 7/2

72 ([505]9

|22 0.825|5h 9/2

5

2 ?0.201|5f 5/2

5

|23 0.741|5h 9/23

2

?0.384|5h 9/25

2 2

band 5?0.19211.3

?

2 ([606]13

|26 0.931|6i 13/2

7

2 2 +0.471|6i 13/2

1

2

|19 0.995|4g 7/27

|20 0.967|4d 5/25

2

+0.131|4g 9/25

0.267

|21 0.935|6i 13/212

([660]1

|22 0.946|6i 13/23

2 2 +0.245|4s 1/2

1

2

Table 4:The theoretically predicted main components of the wavefunctions of the bands 1,2,3,5and 6in

187

Tl in terms of the single-particle levels

band

2??0.981|2212 2??0.981|2212

band 12?

?0.980|221

2

([530]1

272 +0.194|19

1

312

?0.196|191

352

?0.197|191

2??0.964|203

2

2?0.955|2032 band 22?

0.947|203

2

([532]3

292

+0.343|191332 +0.362|191

372

+0.378|191

2?0.815|209

2

+0.231|235

11

2

+0.518|21

7

2

([505]9132

+0.479|217

2

2??0.584|209

2

?0.388|235band 5

2+0.720|23132 +0.350|259

2

+)

2

+

0.580|2313

2

+0.450|259

172 ?0.390|275

2

17

2

?0.298|223212

?0.326|223

25

2

+0.347|223

2

+)

2+0.930|2112

2+?0.924|211

2

2+0.919|2112 2+

0.915|211

2

1

2

3

4

5

6

7

187

Tl

ban d6

[660]1/2

+

cal

exp 49/2

+

45/2

+

41/2

+

37/2

+

33/2

+

29/2

+

25/2

+

21/2

+

17/2

+

ban d5

[606]13/2+

cal

exp 17/2

+

15/2

+

13/2

+

ban d3

[505]9/2

-exp cal

15/2

-13/2

-11/2

-9/2

ban d2

[532]3/2

-cal

exp 37/2

33/2

29/2

-

25/221/2

17/2ban d1[530]1/2-cal exp 35/2

-

31/2

-27/2

-23/2-19/2

-15/2

-185

Tl

ban d6[660]1/2

+

cal

exp 45/2

+41/2

+

37/2

+

33/2+29/2

+25/2

+

21/2

+

17/2

+

ban d3

[505]9/2

cal

exp

11/2

-

9/2-ban d2

[532]3/2

-

cal exp 25/2

-

21/2

-17/2

-

13/2

-

9/2-

183

T l

ban d6[660]1/2

+exp cal

cal

exp

33/2

+

29/2

+25/2+

21/2

+

17/2

+13/2

+

ban d3[505]9/2-11/2

-9/2-

E n e r g y (M e V )

Figure 1:Comparison of calculated energy levels of the rotational bands in 183,185,187

Tl

with the experimental data (taken from Refs.[6,8]).

con?guration,and the 23rd consists of almost purely the |6i 13/213

2

+,π[660]1

2

?(f 7/2)band involves about 10%h 9/2con?guration

and the [532]3

N

j (E cal.

j

?E exp.j )2of the energy spectrum of band 3

in

187

Tl (where N is the number of levels in the band)with respect to the value of γat

several β’s in the upper panel of Fig.2.We also display the variation of the calculated energy spectrum against the value of γat the best ?tted β(?0.162)and the comparison

-1

1

2

3

4 ( M e V

)

0510********

10

10

10

10

10

10

10

E n e r g y (M e V )

Figure 2:Upper panel:parameter γat several axial deformation Lower panel:variation

of the calculated energy β=?0.162against

the value of γand data are taken

from Ref.[6].

with experimental data of the band in the lower panel of Fig.2.The upper panel of Fig.2shows that the variation of the axial deformation parameter β(except for that with angular deformation parameter γin special region)does not a?ect the calculation error χ2so drastically as that of the γhttps://www.sodocs.net/doc/dd13015388.html,bining the upper panel and the lower panel of Fig.2,one can notice clearly that,for zero γ,the calculation error χ2is quite large (about 60)and the calculated level sequence is not consistent with experiments.As the γincreases to 3-5degrees,the calculated level sequence becomes consistent with the experimental one and the χ2decreases to about 10?1.For the value of γin the region 3to 14degrees,the calculation error χ2maintains around 10?1.When γ=15?,the χ2with β=?0.162becomes suddenly the minimum (～10?4)of the χ2(β,γ)and

0.0

0.5

1.0

1.5

( M e V

)

0510********

10

10

10

10

10

E n e r g y (M e V )

Figure 3:Upper panel:parameter γat several axial deformation Lower panel:variation

of the calculated energy β=?0.168against

the value of γand data are taken

from Ref.[8].

the calculated energy well.As γincreases from 15degrees further,the 10?1.Moreover,in the case of β=?0.162,even though the calculated energies of the states with lower angular momentum do not deviate from experimental data obviously,the ones with higher angular momentum do drastically.It is then evident that,when the deformation parameters (β,γ)=(?0.162,15?),the calculated energy spectrum agrees best with experimental data.It indicates that the band 3of

187

Tl is in triaxial oblate deformation.In addition,from

Table 4,we notice that the band 3in 187Tl originates mainly form the 20th single particle orbital.As can be seen from inspecting Table 3,the 20th orbital contains 97.2%of |5h 9/29

2

?(πh 9/2)coupled to a triaxial oblate deformed core.It provides then

( M e V

)

0510********

10

10

10

10

1010

E n e r g y (M e V )

Figure 4:Upper panel:parameter γat several axial deformation Lower panel:variation

of the calculated energy β=?0.164against

the value of γand data are taken

from Ref.[6].

(

M e V

2

)

0510********

10

10

10

10

10

10

E n e r g y (M e V )

Figure 5:Upper panel:parameter γat several axial deformation Lower panel:variation

of the calculated energy β=?0.192against

the value of γand data are taken

from Ref.[6].

a corroboration of the conjecture in Ref.[6].Similar results for the bands3in183,185Tl are obtained,too(the calculation errorsχ2of the energy separations against the value ofγat severalβ’s and the comparison of the calculated energy spectrum withγ∈(0?,29?)and β=?0.168(?0.164)with experimental data are illustrated in Fig.3(4)for183Tl(185Tl) ).The deformation parameters can then be?xed as(?0.168,15?),(?0.164,15?)for the band3of183Tl,185Tl,respectively.These results con?rm the assumption that the band originated from orbital[505]9

2

+shown in Fig.5,one can recognize that the band5([606]13

2

?and [606]13

2?,[532]3

2

+proton con?gurations coupled to a prolate

deformed core.Meanwhile,the negative parity bands built upon the9

2

?con?guration coupled to a core with triaxial oblate deformation(β,γ)=(?0.168,15?),(?0.164,15?),(?0.162,15?),re-

spectively,and the positive parity band on the13

+coupled to a triaxial oblate core with deforma-

2

tion parameters(β,γ)=(?0.192,11.3?).In short,the nuclei183,185,187Tl involve quite rich shape coexistence.Meanwhile our present calculation provides a clue that the triax-ial deformation may arise from the mixing of single particle Nilsson con?gurations.To understand it much better,more investigations are required.

This work was supported partially by the Natural Science Foundation of Guangdong Province with contract No.04011642,partially by the Natural Science Research Foun-dation of the Education Department of Guangdong Province with contract No.Z02069, and partially by the National Natural Science Foundation of China with contract Nos. 10425521,10135030and10075002.One of the authors(YXL)thanks also the support by the Major State Basic Research Development Program under Grant No.G2000077400,the Key Grant Project of Chinese Ministry of Education(CMOE)under contract No.305001, the Foundation for University Key Teacher by the CMOE and the Research Fund for the Doctoral Programme of Higher Education of China with grant No.20040001010.

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