Microscopic projected-quasiparticle description of odd-mass Pb isotopes

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 619 (1997) 129-142

Microscopic projected-quasiparticle description of odd-mass Pb isotopes C.A.P. C e n e v i v a a, L. L o s a n o a, N. T e r u y a a, H. D i a s b a Departamento de Fisica, Universidade Federal da Paratl~a, C.P. 5008, 58051-970 Joao Pessoa, PB, Brazil h Instituto de Ffsica, Universidade de Sao Paulo, C.P 66318, 05315-970 Sdo Paulo, S.P., Brazil Received 10 April 1995; revised 18 February 1997

Abstract

The structure of low-lying levels in odd-mass Pb isotopes (193 ~< A ~< 205) is investigated in terms of a number-projected one- and three-quasiparticle BCS approximation. A surface delta interaction is taken as the residual nucleon-nucleon interaction. Excitation energies, spectroscopic factors, magnetic dipole and electric quadrupole moments are calculated and compared with the experimental data. (~) 1997 Published by Elsevier Science B.V.

1. Introduction

Theoretical studies of the odd-mass Pb isotopes show that level schemes of these nuclei can be described by pure neutron-quasiparticle excitations [ 1-5]. In previous theoretical calculations within a quasiparticle space, the influence of the three-quasiparticle degrees of freedom was considered within the framework of the BCS approximation [6] and also with the quasiparticle multistep shell-model method (QMSM) [7]. The BCS approximation was employed in the description of some individual isotopes without particle number conservation [ 1-4]. In Ref. [1], the low-lying spectrum (up to 1.5 MeV) of 2°sPb was studied to examine the accuracy of this approximation and the adequacy of the SDI as a residual nucleon-nucleon force. The structure of high spin states in the isotopes with 193 ~< A ~< 199 was investigated by Richel et al. [2] and in 197'199pb nuclei by Pautrat et al. [3,4], respectively, with the same approach and residual interaction. The QMSM calculation was restricted to yrast ~ + ~< j~r ~< .~+ and 21 -

j,rr

.~ -

states. The light isotopes 193,195pb, low spin states, and the available data on electromagnetic properties were not examined in Ref. [ 5]. Recent experimental 2

~

~

0375-9474/97/$17.00 (~) 1997 Published by Elsevier Science B.V. All rights reserved. PII S 0 3 7 5 - 9 4 7 4 ( 9 7 ) 0 0 0 5 9 - 6

130

C.A.P Ceneviva et al./Nuclear Physics A 619 (1997) 129-142

studies of odd-mass Pb isotopes provide new data on spectra, particularly for low-lying states, and electromagnetic moments are summarized in Refs. [ 8-15 ]. The main goal of this paper is to give a unified description of the structure of low and high spin states in odd-mass Pb isotopes (193 ~< A ~< 205). The energy spectra, spectroscopic factors, and electromagnetic moments are analysed within the standard BCS approximation in a number-projected one- and three-quasiparticle space ( l q p + 3 q p ) in which the projection is performed after the minimization of the ground state energy (PBCS). The method in which the projection is performed before the minimization (FBCS) was discussed by Mang et al. [16] and by Hara et al. [17]. In comparing the PBCS and FBCS solutions with the exact ones, it was verified that in the intermediate and high pairing force strength (G) regions these two methods approach the exact results. The differences appear in the low-G limit (G < 0.1 MeV), which has a physical significance for studying the rotational spectra of nuclei. The surface delta interaction (SDI) is used for the residual two-body force. The formalism of the PBCS approach is developed in Section 2. The results are discussed in Section 3 and conclusions are drawn in Section 4.

2. Formalism

A detailed description of the PBCS approximation is given in an earlier paper [ 18] and in references given therein. Here only the main formulas are sketched. In order to perform a number projection in the usual BCS approach within the subspace of l q p + 3qp [6], based on the generating function technique [ 19,20], a z-dependent canonical transformation, which defines the quasiparticle operators d t and d, is introduced d~ = o"a

UaC~-- ZVoC

,

dta = o"a1/2 (UaCta __ ZUoC.~) ,

(la) ( lb )

with =

+ z2v )

,

(2)

where c~ (c,~) are particle creation (destruction) operators and a - (ja, ma) and ~ = ( - ) J , ' - m ° ( j ~ , m ~ ) , and ( u ~ , v , ) are the BCS occupation amplitudes. It is easy to see that [eta, de] = ~$a~,

(3)

d~10; z) = 0,

(4)

and

where

I0;Z) = H (Ua o~>0

+ ZUaCtaC~)I0),

(5)

C.A.P Ceneviva et al./Nuclear Physics A 619 (1997) 129-142

131

and 10) represents the particle vacuum. By means of the inverse transformation

Cta --=Ora1/2 (uad t + ZVad-~) ,

(6a)

l/2(uada+zvadt)

(6b)

C a ~ O"a

the shell-model Hamiltonian,

c*.c c,c,,

i

a

(7)

afly6

and the one-body operator,

Ta.

=

~-~ c~c ~ ,

(8)

a~ are rewritten in terms of d (d t) operators, The generating wave functions [20] are rewritten as

Ice; z) = OJa/2Z d~t[0; z ) .

(9a)

for the l qp state and I[ (ab)J1, c] JM; z) = (O'aO'bO'c)J/2 x [Z 3 E I

(J, MljcmclJM)

dtrDIlM l (ab)

Mime

- z2 H ( abJl¢; J) dtjg] IO; Z ) ,

(9b)

for the 3qp state. Here D*J1MI

(ab)-

[OJlM,(ab)]'

= E

(jamajbmblJ1M') dtd~'

(10)

ma mb

H ( abJic; J) = 8j~OSabt~cj~t (UaUaZ2 -- UaUa) --~l,~-I t~j~(ab; c J) (UcVcZ2 - UcUc) ,

(11)

with ~t - (2ja + l) 1/2 and

8jl ( ab, cd) = t~ac~bd -- ( -- )Jl+J"+Jbt~ad t~bc.

(12)

Using the residue theorem

Z" = 2qriS,.l .

(13)

and the anticommutation relation (3) for 2p + 1 particles, the matrix elements of the operators (7) and (8) are easily calculated [21] by means of the generating function properties

C.A.P. Ceneviva et aL/Nuclear Physics A 619 (1997) 129-142

132

= ~

<,P(cr; z)10l~p(/3; z)},

(13a)

(~02p+l (or)1(31~2p+1 ( abJlc; JM) > = 2~i

(¢,(a; z)lOIO(abJlc; JM; z)),

(13b)

(¢2p+~(abJ~c; JM) 10]q'2p+l (deJ~f; J'M')) 1 ~ dz i . t i = 27ri ~ z~p+5(¢ (abJlc;JM;z)IOl~O (deJlf, J M ; z ) ) ,

(13c)

where the contour of integration encircles the origin but not the other points where the transformation (1) becomes singular. In the numerical calculation the integrals

Ik(pq'''t) = ~

d z z - k - I ( ° P ° ' q ' " ° ' t ) H ( u2a+ Z2V2a)

(14)

a>0

are evaluated by means of the finite sums [ 21 ]

L k ( p q . . . t ) = ~1

1+ 2

2 2 Re (Zm)-k(O-pO'q...O',) H ( u2 + ZmUa

~ m=l

'

a>0

15) where

Zm = exp( iTrm/M) ,

16)

and M is a large odd integer number. The eigenvalue problem is solved in the basis composed of the [ ~ 2 p + l ( J M ) ) and [O21,+l (abJ1 c; JM)) states, from which the corresponding eigenfunction for the ith state with total angular momentum J reads

qbi(jM) = X](J)ltp2p+l(JM) ) + ~_~ xi3 (abJlc; J)1~2p+l (abJ, c; JM)),

(17)

abJlc

where X1 and X3 are lqp and 3qp amplitudes, respectively. In the PBCS approximation the lqp and 3qp states with seniority one (J1 = 0 in relation (9b)) are not orthogonal. This is performed by the traditional Schmidt method [22] before the diagonalization of the Hamiltonian. Finally, the matrix elements of electric quadrupole E2 and magnetic dipole M1 operators are expressed in the forms

= A epff ,

(18)


(19)

where %eff is the effective particle charge, and gt and g]ff are, respectively, the orbital and effective spin gyromagnetic ratios. The quantities A, B and C are calculated from the model wave functions (14). Again, the explicit formulas can be found in Ref. [21].

C.A.P. Ceneviva et al./Nuclear Physics A 619 (1997) 129-142

133

3. Results and discussion

3.1. Energy spectra We describe the states of odd-mass Pb isotopes ( Z = 82), with 111 ~< N ~< 123, assuming 2°8pb to be an inert core. The low-lying levels are assumed to come from 3 to 15 neutron holes distributed over the single-particle orbitals 2pl/2, 2p3/2, 1f5/2, I f7/2, 0h9/2, and 0i~3/2, represented by lqp and 3qp states without energy cut-off. The single-particle energies were extracted from experimental results for 2°7Pb [ 23] ), with the values 0.0, 0.90, 0.57, 2.32, 3.41, and 1.63 MeV, respectively. A surface-delta interaction is used as the residual nucleon-nucleon interaction VSDI = -47rG6 ( r i - R ) 6 R)t~(f2i]), with strength G = 0.123 MeV for 2°sPb and 2°3pb nuclei, and G = 0.110 MeV for the lighter isotopes. These values of G are in agreement with the estimate (G ~ 23/A) of Kisslinger and Sorensen [24]. In Fig. 1, we present the experimental and calculated energy spectra for the yrast J~" states in Pb isotopes. The wave functions are listed in Tables 1 and 2. As seen, the experimental schemes of the levels are well reproduced by the calculation for the heavier isotones (2°5'2°3'2°1pb). In the cases of 199'197pb, the agreements between measured and calculated spectra are reasonably good. The largest deviations appear for positive-parity high spin states. For the nuclei 195'193pb the experimental data are more scarce and the agreements are poor. These results can be understood by the systematics of excitation energy of proton two-particle-two-hole (~r(2p-2h)) intruder states in even Pb isotones. The excitation energy of the 0 + intruder states go from 5.23 MeV in 2°8pb to 0.658 MeV in 19°pb [25-28]. This intruder state becomes the lowest excited state in 194pb. Starting from heavier Pb isotopes the trend of the 0 + , 2 + , 4 + ( r r ( 2 p - 2 h ) ) intruder states is to enter the low-energy spectra when approaching 198pb [30]. For the light isotones with mass below A = 196 there is a strong mixing between neutron-quasiparticle states and protoncore excitations (7"r(2p-2h)) [7,30]. This mixing influences the energy and properties of different low-lying states in the 195'193pb isotopes. However, this analysis is out of our model range. Thus, the low-energy spectra of the odd-mass Pb isotones are expected to be more adequately described by means of lqp and 3qp states in proportion to (the increasing) number of neutrons. The full measured and calculated level schemes with excitation energy below 2.0 MeV for 2°5'2°3'2°1pb nuclei are displayed in Fig. 2. Below, we present a short discussion for these three isotopes separately. (a) 2°spb. The theoretical spectrum reproduces the experimental sequence of levels up to about 1.0 MeV, predicting the ½- and 23-- spin assignments for the experimental levels at 0.803 and 0.998 MeV, respectively. The experimental density of levels in the region 1-2 MeV is well reproduced by the present calculation and reinforces the ~L-- and ~ + spin assignments for the measured levels at 1.759 and 1.842 MeV, respectively, as well as predicting a ~ + state at about 1.8 MeV.

(rj-

--

:'

21'

__~13"

---7-

EXP.

0 . 0 - ~

/I-

----3-

1,0 --m

--/3"

'- ---9-'~3'

----~- --''-

~

--/~

--

---11- ----11-

17+ ---9 +

___(331+

\(~-)m~'-

---c.-~s)-

=~',, -"~"

__~.-~'~'

33:

197Pb

~13-

--~Is-

1

---'13+ ----13+

/

---(9)- __/1t

~ /15--~11 + --~"17÷ ~15' "~'(~-+

)-

---13"

)

~-~ 3-+

2t ---9~,11+--11__/9" --'~7-

----t3 ~ ----t3+

---t"

----~ I.i. )t . '' ----17+ 9--/9+ --''11+

:~,,- ~

_-,~

---t231-__//~!2'I

--/9 ~,0./t3 +

----21--~,~-~:

/ ~

~+

~:

---(21-)~2'1:___ ~_~2~_ ~//~_25"___(27)-__~ ~ ! v, .,,,

~-

=:~" =_:~: __/,~:~ 24.

=~13-

t~-~

ImPb

-

----17 +

m/

~ ~ - --(23+'2"5+)-../

-- (~+29 ÷)

ImPb

--~t-~--7-

9-

---7-

----1-

~155"t9+ 11"~ --(19+21+ .d- --(t7+) ~ ;

_. = ~J~ _

~'21 ~ ~ ---['g+i~ __.(23)-

/(27)-

=-(2~*)

___(33)+

--~2~I'

i~Pb

~

----3-

----(1-)

---3- ---3-

----3---~-

----3-

----5-

~ 1 3 - __/55 ----15-1 ----13 + ----(13+) -33 ----3---(3-) =/--t3' ---: ---~---5---~- ---~--~ - "--~ -CALC. EXP. CALC. EXP. CALC. EXP. CALC. EXP. CALC. EXP. CALC. EXP. CALC.

----7-

---9-

--- + - - - 9 ' --~11t3 ---13'

2.0 - - "

--~-I

.~--~,~:_~o~7;~t-

~ /23' ~25 t

27--/-23-

-- -- 21-

~15-

.~33~ =~.~+

--(~+1

_~23 + -~2~'

\27+

~ 33+ ~-29 ~

~IPb

---(29)- -- -:)7" __~-

----

___(33+I

2°sPb

J

!

Fig. 1. Comparison of e×pcrimcntal [ 8 - ] 4 ] and calculated low-lying ,/~ states of odd-mass Pb isotopes. The spins are in 2,/form.

A,

z

w

>.

v

3.0

--t27"3

4.0

--_~

--ct~

5.0

/33~ ~-----29 \27+

~Pb

---(~'}

t-~'~"

"~;~"

~0

~.

"

~"

~

.~

C.A.P Ceneviva et al./Nuclear Physics A 619 (1997) 129-142

135

Table 1 Calculated wave functions of negative-parity states in Pb isotopes. For each nucleus, the lqp and 3qp bases states are denoted by IJc} and [(ja, Jb)Jab, jc}, respectively. Only amplitudes larger than 9% are listed

j[r

ja

Jb

Jab

jc

205 Pb

203 Pb

2Ol Pb

199 P b

197 P b

! 21

1 2

0.991

0.998

0.982

0.984

0.986

321

3 2

0.933

0.979

0.982

0.982

0.982

5_2I

5_ 2

0.965

0.985

0.985

0.987

0.988

0.900

0.633

221

221

l~ 21 13 21

IA 21

17 -TI

19 2 I

5

5

4

3 2 52 52

3

2

5 2 5 2 5 2 3 2

5 2 -3 2 5

52 52 5

4

5 2

5_ 2

4

2 4 2 4

4

7 2 1 ± 2 3 2 3 2 5 2 1 2 ! 2 s

-0.529

-0.302

3 2

2

5_ 2

5_ 2

4

-3 2

0.953 0.914

6

± 2

-5 2

4

72

13 2

13 2 5

4 3

5 2 7

5

4

7

13 2 13 T

13 2 13 T

6

3 2 3 ~

5 2 13 2

-5 2 13 2

4

2 13 T

52 13 T

4

13 T

13 T

8

1

13 2 13 T

1A 2 13 T

6

5_ 2 5 ~

13 T 13 2-

13 T 13 "2

10

13 2

13 2

10

1 2

J3 2 13 T 13 T

13 2 13 T 13 T

8 10

-5 2 3 ~ 3 ~

8

6

8

8

8

8

7 2 -5 2

0.308

0.714

_3 2

2 2

0.856

0.312

-0.619

5 2 5 2

0.487

0.737

3 2

5 2

0.973

-0.313

0.838

0.668

0.359

0.635

0.977

0.986

0.950

0.937

0.984

0.973

0.541 0.679

0.822 -0.474

0.388

0.493 0.625 0.364 0.812 0.907

0.922

0.908

0.524

-0.323

-0.439

7 2 3 ~

-0.524 0.888 0.966 0.816

0.889

0.449

0.386

3 ~ 3 ~

0.580

0.739

0.904

0.304

0.315

0.978 0.960

0.978

0.940 0.905 -0.398

"

-/;-

CALC. ~:(-)

-/;-

~

-

3

"~-7 3

EXP.

J

3

- -

---5-

--

CALC. ~(+)

,++

11.1

llJ Z

IY

(.9

>.

~

-

-3-

----3-

----

EXP.

+

I

3-

-J:,'-

----(1)-

-~

__/~-

7

31-

-

_

r-

+_

CALC. :~(~

--,-

----

--

--

_

=C

----

___(_) __~-

--11-

, :-

-+,

___5----T"

----3-

--

- - ~

+

__

1-, 3-

~m.+pb

(6. T)___/<,,,)-

---e

5

=_11+

--/-t9 + ---21 +

o.o.--,-

0.5

1.0

2.0 -

(b)

I ' 13 I ++

CAI..C. ~(+)

---13÷

11,+

__~_,~,__/

--

--/lg + -_21 +

.j

Z LU

~

(.~

>,,.

m

>+ ~

7 )+

----

----

--__

_~-.

----(~.r)--~(o-)

EXP.

,-

3

(1-)

(3-)

13÷

__~(m t

---(,)-

----7-

o.o + _

o.~

1.0

+

*~'17 +

I +5 - __~1~;. ___.+,11

m

----7

"--(")"

|_Z.. +) I =~(7.'~ +

2"°1---/<,1+7 9 )

(c) I 17+

+

.+

1+ 11+ t3+,

=~++:

=

~

~17+,

15+

13+

~ ~,~'t 131

_-

3-

(1-)

CALC. ~(-)

-_,-

--

----

--

11-

/~3-

7-

/,--~_

~

r

--+-'-.'_//]11-

-=

__~'~15-. '/if,-

--

- -

13+

1°+

13+

CALC. ~:(~)

I+

__///+~_ c ------~11P1r'1 +

3-

~'Pb

Fig. 2. (a) Experimental [8] and calculated spectra of 2°spb. (b) Experimental [9] and calculated spectra of 2°3pb. (c) Experimental [ 10] and calculated spectra of 2mpb. In all figures the spins are in 2J form.

o.o

----3

6

(+)-

7

1-

i"-

---j+-

___q- - -

----

-

--- +-

=-:-

__

~ %

__~3-

o.5

+

(3.1)-

+

Z UJ

=

\(9)

- S , " ~ +-

- - -

--

-+r

--"-

=~-+

'Is

----+.-

---,3-

----4-

-

m~,,1~+

m

7-

~/(13) + "//(3-, 1 - )

B ~

__

--

--

~-

F 'g-) 2oepb1- ----~'~+-

/,(~-, 1-)

.J

+S

>.

2.0

(a)

',L

',o

~0

~C)

+~

~+

+

-~

•

.~

<3,,

C.A.P. Ceneviva et al./Nuclear Physics A 619 (1997) 129-142

137

Table 1 - - c o n t i n u e d

"17

ja

jb

Jab

jc

2O5Pb

21 2 i

13 2 13 2 ~3 2 13 "T

13 2 13 2 13 2 13 "T

10

± 2 _5 2 5_ 2 ~3

0.965

13 2

13 2

12

± 2

0.986

13 2 13 2

13 2 13 2

10

5_ 2 3 2

13 2

13 2

12

! 2

13 2 J3 2 J3 2

13 2 13 2 13 2

10

_5 2 5_ 2 3 2

13 2 T13

13 2 T13

12

13 2

13 2

23 2 1

25-2 I

27 -21

29 -2l

8 10 12

12

12 12

12 12

203 Pb

201 ~

0.783

0.852

0.411

0.394

199Pb

0.938

0.939

0.943

197 Pb

0.947

0.940 0.921

0.958 0.881

0.933

0.379

0.341

0.953 -0.395 0.885

_5 2 3

0.995

_5 2

0.997

0.994

0.988

0.963

0.901 --0.431

0.998

0.998

0.998

0.999

(b) 2°3Pb. The experimental and theoretical number and sequence of levels are practically the same up to 1.3 MeV. The theory suggests ~1- , ~3 - , ~9 - , and 5 - spin assignments for the measured levels at 0.775, 1.088, 1.199, and 1.203 MeV, respectively. At higher excitation energies the theory predicts a number of negative parity states much larger than the experimental results. Below 2.0 MeV the calculation predicts the ~ + and ~ + states as in 2°spb. (c) 2°lpb. The measured and calculated number of levels are approximately the same up to 1.2 MeV. The calculation suggests a ~ - spin assignment for the two experimental states at 0.800 and 0.911 MeV, and 9 - for the two measured levels at 1.014 and 1.186 MeV. The theory predicts a high density of states between 1.2 and 2.0 MeV, especially of positive parity. The discrepancy between calculated and experimental excitation energy for the second 23-- state indicates that it may be necessary to adjust parameters for a more accurate theoretical description of the energy spectrum. This adjustment was not carried out due to the lack of experimental information on the excitation energy of low-lying negative-parity states in the light isotopes, except as displayed in Fig. 1.

3.2. E l e c t r o m a g n e t i c m o m e n t s

The calculations of the magnetic dipole moments ~ were performed with the following gyromagnetic ratios: ~ free ( 1 ) gt = 0 and gesff = u.3gs ; ~ free (2) g~ = 0 and g~ff = v./gs .

C.A.R Ceneviva et al./Nuclear Physics A 619 (1997) 129-142

138 Table 2

C a l c u l a t e d w a v e f u n c t i o n s o f positive-parity states in Pb isotopes. See c a p t i o n o f Table l

j~r

ja

jb

Jab

jc

931+

~5

~I

2

13 -i-

5_ 2 3 2

_5 2 1 2 5_ 2 ~3

2

13 2 13 2 i3 2 13 "T

2 3

1l+ 21

3 2 _5 2 2 5_ 2 3 2 3 2

13 + 21 1_.5+ 21

17 + T1

5_ 2 3 2 5_ 2 2 3 2 5 ~ 3 2 5_ 2 2 1 2

19 + 21

21 + 71 2A + 21

25+ 21 27 + 21 2_29+ 21 33 + 21

2 5_ 2 5 3 2 3 2 5 ~ 2 5_ 2 5_ 2 13 2 5_ 2 13 2 IA 2 13 2 13 2

_5 2 1 2 1 2 5_ 2 5_ 2 3 2

! 2 1 2 5 2 5 2 3 2 1 ~ _l 2 5_ 2 52 3 2 _5 2 1 2 5 5_ 2 5_ 2 5 ~ 5_ 2 7 2 7 2 13 2 7_ 2 13 2 t~ 2 13 2 t3 2

2 2 2 4 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 4 3 4 4 3 4 4 6 5 12 6 12 12 12 12

13 2 13 2 13 2 13 2 13 2 13 2 13 2 13 2 13 2 13 2 13 2 13 2 13 T 13 2 13 2 13 2 13 2 13 2 13 2 13 13 2 13 2 13 7 13 2 1_23 2 13 2 13 2 13 2 13 2 13 2 13 2 13 2

205 Pb

0.787

-0.426

203 Pb

201Pb

0.514

0.314

0.623

0.750

197Pb

0.406

-0.347 -0.387

0.886

199 Pb

0.595

0.403

-0.337 0.466

0.729

0.528

0.441

0.309

-0.320 0.581 -0.305

0.661 -0.377

0.975

0.974

0.979

0.854

0.500

0.396

-0.307

0.445 -0.343 0.610

0.821

0.984

0.989

0.351 -0.307

0.735

0.747 -0.326

0.550 -0.489 0.357

0.858 -0.337

0.520

0.302 -0.308 0.413 -0.438 0.561

0.314

-0.311 0,688

0.798

0.446 -0.439

-0.418

0.402

0.577

0.504

0.504

0.976 0.980

0.895 -0.428

0.913 -0.356 0.896 -0.389

0.958

0.936 -0.321 0.908 -0.355

0.956

0.896 -0.425 0.922

0.333 -0.874

-0.887

-0.318

-0.334

0.316 0.932

0.971

0.873

-0.302

0.952

0.406

0.977

0.849

0.964

--0.488 0.999

0.999

0.999

0.999

0.999

0.999

0.999

0.999

0.999

0.999

1.000

1.000

1.000

1.000

1.000

C.A.P Ceneviva et al./Nuclear Physics A 619 (1997) 129-142

139

Table 3 Comparison between experimental and theoretical electromagnetic moments. The magnetic dipole (/z) and electric quadrupole (Q) are in units n.m, and eb, respectively. The parametrizations ( 1) and (2) are presented in the text /z(nm)

Nucleus 205pb

~Tr

Exp.

Theo. ( 1)

Theo. (2)

Exp.

Theo. ( 1)

Theo. (2)

5_-

+0.7117 4 a

+0.68

+0.95

0.226 37 a

+0.081

+0.163

13+

-0.975 40 a

-0.97

-1.35

0.30 5 a

+0.15

+0.30

25-

-0.845

-1.73

-2.44

0.63 3 a

+0.19

+0.38

21

21 21

14 a

33+

--2.442 83 a

-2.42

-3.39

+0.20

+0.41

5_-

+0.6864 5 a

+0.66

+0.92

+0.095 52 a

+0.046

+0.092

21+

-0.641

21 a

+0.07

+0.10

0.85 3 a

+0.15

+0.31

2

203pb

Q (eb)

2

2

252

-0.738

38 a

-1.10

-I.54

+0.15

0.31

2°lpb

5_-

+0.659

13 b

+0.56

+0.78

--0.009 90 b'c

+0.007

+0.013

199pb

~-

-1.074

1~

-1.03

-I.43

+0.07 7 b

+0.03

2

2

29-

-1.07

29+

-2.4

-0.98

-1.38

+0.18

0.36

-2.1

-3.0

+0.11

0.23

32 -

-1.075 2 a'~l

-1.09

-1.52

-0.08

t3+ 2

--1.105 3 a,d

--1.02

-I.43

+0.47 34 aA

2&2

-0.531

7 a'e

-0.93

-1.30

33+

-2.51

10a'e

-2.42

-3.39

2

2

197pb

2

j95pb

7e

0.05

1e

18a'd

+0.003

+0.005

+0.09

+0.17

+0.14

+0.28

+0.16

+0.33

29+

-2.7

3 f

-2.2

-3.0

+0.09

+0.17

3~+ 2

-2.5

1e

-2.4

-3.4

+0.14

+0.28

2

a From Ref. [31]. b From Ref. [32]. c From Ref. [33]. a From Ref. [34]. e From Ref. I351. l From Ref. 1361.

The electric quadrupole moments, Q, were evaluated with the following effective electric charges (1) eeff= 0.5e and (2) eneft = 1.0e. The available experimental data are compared with the calculated values in Table 3. In the case of magnetic moments, all signs are in agreement with experiment and the magnitudes can be reproduced by the theory if the effective parameter is adjusted, except for the ~ + state in 2°3pb. The absence of experimental data on the magnetic dipole moments of the ~,+ state in the neighboring isotopes makes it difficult to understand the discrepancy between measured and calculated values for 2°3pb. For the electric quadrupole moments the agreement with experiment is only reasonable. Two calculated and measured moments have discrepant signs, for the ~1 state in

2°lpb and for the 31 state in 197pb. However, the magnitudes are very small (,-~ 10 -3) and a small variation in the parametrization can change the theoretical predictions. The other predicted values are, in general, smaller than those measured. To fit the experimental data it is necessary to increase the effective neutron charge even more. It means that

140

C.A.P. Ceneviva et al./Nuclear Physics A 619 (1997) 129-142

Table 4 Calculated magnetic dipole and electric quadrupole moments for some low-lying levels in Pb isotopes. See caption of Table 3 205Pb Jff

txl

tz2

201Pb Q2

tl,2

i2 -1

0.31

~21-

-0.87

5_2

0.68

0.95 0.163

72 -

0.86

1.21 0.116 -0.26 --0.36

1.23

1.72 0.143

29 -

0.44 -

IZl

197Pb Q2

-0.19 -0.27

-1.22 0.110 -1.36 -1.91 0.56

-

tz2

0.26

0.36

0.085 -1.09 -1.52 0.014

0.60

l,Zl

0.31

~2

Q2

0.44

0.005 -0.95 -1.34 -0.049

0.84 -0.071

0.031 -1.10 -1.54

0.68

0.95 -0.105

0.182 -1.07 -1.50 0.88

0.182

0.118 --0.93 --I.30

0.283 --0.96 --1.34

0.112

232

-2.13 -2.98 0.412 -1.18

-1.65

0.181 -1.85 -2.59

0.251 -1.83

-2.57

0.167

252

-1.74

-2.43 0.382 -0.93

-1.31

0.228 -1.96 -2.75

0.257 -2.10 -2.94

0.175

27~-

-1.38

-1.94 0.372 -1.38 -1.94

0.349 -1.59 -2.22

0.267 -1.81

-2.54

0.080

29-1.15 -1.62 0.540 -1.14 -1,59 2 1~+ -0.97 -1.35 0.301 -1.47 -2~06 2

0.392 -1.13 -1.58

0.212 -1.15 -1.61

0.039

0.238 -1.02 -1.43

0.171 -0.97 -1.36

0.092

2t+ 2

0.216 -0.31

-0.43

0.108 -1.54 -2.16

0.038

0.322 -1.67

0.05

0.07 -1.52

-0.96 -0.029

0.63

0.167

--1.37 --1.92 0.169 --0.76 --1.06

0.07 0.338

1.24 -0.023 -0.68

Q2

2~2

0.05

0.89

0.78

#l

193Pb

23 + T

-1.04

-1.45 0.451 -1.09

25+ 2 27+ 2

-1.95

-2.73 0.420 -1.94 -2.71

29 +

-2.13

-2.98 0.255 -2.13 -2.98

0.239 -2.13 -2.98

0.205 -2.13 -2.98

0.130

-2.42

-3.39 0.407 -2.42 -3.39

0.382 -2.42 -3.39

0.327 -2.42 -3.39

0.208

T ~+ 2

-1.98 -2.78 0.155 -1.98

-2.78

-2.33

0.067 -1.69 -2.36

0.038

0.394 -1.83 -2.57

0.124 -1.84 -2.57

0.073

0.145 -1.98

0.124 -1.98 -2.78

0.079

-2.78

the quadrupolar p h o n o n excitations o f the core may be considered for a m o r e c o m p l e t e description o f the properties o f the l o w - l y i n g states in Pb isotopes. In v i e w o f the g o o d a g r e e m e n t obtained for the m a g n e t i c m o m e n t s , and the reasonable theoretical estimates for the electric ones, in Table 4, the e l e c t r o m a g n e t i c m o m e n t s calculated tbr s o m e o f the l o w e r J ~ states in 2°5'2°1,197,193pb isotopes are displayed. For the other isotopes, m o s t o f the results fall between the corresponding values for the n e i g h b o r i n g nuclei. H a v i n g in m i n d the discussion above, these predictions can serve as a g u i d e for future measurements. 3.3. S p e c t r o s c o p i c f a c t o r s

T h e m e a s u r e d and calculated pickup spectroscopic factors are c o m p a r e d in Table 5. A l t h o u g h experimental data are available only for 2°sPb, the predictions for the 2°3,2°1pb isotopes are reported also. This table shows, in general, reasonable a g r e e m e n t between e x p e r i m e n t and theory. The fragmentation o f the single-particle configurations in the l o w - l y i n g states are w e l l r e p r o d u c e d and the differences in energy are <~ 80 keV. O n c e again, the c o m p a r i s o n indicates j r = l -

and -32- for the measured levels at 0.803 and

C.A.P Ceneviva et al./Nuclear Physics A 619 (1997) 129-142

141

Table 5 Comparison between experimental [ 37,38 ] and calculated Sj pickup spectroscopic factors for low-lying states of 2o5,2o3,2olPb isotopes. The excitation energies are displayed only for 2°spb 205Pb

Jff

203Pb

2olPb

Sj

Sj

Calc.

Calc.

Calc.

5.6

5.21

4.38

3.24

0.15

0.62

0.02

0.02

0.17

0.02

0.02

Sj

Energy (keV)

Exp. 521 5-22

Calc. 0

0

(d, t)

750

683

1265

1018

~2

11

1.67

0.76

0.55

0.32

L22

~800

858

0.03

0.01

0.01

0.01

21

,,~260

287

4.69

3.42

3.41

3.08

~580

590

O. 13

0.26

0.00

0.01

0.37

0.06

0.03

_523 i-

2

998

948

721

703

651

0.01

0.01

0.09

722

1043

986

0.19

0.16

0.00

7-

1575

1434

0.01

0.36

5.25

7--24

~ 1610

1685

0.71

0.67

0.73

0.29

7--

~1770

1748

5.27

6.38

5.32

0.43

221

980

898

0.61

0.04

0.00

0.00

222-

1499

1258

0.01

0.02

0.01

1010

1030

13.23

12.88

12.81

_

1796

0.83

0.42

0.19

~3

23

25

IA21 t3+ 22

11.5

0.998 MeV, respectively, and J~ = 9 - and 7 - for the experimental levels at 1.449 and 1.575 MeV, respectively, in 2°sPb.

4. Conclusions Within the framework of the projected lqp + 3qp calculations using SDI as the residual interaction, the available data on the energy spectrum, electric and magnetic moments, and spectroscopic factors were examined. For the first time, a unified description including low and high spins states of odd-mass lead isotopes is presented. We can state that the present model describes the main features of the experimental level schemes of the low-lying states. For a m o ~ accurate description of the light isotope spectra it is necessary to obtain experimental information about single-particle states in these nuclei. Significant improvement is obtained with respect to earlier lqp + 3qp calculations [ 1-4] carried out without number-projected states. The core correlations not included due to the proton two-particle-two-hole excitations may affect in a more

142

CA.P Ceneviva et al./Nuclear Physics A 619 (1997) 129-142

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