69Ga in the Alaga model

Volume 39B. number 4

PHYSICS

69Ga

IN THE

LETTERS

ALAGA

15 May 1972

MODEL

V. PAAR

The Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark and Institute Ruder BogkoviS, Zagreb, Yugoslavia

Received 15 March 1972 ]'he 69Ga isotot~ is described in the model of coupling a three-particle proton valence-shell cluster to the low-frequency quadrupole vibrational field. The nuclei 65, 67, 6 9 G a a r e r a t h e r s i m i l a r ; ;Jl t h e r e s e e m s to e x i s t a o n e - t o - o n e c o r r e s p o n d e n c e between the l o w - l y i n g l e v e l s . In this l e t t e r , we s h a l l c o n c e n t r a t e our attention on 69Ga, which was most frequently studied, both e x p e r i m e n t a l l y [1-10] and t h e o r e t i c a l l y [2, 11-15 ]. It was d e s c r i b e d by introducing p a r t i c l e - p h o n o n [2], q u a s i p a r t i c l e phonon [ 11,15 ], q u a s i p a r t i c l e - a n h a r monic phonon [ 12, 13 ], and q u a s i - p a r t i c l e - p h o n o n coupling with the additional d i p o l e - d i p o l e i n t e r a c t i o n t e r m [14]. The o n e - p a r t i c l e phonon l : i c t u r e [2] p r e d i c t s the wrong sign of the g r o u n d - s t a t e q u a d r u p o l e m o m e n t , while the one p ~ l 2 h o l e - p h o n o n (or uns p e c i f i e d c o r e ) model f a i l s to account for the l a r g e s p e c t r o s c o p i c f a c t o r s found e x p e r i m e n t a l l y [6-8] for the two lowest e x c i t e d s t a t e s . T h e s e

p r o n o u n c e d e x p e r i m e n t a l p r o p e r t i e s r e v e a l that one has to account both for the h o l e - l i k e c h a r a c t e r of the g r o u n d - s t a t e (3/2~,) and p a r t i c l e - l i k e c h a r a c t e r of the f i r s t (1/2[) and s e c o n d ( 5 / 2 ~ ) e x c i t e d state. This c o u l d p o s s i b l y be a c h i e v e d in a o n e q u a s i p a r t t c l e a p p r o a c h . However, the K i s s l i n g e r S o r e n s e n p a r a m e t r i z a t i o n [11] s e e m s not to be a p p r o p r i a t e , s i n c e the 3 / 2 ~ s t a t e t u r n s out to be p a r t i c l e - l i k e , what r e s u l t s in the negative g r o u n d s t a t e quadrupole m o m e n t [11]*. On the o t h e r hand, the c a l c u l a t i o n f r o m ref. [14] a c c o u n t s for the p o s i t i v e g r o u n d - s t a t e quadrupole m o m e n t and l a r g e s p e c t r o s c o p i c f a c t o r s , but both the o c c u p a tion n u m b e r s and q u a s i - p a r t i c l e e n e r g i e s of the t h r e e s t a t e s involved have been t r e a t e d a s f r e e parameters. A n a t u r a l d e s c r i p t i o n of the Z = 31 nuclei i s p r o v i d e d by coupling a t h r e e - p a r t i c l e v a l e n c e p r o t o n shell (28-50) c l u s t e r to the l o w - f r e q u e n c y q u a d r u p o l e v i b r a t i o n a l f i e l d (Alaga model) [17].

For the particle-Like state, the quadrupole moment is negative, while for the hole-like state it is positive.

2 I E{MeVI

-

-

_

_

-

-

5/2?/2-

~.--

1/2T/2" /5/2.7]]/2_

/0/2'( 3~'2,,5 ~ ) ,I/2

3r~- - 1 / 2 -

~9/2"3/2'g/2('Y2/~1/2) -

~

~:

j 53/2/2-

'[3/2; 5/2-)

"/2-

A ~2~ 5./2-)

/ 7 . --

5.-

-

3/2E1/2°

_

"~'; " ' ''~"

i __--

- - - -

- - "

y,~7/25/2'7/23£2-

-

_ -

9/2* 112-

112-

"g/2-

"/2-

5/2-

- - ? / 2 - 3/2-

1/2-

,3/2-

I'',I?/2 -

3/2,9/2~

.5/2- (`7/2)

"9/2-

~ /2-

5/2,7/2 7/2 -

-

-

3/2"

,12- --~,2-

5/2--'712-

-

5/'2,?~(9/2) "//2,5/2 1 '~.3/2-

7,

l

.___~/2-

3/2-

-

(1/27 5/217/2")

3/2-

5/2-

",,2

;'/2-

3/2-

--51,-

__5/2-

112~ 3 1 2 -

1/2"

-

-

(1/2;3/27 5J2-) 1/2-

-

-

3/25/'2-

3/2-

5/2-

5/2~

I/2-

112"

13,,'z 7/2-

--1,2-

-

-

-

-

-

-

--'7/2"

--.112-

_ _ - -

5123/'2-

5/2-

-

-

5/2-

1i2-

-

-

1/2-

3/2"

-

-

3/2-

5/2-(3/2"~ 7/2-) 3/2-

- - 5 / 2 -

-

1/2"

-

-

3/2-

512-

5/2-

-

-

5/2-

1/2"

-

-

3/2-

69GA KISSLiNGER, SOIII~NSEN (11]

1/2"

OSG& OONO, K]SSLINGER, KUMAR [12]

-

-

1,2-

3/2-

6oGa KISSLINOER, KUMAR [13J

-

-

3/2-

69(~ilt TEMPIERLEY* MC.DANIELS* WELLS [2]

I/2-

3/2-

69Ga PARADE LLIS, NONTZEA$ [ l & ]

-

-

3/2-

-

-

312-

69Of I

69Ga

6';'Ga

PRESENT CALCULATION

EXPERIMENT

EXPERIMENT

6'TO& PRESENT CALCULATION

Fig.1. Experimental 69Ga spectrum and the results of the available calculations. For comparison, 67Ga ~,pectra are drawn on the right side. 466

PHYSICS

VoLume 39B, number 4

15 May 1972

LETTERS

Table 1 Wave functions of a few low-lying states in 69Ga. The basis states are I[(/~,, l~ ,)Jo,lM'],~]J, NR;l ~. Here lj are the single-particle quantum numbers, Jo is the intermediate and J the total affgular momentum of the antisymmetrized three-particle state. N and R represent the number of phonons and the anguJar mo~rentum of this N-phonon state, respectively. For the ease l!, = ln., = l.mm = l., j<~, the basis states are [(lj)3}J, NR; l). Only amplitu3es larger than J J ? .7 4% are listed.

3/21

1/21

I[(p,~/~)22, Pi/2]5/2, 12)

0.220

If(P3/2)20, Pl/2]l/2,00>

0.S01

[[(P3/2)3]3/2, 12> i[(Pl/2)20,P3/213/2, 00~ I[{P3/2)3]3/2.00> ][ (f5/2)20' P3/2"]3/2' 00)

0.207 0.256 0.817 0.312

I[(P3/2)22, Pl/215/2,12~ I[(f5/2)20,Pl/21 1/2,00} If(P3/2)3] 3/2, 12} I[(P3/2)20, f5/2]5/2, 12)

-0.215 0.250 0.229 -0.313

5/21 I[ (P3/2)20, f5/2 ]5/2, 12> -0.238 I[(p3/ 2)20,Pi / 211/2,12/~ -0.228 [[(Pl/2)20, f5/215/2, 00> 0.206 ![(P3/2)20, f5/2]~5/2, 00) 0.790 I[(f5/2)315/2, 00> 0.212

3/22 [ (pl/,2)20, P3/213/2, 12) i [ (P3/2) 3]3/2,12) I[(f5/2)20, P3/2}3/2, 12> I[(Pl/2)20, P3/213/2, 00> [[(f5/2)20, P3/213/2, 00> [[(P3/2)22, Pl/2]3/2,00, ~ If (P3/2)20, pl/2] 1/2, 12}

-0.228 -0.,t24 -0.203 0.333 0.285 0.442 0.316

5/22 ][(P3/2) 313/2, 12} I[(Pl/2)20, P3/213/2, 12) I[ (P3/2)20, Pl/2] I/2, 12} I[(P3/2)22, Pi/215/2, 00~

-0.565 -0.243 -0.215 0.618

1/~ 2

] [(P3/2)313/2, 12> -0.544 I[ (P3/2)20, f5/215/2,12> -0.266 {[(Pl/2P3/2 )2, f5/211/2,00>-0.240 1[(P3/2)22, f5/2]1/2, 00> 0.574 I[(f5/2)20,p3/213/2,12) -0.256 7/21 ][(P3/2 )3 3/2, 12} [:(P3/2)20, f5/2]5/2, 12)

0.6,t9 0.210

i [(f5/2)~,

P3/2 )3/2,12>

0.210

I[(P3/2 ) 2, f5/217/2, 00)

0.527

The three lowest states are then expected to arise f r o m .the I[(P3/2)3123-, O0;23-),5[ [(P3/2)20, Pl/21 ~, 00;~ ) and I[(p ~/~)2 0, f5/2] ~,00;~) configurations. In the zeroth-oY(fer approximation the ground state is hole-like, so the corresponding quadrumoment is positive. On the other hand, the ~ ole 8Zn ground-state is arising from the configuration [(po ~,)20, 00;0) consequently, large spectroscopic /~f~c~torsof the three lowest states of 69Ga are predicted for stripping reactions. Higher states, based on phonon multiplets and/or broken pairs, should be excited more weakly. The electric quadrupole moments and E2 transitions between the low-lying states are increased due to the coherent particle and collective contributions. The following group of states is the quadrup-

let 1/22, 3/22, 5/2 2 and 7/21, arising in zerothorder approximation from the one-phonon multiplet[(p:l 2 )3] 3/2,12;I= ½, ~ ~, 72>, which is

3/

based on the groundstate. Experiments give rather unambiguous candidates for the 1/22 , 3/22, and 7/21 members [1-9], while the situation with the counterpart of the calculated 5/22 state is not quite clear. Our model predicts a very small spectroscopic factor for the 5/22 state. The calculation for 69Ga was performed in the following parametrization: c(p 2) - ~(P3 2) = 0.9 MeV, ~(f"/")o.~.- ~(P3/2}0,4 = t 2 1.4MeV,C=0.3, = . • MeV, ~w Here ~(lj)are the single-particle energies, kw 2 is the ph0non-energy, G the pairing strength of the residual particle-particleinteraction and a the particle-field coupling strength. In fig.l,

467

PHYSICS

Volume 39B, number 4

LETTERS

15 May 1972

Table 2 E l e c t r o m a g n e t i c p r o p e r t i e s of 69Ga. The available theoretical and e x p e r i m e n t a l r e s u l t s are compared. The effective char s of the p r e s e n t calculation a r e : eP = 2, e~b=2.7, and the gyromagnetie ratio~. : g R = Z / A , g l = 1, g ~ = 2.35. The eg~fective proton charge ePff was tak~ffin accoe~anee w~th refs. [18-20], the e f f e c t i v e ' v i b r a t o r c~arge c o r r e s p o n d s approximately to the value obL4med from B(E2) (01~ 21) (~)°Ni) ; the gR and g l " b a r e " values have been used, and the effective g s is appreciably reduced d u e t o quenching [18]. B(E2)(eb)2 [ 15}

[ 14]

Present calc.

Exp. 0.016[ I]

1/21-.3/21 . . . . .

0.077

0.026

~).015

5/2~-~ 3/2]

0.005

0.002

0.004

5/2~ I/2]

0.037

0.006

0.026

-

0.026

0.034

0.017

0.084

0.000

0.031

0.078

0.025

3/2~3/2~ 3/2~ ~ l/2~ .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Pre sent

[ii] -0.19

0.120

0.183

0.000

0.000 -

.

Q (eb)

[14]

ea[c.

+0.I0

+0.20

1/2i

~_(_n._m_..)_ _ Pre sent

Exp.

[14] ,0.19 [10]

2.1.4

ea|e. 1.97

Exp.

2.02110]

2.28

5/21

-0.40

1.92

3/22

-0.19

1.05

theoretical s p e c t r a from available calculations a r e compared with the p r e s e n t r e s u l t and exp e r i m e n t . Results for 67Ga a r e given for c o m parison. Table I contains the calculated wave functions. Table 2 c o m p a r e s the available theo r e t i c a l and experimental electromagnetic p r o p erties. In t h e p r e s e n t m o d e l , t h e a n h a r m o n i c s t r u c t u r e of t h e n e i g h b o u r i n g e v e n n u c l e i a s w e l l a s t h e e x p l i c i t a p p e a r a n c e of t h e b r o k e n a n d p r o moted pairs are accounted for. The latter corr e l a t i o n s a r e n e g l e c t e d in t h e q u a s i p a r t i c l e anharmonic phonon approach. T h e f l e x i b i l i t y of t h e m o d e l on t h e o n e h a n d , a n d t h e l a c k of e x p e r i m e n t a l d a t a on t h e o t h e r , l e a v e s o m e f r e e d o m in p a r a m e t r i z a t t o n , so the p r e s e n t p a r a m e t r i z a t i o n i s not e x p e c t e d to b e t h e o p t i m a l one. F o r t h a t r e a s o n , t h e m e a s u r e m e n t s of s o m e m o r e p r o p e r t i e s of 6 9 G a ( e s p e c i a l ly e l e c t r o m a g n e t i c ) , w o u l d b e d e s i r a b l e . The present model has also been rather succ e s s f u l l y a p p l i e d to odd M n (Z=25) [20], Ag (Z=47) [20], I(Z=53) [20] a n d A u ( Z = 7 9 ) [ 1 7 - 1 9 ] i s o t o p e s . T h e a u t h o r e x p r e s s e s h i s d e e p g r a t i t u d e to P r o f e s s o r G. A l a g a f o r h i s c o n t i n u o u s i n t e r e s t during this work and for clarifying discussions; h e i s i n d e b t e d to t h e N i e l s B o h r I n s t i t u t e f o r e x cellent working conditions. References

[1] L.W. Fagg, G.H. Geer and G.A.Wolicki, Phys. Rev. 104 (1956) 1073. 468

B(MI)(n.m.) 2 Present [ 14] calc.

[2] J. K. Temperley, D.K. McDaniels and D. V. Wells, Phys. Rev. 139 (1967) Bl125. [3] W.H. Zoller, G. E. Gordon and W. B. Waiters, Nuel. Phys. A124 (1969)15. [4] S. Raman and G. Couch, Phys. Roy. C1(1970)744. [5] H. Langhoff and L. F r e v e r t , Nucl. Phys. Al11(1968) 225. [6] V. V. Okorokov et al., Yadern. Fiz. 8(1968)668. [7] R.G. Couch, J . A . Biggerstaff, F . G . P e r e y and S. Raman, Phys. Rev. 2C (1970) 149. [8] B. Zeidman, R.H. Siemssen and L. L. Lee J r . , Bull. Am. Phys. Soc. 10(1965)1126. [9] D. E. Velkley, K. C. Chung, A. Mitter, J. D. B r a n d e n b e r g e r and M. T. M c E [ [ i s t r e m , Phys. Roy. 179 (1969) 1090. [10] K. Way, ed., Nucl. Data B1(1966), B2(1968). [11] L. S. K i s s i i n g e r and R. A. Sorensen, Rev. Mod. Phys. 35(1963)853. [12] P. Bond, L. S. K i s s l i n g e r and K. Kumar, as quoted in in ref. [ 7]. [ 13] L. S. Kisslinger and K. Kumar, Phys. Rev. L e t t e r s 19 (1967) 1239. [ 14] T. P a r a d e i l i s and S. Hontzeas, Can. J. Phys. 49 (1971) 1750. [15] B. S. Reehal and R. A. Sorensen. Phys. Rev. C2(1970) 819. [16] A. Bohr and B. R. Motte[son, Mat. Fys. Medd. Dan. Vid. Seisk. 27(1953)No. 16. [ 17] G. Alaga, Bull. Am. Phys. Soc. 4(1959)359. [18] G. Alaga and G. lalongo, Nue[. Phys. A97(1967)600; G. Alaga, Rendieonti Scuola Internazionale, Varenna, XL Corse (1967)p. 28; G. lalongo, Thesis, New York University, 1966. [19] G. Alaga and V. Paar, to be published. [20} V. P a a r , to be published.