Volume 35B, number 3 PHYSICS LETTERS 24 May 1971

ON T H E N A T U R E O F T H E I S O M E R I C S T A T E IN T H E D O U B L Y - E V E N N : 82 N U C L E I

K. HEYDE, M. WAROQUIER * and G. Vanden BERGHE **

Instituut voor Nucleaire Wetenschappen, Rijksuni~ersiteit . Proeftuinslraat 40. 9000 Gent. Belgiunz

Received 10 March 1971

The i somer ic state in the doubly-even single closed shell nuclei with N= 82 neutrons has been studied in a quas i -par t ic le model. The lifetime is calculated and compared to the experimental data. Some com- ponents of the wave-function of the i somer ic state are compared with experimental data from one-nucleon t r ans fe r react ions.

In the d o u b l y - e v e n N = 82 s i n g l e - c l o s e d - s h e l l n u c l e i , an i s o m e r i c s t a t e h a s b e e n s t ud i ed [1-7] wi th h a l f - l i f e r a n g i n g f r o m 162 n s e c in 134Te up to 16 /zsec in 142Nd. T h i s i s o m e r i c s t a t e i s a l - w a y s a s s o c i a t e d wi th the e m i s s i o n of t h r e e E2 g a m m a r a y s in d e c a y i n g to the g r o u n d - s t a t e , s u g g e s t i n g a J ~ = 6 + - 4 + ~ 2 + ~ 0 + c a s c a d e .

We t r y to g ive a d e s c r i p t i o n of the p o s i t i o n

* Aspirant of the "Nationaal Fonds voor Wetensehap- pelijk Onderzoek".

** Research worker of the "Interunivers i ta i r Instituut voor Kernwetensehappen".

and n a t u r e of t h i s i s o m e r i c l e v e l in a q u a s i - p a r t i c l e d e s c r i p t i o n of t h e s e d o u b l y - e v e n nuc le i .

F r o m the o d d - p r o t o n N = 82 nuc le i and the knowledge of the s i n g l e - p a r t i c l e s t a t e s [8], it w a s p o s s i b l e t h r o u g h use of the i n v e r s e - g a p e q u a t i o n s [9], to d e t e r m i n e a s e t of s i n g l e - p a r - t i c l e s e l f - e n e r g i e s [10,11]. As t w o - b o d y f o r c e , a g a u s s i a n i n t e r a c t i o n of the f o r m V = V o x exp ( - /3 r2 ) (P S + t P T ) with t r i p l e t - t o - s i n g l e t r a t i o t = +0.2 and r a n g e p a r a m e t e r ~ = 0 . 3 2 5 f m -2 has b e e n used . T h e s e v a l u e s f o r the p a r a m e t e r s g ive a good d e s c r i p t i o n of the odd-A and d o u b l y - e v e n e n e r g y s p e c t r a [11]. The e n e r g y m a t r i c e s have

EIMeV}~

2.0

1.0

Jff

sz l e 5~ s6Ba .~ ,e s2:~m theory exp.

j ~ ---f "6..~-6:

_ 2 ~k

/ , / /

0 + 0"

Fig. 1. The theoret ical and experimental level schemes for the doubly-even N= 82 nuclei for the f i rs t J y = 0 +, 2 +, 4 + and 6 + levels.

211

Volume 35B, number 3 ] ? H Y S I C S L E T T E R S 24 May 1971

Table 1 The l ifet ime for the i somer i c jTr = 6 + level as calculated together with the exper imenta l half- l ife. Also the conve r -

sion coefficient and the gamma half- l i fe for the E2 6 + ~ 4 + as well as the t rans i t ion energy are given.

EN (MeV) T½(~) (sec.) ol T T½ (theory) T½ (exp.) (s) (s)

134Te 0.115 0.434 × 10 -6 1.39 0.181 x 10 -6 0.162 × 10 -6

136Xe 0.197 0.361 x 10 -6 0.19 0.303 × 10 -6 2.8 × 10 -6

138Ba 0.192 1.46 × 10 -6 0.19 1.23 x 10 -6 0.8 x 10 -6

140Ce 0.025 988 x 10 -1 676.0 1460 x 10 -6 6.24 x 10 -6

142Nd 0.108 2.58 × 10 -4 1.65 97.5 x 10 -6 16.0 x 10 -6

144Sm 0.021 3.13 x 10 -2 2233.0 14.0 x 10 -6 0.2-20.0 × 10-6

Table 2 The impor tan t configurat ions for the JY - 6 + and 4 + levels and the i r amplitudes. In the case of 140Ce(1) means the

calculat ion with t r i p l e t - t o - s ing l e t ra t io t = + 0.2 where (2) means t = + 1.0.

j~" _ 4 + 134Te 136Xe 138Ba 140Ce(1) 140Ce(2) 142Nd 144Sm

(2d5/2) 2 0.038 0.075 0.132 0.614 0.657 0.939 0.952

(lg7/2) 2 0.991 0.983 0.968 0.623 0.617 0.186 0.151

(2d5/2 lg7/2) 0.085 0.136 0.187 0.452 0.407 0.191 0.184

j~" = 6 +

( lg7/2)2 0.988 0.974 0.935 0.306 0.411 0.222 0.169

(2d5/2 lg7/2) 0.152 0.223 0.353 0.952 0.911 0.975 0.985

b e e n c o n s t r u c t e d and d i a g o n a l i z e d in an o r t h o - n o r m a l i z e d two q u a s i - p a r t i c l e (2QP) b a s i s a f t e r p r o j e c t i n g out t he s p u r i o u s J ~ = 0 + s t a t e [12]. In t he c a s e of 1 3 4 T e , a n e x a c t s h e l l - m o d e l c a l c u l a - t i o n cou ld b e p e r f o r m e d by p u t t i n g t he o c c u p a t i o n p r o b a b i l i t i e s of t he s i n g l e - p a r t i c l e p r o t o n s t a t e s v 2 = 0 .

The l e v e l p o s i t i o n of the f i r s t J 7 ~ = 0 +, 2 +, 4 + and 6 + l e v e l s a s c a l c u l a t e d i s s h o w n in f ig. 1, t o g e t h e r w i t h t he c o r r e s p o n d i n g e x p e r i m e n t a l l e v e l s . To t e s t a l s o t he w a v e - f u n c t i o n s , t h e l i f e - t i m e of t h e J ~ = 6 + l e v e l h a s b e e n c a l c u l a t e d and a c c o u n t s v e r y w e l l f o r t h e i s o m e r i c s t a t e . In a l l c a s e s , c o r r e c t i o n f o r E2 c o n v e r s i o n h a s b e e n p e r f o r m e d t h r o u g h u s e of t he i n t e r p o l a t i o n p r o - g r a m a s g i v e n by H a g e r and S e l t z e r [13]. In t a b l e 1, t h e s e c o n v e r s i o n c o e f f i c i e n t s a r e g i v e n , t o - g e t h e r w i th the g a m m a h a l f - l i f e , t he t o t a l e x - p e r i m e n t a l and t h e o r e t i c a l h a l f - l i f e .

In s o m e c a s e s , i n s t e a d of t he o c c u p a t i o n p r o - b a b i l i t i e s a s o b t a i n e d f r o m t he IGE p r o c e d u r e , e x p e r i m e n t a l v a l u e s [14] h a v e b e e n u s e d in o r d e r to look f o r t he i n f l u e n c e on t he h a l f - l i f e . T h i s d e p e n d e n c e i s s e e n to b e v e r y i m p o r t a n t in 136Xe

and 140Ce w h e r e t he v a l u e of B(E2 ; 6 + ~ 4 +) i s no t s t a b l e a g a i n s t s m a l l v a r i a t i o n s in e i t h e r t he o c c u p a t i o n p r o b a b i l i t i e s o r t he w a v e - f u n c t i o n s . A s a g e n e r a l p o i n t , we r e m a r k t h a t t he J ~ = 4 + l e v e l h a s m a i n l y t h e ~ l g / )2 c o n f i g u r a t i o n , bu t 7 ,2 c h a n g e s n a t u r e in l~UCe w h e r e i t b e c o m e s a m i x t u r e of ( l g )2, (2d5/2)2 and (2d / l g ) 7 /2 5,.2 7/2 w i t h r e s p e c t i v e l y 0 .623 , '0.614 and 0.452 a s a m - p l i t u d e s ; t h i s b e c a u s e the 1QP e n e r g i e s f o r the l g 7 / 2 and 2 d . ~ / ) p r o t o n s t a t e s a r e c r o s s i n g in 140'Ce. F o r i : t~Nd, 144Sm the (2d 5/2) 2 c o n f i g u r a - t i o n i s the m a i n c o m p o n e n t . As to t he J ~ = 6 + l e v e l , we h a v e a ( l g7 /2 )2 c h a r a c t e r up to 140Ce w h e r e t he (2d5 /2 l g 7 / 2 ) c o n f i g u r a t i o n b e c o m e s t he i m p o r t a n t p a r t . The p r e c i s e a m p l i t u d e s f o r t h e s e i m p o r t a n t c o n f i g u r a t i o n s a r e g i v e n in t a b l e 2.

T h r o u g h u s e of o n e - n u c l e o n t r a n s f e r r e a c t i o n s to l e v e l s J ~ of t h e d o u b l y - e v e n nuClei , one h a s a p o s s i b i l i t y of o b t a i n i n g the ( l g7 /2 ) 2, (2d5/2) 2 and (2d5 /21gT/2 ) a m p l i t u d e s of t h e s e s t a t e s . The s p e c t r o s c o p i c f a c t o r Sl(~,J) f o r a s t r i p p i n g - r e a c t i o n f r o m an o d d - A n u c l e u s (wi th g r o u n d - s t a t e sp in 9 ) to a n e x c i t e d s t a t e in the d o u b l y -

212

Volume 35B, number 3 P H Y S I C S

even n u c l e u s wi th a n g u l a r m o m e n t u m J ~ , can be w r i t t e n a s :

Sl(~,J) = ~ Sj(~,J) j :l~½

with

x (5 aj5 b~- (-) J+J+~Sa~Sbj)l 2

in which ~i)(ab,J) a r e the a m p l i t u d e s fo r the ith l e v e l wi th sp in J ~ f o r the v a r i o u s c o m p o n e n t s ,

In 134Te,_ the t r a n s i t i o n is n e a r l y a n u r e 136Xe h o w e v e r , ( lg7/2)~+~4+ E2 t r a n s i t i o n . In

the r e s u l t i s not s ing le . The e x p e r i m e n t a l va lue g - -

as m e a s u r e d by C a r r a z et al. [2] g i v e s 2.8 g s e c which does not fo l low a s m o o t h b e h a v i o u r f r o m 0 . 1 6 2 # s e c i n 1 3 4 T e t o 0.8 g s e c in 138Ba. Th i s l i t t l e bump is r e p r o d u c e d if the occupa t ion p r o - b~b i l i ty of the lg7 /2 p r o t o n s t a t e i s taken as Ylg7/2 = 0•5 in a g r e e m e n t wi th a p u r e s h e l l - m o d e l p i c t u r e f o r 136Xe with a va lue of T~ = 2•67 g s e c . T h e e x p e r i m e n t a l va lue g i v e s V~g7~ 2 = 0•437

2 0 005 [14]. If Ylg • = 0.5 i s cons id /e red , the

• ~ / 2 . l a r g e t e r m c o m i n g ~ r o m the (lg7/2)6+_~4+ t r a n s i - t ion b e c o m e s z e r o th rough the o c c u r r e n c e of the

2 2 f a c t o r (UlgT/p.- b'lg~/2) in the e x p r e s s i o n of B(E2; 6 + ~ '4 -+) and '~n ly s m a l l t e r m s add up to a v a l u e of B(E2; 6 + ~ 4 + ) = 0.679 e2fm 4. F o r 138Ba, the ( lg7/2)6+~4 + 2QP t r a n s i t i o n is aga in the i m p o r t a n t p a r t and g i v e s a l s o a s t ab l e r e s u l t a g a i n s t s m a l l c h a n g e s in the a m p l i t u d e s and o c - cupa t ion p r o b a b i l i t i e s . F o r 138Ba, the g o o d n e s s of the ( lg7 /2) 2 c o n f i g u r a t i o n has b e e n t e s t e d in a 139La(d, 3He) 138Ba r e a c t i o n [15] wi th as a m p l i - tude f o r th i s componen t in the J U = 4 + s ta te ; 1 . 1 5 " . F o r the 6 l e v e l , the ( lg7/2)2 and (2d5/2 lg7 /2) a m p l i t u d e s a r e m e a s u r e d as 0.882 and 0.453 which a g r e e v e r y we l l wi th the t h e o r e - t i c a l v a l u e s of 0.935 and 0.353• A c c i d e n t a l c a n - c e l l a t i o n e f f e c t s b e t w e e n the c o n t r i b u t i o n s to B(E2; 6 + ~ 4 +) f r o m c o n f i g u r a t i o n s in 140Ce g i v e s aga in an u n s t a b l e va lue fo r the ha l f - l i f e • In changing the t r i p l e t - t o - s i n g l e t r a t i o to t = 1.0 and a d j u s t i n g V o t ohave a good r e p r o d u c t i o n of the t h e o r e t i c a l l e v e l s c h e m e , a va lue of 7½ = 9.71 ~ s e c is obtained• A l s o us ing the e x p e r i m e n - t a l da ta of Wi lden tha l [14] on the occupa t ion p r o - b a b i l i t i e s wi th the o r i g i n a l ( t = +0•2) w a v e - func t ions g ive s T_~ = 19•2 g s e c . In the o n e - n u c l e o n

2

t r a n s f e r r e a c t i o n , only the (2d5/2 lg7 /2) a m p l i t u d e in JU = 6 + could be e x t r a c t e d with 1.25 as r e su l t • T h i s a l s o p r o v e s the chang ing s t r u c t u r e of the

* Values l a rger than 1.0 for the amplitudes can be ob- tained as the e r r o r on the spectroscopic factors is about 30-40%.

L E T T E R S 24 May 1971

i s o m e r i c JU = 6 + l e v e l in p a s s i n g 140Ce. In the c a s e of 142Nd and 144Sm, aga in s t ab l e

r e s u l t s o c c u r f r o m the (2d5/2 lg7/2)6+ ~ (2d5/2)~+, - and (2d5/2 lg7/2)~+:_~+ con t r i bu t i ons to the B(E2) value• A l s o on l~2Nd, the s t r i p p i n g r e a c t i o n 1 4 1 p r ( 3 H e , d)142Nd has been p e r f o r m e d [15] wi th fo r the JU = 4 + s ta te as main c o n f i g u r a t i o n (2d5/2) 2 and a m p l i t u d e 1•138. In 144Sm, the c a l - cu la t ed va lue of T~_ = 14 # s e c f i t s in the v e r y l a r g e e x p e r i m e n t a l e r r o r .

So we can conc lude that the i s o m e r i c l e v e l in a l l N=82 s i n g l e - c l o s e d she l l nuc l e i i s due to a JU = 6 + 2QP con f igu ra t i on which is ma in ly ( lg7/2) 2 up to 140Ce, but then changes in a (2d5/21~7/2) con f igu ra t ion . T h i s c o n c l u s i o n is f u r t h e r suppor t ed by the o n e - n u c l e o n s t r i p p i n g and p i c k - u p r e a c t i o n da ta of the Mich igan group [15]. A l so , the s a t i s f a c t o r y d e s c r i p t i o n of the h a l f - l i v e s po in t s to th is kind of de sc r i p t i on •

The a u t h o r s would l ike to thank Pro f . J . L. V e r h a e g h e fo r his i n t e r e s t in th i s work . They a r e g r a t e f u l to B. H. Wi lden tha l fo r c o m m u n i c a t i n g h i s m o s t r e c e n t da ta on the e v e n - e v e n N = 8 2 nuc l e i and f r o m the Mich igan e x p e r i m e n t a l group• They a l s o l ike to e x p r e s s t h e i r g r a t i t u d e to P ro f . C. C. G r o s j e a n and his s taf f f o r ex tend ing computing facilities on the IBM 360/30 ordinator of the "Rekenlaboratorium RUG".

References [1] w. John, F.W. Guy and J. J. Wesolowski, Phys. Rev.

C, vol. 2 (1970) 1451. [2] L. C. Carraz , J. Blaehot, E. Monnand and A. Moussa

Nucl. Phys. A158 (1970) 403. [3] L. C. Carraz , J. Blachot, E. Monnand et A. Moussa,

Compt. Rendus, sdrie B, tome 271 (1970) 545. [4] H. W. Baer, J. J. Reidy and M• L. Wiedenbeck,

Nucl. Phys. A l l 3 (1968) 33. [5] H. Krehbeil, Phys. Letters 13 (1964) 65. [6] J. Kownaeki, Z. Sujkowski, L.E. Frdberg, H. Hyde

and J. Adam, Contrib. to Conf. on Propert ies of nuclear states (Addendum, Montreal, 1969) p. 4.

[7] R. Arlt et al., Dubna preprint P6-4246 (1969), [8] W. H. Wildenthal, E. Newman and R. L. Auble. Phys.

Letters 27B (1968) 628. [9] V. Gillet, B. Giraud and M. Rho, Nucl. Phys. A103

(1967) 257. [10] K. Heyde and M. Waroquier, Phys. I_~tters 29B

(1969) 147. [11] M. Waroquier and K. Heyde, Nucl. Phys., to be

published• [12] M. Waroquier and K. Heyde, Nucl. Phys. A144

{1970) 481• [13] R. S. Hager and E. C. Seltzer , Nuclear Data A4

(1968) 1. [14] B. H. Wildenthal, private communication and Phys.

Rev., to be published. [15] W. Jones, L• Borgmann, W. C. Parkinson and J.

Bardwick, private communication from B. H. Wildenthal.

213