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Semimicroscopic description of even Hg spectra
Volume 22, number 5
PHYSICS
SEMIMICROSCOPIC
LETTERS
DESCRIPTION
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Hg
SPECTRA
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G. A L A G A ** and G. IALONGO University o f California, L a w r e n c e Radiation Laboratory, B e r k e l e y and Institut de Physique Nucl~aire, Division de Physique ThSorique-Orsay and N e w Y o r k University, N e w Y o r k Received 15 August 1966
A qualitative description of the properties of the low lying states of the even even nuclei around lead and tin isotopes is proposed in t e r m s of protons (holes) coupled to a harmonic vibrator.
In the p r e s e n t c o m m u n i c a t i o n we would l i k e to r e p o r t s o m e r e s u l t s f o r the e v e n m e r c u r y i s o t o p e s o b t a i n e d u s i n g the m o d e l of two p r o t o n h o l e s c o u p l e d to a v i b r a t o r [1]. The m o t i v a t i o n f o r s u c h a d e s c r i p t i o n is g i v e n by the e x p e r i m e n t a l r e s u l t s [2] that in the e v e n e v e n n u c l e i a r o u n d P b , Sn, e t c . s o m e of the low l y i n g s t a t e s of t h e s e n u c l e i s h o w a " v i b r a t i o n l i k e " s t r u c t u r e . F r o m t h e . e x p e r i m e n t s , ft~rther~ m o r e , is a p p a r e n t that the d e v i a t i o n s f r o m the h a r m o n i c v i b r a t i o n s s e e m to be on a v e r a g e 20-30%. T y p i c a l d e v i a t i 0 r ~ a r e in the l e v e l p o s i t i o n s , the v i o l a t i o n oi the s i m p l e h a r m o n i c v i b r a t o r s e l e c t i o n and i n t e n s i t y r u l e s [3], p o s s i b l e d i f f e r e n c e s in the m a g n e t i c m o m e n t s of the e x c i t e d s t a t e s [4] and the e x i s t e n c e of the q u a d r u p o l e m o m e n t s in the e x c i t e d s t a t e s [5]. In a d d i t i o n to t h e s e d e v i a t i o n s in s o m e n u c l e i new l e v e l s h a v e b e e n found in the r e g i o n of the two phonon s t a t e s which do not s h o w v i b r a t i o n a l c h a r a c t e r i s t i c s . The c h a r a c t e r i s t i c s o f the h i g h e r e x c i t e d s t a t e s s e e m to be a l a r g e n u m b e r of l e v e l s w h o s e s t r u c t u r e is only p a r f i a l l y r e s o l v e d [2]. The p o s i t i o n of the f i r s t e x c i t e d 2~ s t a t e s ii~ l e a d and tin ~ s o t o p e s c o m p a r e d to t h o s e of A + 2 and A ± 4 2 3 is a p p r o x i m a t e l y 2 : 1 and 3 : 1 r e s p e c t i v e l y . We a s s u m e in the c a l c u l a t i o n s that the two p r o t o n h o l e s of the Hg i s o t o p e s a r e d i s t r i b u t e d in the s i n g l e h o l e s h e l l m o d e l s t a t e s dzl - s½ = = 0.350, h~ s !2 = 1.343 and dai - s~1 = 1.673 M e V r e s p e c t i v e l y [6]. The i n t e r a c t i o n b e t w e e n the two h o l e s is a s s u m e d to be a p a i r i n g f o r c e (G =0.1 -
* Part of this work was performed under AEC contract No W-7405-eng-48 and part supported by the U. S. Army Research Office (Durham). ** On leave of absence from the University of Zagreb and the Rudjer Bogkovid Institute. Zagreb.
M e V ) . The i n t e r a c t i o n of the p r o t o n h o l e s with the v i b r a t o r is taken in the f o r m of the h o l e s u r f a c e i n t e r a c t i o n [7] (a = (k>/(~fw/87rC2)½ = 0.6 MeV with ~w = 1.027 MeV, C 2 = 545 MeV and (k> = 70.5 MeV). As we do not e x p e c t the p e r t u r b a t i o n t r e a t m e n t to be v a l i d in this c a s e we p e r f o r m a d i a g o nalisation. T h e c a l c u l a t i o n was p e r f o r m e d in f o u r s t e p s in o r d e r to s t u d y the ~ ) r o p e r t i e s of the m o d e l and to o b t a i n r e s u l t s to be c o m p a r e d to the e x p e r i m e n t s . In the f i r s t c a s e we took the s ±z and d~ and v i b r a t-0r s t a t e s up to two p h o n o n s . C a s e 2 is c a s e 1 with the t h r e e phonon s t a t e s added. C a s e 3 is c a s e 1 w i t h the a d d i t i o n of c o n f i g u r a t i o n s h~z and d~. C a s e 4 is c a s e 1 with t h r e e p h o n o n s and h~ and d~z added. The energy levels resulting from our treatm e n t a r e s h o w n in f i g . 1 and s h o u l d be c o m p a r e d to the e x p e r i m e n t a l l e v e l s in f i g . 2. The c a l c u l a t i o n of the B ( E 2 ) ' s r e l a t i v e l y to the g r o u n d s t a t e t r a n s i t i o n (B(E2) =-0.217 e2 × 10-48 c m 4 = " = 12 B ( E 2 ) s p ) , s h o w that the 21 41 22 and 02 a r e the l e v e l s of the " v i b r a t i o n l i k e " s t r u c t u r e . 23 and 11 a r e l e v e l s , o f d i f f e r e n t c h a r a c t e r . The 6 b l l e c t i v e n a t u r e of the s t a t e s and the a p p r o x i m a t e i n t e n s i t y r u l e s s h o w up in the c o h e r e n t c o n t r i b u t i o n of the h o l e s and the v i b r a t o r to the e n h a n c e m e n t of the E(2) t r a n s i t i o n p r o b a b i l i t i e s of the c o u p l e d s y s t e m . The q u a d r u p o l e r e d u c e d t r a n s i t i o n p r o b a b i l i t i e s 21 --* 01 , 22 -'-* 21 and 41 --* 21 a r e c o n s i d e r a b l y e n h a n c e d w h i l e 02 ---* 21 is of the s p i n p a r t i c l e m a g n i t u d e [B(E2) (21 ---* 0 1 ) / ( B ( E 2 ) (22 -* 21) ~ 3]. The a p p r o x i m a t e s e l e c t i o n r u l e s f o r the 22 --* 01, 22 -'* 41, 23 --* 01, 23 -* 21, 23 '-* 41 and 23 --* 22 t r a n s i t i o n s a r e a r e s u l t of the " a p p r o x i m a t e c a n c e l l a t i o n s " of the c o n t r i b u t i o n s of the h o l e s and v i b r a t o r . T h i s i n d i c a t e s 619
Volume 22, n u m b e r 5
PHYSICS
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15 September 1966
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LETTERS
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Fig. 1. Calculated positive parity states in 198Hg for cases 1, 2, 3, 4 (a =0.6). t h a t t h e e n e r g i e s a n d t h e i n t e n s i t i e s of t h e " a l lowed" collective B(E2)components will be rather i n d e p e n d e n t of t h e c o n f i g u r a t i o n s a n d r e a s o n a b l y well reproduced by our calculations. The transitions violating the "selection rules" and the prop e r t i e s of t h e n o n c o l l e c t i v e s t a t e s w i l l d e p e n d m o r e s e n s i t i v e l y o n t h e d e t a i l s of n u c l e a r d y n a m i c s *. T h e c a l c u l a t e d q u a d r u p o l e m o m e n t of t h e 620
f i r s t e x c i t e d s t a t e 21 in 198Hg r e l a t i v e to t h e 21 ~ 01 t r a n s i t i o n s i s Q = 0.5 e × 1 0 - 2 4 c m 2. A value intermediate between the single particle and the rotational quadrupole moment and almost the * Calculations with m o r e r e a l i s t i c forces between the proton holes have been p e r f o r m e d recently by A. Covello and G. S a r t o r i s i l l u s t r a t i n g the points mentioned (private communication).
Volume 22, number 5 1365
PHYSICS LETTEt~S
'2+ 1422
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1255
lOS4 1039
4+ lOgO Z+ 1047
426
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Fig. 2. Experimentally known lowest lying states in Hg isotopes. An alternative assignment as 0+ of the 1.029 MeV level in 200Hg has been preposed [2]. s a m e as that of the 197Au in the ground state (Q = 0.56e ×10 -24 cm2). This s u g g e s t s in addition that our coupled s y s t e m might be d e f o r m e d and have "rotation like" s t r u c t u r e . This is also a p p a r e n t f r o m our fig. 1 in which the 61 state is around 2 MeV. Indicating that at this stage it is difficult to tell the "rotational" f r o m the " v i b r a tional" s t r u c t u r e in this region. Recently a s e a r c h for the high spin levels in Hg isotopes [8] r e s u l t i n g f r o m the e l e c t r o n capture from the 7 + i s o m e r i c state of even thalium isotopes has failed. This c e r t a i n l y i n d i c a t e s that the 61 state in Hg isotopes is much higher or, as in our d e s c r i p t i o n , has a different s t r u c t u r e f r o m thalium i s o m e t r i c s t a t e s . This e x p e r i m e n t together with our c a l c u l a t e d quadrupole m o m e n t s for the 41 and 61, which a r e positive and of the s a m e o r d e r of magnitude as that for the ground state, gives s e r i o u s support for the e x i s t e n c e of a " r o t a t i o n a l band" in Hg isotopes. The f a i l u r e to detect the high spin s t a t e s in Hg isotopes we a t t r i b u t e to K - f o r b i d d e n n e s s . E v e n f o r a c o n s t r u c tive e s t i m a t e the r e t a r d a t i o n factor is of the o r d e r of magnitude 104 - 105. The quadrupole m o m e n t s of 22 and 23 s t a t e s a r e slightly s m a l l e r than that of 21 and negative. The B(M1) for the magnetic component of the t r a n s i t i o n s 22 ~ 21 and 23 --- 22 a r e r e t a r d e d a few hundred t i m e s c o m p a r e d to the single p a r t i cle value while 23 ---21 t r a n s i t i o n is of the single p a r t i c l e magnitude. The magnetic m o m e n t of the 21 state of T48Hg has been m e a s u r e d [4] to be = 0.76 n . m . F r o m our calculation it s e e m s that a l m o s t its e n t i r e c o n t r i b u t i o n is f r o m the o r b i t a l motion of the two holes, so the c o n t r i b u t i o n of the v i b r a t o r to the magnetic m o m e n t is s m a l l , gR ~ 0. The r e t a r d a t i o n of the 22 ~ 21 and 23 --- 22 magnetic components is again a r e s u l t of the app r o x i m a t e c a n c e l l a t i o n of the o r b i t a l and spin cont r i b u t i o n s of the p r o t o n holes. In the 23 ~ 21 t r a n s i t i o n they add up and give the m e n t i o n e d r e -
15 September 1966
s u i t s . The m a g n e t i c m o m e n t of 2 2 state is a p p r e ciably different f r o m the magnetic m o m e n t of the ground state b e c a u s e of the " c a n c e l l a t i o n " of the spin and o r b i t a l c o n t r i b u t i o n to the magnetic m o ment. B e s i d e s the positive p a r i t y s t a t e s we also obtain low-lying s t a t e s of negative p a r i t y m i s s i n g in fig. 1. F o r the case 4 they should be at the e n e r gies E 5_ = - 0 . 1 1 5 , E 6_=-0.099, E 4_ = 0 . 2 1 5 a n d E 7- = 0.215 MeV, r e s p e c t i v e l y . F o r 198Hg they should be below and around 2 MeV. A doublet of such states has been found e x p e r i m e n t a l l y [8] at 1.6 MeV. It is p r o b a b l y the 51 ~ 71 doublet, b e cause o u r calculated B(E2) is for the 7~ ~ 51 t r a n s i t i o n of collective type c o n t r a r y to the 6~ -~ 5~ t r a n s i t i o n which is below single p a r t i c l e value. At the s a m e time this e x p e r i m e n t s e e m s to indicate that 71 and 7 + i s o m e r i c state in thalium have s i m i l a r s t r u c t u r e b e c a u s e of r e l a t i v e l y l a r g e l o g f t ~ 6.2 value. The 5~ ~ 4~ E(1) t r a n sition is also expected to be s l o w e r on the account of K - f o r b i d d e n n e s s . We hope that this conclusion will not change s e r i o u s l y by including octupole v i b r a t i o n s with ~w ~ 2.6 MeV in our p i c t u r e . F r o m the foregoing d i s c u s s i o n it is a p p a r e n t that b e s i d e s of a l r e a d y mentioned p r o p e r t i e s of 198Hg we p r e d i c t for the 22 ~ 21 t r a n s i t i o n a mixed E(2) +M(1) while for 23 ~ 21 t r a n s i t i o n a p u r e M(1) c h a r a c t e r . 23 --* 21 is of single p a r t i cle magnitude. The 23 ~ 22 t r a n s i t i o n a p p e a r s in o u r d e s c r i p t i o n r a t h e r weak and for us hard to decide on its c h a r a c t e r . We might j u s t say that our r e s u l t s a r e in qualitative a g r e e m e n t with the e x p e r i m e n t s [2]. We have also p e r f o r m e d c a l c u l a t i o n s for odd A thalium and 'gold isotopes coupling one and t h r e e holes to the v i b r a t o r . We find the r e s u l t s in good a g r e e m e n t with the e x p e r i m e n t s and the r e s u l t s r e p o r t e d here [9]. F r o m this, it s e e m s that the method used in these c a l c u l a t i o n s being r a t h e r s i m p l e might be useful in d e s c r i b i n g p r o p e r t i e s of nuclei around the single closed s h e l l s . In the s a m e way as we find collective and two p a r t i c l e s t a t e s in Hg as well as m i x t u r e s of them, we find collective and one p a r t i c l e s t a t e s in T1 and also evidence for a s i m i l a r level s t r u c t u r e in Au i s o topes. Due to our s c h e m a t i c t r e a t m e n t of the n e u t r o n holes coupling to the p r o t o n s h e l l s "effective neut r o n s " f r o m which we took only the collective s t a t e s , we expect that our p i c t u r e will b r e a k down for higher excited s t a t e s . We a r e m i s s i n g the single n e u t r o n e x c i t a t i o n s . The higher excited s t a t e s in fig. 1 a r e not expected to r e p r e s e n t all the s t a t e s and they a r e given for the sake of c o m parison. 621
Volume 22, number 5
PHYSICS
In t h e m i c r o s c o p i c d e s c r i p t i o n of t h e s e n u c l e i [10] t h e p r o t o n n e u t r o n i n t e r a c t i o n of t h e " v a l e n t " n e u t r o n s with the p r o t o n c l o s e d s h e l l s p r o v i d e a m o r e r e a l i s t i c d e s c r i p t i o n of t h e " e f f e c t i v e n e u t r o n s " . The i n t e r a c t i o n of t h e " e f f e c t i v e n e u t r o n s " w i t h t h e " v a l e n t " p r o t o n s on t h e o t h e r h a n d g i v e the i n t e r f e r e n c e e f f e c t s we have b e e n d i s c u s s i n g previously. Due to the d i f f i c u l t i e s in c a r r y i n g out t h i s p r o g r a m t h e m o r e d e t a i l e d c o m p a r i s o n of t h e r e s u l t s of t h e two a p p r o a c h e s i s n o t p o s s i b l e a t p r e s e n t . T h e c o m p u t a t i o n s h a v e b e e n c a r r i e d out a t t h e c o m p u t a t i o n d e p a r t m e n t U C L R L with the p r o g r a m w r i t t e n b y R. G o l d s t e i n w h o s e h e l p i s g r a t e f u l l y acknowledged. D i s c u s s i o n s w i t h t h e s u m m e r v i s i t o r s of 1965 in B e r k e l e y , A. B o h r a n d B. R. M o t t e l s o n in C o p e n h a g e n a n d R. A r v i e u , L o m b a r d a n d M. V ~ n ~ r o n i in O r s a y h a v e b e e n v e r y s t i m u l a t i n g .
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References 1. N. Bijedi~, P h y s i c s L e t t e r s 16 (1965) 47; N. Bijedi~, Thesis, University of Zagreb 1964 (unpublished) ; G. Alaga and G. Ialongo, UCRL - 16385 (1965); G. Ialongo and G. Alaga, Bull. Am. Phys. Soc. 11 (1966) 102. 2. Nuclear Data Cards B. Jung and G. Anderson, Nucl. Phys. 15 (1960) 108; L. V. Groshev, A.M. Demidov, V.A. Ivanov, V.N. Lutsenko and V. I. Pelekhov, Izvest. Akad. Nauk SSR, Ser. Phys. 27 (1963) 1377; R. E. Segel, R.K. Smither and R. T. Carpenter, Phys. Rev. 133 (1964) B583; M. Sakai, H. Ikegami, T. Yamazaki and K. Saito, Nucl. Phys. 65 (1965) ;
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M. Sakai, T. Yamazaki and E. Jiri, INS-Report 77, Tokyo, 1965; B. P. Maier, U. Gruber, H. R. Koch and O. W. B. Schult, Z, Phys. 185 (1965) 478; J. A. M. de villiers, C.A. Engelbrecht, W.G. Vonaeh and A. B. Smith, Neutron Scattering from Au, Hg and T1 (1965) (preprint). J. J. Kraushaar and M. Goldhaber, Phys. Rev. 89 (1953) 1081; A. de Shalit and M. Goldhaber, Phys. Rev. 92 (1953) 1211. L. Keszthelyi, I. Berkes, I. D~zsi, B. Moln~r and L. Pocs, Fhysics Letters 8 (1964) 195; J.A. Cameron, Can. J. Phys. 42 (1964) 1681; H. J. K~Srner and U. Ortabasi, Nucl. Phys. 70 (1965) 28; Y. K. Agarwal, C. V. K. Baba and S. K. Bhattacherjee~ Nucl. Phys. 79 (1966) 437. J. de Boer, R. G. Stokstad, G. D. Symons and A. Winther, Phys. Rev. Letters 14 (1965) 564; I. Hall, private communication. S. Hinds, R. Middleton, J.H. Bjerregaard, O. Hansen and O. Nathan, Physics Letters 17 (1965) 302; D. Eccleshall and M. J. L. Yates, Physics Letters 19 (1965) 301. A. Bohr, Mat. Fys. Medd. Dansk. Vid. Selsk. 28 no 4 (1952); A. Bohr and B. R Mottelson, Mat. Fys. Medd. Dansk. Vid. Selsk. 27 no 16 (1953); D. C. Choudhury, Mat. Fys. Medd. Dansk. Vid. Selsk. 28 no 4 (1954); G. Scharf-Goldhaber and J. Weneser, Phys. Rev. 98 (1955) 212; D. G. Raz. Phys. Rev. 114 (1958) 1116. R. F. Petry, R.A. Neumann and J. S. Evans, P h y s i c s L e t t e r s 21 (1966) 541. G. Alaga, Bull. Am. Phys. Soc. 11, V. 4 (1959) 359; G. Ialongo, Thesis, New York University (1966). L.S. K i s s l i n g e r and R. S. S!trensen, Rev. Mod. Phys. 35 (1963) 853; R. Arvieu and M. V~n~roni, Compt. Rend. 250 (1960) 992, 2155; T. Tamura and T. Udagawa, Phys. Rev. L e t t e r s 15 (1965) 765.
PHYSICS
SEMIMICROSCOPIC
LETTERS
DESCRIPTION
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Hg
SPECTRA
*
G. A L A G A ** and G. IALONGO University o f California, L a w r e n c e Radiation Laboratory, B e r k e l e y and Institut de Physique Nucl~aire, Division de Physique ThSorique-Orsay and N e w Y o r k University, N e w Y o r k Received 15 August 1966
A qualitative description of the properties of the low lying states of the even even nuclei around lead and tin isotopes is proposed in t e r m s of protons (holes) coupled to a harmonic vibrator.
In the p r e s e n t c o m m u n i c a t i o n we would l i k e to r e p o r t s o m e r e s u l t s f o r the e v e n m e r c u r y i s o t o p e s o b t a i n e d u s i n g the m o d e l of two p r o t o n h o l e s c o u p l e d to a v i b r a t o r [1]. The m o t i v a t i o n f o r s u c h a d e s c r i p t i o n is g i v e n by the e x p e r i m e n t a l r e s u l t s [2] that in the e v e n e v e n n u c l e i a r o u n d P b , Sn, e t c . s o m e of the low l y i n g s t a t e s of t h e s e n u c l e i s h o w a " v i b r a t i o n l i k e " s t r u c t u r e . F r o m t h e . e x p e r i m e n t s , ft~rther~ m o r e , is a p p a r e n t that the d e v i a t i o n s f r o m the h a r m o n i c v i b r a t i o n s s e e m to be on a v e r a g e 20-30%. T y p i c a l d e v i a t i 0 r ~ a r e in the l e v e l p o s i t i o n s , the v i o l a t i o n oi the s i m p l e h a r m o n i c v i b r a t o r s e l e c t i o n and i n t e n s i t y r u l e s [3], p o s s i b l e d i f f e r e n c e s in the m a g n e t i c m o m e n t s of the e x c i t e d s t a t e s [4] and the e x i s t e n c e of the q u a d r u p o l e m o m e n t s in the e x c i t e d s t a t e s [5]. In a d d i t i o n to t h e s e d e v i a t i o n s in s o m e n u c l e i new l e v e l s h a v e b e e n found in the r e g i o n of the two phonon s t a t e s which do not s h o w v i b r a t i o n a l c h a r a c t e r i s t i c s . The c h a r a c t e r i s t i c s o f the h i g h e r e x c i t e d s t a t e s s e e m to be a l a r g e n u m b e r of l e v e l s w h o s e s t r u c t u r e is only p a r f i a l l y r e s o l v e d [2]. The p o s i t i o n of the f i r s t e x c i t e d 2~ s t a t e s ii~ l e a d and tin ~ s o t o p e s c o m p a r e d to t h o s e of A + 2 and A ± 4 2 3 is a p p r o x i m a t e l y 2 : 1 and 3 : 1 r e s p e c t i v e l y . We a s s u m e in the c a l c u l a t i o n s that the two p r o t o n h o l e s of the Hg i s o t o p e s a r e d i s t r i b u t e d in the s i n g l e h o l e s h e l l m o d e l s t a t e s dzl - s½ = = 0.350, h~ s !2 = 1.343 and dai - s~1 = 1.673 M e V r e s p e c t i v e l y [6]. The i n t e r a c t i o n b e t w e e n the two h o l e s is a s s u m e d to be a p a i r i n g f o r c e (G =0.1 -
* Part of this work was performed under AEC contract No W-7405-eng-48 and part supported by the U. S. Army Research Office (Durham). ** On leave of absence from the University of Zagreb and the Rudjer Bogkovid Institute. Zagreb.
M e V ) . The i n t e r a c t i o n of the p r o t o n h o l e s with the v i b r a t o r is taken in the f o r m of the h o l e s u r f a c e i n t e r a c t i o n [7] (a = (k>/(~fw/87rC2)½ = 0.6 MeV with ~w = 1.027 MeV, C 2 = 545 MeV and (k> = 70.5 MeV). As we do not e x p e c t the p e r t u r b a t i o n t r e a t m e n t to be v a l i d in this c a s e we p e r f o r m a d i a g o nalisation. T h e c a l c u l a t i o n was p e r f o r m e d in f o u r s t e p s in o r d e r to s t u d y the ~ ) r o p e r t i e s of the m o d e l and to o b t a i n r e s u l t s to be c o m p a r e d to the e x p e r i m e n t s . In the f i r s t c a s e we took the s ±z and d~ and v i b r a t-0r s t a t e s up to two p h o n o n s . C a s e 2 is c a s e 1 with the t h r e e phonon s t a t e s added. C a s e 3 is c a s e 1 w i t h the a d d i t i o n of c o n f i g u r a t i o n s h~z and d~. C a s e 4 is c a s e 1 with t h r e e p h o n o n s and h~ and d~z added. The energy levels resulting from our treatm e n t a r e s h o w n in f i g . 1 and s h o u l d be c o m p a r e d to the e x p e r i m e n t a l l e v e l s in f i g . 2. The c a l c u l a t i o n of the B ( E 2 ) ' s r e l a t i v e l y to the g r o u n d s t a t e t r a n s i t i o n (B(E2) =-0.217 e2 × 10-48 c m 4 = " = 12 B ( E 2 ) s p ) , s h o w that the 21 41 22 and 02 a r e the l e v e l s of the " v i b r a t i o n l i k e " s t r u c t u r e . 23 and 11 a r e l e v e l s , o f d i f f e r e n t c h a r a c t e r . The 6 b l l e c t i v e n a t u r e of the s t a t e s and the a p p r o x i m a t e i n t e n s i t y r u l e s s h o w up in the c o h e r e n t c o n t r i b u t i o n of the h o l e s and the v i b r a t o r to the e n h a n c e m e n t of the E(2) t r a n s i t i o n p r o b a b i l i t i e s of the c o u p l e d s y s t e m . The q u a d r u p o l e r e d u c e d t r a n s i t i o n p r o b a b i l i t i e s 21 --* 01 , 22 -'-* 21 and 41 --* 21 a r e c o n s i d e r a b l y e n h a n c e d w h i l e 02 ---* 21 is of the s p i n p a r t i c l e m a g n i t u d e [B(E2) (21 ---* 0 1 ) / ( B ( E 2 ) (22 -* 21) ~ 3]. The a p p r o x i m a t e s e l e c t i o n r u l e s f o r the 22 --* 01, 22 -'* 41, 23 --* 01, 23 -* 21, 23 '-* 41 and 23 --* 22 t r a n s i t i o n s a r e a r e s u l t of the " a p p r o x i m a t e c a n c e l l a t i o n s " of the c o n t r i b u t i o n s of the h o l e s and v i b r a t o r . T h i s i n d i c a t e s 619
Volume 22, n u m b e r 5
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15 September 1966
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Fig. 1. Calculated positive parity states in 198Hg for cases 1, 2, 3, 4 (a =0.6). t h a t t h e e n e r g i e s a n d t h e i n t e n s i t i e s of t h e " a l lowed" collective B(E2)components will be rather i n d e p e n d e n t of t h e c o n f i g u r a t i o n s a n d r e a s o n a b l y well reproduced by our calculations. The transitions violating the "selection rules" and the prop e r t i e s of t h e n o n c o l l e c t i v e s t a t e s w i l l d e p e n d m o r e s e n s i t i v e l y o n t h e d e t a i l s of n u c l e a r d y n a m i c s *. T h e c a l c u l a t e d q u a d r u p o l e m o m e n t of t h e 620
f i r s t e x c i t e d s t a t e 21 in 198Hg r e l a t i v e to t h e 21 ~ 01 t r a n s i t i o n s i s Q = 0.5 e × 1 0 - 2 4 c m 2. A value intermediate between the single particle and the rotational quadrupole moment and almost the * Calculations with m o r e r e a l i s t i c forces between the proton holes have been p e r f o r m e d recently by A. Covello and G. S a r t o r i s i l l u s t r a t i n g the points mentioned (private communication).
Volume 22, number 5 1365
PHYSICS LETTEt~S
'2+ 1422
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lOS4 1039
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Fig. 2. Experimentally known lowest lying states in Hg isotopes. An alternative assignment as 0+ of the 1.029 MeV level in 200Hg has been preposed [2]. s a m e as that of the 197Au in the ground state (Q = 0.56e ×10 -24 cm2). This s u g g e s t s in addition that our coupled s y s t e m might be d e f o r m e d and have "rotation like" s t r u c t u r e . This is also a p p a r e n t f r o m our fig. 1 in which the 61 state is around 2 MeV. Indicating that at this stage it is difficult to tell the "rotational" f r o m the " v i b r a tional" s t r u c t u r e in this region. Recently a s e a r c h for the high spin levels in Hg isotopes [8] r e s u l t i n g f r o m the e l e c t r o n capture from the 7 + i s o m e r i c state of even thalium isotopes has failed. This c e r t a i n l y i n d i c a t e s that the 61 state in Hg isotopes is much higher or, as in our d e s c r i p t i o n , has a different s t r u c t u r e f r o m thalium i s o m e t r i c s t a t e s . This e x p e r i m e n t together with our c a l c u l a t e d quadrupole m o m e n t s for the 41 and 61, which a r e positive and of the s a m e o r d e r of magnitude as that for the ground state, gives s e r i o u s support for the e x i s t e n c e of a " r o t a t i o n a l band" in Hg isotopes. The f a i l u r e to detect the high spin s t a t e s in Hg isotopes we a t t r i b u t e to K - f o r b i d d e n n e s s . E v e n f o r a c o n s t r u c tive e s t i m a t e the r e t a r d a t i o n factor is of the o r d e r of magnitude 104 - 105. The quadrupole m o m e n t s of 22 and 23 s t a t e s a r e slightly s m a l l e r than that of 21 and negative. The B(M1) for the magnetic component of the t r a n s i t i o n s 22 ~ 21 and 23 --- 22 a r e r e t a r d e d a few hundred t i m e s c o m p a r e d to the single p a r t i cle value while 23 ---21 t r a n s i t i o n is of the single p a r t i c l e magnitude. The magnetic m o m e n t of the 21 state of T48Hg has been m e a s u r e d [4] to be = 0.76 n . m . F r o m our calculation it s e e m s that a l m o s t its e n t i r e c o n t r i b u t i o n is f r o m the o r b i t a l motion of the two holes, so the c o n t r i b u t i o n of the v i b r a t o r to the magnetic m o m e n t is s m a l l , gR ~ 0. The r e t a r d a t i o n of the 22 ~ 21 and 23 --- 22 magnetic components is again a r e s u l t of the app r o x i m a t e c a n c e l l a t i o n of the o r b i t a l and spin cont r i b u t i o n s of the p r o t o n holes. In the 23 ~ 21 t r a n s i t i o n they add up and give the m e n t i o n e d r e -
15 September 1966
s u i t s . The m a g n e t i c m o m e n t of 2 2 state is a p p r e ciably different f r o m the magnetic m o m e n t of the ground state b e c a u s e of the " c a n c e l l a t i o n " of the spin and o r b i t a l c o n t r i b u t i o n to the magnetic m o ment. B e s i d e s the positive p a r i t y s t a t e s we also obtain low-lying s t a t e s of negative p a r i t y m i s s i n g in fig. 1. F o r the case 4 they should be at the e n e r gies E 5_ = - 0 . 1 1 5 , E 6_=-0.099, E 4_ = 0 . 2 1 5 a n d E 7- = 0.215 MeV, r e s p e c t i v e l y . F o r 198Hg they should be below and around 2 MeV. A doublet of such states has been found e x p e r i m e n t a l l y [8] at 1.6 MeV. It is p r o b a b l y the 51 ~ 71 doublet, b e cause o u r calculated B(E2) is for the 7~ ~ 51 t r a n s i t i o n of collective type c o n t r a r y to the 6~ -~ 5~ t r a n s i t i o n which is below single p a r t i c l e value. At the s a m e time this e x p e r i m e n t s e e m s to indicate that 71 and 7 + i s o m e r i c state in thalium have s i m i l a r s t r u c t u r e b e c a u s e of r e l a t i v e l y l a r g e l o g f t ~ 6.2 value. The 5~ ~ 4~ E(1) t r a n sition is also expected to be s l o w e r on the account of K - f o r b i d d e n n e s s . We hope that this conclusion will not change s e r i o u s l y by including octupole v i b r a t i o n s with ~w ~ 2.6 MeV in our p i c t u r e . F r o m the foregoing d i s c u s s i o n it is a p p a r e n t that b e s i d e s of a l r e a d y mentioned p r o p e r t i e s of 198Hg we p r e d i c t for the 22 ~ 21 t r a n s i t i o n a mixed E(2) +M(1) while for 23 ~ 21 t r a n s i t i o n a p u r e M(1) c h a r a c t e r . 23 --* 21 is of single p a r t i cle magnitude. The 23 ~ 22 t r a n s i t i o n a p p e a r s in o u r d e s c r i p t i o n r a t h e r weak and for us hard to decide on its c h a r a c t e r . We might j u s t say that our r e s u l t s a r e in qualitative a g r e e m e n t with the e x p e r i m e n t s [2]. We have also p e r f o r m e d c a l c u l a t i o n s for odd A thalium and 'gold isotopes coupling one and t h r e e holes to the v i b r a t o r . We find the r e s u l t s in good a g r e e m e n t with the e x p e r i m e n t s and the r e s u l t s r e p o r t e d here [9]. F r o m this, it s e e m s that the method used in these c a l c u l a t i o n s being r a t h e r s i m p l e might be useful in d e s c r i b i n g p r o p e r t i e s of nuclei around the single closed s h e l l s . In the s a m e way as we find collective and two p a r t i c l e s t a t e s in Hg as well as m i x t u r e s of them, we find collective and one p a r t i c l e s t a t e s in T1 and also evidence for a s i m i l a r level s t r u c t u r e in Au i s o topes. Due to our s c h e m a t i c t r e a t m e n t of the n e u t r o n holes coupling to the p r o t o n s h e l l s "effective neut r o n s " f r o m which we took only the collective s t a t e s , we expect that our p i c t u r e will b r e a k down for higher excited s t a t e s . We a r e m i s s i n g the single n e u t r o n e x c i t a t i o n s . The higher excited s t a t e s in fig. 1 a r e not expected to r e p r e s e n t all the s t a t e s and they a r e given for the sake of c o m parison. 621
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In t h e m i c r o s c o p i c d e s c r i p t i o n of t h e s e n u c l e i [10] t h e p r o t o n n e u t r o n i n t e r a c t i o n of t h e " v a l e n t " n e u t r o n s with the p r o t o n c l o s e d s h e l l s p r o v i d e a m o r e r e a l i s t i c d e s c r i p t i o n of t h e " e f f e c t i v e n e u t r o n s " . The i n t e r a c t i o n of t h e " e f f e c t i v e n e u t r o n s " w i t h t h e " v a l e n t " p r o t o n s on t h e o t h e r h a n d g i v e the i n t e r f e r e n c e e f f e c t s we have b e e n d i s c u s s i n g previously. Due to the d i f f i c u l t i e s in c a r r y i n g out t h i s p r o g r a m t h e m o r e d e t a i l e d c o m p a r i s o n of t h e r e s u l t s of t h e two a p p r o a c h e s i s n o t p o s s i b l e a t p r e s e n t . T h e c o m p u t a t i o n s h a v e b e e n c a r r i e d out a t t h e c o m p u t a t i o n d e p a r t m e n t U C L R L with the p r o g r a m w r i t t e n b y R. G o l d s t e i n w h o s e h e l p i s g r a t e f u l l y acknowledged. D i s c u s s i o n s w i t h t h e s u m m e r v i s i t o r s of 1965 in B e r k e l e y , A. B o h r a n d B. R. M o t t e l s o n in C o p e n h a g e n a n d R. A r v i e u , L o m b a r d a n d M. V ~ n ~ r o n i in O r s a y h a v e b e e n v e r y s t i m u l a t i n g .
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References 1. N. Bijedi~, P h y s i c s L e t t e r s 16 (1965) 47; N. Bijedi~, Thesis, University of Zagreb 1964 (unpublished) ; G. Alaga and G. Ialongo, UCRL - 16385 (1965); G. Ialongo and G. Alaga, Bull. Am. Phys. Soc. 11 (1966) 102. 2. Nuclear Data Cards B. Jung and G. Anderson, Nucl. Phys. 15 (1960) 108; L. V. Groshev, A.M. Demidov, V.A. Ivanov, V.N. Lutsenko and V. I. Pelekhov, Izvest. Akad. Nauk SSR, Ser. Phys. 27 (1963) 1377; R. E. Segel, R.K. Smither and R. T. Carpenter, Phys. Rev. 133 (1964) B583; M. Sakai, H. Ikegami, T. Yamazaki and K. Saito, Nucl. Phys. 65 (1965) ;
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M. Sakai, T. Yamazaki and E. Jiri, INS-Report 77, Tokyo, 1965; B. P. Maier, U. Gruber, H. R. Koch and O. W. B. Schult, Z, Phys. 185 (1965) 478; J. A. M. de villiers, C.A. Engelbrecht, W.G. Vonaeh and A. B. Smith, Neutron Scattering from Au, Hg and T1 (1965) (preprint). J. J. Kraushaar and M. Goldhaber, Phys. Rev. 89 (1953) 1081; A. de Shalit and M. Goldhaber, Phys. Rev. 92 (1953) 1211. L. Keszthelyi, I. Berkes, I. D~zsi, B. Moln~r and L. Pocs, Fhysics Letters 8 (1964) 195; J.A. Cameron, Can. J. Phys. 42 (1964) 1681; H. J. K~Srner and U. Ortabasi, Nucl. Phys. 70 (1965) 28; Y. K. Agarwal, C. V. K. Baba and S. K. Bhattacherjee~ Nucl. Phys. 79 (1966) 437. J. de Boer, R. G. Stokstad, G. D. Symons and A. Winther, Phys. Rev. Letters 14 (1965) 564; I. Hall, private communication. S. Hinds, R. Middleton, J.H. Bjerregaard, O. Hansen and O. Nathan, Physics Letters 17 (1965) 302; D. Eccleshall and M. J. L. Yates, Physics Letters 19 (1965) 301. A. Bohr, Mat. Fys. Medd. Dansk. Vid. Selsk. 28 no 4 (1952); A. Bohr and B. R Mottelson, Mat. Fys. Medd. Dansk. Vid. Selsk. 27 no 16 (1953); D. C. Choudhury, Mat. Fys. Medd. Dansk. Vid. Selsk. 28 no 4 (1954); G. Scharf-Goldhaber and J. Weneser, Phys. Rev. 98 (1955) 212; D. G. Raz. Phys. Rev. 114 (1958) 1116. R. F. Petry, R.A. Neumann and J. S. Evans, P h y s i c s L e t t e r s 21 (1966) 541. G. Alaga, Bull. Am. Phys. Soc. 11, V. 4 (1959) 359; G. Ialongo, Thesis, New York University (1966). L.S. K i s s l i n g e r and R. S. S!trensen, Rev. Mod. Phys. 35 (1963) 853; R. Arvieu and M. V~n~roni, Compt. Rend. 250 (1960) 992, 2155; T. Tamura and T. Udagawa, Phys. Rev. L e t t e r s 15 (1965) 765.