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Microscopic models for vibrational states in even nickel isotopes
Volume 19, number 2
PHYSICS LETTERS
MICROSCOPIC MODELS FOR IN EVEN NICKEL
VIBRATIONAL ISOTOPES *
1 October 1965
STATES
L. S. HSU and J. B. FRENCH Department of Physics and Astronomy, University of Rochester, Rochester, New York and Oak Ridge National Laboratory, Oak Ridge, Tennessee
Received 1 September 1965
In o r d e r to study the r e l a t i o n s h i p between v a r i o u s m i c r o s c o p i c m o d e l s for l o w - l y i n g s t a t e s of v i b r a t i o n a l n u c l e i we have used v a r i o u s methods ( s h e l l - m o d e l , s e n i o r i t y , T - D and R.P.A. a p p r o x i mations) to c a l c u l a t e the low-lying s p e c t r a for even nickel isotopes (58Ni, 60Ni, 62Ni and 64Ni). The r e s u l t s for 62Ni and 64Ni a r e shown in figs. 1 and 2. The active o r b i t s a r e 2p~e3, lf~, 21~2 for n e u t r o n s only. The s i n g l e - p a r t i c l e e n e r g i e s were t a k e n to be 0, 0.78 and 1.08 MeV r e s p e c t i v e l y . The two-body i n t e r a c t i o n used was an S - s t a t e i n t e r a c t i o n * * produced, via a l e a s t - s q u a r e s a n a l y s i s to e n e r g y l e v e l s , by Pandya and Soga [1] u s i n g the s h e l l - m o d e l p r o g r a m s of Cohen et al. [2]. It gives a r e a s o n a b l e fit (mean s q u a r e d e v i a tion 0.035 MeV2) to 27 known e n e r g i e s and, when s u p p l e m e n t e d by a monopole i n t e r a c t i o n -0.42 n(n-1) MeV, where n is the n u m b e r o p e r a t o r , gives a s a t i s f a c t o r y t r e a t m e n t for the binding e n e r g i e s . B e s i d e s that, with an effective c h a r g e of 1.55 e, the s h e l l - m o d e l BE2 values between the ground state and the f i r s t excited state and the E2, M1 m i x i n g r a t i o between the f i r s t and second 2 + s t a t e s a g r e e well with e x p e r i m e n t . We b e l i e v e t h e r e f o r e that the i n t e r a c t i o n used is a r e a s o n a b l e one and that the c o n c l u s i o n s which we shall draw about the models a r e liable to be valid. Among the m o d e l s , TDn and R P n r e f e r to the T a m m - D a n c o f f and r a n d o m - p h a s e a p p r o x i m a t i o n s whose b a s i s o p e r a t o r s do not involve m o r e than n q u a s i - p a r t i c l e c r e a t i o n or d e s t r u c t i o n o p e r a t o r s . We have c o n s i d e r e d n = 2 and 4, with the s p u r i o u s * Supported in part by the U.S. Atomic Energy Commission. ** The parameters are Ino = -9,73, -13.33, -1.54 and -0.677_MeV for n = 0, 1, 2 and 3 respectively, where I 1 = j R 2 n l ( r ) V(r)r 2 dr.
s t a t e s a r i s i n g from p a r t i c l e - n u m b e r - n o n - c o n s e r v a t i o n e l i m i n a t e d by the Schmidt p r o c e s s . In P T D n (the p r o j e c t e d T-D), we have p r o j e c t e d and r e n o r m a l i z e d the p a r t of the TDn wave f u n c tions which have the p r o p e r n u m b e r of p a r t i c l e s . The s e n i o r i t y a p p r o x i m a t i o n s (vn) work in the p a r t i c l e space (as opposed to the q u a s i - p a r t i c l e space); they c o r r e s p o n d to the u s u a l s h e l l - m o d e l c a l c u l a t i o n s u s i n g a r e p r e s e n t a t i o n which d i s p l a y s the s e n i o r i t y of the p a r t i c l e s in s e p a r a t e o r b i t s , the v e c t o r space being then t r u n c a t e d so that the sum of the individual s e n i o r i t i e s is not g r e a t e r than n. We have taken n = 2, 4. F i n a l l y s.m. r e f e r s to a c o m p l e t e s h e l l - m o d e l c a l c u l a tion which was made by u s i n g the O R N L - R o c h e s t e r s h e l l - m o d e l computing p r o g r a m s . Since these p r o g r a m s employ an i n d i v i d u a l - o r b i t s y m p l e c t i c s y m m e t r y (or s e n i o r i t y for identical p a r t i c l e s ) r e p r e s e n t a t i o n they w e r e used also for the vn calculations. It is found that in all c a s e s the R P n and TDn excitation o p e r a t o r s a r e a l m o s t identical (the total weight of the " b a c k w a r d - g o i n g " p a r t s being ~ 1%). T h e s e two models should then, for the s a m e n, give c l o s e l y s i m i l a r r e s u l t s for e s s e n t i a l l y all p u r p o s e s ; this shows up for example in the e n e r g i e s . All the models give a s a t i s f a c tory t r e a t m e n t for the f i r s t 2 + state. The s q u a r e d o v e r l a p s with s h e l l - m o d e l wave f u n c tions for the c a s e s where c o m p a r i s o n is p o s s i b l e (PTDn, vn) lie between 85% and 99.6%. Above the f i r s t 2 + state the n = 2 models all p r e d i c t a 0+-2+-4 + t r i p l e t , whose m e m b e r s a r e s o m e t i m e s widely spaced. We m u s t however not r e g a r d these as a two-phonon t r i p l e t since, for example in the T a m m - D a n c o f f a p p r o x i m a t i o n s , a 2-phonon state m u s t involve 4 q u a s i p a r t i c l e s (since, as shown below, the 1-phonon state is well d e s c r i b e d by 2
135
Volume 19, number 2
PHYSICS
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Fig. 1. Spectra of 62Ni calculated via various approximations. Only a few low-lying s t a t e s a r e shown in the PTD2 and PTI)4 s p e c t r a . The dotted line in the TD4 c a s e shows the position to which the f i r s t 0+ state moves when one allows admixing between the vacuum s t a t e and the 0+ f o u r - q u a s i - p a r t i c l e s t a t e s . quasiparticles). The n = 4 models and the comp l e t e c a l c u l a t i o n s h o w f u r t h e r s t a t e s in t h e 2 - 3 M e V r e g i o n ; t h e q u e s t i o n s t h e n a r e w h e t h e r t h e 0 +2 + - 4 + n = 2 s t a t e s c a n b e s t i l l i d e n t i f i e d in t h e extended models and whether new states which app e a r c a n b e r e g a r d e d a s t w o - p h o n o n s t a t e s . It t u r n s out t h a t t h e n = 2 s t a t e s c a n b e i d e n t i f i e d f a i r l y w e l l ( a s d e t e r m i n e d b y t h e e n e r g i e s a n d , in cases where they are pertinent, the overlaps) w i t h t h e f i r s t 4 +, t h e s e c o n d 0 + a n d t h e t h i r d 2 + s h e l l - m o d e l s t a t e s . T h e s e c o n d (n = 4 o r s . m . ) 2 + 136
state is then a candidate for a two-phonon state a s i s a l s o t h e s e c o n d 4 + s t a t e but, a t l e a s t f o r 62Ni, t h e r e i s no n a t u r a l c a n d i d a t e f o r t h e t w o phonon 0+ state. As far as the general comparisons with the shell-model are concerned, the vn approximation is better than the PTDn w h i c h in t u r n i s b e t t e r t h a n t h e T D n o r R P n a p proximations. It i s f o u n d t h a t t h e e l e c t r i c q u a d r u p o l e t r a n s i t i o n s f r o m t h e s e c o n d 2 + to t h e f i r s t 2 + a n d t h a t f r o m t h e f i r s t 2 + to t h e g r o u n d s t a t e a r e
Volume 19, number 2
PHYSICS LETTERS
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1 October 1965
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Fig. 2. Spectra for 64Ni. For this case v4 is identical with the complete shell-model calculation. e n h a n c e d by a f a c t o r of ten c o m p a r e d with the s i n g l e - p a r t i c l e e s t i m a t e s . In fact, a l m o s t a ll the e l e c t r i c q u a d r u p o l e s t r e n g t h f r o m the ground s t a t e g o e s to the f i r s t 2 + s t a t e , which t h e r e f o r e is a l m o s t i d e n t i c a l with Q2(e.m.) ~ o , w h e r e Q2(e.m.) is the E2 o p e r a t o r and ~ o the g r o u n d sta t e function. The s q u a r e s of the p e r t i n e n t wave function o v e r l a p s a r e about 95% e x c e p t f o r 64Ni fo r which the f i g u r e if 85%. Note that, b e c a u s e of this, the a b s e n c e of a 2 + ~ 0 + c r o s s o v e r t r a n s i t i o n i s not e v i d e n c e that the s e c o n d 2 + s t a t e i s a two-phonon state. C o n s i d e r now the p o s s i b l e two-phonon n a t u r e
of this state. D i s c a r d i n g the v e r y s m a l l f o u r q u a s i - p a r t i c l e a d m i x t u r e in the f i r s t 2 + state, and r e n o r m a l i z i n g t h e r e s u l t i n g v e c t o r , y i e l d s a wave function whose o v e r l a p with the f i r s t ex ci t ed st at e in the TD2 a p p r o x i m a t i o n is l a r g e r than 99.6%. We t h e r e f o r e take Q2, the e x c i t a t i o n o p e r a t o r (or " q u a s i - b o s o n " c r e a t i o n o p e r a t o r ) f o r the f i r s t e x c i t e d s t a t e in the TD2 a p p r o x i m a tion, c o n s t r u c t and r e n o r m a l i z e the v e c t o r s (Q2 × Q2)0, 2, 4 ~ o ' and expand t h e m in t e r m s of the TD4 e i g e n v e c t o r s . It i s found that 46 to 83% of the 2 + and 4 + two-phonon s t r e n g t h s go to the s e c o n d 2 + and 4 + TD4 s t a t e s while the 0 + two137
Volume 19, number 2
PHYSICS LETTERS
phonon state has a c e n t r o i d at ~ 5 MeV and cannot be identified with any l o w - l y i n g 0 + TD4 s t a t e s . The c o n c l u s i o n t h e r e f o r e i s that it is not u n r e a s o n a b l e to think of the 2+ and 4 + m e m b e r s of the t r i p l e t as being l a r g e l y two-phonon s t a t e s but this is not at all t r u e for the 0 + m e m b e r . Apart f r o m phonons we might ask whether in the r e g i o n above the f i r s t 2 + state we can identify some other d o u b l e - e x c i t a t i o n type of s t r u c t u r e . To c o n s i d e r this we have, s e p a r a t e l y for each nucleus, a t t e m p t e d to c o n s t r u c t , by m e a n s of a least squares analysis, a self-adjoint single-particle quadrupole excitation o p e r a t o r , which when acting on the s h e l l - m o d e l ground state p r o d u c e s the f i r s t 2+ state. The a t t e m p t is s u c c e s s f u l , the c o r r e s p o n d i n g s q u a r e d o v e r l a p s being at l e a s t 95%; the o p e r a t o r s produced a r e not v e r y different f r o m the e l e c t r o m a g n e t i c t r a n s i t i o n o p e r a t o r . It t u r n s out now that the 0 + d o u b l e - e x c i t a t i o n can be well identified with the second 0 + state (86- 94%), the 2 + d o u b l e - e x c i t a t i o n f a i r l y well with the second 2 + state (41- 87%) while the 4 + d o u b l e - e x c i t a t i o n s t r e n g t h is d i s t r i b u t e d m a i n l y between the f i r s t and second 4 + s t a t e s . A p a r t f r o m the splitting of the 4 + s t r e n g t h into two p a r t s , we see than that we can indeed identify
1October 1965
a " t r i p l e t " of d o u b l e - q u a d r u p o l e - e x c i t a t i o n states, the f u n d a m e n t a l excitation however being s e l f adjoint and quite different f r o m a phonon excitation (which is d e s c r i b e d by a n o n - s e l f - a d j o i n t quasi-boson creation operator). We intend to extend these c a l c u l a t i o n s to c a s e s where there a r e active p r o t o n s and n e u t r o n s , and to c o n s i d e r the s o u r c e of the effective charge by allowing excitations f r o m the closed s h e l l s . We intend also to study s e v e r a l a s s o c i a t e d p r o b l e m s by application of s u m - r u l e techniques. The authors wish to thank Dr. E. C. Halbert for her cooperation and help in connection with the c a l c u l a t i o n s , Dr. S. P. P a n d y a for his advice conc e r n i n g the b e s t i n t e r a c t i o n to be used, and Dr. T a r o T a m u r a and Mr. S. S. M. Wong for helpful conversations.
References
1. S. Pandya and M.Soga, unpublished (1964). 2. S. Cohen, R. Lawson, M. Macfarlane and M. Soga, Physics Letters 9 (1964) 180.
*****
ISOBARIC
ANALOGUE
STATES
VIA
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AND
(p, n) ON 1 4 4 S m
*
C. F. MOORE and R. K. JOLLY Department of P h y s i c s , Florida State University, Tallahassee , Florida
Received 14 September 1965
The o b s e r v a t i o n of i s o b a r i c analogue s t a t e s as r e s o n a n c e s in a highly excited compound n u c l e u s by Fox et al. [1] has evoked a g r e a t deal of i n t e r e s t in o b s e r v i n g such s t a t e s in v e r y heavy nuclei. Recent work of Moore et al. [2] on Mo isotopes shows that m e a s u r e m e n t of these r e s o n a n c e s at a few chosen a n g l e s can be u s e d for s p e c t r o s c o p i c a s s i g n m e n t s of the s t a t e s of the n e u t r o n plus t a r get s y s t e m . F u r t h e r a t t e m p t s a r e c u r r e n t l y u n d e r way a t F l o r i d a State U n i v e r s i t y to e x p l o r e the r a n g e of a p p l i c a b i l i t y of this new s p e c t r o s c o p i c tool to v e r y heavy nuclei. However, i n v e r y heavy n u c l e i one r u n s into the following p r a c t i c a l difficulties: (i) the Coulomb b a r r i e r s a r e r a t h e r high and this may involve u s i n g the tandem Van de Graaff 138
a c c e l e r a t o r close to the l i m i t of its t e r m i n a l voltage. E l e c t r i c a l breakdowns may l i m i t o b s e r vation of the useful r a n g e of excitation function; (ii) the heavy compound s y s t e m at high excitation e n e r g i e s may have s e v e r a l open d e - e x c i tation c h a n n e l s r e s u l t i n g in a b r o a d e n i n g of the o b s e r v e d r e s o n a n c e s . This p r o b l e m is f u r t h e r a g g r a v a t e d in heavy n u c l e i due to i n c r e a s e in level d e n s i t y with i n c r e a s i n g m a s s and excitation energy. In the case of 144Sm the f i r s t factor was a l i m i t a t i o n as the e n e r g y of i n c i d e n t p a r t i c l e s could not be taken any higher than 10.8 MeV at * Supported in part by the Air Force Office of Scientific Research.
PHYSICS LETTERS
MICROSCOPIC MODELS FOR IN EVEN NICKEL
VIBRATIONAL ISOTOPES *
1 October 1965
STATES
L. S. HSU and J. B. FRENCH Department of Physics and Astronomy, University of Rochester, Rochester, New York and Oak Ridge National Laboratory, Oak Ridge, Tennessee
Received 1 September 1965
In o r d e r to study the r e l a t i o n s h i p between v a r i o u s m i c r o s c o p i c m o d e l s for l o w - l y i n g s t a t e s of v i b r a t i o n a l n u c l e i we have used v a r i o u s methods ( s h e l l - m o d e l , s e n i o r i t y , T - D and R.P.A. a p p r o x i mations) to c a l c u l a t e the low-lying s p e c t r a for even nickel isotopes (58Ni, 60Ni, 62Ni and 64Ni). The r e s u l t s for 62Ni and 64Ni a r e shown in figs. 1 and 2. The active o r b i t s a r e 2p~e3, lf~, 21~2 for n e u t r o n s only. The s i n g l e - p a r t i c l e e n e r g i e s were t a k e n to be 0, 0.78 and 1.08 MeV r e s p e c t i v e l y . The two-body i n t e r a c t i o n used was an S - s t a t e i n t e r a c t i o n * * produced, via a l e a s t - s q u a r e s a n a l y s i s to e n e r g y l e v e l s , by Pandya and Soga [1] u s i n g the s h e l l - m o d e l p r o g r a m s of Cohen et al. [2]. It gives a r e a s o n a b l e fit (mean s q u a r e d e v i a tion 0.035 MeV2) to 27 known e n e r g i e s and, when s u p p l e m e n t e d by a monopole i n t e r a c t i o n -0.42 n(n-1) MeV, where n is the n u m b e r o p e r a t o r , gives a s a t i s f a c t o r y t r e a t m e n t for the binding e n e r g i e s . B e s i d e s that, with an effective c h a r g e of 1.55 e, the s h e l l - m o d e l BE2 values between the ground state and the f i r s t excited state and the E2, M1 m i x i n g r a t i o between the f i r s t and second 2 + s t a t e s a g r e e well with e x p e r i m e n t . We b e l i e v e t h e r e f o r e that the i n t e r a c t i o n used is a r e a s o n a b l e one and that the c o n c l u s i o n s which we shall draw about the models a r e liable to be valid. Among the m o d e l s , TDn and R P n r e f e r to the T a m m - D a n c o f f and r a n d o m - p h a s e a p p r o x i m a t i o n s whose b a s i s o p e r a t o r s do not involve m o r e than n q u a s i - p a r t i c l e c r e a t i o n or d e s t r u c t i o n o p e r a t o r s . We have c o n s i d e r e d n = 2 and 4, with the s p u r i o u s * Supported in part by the U.S. Atomic Energy Commission. ** The parameters are Ino = -9,73, -13.33, -1.54 and -0.677_MeV for n = 0, 1, 2 and 3 respectively, where I 1 = j R 2 n l ( r ) V(r)r 2 dr.
s t a t e s a r i s i n g from p a r t i c l e - n u m b e r - n o n - c o n s e r v a t i o n e l i m i n a t e d by the Schmidt p r o c e s s . In P T D n (the p r o j e c t e d T-D), we have p r o j e c t e d and r e n o r m a l i z e d the p a r t of the TDn wave f u n c tions which have the p r o p e r n u m b e r of p a r t i c l e s . The s e n i o r i t y a p p r o x i m a t i o n s (vn) work in the p a r t i c l e space (as opposed to the q u a s i - p a r t i c l e space); they c o r r e s p o n d to the u s u a l s h e l l - m o d e l c a l c u l a t i o n s u s i n g a r e p r e s e n t a t i o n which d i s p l a y s the s e n i o r i t y of the p a r t i c l e s in s e p a r a t e o r b i t s , the v e c t o r space being then t r u n c a t e d so that the sum of the individual s e n i o r i t i e s is not g r e a t e r than n. We have taken n = 2, 4. F i n a l l y s.m. r e f e r s to a c o m p l e t e s h e l l - m o d e l c a l c u l a tion which was made by u s i n g the O R N L - R o c h e s t e r s h e l l - m o d e l computing p r o g r a m s . Since these p r o g r a m s employ an i n d i v i d u a l - o r b i t s y m p l e c t i c s y m m e t r y (or s e n i o r i t y for identical p a r t i c l e s ) r e p r e s e n t a t i o n they w e r e used also for the vn calculations. It is found that in all c a s e s the R P n and TDn excitation o p e r a t o r s a r e a l m o s t identical (the total weight of the " b a c k w a r d - g o i n g " p a r t s being ~ 1%). T h e s e two models should then, for the s a m e n, give c l o s e l y s i m i l a r r e s u l t s for e s s e n t i a l l y all p u r p o s e s ; this shows up for example in the e n e r g i e s . All the models give a s a t i s f a c tory t r e a t m e n t for the f i r s t 2 + state. The s q u a r e d o v e r l a p s with s h e l l - m o d e l wave f u n c tions for the c a s e s where c o m p a r i s o n is p o s s i b l e (PTDn, vn) lie between 85% and 99.6%. Above the f i r s t 2 + state the n = 2 models all p r e d i c t a 0+-2+-4 + t r i p l e t , whose m e m b e r s a r e s o m e t i m e s widely spaced. We m u s t however not r e g a r d these as a two-phonon t r i p l e t since, for example in the T a m m - D a n c o f f a p p r o x i m a t i o n s , a 2-phonon state m u s t involve 4 q u a s i p a r t i c l e s (since, as shown below, the 1-phonon state is well d e s c r i b e d by 2
135
Volume 19, number 2
PHYSICS
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Fig. 1. Spectra of 62Ni calculated via various approximations. Only a few low-lying s t a t e s a r e shown in the PTD2 and PTI)4 s p e c t r a . The dotted line in the TD4 c a s e shows the position to which the f i r s t 0+ state moves when one allows admixing between the vacuum s t a t e and the 0+ f o u r - q u a s i - p a r t i c l e s t a t e s . quasiparticles). The n = 4 models and the comp l e t e c a l c u l a t i o n s h o w f u r t h e r s t a t e s in t h e 2 - 3 M e V r e g i o n ; t h e q u e s t i o n s t h e n a r e w h e t h e r t h e 0 +2 + - 4 + n = 2 s t a t e s c a n b e s t i l l i d e n t i f i e d in t h e extended models and whether new states which app e a r c a n b e r e g a r d e d a s t w o - p h o n o n s t a t e s . It t u r n s out t h a t t h e n = 2 s t a t e s c a n b e i d e n t i f i e d f a i r l y w e l l ( a s d e t e r m i n e d b y t h e e n e r g i e s a n d , in cases where they are pertinent, the overlaps) w i t h t h e f i r s t 4 +, t h e s e c o n d 0 + a n d t h e t h i r d 2 + s h e l l - m o d e l s t a t e s . T h e s e c o n d (n = 4 o r s . m . ) 2 + 136
state is then a candidate for a two-phonon state a s i s a l s o t h e s e c o n d 4 + s t a t e but, a t l e a s t f o r 62Ni, t h e r e i s no n a t u r a l c a n d i d a t e f o r t h e t w o phonon 0+ state. As far as the general comparisons with the shell-model are concerned, the vn approximation is better than the PTDn w h i c h in t u r n i s b e t t e r t h a n t h e T D n o r R P n a p proximations. It i s f o u n d t h a t t h e e l e c t r i c q u a d r u p o l e t r a n s i t i o n s f r o m t h e s e c o n d 2 + to t h e f i r s t 2 + a n d t h a t f r o m t h e f i r s t 2 + to t h e g r o u n d s t a t e a r e
Volume 19, number 2
PHYSICS LETTERS
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1 October 1965
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Fig. 2. Spectra for 64Ni. For this case v4 is identical with the complete shell-model calculation. e n h a n c e d by a f a c t o r of ten c o m p a r e d with the s i n g l e - p a r t i c l e e s t i m a t e s . In fact, a l m o s t a ll the e l e c t r i c q u a d r u p o l e s t r e n g t h f r o m the ground s t a t e g o e s to the f i r s t 2 + s t a t e , which t h e r e f o r e is a l m o s t i d e n t i c a l with Q2(e.m.) ~ o , w h e r e Q2(e.m.) is the E2 o p e r a t o r and ~ o the g r o u n d sta t e function. The s q u a r e s of the p e r t i n e n t wave function o v e r l a p s a r e about 95% e x c e p t f o r 64Ni fo r which the f i g u r e if 85%. Note that, b e c a u s e of this, the a b s e n c e of a 2 + ~ 0 + c r o s s o v e r t r a n s i t i o n i s not e v i d e n c e that the s e c o n d 2 + s t a t e i s a two-phonon state. C o n s i d e r now the p o s s i b l e two-phonon n a t u r e
of this state. D i s c a r d i n g the v e r y s m a l l f o u r q u a s i - p a r t i c l e a d m i x t u r e in the f i r s t 2 + state, and r e n o r m a l i z i n g t h e r e s u l t i n g v e c t o r , y i e l d s a wave function whose o v e r l a p with the f i r s t ex ci t ed st at e in the TD2 a p p r o x i m a t i o n is l a r g e r than 99.6%. We t h e r e f o r e take Q2, the e x c i t a t i o n o p e r a t o r (or " q u a s i - b o s o n " c r e a t i o n o p e r a t o r ) f o r the f i r s t e x c i t e d s t a t e in the TD2 a p p r o x i m a tion, c o n s t r u c t and r e n o r m a l i z e the v e c t o r s (Q2 × Q2)0, 2, 4 ~ o ' and expand t h e m in t e r m s of the TD4 e i g e n v e c t o r s . It i s found that 46 to 83% of the 2 + and 4 + two-phonon s t r e n g t h s go to the s e c o n d 2 + and 4 + TD4 s t a t e s while the 0 + two137
Volume 19, number 2
PHYSICS LETTERS
phonon state has a c e n t r o i d at ~ 5 MeV and cannot be identified with any l o w - l y i n g 0 + TD4 s t a t e s . The c o n c l u s i o n t h e r e f o r e i s that it is not u n r e a s o n a b l e to think of the 2+ and 4 + m e m b e r s of the t r i p l e t as being l a r g e l y two-phonon s t a t e s but this is not at all t r u e for the 0 + m e m b e r . Apart f r o m phonons we might ask whether in the r e g i o n above the f i r s t 2 + state we can identify some other d o u b l e - e x c i t a t i o n type of s t r u c t u r e . To c o n s i d e r this we have, s e p a r a t e l y for each nucleus, a t t e m p t e d to c o n s t r u c t , by m e a n s of a least squares analysis, a self-adjoint single-particle quadrupole excitation o p e r a t o r , which when acting on the s h e l l - m o d e l ground state p r o d u c e s the f i r s t 2+ state. The a t t e m p t is s u c c e s s f u l , the c o r r e s p o n d i n g s q u a r e d o v e r l a p s being at l e a s t 95%; the o p e r a t o r s produced a r e not v e r y different f r o m the e l e c t r o m a g n e t i c t r a n s i t i o n o p e r a t o r . It t u r n s out now that the 0 + d o u b l e - e x c i t a t i o n can be well identified with the second 0 + state (86- 94%), the 2 + d o u b l e - e x c i t a t i o n f a i r l y well with the second 2 + state (41- 87%) while the 4 + d o u b l e - e x c i t a t i o n s t r e n g t h is d i s t r i b u t e d m a i n l y between the f i r s t and second 4 + s t a t e s . A p a r t f r o m the splitting of the 4 + s t r e n g t h into two p a r t s , we see than that we can indeed identify
1October 1965
a " t r i p l e t " of d o u b l e - q u a d r u p o l e - e x c i t a t i o n states, the f u n d a m e n t a l excitation however being s e l f adjoint and quite different f r o m a phonon excitation (which is d e s c r i b e d by a n o n - s e l f - a d j o i n t quasi-boson creation operator). We intend to extend these c a l c u l a t i o n s to c a s e s where there a r e active p r o t o n s and n e u t r o n s , and to c o n s i d e r the s o u r c e of the effective charge by allowing excitations f r o m the closed s h e l l s . We intend also to study s e v e r a l a s s o c i a t e d p r o b l e m s by application of s u m - r u l e techniques. The authors wish to thank Dr. E. C. Halbert for her cooperation and help in connection with the c a l c u l a t i o n s , Dr. S. P. P a n d y a for his advice conc e r n i n g the b e s t i n t e r a c t i o n to be used, and Dr. T a r o T a m u r a and Mr. S. S. M. Wong for helpful conversations.
References
1. S. Pandya and M.Soga, unpublished (1964). 2. S. Cohen, R. Lawson, M. Macfarlane and M. Soga, Physics Letters 9 (1964) 180.
*****
ISOBARIC
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C. F. MOORE and R. K. JOLLY Department of P h y s i c s , Florida State University, Tallahassee , Florida
Received 14 September 1965
The o b s e r v a t i o n of i s o b a r i c analogue s t a t e s as r e s o n a n c e s in a highly excited compound n u c l e u s by Fox et al. [1] has evoked a g r e a t deal of i n t e r e s t in o b s e r v i n g such s t a t e s in v e r y heavy nuclei. Recent work of Moore et al. [2] on Mo isotopes shows that m e a s u r e m e n t of these r e s o n a n c e s at a few chosen a n g l e s can be u s e d for s p e c t r o s c o p i c a s s i g n m e n t s of the s t a t e s of the n e u t r o n plus t a r get s y s t e m . F u r t h e r a t t e m p t s a r e c u r r e n t l y u n d e r way a t F l o r i d a State U n i v e r s i t y to e x p l o r e the r a n g e of a p p l i c a b i l i t y of this new s p e c t r o s c o p i c tool to v e r y heavy nuclei. However, i n v e r y heavy n u c l e i one r u n s into the following p r a c t i c a l difficulties: (i) the Coulomb b a r r i e r s a r e r a t h e r high and this may involve u s i n g the tandem Van de Graaff 138
a c c e l e r a t o r close to the l i m i t of its t e r m i n a l voltage. E l e c t r i c a l breakdowns may l i m i t o b s e r vation of the useful r a n g e of excitation function; (ii) the heavy compound s y s t e m at high excitation e n e r g i e s may have s e v e r a l open d e - e x c i tation c h a n n e l s r e s u l t i n g in a b r o a d e n i n g of the o b s e r v e d r e s o n a n c e s . This p r o b l e m is f u r t h e r a g g r a v a t e d in heavy n u c l e i due to i n c r e a s e in level d e n s i t y with i n c r e a s i n g m a s s and excitation energy. In the case of 144Sm the f i r s t factor was a l i m i t a t i o n as the e n e r g y of i n c i d e n t p a r t i c l e s could not be taken any higher than 10.8 MeV at * Supported in part by the Air Force Office of Scientific Research.