- Email: [email protected]
Collective correlations in the giant resonance of 60Ni
Volume 23, number I0
PHYSICS L E T T E R S
s u l t s so that e x p e r i m e n t a l and t h e o r e t i c a l l e v e l s f o r T = 0, J Y = 2- a g r e e . The_splitting is due ent i r e l y to the T a l m i i n t e g r a l I l l s whose value was taken as -1.1 MeV. The r e s u l t s a r e s i m i l a r to t h o s e for one p a r t i c l e spin o r b i t coupling of [5] as w e l l as to those of B a r r e t t [8]. The p r e s e n t a n a l y s i s is being continued to include a m u c h l a r g e r n u m b e r of quanta [9] and a l s o a m o r e r e a l i s t i c type of f o r c e . The only t h e o r e t i cal p r o b l e m r e m a i n i n g is to find an unambiguous way of d e r i v i n g T a l m i i n t e g r a l s f r o m f o r c e s with hard cores. We a r e indebted to P r o f . R. K, Sheline f o r pointing out the i n t e r e s t of the p r e s e n t p r o b l e m and f o r supplying i n f o r m a t i o n on the e x p e r i m e n t a l data.
COLLECTIVE
CORRELATIONS
IN
5December 1966
References 1. C.Werntz, Phys. Rev.133 (1964) B19. 2. W.E. Meyerhof and J. McElearney, Nucl. Phys. 74 (1965) 533; D. Yu and W. E. Meyerhof, Nuel. Phys. 80 (1966) 481. 3. R.Froseh, R.E.Rand, M.R.Yearian, H.L.Crannell and L.R.Suelzle, Phys. Letters 19 (1965) 155. 4. T.A.Tombrello, Phys. Rev. 138 (1965)B40. 5. A. De-Shalit and J.D. Walecka, Phys. Rev. 147 (1966) 763. 6. P.Kramer and M. Moshinsky, Nucl. Phys. 82 (1966) 241. 7. H.A.Jahn, Proc. Roy. Soc. London 205A (1951) 192. 8. B.R. Barrett, Phys. Rev., to be published. 9. P . K r a m e r and M. Moshinsky, Group theory of harmonic oscillators and nuclear structure, to appear in: Group theory and applications, ed. E. M. Loebl (Academic P r ess 1967).
THE
GIANT
RESONANCE
OF
60Ni
*
J. B. SEABORN, D. DRECHSEL, H. ARENH()VEL and W. GREINER
Institut j~r Theoretische Physik der Universit~tt Frankfurt, Frankfurt am Main, Germany Received 3 November 1966
The theory of collective correlations is applied to the giant dipole resonances of 60Ni. The results are in good agreement with (% n) measurements. The theory is also compared with the pure particle-hole model as well as with the dynamic collective theory.
The d y n a m i c c o l l e c t i v e t h e o r y of n u c l e a r exc i t a t i o n s [1, 2] has m e t with c o n s i d e r a b l e s u c c e s s in explaining i n t e r m e d i a t e s t r u c t u r e o b s e r v e d in n u c l e a r p h o t o - a b s o r p t i o n c r o s s s e c t i o n s . The t h e o r y is b a s e d on the a s s u m p t i o n that in addition to p e r f o r m i n g the giant dipole v i b r a t i o n s d e s c r i b e d by the h y d r o d y n a m i c model, the n u c l e u s a l s o ch a nge s shape. The s t r u c t u r e in the giant dipole r e s o n a n c e is a t t r i b u t e d to the i n t e r a c t i o n of t h e s e two c o l l e c t i v e m o d e s (i.e. giant dipole and s u r f a c e o s c i l l a t i o n s ) . In s pi te of the s u c c e s s e s t h e r e s t i l l r e m a i n d i s c r e p a n c i e s between theory and exp e r i m e n t , p a r t i c u l a r l y , on the low e n e r g y s i d e of the giant r e s o n a n c e w h e r e a c o m p l e t e l y c o l l e c t i v e d e s c r i p t i o n ( a s s u m i n g only q u a d r u p o l e s u r f a c e v i b r a t i o n s ) p r e d i c t s no s t r u c t u r e at all [2]. R e cently, in an e f f o r t to i m p r o v e the a g r e e m e n t with e x p e r i m e n t , an ex te n s i o n of the t h e o r y was f o r m u l a t e d in which the giant dipole v i b r a t i o n s w e r e t r e a t e d as s i n g l e - p a r t i c l e e x c i t a t i o n s w h i le 576
the c o l l e c t i v e d e s c r i p t i o n was r e t a i n e d f o r the (quadrupole) s u r f a c e o s c i l l a t i o n s [3]. C a l c u l a t i o n s w e r e c a r r i e d out f o r 12C and n e a r l y q u a n t i t a t i v e a g r e e m e n t was obtained for the p o s i t i o n s and r e l a t i v e s t r e n g t h s of the t h r e e m a j o r peaks o b s e r v e d in the e x p e r i m e n t a l c r o s s section. A m o r e r i g o r ous t e s t of the t h e o r y would be p r o v i d e d by n u c l e i which show a g r e a t deal m o r e s t r u c t u r e in the a b s o r p t i o n c r o s s s e c t i o n than does 12C. E x p e r i m e n t a l (~, n) c r o s s s e c t i o n m e a s u r e m e n t s [4] ind i c a t e that n i c k e l is a good e x a m p l e and the p u r pose of the p r e s e n t note is to r e p o r t the r e s u l t s of the application of the extended t h e o r y to 60Ni. The c a l c u l a t i o n a l p r o c e d u r e outlined in ref. 3 is followed h e r e with one ex cep t i o n which we now d i s c u s s . In ref. 3 the ex t en si o n of the c o m p l e t e l y c o l l e c t i v e d e s c r i p t i o n to one in which the giant * This work is supported by the Deutsche Forschungsgemeinschaft.
Volume 23, number 10
PHYSICS LETTERS
dipole vibrations are described as single-particle e x c i t a t i o n s was a c c o m p l i s h e d by r e q u i r i n g that the dipole o p e r a t o r s ph
=
~ 3 /~v
v) all a v ,
(1)
and
D[:o]ll = Me(a[1] + Ml[ot[1] X ot[2]][1]),
(2)
in the s i n g l e - p a r t i c l e and c o l l e c t i v e p i c t u r e s be the s a m e . However, the s i n g l e - p a r t i c l e b a s i s functions (labelled by g , v in eq. (1)) a r e c a l c u l a t e d in a s p h e r i c a l potential well. M o r e o v e r , the d i p o l e v i b r a t i o n a m p l i t u d e s ot [1] a l s o r e f e r to a s p h e r i c a l n u c l e a r s y s t e m . Thus it a p p e a r s to be n a t u r a l to a s s o c i a t e D[~ with only that p a r t of D[:o]ll a p p r o p r i a t e to a n u n d e f o r m e d nucleus, i.e.
5December 1966
t r o n s and p r o t o n s . The p a r a m e t e r s of the nucleonnucleon f o r c e for 60Ni a r e taken to be those obtained by G i l l e t [5] for 12C with the r a n g e p a r a m e t e r p r o p e r l y a d j u s t e d for the d i f f e r e n t m a s s . The s t r e n g t h Vo (= -47 MeV) was a d j u s t e d so a s to r e p r o d u c e the e x p e r i m e n t a l e n e r g y of the m a i n d i pole peak. In the c o l l e c t i v e m o d e l of the g i a n t r e s o n a n c e the e n e r g y of the giant dipole v i b r a t i o n is given by [ I , 2] ~_ EG'R" -
SONi #
..[ t(,tt
30
~
,o
}
•
[
l
.! l
/5
} 20
Configuration
(2p;~)-12d~
9.76
2_1
2
(2pt) 3s½ (2p~) ~ 2 d I (if,) 2 -1
2
(lf?) 2dfi ~-1 z (lfI) lg I (ld~) ~ 2p! 2 1 (ld~)if i2 2
1
2
(ida)- 2p~ (2s½) - 2p~ (2sl) -12pl (ld~)- lf~ 2 1 2 (ldfi) - 2p~ 2
2
t 25
EtMeV)
90 80 7O 60 Fig. Ib 40 30
[
tO 10
Energy (MeV)
Fig. lo
l
23
Table 1 Particle-hole energies for 60Ni.
(3)
,
4O
/~plh]= MoOtI l l , w h e r e M o i s a c o n s t a n t c a l c u l a t e d in the c o l l e c t i v e m o d e l [2]. This i s the i n t e r p r e t a t i o n of the dipole o p e r a t o r s e m p l o y e d in the p r e s e n t work. We view the ground s t a t e of 60Ni a s c o n s i s t i n g of a c l o s e d l f i p r o t o n s h e l l and a c l o s e d 2p~ neut r o n s u b s h e l l . ~ E n e r g i e s for the s i n g l e - p a r ~ c l e s t a t e s of 60Ni w e r e c a l c u l a t e d in a W o o d s - S a x o n well. The r e l e v a n t p a r t i c l e - h o l e c o n f i g u r a t i o n s and e n e r g i e s a r e given in t a b l e 1. We a s s u m e that for a given configuration the p a r t i c l e - h o l e e n e r g i e s a r e the s a m e for both neu-
CK ~- A ~
208 R
,
,
,5
I ]
~
E~Mov~
70
50
11.22 12.82 lO.O7
40
14.33 18.20 12.84 11.96 10.78 10.43 12.49 15.37 14.19
io
Fig. I¢
3(,; 25
sO
[rI;0I
s,
Fig. I. (a) HeIative dipole strengths of eigenstates of 60Ni with collective correlations. The experimental points are from the (% n) cross section m e a s u r e m e n t s of ref. 4. (b) Dipole strengths of states obtained in the usual partiele-hole calculation (i.e. without collective correlations).
(c) Relative strengths of dipole states given by the dynamic collective theory. All dipole strengths are given the same (arbitrary) units. The scale in (b) and (e) has been reduced by one half compared to that in (a). 577
Volume 23, number 10
PHYSICS
w h e r e R i s the n u c l e a r r a d i u s , M the n u c l e o n m a s s , and K the s y m m e t r y e n e r g y of the B e t h e - W e i z a c k e r m a s s f o r m u l a . T h i s K a p p e a r s a s a p a r a m e t e r in the c a l c u l a t i o n of r e f . 3. Its v a l u e is f i x e d in the p r e s e n t c a l c u l a t i o n by s e t t i n g the r i g h t hand s i d e of eq. (3) e q u a l to the e x p e r i m e n t a l e n e r g y of the g i a n t r e s o n a n c e (16.5 M e V f o r n i c k e l ) , i.e. K = K(Vo) is d e t e r m i n e d by the s t r e n g t h of the n u c l e o n n u c l e o n f o r c e . T h i s g i v e s a v a l u e of K ~ 17 MeV. T h e p a r a m e t e r s f o r the c o l l e c t i v e q u a d r u p o l e o s c i l l a t o r a r e rio = 0.211 and ~w 2 = 1.333 MeV. They a r e t a k e n f r o m the low e n e r g y e x p e r i m e n t a l s p e c t r u m [6]. The r e l a t i v e s t r e n g t h s of the d i p o l e s t a t e s obt a i n e d f r o m the d i a g o n a l i z a t i o n of the e n e r g y m a t r i x a r e p r e s e n t e d in fig. l a . A l s o , f o r c o m p a r i son, the e x p e r i m e n t a l (~, n) c r o s s s e c t i o n [4] f o r n a t u r a l n i c k e l (68% 58Ni and 26% 60Ni) is shown. Of c o u r s e , the c a l c u l a t e d s t r e n g t h s should be c o m p a r e d w i t h the t o t a l a b s o r p t i o n c r o s s s e c t i o n f o r the p u r e A = 60 i s o t o p e . T h i s is p a r t i c u l a r l y t r u e f o r n i c k e l i n a s m u c h a s the (:y, p) c r o s s s e c t i o n is e x p e c t e d to be of the s a m e o r d e r a s the (11, n) *. Up u n t i l now, h o w e v e r , no s u c h m e a s u r e m e n t s h a v e b e e n r e p o r t e d . N e v e r t h e l e s s , the n u m b e r of s t r o n g d i p o l e s t a t e s p r e d i c t e d for 60Ni, t h e i r e n e r g i e s , and t h e i r r e l a t i v e s t r e n g t h s a r e in s t r i k i n g l y good a g r e e m e n t w i t h the a v a i l a b l e e x p e r i m e n t a l data. M o r e o v e r , it is r e m a r k a b l e t h a t s o m u c h s t r u c t u r e can b e n e a r l y q u a n t i t a t i v e l y a c c o u n t e d f o r w i t h e s s e n t i a l l y o n l y one a d j u s t a b l e p a r a m e t e r , n a m e l y the s t r e n g t h Vo of the n u c l e o n * We are grateful to E.Hayward for bringing this point to our attention.
578
LETTERS
5 December 1966
n u c l e o n f o r c e . T h e r e s u l t s of the p u r e p a r t i c l e h o l e c a l c u l a t i o n (i.e. w i t h o u t c o l l e c t i v e c o r r e l a tions) a r e shown in fig. lb. F i g . l c s h o w s the r e s u l t s of the d y n a m i c c o l l e c t i v e t h e o r y (the giant resonances are treated collectively). Comparison of the t h r e e p i c t u r e s i n d i c a t e s that the s p e c i a l f e a t u r e s c o n t a i n e d in f i g s . l b and l c (pure p a r t i c l e - h o l e s t r u c t u r e and p u r e c o l l e c t i v e s t r u c t u r e , r e s p e c t i v e l y ) are" a l s o p r e s e n t in fig. l a . Thus the e x t e n s i o n of the d y n a m i c c o l l e c t i v e t h e o r y i n t r o d u c e s a d d i t i o n a l s t r u c t u r e in the ~ i a ~ t r e s o n a n c e of 60Ni in a g r e e m e n t w i t h e x p e r i m e n t , and at the s a m e t i m e p r e s e r v e s the g e n e r a l f e a t u r e s of the o r i g i n a l ( c o m p l e t e l y c o l l e c t i v e ) t h e o r y a s w e l l as t h o s e of the p u r e p a r t i c l e - h o l e d e s c r i p t i o n . We w i s h to thank Dr. R. C a s w e l l f o r the u s e of a c o m p u t e r c o d e f o r c a l c u l a t i n g the e n e r g i e s of the s i n g l e - p a r t i c l e s t a t e s .
References
1. M.Danos and W. Greiner, Ptlys. Rev. 134 (1964) B284; The r)hotonuclear effect in heavy deformed nuclei, to be published in Phys. Rev. 2. H . J . W e b e r , M.Huber a n d W . G r e i n e r , Z.Physik 192 (1966) 182,223; M.G.Huber, H . J . W e b e r , M.Danos and W.Grein, r, Phys. Letters 15 (1965) 529 and ~o be published. 3. D.Drechsel, J . B . S e a b o r n and W. Greiner, Phys. Rev. Letters 17 (1966) 488. 4. G.Bacin, G.C.Bonazzola, B.Minetti, C.Mo!ino, L. Pasquatini and G. Pirogino, Nucl. Phys. 67 {1965) 178. 5. V.Gillet, thesis, University of P a r i s (1962). 6. P . H . S t e l s o n and L.Grodzins, Nucl. Data 1A (1965) 29.
PHYSICS L E T T E R S
s u l t s so that e x p e r i m e n t a l and t h e o r e t i c a l l e v e l s f o r T = 0, J Y = 2- a g r e e . The_splitting is due ent i r e l y to the T a l m i i n t e g r a l I l l s whose value was taken as -1.1 MeV. The r e s u l t s a r e s i m i l a r to t h o s e for one p a r t i c l e spin o r b i t coupling of [5] as w e l l as to those of B a r r e t t [8]. The p r e s e n t a n a l y s i s is being continued to include a m u c h l a r g e r n u m b e r of quanta [9] and a l s o a m o r e r e a l i s t i c type of f o r c e . The only t h e o r e t i cal p r o b l e m r e m a i n i n g is to find an unambiguous way of d e r i v i n g T a l m i i n t e g r a l s f r o m f o r c e s with hard cores. We a r e indebted to P r o f . R. K, Sheline f o r pointing out the i n t e r e s t of the p r e s e n t p r o b l e m and f o r supplying i n f o r m a t i o n on the e x p e r i m e n t a l data.
COLLECTIVE
CORRELATIONS
IN
5December 1966
References 1. C.Werntz, Phys. Rev.133 (1964) B19. 2. W.E. Meyerhof and J. McElearney, Nucl. Phys. 74 (1965) 533; D. Yu and W. E. Meyerhof, Nuel. Phys. 80 (1966) 481. 3. R.Froseh, R.E.Rand, M.R.Yearian, H.L.Crannell and L.R.Suelzle, Phys. Letters 19 (1965) 155. 4. T.A.Tombrello, Phys. Rev. 138 (1965)B40. 5. A. De-Shalit and J.D. Walecka, Phys. Rev. 147 (1966) 763. 6. P.Kramer and M. Moshinsky, Nucl. Phys. 82 (1966) 241. 7. H.A.Jahn, Proc. Roy. Soc. London 205A (1951) 192. 8. B.R. Barrett, Phys. Rev., to be published. 9. P . K r a m e r and M. Moshinsky, Group theory of harmonic oscillators and nuclear structure, to appear in: Group theory and applications, ed. E. M. Loebl (Academic P r ess 1967).
THE
GIANT
RESONANCE
OF
60Ni
*
J. B. SEABORN, D. DRECHSEL, H. ARENH()VEL and W. GREINER
Institut j~r Theoretische Physik der Universit~tt Frankfurt, Frankfurt am Main, Germany Received 3 November 1966
The theory of collective correlations is applied to the giant dipole resonances of 60Ni. The results are in good agreement with (% n) measurements. The theory is also compared with the pure particle-hole model as well as with the dynamic collective theory.
The d y n a m i c c o l l e c t i v e t h e o r y of n u c l e a r exc i t a t i o n s [1, 2] has m e t with c o n s i d e r a b l e s u c c e s s in explaining i n t e r m e d i a t e s t r u c t u r e o b s e r v e d in n u c l e a r p h o t o - a b s o r p t i o n c r o s s s e c t i o n s . The t h e o r y is b a s e d on the a s s u m p t i o n that in addition to p e r f o r m i n g the giant dipole v i b r a t i o n s d e s c r i b e d by the h y d r o d y n a m i c model, the n u c l e u s a l s o ch a nge s shape. The s t r u c t u r e in the giant dipole r e s o n a n c e is a t t r i b u t e d to the i n t e r a c t i o n of t h e s e two c o l l e c t i v e m o d e s (i.e. giant dipole and s u r f a c e o s c i l l a t i o n s ) . In s pi te of the s u c c e s s e s t h e r e s t i l l r e m a i n d i s c r e p a n c i e s between theory and exp e r i m e n t , p a r t i c u l a r l y , on the low e n e r g y s i d e of the giant r e s o n a n c e w h e r e a c o m p l e t e l y c o l l e c t i v e d e s c r i p t i o n ( a s s u m i n g only q u a d r u p o l e s u r f a c e v i b r a t i o n s ) p r e d i c t s no s t r u c t u r e at all [2]. R e cently, in an e f f o r t to i m p r o v e the a g r e e m e n t with e x p e r i m e n t , an ex te n s i o n of the t h e o r y was f o r m u l a t e d in which the giant dipole v i b r a t i o n s w e r e t r e a t e d as s i n g l e - p a r t i c l e e x c i t a t i o n s w h i le 576
the c o l l e c t i v e d e s c r i p t i o n was r e t a i n e d f o r the (quadrupole) s u r f a c e o s c i l l a t i o n s [3]. C a l c u l a t i o n s w e r e c a r r i e d out f o r 12C and n e a r l y q u a n t i t a t i v e a g r e e m e n t was obtained for the p o s i t i o n s and r e l a t i v e s t r e n g t h s of the t h r e e m a j o r peaks o b s e r v e d in the e x p e r i m e n t a l c r o s s section. A m o r e r i g o r ous t e s t of the t h e o r y would be p r o v i d e d by n u c l e i which show a g r e a t deal m o r e s t r u c t u r e in the a b s o r p t i o n c r o s s s e c t i o n than does 12C. E x p e r i m e n t a l (~, n) c r o s s s e c t i o n m e a s u r e m e n t s [4] ind i c a t e that n i c k e l is a good e x a m p l e and the p u r pose of the p r e s e n t note is to r e p o r t the r e s u l t s of the application of the extended t h e o r y to 60Ni. The c a l c u l a t i o n a l p r o c e d u r e outlined in ref. 3 is followed h e r e with one ex cep t i o n which we now d i s c u s s . In ref. 3 the ex t en si o n of the c o m p l e t e l y c o l l e c t i v e d e s c r i p t i o n to one in which the giant * This work is supported by the Deutsche Forschungsgemeinschaft.
Volume 23, number 10
PHYSICS LETTERS
dipole vibrations are described as single-particle e x c i t a t i o n s was a c c o m p l i s h e d by r e q u i r i n g that the dipole o p e r a t o r s ph
=
~ 3 /~v
v) all a v ,
(1)
and
D[:o]ll = Me(a[1] + Ml[ot[1] X ot[2]][1]),
(2)
in the s i n g l e - p a r t i c l e and c o l l e c t i v e p i c t u r e s be the s a m e . However, the s i n g l e - p a r t i c l e b a s i s functions (labelled by g , v in eq. (1)) a r e c a l c u l a t e d in a s p h e r i c a l potential well. M o r e o v e r , the d i p o l e v i b r a t i o n a m p l i t u d e s ot [1] a l s o r e f e r to a s p h e r i c a l n u c l e a r s y s t e m . Thus it a p p e a r s to be n a t u r a l to a s s o c i a t e D[~ with only that p a r t of D[:o]ll a p p r o p r i a t e to a n u n d e f o r m e d nucleus, i.e.
5December 1966
t r o n s and p r o t o n s . The p a r a m e t e r s of the nucleonnucleon f o r c e for 60Ni a r e taken to be those obtained by G i l l e t [5] for 12C with the r a n g e p a r a m e t e r p r o p e r l y a d j u s t e d for the d i f f e r e n t m a s s . The s t r e n g t h Vo (= -47 MeV) was a d j u s t e d so a s to r e p r o d u c e the e x p e r i m e n t a l e n e r g y of the m a i n d i pole peak. In the c o l l e c t i v e m o d e l of the g i a n t r e s o n a n c e the e n e r g y of the giant dipole v i b r a t i o n is given by [ I , 2] ~_ EG'R" -
SONi #
..[ t(,tt
30
~
,o
}
•
[
l
.! l
/5
} 20
Configuration
(2p;~)-12d~
9.76
2_1
2
(2pt) 3s½ (2p~) ~ 2 d I (if,) 2 -1
2
(lf?) 2dfi ~-1 z (lfI) lg I (ld~) ~ 2p! 2 1 (ld~)if i2 2
1
2
(ida)- 2p~ (2s½) - 2p~ (2sl) -12pl (ld~)- lf~ 2 1 2 (ldfi) - 2p~ 2
2
t 25
EtMeV)
90 80 7O 60 Fig. Ib 40 30
[
tO 10
Energy (MeV)
Fig. lo
l
23
Table 1 Particle-hole energies for 60Ni.
(3)
,
4O
/~plh]= MoOtI l l , w h e r e M o i s a c o n s t a n t c a l c u l a t e d in the c o l l e c t i v e m o d e l [2]. This i s the i n t e r p r e t a t i o n of the dipole o p e r a t o r s e m p l o y e d in the p r e s e n t work. We view the ground s t a t e of 60Ni a s c o n s i s t i n g of a c l o s e d l f i p r o t o n s h e l l and a c l o s e d 2p~ neut r o n s u b s h e l l . ~ E n e r g i e s for the s i n g l e - p a r ~ c l e s t a t e s of 60Ni w e r e c a l c u l a t e d in a W o o d s - S a x o n well. The r e l e v a n t p a r t i c l e - h o l e c o n f i g u r a t i o n s and e n e r g i e s a r e given in t a b l e 1. We a s s u m e that for a given configuration the p a r t i c l e - h o l e e n e r g i e s a r e the s a m e for both neu-
CK ~- A ~
208 R
,
,
,5
I ]
~
E~Mov~
70
50
11.22 12.82 lO.O7
40
14.33 18.20 12.84 11.96 10.78 10.43 12.49 15.37 14.19
io
Fig. I¢
3(,; 25
sO
[rI;0I
s,
Fig. I. (a) HeIative dipole strengths of eigenstates of 60Ni with collective correlations. The experimental points are from the (% n) cross section m e a s u r e m e n t s of ref. 4. (b) Dipole strengths of states obtained in the usual partiele-hole calculation (i.e. without collective correlations).
(c) Relative strengths of dipole states given by the dynamic collective theory. All dipole strengths are given the same (arbitrary) units. The scale in (b) and (e) has been reduced by one half compared to that in (a). 577
Volume 23, number 10
PHYSICS
w h e r e R i s the n u c l e a r r a d i u s , M the n u c l e o n m a s s , and K the s y m m e t r y e n e r g y of the B e t h e - W e i z a c k e r m a s s f o r m u l a . T h i s K a p p e a r s a s a p a r a m e t e r in the c a l c u l a t i o n of r e f . 3. Its v a l u e is f i x e d in the p r e s e n t c a l c u l a t i o n by s e t t i n g the r i g h t hand s i d e of eq. (3) e q u a l to the e x p e r i m e n t a l e n e r g y of the g i a n t r e s o n a n c e (16.5 M e V f o r n i c k e l ) , i.e. K = K(Vo) is d e t e r m i n e d by the s t r e n g t h of the n u c l e o n n u c l e o n f o r c e . T h i s g i v e s a v a l u e of K ~ 17 MeV. T h e p a r a m e t e r s f o r the c o l l e c t i v e q u a d r u p o l e o s c i l l a t o r a r e rio = 0.211 and ~w 2 = 1.333 MeV. They a r e t a k e n f r o m the low e n e r g y e x p e r i m e n t a l s p e c t r u m [6]. The r e l a t i v e s t r e n g t h s of the d i p o l e s t a t e s obt a i n e d f r o m the d i a g o n a l i z a t i o n of the e n e r g y m a t r i x a r e p r e s e n t e d in fig. l a . A l s o , f o r c o m p a r i son, the e x p e r i m e n t a l (~, n) c r o s s s e c t i o n [4] f o r n a t u r a l n i c k e l (68% 58Ni and 26% 60Ni) is shown. Of c o u r s e , the c a l c u l a t e d s t r e n g t h s should be c o m p a r e d w i t h the t o t a l a b s o r p t i o n c r o s s s e c t i o n f o r the p u r e A = 60 i s o t o p e . T h i s is p a r t i c u l a r l y t r u e f o r n i c k e l i n a s m u c h a s the (:y, p) c r o s s s e c t i o n is e x p e c t e d to be of the s a m e o r d e r a s the (11, n) *. Up u n t i l now, h o w e v e r , no s u c h m e a s u r e m e n t s h a v e b e e n r e p o r t e d . N e v e r t h e l e s s , the n u m b e r of s t r o n g d i p o l e s t a t e s p r e d i c t e d for 60Ni, t h e i r e n e r g i e s , and t h e i r r e l a t i v e s t r e n g t h s a r e in s t r i k i n g l y good a g r e e m e n t w i t h the a v a i l a b l e e x p e r i m e n t a l data. M o r e o v e r , it is r e m a r k a b l e t h a t s o m u c h s t r u c t u r e can b e n e a r l y q u a n t i t a t i v e l y a c c o u n t e d f o r w i t h e s s e n t i a l l y o n l y one a d j u s t a b l e p a r a m e t e r , n a m e l y the s t r e n g t h Vo of the n u c l e o n * We are grateful to E.Hayward for bringing this point to our attention.
578
LETTERS
5 December 1966
n u c l e o n f o r c e . T h e r e s u l t s of the p u r e p a r t i c l e h o l e c a l c u l a t i o n (i.e. w i t h o u t c o l l e c t i v e c o r r e l a tions) a r e shown in fig. lb. F i g . l c s h o w s the r e s u l t s of the d y n a m i c c o l l e c t i v e t h e o r y (the giant resonances are treated collectively). Comparison of the t h r e e p i c t u r e s i n d i c a t e s that the s p e c i a l f e a t u r e s c o n t a i n e d in f i g s . l b and l c (pure p a r t i c l e - h o l e s t r u c t u r e and p u r e c o l l e c t i v e s t r u c t u r e , r e s p e c t i v e l y ) are" a l s o p r e s e n t in fig. l a . Thus the e x t e n s i o n of the d y n a m i c c o l l e c t i v e t h e o r y i n t r o d u c e s a d d i t i o n a l s t r u c t u r e in the ~ i a ~ t r e s o n a n c e of 60Ni in a g r e e m e n t w i t h e x p e r i m e n t , and at the s a m e t i m e p r e s e r v e s the g e n e r a l f e a t u r e s of the o r i g i n a l ( c o m p l e t e l y c o l l e c t i v e ) t h e o r y a s w e l l as t h o s e of the p u r e p a r t i c l e - h o l e d e s c r i p t i o n . We w i s h to thank Dr. R. C a s w e l l f o r the u s e of a c o m p u t e r c o d e f o r c a l c u l a t i n g the e n e r g i e s of the s i n g l e - p a r t i c l e s t a t e s .
References
1. M.Danos and W. Greiner, Ptlys. Rev. 134 (1964) B284; The r)hotonuclear effect in heavy deformed nuclei, to be published in Phys. Rev. 2. H . J . W e b e r , M.Huber a n d W . G r e i n e r , Z.Physik 192 (1966) 182,223; M.G.Huber, H . J . W e b e r , M.Danos and W.Grein, r, Phys. Letters 15 (1965) 529 and ~o be published. 3. D.Drechsel, J . B . S e a b o r n and W. Greiner, Phys. Rev. Letters 17 (1966) 488. 4. G.Bacin, G.C.Bonazzola, B.Minetti, C.Mo!ino, L. Pasquatini and G. Pirogino, Nucl. Phys. 67 {1965) 178. 5. V.Gillet, thesis, University of P a r i s (1962). 6. P . H . S t e l s o n and L.Grodzins, Nucl. Data 1A (1965) 29.