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Nuclear binding energy and quadrupole-quadrupole interaction
Volume 19, number 5
PHYSICS LETTERS
15 November 1965
We should e m p h a s i z e finally that although some r e s i d u a l i n t e r a c t i o n s involving additional c o n f i g u r a t i o n s have b e e n ignored in this d i s c u s sion, t h e i r influence is not expected to modify d r a s t i c a l l y our p r e s e n t r e s u l t s . This expectation has b e e n c o n f i r m e d by m o r e s y s t e m a t i c c a l c u l a t i o n s to be r e p o r t e d e l s e w h e r e [7].
t e r i s t i c of analog e x c i t a t i o n s i.e., r e l a t i v e l y n a r r o w total width, h i n d e r e d p r o b a b i l i t y for g r o u n d - s t a t e n e u t r o n e m i s s i o n , etc. It will also be a s s o c i a t e d with a " p a r e n t " analog in 90y, a highly c o h e r e n t n e u t r o n - p a r t i c l e - p r o t o n - h o l e state at an e n e r g y (obtained by s u b t r a c t i n g the a p p r o p r i a t e Coulomb energy) of ~ 8 MeV in 9 0 y i.e., unbound by ~ 1 MeV. The o b s e r v a t i o n of t h i s state in e.g. s - w a v e n e u t r o n i n t e r a c t i o n s with 8 9 y will be of i n t e r e s t . Additional e x a m p l e s a r e provided by the analog (J = 1-) s t a t e s o b s e r v e d in (p,p) [4], (p, n) [4], or (P,Y0) [5] r e a c t i o n s at 14.4, 15.7 and 16.3 MeV. Our c o n s i d e r a t i o n s p r e d i c t t h e i r lower (in e n e r g y and isospin) c o u n t e r p a r t s to lie ~ 5 MeV below i.e., at 9.4, 10.7 and 11.3 MeV. We c o m p a r e this c o n c l u s i o n with the e l a s t i c y - r a y e x p e r i m e n t s of Axel et al. (6) in the 9-12 MeV r e g i o n in Zr. A s i g n i f i c a n t c o n c e n t r a t i o n of r a d i a t i v e s t r e n g t h has been r e p o r t e d by these a u t h o r s at excitation e n e r g i e s of 9.1, 10.4 and 11.5 MeV. The c o m p a r i s o n with our e s t i m a t e s ~s v e r y s a t i s f a c t o r y although the a p p r o x i m a t e n a t u r e of eq. (2) should be r e m e m b e r e d . The o s c i l l a t o r s t r e n g t h exhausted in this i n t e r v a l is r e p o r t e d by Min [6] to be ~ 2 % of the c l a s s i c a l s u m - r u l e limit. F r o m eq. (4) the o s c i l l a t o r s t r e n g t h of the T + 1 s t a t e s can then be obtained which in t u r n gives an e s t i m a t e of ~160 eV for the o v e r a l l r a d i a t i v e width F T + 1 in the 14-16.5 MeV region. The c o m p a r i s o n with the sum of the l o w e r - l i m i t e s t i m a t e s of 40, 0 and 85 eV obt a i n e d by Black and T a n n e r [6] for the r a d i a t i v e widths of the 14.4, 15.7 and 16.3 MeV l e v e l s r e s p e c t i v e l y , is again encouraging.
We wish to thank Drs. A. K. K e r m a n , F . R . M e t z g e r and S. M. Shafroth for s t i m u l a t i n g d i s cussions~
References 1. See e.g., G.E. Brown, Unified theory of nuclear models, {North-Holland Publ. Comp., Amsterdam). 2. The properties of analog states were reviewed by J. D. Anderson, J.B. French and J. D. Fox in Nuclear spectroscopy with direct reactions {proceedings), ANL-6878(1964). 3. R.Nathans and P. F. Yergin, Phys. Rev. 98 (1955} 1296; P. F. Yergtn and B. P. Fabricand, Phys. Rev. 104 (1956) 1334. 4. J.D. Fox, C.F. Moore and D. Robson, Phys. Rev. Letters 12 (1964) 198. 5. J.L. Black and N. W. Tanner, Physics Letters 11 (1964) 135; See also J. D. Fox, C.F. Moore and D. Robson, Bull. Am. Phys. Soc. 10 (1965) 52. 6. P.Axel, K.Min, N.Stein and D.C.Sutton, Phys. Rev. Letters 10 (1963) 299; K.Min, Ph.D.Thesis, University of Illinois (1963). We wish to thank Dr. Axel for providing us with a copy of Min's thesis as well as for discussions. 7. B.Goulard and R.H.Venter, to be published.
NUCLEAR BINDING ENERGY AND QUADRUPOLE-QUADRUPOLE INTERACTION A. ARIMA and M. NOMURA
Department of Physics, Faculty of Science, University of Tokyo and H. KAWARADA
Department of Applied Physics, Faculty of Engineering, University of Tokyo Received 12 October 1965
Many i n t e r e s t i n g a t t e m p t s have been made to study n u c l e a r collective motion. It has b e e n 400
pointed out that a l o n g - r a n g e i n t e r a c t i o n plays an i m p o r t a n t role in the collective v i b r a t i o n and
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PHYSICS LETTERS
15 November 1965
(MeV) 1.5
(Mev) 0.5 a(Z,N) 0
0.5 -0.5
0
-I
-I.5 ~ 0
I
I
2
4
I
6
I
[
I
I
8
I0
12
14
& ;o ;, ¢8 ,& ,& ,io ,;, Neutron number
< j'221 v, lj'22>-= 1 (MeV)
Fig. 1. Excitation energy of states with the spin two. A pure configuration (jj~)n is taken, and states are a s sumed to have pure seniority 2, which must be broken when the Q-Q interaction lffenerally v2} is introduced. The coupling constant K2 is empirically about -5 -t -10 MeV. rotation. This interaction is frequently approxim a t e d by the q u a d r u p o l e - q u a d r u p o l e one which was i n t r o d u c e d f i r s t by E l l i o t t in h i s c l a s s i c p a p e r s [1]. In o r d e r to e x p l a i n the e x c i t a t i o n e n e r g y of 2 + v i b r a t i o n a l s t a t e s in e v e n - e v e n nuclei, the Q-Q i n t e r a c t i o n m u s t be a t t r a c t i v e [2]. T a l m i , h o w e v e r , r a i s e d an i n t e r e s t i n g question c o n c e r n i n g the sign of t h i s i n t e r a c t i o n [3]. A c c o r d i n g to the s h e l l m o d e l , the binding e n e r g y i s e x p r e s s e d in t e r m s of the s i m p l e f o r m u l a which i s q u a d r a t i c in n u m b e r of n u c l e o n s n outside the c l o s e d shell. T a l m i found that a t e r m p r o p o r t i o n a l to n 2 i s r e p u l s i v e , which l e a d s h i m to conclude that a long r a n g e i n t e r a c t i o n including the Q-Q i n t e r a c t i o n m u s t be r e p u l s i v e . F u r t h e r m o r e , he s u g g e s t e d that the s i n g l e c l o s e d s h e l l nucleus can n e v e r be d e f o r m e d owing to the r e p u l s i v e n a t u r e of the Q-Q i n t e r action. It i s , h o w e v e r , the p u r p o s e of t h i s s h o r t note to show that the Q-Q i n t e r a c t i o n m u s t be a t t r a c t i v e b e c a u s e of the r e p u l s i v e n 2 t e r m . L e t us t a k e the p u r e c o n f i g u r a t i o n ]~ and the p u r e S e n i o r i t y coupling s c h e m e ; the g r o u n d s t a t e of an e v e n - e v e n nucleus h a s s e n i o r i t y z e r o and that of an odd n u c l e u s h a s s e n i o r i t y one. The e x p e c t a t i o n value of the t w o - b o d y i n t e r a c t i o n i s e a s i l y obtained a s follows [4]
Fig. 2. Ot(Z,N). The coefficient of the n2 term in the observed binding energies is one-eighth of a(Z, N). n
j vU Ijn v=O J=O> = (1)
-
n(2j+
I,,Ij2oo> +
,
where
1Xj2jI VIj2j),
and
< j 2 2 j I V Ij22j> = = -2
(2) ~
(2X+
1)W(jjjj;Xd)(jjX[ V[jjX).
X=even
T h i s V i s j u s t the p a r t i c l e - h o l e i n t e r a c t i o n [4]. We c o n s i d e r only the c a s e of e v e n - e v e n nuclei h e r e , but the s a m e a r g u m e n t s holds in the c a s e of odd nucleus. Now V i s d i v i d e d into t h r e e p a r t s , V = V0 + V I + V2
(S)
where
V 0 = vO(rl, r2) = 2 f f v ( n 2 ) d e d ~ and f d ~ m e a n s averaging over the spin coordinate, V 1 is the odd tensor interaction and V 2 is the even tensor interaction [5], the rank of which is larger than zero. B y this separation the matrix element (2) turns out to be very simple,
= = 2j¢j2OOlVo ~200> + + -
(2')
v2lj20o>. 401
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PHYSICS LETTERS
80 ~
v
e
d
nuclei
§ 70 c
.
.
.
.
.
.
.
.
.
.
.
60 50
t
90
I
ii10
I00 Neutron number
*
I
120 126
Section of the binding energy ~o ~,.~(.ilrfoce at certain : 2 ) / . ~ / uJ ad CO'~"
"~...~.~'" 82
90
I 0 II Neutron number
(b) 120 126
Fig. 3. (a) A schematic representation of one region of the nuclear periodic table. Nuclei are deformed inside an ellipsoid and spherical outside it. (b) Section of the binding energy surface at certain Z of the above region.
A s p i n - d e p e n d e n t i n t e r a c t i o n a. aV(n) i s an e x a m ple of the odd t e n s o r i n t e r a c t i o n , and the u s u a l l o c a l W i g n e r i n t e r a c t i o n is a l i n e a r c o m b i n a t i o n of V0 and V2 type i n t e r a c t i o n s . Substituting t h i s r e l a t i o n into (1), we obtain
-if/_.] -- ~,,l.2oo ~Vl
(4) 2j + 1 , "2^^ I¢200> + 2-T:T_ 1 u uo 1' "~21J2oo~± ' '~ +
+ ½.(.- 1){(Poo Ivot,Poo> - 2 ~ _ 1cj2oo I v21 ./2oo)}. In the r i g h t hand s i d e of (4), the f i r s t t e r m shows the p a i r i n g p r o p e r t y and the s e c o n d the long r a n g e c o r r e l a t i o n s . It m u s t be n o t i c e d that the s i g n of the c o n t r i b u t i o n coming f r o m V0 to the long r a n g e c o r r e l a t i o n s is d i f f e r e n t f r o m t h a t of V2. The l o n g e s t r a n g e p a r t V0 which i s the m o n o p o l e - m o n o p o l e i n t e r a c t i o n m u s t be r e p u l s i v e without V2. A u s u a l c e n t r a l f o r c e which i s u s e d by the s h e l l m o d e l c a l c u l a t i o n t e n d s to V0 in the long r a n g e l i m i t . T a l m i ' s c o n c l u s i o n i s c o m p l e t e l y c o r r e c t in t h i s point. If we t a k e a l i n e a r c o m b i n a t i o n of the s h o r t r a n g e i n t e r a c t i o n 6(r) and the Q-Q one, the 402
f o r m e r c o n t r i b u t e s only to the p a i r i n g e n e r g y and the l a t t e r to both the p a i r i n g e n e r g y and the long r a n g e c o r r e l a t i o n . H e r e the Q-Q i n t e r a c t i o n m u s t be a t t r a c t i v e to r e p r o d u c e the r e p u l s i v e n2 d e p e n dence of the n u c l e a r binding e n e r g y . G e n e r a l l y , h o w e v e r , we can s a y only that the quantity (j200 ]V01j200) - 2 j - - ~ (j200 IV21j200)
O4 qi"
z 1"-4 Ld ad
15 November 1965
(5)
m u s t be r e p u l s i v e . F o r the 6(r) i n t e r a c t i o n , t h i s quantity v a n i s h e s , so that the 5(r) i n t e r a c t i o n b e l o n g s to the V 1 type. This condition (5) g i v e s v e r y i n t e r e s t i n g condition on the c e n t r a l W i g n e r i n t e r a c t i o n . A c c o r d i n g to t h i s condition, a l o c a l W i g n e r i n t e r a c t i o n such a s Yukawa and G a n s s i a n m u s t be r e p u l s i v e . On the other hand a s p i n dependent p o t e n t i a l (¢rcr)V(r) m u s t be a t t r a c t i v e . A s s u m i n g the s a m e s i m p l e c o n f i g u r a t i o n in, we can c a l c u l a t e an e x c i t a t i o n e n e r g y ~ E j ( n ) of a s t a t e with the s e n i o r i t y two and the a n g u l a r m o m e n t u m J,
AEj(n) = -(j200 I V1} j200) + ( j 2 2 J t V l l j 2 2 j ) + _(j200 }V21j200 > _ ( 2 ~ _2 (22)~( 2- 4£). 6),J (~200 IV2lJ 200) +
1 + 2(2a6) [(2fl-4)Dj-4(j22JtV21j22J] + (2 e - 2n) 2 + 2 ( 2 a - 4 ) ( m e - 6) [ 2__~___ 2(j2004
(6)
}v2]y200> +
+{ 2 (j22J IV21j22j)-Dj}] w h e r e 2~2 : 2j+ 1 a n d D j =-' ( j 2 j I' V 2 1 j 2 j ) + + (j2jI~221j2j>. it i s i n t e r e s t i n g to o b s e r v e that A E j i s a c o n s t a n t without V2, while V2 g i v e s a q u a d r a t i c dependence of A E j on n. G e n e r a l l y , even in the single c l o s e d s h e l l nuclei, f o r e x a m p l e the Sn i s o t o p e s , A E 2 of a nucleus with the s h e l l half f i l l e d i s l o w e r than that of nucleus with a few p a r t i c l e s (or holes) outside (or inside) the c l o s e d s h e l l [6]. To e x p l a i n t h i s situation, we obtain the following condition
D2- {2 + 2--5~__2<¢200 Iv2 LjZ00>} < 0. (7) If the Q-Q i n t e r a c t i o n is t a k e n a s the i n t e r a c t i o n V2, 22 1 V 2 = kr 1 r 2 P 2 ( c o s ~ co), (8) t h i s condition b e c o m e s 10 k[1 - ~2(2~2- 2) +lOW(jjjj;22)] < 0 .
(9)
The quantity in the b r a c k e t i s a l w a y s p o s i t i v e f o r
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j >/3, and v a n i s h e s f o r j = ~ when the s e n i o r i t y i s a l w a y s good quantum n u m b e r [7]. T h e r e f o r e the condition (9) i s s a t i s f i e d f o r ] >I ~z when k i s n e g a t i v e o r V2 i s a t t r a c t i v e . A n o t h e r i n t e r e s t i n g thing i s o b s e r v e d in (6). F r e q u e n t l y it i s pointed out that the s h e l l m o d e l c a l c u l a t i o n s of a few nucleon s y s t e m g i v e s h i g h e r AE 2 when the Q-Q i n t e r a c t i o n i s m o r e a t t r a c t i v e . On the o t h e r hand the r a n d o m p h a s e method of t r e a t i n g m a n y nucleon s y s t e m g i v e s l o w e r AE 2. T h i s s e e m s to be a p u z z l e . H o w e v e r , t h i s p u z z l e i s s o l v e d by studying the equation (6). F o r s i m p l i c i t y l e t u s a s s u m e 2~ >>1. O m i t t i n g the q u a n t i t i e s of the o r d e r of 1 / 2 ~ , we obtain A E 2 ( , ) = -(]200 IV1 I]200> + (j222 IV1[]222> +
-(]200 1V21]200>
+ (1 - n / ~ ) 2 ( j 2 2 2 1V21]222> + + (n/n)(1-./Ze)D 2.
When
(10)
n = 2,
~E2(2 ) = -
]Vllj200> + <]222 ]V1]j222 > +
-(]200 Iv21]200> + (]222 In t h i s [(j200 means action
Iv21j222>.
(11)
equation we can e a s i l y show that Iv2]j2o0>l > 1<]222 ]V2]j222>], which that the s t r o n g e r a t t r a c t i v e Q-Q i n t e r g i v e s a l a r g e r AE2(2). When n =
15 November 1965
o b s e r v e d binding e n e r g y BE(Z,/7) [10], ~(Z,N)
= BE(Z,N+2)
-
2BE(Z,N)+BE(Z,N- 2)
w h e r e Z and N a r e not n e c e s s a r i l y the m a g i c n u m b e r s . T h i s a i s constant ff the binding e n e r g y can be a p p r o x i m a t e d b y the q u a d r a t i c f o r m u l a of n. The i n t e r e s t i n g r e s u l t s :rare shown in fig. 2. This a i s a l w a y s r e p u l s i v e b o t h in the s p h e r i c a l r e g i o n and d e f o r m e d r e g i o n . In the t r a n s i t i o n a l r e g i o n s (N = 90 and N ~ 114), a b e c o m e s suddenly a t t r a c t i v e . T h i s happens p r o b a b l y b e c a u s e t h e r e a r e two s u r f a c e s of the binding e n e r g y [11]. One of t h e m b e l o n g s to the s p h e r i c a l shape and the o t h e r to the d e f o r m e d shape. Both c u r v e s of the binding e n e r g y a r e a p p r o x i m a t e d by the q u a d r a t i c f o r m u l a e of n. H o w e v e r , when the c o m p e t i t i o n o c c u r s , the r e a l binding e n e r g y i s on the envelope of two k i n d s of e n e r g y c u r v e s . T h e r e f o r e it m u s t be s t r e s s e d that only in the t r a n s i t i o n a l r e g i o n a can be attractive. The a u t h o r s would like to e x p r e s s t h e i r h e a r t i l y thanks to P r o f e s s o r S. Y o s h i d a and Dr. K. I k e d a f o r s t i m u l a t i n g d i s c u s s i o n s . T h e y a r e a l s o v e r y much indebted to P r o f e s s o r M. Kawai for his criticism.
a E 2 ( e ) = -(/200 I v 1 I]200> + + (12) - ÷ I D 2 , where n 2 =4k0U ¢t 3)2 and <)200 V211%0> = = IIQ II]>z- For j> the m o r e strongly
a t t r a c t i v e Q-Q i n t e r a c t i o n r e s u l t s in a s m a l l e r t~E2(~). G e n e r a l l y when n o r 2 ~ - n i s s m a l l , the s t r o n g e r k, the l a r g e r A E 2. On the c o n t r a r y , the s t r o n g e r K, the s m a l l e r ~ E 2 when 12~ - 2n [ i s s m a l l . T h i s f a c t e x p l a i n s c l e a r l y why the m o r e s t r o n g l y a t t r a c t i v e Q-Q i n t e r a c t i o n c a u s e s h i g h e r t~E2 in a n u c l e u s n e a r c l o s e d s h e l l s , while in the m i d d l e s h e l l the m o r e a t t r a c t i v e Q-Q i n t e r a c t i o n g i v e s a l o w e r AE 2. In the m i d d l e of the s h e l l , a too s t r o n g Q-Q i n t e r a c t i o n can m a k e AE2 n e g a t i v e , when of c o u r s e the s e n i o r i t y s c h e m e m u s t be b r o k e n s t r o n g l y . When t h i s happens, d e f o r m a tion m a y be p o s s i b l e even in the s i n g l e c l o s e d s h e l l n u c l e i [8]. F r e q u e n t l y it i s e x p e c t e d that a q u a d r a t i c t e r m in the n u c l e a r binding e n e r g y b e c o m e s a t t r a c t i v e when a n u c l e u s i s d e f o r m e d [9]. We c a l c u l a t e the following q u a n t i t i e s by using the
References 1. J . P . Elliott, Proc.Roy.Soc.A245 (1958) 128, 562. 2. T. Tamura and T. Udagawa, Prog. Theor. Phys. 26 (1961) 947. 3. I. Talmi, Rev. Mod. Phys. 32 (1962) 704. 4. A. Arima and H. Kawarada, J. Phys. Soc. Japan 19 (1964) 1768. 5. A. de-Shalit and I. Talmi, Nuclear shell theory (Academic P r e s s , New York and London, 1963). 6. K.Way, N.B.Gove, C.L.McGinnis and R.Nakasima, Nuclear data sheets, (National Research Council, Washington D.C.). 7. C .Schwartz and A .de-Shalit. Phys. Rev. 94 (1954) 1257.
8. M. Baranger and K. Kumar, Nuclear Physics 62 (1965) 113. 9. G.E. Brown, Unified nuclear theory (North-Holland Publ. Co., Amsterdam, 1964). 10. I. Konig, J. Mattauch and A. Wapstra, Nuclear Phys. 31 (1962) 18. 11. R.A. Demirkhanov, V.V. Dorokhov and V. G. Soloviev, preprint.
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We should e m p h a s i z e finally that although some r e s i d u a l i n t e r a c t i o n s involving additional c o n f i g u r a t i o n s have b e e n ignored in this d i s c u s sion, t h e i r influence is not expected to modify d r a s t i c a l l y our p r e s e n t r e s u l t s . This expectation has b e e n c o n f i r m e d by m o r e s y s t e m a t i c c a l c u l a t i o n s to be r e p o r t e d e l s e w h e r e [7].
t e r i s t i c of analog e x c i t a t i o n s i.e., r e l a t i v e l y n a r r o w total width, h i n d e r e d p r o b a b i l i t y for g r o u n d - s t a t e n e u t r o n e m i s s i o n , etc. It will also be a s s o c i a t e d with a " p a r e n t " analog in 90y, a highly c o h e r e n t n e u t r o n - p a r t i c l e - p r o t o n - h o l e state at an e n e r g y (obtained by s u b t r a c t i n g the a p p r o p r i a t e Coulomb energy) of ~ 8 MeV in 9 0 y i.e., unbound by ~ 1 MeV. The o b s e r v a t i o n of t h i s state in e.g. s - w a v e n e u t r o n i n t e r a c t i o n s with 8 9 y will be of i n t e r e s t . Additional e x a m p l e s a r e provided by the analog (J = 1-) s t a t e s o b s e r v e d in (p,p) [4], (p, n) [4], or (P,Y0) [5] r e a c t i o n s at 14.4, 15.7 and 16.3 MeV. Our c o n s i d e r a t i o n s p r e d i c t t h e i r lower (in e n e r g y and isospin) c o u n t e r p a r t s to lie ~ 5 MeV below i.e., at 9.4, 10.7 and 11.3 MeV. We c o m p a r e this c o n c l u s i o n with the e l a s t i c y - r a y e x p e r i m e n t s of Axel et al. (6) in the 9-12 MeV r e g i o n in Zr. A s i g n i f i c a n t c o n c e n t r a t i o n of r a d i a t i v e s t r e n g t h has been r e p o r t e d by these a u t h o r s at excitation e n e r g i e s of 9.1, 10.4 and 11.5 MeV. The c o m p a r i s o n with our e s t i m a t e s ~s v e r y s a t i s f a c t o r y although the a p p r o x i m a t e n a t u r e of eq. (2) should be r e m e m b e r e d . The o s c i l l a t o r s t r e n g t h exhausted in this i n t e r v a l is r e p o r t e d by Min [6] to be ~ 2 % of the c l a s s i c a l s u m - r u l e limit. F r o m eq. (4) the o s c i l l a t o r s t r e n g t h of the T + 1 s t a t e s can then be obtained which in t u r n gives an e s t i m a t e of ~160 eV for the o v e r a l l r a d i a t i v e width F T + 1 in the 14-16.5 MeV region. The c o m p a r i s o n with the sum of the l o w e r - l i m i t e s t i m a t e s of 40, 0 and 85 eV obt a i n e d by Black and T a n n e r [6] for the r a d i a t i v e widths of the 14.4, 15.7 and 16.3 MeV l e v e l s r e s p e c t i v e l y , is again encouraging.
We wish to thank Drs. A. K. K e r m a n , F . R . M e t z g e r and S. M. Shafroth for s t i m u l a t i n g d i s cussions~
References 1. See e.g., G.E. Brown, Unified theory of nuclear models, {North-Holland Publ. Comp., Amsterdam). 2. The properties of analog states were reviewed by J. D. Anderson, J.B. French and J. D. Fox in Nuclear spectroscopy with direct reactions {proceedings), ANL-6878(1964). 3. R.Nathans and P. F. Yergin, Phys. Rev. 98 (1955} 1296; P. F. Yergtn and B. P. Fabricand, Phys. Rev. 104 (1956) 1334. 4. J.D. Fox, C.F. Moore and D. Robson, Phys. Rev. Letters 12 (1964) 198. 5. J.L. Black and N. W. Tanner, Physics Letters 11 (1964) 135; See also J. D. Fox, C.F. Moore and D. Robson, Bull. Am. Phys. Soc. 10 (1965) 52. 6. P.Axel, K.Min, N.Stein and D.C.Sutton, Phys. Rev. Letters 10 (1963) 299; K.Min, Ph.D.Thesis, University of Illinois (1963). We wish to thank Dr. Axel for providing us with a copy of Min's thesis as well as for discussions. 7. B.Goulard and R.H.Venter, to be published.
NUCLEAR BINDING ENERGY AND QUADRUPOLE-QUADRUPOLE INTERACTION A. ARIMA and M. NOMURA
Department of Physics, Faculty of Science, University of Tokyo and H. KAWARADA
Department of Applied Physics, Faculty of Engineering, University of Tokyo Received 12 October 1965
Many i n t e r e s t i n g a t t e m p t s have been made to study n u c l e a r collective motion. It has b e e n 400
pointed out that a l o n g - r a n g e i n t e r a c t i o n plays an i m p o r t a n t role in the collective v i b r a t i o n and
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15 November 1965
(MeV) 1.5
(Mev) 0.5 a(Z,N) 0
0.5 -0.5
0
-I
-I.5 ~ 0
I
I
2
4
I
6
I
[
I
I
8
I0
12
14
& ;o ;, ¢8 ,& ,& ,io ,;, Neutron number
< j'221 v, lj'22>-
Fig. 1. Excitation energy of states with the spin two. A pure configuration (jj~)n is taken, and states are a s sumed to have pure seniority 2, which must be broken when the Q-Q interaction lffenerally v2} is introduced. The coupling constant K
Fig. 2. Ot(Z,N). The coefficient of the n2 term in the observed binding energies is one-eighth of a(Z, N). n
-
n(2j+
I,,Ij2oo> +
where
1Xj2jI VIj2j),
and
< j 2 2 j I V Ij22j> = = -2
(2) ~
(2X+
1)W(jjjj;Xd)(jjX[ V[jjX).
X=even
T h i s V i s j u s t the p a r t i c l e - h o l e i n t e r a c t i o n [4]. We c o n s i d e r only the c a s e of e v e n - e v e n nuclei h e r e , but the s a m e a r g u m e n t s holds in the c a s e of odd nucleus. Now V i s d i v i d e d into t h r e e p a r t s , V = V0 + V I + V2
(S)
where
V 0 = vO(rl, r2) = 2 f f v ( n 2 ) d e d ~ and f d ~ m e a n s averaging over the spin coordinate, V 1 is the odd tensor interaction and V 2 is the even tensor interaction [5], the rank of which is larger than zero. B y this separation the matrix element (2) turns out to be very simple,
(2')
v2lj20o>. 401
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80 ~
v
e
d
nuclei
§ 70 c
.
.
.
.
.
.
.
.
.
.
.
60 50
t
90
I
ii10
I00 Neutron number
*
I
120 126
Section of the binding energy ~o ~,.~(.ilrfoce at certain : 2 ) / . ~ / uJ ad CO'~"
"~...~.~'" 82
90
I 0 II Neutron number
(b) 120 126
Fig. 3. (a) A schematic representation of one region of the nuclear periodic table. Nuclei are deformed inside an ellipsoid and spherical outside it. (b) Section of the binding energy surface at certain Z of the above region.
A s p i n - d e p e n d e n t i n t e r a c t i o n a. aV(n) i s an e x a m ple of the odd t e n s o r i n t e r a c t i o n , and the u s u a l l o c a l W i g n e r i n t e r a c t i o n is a l i n e a r c o m b i n a t i o n of V0 and V2 type i n t e r a c t i o n s . Substituting t h i s r e l a t i o n into (1), we obtain
(4) 2j + 1 , "2^^ I¢200> + 2-T:T_ 1 u uo 1' "~21J2oo~± ' '~ +
+ ½.(.- 1){(Poo Ivot,Poo> - 2 ~ _ 1cj2oo I v21 ./2oo)}. In the r i g h t hand s i d e of (4), the f i r s t t e r m shows the p a i r i n g p r o p e r t y and the s e c o n d the long r a n g e c o r r e l a t i o n s . It m u s t be n o t i c e d that the s i g n of the c o n t r i b u t i o n coming f r o m V0 to the long r a n g e c o r r e l a t i o n s is d i f f e r e n t f r o m t h a t of V2. The l o n g e s t r a n g e p a r t V0 which i s the m o n o p o l e - m o n o p o l e i n t e r a c t i o n m u s t be r e p u l s i v e without V2. A u s u a l c e n t r a l f o r c e which i s u s e d by the s h e l l m o d e l c a l c u l a t i o n t e n d s to V0 in the long r a n g e l i m i t . T a l m i ' s c o n c l u s i o n i s c o m p l e t e l y c o r r e c t in t h i s point. If we t a k e a l i n e a r c o m b i n a t i o n of the s h o r t r a n g e i n t e r a c t i o n 6(r) and the Q-Q one, the 402
f o r m e r c o n t r i b u t e s only to the p a i r i n g e n e r g y and the l a t t e r to both the p a i r i n g e n e r g y and the long r a n g e c o r r e l a t i o n . H e r e the Q-Q i n t e r a c t i o n m u s t be a t t r a c t i v e to r e p r o d u c e the r e p u l s i v e n2 d e p e n dence of the n u c l e a r binding e n e r g y . G e n e r a l l y , h o w e v e r , we can s a y only that the quantity (j200 ]V01j200) - 2 j - - ~ (j200 IV21j200)
O4 qi"
z 1"-4 Ld ad
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(5)
m u s t be r e p u l s i v e . F o r the 6(r) i n t e r a c t i o n , t h i s quantity v a n i s h e s , so that the 5(r) i n t e r a c t i o n b e l o n g s to the V 1 type. This condition (5) g i v e s v e r y i n t e r e s t i n g condition on the c e n t r a l W i g n e r i n t e r a c t i o n . A c c o r d i n g to t h i s condition, a l o c a l W i g n e r i n t e r a c t i o n such a s Yukawa and G a n s s i a n m u s t be r e p u l s i v e . On the other hand a s p i n dependent p o t e n t i a l (¢rcr)V(r) m u s t be a t t r a c t i v e . A s s u m i n g the s a m e s i m p l e c o n f i g u r a t i o n in, we can c a l c u l a t e an e x c i t a t i o n e n e r g y ~ E j ( n ) of a s t a t e with the s e n i o r i t y two and the a n g u l a r m o m e n t u m J,
AEj(n) = -(j200 I V1} j200) + ( j 2 2 J t V l l j 2 2 j ) + _(j200 }V21j200 > _ ( 2 ~ _2 (22)~( 2- 4£). 6),J (~200 IV2lJ 200) +
1 + 2(2a6) [(2fl-4)Dj-4(j22JtV21j22J] + (2 e - 2n) 2 + 2 ( 2 a - 4 ) ( m e - 6) [ 2__~___ 2(j2004
(6)
}v2]y200> +
+{ 2 (j22J IV21j22j)-Dj}] w h e r e 2~2 : 2j+ 1 a n d D j =-' ( j 2 j I' V 2 1 j 2 j ) + + (j2jI~221j2j>. it i s i n t e r e s t i n g to o b s e r v e that A E j i s a c o n s t a n t without V2, while V2 g i v e s a q u a d r a t i c dependence of A E j on n. G e n e r a l l y , even in the single c l o s e d s h e l l nuclei, f o r e x a m p l e the Sn i s o t o p e s , A E 2 of a nucleus with the s h e l l half f i l l e d i s l o w e r than that of nucleus with a few p a r t i c l e s (or holes) outside (or inside) the c l o s e d s h e l l [6]. To e x p l a i n t h i s situation, we obtain the following condition
D2- {2
(9)
The quantity in the b r a c k e t i s a l w a y s p o s i t i v e f o r
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j >/3, and v a n i s h e s f o r j = ~ when the s e n i o r i t y i s a l w a y s good quantum n u m b e r [7]. T h e r e f o r e the condition (9) i s s a t i s f i e d f o r ] >I ~z when k i s n e g a t i v e o r V2 i s a t t r a c t i v e . A n o t h e r i n t e r e s t i n g thing i s o b s e r v e d in (6). F r e q u e n t l y it i s pointed out that the s h e l l m o d e l c a l c u l a t i o n s of a few nucleon s y s t e m g i v e s h i g h e r AE 2 when the Q-Q i n t e r a c t i o n i s m o r e a t t r a c t i v e . On the o t h e r hand the r a n d o m p h a s e method of t r e a t i n g m a n y nucleon s y s t e m g i v e s l o w e r AE 2. T h i s s e e m s to be a p u z z l e . H o w e v e r , t h i s p u z z l e i s s o l v e d by studying the equation (6). F o r s i m p l i c i t y l e t u s a s s u m e 2~ >>1. O m i t t i n g the q u a n t i t i e s of the o r d e r of 1 / 2 ~ , we obtain A E 2 ( , ) = -(]200 IV1 I]200> + (j222 IV1[]222> +
-(]200 1V21]200>
+ (1 - n / ~ ) 2 ( j 2 2 2 1V21]222> + + (n/n)(1-./Ze)D 2.
When
(10)
n = 2,
~E2(2 ) = -
]Vllj200> + <]222 ]V1]j222 > +
-(]200 Iv21]200> + (]222 In t h i s [(j200 means action
Iv21j222>.
(11)
equation we can e a s i l y show that Iv2]j2o0>l > 1<]222 ]V2]j222>], which that the s t r o n g e r a t t r a c t i v e Q-Q i n t e r g i v e s a l a r g e r AE2(2). When n =
15 November 1965
o b s e r v e d binding e n e r g y BE(Z,/7) [10], ~(Z,N)
= BE(Z,N+2)
-
2BE(Z,N)+BE(Z,N- 2)
w h e r e Z and N a r e not n e c e s s a r i l y the m a g i c n u m b e r s . T h i s a i s constant ff the binding e n e r g y can be a p p r o x i m a t e d b y the q u a d r a t i c f o r m u l a of n. The i n t e r e s t i n g r e s u l t s :rare shown in fig. 2. This a i s a l w a y s r e p u l s i v e b o t h in the s p h e r i c a l r e g i o n and d e f o r m e d r e g i o n . In the t r a n s i t i o n a l r e g i o n s (N = 90 and N ~ 114), a b e c o m e s suddenly a t t r a c t i v e . T h i s happens p r o b a b l y b e c a u s e t h e r e a r e two s u r f a c e s of the binding e n e r g y [11]. One of t h e m b e l o n g s to the s p h e r i c a l shape and the o t h e r to the d e f o r m e d shape. Both c u r v e s of the binding e n e r g y a r e a p p r o x i m a t e d by the q u a d r a t i c f o r m u l a e of n. H o w e v e r , when the c o m p e t i t i o n o c c u r s , the r e a l binding e n e r g y i s on the envelope of two k i n d s of e n e r g y c u r v e s . T h e r e f o r e it m u s t be s t r e s s e d that only in the t r a n s i t i o n a l r e g i o n a can be attractive. The a u t h o r s would like to e x p r e s s t h e i r h e a r t i l y thanks to P r o f e s s o r S. Y o s h i d a and Dr. K. I k e d a f o r s t i m u l a t i n g d i s c u s s i o n s . T h e y a r e a l s o v e r y much indebted to P r o f e s s o r M. Kawai for his criticism.
a E 2 ( e ) = -(/200 I v 1 I]200> +
a t t r a c t i v e Q-Q i n t e r a c t i o n r e s u l t s in a s m a l l e r t~E2(~). G e n e r a l l y when n o r 2 ~ - n i s s m a l l , the s t r o n g e r k, the l a r g e r A E 2. On the c o n t r a r y , the s t r o n g e r K, the s m a l l e r ~ E 2 when 12~ - 2n [ i s s m a l l . T h i s f a c t e x p l a i n s c l e a r l y why the m o r e s t r o n g l y a t t r a c t i v e Q-Q i n t e r a c t i o n c a u s e s h i g h e r t~E2 in a n u c l e u s n e a r c l o s e d s h e l l s , while in the m i d d l e s h e l l the m o r e a t t r a c t i v e Q-Q i n t e r a c t i o n g i v e s a l o w e r AE 2. In the m i d d l e of the s h e l l , a too s t r o n g Q-Q i n t e r a c t i o n can m a k e AE2 n e g a t i v e , when of c o u r s e the s e n i o r i t y s c h e m e m u s t be b r o k e n s t r o n g l y . When t h i s happens, d e f o r m a tion m a y be p o s s i b l e even in the s i n g l e c l o s e d s h e l l n u c l e i [8]. F r e q u e n t l y it i s e x p e c t e d that a q u a d r a t i c t e r m in the n u c l e a r binding e n e r g y b e c o m e s a t t r a c t i v e when a n u c l e u s i s d e f o r m e d [9]. We c a l c u l a t e the following q u a n t i t i e s by using the
References 1. J . P . Elliott, Proc.Roy.Soc.A245 (1958) 128, 562. 2. T. Tamura and T. Udagawa, Prog. Theor. Phys. 26 (1961) 947. 3. I. Talmi, Rev. Mod. Phys. 32 (1962) 704. 4. A. Arima and H. Kawarada, J. Phys. Soc. Japan 19 (1964) 1768. 5. A. de-Shalit and I. Talmi, Nuclear shell theory (Academic P r e s s , New York and London, 1963). 6. K.Way, N.B.Gove, C.L.McGinnis and R.Nakasima, Nuclear data sheets, (National Research Council, Washington D.C.). 7. C .Schwartz and A .de-Shalit. Phys. Rev. 94 (1954) 1257.
8. M. Baranger and K. Kumar, Nuclear Physics 62 (1965) 113. 9. G.E. Brown, Unified nuclear theory (North-Holland Publ. Co., Amsterdam, 1964). 10. I. Konig, J. Mattauch and A. Wapstra, Nuclear Phys. 31 (1962) 18. 11. R.A. Demirkhanov, V.V. Dorokhov and V. G. Soloviev, preprint.
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