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Some evidence of regge trajectory
~olume 3, number 1
PHYSICS
SOME
EVIDENCE
OF
LETTERS
REGGE
15 November 1962
TRAJECTORY
S. IWAO Istituto di Fisiea dell' UniversitY, Genova, Italy, and Istituto di Fisica Nucleare, Sezione di Genova and G. A. VIANO Istituto di Fisica dell' Universit~t, Genova, Italy Received 19 October 1962
Regge has shown, in p o t e n t i a l s c a t t e r i n g , that the s c a t t e r i n g a m p l i t u d e h a s s i n g u l a r i t i e s c o r r e sponding to the i n t e g e r v a l u e s of the a n g u l a r m o m e n t a 1). We want to show in t h i s p a p e r s o m e t y p i c a l exa m p l e of the Regge t r a j e c t o r y i n the low e n e r g y a l p h a - a l p h a s c a t t e r i n g and d i s c u s s the p h y s i c a l s i g n i f i c a n c e of the r e s u l t . We s h a l l f i r s t d i s c u s s the r e s u l t s o b t a i n e d f r o m the r e a l p a r t of the a n g u l a r m o m e n t u m and the r e lated q u a n t i t i e s . The r e a l p a r t of the a n g u l a r m o m e n t u m a s a t i s f i e s the following r e l a t i o n d [9(~ + I)] = dE
2MR2
(I)
2MR 2 =
(3.32 + 0.20) x 10 -11 cm ,
C = 0.65 + 0 . 2 6 . The r a d i u s R and the r e a l p a r t of the a n g u l a r m o m e n t u m at z e r o e n e r g y ~(0) a r e given to be R = (4'.19 + 0.20" - 0.02. 0.24) x 10_13 c m , and ~(0) = 0.44 +- 0.11 The r a d i u s o b t a i n e d above will be c o m p a r e d with those f r o m o t h e r s o u r c e s . If we a s s u m e that Be 8 in i t s ground s t a t e i s the S-wave bound state of two a l p h a s , the r a d i u s of the Be 8, RBe, i s given by 1
RBe = (2MIEBel)-~ = 1.05 × 10 -12 cm ,
where E, M and R are the kinetic energy, the r e duced mass and the radius of the effective interaction. Eq. (I) holds, in general, for two spinless particle interaction in the non-relativistic problem. We shall start from the lower three states of Be 8 nucleus by making use of eq. (I). There are three known states with the spins and parities 0+, 2+ and 4+. All these states have isospin 0. The r e lative kinetic energies of two alphas in these states are 0.094, 2.99 and 11.79 MeV, respectively 2,3). As we are going to discuss below, these states are convenient for our study. Their predominant decay modes are into two alphas. The alpha particle is a strongly bound state of four nucleons with isospin 0, angular momentum 0, parity +, and binding energy 28.30 MeV. We may consider Be8 as a bound state of two alphas in the low energy region considered above. Condition (1) is almost satisfied by the states discussed above. If we assume strict linearity, the analytic continuation of the real part of the angular momenttu'n is performed simply by
~(~ + 1) = 2 M R 2 E
t h r e e p o i n t s by m a k i n g u s e of the two p a r a m e t e r e q u a t i o n given above. We get
+ C ,
(2)
w h e r e C is a c o n s t a n t . We shall m a k e a l e a s t s q u a r e s fit to the o b s e r v e d
where EBe is the binding energy of the two alphas. The radius obtained in this way is about 2.5 times larger than that from the Regge trajectory. If we assume that the Be 8 has a dumb-bell structure of two alphas for low energies, the rotational model 4) gives the same radius as that obtained from the Regge trajectory. The open structure of the Be 8 nucleus is due to the small binding energy of two alphas in its ground state. In the current treatment the reduced widths were tabulated assuming the radius R = (4-5) × 10-13 cm (ref. 2)). As we have seen above the method based on Regge explains well the radius of the effective interaction of the alpha-alpha scattering. If the r e a l p a r t of the a n g u l a r m o m e n t u m of a l p h a - a l p h a s c a t t e r i n g i s c o n t r o l l e d by the v a c u u m pole in the m o m e n t u m t r a n s f e r c h a n n e l , ~(0) i s expected to be 1 5). The v a l u e o b t a i n e d by u s i s about half of that value. The i m a g i n a r y p a r t of the a n g u l a r m o m e n t u m s a t i s f i e s the following r e l a t i o n .
B(E r ) r ( E r ) = ~i~(Er)/dE ,
(3)
where Er and r ( E r ) are the resonance energy and
Volume 3, number 1
PHYSICS
Table 1 Imaginary part of the angular momenta for two alpha interaction. ~
0.094 2.99 11.79
0 2 4
(2.6 _+ 1.0) × 10-3 0.24 + 0.10 0.77
We s e e f r o m t a b l e 1 that the i m a g i n a r y p a r t of the a n g u l a r m o m e n t u m in the l o w e s t e n e r g y s t a t e i s v e r y s m a l l , a s e x p e c t e d f r o m the R e g g e t h e o r y . In o r d e r to d e t e r m i n e the e n e r g y d e p e n d e n c e of /3 we s h a l l expand it in the p o w e r s e r i e s of E.
B(E) = a + bE + cE2 + . . . .
(4)
T h e c a l c u l a t e d t h r e e c o n s t a n t s a r e g i v e n by a = 0.56 × 10 - 2 , b = 0.17 x 10-11 c m and c = - 0.72 x 10-24 c m 2 . T h e s e n u m b e r s show that B m a y h a v e l i n e a r e n e r g y d e p e n d e n c e in the e n e r g y r e g i o n c o n s i d e r e d above. Th e e x p e r i m e n t a l e r r o r s of the r e s o n a n c e widths a r e so l a r g e that the t h r e e p a r a m e t e r fit m a d e h e r e is a p r o v i s i o n a l one. We a r e now going to d i s c u s s the t r a j e c t o r y with odd J p a r i t y . T h e r e a r e two s t a t e s with s p i n s and p a r i t i e s 1+ and 3 +. T h e y m a y h a v e i s o s p i n 1. T h e r e l a t i v e k i n e t i c e n e r g i e s of two a l p h a s in t h e s e s t a t e s a r e 17.73 and 19.31 MeV. T h e 1 + and 3 + s t a t e s d e c a y into 7, P and n , p r e s p e c t i v e l y . The g r a d i e n t of the t r a j e c t o r y d e f i n e d by eq. (1) i s 2MR 2= 5.84 x 10-11 c m , which i s about two t i m e s b i g g e r than that of the t r a j e c t o r y with the e v e n J p a r i t y . T h e r e d u c e d m a s s of the n u c l e o n and the r e s i d u a l n u c l e u s i s a l i t t l e s m a l l e r than one half of the r e d u c e d m a s s of two a l p h a s . T h e s e f a c t s g i v e about t w i c e a s l a r g e an i n t e r a c t i o n r a d i u s c o m p a r e d to the r e a c t i o n r a d i u s 1.4 (A~ + 1) × 10-13 c m . It i s r a t h e r h a r d to u n d e r s t a n d the a n o maly obtained here. We found a l a r g e d i f f e r e n c e b e t w e e n the g r a d i e n t s of the e v e n and the odd J p a r i t y t r a j e c t o r i e s . E x p e r i m e n t a l l y the d e c a y m o d e of the odd J p a r i t y s t a t e s i s not into two a l p h a s but into n u c l e o n p l u s n u c l e u s (the s m a l l g a m m a width w i l l be n e g l e c t e d -
10
15 November 1962
in the f o l l o w i n g d i s c u s s i o n ) . T h e d i f f e r e n c e of the g r a d i e n t s m a y be e x p l a i n e d by the f o l l o w i n g c o n siderations: 1. Th e spin d e p e n d e n c e of the n u c l e o n - n u c l e u s i n t e r a c t i o n w i l l m o d i f y eq. (1). 2. Th e a l p h a - a l p h a p o t e n t i a l i s q u i t e d i f f e r e n t f r o m the n u c l e o n - n u c l e u s p o t e n t i a l . F i n a l l y we want to m a k e a c o m m e n t on a f u t u r e study. If we s u b t r a c t the pole t e r m s , we can ap p l y the N / D m e t h o d to the a l p h a - a l p h a s c a t t e r i n g 6). L e t us c a l l the s c a t t e r i n g a m p l i t u d e of the a l p h a al p h a s c a t t e r i n g in s - c h a n n e l A ( s , z ) . We get
t he t o t a l width at that e n e r g y r e s p e c t i v e l y . T h e i m a g i n a r y p a r t s of the a n g u l a r m o m e n t a in the t h r e e r e s o n a n c e e n e r g i e s a r e g i v e n in t a b l e 1.
E r (MeV)
LETTERS
d(s,~) -- ~ dl(s,~) Pl(-z) \- A~Ub(s,z) -~ ~ Cl(S) s i n n ~ ( ( s ) + /-~
(5)
-
w h e r e s i s the s q u a r e of the c e n t e r of m a s s m o m e n t u m q, s : q2, z = c o s 0, l = a + i/3, ci(s), Pl(x) and A~Ub(s, z) a r e the a p p r o p r i a t e c o e f f i c i e n t s , the L e g e n d r e function of the c o m p l e x a n g u l a r m o m e n t u m and the s u b t r a c t e d a m p l i t u d e . T h e f u n c t i o n A~Ub(s,z) has no p o l e s in the e n e r g y p l a n e . T h u s we d e f i n e Nl(S, z) and Dl(S ,z ) a s
gl(s , z) : d~Ub(s, z) Dl(S, z) .
(6)
Th e s t a n d a r d m e t h o d m a y ap p l y to the r e l a t i o n o b tained above. In t h i s p a p e r we h a v e st u d i ed on the a n a l y t i c a l p r o p e r t i e s of the a n g u l a r m o m e n t a of the a l p h a al p h a s c a t t e r i n g f r o m the r e s o n a n c e s t a t e s of Be 8. T h u s d i r e c t study of the a l p h a - a l p h a s c a t t e r i n g i s n e c e s s a r y in f u r t h e r study of the p r o b l e m .
References 1) T.Regge, Nuovo Cimento 14 (1959) 951:18 (1960} 947. 2) F.Ajzenberg-Selove and T. Lauritsen, Nuclear Phys. 11 (1959) 1. 3) L.A. Konig, J . H . E . Mattauch and A. H. Wapstra, Nuclear Phys. 31 (1962) 18. 4) A.Bohr, Dan. Mat. Fys. Medd. 26 (1952) hr. 14. 5) G.F.Chew, UCRL-10058, 8 February 1962. See also papers cited in this paper. 6) G. F. Chew, S-matrix theory of strong interactions (Benjamin Inc., New York, 1961)
PHYSICS
SOME
EVIDENCE
OF
LETTERS
REGGE
15 November 1962
TRAJECTORY
S. IWAO Istituto di Fisiea dell' UniversitY, Genova, Italy, and Istituto di Fisica Nucleare, Sezione di Genova and G. A. VIANO Istituto di Fisica dell' Universit~t, Genova, Italy Received 19 October 1962
Regge has shown, in p o t e n t i a l s c a t t e r i n g , that the s c a t t e r i n g a m p l i t u d e h a s s i n g u l a r i t i e s c o r r e sponding to the i n t e g e r v a l u e s of the a n g u l a r m o m e n t a 1). We want to show in t h i s p a p e r s o m e t y p i c a l exa m p l e of the Regge t r a j e c t o r y i n the low e n e r g y a l p h a - a l p h a s c a t t e r i n g and d i s c u s s the p h y s i c a l s i g n i f i c a n c e of the r e s u l t . We s h a l l f i r s t d i s c u s s the r e s u l t s o b t a i n e d f r o m the r e a l p a r t of the a n g u l a r m o m e n t u m and the r e lated q u a n t i t i e s . The r e a l p a r t of the a n g u l a r m o m e n t u m a s a t i s f i e s the following r e l a t i o n d [9(~ + I)] = dE
2MR2
(I)
2MR 2 =
(3.32 + 0.20) x 10 -11 cm ,
C = 0.65 + 0 . 2 6 . The r a d i u s R and the r e a l p a r t of the a n g u l a r m o m e n t u m at z e r o e n e r g y ~(0) a r e given to be R = (4'.19 + 0.20" - 0.02. 0.24) x 10_13 c m , and ~(0) = 0.44 +- 0.11 The r a d i u s o b t a i n e d above will be c o m p a r e d with those f r o m o t h e r s o u r c e s . If we a s s u m e that Be 8 in i t s ground s t a t e i s the S-wave bound state of two a l p h a s , the r a d i u s of the Be 8, RBe, i s given by 1
RBe = (2MIEBel)-~ = 1.05 × 10 -12 cm ,
where E, M and R are the kinetic energy, the r e duced mass and the radius of the effective interaction. Eq. (I) holds, in general, for two spinless particle interaction in the non-relativistic problem. We shall start from the lower three states of Be 8 nucleus by making use of eq. (I). There are three known states with the spins and parities 0+, 2+ and 4+. All these states have isospin 0. The r e lative kinetic energies of two alphas in these states are 0.094, 2.99 and 11.79 MeV, respectively 2,3). As we are going to discuss below, these states are convenient for our study. Their predominant decay modes are into two alphas. The alpha particle is a strongly bound state of four nucleons with isospin 0, angular momentum 0, parity +, and binding energy 28.30 MeV. We may consider Be8 as a bound state of two alphas in the low energy region considered above. Condition (1) is almost satisfied by the states discussed above. If we assume strict linearity, the analytic continuation of the real part of the angular momenttu'n is performed simply by
~(~ + 1) = 2 M R 2 E
t h r e e p o i n t s by m a k i n g u s e of the two p a r a m e t e r e q u a t i o n given above. We get
+ C ,
(2)
w h e r e C is a c o n s t a n t . We shall m a k e a l e a s t s q u a r e s fit to the o b s e r v e d
where EBe is the binding energy of the two alphas. The radius obtained in this way is about 2.5 times larger than that from the Regge trajectory. If we assume that the Be 8 has a dumb-bell structure of two alphas for low energies, the rotational model 4) gives the same radius as that obtained from the Regge trajectory. The open structure of the Be 8 nucleus is due to the small binding energy of two alphas in its ground state. In the current treatment the reduced widths were tabulated assuming the radius R = (4-5) × 10-13 cm (ref. 2)). As we have seen above the method based on Regge explains well the radius of the effective interaction of the alpha-alpha scattering. If the r e a l p a r t of the a n g u l a r m o m e n t u m of a l p h a - a l p h a s c a t t e r i n g i s c o n t r o l l e d by the v a c u u m pole in the m o m e n t u m t r a n s f e r c h a n n e l , ~(0) i s expected to be 1 5). The v a l u e o b t a i n e d by u s i s about half of that value. The i m a g i n a r y p a r t of the a n g u l a r m o m e n t u m s a t i s f i e s the following r e l a t i o n .
B(E r ) r ( E r ) = ~i~(Er)/dE ,
(3)
where Er and r ( E r ) are the resonance energy and
Volume 3, number 1
PHYSICS
Table 1 Imaginary part of the angular momenta for two alpha interaction. ~
0.094 2.99 11.79
0 2 4
(2.6 _+ 1.0) × 10-3 0.24 + 0.10 0.77
We s e e f r o m t a b l e 1 that the i m a g i n a r y p a r t of the a n g u l a r m o m e n t u m in the l o w e s t e n e r g y s t a t e i s v e r y s m a l l , a s e x p e c t e d f r o m the R e g g e t h e o r y . In o r d e r to d e t e r m i n e the e n e r g y d e p e n d e n c e of /3 we s h a l l expand it in the p o w e r s e r i e s of E.
B(E) = a + bE + cE2 + . . . .
(4)
T h e c a l c u l a t e d t h r e e c o n s t a n t s a r e g i v e n by a = 0.56 × 10 - 2 , b = 0.17 x 10-11 c m and c = - 0.72 x 10-24 c m 2 . T h e s e n u m b e r s show that B m a y h a v e l i n e a r e n e r g y d e p e n d e n c e in the e n e r g y r e g i o n c o n s i d e r e d above. Th e e x p e r i m e n t a l e r r o r s of the r e s o n a n c e widths a r e so l a r g e that the t h r e e p a r a m e t e r fit m a d e h e r e is a p r o v i s i o n a l one. We a r e now going to d i s c u s s the t r a j e c t o r y with odd J p a r i t y . T h e r e a r e two s t a t e s with s p i n s and p a r i t i e s 1+ and 3 +. T h e y m a y h a v e i s o s p i n 1. T h e r e l a t i v e k i n e t i c e n e r g i e s of two a l p h a s in t h e s e s t a t e s a r e 17.73 and 19.31 MeV. T h e 1 + and 3 + s t a t e s d e c a y into 7, P and n , p r e s p e c t i v e l y . The g r a d i e n t of the t r a j e c t o r y d e f i n e d by eq. (1) i s 2MR 2= 5.84 x 10-11 c m , which i s about two t i m e s b i g g e r than that of the t r a j e c t o r y with the e v e n J p a r i t y . T h e r e d u c e d m a s s of the n u c l e o n and the r e s i d u a l n u c l e u s i s a l i t t l e s m a l l e r than one half of the r e d u c e d m a s s of two a l p h a s . T h e s e f a c t s g i v e about t w i c e a s l a r g e an i n t e r a c t i o n r a d i u s c o m p a r e d to the r e a c t i o n r a d i u s 1.4 (A~ + 1) × 10-13 c m . It i s r a t h e r h a r d to u n d e r s t a n d the a n o maly obtained here. We found a l a r g e d i f f e r e n c e b e t w e e n the g r a d i e n t s of the e v e n and the odd J p a r i t y t r a j e c t o r i e s . E x p e r i m e n t a l l y the d e c a y m o d e of the odd J p a r i t y s t a t e s i s not into two a l p h a s but into n u c l e o n p l u s n u c l e u s (the s m a l l g a m m a width w i l l be n e g l e c t e d -
10
15 November 1962
in the f o l l o w i n g d i s c u s s i o n ) . T h e d i f f e r e n c e of the g r a d i e n t s m a y be e x p l a i n e d by the f o l l o w i n g c o n siderations: 1. Th e spin d e p e n d e n c e of the n u c l e o n - n u c l e u s i n t e r a c t i o n w i l l m o d i f y eq. (1). 2. Th e a l p h a - a l p h a p o t e n t i a l i s q u i t e d i f f e r e n t f r o m the n u c l e o n - n u c l e u s p o t e n t i a l . F i n a l l y we want to m a k e a c o m m e n t on a f u t u r e study. If we s u b t r a c t the pole t e r m s , we can ap p l y the N / D m e t h o d to the a l p h a - a l p h a s c a t t e r i n g 6). L e t us c a l l the s c a t t e r i n g a m p l i t u d e of the a l p h a al p h a s c a t t e r i n g in s - c h a n n e l A ( s , z ) . We get
t he t o t a l width at that e n e r g y r e s p e c t i v e l y . T h e i m a g i n a r y p a r t s of the a n g u l a r m o m e n t a in the t h r e e r e s o n a n c e e n e r g i e s a r e g i v e n in t a b l e 1.
E r (MeV)
LETTERS
d(s,~) -- ~ dl(s,~) Pl(-z) \- A~Ub(s,z) -~ ~ Cl(S) s i n n ~ ( ( s ) + /-~
(5)
-
w h e r e s i s the s q u a r e of the c e n t e r of m a s s m o m e n t u m q, s : q2, z = c o s 0, l = a + i/3, ci(s), Pl(x) and A~Ub(s, z) a r e the a p p r o p r i a t e c o e f f i c i e n t s , the L e g e n d r e function of the c o m p l e x a n g u l a r m o m e n t u m and the s u b t r a c t e d a m p l i t u d e . T h e f u n c t i o n A~Ub(s,z) has no p o l e s in the e n e r g y p l a n e . T h u s we d e f i n e Nl(S, z) and Dl(S ,z ) a s
gl(s , z) : d~Ub(s, z) Dl(S, z) .
(6)
Th e s t a n d a r d m e t h o d m a y ap p l y to the r e l a t i o n o b tained above. In t h i s p a p e r we h a v e st u d i ed on the a n a l y t i c a l p r o p e r t i e s of the a n g u l a r m o m e n t a of the a l p h a al p h a s c a t t e r i n g f r o m the r e s o n a n c e s t a t e s of Be 8. T h u s d i r e c t study of the a l p h a - a l p h a s c a t t e r i n g i s n e c e s s a r y in f u r t h e r study of the p r o b l e m .
References 1) T.Regge, Nuovo Cimento 14 (1959) 951:18 (1960} 947. 2) F.Ajzenberg-Selove and T. Lauritsen, Nuclear Phys. 11 (1959) 1. 3) L.A. Konig, J . H . E . Mattauch and A. H. Wapstra, Nuclear Phys. 31 (1962) 18. 4) A.Bohr, Dan. Mat. Fys. Medd. 26 (1952) hr. 14. 5) G.F.Chew, UCRL-10058, 8 February 1962. See also papers cited in this paper. 6) G. F. Chew, S-matrix theory of strong interactions (Benjamin Inc., New York, 1961)