Square-well potential and three-nucleon bound-state properties

Volume 34B, number 3

PHYSICS LETTERS

SQUARE-WELL THREE-NUCLEON R. VAN

POTENTIAL BOUND-STATE

WAGENINGEN

15February 1971

AND PROPERTIES

and G. ERENS*

Natuurkundig Laboratorium der Vrije Universiteit, A m s t e r d a m , The Netherlands

Received 14 September 1970

The fact that several three-nucleon bound-state properties calculated with square-well potentials are very close to the corresponding values for repulsive-core potentials, is attributed to the comparatively strong attraction at large distances.

In a r e c e n t l e t t e r P a s k [1] has c o m m e n t e d on the r o l e of n o n - c e n t r a l i n t e r a c t i o n s in l o w - e n e r g y n u c l e a r p h y s i c s , e s p e c i a l l y in the t h r e e - n u c l e o n s y s t e m s . He v e r y c l e a r l y d e m o n s t r a t e s that such i n t e r a c t i o n s a r e c r u c i a l to a c o r r e c t u n d e r s t a n d ing of the n u c l e a r d y n a m i c s . In p a r t i c u l a r Pask points out that, although s o m e p r o p e r t i e s of the t h r e e - n u c l e o n s y s t e m can be roughly fitted by c e n t r a l f o r c e s only, as shown by B r a y s h a w and Buck [2], this does not mean that the t h r e e - n u cleon s y s t e m cannot be used to d i s c r i m i n a t e b e tween different t wo -n u c le o n p o t e n ti a l s , nor that such s i m p l e f o r c e s tell us s o m e th in g about the m a t h e m a t i c a l p r o p e r t i e s of m o r e r e a l i s t i c i n t e r actions. H o w e v e r , the a r g u m e n t s of Pask do not r e solve the intriguing question why such a good fit can be obtained at all with a s q u a r e - w e l l potential without any repulsion. In p a r t i c u l a r we would like to know why the r e s u l t s for the s q u a r e - w e l l potential a r e so s u r p r i s i n g l y c lo s e to the c o r r e s p o n d i n g r e s u l t s obtained with r e p u l s i v e - c o r e p ot e nt i al s , like the (I, I I I ) p o t e n t i a l of Malfliet and Tjon [3]. That this is so can be c l e a r l y seen f r o m the r e s u l t s given in table 1 and fig. 1. The se w e r e calculated using a v a r i a t i o n a l method with a h y p e r s p h e r i c a l function b a s i s . D e t a i ls about the c a l c u l a t i o n s can be found in the t h e s i s of E r e n s [4]. The question b e c o m e s even m o r e i n t e r e s t i n g when we look at the r e s u l t s calculated with a tt r a c t i v e Yukawa ((II, IV) of ref. [3]) and exponential (Ohmura, ref. [5]) p o t e n ti a l s , also given in table 1 and fig. 1. We than s e e that all s q u a r e * Present address: Department of Physics, University of South-Africa, P.O. Box 392, Pretoria, SouthAfrica. 184

well r e s u l t s a r e much c l o s e r to those f o r the s o f t - c o r e potential than to those f o r the other two potentials. All four potentials a r e adjusted to roughly the s a m e l o w - e n e r g y n u c l e o n - n u c l e o n data. Of c o u r s e , none of the p o t e n t i a l s used is r e a l i s t i c , since they do not contain n o n - c e n t r a l p a r t s , and since t h r e e of them do not have r e p u l si v e c o r e s to account f o r the high e n e r g y s c a t t e r i n g data. H o w e v e r , the p u r p o s e of this note is to find a r e a s o n f o r the unexpected r e sults for the s q u a r e well. In o r d e r to explain this apparently anomalous b e h a v i o u r of the s q u a r e well we must look f o r a p r o p e r t y which it s h a r e s with the r e p u l s i v e - c o r e potential, but not with the other two potentials. The only such p r o p e r t y we have been able to find is that both the s q u a r e - w e l l potential and the r e p u l s i v e - c o r e potential a r e s t i l l r a t h e r s t r o n g l y a t t r a c t i v e at i n t e r - p a r t i c l e d i s t a n c e s of the o r d e r of 2 fro, w h e r e a s f o r the Yukawa and exponential potentials the a t t r a c t i o n is much l e s s t h e r e . We t h e r e f o r e su g g est that the b o u n d - s t a t e p r o p e r t i e s of the t h r e e - n u c l e o n s y s t e m s a r e d et e r m i n e d not so much by the ab sen ce or p r e s ence of r e p u l s i o n n e a r the origin, but by the v a l u e s of the i n t e r - p a r t i c l e d i s t a n c e s for which the t w o - n u cl eo n potential is a t t r a c t i v e and s t i l l has an a p p r e c i a b l e strength. Indeed, by i n s p e c t i n g the a t t r a c t i v e Yukawa, exponential and s q u a r e - w e l l potential in this o r d e r , we see that the r e g i o n w h e r e the a t t r a c tion is st i l l f a i r l y s t r o n g m o v es outward, and that the shift is c o n s i d e r a b l y l a r g e r when going f r o m the exponential to the s q u a r e - w e l l potential than f r o m the Yukawa to the exponential potential. It is just in this s a m e o r d e r that each quantity in table 1 and fig. 1 e i t h e r i n c r e a s e s or d e c r e a s e s monotonically f o r t h ese t h r e e a t t r a c t i v e potentials;

Volume 34B, n u m b e r 3

PHYSICS

LETTERS

15 F e b r u a r y 1971

Table 1 Various bound-state p r o p e r t i e s of t h r e e - n u c l e o n s y s t e m s calculated for four different two-nucleon potentials Binding energy (Me V) 3He ~H

3He

3H

11.23 10.42 8.44 8.11

0.80 1.32 2.13 1.82

0.66 1.15 1.94 1.68

Potential Attr. Yukawa Attr. exp. square well s o ~ - c o r e Yukawa

12.11 11.24 9.12 8.77

% S' state

Charge radius (fm) 3He ~H 1.56 1.64 1.84 1.89

Coulomb energy difference (Me V)

1.43 1.49 1.67 1.72

0.883 0.817 0.683 0.655

h

I F'h,3H,,I

Squor¢

Well

t

0.01 S o f t - core

YJ

0 0.001

0

&

8 q2 ( fro-2 )

12

16

Fig. 1. The charge f o r m f a c t o r of 3He. The c u r v e s are obtained using t h r e e - n u c l e o n wave functions generated by the two-nucleon potentials indicated. E x p e r i m e n t a l points are included. A s i m i l a r picture is obtained for 3H.

and again the (absolute) difference is larger between the square-well and exponential potential results than between the exponential and Yukawa p o t e n t i a l r e s u l t s . On t h e o t h e r h a n d t h e r e g i o n of c o m p a r a t i v e l y s t r o n g a t t r a c t i o n d o e s n o t s h i f t very much farther when going from the squarew e l l p o t e n t i a l to t h e r e p u l s i v e - c o r e p o t e n t i a l , a n d indeed the calculated quantities are quite close t o e a c h o t h e r f o r t h e s e two p o t e n t i a l s . Our suggestion may be tested somewhat more d i r e c t l y b y l o o k i n g a t t h e p r o b a b i l i t y t h a t two particles are a certain distance r apart, avera g e d o v e r a l l p o s s i b l e p o s i t i o n s of t h e t h i r d

i

0.8

i

II.~ 1 212

131.2

i

~.0

r( f m )

Fig. 2. The probability that two of the t h r e e p a r t i c l e s in the t h r e e - b o s o n model will be a distance r apart averaged over the position of the t h i r d particle. particle. For the three-identical-boson model this distribution can be easily calculated approxim a t e l y f r o m t h e e q u i v a l e n t t w o - b o d y m e t h o d [6]. T h e s e d i s t r i b u t i o n s a r e d e p i c t e d in fig. 2. A g a i n it i s s e e n t h a t a l l f e a t u r e s h a v e t h e s a m e s e q u e n c e . At s m a l l d i s t a n c e s t h e Y u k a w a a n d e x p o n e n t i a l p o t e n t i a l give m u c h l a r g e r p r o b a b i l i t i e s t h a n the s q u a r e - w e l l a n d r e p u l s i v e - c o r e p o t e n tials, for large distances the reverse is true. M o r e o v e r t h e m a x i m a of t h e f i r s t two p o t e n t i a l s lie at significantly smaller distances than those f o r t h e l a s t two p o t e n t i a l s . On t h e b a s i s of t h e a b o v e e v i d e n c e we e x p e c t t h a t t h e r e s u l t s f o r e.g. a G a u s s a n d S a x o n W o o d s f o r m of i n t e r a c t i o n t a k e , i n t h i s o r d e r , a n i n t e r m e d i a t e p o s i t i o n b e t w e e n t h o s e f o r the exponential and square-well potentials. Furthermore, it would b e d e s i r a b l e t o do 185

Volume 34B, number 3

PHYSICS

c a l c u l a t i o n s with the " s u p e r s o f t - c o r e " p o t e n t i a l s d e v e l o p e d by S p r u n g and S r i v a s t a v a [7], w h i c h fit the 'S o pp and np p h a s e s h i f t s up to 400 MeV o r h i g h e r . T h e y h a v e only a v e r y w e a k r e p u l s i o n n e a r the o r i g i n , but a r e quite s t r o n g l y a t t r a c t i v e at r a t h e r l a r g e d i s t a n c e s . Such an i n v e s t i g a t i o n would be p a r t i c u l a r l y i n t e r e s t i n g s i n c e , on the one hand, t h e r e is a s t r i k i n g q u a l i t a t i v e , and in some cases even semi-quantitative agreement b e t w e e n the half o f f - s h e l l b e h a v i o u r of t h o s e p o t e n t i a l s and of the 'S o R e i d p o t e n t i a l (cf. f i g s . 6 - 1 0 of r e f . [7]), w h e r e a s , on the o t h e r hand, B r a y s h a w and B u c k [2] h a v e s t a t e d that t h e r e is a s i m i l a r i t y b e t w e e n the half o f f - s h e l l b e h a v i o u r of the t w o - n u c l e o n t m a t r i x of the s q u a r e w e l l and that of the 'S o R e i d p o t e n t i a l . Now F i e d e l d e y [8] h a s shown that the t h r e e - b o s o n b i n d i n g e n e r g y is a l m o s t the s a m e f o r r a n k two s e p a r a b l e p o t e n t i a l s w i t h the s a m e o n - s h e l l and half o f f - s h e l l b e h a v i o u r which, t a k e n t o g e t h e r with the a b o v e f a c t s , would i m p l y that at l e a s t f o r c e n t r a l p o t e n t i a l s it w i l l be h a r d to d i s c r i m i n a t e b e t w e e n c e r t a i n q u i t e d i f f e r e n t r a d i a l f o r m s of the p o t e n t i a l e v e n w i t h the h e l p of t h r e e - n u c l e o n b o u n d - s t a t e p r o p erties. M o r e g e n e r a l l y , in c o n n e c t i o n with o u r s u g g e s t i o n , it would be w o r t h w h i l e to i n v e s t i g a t e w h e t h e r the (half) o f f - s h e l l t m a t r i x e l e m e n t s a r e s e n s i t i v e to the d i s t a n c e s f o r which the a t -

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t r a c t i o n is s t i l l a p p r e c i a b l e . F i n a l l y we would like to e m p h a s i z e a g a i n that the p o t e n t i a l s u s e d in this p a p e r a r e by no m e a n s r e a l i s t i c , so that no s e r i o u s c o m p a r i s o n with e x p e r i m e n t a l r e s u l t s can be m a d e . On the o t h e r hand we e x p e c t that the e f f e c t d i s c u s s e d a b o v e w i l l s t i l l be p r e s e n t when n o n - c e n t r a l t e r m s a r e i n c l u d e d in the i n t e r a c t i o n . One of us (G. E.) g r a t e f u l l y a c k n o w l e d g e s the f i n a n c i a l s u p p o r t of the South A f r i c a n C o u n c i l f o r S c i e n t i f i c and I n d u s t r i a l R e s e a r c h , and of the Shell C o m p a n y (South A f r i c a ) .

References [1] C. Pask, Phys. Letters 33B (1970) 325. [2] D. D. Brayshaw and B. Buck, Phys. Rev. Letters 24 (1970) 733. [3] R. A. Malfliet and J. A. Tjon, Nucl. Phys. A127 (1969) 161. [4] G. Erens, Bound-state properties of a model t h r e e nucleon system calculated with a hypersphericai function basis, Thesis Vrije Universiteit, A m s t e r dam, 1970. [5] T. Ohmura, Progr. Theor. Phys. 41 (1969) 419. [6] A. R. Bodmer and S. All, Nucl. Phys. 56 (1964) 657. [7] D. W. L. Sprung and M. K. Srivastava, Nucl. Phys. A139 (1969) 605. [8] H. Fiedeldey, Phys. Letters 30B (1969) 603.