An isomeric state of the ΛHe7 hyperfragment

Volume 2, number 1

AN

PHYSICS

ISOMERIC

STATE

OF

LETTERS

THE

1August 1962

AHe 7 HYPERFRAGMENT

L. R. B. E L T O N ** Institute for Theoretical Physics, University of Copenhagen, Denmark Received 11 July 1962

R has been p r o p o s e d 1) that t h e r e e x i s t s an i s o m e r i c state hHe 7 . , which A - d e c a y s in p r e f e r e n c e to going to the g r o u n d state t h r o u g h a T - t r a n s i t i o n . The e s t i m a t e d excitation e n e r g y of this state is 1.8 + 0.6 MeV. If this i n t e r p r e t a t i o n i s c o r r e c t , then the binding e n e r g y of the A - h y p e r o n in the ground state of ^He7 is 5.1 + 0.4 MeV. This is in a c c o r d with the c o r r e s p o n d i n g quantity f o r hLi7, which is 5.5 + 0.3 MeV 2), and f o r ABe7, w h e r e it is 5.9 C 0.6 MeV 3). The e s t i m a t e d excitation e n e r g y of AHe7* is in v e r y good a g r e e m e n t with that f o r the f i r s t excited state of He 6, which is 1.71 + 0 . 0 1 MeV 4). We shall t h e r e f o r e a s s u m e that f o r the p u r p o s e of a r a d i a t i v e t r a n s i t i o n t h r A - h y p e r o n in ^He7 a c t s m e r e l y a s a s p e c t a t o r , a p a r t f r o m giving a g e n e r a l l y g r e a t e r binding to the nucleons. T h e r e is s o m e u n c e r t a i n t y w h e t h e r the f i r s t excited state of He 6 a c t u a l l y d e c a y s to the ground state b y m e a n s of a r a d i a t i v e t r a n s i t i o n and a d e c a y t h r o u g h b r e a k - u p , He 6 -- He 4 + 2 n, is c e r t a i n l y favoured441). However~ this d e c a y is e n e r g e t i c a l l y not p o s s i b l e f o r ^HeT. A s the ground snd f i r s t excited s t a t e s of He6 a r e 0 + and 2 + r e s p e c t i v e l y , we s h a h a s s u m e an E 2 - t r a n s i t i o n a l s o f o r the ~,-decay of AHe 7 . , and we shall show that the l i f e t i m e f o r this t r a n s i t i o n s u b s t a n t i a l l y e x c e e d s that of the /~-particle, so that the i s o m e r i c state d e c a y s by A - d e c a y a s h a s been c o n j e c t u r e d . The p r o b a b i l i t y f o r such a t r a n s i t i o n with e m i s sion of a quantum t/oa is 5)

__l V ,t,, x5

T(E2) - 5 M - ~ --~ (Q~)2 ,

ling, so that we have ~/(L = 2, S = 0, T = 1, T 3 = -1), ~f(L = 0, S = 0, T = 1, T 3 = -1). A s t h e r e a r e only n e u t r o n s outside the c l o s e d 1 s - s h e l l , the t r a n s i t i o n m u s t be due e n t i r e l y to c o r e p o l a r i z a t i o n . This is often s i m u l a t e d by giving the e x t r a - c o r e p a r t i c l e s an additional effective c h a r g e 6). If we a p p r o x i m a t e the n u c l e a r wavefunctions b y t h e i r lowest t e r m s

¢ i = ( l s ) 4 ( l s A ) ( l P ) 2,

L = 2,

S = 0,

(3)

Cf = ( l s ) 4 ( l s A ) ( l p ) 2,

L = 0,

S = 0,

(4)

and u s e h a r m o n i c o s c i l l a t o r wavefunctions with a s p r i n g constant a, then 1

( Q~2) = - (~-r) -~ qea 2 ,

(5)

w h e r e qe is the effective c h a r g e on e a c h of the two 1 p - n e u t r o n s . That the m a t r i x e l e m e n t is i n d e p e n dent of M is p h y s i c a l l y obvious. P r e s c r i p t i o n s given f o r q 6) a r e c e r t a i n l y not valid f o r a n u c l e u s a s light a s AHeT, and we t h e r e f o r e e s t i m a t e the c o r e p o l a r i z a t i o n by c a l c u l a t i n g the effect of a t w o - b o d y i n t e r a c t i o n V12 between one of the l p - n e u t r o n s and one of the I s - n u c l e o n s which r a i s e s the l a t t e r to a h i g h e r level in the o s c i l l a t o r well. In f i r s t o r d e r p e r t u r b a t i o n t h e o r y the only configuration that c o n t r i b u t e s to (1) is that in which a i s - p r o t o n is r a i s e d to the l d - l e v e l . We t h e r e f o r e put ~bi = ¢ i + ~ c/x ~aiX,

(1)

Cf = Cf + ~2 ~f2 ,

(6)

where where

,4

(Qf) = ek~=lqkr2k Y~M* (ek,lPk) Ib] i~i dr1...dr A

¢0iX : (ls)4(Ish)(1p)2=k(1s)-1(1d), L = 2, S = 0, (7) , (2)

x = wlc and eqk is the charge of the k-th particle. W e shall a s s u m e the nuclear wavefunctions to be given by shell model wavefunctions in L-S coup* Research supported by the National Institutefor Research in Nuclear Science, Great Britain. ** On leave of absence from Battersea College of Technology, London.

~f2 = ( l s ) 4 ( l s A ) ( l P ) L 2 ( l s ) - I ( l d ) ,

L = 0, S= 0,

(8)

and

2If°aN cik =


?Z : @Sz I V12 I/).

(9)

Here/7o~N = li/Ma 2 is the distance between the energy levels in the well and M is the m a s s of a nucleon. For the two-body interaction w e take

41

Volume 2, number 1

PHYSICS

V12 = - (Voc + Vo(~ a l " a2)exp ~-'r212/r2~o' ,

(10)

w h e r e the p a r a m e t e r s a r e o b t a i n e d f r o m low e n e r g y n - p s c a t t e r i n g data 7). The c o n f i g u r a t i o n a d m i x t u r e c o e f f i c i e n t s have been e v a l u a t e d by m e a n s of the t e c h n i q u e s of second q u a n t i z a t i o n 8), w h e r e we have to c o n s i d e r the g r a p h s of fig. 1, t o g e t h e r with the c o r r e s p o n d i n g g r a p h s i n v o l v i n g exchange of the two l p - n u c l e o n s . It will be n o t i c e d that ~/k and q~. c o n t a i n c o n f i g u r a t i o n s which do not e v e n t u a l l y c~)ntribute to . They do, h o w e v e r , c o n t r i b u t e to the c o e f f i c i e n t s c i k and cf2. A f t e r a lengthy c a l c u l a t i o n we obtain _1

1

1 (23__)e (5~ a2
foo

Rld(1) R l p ( 2 ) f 2 ( r l , r 2

O

) R l s ( 1 ) Rlp(2)

O

× r 2r 2dr 1dr2,

WE =

fF O

nlp(1) Rld(2)fl(rl,r

(12)

2) R l s ( 1 ) Rlp(2)

O

xr 2r 2dr ldr 2.

(13)

Here the R ( r ) a r e the r a d i a l w a v e f u n c t i o n s and 4~

V12 = ~ J l ~

~ 'r

,

1 r2) P/(cos ~12)"

(14)

Methods for evaluating integrals of the above type have been given by Talmi 9L The spring constant for the Ip-shell is 10) a = 1.6 + 0.1 fm, but as the h-hyperon gives some added b i n d i n g , we take a = 1.5 f m , c o r r e s p o n d i n g to h~0N = 19 MeV. The choice i s not c r i t i c a l for the f i n a l r e s u l t . Then C/o = - 0.140 ,

cf2 = - 0.045 ,

(i5) = . 0.040 ea 2 . Hence T(E2) = 9.0 x 108 sec -1 ,

(16)

so that the l i f e t i m e f o r the ~ , - t r a n s i t i o n i s ~'~, = 11 x 10-10 sec. A s the l i f e t i m e of a A - h y p e r o n i s rA = 2.5 x 10 -10 s e c , the s t a t e will d e c a y by Adecay. C o m p a r i n g (15) with (5) we a l s o s e e that the s o - c a l l e d effective c h a r g e i s qe = 0.045 e f o r each 1 p - n e u t r o n . F r o m the s i z e of Cio and cf2 it i s a p p a r e n t that h i g h e r o r d e r t e r m s will not c o n t r i bute s i g n i f i c a n t l y . A s we have u s e d h a r m o n i c o s c i l l a t o r w a v e f u n c -

42

1August 1962 N

N N P

N

PN

NN NN

NN

NN

NN NP

PN

NN

Fig. 1. Graphs for matrix elements (9). The top line gives the direct terms and the bottom line the exchange terms. N and P denote neutron and proton, and s, p and d give the angular momentum of the particles. Spin directions have not been marked.

(11)

w h e r e Cio and cf2 can be e x p r e s s e d in t e r m s of the d i r e c t and exchange r a d i a l i n t e g r a l s WD =

LETTERS

t i o n s , we have b e e n able to i g n o r e the effect of the s p u r i o u s c e n t r e of m a s s m o t i o n in the p o t e n t i a l well 11,12). F o r a s q u a r e well the effect i s c a n c e l l e d 12) by giving each l p - n e u t r o n an effective c h a r g e qe = - O. 25 Z e / A 2, which in our case a m o u n t s to q = - 0.01. Thus f o r a r e a l i s t i c p o t e n t i a l the effect will i n c r e a s e the y - l i f e t i m e v e r y slightly. F i n a l l y , it i s c l e a r that the long l i f e t i m e of h:~e7* i s due to the fact that a l l e x t r a - c o r e n u c l e o n s a r e n e u t r o n s and that the s - c o r e i s p a r t i c u l a r l y difficult to d e f o r m . It i s t h e r e f o r e not l i k e l y that any other h y p e r f r a g m e n t -- except p r e s u m a b l y AHe6 -- can exist in an isomeric state. My thanks are due to Professor G. E. Brown for drawing my attention to this problem; and to him, to Professors N.Austern and J. S. Blair, and to Dr. B. R. Easlea for most useful discussions. I am grateful to Professors Niels and Aage Bohr and the Copenhagen Institute for Theoretical Physics for hospitality, and to the Danish Government for the award of a State Scholarship. References 1) J. Pniewski and M. Danysz, Physics Letters 1 (1962} 142. 2) N.Crayton et al., EFINS 61-62, quoted in ref. 1}. 3) S. J. St. Lorant and S. Lokanathan, Physics Letters 1 (1962} 223. 4) F.Ajzenberg-Selove and T. Lauritsen, Nuclear Phys. 11 (1959) 1 -5) J. M. Blatt and V.F.Weisskopf, Theoretical nuclear physics (Wiley, 1962), p. 595. 6) B. R. Mottelson, The many-bodyproblem (Dunod, 1959), p. 283. 7) L.R.B. Elton, Introductorynuclear theory (Pitman, 1959), p. 81. 8) G. E. Brown, L. Castillejo and J.A. Evans, Nuclear Phys. 22 (]961) 1. 9) I. Talmi, Helv. Phys. Acta 25 (1952) 185. i0) L.R.B. Elton, Nuclear sizes (Oxford, 1961), p. 23. 11) J. P. Elliott and T. H. R. Skyrme, Nuovoeimento4 (1956) 164. 12) S.Gartenhausand C.Schwartz, Phys. Rev. 108 (1957) 482.