Pion absorption in the oxygen isotopes

Volume 32B, number 7

PHYSICS

PION

ABSORPTION

IN

LETTERS

THE

31 August 1970

OXYGEN

ISOTOPES

K. CHUNG and M. G. H U B E R Institute for Theoretical Physics, Erlangen, Germany B. B L U M Institute for Theoretical Physics, Frankfurt, Germany

and M. DANOS National Bureau of Standards, Washington, D.C., USA Received 10 July 1970

The absorption rates for bound pions in 160 and 180 are calculated on the basis of an independent particle model modified by short range nucleon-nucleon correlations. The ratio of the l s and 2p level widths can be explained quantitatively by assuming an exchange of high momenta (/~q ~ 300 MeV/c) between otherwise independently moving nucleons. It turns out that the detailed properties of the two d-neutrons in 180 are of vital importance to account for the observed differences of the absorption rates.

In a r e c e n t e x p e r i m e n t by B a c k e n s t o s s et al. [1] the p r o p e r t i e s of the p i c n i c I s and 2p l e v e l s of 160 and 180 h a v e b e e n s t u d i e d ( s e e fig. 1). F o r t h e r a t i o s E and ~ of t h e e n e r g y s h i f t s and w i d t h s the f o l l o w i n g v a l u e s h a v e b e e n r e p o r t e d : E l s = ~ l s (180) / ~ l s (160) = 1.27 Vls = Fls (180)/Fls

~2p = r 2 p ( 1 8 0 ) / r 2p (160) = 0 . 8 1 ± 0.20 (1.c) T h e s e r e s u l t s a r e d i f f i c u l t to u n d e r s t a n d in t e r m s of an o p t i c a l p o t e n t i a l f o r t h e p i o n - n u c l e u s i n t e r a c t i o n [2]; t h e t h r e e e x p e r i m e n t a l n u m b e r s of eq. (1) s e e m to c o n t r a d i c t e a c h o t h e r : t h e v a l u e l s > 1 i n d i c a t e s that the r e a l p a r t of t h i s p o t e n t i a l i s m o r e s t r o n g l y r e p u l s i v e in 180 than in 1 6 0 . C o r r e s p o n d i n g l y the p r o b a b i l i t y P of f i n d i n g a pion i n s i d e t h e n u c l e u s is r e d u c e d f o r a I s pion

[3]: (2 .a)

F o r a 2 p - p i o n , h o w e v e r , t h e v a l u e of P i s i n c r e a s e d *: P 2 p ( 1 8 0 ) / P 2 p (160) = 1.09

(2.b)

* This effect is essentially due to the larger radius of 180; the 2p-wave function is almost unaffected by the strong pion-nucleus repulsion. 536

ss

2p

(1.a)

(160) = 1 . 1 5 ~ 0 . 1 5 (1.b)

Pz s ( 1 8 0 ) / p is (16o) = 0.80

E keY|

180

'a'"'m"'"1111

llillllllll[~--

"F2p = 38 eV

T2p = 4.7 eV

ri~

"1~1s=7.Se key

=8.67 key

160

21C

220 I A I s : 2 3 . ? t keV

230 is 2/,O

Coulomb

interQction

AIs:18.63 keY

-~

L Experiment

Experiment

Coul~r~b interoction

Fig. 1. The energies of the l s and the 2p Levels of the picnic atoms 160 and 180 are shown schematically. The Coulomb-effects are compared with the experimental results. T h e r e f o r e , one e x p e c t s V l s < 1 and V2p > 1. T h i s i s in a p p a r e n t c o n t r a s t to the e x p e r i m e n t a l r e s u l t s of eq. (l.b) and ( l . c ) . T h e s e d a t a a r e e v e n m o r e p u z z l i n g in t h e f r a m e w o r k of an i n d e p e n d e n t p a r t i c l e m o d e l : the y " can be a b s o r b e d only by a p r o t o n . S i n c e t h e two a d d i t i o n a l n e u t r o n s do not g r e a t l y a f f e c t t h e p r o t o n w a v e f u n c t i o n s , any c h a n g e of the a b s o r p t i o n w i d t h s o p p o s i t e to t h o s e e x p r e s s e d in eq. (2) c a n n o t be u n d e r s t o o d in s u c h a m o d e l . It a l s o

Volume 32B, number 7

PHYSICS LETTERS

31 August 1970

Table 1. The experimental values of the l s and the 2p absorption rates W1. and W2n l in 160 and 180 are compared with calculations including ("correlated pair") and not inelusing (~shell model") the effects of the short range two-body correlations. The correlations are characterized by the parameter ~ (see eq. 4). Absorption rates (sec -1) Wls ( 1 8 0 ) Wls(160) W2p(180) W2p(160) J

3.4 x 1012

7.0 x 1012

/~q = 240 MeV/c

5.1 x 1018

4.8 x 1018

2.3 x 1015

2,5 x 1015

//q = 300 MeV/c

8.5 × 1018

6.5 x 1018

3.1 x 1015

3.3 x 1015

//q = 400 MeV/c

3.2 x 1018

3.0 x 1018

8.7 x 1014

9.0 x 1014

Experiment

1.38 x 1019

1.2 x 1019

4.8 x 1015

6.0 x 1015

shell mode!

8.1 x 1010

8.8 x 1010

correlated pair

should be noted, that the e x p e r i m e n t a l a b s o r p t i o n r a t e s a r e by f a r too l a r g e to be accounted for by independently moving n u c l e o n s (see table 1). Although the pion is a s s u m e d to i n t e r a c t only with one n u c l e o n at a t i m e , the a b s o r p t i o n p r o c e s s leads p r e d o m i n a n t l y to the e m i s s i o n of two n u c l e o n s . This effect i s a p p a r e n t l y due to the high m o m e n t u m c o m p o n e n t s of the wave function r e s u l t i n g f r o m the s h o r t r a n g e p a r t of the twobody i n t e r a c t i o n between the n u c l e o n s . A r e c e n t paper [4] i n v e s t i g a t e d which m o m e n tum components m u s t be p r e s e n t in the (initial and final state) nucleon wave function in o r d e r to explain the l s and 2p a b s o r p t i o n widths in 160. When expanding the J a s t r o w c o r r e l a t i o n factor [5], F(rii) = 1-f(rij) , in t e r m s of m o m e n t u m eigenfun~tions f(rl2)

= f dq o~(q) jo(qrl2)

(3)

a component j o ( q r l 2 ) d e s c r i b e s the exchange of m o m e n t u m ~/q between the n u c l e o n 1 and 2, and ¢o(q) d e t e r m i n e s the p r o b a b i l i t y of such an exchange. By putting

w(q)

= 5(q-~),

(4)

it was found that the 160 data can be explained with 250 Mev/c ~< r ~ ~< 400

MeV/c.

(5)

The s a m e v a l u e s a r e also r e q u i r e d to fit e l a s tic e l e c t r o n s c a t t e r i n g data [6]. In the p r e s e n t note we r e p o r t on the r e s u l t s of applying a s i m i l a r a n a l y s i s to the case of 180. The s i n g l e p a r t i c l e wave functions a r e d e t e r m i n e d by a Wood-Saxon p o t e n t i a l s , both in the i n i t i a l and the final state. F o r the i n t e r a c t i o n the n o n r e l a t i v i s t i c G a l i l e a n - i n v a r i a n t (ps-pv) i n t e r a c t i o n H a m i l t o n i a n [4]

Hint = f ~c (~. ~.

V~ - ~ur . ¢ ~ .

VN )

(6)

was used. H e r e ¢ and r denote the spin and i s o spin o p e r a t o r acting on a nucleon, ~ and M the m a s s of the pion and the nucleon, r e s p e c t i v e l y . F r o m p i o n - n u c l e o n s c a t t e r i n g data the coupling c o n s t a n t f is known to be

f2/41r~ic

= 0.083 + 0.005

The pion wave function ~blrwas obtained a s the solution of the K l e i n - G o r d o n equation taking into account the n u c l e a r finite size and the v a c u u m p o l a r i z a t i o n effects. The r e a l p a r t of the s t r o n g i n t e r a c t i o n of the pion with the n u c l e u s was d e s c r i b e d by a r e p u l s i v e potential which has been fitted to the o b s e r v e d energy shifts of the pionic a t o m s [3]. The a b s o r p t i o n p r o c e s s i s d e s c r i b e d by a two body m a t r i x e l e m e n t ; fdqw(q)<~a,(1)~k/3' (2)IHint (1) ]Jo(qrl2)@a(1)~/fl(2)>

(7)

+ fdqw(q)(~,(1)~k~,(2)Jo(qrl2)fHint

(1) [ ~ot (1)~kfl(2))

By putting eq. (4) in eq. (7) the m a t r i x e l e m e n t is given by: M~ = (~ot, (1)~kfl,(2)IHint(1)Ijo(Orl2)~(1)~hfl(2))

(8) + (~k~, (1)~kfl,(2)Jo(~r12)IHint (1) I@~(1)~h/3(2)> The c a l c u l a t i o n s have been p e r f o r m e d for v a r i o u s values of the p a r a m e t e r ~. The r e s u l t s a r e given in table 1. As can be seen, the p r o p e r o r d e r of magnitude has been reproduced; the r e m a i n i n g d i s c r e p a n c y between m e a s u r e d and c a l culated capture r a t e s is roughly a factor of two. It s e e m s to be p a r t l y due to the u n c e r t a i n t i e s in the pionic wave function, q57r, inside of the nu537

Volume 32B, number 7

PHYSICS LETTERS

cleus. F u r t h e r m o r e the energy dependent optical potential f o r the continuum wave functions m a y be m o r e r e a l i s t i c . Th e r e l a t i v e widths, h o w e v e r , a r e in good a g r e e m e n t with the e x p e r i m e n t : I s (calc.) = 1.20 + 0.10

(9.a)

72p (calc.) = 0.93 + 0.03

(9.b)

The r a n g e of the t h e o r e t i c a l v a l u e s indicated in eqs. (9.a) and (9.b) is a s s o c i a t e d with the r a n g e of v a l u e s of ~ given by (5). The r e s u l t s obtained r e f l e c t the d e t a i l s of the s t r u c t u r e of the two nuclei: the two additional neutrons in the dshe l l of 180 cannot d i r e c t l y c o n t r i b u t e to the a b s o r p t i o n p r o c e s s ; h o w e v e r , the n u m b e r of p r o t o n - n e u t r o n p a i r s has i n c r e a s e d . It t u r n s out, that the two-body m a t r i x e l e m e n t containing a ds h e l l neutron is l a r g e r than a v e r a g e in the c a s e of the a b s o r p t i o n of a l s pion and s m a l l e r in the c a s e of a 2p pion. T h i s can be u n d e r s t o o d as follows: a d - n u c l e o n in a c o r r e l a t e d p a i r p r o v i d e s m o r e r e l a t i v e m o m e n t u m than a p- or s nucleon; c o r r e s p o n d i n g l y the m o m e n t u m m i s m a t c h is s m a l l e r and the a b s o r p t i o n p r o b a b il it y l a r g e r . T h i s ap p l i es to l s - p i o n s which a r e d i s t r i b u t e d o v e r the whole v o l u m e . A 2p-pion, howe v e r , is c o n c e n t r a t e d at the n u c l e a r s u r f a c e ; h e r e the o v e r l a p between p r o t o n s and the d - s h e l l n e u t r o n s is r e d u c e d and the e f f e c t of the c o r r e lations is t h e r e f o r e l e s s i m p o r t a n t . F r o m the p r e s e n t a n a l y s i s the following conc l u s i o n s can be drawn:

538

31 August 1970

1. The a s s u m p t i o n of c o r r e l a t e d n u c l e o n - n u cleon p a i r s inside of the nucleus p r o v e s to be s u c c e s s f u l in explaining the o b s e r v e d a b s o r p t i o n rates; 2. the a b s o r p t i o n r a t e is quite s e n s i t i v e to the d e t a i l s in the wave function of the nucleons i n v o l v ed , e s p e c i a l l y to the m o m e n t u m d i s t r i b u t i o n of the bound nucleons. This is an indication that f r o m pion a b s o r p t i o n it s e e m s to be p o s s i b l e to extract detailed nuclear structure information The a u t h o r s want to thank G. B a c k e n s t o s s , CERN, f o r suggesting this p r o b l e m and f o r h e l p ful d i s c u s s i o n s . One of us (K. Chung) a c k n o w l edges the p o s s i b i l i t y to w o r k at CERN as a v i s i t o r , and the kind hospitality extended to h i m there.

References [1] G. Hackenstoss et al., Phys. Letters 25B (1967) 365, Phys. Letters 29B (1969) 140; G. Backenstoss, Proc. Third Inter. Conf. on High energy physics and nuclear structure, to be pub[ished. [2] M.Krell and T.Ericson, Nucl. Phys. H l l (1969) 521. [3] B.Blum, Diplom Thesis (Frankfurt/M 1968) and to be published. [4] K. Chung, M.Danos and M.G.Huber, Phys. Letters 29B (1969) 265. [5] R.Jastrow, Phys. Rev. 98 (1955) 1479. [6] S.T.Tuan, L.E.Wright andM.G.Huber, Phys. Rev. Letters 23 (1969) 174; L.E.Wright, S.T. Tuan and M.G.Huber, Lettere aI Nuovo Cimento 13 (1970) 253.