Δ excitations in nuclei

Volume 245, number 1

PHYSICS LETTERS B

2 August 1990

A excitations in nuclei T. U d a g a w a a.b, S.-W. H o n g b a n d F. Osterfeld b a Department of Physics, University of Texas, Austin, TX 78712, USA b Institutefffr Kernphysik, Kernforschungslange Jiilich GmbH, Postfach 1913, D-5170 Jiilich, FRG Received 30 April 1990; revised manuscript received 3 May 1990

An approach is proposed for the investigation of delta (A) excitations in nuclei induced by intermediate energy charge exchange reactions. The approach is based on the isobar-hole model which treats the dynamics of the A in the nucleus explicitly. The distortion effects of the projectile wave functions and the A-hole correlations in the nucleus are included properly. It is found that the A-hole correlations mediated by the energy-dependent n-exchange interaction of the spin longitudinal channel shifts the Apeak position downwards in energy by ~ 30 MeV. The origin of this shift is explained in detail.

Re~ntly, the (p, n) [ 1 ], the (3He, t) [2-7], and the (d, 2p) [5,6,8,9] charge exchange reactions at intermediate energies have revealed a systematic downward energy shift of the A(1232) resonance peak position for targets with A/> 12 with respect to that observed for a proton target. In contrast to this, in the case of 7 absorption [ 10 ] and inelastic electron scattering experiments [ 11-13 ], the A peak does not show such a pronounced displacement. The electromagnetic probes excite the A transversely, i.e., by the transition operator S × q T ( S and T are the spin and isospin transition operators, respectively), while the hadronic probes measure both the transverse (TR) and the longitudinal (LO) spin-isospin response. Therefore, it has been speculated that the shift of the A peak might be due to a nuclear medium effect in the isovector spin LO ( S . q T ) channel [6,14,15]. That is, if the delta particle-nucleon hole (AN- ~) residual interaction becomes strongly attractive at large momentum transfers Iq l ~ 1-2 f m - ~in this channel, then this attraction might lead to a lowering of the A mass produced in the target. Along this line of reasoning, no shift of the A-peak position is to be observed with the electromagnetic probes. In this letter we shall show that the LO response is indeed shifted downwards in energy and that this shift is caused by the energy-dependent n-exchange interaction. First, we present our approach for the description of A excitations in nuclei, in which the distortion ef-

fects on the projectile wave functions and the nuclear medium effects on the A are taken into account properly. The approach is based on the isobar-hole model which has been successfully used in the description of pion-nuleus scattering [ 16,17 ] and 7 absorption [ 17,18 ]. The A is assumed to move in a complex onebody potential, the imaginary part of which simulates the spreading effects. The A N - ~correlations are also included and are explicitly treated in a coupled channels (CC) formalism [19]. The model for the description of A excitations in nuclei is constructed as follows. The A is created in the charge exchange process A + a ~ (B + A) + b. Here A (B) and a (b) denote the target (residual nucleus) and projectile (ejectile), respectively. The reaction is assumed to proceed via a one-step direct process and is treated by the distorted wave impulse approximation (DWIA). The A thus excited then interacts with the hole-nucleus B via a complex one-body potential and the A N - ~ residual interaction. The effect of the latter is calculated within the T a m m - D a n c o f f approximation. The inclusive cross section for the process may be given as d2cr EaEbEAEB+ a kb Im(-(plGIp)), d E b d l 2 b - (2gh2c2W) 2 ka (1) where Ei is the total energy of particle i ( i = a, b, A,

0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )

1

Volume 245, number 1

PHYSICS LETTERS B

and B + A ) , k~ (kb) is the wave number o f a (b), and W is the total energy of all particles in the center of momentum system. I9) is the doorway state excited by the reaction and G is the Green's function that describes the propagation of the B + A system. ]9) is explicitly given by 19> = ( z A - } ~ I/NN,Na IXa~+ >~0a~0A> ,

(2)

where Xa~+~ and Zb( ~* are the projectile distorted wave functions in the incident and exit channel, respectively, ~0~and q~bare the intrinsic wave functions of a and b, and ~0Ais the initial target wave function assumed to be of spin-parity, IA=0 +. The effective N N - , N A transition operator is denoted by tNN,Naand its explicit form will be specified later. The round bra ( Ion the right-hand side ofeq. (2) denotes the integration with respect to the projectile coordinates only. For G, we make the following ansatz [ 16 ]: G=

1

E + iF,~/2- HB - T/, - UA -- VNA,N~ "

(3)

Here, E is the excitation energy of the B + A system, Fz is the energy-dependent free decay width of the A, and T6 and L~ are the kinetic energy operator and the A-nucleus potential, respectively. HB is the hamiltonian of nucleus B, and VNa.NAis the residual interaction describing the A N - ~correlations. The central problem is then to evaluate I~ ) -= G I P) in ( 1 ), which we rewrite as Ig ) ( = G I O ) ) = G o I A ) ,

(4)

IA) = IP) + VNA,NAGo IA) .

(5) (6}

Here, Go is the (optical model) Green's function obtained from G by neglecting VNa.Na. Once IA) is known, I~') is simply obtained from (4). Thus the problem is now to solve eq. (6). To do this, we expand I A ) ( a n d similarly I P ) ) in terms of the basic channel wave function [ YaO~]JA.I a s , A ) = ~ I Aah(r) I [ Y a ~ ] j M ) ,

expansions into eq. (6), one can derive a system of inhomogeneous coupled channels (CC) integral equations for the radial wave functions AAh(r). There are Nc coupled equations corresponding to the Nc Ah pairs. We use the Lanczos method [20 ] to solve the CC equations [ 19 ]. The A-nucleus optical potential, U6, in Go is taken as a complex Woods-Saxon potential with the radius parameter R = 1.1A 1/3 fm, diffuseness a = 0 . 5 3 fm, and the depths VA= - 2 5 MeV and W A = - 4 0 MeV for the real and imaginary potential, respectively [ 16,21 ]. The real part, Va, is assumed to be the sum of the A-nucleus single particle potential (depth= - 5 5 MeV) and of the real part of the A spreading potential ( s t r e n g t h = + 3 0 MeV) [21]. W~x represents the imaginary part of the A spreading potential [ 21 ]. The free decay width FA in Go is parameterized in the usual form [ 22 ]. l~a,h6 is assumed to consist of the n and 9-exchange potentials with an additional short-range interaction. In the P exchange we keep only the tensor interaction and drop the central part assuming that the latter can be effectively included in the short-range interaction. In the momentum representation, VNa.Na may be given as a sum of LO and TR components [23] VN~,NA = [ 1~,LN~,N~(S~ "4) (S~'0) + vTs,Na(S, X~)" (S] X4)] T~ "T~,

(8)

where V LNa,N~X= 4~hc

1

Go = E + i F e , / 2 - H B - TA -- U,~ '

2 August 1990

(7)

Ah

where Ya is the spin-angle wave function of the A and 0h is the eigenfunction of Hu, J and M being the total angular momentum and its projection. Inserting these

['f,(t)2

×(

+

~ t g'AA f ~(_)

2 f z ( t ) Vp) ,

3 m2

In (9), the f ( t = ~ o Z - q 2) are the meson-baryon vertex factors [17], which we assume to be f ( t ) = fNa(A2-m2)/(A2-t) (i=m9), and vi=q2/ ( t - m ] ) ( i = =, P). Here, mi andA, are the mass and cut-off mass of meson i, respectively. In the present calculations, the minimal g5,6 that cancels out the afunction piece of the =-exchange potential is used. Then the Landau-Migdal parameter gS,a ~0.3 (in units o f JrcNN ~---47r~lCf=NNfrcNN/m=2 ~ 1600 MeV fm 3). Note that this parameter depends on the choice of

Volume 245, number 1

PHYSICS LETTERS B

U6. Its accurate value is finally fixed from the requirement to reproduce the peak position of the A resonance in the medium. The other parameters in VNa.N6 were fixed as follows: f~NA=0.324, f 2oNA = 16.63, m~=0.14 GeV, rap=0.77 GeV¢I~=0.78 GeV, and Ap = 2 GeV. For the tNN,N~ transition operator the following simple ansatz is used:

2 August 1990

0.60-

p (p,n)A" 0.50 >. O.40-

(a)

Tp = 800 MeV

--

0 = 0°

deltQ function (0'Na =0.33)

-- one pion exohonge x 0,33

E "~0.30 ~.0.20 ~b 0.10-

INN'Na=g'NAJnN&( A ~ - t

"//

2

~

I

0.00

x [(~,-~)

(s~-~) +

(~,

xO-(s~ x#)]

T, .1"~ (10)

p(3He, t)A" %-

1.00

T3,e= 2 GeV

0.80-

with J~N~= 4~hCf~NNf~N~/m2~ 800 MeV fm 3, g ~ = 0.335, and A~ = 650 MeV. We made this choice since studies of the tNN.N~ interaction by means of the (d, 2p) reaction [9,24] indicated that there is a strong TR component in tNN,N£x with the ratio T R / L O = 2/1. In addition, these experiments suggested that tNN,N ~ is nearly constant in the (co, q) range relevant to the A-resonance region. We have reanalysed the p(d, 2p)A ° data of Ellegaard et al. [9] using the interaction ofeq. (10) [24]. The result is that this interaction can, indeed, simultaneously describe the inclusive cross section and the tensor analysing power data [9,24]. In fig. 1a we show the zero degree spectrum of the basic reaction p(p, n)A ++ at E = 8 0 0 MeV incident energy (full curve) obtained with the interaction of eq. (10). Both the shape and the magnitude of the experimental cross section [ 1 ] are reproduced well. For comparison we also give the results obtained with the one-pion-exchange interaction. This interaction can describe the shape of the cross section, but it fails in reproducing the tensor analysing power data of the p(d, 2p)A ° reaction [9,24]. In fig. lb we show a similar analysis of the p(3He, t)A ++ reaction at E = 2 GeV and zero degrees. The theoretical cross sections were calculated with the magnetic transition form factor f(oA q) = e x p [ 0 . 4 9 ( q 2 - ~ o 2 ) ] which was taken from experiment [2,4]. As can be seen from fig. lb, the tNN.N~interaction o f e q . (10) describes the p(gHe, t)A ++ data rather well. In particular, it gives a much better description of the data than the one-pion-exchange interaction (dashed curve ). The latter produces a cross section which peaks at too high an excitation energy.

e = 0°

~b/ --

della function (g'U& =0.33) - one pion exchQnge x 0.5

0.60-

~o.4o-

~b

0.200.00

i

100

i

200

500 400 (.O L (MeV)

500

600

700

Fig. 1. Zero degree spectra of the reactions p (p, n)A ++ at E= 800 MeV (a) and p(3He, t)& ++ at E=2 GeV (b). The (p, n) data are taken from ref. [ 1] and the (3He, t) data are taken from ref. [ 3 ]. For each reaction two different theoretical cross sections are compared to the data: One of them is calculated with the tNN.NA interaction of eq. (10) (full curve) and the other with the onepion exchange potential (dashed curve). The cross section due to the latter was renormalized as indicated in the figure. We remark that with the interaction of eq. (10) there is no need for a reaction mechanism involving the A excitation of the 3He projectile. This mechanism was advertised by Oset et al. [25 ] to explain the discrepancy between theory and experiment in case that the one-pion-exchange potential is used for /'NN,N~ (dashed curve). The calculations for the 12C(p, n) and ~2C(3He, t) reactions were performed by using the optical potentials given in table 1 and a pure shell-model configuration for the ~2C ground state wave function. The AN-~ -model space included all s- and p-hole states and all A orbitals with orbital angular m o m e n t u m la~< 8. With this model space, a typical value of Nc turned out to be 24. Fig. 2a shows the results for the ~2C(p, n) reaction in comparison with the data [ 1 ]. The full and long

Volume 245, number 1

PHYSICS LETTERS B

2 August 1990

Table 1 The optical potential parameters used in the DWIA calculations. V

p n 3He t

(MeV)

4.70 - 3.27 4.70 - 3.27

r (fm)

a (fro)

W (MeV)

r~ (fm)

ai (fm)

0.900 1.125 0.900 1.125

0.530 0.976 0.530 0.976

-40.0 - 20.0 - 72.0 - 35.0

0.931 1.125 0.931 1.125

0.568 0.592 0.568 0.592

target is shifted by about 70 MeV relative to that of the proton target (see fig. 1 ). This shift of 70 MeV has three different origins: The first is simple kinematics. The second is due to the A-spreading potential, W,x, which produces a relatively large cross section on the low excitation energy side of the A resonance in nuclei. Both effects c o m b i n e d with the b r o a d e n i n g of the A resonance in nuclei due to the Fermi m o t i o n lead to a trivial shift of the A b u m p in t2C by ~ 40 MeV relative to the A b u m p of the proton target. After consideration of these two effects, 30 MeV of the observed total shift of 70 MeV still r e m a i n unexplained. This additional shift is produced by the A-hole correlation effect shown in fig. 2a. The og-dependence of the n-exchange interaction plays a crucial role in obtaining this shift. To demonstrate this, let us define AEj= (t/IjI VNA,NA[~IlJ), where ~,j ((~usl~'j) = 1) is q/with the definite J~. I q/J) is the conjugate state of I~uj). AEj is a measure of the energy shift of the J~ state. It may be split into LO and T R c o m p o n e n t s as

0.80 -

0.70-

12C (p,n)

(o)

Tp = B00 MeV

--Correlated

~'0.600 = 0 o

Uncorreloted

-

..~.0.50-

~,.~

--- TR

~0.400.30-

"

-

/:,

,o

-

0.200.I0-

0.00 0.70-

~0.60-

,,,~0.50-

i

i

i

i

i

12C (p,n)

i

(b)

Tp,= 800 MeV 0=0

°

"~0.400.30-

0,200,10-

0.00 100

200

300 400 (.0L (MeV)

s00

600

700

Fig. 2. Zero degree neutron spectra for the reaction ]2C(p, n) at E= 800 MeV. The data are taken from ref. [ l ]. The theoretical cross sectionswere multiplied with a normalisationfactor N= 1.5. (a) The theoretical spectra calculated with (full curve ) or without (long dashed curve) AN-~ correlations. Also the longitudinal and transverse cross sectionsare shown separately for the case where correlations are included. (b) Contributions from different multipoles J* to the cross section. dashed curves represent the cross sections calculated with and without correlations, respectively. Both cross sections were multiplied with a n o r m a l i s a t i o n factor N = 1.5 to reproduce the m a g n i t u d e of the experimental data. One can see a strong energy shift in the A-peak position between the correlated a n d uncorrelated calculations. This energy shift a m o u n t s to ~ 30 MeV. We remark that the A-peak position for the J2C

1

A E j _ ( 2 n ) 3 J dq q2[ V~,,.NaML(q)

+ VTNa,N,xMT(q) ] ,

( 11)

where M E ( q ) = f dO ( - ~ j l e x p ( - i q ' r ) S ' ~

× (Op-Tt~=+,St.~exp(iq.r')l~uj)

Tu=_, 10) .

(12)

The expression for M T ( q ) is obtained from ME(q) by replacing S ' ~ with S)<~. In fig.3 we show VNA,NA L plotted versus q for O)~ab=250 MeV. Note that L VN~t,NA has a singularity at qpo~e= (/12m ~ ) 1 / 2 ~ 0 . 8 4 f m - ' . VNA,N L A is repulsive for q < qpo~e, but attractive for q > qpo~e. For a finite nucleus, the t h r e e - m o m e n t u m q is not a conserved q u a n t i t y in intermediate A N - ~ scatterings. Therefore, the integral

Volume 245, number 1

PHYSICS LETTERS B

2 August 1990

1.60-

r-

--I~ .

/ / ~/

/I/"

.

.

.

....

60

"'-.

,,~

(a)

1.40=

250

MeV

~jectile

~ '1.20-

, ( x os4 )

~I.00~

1+, L : O

~0.803

l , f

0.60~

0.40

0.20-

I

.00

,

.50

i/i T 1.00

1.50

0.00

i

-

r

2.00

,

2.50

I

3.00

I

3.50

1.40-

4.00

q (fr.-") Fig. 3. The AN-~ residual interaction in the spin longitudinal channel (full curves) and the longitudinal transition density for the J~= 1÷, L = 0 state (dashed curve) as functions of momentum transfer q.

~

12C(SHe,t)

I

(b)

T3,,= 2 GeM

'1.20-

e= o°

1.00-

- - - Distoded wave - - - Plane wave x 0.25

~0.80-

0.60-

~ 0.40-

over q appears in eqs. ( 11 ) and ( 12 ). In fig. 3 the real part of q2Mc(q) is also shown for the J~= 1 +, L = 0 state (dashed curve). As seen, the peak appears in the attractive region. This reflects the fact that t < 0. By folding q2ML(q) with VNzx.Na c one obtains a net attractive energy shift. This happens for all multipoles in the LO channel, leading to the downward shift of the A-peak position. Fig. 2b shows the contributions to the cross section from different multipoles J~. The lower spin states undergo the bigger energy shift. A large number o f J ~ is needed to obtain convergence in the cross section. We include all the multipoles up to J~ = 9 +. Similar energy shifts for different multipoles J~ were also found in a partial wave analysis of the total p i o n - n u cleus cross section for pion energies in the A-resonance region [26 ]. In fig. 4a we compare the theoretical cross sections for the ~2C(3He, t) reaction with the experiment. The full curves represent the cross sections including A N correlations, finite size effects of the projectile, and the distortion effects. The theoretical cross sections were multiplied by a normalisation factor N = 1.5 to reproduce the magnitude of the experimental data. The finite size of the projectile was parameterized by the magnetic transition form factorf(co, q) = e x p [ 0 . 4 9 ( q 2 - o 9 2 ) ] which was taken from experiment [2,4]. This form factor causes a trivial shift of the peak position towards lower excitation energies. This

0.200.00

I

I

[

I

100

200

300

400

~L (MeV)

500

600

700

Fig. 4. Zero degree triton spectra for the reaction ~2C(3He, t) at E = 2 GeV. The data are taken from ref. [3]. The theoretical cross sections were multiplied with a normalisation factor N = 1.5. The full curve represents our final results. The long dashed line is the cross section calculated by neglecting the projectile form factor, and multiplied by 0.34. The longitudinal and transverse responses are also shown separately. (b) Comparison of DWIA and PWIA cross section calculations to each other and to the data. The PWlA cross section was renormalized by a factor 0.25.

can be seen from a comparison of the full and the long dashed curves in fig. 4a. The latter curve is calculated for a point projectile, i.e., neglecting the projectile form factor. Also in fig. 4a the LO and TR cross sections are shown separately. They have a similar structure as the corresponding cross sections in the (p, n) reaction. In fig. 4b a D W I A calculation of the ~zC(3He, t) reaction is compared to a plane wave impulse approximation ( P W I A ) calculation and to the data. The two theoretical cross sections differ only in magnitude, but agree in shape. This means that the distortions at intermediate incident energies mainly lead to a pure absorption of flux, reducing the magnitude of the cross section but leaving the shape of the cross section unchanged. If there is at all an effect of dis-

Volume 245, number 1

PHYSICS LETTERS B

t o r t i o n s o n the r e s o n a n c e shape, t h e n it is a small upw a r d energy shift o f the D W I A cross section relative to the P W I A result. We have also p e r f o r m e d calculations a s s u m i n g the tNN,N~ i n t e r a c t i o n o f E s b e n s e n a n d Lee [22 ]. T h e A peak o f the J2C (3He, t) reaction is again well reproduced, b u t in this case the excitation is o f pure LO character o n c o n t r a r y to the e x p e r i m e n t a l observation o f the ratio o f L O : T R ~ 1 : 2. In the~2C(p, n ) reaction the calculated A peak appears n o w at too low a n excitation energy as is expected from the LO cross section s h o w n in fig. 2a. M o r e o v e r , with this interaction we c a n n o t r e p r o d u c e the (3He, t) d a t a o f A b l e e v et al. [ 7 ] m e a s u r e d at the i n c i d e n t energies o f 4 a n d 1 1 GeV. O n the other h a n d , with the tNN.NA o f eq. ( 1 0 ) we can. In s u m m a r y , we have s h o w n that the shift o f the Apeak p o s i t i o n o b s e r v e d in the (p, n ) a n d (3He, t) reactions at i n t e r m e d i a t e i n c i d e n t energies is due to the strongly attractive correlations in the LO s p i n - i s o p i n c h a n n e l . T h i s a t t r a c t i o n comes f r o m the energy-dep e n d e n t ~-exchange interaction. To o b t a i n sufficient a t t r a c t i o n we n e e d a g ~ p a r a m e t e r o f a b o u t ] ( i n u n i t s o f J~ZXA----47~hcf,~Naf~NA/m2~1600 M e V f m 3) which c o r r e s p o n d s to m i n i m a l short-range correlations. N o significant energy shift is f o u n d in the T R s p i n - i s o s p i n response. This is in a g r e e m e n t with what is o b s e r v e d in the electroexictation o f the A. F i n a l l y we r e m a r k that for b o t h the (p, n ) a n d the (3He, t) reactions there r e m a i n s o m e d i s c r e p a n c i e s b e t w e e n the calculated a n d m e a s u r e d spectra in the lower o~ region. Note in this o~ region n u c l e o n - p a r t i c l e - n u cleon-hole excitations can c o n t r i b u t e to the cross section, which are n o t i n c l u d e d in the p r e s e n t calculations. T h e i n c l u s i o n o f N N -~ excitations is in progress. F u r t h e r m o r e , in case o f the (3He, t) reaction the s p i n - l o n g i t u d i n a l a n d s p i n - t r a n s v e r s e projectile form factors m a y be different. In the p r e s e n t work both form factors were a s s u m e d to be equal. The s p i n - l o n g i t u d i n a l s p i n - t r a n s v e r s e form factors are different if there is a d-state a d m i x t u r e to the 3He a n d t r i t o n (ls½)3 wave functions. In this case the spinl o n g i t u d i n a l form factor can b e c o m e larger t h a n the t r a n s v e r s e o n e l e a d i n g to a n a d d i t i o n a l energy shift in the (3He, t) reaction [ 16 ]. F u r t h e r spin-flip trans-

2 August 1990

fer e x p e r i m e n t s are n e e d e d to solve this p r o b l e m . T h e authors acknowledge valuable discussions with C. G a a r d e . T h e work is s u p p o r t e d in part by the U S D e p a r t m e n t o f Energy.

References [ 1 ] D.A. Lind, Can. J. Phys. 65 (1987) 637; C.G. Cassapakis et al., Phys. Len. B 63 (1976) 35; B.E. Bonner et al., Phys. Rev. C 18 (1978) 1418. [ 2 ] C. Ellegaard et al., Phys. Lett. B 154 ( 1985 ) 110. [3] D. Contardo et al., Phys. Lett. B 168 (1986) 331. [4] I. Bergqvist et al., Nucl. Phys. A 469 (1987) 648. [5] C. Ellegaard, Can. J. Phys. 65 (1987) 600. [ 6 ] C. Gaarde, Nucl. Phys. A 478 (1988) 475c. [7] V.G. Ableev et al., Soy. Phys. JETP. Lett. 40 (1984) 763. [ 8 ] C. Ellegaard et al., Phys. Rev. Len. 59 (1987) 974. [9] C. Ellegaard et al., Phys. Lett. B 231 (1989) 365. [10] B. Mecking, Proc. Intern. conf. on Nuclear physics with electromagnetic interactions (Mainz 1979), Lecture Notes in Physics, Vol. 108 (1979) p. 382. [ 11 ] P. Barreau et al., Nucl. Phys. A 402 ( 1983 ) 515. [ 12] J.S. O'Connell et al., Phys. Rev. Lett. 53 (1984) 1627. [13] R.M. Sealock el al., Phys. Rev. Lett. 62 (1989) 1350. [ 14] G. Chanfray and M. Ericson, Phys. Lett. B 141 (1984) 163. [15]J. Delorme and P.A.M. Guichon, preprint, Institut de Physique Nucl6aire, Lyon-l; private communication. [ 16] M. Hirata, J.H. Koch, F. Lenz and E.J. Moniz, Phys. Len. B70 (1977) 281;Ann. Phys. (NY) 120 (1979) 205. [ 17 ] E. Oset, H. Toki and W. Weise, Phys. Rep. 83 (1982) 281. [ 18] J.H. Koch, E.J. Moniz and N. Ohtsuka, Ann. Phys. (NY) 154 (1984) 99. [ 19] T. Udagawa, S.-W. Hong and F. Osterfeld, Proc. 5th Intern. Conf. on Nuclear reaction mechanisms, ed. E. Gadioli (Universitfi di Milano, 1988 ) p. 606. [20] R.R. Whitehead et al., Advances in Nuclear Physics, eds. M. Baranger and E. Vogt, Vol. 9 (Plenum, New York, 1977 ) p. 123. [21 ] Y. Horikawa, M. Thies and F. Lenz, Nucl. Phys. A 345, (1980) 386. [ 22 ] H. Esbensen and T.-S.H. Lee, Phys. Rev. C 32 (1985) 1966. [23 ] M.R. Anastasio and G.E. Brown, Nucl. Phys. A 285 (1977) 516. [24] P. Oltmanns, F. Osterfeld, T. Udagawa and S.W. Hong, to be published. [25] E. Oset, E. Shiino and H. Toki, Phys. Lett. B 224 (1989) 249. [26] W. Weise, Nucl. Phys. A 278 (1977) 402; E. Oset and W. Weise, Nucl. Phys. A 319 ( 1979 ) 477; A 322 (1979) 365.