Simultaneous study of the 11Li and 10Li nuclei in a microscopíc cluster model

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 626 (1997) 647-668

Simultaneous study of the llLi and l°Li nuclei in a microscopic cluster model E Descouvemont 1 Physique Nuclgaire Th~orique et Physique Math~matique, CP229, Universitg Libre de Bruxelles, B- 1050 Bruxelles, Belgium

Received 12 August 1997; accepted 17 September 1997

Abstract The 11Li and ~°Li nuclei are investigated in the Generator Coordinate Method involving 9Li+n+n and 9Li+n cluster configurations, respectively. The 9Li nucleus is described in the shell model with all p-shell wave functions allowed by the Pauli principle. This procedure yields many 9Li states, defined by a flexible mixing of shell model basis states. Two different nucleonnucleon interactions are used. We compute some spectroscopic properties of llLi, such as radii, quadrupole moments and densities. The model predicts a 1/2- excited state, but no positive-parity state. With the same conditions of calculation, we study the l°Li system. We find a l + assignment for the ground state. We present the s phase shifts, and discuss the possible existence of a low-energy virtual s state. The properties of such a state are shown to be partly model dependent. @ 1997 Elsevier Science B.V. PACS: 21.60.Gx, 21.10.Gv, 27.20.+n

I. Introduction Light nuclei close to the drip lines are widely investigated [ 1-4]. These exotic nuclei present new properties, such as a halo structure, which deserve many theoretical studies. The l tLi nucleus was the first candidate for a halo enhancement of the interaction cross section with heavy targets lead coworkers [5,6] to the hypothesis of a 9 L i + n + n structure, with an extension of the external neutrons. J Mattre de Recherches FNRS. 0375-9474/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PII S0375-9474(97)00504-6

experimental and system. A strong Tanihata and his important spatial

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P. Descouvemont/Nuclear Physics A 626 (1997) 647-668

Since more than ten years, a considerable amount of work has been devoted to the l lLi nucleus. Different kinds of experiments have been performed to investigate its structure; for example: (i) Breakup of l JLi, which shows a narrow peak in the momentum distribution of 9Li fragments [7] ; (ii) Elastic scattering of llLi and 9Li with protons [8]; (iii) Measurements of the quadrupole and magnetic moments [9] of l lLi, which are quite similar to the 9Li values; (iv) /3 decay measurements of 11Li to high-excited states of llBe [ 10]; (v) Investigation of the I ILi spectrum with different techniques [ 11-13 ] ; a dipole nature of the first excited state has been recently suggested [ 14]. In the theoretical point of view, the l lLi nucleus has been studied in many different ways: the Faddeev Method [ 15], the standard shell model [ 16] and the cluster-orbital shell model [ 17], different versions of the three-body model [18,19], the Hartree-Fock model [20], and the Antisymmetrized Molecular Dynamics (AMD) method [21]. All these works conclude on a halo structure of z~Li, mainly due to the low binding energy of the two external neutrons. This property requires that theoretical models be able to describe the wave functions not only at small core-neutron distances, but also at larger distances, where the neutron density is much larger than in non-halo nuclei. We refer the reader to Refs. [ 22,4,3 ] for further references on experimental and theoretical works. Interest for the 11Li nucleus has been still revived by the suggestion [23] that a low-energy virtual s state in l°Li is necessary to explain the 9Li momentum distribution of l lLi breakup. Several experiments have been devoted to the l°Li spectrum, and to the search for a low energy s state [24-31], but the situation is still unclear. On the theoretical side, the existence of 1÷ and/or 2 + low-lying states seems now well established, but the presence of an s resonance is predicted by some models [32,19] and contradicted by others [27,33,34]. In the present work, we aim at investigating the I I Li and l°Li nuclei in the multicluster Generator Coordinate Method (GCM). The input of the model is a nucleon-nucleon interaction, and a cluster structure for llLi (= 9Li+n+n) and l°Li (= 9Li+n). Notice that the llLi and l°Li basis not only involve 9Li cluster in its ground state, but also in different excited states. Core excitations have been shown to be important in halo nuclei [35,19], and therefore should be taken into account for a reliable description of l lLi and l°Li. The multicluster GCM has been applied to different halo nuclei, such as 6He [36] or 1aBe [35]. In this microscopic model, a halo structure is not a priori included; the basis wave functions involve many spatial configurations, and the structure of the nucleus is determined by the A-nucleon Schr6dinger equation. The originality of the present work is to perform a simultaneous study of llLi and l°Li, with identical nucleon-nucleon forces. These forces involve one parameter which is fitted on the well known i i Li binding energy. The l°Li study is therefore completely parameter free. The first goal of this work is a detailed analysis of the 11Li nucleus in the GCM approach. This investigation allows us to test the model for the subsequent application to l°Li, but also to complement the knowledge of 11Li. In parallel with ground state

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properties, we analyze some excited states. The second step is the application of the model to the l°Li system, which is known to have no bound states. Consequently, a theoretical description of l°Li requires a correct treatment of boundary conditions. In the present study, scattering asymptotic conditions are satisfied with the use of the microscopic R-matrix method (MRM). This method has been shown to be an efficient complement to the GCM approach: bound-state, resonance and scattering properties of a system can be determined in a consistent way [37]. The paper is organized as follows. In Section 2 we briefly present the microscopic model, and the conditions of the calculation. The 9Li wave functions are analyzed. In Section 3 we describe different properties of lILi: r.m.s, radii, quadrupole moment, neutron and proton densities, etc. The l°Li nucleus is discussed in Section 4, and conclusion remarks are given in Section 5.

2. The microscopic model 2.1. 9Li wave functions

The llLi (l°Li) wave functions are described in the three (two) cluster model, involving 9Li as a core of the system, and two (one) external neutrons. In this model, the core wave functions are defined in the harmonic oscillator model involving all possible configurations in the p-shell. In this way, core excitations, expected to be important in halo nuclei [ 19], are consistently introduced. Such a multicluster model has been already applied to other systems, such as 14Be for example [35]. Details about the method are given in Refs. [35,38], and we just focus here on the peculiarities of the 9Li nucleus. Let us consider the 90 Slater determinants ¢i describing 9Li states in the p shell. Projection on total spin I, intrinsic spin S, angular momentum L and isospin T is achieved by diagonalization of the (12, I z , S 2, L 2, T 2) operators in basis &i. Internal wave functions of 9Li are therefore given by 90

cSLT = ~

Iv dcSLZiq~i ,

( 1)

i=1

where coefficients d I~ are obtained from diagonalization of the operators mentioned above, and where c is an additional quantum number used to distinguish states with identical ( I u S L T ) values. Let us notice that basis states ( 1 ) (or, in other words, coefficients d I~) do not depend on the Hamiltonian H. The energy spectrum and wave functions of 9Li are obtained from the eigenvalue problem 1

1

1

1

(¢c,s,L,~-,I H - E~, ]dPcSLr)Dr,cSLr = O, cSLT

(2)

P. Descouvemont/Nuclear Physics A 626 (1997) 647-668

650

\ 0

1~4\

'

116

'

1.'8

1.4

'

1.6

b (fm)

1.8

b (fro)

-5 -20

2>

-10

q)

:5 LLI

-15

3/2,1

-20

3/2,2 1/2,0 1/2,3 1/2,2 1/2,1

1=7/2

-25

-25

1=5/2 1=1/2 -30

Fig. 1. Binding energy of 9Li as a function of the oscillator parameter b. On the left panel, St) = 0 and the curves are labeled by (S, L). The curves with (S, L) = ( 1 / 2 , 0 ) and ( 3 / 2 , 0 ) are identical. On the right panel, S0 = 40 MeV fm 5 and the curves are labeled by the total spin 1. In each case, the first eigenvalue is shown.

Table 1 Components and amplitudes C~Lr [Eq. ( 3 ) ] in the 9Li wave functions; n is the number of c values. The symbol ( * ) indicates the basis functions dropped in the 9 L i + n and 9 L i + n + n description I

T

S

L

n

a

b

c

d

1/2

3/2

1/2 1/2 3/2 3/2 1/2 1/2 1/2 3/2 3/2 3/2 1/2 1/2 1/2 3/2 3/2 1/2 3/2

0 l 1 2 l I 2 0 1 2 ! 2 3 1 2 3 2

l 2 l l 1 2 2 1 1 1 1 2 1 1 1 1 1

9.0 80.0 0.2 10.8 0 77.7 8.4 5.6 2.4 5.9 0 89.7 8.0 1.4 0.9 89.3 10.7

6.9 83.8 0.2 9.1 0 81.8 7.2 4.5 1.7 4.8 0 90.5 7.9 1.0 0.6 92.0 8.0

19.2 66.8 0.2 13.8 0 64.3 10.3 12.0 4.6 8.8 0 91.5 4.7 2.6 1.2 73.9 26.1

19.1 64.9 0.2 15.8 0 61.6 ll.3 11.6 5.4 10.1 0 89.4 6.1 3.1 1.4 74.5 25.5

3/2

5/2

7/2

5/2 3/2

5/2 1/2

1/2

V2 interaction with m = 0.6, So = 40 MeV fm 5. b V2 interaction with m = 0.765, &) = 39 MeV fm 5. Minnesota interaction with u = 1, So = 40 MeV fm 5. d Minnesota interaction with u = 0.804, So = 51 M e V f m 5.

* *

* *

* * *

P. Descouvemont/Nuclear PhysicsA 626 (1997) 647-668

651

where 3/refers to the excitation level in partial wave 1. In Table 1 we give the different sets of (SLT) values allowed by the Pauli principle. In Fig. 1 we show the first eigenvalue of H as a function of the oscillator parameter b. In this preliminary study, the Volkov V2 [39] interaction is used, with a standard value for the Majorana parameter m = 0.6. On the left panel, the spin-orbit force is not included, and S and L are therefore good quantum numbers. It is seen that the lowest configuration corresponds to S = 1/2, L = 1 and is fairly close to the S = 1/2, L = 2 configuration. The curves are nearly parallel tbr reasonable choices of the oscillator parameter, and the minima are located near b = 1.7 fm, except for the S = 3/2, L = 1 whose minimum is located beyond 1.8 fm. When a spin orbit force [40] is introduced (a standard amplitude So = 40 M e V f m 5 is adopted here), S and L are no more good quantum numbers, and the eigenvalues E( (see Eq. ( 2 ) ) are plotted on the right panel of Fig. 1. The I = 3/2 curve, which corresponds to the 9Li ground state, presents a minimum near b = 1.5 fro, but a reasonable energy splitting with I = 1/2 ( ~ 2.7 MeV) requires larger values, close to 1.75 fro, where the other partial waves present their own minimum. In view of this first analysis, and also of the 9Li spectroscopic properties (see below) an oscillator parameter b = 1.75 fm is adopted. This value represents a compromise between different requirements, such as the energy spectrum, r.m.s, radius or quadrupole moment. In order to evaluate the sensitivity with the nucleon-nucleon interaction, we also consider the Minnesota force [41], involving parameter u whose standard value is u = 1. Parameters m (for the Volkov force) and u (for the Minnesota force) are allowed to slightly vary around their standard value; reasonable choices for the spin orbit amplitude So range between So ~ 2 5 - 6 0 MeV fm 5, according to the system. In the study of I ILi, these degrees of freedom will be used to fix some basic properties of the system. A natural requirement is the binding energy of IILi (0.30 MeV, see Ref. [42] ); a second requirement is chosen as the excitation energy of 9Li (2.69 MeV, see Ref. [ 4 3 ] ) which, in multichannel models, must be as accurate as possible. These conditions are fulfilled with parameter sets (m = 0.765, So = 39 M e V f m 5) for the V2 force and (u = 0.804, So = 51 M e V f m 5) for the Minnesota lorce. The spectrum of 9Li is shown in Fig. 2. Experimental energies of the ground state and first excited state (assumed to be 1 / 2 - ) have been fitted by the nucleon-nucleon interaction. The shell model predicts three additional low-energy states with an identical ordering for both interactions. Energies predicted by the Minnesota potential are in very nice agreement with the experimental spectrum. The analysis of the 9Li wave functions is complemented by the weight of the different (SLT) components. In Table 1 we give the amplitudes

C~cr = ~

, IDj.csLr I2 ,

(3)

C

for the first eigenvalue of partial wave I, and for different turns out that qualitatively, amplitudes (3) do not strongly the dominant component remaining unchanged for each 1 1 = 3 / 2 - and in the first excited state I = 1 / 2 - , the (S =

nucleon-nucleon forces. It depend on the interaction, value. In the ground state 1/2, L = 1) component is

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P Descouvemont/Nuclear Physics A 626 (1997) 647-668

eLi -

-

7-

m

3 -

>

-

7-

4

o.)

- - 5 -

x Ld

- - i -

- - I -

- - 3 -

- - 3 -

Minnesoto

V2

m

3

-

exp.

Fig. 2. Energy spectrum of the 9Li nucleus.The states are labeled by 21. Table 2 Spectroscopic properties of the 9Li nucleus

•(fm) ~ (fro) ~ (fm) Q (e fm2) B(E2,3/2- ~ l/2-)(e2fm 4)

V2

Minnesota

2.41

2.41

2.41 -b 0.02 a, 2.32 q- 0.02 t'

2.26

2.26

2.30 4- 0.02 a, 2.18 4- 0.02 b 2.50 4- 0.02 a, 2.39 4- 0.02 b

2.47

2.47

-2.49

-2.25

3.15

2.75

Experiment

IQ[ = 2.74 4- 10c

a Ref. 151. h Ref. [61. c Ref. [9]. dominant, as suggested by Fig. 1. The S = 3 / 2 components represent about 10% of the wave functions. The (S = 1/2, L = 2) and (S = 1/2, L = 3) terms represent about 90% of the I = 5 / 2 - and 7 / 2 - wave functions respectively. Spectroscopic properties of 9Li are displayed in Table 2. The difference between the proton and neutron radii is supported by experiment. The quadrupole moment Q is found negative, and slightly dependent on the nucleon-nucleon interaction. Improvement with the experimental data would require a larger oscillator parameter. In the simple (¢rp3/2) 1(v P3/2) 4 description of 9Li, the quadrupole moment would be Q = - e b 2 = - 3 . 0 6 e fm 2 for any choice of the force. The present value Q = - 2 . 4 9 e fm z shows that solving the Schr6dinger equation in the full p shell yields a rather different description of 9Li. The E2 reduced transition probability from the ground state to the 1 / 2 - excited state is predicted close to 3 e 2 fm 4.

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2.2. Three-cluster 9Li+n+n wave functions Let us consider 9Li wave functions (1) located at $1, and two spinors ~n, located at $2 and $3, for the external neutrons. A non-projected llLi wave function reads, in the three-cluster GCM: ,4~11Vl [ (Xb/lb,lP2P3 ( S l , $2, $ 3 ) ~-* A .,,.gLi t S l )q~n 2 ( $ 2 )

qgnv3 ( $ 3 ) ,

(4)

vl is given by ( l ) (we drop the cSLT indices for clarity), and v2 where function ,.fill WOLi and v3 represent the spin orientations of the external neutrons. Notice that all possible (cSLT) sets are contained in (4). Up to a trivial c.m. factor, which exactly factorizes when the oscillator parameters are equal, this basis wave function is a Slater determinant. Projection over total angular momentum J and on parity 0r yields projected GCM states [44]:

o s ~ , ~ v ~ x ( R~, R2, ,~) = ~I

/E)~K(~)~J(~)(1

~ +77.p)(lbl, vtwv~(Sl,S2, S3)d~ ,

(5)

where ~ 1 ( / 2 ) and D~K(12) are the rotation operator and the Wigner function respectively, both depending on the Euler angles 12, and P is the parity operator. The definition of the generator coordinates (Rl, R2, a ) is given in Fig. 2 of Ref. [35]: Rl is the distance between 9Li and the center of mass of the neutrons, separated by R2; a is the angle between both directions. In Eq. (5) K is the spin projection over the intrinsic z axis that we choose along R~. In order to save computer times, only the lowest value K = 1/2 is introduced. The llLi Hamiltonian is then diagonalized in basis (5), yielding l lLi wave functions XP'JM'rr= Z

f .cJqr t R l'R2'°tJ'*V" _ , .~JM~r ~ ( R i , R 2 , a ) JJllvlV2V~ / I VlP2P3~

dRidR2da '

(6)

/IVlV2V3

where the generator functions fJ~" are obtained from the Hill-Wheeler equation [45] involving the Hamiltonian and overlap kernels. As shown in Ref. [35], the calculation of these kernels requires a numerical evaluation of three-dimensional integrals involving matrix elements between unprojected Slater determinants (4). In practice, the integrations over the generator coordinates are replaced by sums over finite sets of (R1, R2, a) values. The use of Slater determinants allows a systematic calculation and is well adapted to fully numerical approaches. However, the large amount of computer times required in the present case forces us to adopt some limitations in the choice of the number of generator coordinates.

3. The llLi nucleus

3.1. Energy surfaces and spectrum In Fig. 3 we present the energy surfaces of the llLi ground state, for the Volkov and Minnesota interactions. Here and in the following, the basis has been reduced, in

654

P Descouvemont/Nuclear Physics A 626 (1997) 647-668

V2

Minnesota

6

6

5

5

E

.%

4 Cxl

rY

J=3/2 -

3

/

2 1

i

4 3 2 1 0

2 R1

1

2

1

3

fm)

Fig. 3. Energy surfaces (with respect to the 9Li energy) of the ULi ground state with the V2 and Minnesota potentials. The curves are plotted by steps of 0.5 MeV and the lowest energy is indicated.

order to save computer times. From Table 1 it appears that some 9Li components are negligible, and are therefore taken out from the total 11Li basis. These components are indicated by a star ( . ) in Table 1. We have checked on simplified simulations that this approximation yields non-significant changes on the final results. The energy surfaces EJ~(R1, R2) are defined by the eigenvalue problem: Z

JTr I;~.~..~½(R1,R2,a)IH-EJ~r(RI,R2)I~Iww2~½(RI,R2,ce))=O,

Jrr /tpJTr ci~,~v2v3t

(7)

11/Jl/J2/~3

for fixed values of R1 and R2. Notice that we take a single value for the a angle, chosen according to the symmetry of the system ( a = 90°). Fig. 3 shows that qualitatively the energy surfaces relative to the ground state are similar for both interactions. However, the precise location of the minima is somewhat different (R1 = 3.6 fm, R2 = 4.0 fm for V2, and R1 = 2.4 fm, R2 = 4.5 fm for Minnesota). Both interactions yield a large distance between the external neutrons, which rules out the existence of a dineutron in l lLi. The energy surface is flatter with the Minnesota force than with the Volkov force. In Fig. 4 we consider the 1 / 2 - and 1/2 + partial waves, with the V2 interaction. The 1 / 2 - surface presents a minimum at Rl = 4.2 fm, R2 = 2.0 fm indicating again a large distance between the 9Li core and the external neutrons, but the picture of a 9Li÷dineutron structure would not be unrealistic. The existence of a pronounced minimum supports a 1 / 2 - state in the llLi spectrum (see below). On the contrary, the 1/2 + surface does not contain any minimum and therefore suggests that no 1/2 + state should be expected in a 9Li+n+n model. We have investigated the 3/2 +, 5/2 + and 5 / 2 - partial waves which are similar to the 1/2 + partial wave, and therefore are not shown here.

P. Descouvemont/Nuclear Physics A 626 (1997) 647-668

J=1/2-

655

J=1/2 +

6

9

6 5

E Oq

4

4

3

3

Od

2

4

2

1

1 I

0

1

2

3

4

5

I

i

i

i

i

6

1

2

3

4

/

5

i

6

Fig. 4. Energy surfaces with the V2 force of the 1/2- and 1/2 + partial waves in nLi. The energy step is 0.5 MeV and 1 MeV for J = I/2- and J = 1/2 + respectively. Let us now consider the l lLi spectrum with several configurations. As mentioned in Section 2, the length of numerical calculation forces us to limit the configuration space. A single angle c~ = 90 °, corresponding to the symmetry of the system, has been adopted. For the generator coordinate RI, we select five values ranging from 1.5 fm to 7.5 fm with a mesh of 1.5 fm; three R2 values are chosen equal to 2, 4, and 6 fm (for R2 = 6 fm, only RI = 3 fm is adopted). According to Fig. 3, this set of generator-coordinator values should cover reasonably well the three-body configuration space. The G C M spectrum of llLi is shown in Fig. 5 with experimental data [ 1 1 - 1 3 ] . As suggested by the energy surfaces, the model predicts a 1 / 2 - excited state, but no state with other JTr values. The I / 2 - excitation energy is rather sensitive to the interaction, and no definite assignment can be done. However, a recent experiment of Korsheninnikov et al. [ 14] concludes on a positive parity for the experimental 1.3 MeV state, also observed in Refs. [11,13]. Consequently the 1 / 2 - GCM state is a possible candidate for the second excited state (2.47 + 0.07 MeV in Ref. [ 12], 3.0 + 0.2 MeV in Ref. [ 13]).

3.2. Ground-state properties

Spectroscopy of the llLi ground state has been investigated by many authors (see references in Ref. [ 3 ] ). Here, we want to emphasize on peculiarities of the microscopic approach, and to complement theoretical calculations by some new results. Table 3 shows the matter, proton and neutron radii of the I tLi ground state. For the V2 interaction, the r.m.s, radius is very close to the lower limit of the experimental value. However, the Minnesota interaction yields too small r.m.s, radius. The same

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656

11Li 4 © m

3

1-

X

2 m

- - , 3 -

0

1-

- - 3 -

Minnesota

- - 3 -

V2

- - 3 -

Koboyoshi

- - 3 -

Bohlen

Korsheninnikov

Fig. 5. I I Li spectrum with the Minnesota and Volkov interactions (the states are labeled by 2J). Experimental data are taken from Refs. [ 11-13]. The dashed line indicates the threshold. Table 3 Spectroscopic properties of the lJ Li nucleus V2

Minnesota

Experiment

(fm)

2.95

2.84

3 . 1 0 + 0 . 1 7 a, 3.011 b

(k/~p

(fm)

2.35

2.32

2.88 4- 0.11 a, 2.235 b

(~n

(fm)

3.21 ::t: 0.17 a, 3.255 h

Q ( e f m 2) B(E2, 3 / 2 - ~

1 / 2 - ) ( e 2 fm 4)

3.00

2.85

-3.07

-3.13

3.66

2.92

IQI = 3.12 4- 0.45 c

" Ref. 16 l. b Ref. 1461. c Ref. 191.

conclusions hold for the neutron radius, which is a measurement of the halo structure. The experimental value of Ref. [ 6 ] for the proton radius is surprising, and contradicts the picture of llLi as a 9Li core and two external neutrons. Such a description would require similar proton radii for 9Li and llLi. The recent experiment of Penionzhkevich [46] gives smaller matter and proton radii, rather close to the GCM values. The validity of the model is supported by the quadrupole moment of 11Li, which is found very close to the experimental value [9]. The difference with the 9Li quadrupole moment (see Table 2) mainly arises from core-polarization effects or, in other words, from excited configurations. The E2 reduced transition probability to the 1 / 2 - excited state is found similar to the core values (see Table 2). Densities of the i iLi ground state provide a valuable information for its structure. The density is defined as

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657

9Li

0.25 0.20

A ,"3

I

E

0.15

~

Li (Mn)

'~, 't~l~Li(V2) 1Li

(Mn)

q.-

o.. 0.10 0.05

',, \\

0

,,,, \ \ \

\\'~. 1

i

2 3 4 5 6 7 8 9

,(2),

,

,

•

0

r--4

9Li

4

1 2 3 4 5 6 7 8 9

(fro) Fig. 6. Neutron (full lines) and proton (dashed lines) monopole densities of shows integrals lo(r), defined by (10).

llLi

pJMlr(r) =(al~JM~r Ai~=lt~(ri-Rcm-r) (½÷eti3) ~ItJMTrI ,

and 9Li. The lower panel

(8)

where ri and ti denote the space and isospin coordinates of nucleon i, and Rcm is the c.m. coordinate. The coefficient E is equal to -1 for the proton density and +1 for the neutron density. The GCM densities are computed as explained in Ref. [47]; center of mass effects are exactly removed. Density (8) is written in a multipole expansion as

pJM~(r) = ~-~(JMAO[JM)pJ~(r)Y°( ~r) , A

(9)

where p ~ ( r ) are the multipole densities. In Fig. 6 we present the monopole densities of l ILi, and compare them to the core densities. 2 As expected, the proton densities are very similar. However, as indicated in the logarithmic plot, the neutron densities are 2 Numerical values are available upon request.

P. Descouvemont/NuclearPhysics A 626 (1997) 647-668

658

r (fro) ,

4

2

8

6

-0.01, I

-0.02

C.

// ,9

-0.0,_3

~/gLi

-0.04

I~

.

i,/

.

10-~2

2 i

4 i

i

6 i

IL'X

10-'

i

.

.

2

.

4

.

6

o,2)

8

8 i

i

i

m '

~

E

9L i

N--

.

-,...3 -4

.

.

.

.

.

.

',-........~_ t ~ ~11Li (Mn) ................... 't--U ~ . / Li (V2)

Fig. 7. Neutron (full lines) and proton (dashed lines) quadrupole densities of ULi and 9Li. The lower panel shows integrals 12(r), defined by (11 ). The ULi proton densities are very similar to the 9Li proton density and therefore are not shown.

quite different (at the scale of this figure, the neutron and proton densities of 9Li are undistinguishable). Investigation of monopole densities is complemented by function I0 defined as r

Io(r)

V/-~ / pO( S)S 2 ds,

(10)

,J

o which is normalized in such as way that it tends to the nucleon number when r tends to infinity. This integral provides a qualitative insight on the internal and external contributions o f the wave function. Let us define as rl/2 the r-value where the integrated density (10) reaches one half of its maximum. For protons and neutrons in 9Li, we have rl/2 .-~ 2.0 fm, whereas neutrons in llLi are characterized by rl/2 ~ 2.5 fm which provides a further confirmation of the halo structure in ]]Li. Fig. 7 shows the quadrupole densities of 9Li and 11Li. The proton densities present a narrow (negative) peak near r = 1.5 fm, and then fastly decrease to zero. The neutron densities are much smaller in the inner part, but extend to large r-values. Integration of

P. Descouvemont/Nuclear Physics A 626 (1997) 647-668

659

M=1/2

neutrons

protons

Fig. 8. Total proton and neutron densities of the 11Li ground state [see Eq. (9)].

the quadrupole density yields the quadrupole moment. Consequently, we show in Fig. 7, the integral 12, defined as

12(r) =

7

~201~23-)e

p2(S)s4ds,

(11)

0

whose limit is I2(oc) = Q. As concluded from the monopole densities, integration of (11) for neutron components requires the knowledge of p2 up to large r-values ( ~ 8 fm). This effect is here strengthened by a further s 2 term in integral (11 ). In Fig. 8 we give the total densities (9) in the (xz) plane for M = 1/2 and M = 3/2. These densities combine the monopole and quadrupole terms. As expected from Fig. 6, the proton densities present their maximum for x = z = 0. The asymmetry between the x and z directions is responsible for the quadrupole moment. On the contrary, neutron densities present a minimum for x = z = 0, which compensates the enhancement at large distances. Densities corresponding to M = 1/2 and M = 3/2 are quite similar for neutrons since p2 is always small with respect to P0. Fig. 9 presents the 3 / 2 - --~ 1 / 2 - transition densities for 9Li and llLi. As the quadrupole density of the ground state, these transition densities present a narrow peak near 1.5 fm. Integration over r provides the B(E2) values given in Table 3. Notice that again the transition densities of ~~Li extend much further than densities of 9Li. Although densities of 9Li are larger in the inner region, integration up to infinity provides a larger B(E2) in llLi (see Table 3).

660

P Descouvemont/Nuclear Physics A 626 (1997) 647-668

0.06

/' i

I

//--,., 9Li

v

i-" oq

0.04

//

"',';', 1 1 .

1

2

I (-q

0.02 (-q

o__ 3

4

5

6

r

Fig. 9. Neutron (full lines) and proton (dashed lines) 3/2- ~ 1/2- transition densities for ULi and 9Li.

4. The l°Li system 4.1. Energy curves and spectrum

Interest for the l°Li unstable nucleus has been raised by Thompson et al. [23], who suggested that the 9Li momentum distribution can be accounted for by assuming the presence of a low-lying Sl/2 virtual state in the 9Li+n system. A possible parity inversion in the low-lying part of the l°Li spectrum was already suggested by Barker and Hickey [48] from a mass systematics near A = 10. Several experiments have been done to improve the knowledge of the l°Li spectrum [24-31], but the situation remains ambiguous. In parallel, different theoretical models [33,27,32,49,19,34] have been applied to the 9Li+n system. A brief discussion of the current situation will be presented in Section 4.2, at the light of the present results. Before studying the l°Li spectrum, we present in Fig. 10 the 9Li÷n energy curves (see for example Ref. [50] for definition). The energy curves can not be considered as genuine nucleus-nucleus potentials, but they provide a valuable information on the physics of the system. For the l°Li investigation, the conditions of the calculation are identical to those of i i Li. In particular, the nucleon-nucleon interaction is unchanged; this is important to have a reasonable predictive power on the ~°Li structure. Positive-parity energy curves present a minimum near 4.5 fm, and bound states or narrow resonances can be expected. In negative parity, the 0 - and 3 - energy curves are repulsive, but the 1- and 2 - (g = 0) energy curves present a very shallow minimum at large distances. In Table 4 we present the l°Li properties obtained in the present model. As suggested by the energy curves, 0 +, 1+ and 2 + partial waves present a resonance (or bound state). On the contrary, no negative parity state is found. Except for the 2 + resonance with the Volkov force, the theoretical states are overbound. This result is normal and simply arises from the variational principle. The llLi system requires three generator

661

P Descouvemont/Nuclear Physics A 626 (1997) 647-668 _

0-

6

co

8 3+

4

6 4

W

2 2 0+

6

I

; '1;

2

4

6

I

I

8

I

10

-2 R Fig. 10. 9Li+n energy curves with the V2 potential.

Table 4 mLi properties. Dimensionless reduced widths (calculated for a channel a = 9.5 fm) are expressed in % and energies in MeV. Experimental data are taken from Ref. [271 j~r E( 1+) 02( 1+) E(2 +) 02(2 + ) E ( 0 +) F ( 0 +)

V2

Minnesota

Experiment

-0.30 7.8 0.86 22.5 1.9 0.3

-1.57 2.7 -0.69 4.1 2.0 0.2

0.42 4- 0.05 12.3 0.80 4- 0.06 14.6

coordinates, or three degrees of freedom, whereas l°Li is described with a single degree of freedom. Since a limited choice of generator coordinates must be adopted, the l°Li wave functions are closer to the exact solution of the Schr6dinger equation than the l lLi wave functions. Owing to the identical nucleon-nucleon forces for both systems, J°Li energies are overbound. However, we do not aim here at giving precise energies of resonances, but we try to analyze the spin assignments which are still unclear. If one takes account of the overbinding discussed before, the GCM definitely predicts positive parity states in the low-energy spectrum. The dimensionless reduced widths are not too far from experiment. If the energy dependence of the reduced width is much weaker than this of the total width, an enhancement is obtained for increasing energies. The model also predicts a narrow 0 + resonance at higher energy, in agreement with the RGM study of Wurzer and Hofmann [34]. In Section 4.2 we investigate in more detail negative-parity partial waves.

662

P. Descouvemont/Nuclear Physics A 626 (1997) 647-668

4.2. What is a virtual neutron s state?

Properties of a bound state are defined without ambiguity. However, energies and widths of a resonance are, in some extent, model dependent [51]. The situation is even worse for s-wave neutron resonance where Coulomb barrier does not exist. Recent works on the l°Li nucleus have suggested such a low energy virtual state, based on a negative scattering length. Before discussing the G C M results, we want to illustrate properties of neutron s resonances, by investigating the 9Li÷n system in the potential model. This simple approximation neglects the structure of 9Li, but provides a fairly intuitive approach. Let us consider the Gaussian potential V ( p ) = Vo e x p ( - ( p / p o ) 2) ,

(12)

depending on the 9Li÷n relative distance p. With ~ = - 7 . 8 MeV, and P0 = 2.55 fm, this potential is known [23] to present a low energy resonance. In the following, we keep P0 = 2.55 fm and we allow I,~ to be modified. For each potential, we determine the bound state energy, if it exists, and the scattering length as defined by 1 as = - ~01im~ tan 8(k) ,

(13)

where k is the wave number and 6 the phase shift. In Fig. 11 we present the scattering length as and the bound-state energy E0 as a function of Vo. It is seen that near V0 = - 10 MeV, as presents a fast variation, and becomes negative above V0 ,~ - 10 MeV. The potential presents a bound state whose energy is defined within the numerical accuracy. However, for V0 ~> - 1 0 MeV, i.e. in the virtual-state regime, a bound state is not present, and the resonance energy should be obtained from the phase shift 6. We parametrize 6 with a one-level R-matrix approximation: F(E)/2 6 ( E ) ~ arctan - Eo- E

ka,

(14)

which defines the resonance energy E0 and width F ( E ) . In (14), - k a is the hard sphere phase shift, and F ( E ) is related to the resonance width F0 by F( E) = ro ( E/Eo)1/2 .

(15)

If 6 ( E ) is known within the numerical accuracy, it is important to notice that Eo and F0 depend on the assumption (14). The main input is the channel radius a, but higher-order terms might also be considered. For each %, we have parametrized the phase shifts with Eq. (14), and the resulting Eo and /I0 are presented in Fig. 11 for a = 3,5, and 7 fm. The accuracy of the fit (14) is always better than a few degrees for energies between 0 and 3 MeV. Clearly, energies and widths sensitively depend on the channel radius, even if they provide the same phase shifts and scattering lengths. For V0 = - 7 . 8 MeV, which is used in Ref. [23], E0 = 1.3, 0.43, and 0.21 MeV for a = 3, 5, and 7 fm respectively although they correspond to

P. Descouvemont/Nuclear Physics A 626 (1997) 647-668

> q)

~

24 16~

Lo -15 > (1)

. -10

0

IJ_l

_2 Lj74-sf---

'~=3 fm

a=5 fm

~

a=7 fm -5

61

0

. ~ = 3

~ o

663

fm

/

a=7 fm

o

60 E 40 20 09 0-(3 -20 -40

Vo (MeV) Fig. 11. Scattering lengths as, energy E0 and total width I'o [see Eq. (14)] for the 9Li+n s wave in the Gaussian potential model (12). the same scattering length. This typical example shows how difficult are the comparison between different works, and the interpretation of negative scattering lengths. 4.3. G C M results f o r s - w a v e states

Let us now turn to the GCM phase shifts presented in Fig. 12. Since the nucleonnucleon interaction might be parity dependent we complement our study with other conditions of calculation. Standard parameters (m = 0.6 for V2, and u = 1.0 for Minnesota) have also been used. These values strongly overbound the llLi and l°Li ground states, but cannot be completely disregarded for negative-parity partial waves. As in Section 4.2, the G C M phase shifts have been fitted by (14), and the resulting energies and widths are displayed in Table 5, where the scattering lengths are also shown. Let us first discuss J = 2 - . With the llLi interaction, the scattering length is as = +1.91 fm for V2 and as = - 0 . 9 9 fm for Minnesota. As observed before, the corresponding energies and widths are strongly dependent upon the channel radius a. In any

664

P. Descouvemont/Nuclear Physics A 626 (1997) 647-668 m

J -- 1 / O3 q) q)

0

I

....

Mn

~

.............. ' ........

t-

~

........ 5 .......... V2 (m=0.6)

I

L_

(u=l)

' ............

- - Mn ( u = 0 . 8 0 4 )

C)3 q) q:D

V2 ( m = 0 . 7 6 5 )

135 '4---

C(/3

90

(1) 03 ID {-

45

J

=

2

..........................................................

O...

....................................

i

0 '

~

I

1.0

Mn ( u : l )

.........[.................r................ ~............

1.5

~

V2 ( m = 0 . 6 ) Mn (u=0.804-) V2 ( m = 0 . 7 6 5 )

Ecru ( M e V ) Fig. 12. J = 1- and J = 2 - 9Li-Fn GCM phase shifts for different nucleon-nucleon interactions. The full curves correspond to the 1]Li forces, while the dashed curves correspond to the modified interactions (see text). Table 5 Scattering lengths as (in fm) of the 9LH-n system and R-matrix parameters (in MeV) of the GCM phase shifts a = 5 fm

as

J=2-

J=l-

V2 V2 (m = 0.6) Minn.

1.91 -4.05 -0.99

Minn. (u = 1)

90.4

V2 V2 (m = 0.6) Minn. Minn. (u = 1)

1.95 -2.20 0.13 -6.10

a = 7 fm

a=9fm

Eo

Fo

Eo

Fo

Eo

Fo

9.2 L 1.96 3.16

36.0 10.34 14.04

2.61 0.72 1.15

8.94 1.78 4.12

1.18 0.35 0.56

3.78 1.10 1.76

1.20 0.47 0.74 0.26

3.84 1.50 2.34 0.86

E0 = -0.003 9.73 2.85 4.69 1.49

38.6 14.44 20.6 8.46

2.67 0.97 1.54 0.56

9.16 3.68 5.45 2.26

case, r e s o n a n c e s are very b r o a d , w i t h a w i d t h larger t h a n t h e energy. W i t h m = 0.6, e n e r g i e s o b t a i n e d w i t h t h e V o l k o v f o r c e are lower; t h e M i n n e s o t a i n t e r a c t i o n w i t h u = 1 yields a weakly bound (E = -0.003

MeV)

state. T h e s i t u a t i o n for J = 1 - is similar,

but t h e e n e r g i e s are a l w a y s s h i f t e d to h i g h e r values. F r o m t h e s e results, w e c o n c l u d e t h a t t h e e x i s t e n c e or a b s e n c e o f a v i r t u a l s state is r a t h e r m o d e l d e p e n d e n t , a n d t h a t a special a t t e n t i o n m u s t b e p a i d o n the d e f i n i t i o n o f t h e r e s o n a n c e p r o p e r t i e s . S i n c e a g i v e n p h a s e s h i f t c a n b e fitted w i t h d i f f e r e n t R - m a t r i x p a r a m e t e r s , it s e e m s t h a t t h e s c a t t e r i n g l e n g t h d o e s n o t p r o v i d e e n o u g h i n f o r m a t i o n to conclude on s-wave properties.

P. Descouvemont/Nuclear Physics A 626 (1997) 647-668

665

Table 6 Spin assignmentsto the l°Li ground and first excited states Ref. Exp.

Theory

125 ] [26] [27 ] 128 ] 1291 1311 1301 [481 [27] 1491 [331 I321 [34] [191 1311 Present

gs/first exc. Method

su2 sl/27 Pl/z/Pl/2 ?/Pl/2 su2 1+/2 +

sj/2/pl/2 22+/1 + 2-/1 + 1+/2 + 2-/1 + 1+/2 + (s)/1 + 1+/2 + 1+/2 +

~'- absorptionby Il B Neutron decay of t°Li produced by fragmentation 9Be(13C,12N) l°Li and 13C(lnc,17F)l°Li reactions Momentum spectra of the IIB(7Li,8B)l°Li reactionat 130 MeV Stripping reactionsof IIBe and IILi with a carbon target I°Be(12C,12N)l°Li and 9Be(13C,12N)l°Li reactions Invariantmass spectroscopy Mass systematics in A = 10 - 12 nuclei RPA method includingpairing correlations Systematics of light exotic nuclei in the shell model Orthogonality ConditionModel Shell model with isospin dependentkinetic energiesfor sd-shell orbits MultichannelResonatingGroup Method Core + neutron potential model QRPA calculation MultichannelGeneratorCoordinate Method

4.4. Discussion of the ground-state nature Many experimental and theoretical works have been devoted to the t°Li spectrum. The resulting spin assignments of the ground state are summarized in Table 6. Let us first discuss the experimental results. After the pioneering work of Wilcox et al. [24] who observed a broad resonance near Ecru -- 0.8 MeV, Amelin et al. [25] gave a g = 0 assignment to the ground state near Ecm = 0.2 MeV. Kryger et al. [26] investigated the velocity spectrum of 9 L i + n systems produced in the fragmentation of an tSo beam. These authors interpret a narrow central peak as an evidence, either for a g = 0 ground state, or for a resonance near 2.5 MeV. From 9Be(13C,12N)l°Li and 13C(14C,17F)l°Li spectra, Bohlen et al. [27] conclude on a 1+/2 + assignment for the ground and first excited states; the authors reach the same conclusion with a RPA calculation. This interpretation was then contradicted by Young et al. [28] who measured momentum spectra in the 1~B(7Li,8B)I°Li reaction; Young et al. support the result of Kryger et al. [26] but do not find a clear evidence for an s assignment. Zinser et al. [29] used stripping reactions of JiBe and ~Li on a carbon target. These authors fit the momentum distribution using different scattering lengths for the 9Li+n system. The negative value ( ~<-20 fm) is considered by Zinser et ai. as an evidence for a g = 0 state below 50 keV. However, we have shown that the link between scattering length and resonance energy is not clear. Bohlen et al. [31] remeasured the l°Be(12C,12N)l°Li and 9Be(13c,lzN)t°Li spectra with improved experimental setup. These authors do not find evidence for a lowenergy s state, and conclude on a 1+/2 + assignment for the ground and first excited states. Very recently, Zinser et al. [30] used the invariant-mass spectroscopy of l°Li to investigate the l°Li spectrum. These authors find evidence for an s wave ground state near 0.21 MeV, and for a resonance with a p wave nature at 0.62 MeV. From

666

p Descouvemont/Nuclear Physics A 626 (1997) 647-668

these results, it turns out that there are several indications for an s-wave state close to the threshold, but other interpretations are possible. Obviously, available experimental data are not internally consistent, and further measurements are needed to clarify the situation. The first suggestion for parity inversion based on theoretical grounds has been done by Barker and Hickey [48], who performed a mass systematics near A = 10. Microscopic [ 34], semi-microscopic [ 33 ] cluster models and RPA [ 27,31 ] calculations all agree on a 1+/2 + assignment (Ref. [27] actually suggests 2+/1+). In the OCM calculation of Kat6 and Ikeda [33], and in the RGM calculation of Wurzer and Hofmann [34], the authors stress the importance of 9Li*+n configurations. Shell-model approaches [49,32] provide a 2 - / 1 + assignment, but the treatment of boundary conditions is not fully satisfactory. A recent calculation by Nunes et al. [ 19] in the potential model suggests a s wave for the ground state. Nunes et al. find a scattering length of - 1 9 fm ( J = 1 - ) and - 2 5 fm ( J = 2 - ) , that they assign to a resonance slightly above the threshold. The present microscopic model, where the interaction is fixed from the l lLi nucleus, confirms previous microscopic or semi-microscopic approaches: a possible s state should not be the ground state of l°Li. Anyway, further improvements of theoretical models are still required before drawing definite conclusions.

5. Conclusion

A multicluster model is well adapted to the description of I ILi and l°Li. The halo structure of l lLi has been reproduced by the present calculation, which yields r.m.s. radii and quadrupole moments in reasonable agreement with experiment. Owing to the complexity of the numerical calculations, some simplifying assumptions must be done on the 9Li basis, and on the 9Li+n+n configuration space. However, the present model has been shown to reproduce fairly well the 9Li low-energy spectrum, and some spectroscopic properties of 11Li. We have computed monopole and quadrupole densities of the ground state, and transition densities to a theoretical 1 / 2 - excited state. This state might be assigned to the second excited state observed by Korsheninnikov et al. [13] since the first excited state seems to have a positive parity [ 14]. The l lLi study is the starting point for the l°Li investigation, performed with the same GCM basis, and the same nucleon-nucleon interaction. This is essential to have some predictive power on the l°Li properties. Under these conditions, the I°Li states are overbound, as explained in Section 4.1. However, this problem is not crucial as long as the ordering of the levels is concerned. The GCM predicts a 1+ assignment for the ground-state, and a 2 + first excited state, as observed by Bohlen et al. [31 ] in a recent experiment. The existence of a virtual s wave remains open to discussion. We have shown that, in the theoretical viewpoint, the definition of energy and width must be carefully stated, and that differences between models may arise from different definitions. We have

P. Descouvemont/Nuclear Physics A 626 (1997) 647-668

667

c o n s i d e r e d a p o s s i b l e p a r i t y effect in l°Li b y m o d i f y i n g t h e n u c l e o n - n u c l e o n i n t e r a c t i o n for n e g a t i v e - p a r i t y partial waves. N o d e f i n i t e c o n c l u s i o n can b e d r a w n for a s r e s o n a n c e but the m o d e l s u g g e s t s t h a t it s h o u l d n o t b e t h e g r o u n d state. T h e t h e o r e t i c a l a n d e x p e r i m e n t a l are still unclear, a n d f u r t h e r w o r k is desired to i m p r o v e o u r u n d e r s t a n d i n g o f the l°Li puzzle.

Acknowledgements T h i s text p r e s e n t s r e s e a r c h results o f the B e l g i a n p r o g r a m o n i n t e r u n i v e r s i t y attraction p o l e s i n i t i a t e d b y the B e l g i a n - s t a t e F e d e r a l S e r v i c e s for Scientific, Technical a n d C u l t u r a l Affairs.

References II] 121 131 141 151

A.C. Mueller and B.M. Sherrill, Ann. Rev. Nucl. Part. Sc. 43 (1993) 529. P.G. Hansen, Nucl. Phys. A 588 (1995) lc. I. Tanihata, J. Phys. G 22 (1996) 157. B. Jonson, Nucl. Phys. A 574 (1994) 151c. 1. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida, N. Yoshikawa, K. Sugimoto, O. Yamakawa, T. Kobayashi and N. Takahashi, Phys. Rev. Lett. 55 (1985) 2676. [ 61 1. Tanihata, T. Kobayashi, O. Yamakawa, S. Shimoura, K. Ekuni, K. Sugimoto, N. Takahashi, T. Shimoda and H. Sato, Phys. Lett. B 206 (1988) 592. 171 N.A. On-, N. Anantaraman, S.M. Austin, C.A. Bertulani K. Hanold, J.H. Kelley, R.A. Ka'yger, D.J. Morrissey, B.M. Sherril, G.A. Souliotis, M. Steiner, M. Thoenn, J.S. Winfield, J.A. Winger and B.M. Young, Phys. Rev. C 51 (1995) 3116. [81 S. Hirenzaki, H. Toki and 1. Tanihata, Nucl. Phys. A 552 (1993) 57. [ 91 E. Arnold, J. Bonn, A. Klein, P. Lievens, R. Neugart, M. Neuroth, E.W. Otten, H. Reich and W. Widdra, Z. Phys. A 349 (1994) 337. II01 M.J.G. Borge, L. Grigorenko, D. Guillemaud-Mueller, P. Hornshoj, E Humbert, B. Jonson, T.E. Leth, G. Martfnez Pinedo, I. Mukha, T. Nilsson, G. Nyman, K. Riisager, G. Schrider, M.H, Smedberg, O. Tengblad and M.V. Zhukov and ISOLDE Collaboration, Nucl. Phys. A 613 (1997) 199. l 111 T. Kobayashi, Nucl. Phys. A 538 (1992) 343c. 121 H.G. Bohlen, R. Kalpakchieva, D.V. Aleksandrov, B. Gebauer, S.M. Grimes, T. KirM. von Lucke-Petsch, T.N. Massey, I. Mukha, W. von Oertzen, A.A. Oglobin, A.N. OC. Seyfert, T. Stolla, M. Wilpert and T. Wilpert, Z. Phys. A 351 (1995) 7. 131 A.A. Korsheninnikov, E. Yu. Nikolskii, T. Kobayashi, A. Ozawa, S. Fukuda, E.A.S. Momota, B.G. Novatskii, A.A. Oglobin, V. Pribora, I. Tanihata and K. Yoshida, Phys. Rev. C 53 (1996) R537. 14 ] A.A. Korsheninnikov, E.A. Kuzmin, E.Yu. Nikolskii, O.V. Bochkarev, S. Fukuda, S.A. Goncharov, S. lto, T. Kobayashi, S. Momota, B.G. Novatskii, A.A. Oglobin, AV. Probora, 1. Tanihata and K. Yoshida, Phys. Rev. Lett. 78 (1997) 2317. 15 D.V. Fedorov, E. Gan.ido and A.S. Jensen, Phys. Rev. C 51 (1995) 3050. 16 L.D. Skouras, H. Miither and M.A. Nagarajan, Nucl. Phys. A 561 (1993) 157. [17 Y. Tosoka and Y. Suzuki, Nucl. Phys. A 512 (1990) 46. 118 N. Vinh Mau and J.C. Pacheco, Nucl. Phys. A 607 (1996) 163. [ 19 EM. Nunes, l.J. Thompson and R.C. Johnson, Nucl. Phys. A 596 (1996) 171. 120 H. Sato and Y. Okuhara, Phys. Rev. C 34 (1986) 2171. [21 H. Horiuchi, Y. Kanada-En'yo and A. Ono, Z. Phys. A 349 (1994) 279. 1221 T. Kobayashi, Nucl. Phys. A 553 (1993) 465c. 1231 l.J. Thompson and M.V. Zhukov, Phys. Rev. C 49 (1994) 1904.

668

P Descouvemont/Nuclear Physics A 626 (1997) 647-668

1241 K.H. Wilcox, R.B. Weisenmiller, G.J. Wozniak, N.A. Jelley, D. Ashery and J. Cemy, Phys. Lett. B 59 (1975) 142. [25] A.I. Amelin, M.G. Gomov, Yu. B. Gurov, A.L. ll'in, P.V. Morokhov, V.A. Pechkurov, V.I. Savel'ev, F.M. Sergeev, S.A. Smirnov, B.A. Chemyshev, R.R. Shafigullin and A.V. Shishkov, Sov. J. Nucl. Phys. 52 (1990) 501. [261 R. Kryger, A. Azhari, A. Galonsky, J.H. Kelley, R. Pfaff, E. Ramakrishnan, D. Sackett, B.M. Sherril, M. Thoennessen, J.A. Winger and S. Yokoyama, Phys. Rev. C 47 (1993) 2439. [271 H.G. Bohlen, B. Gebauer, M. Von Lucke-Petsch, W. von Oertzen, A.N. Ostrowski, M. Wilpert, Th. Wilpert, H. Lenske, D.V. Alexandrov, A.S. Demyanova, E. Nikolskii, A.A. Korsheninnikov, A.A. Oglobin, R. Kalpakchieva, Y.E. Penionzhkevich and S. Piskor, Z. Phys. A 344 (1993) 381. [ 281 B.M. Young, W. Benenson, J.H. Kelley, N.A. Orr, R. Pfaff, BM. Sherril, M. Steiner, M. Thoennessen, J.S. Winfield, J.A. Winger, S.J. Yennello and A. Zeller, Phys. Rev. C 49 (1994) 279. [291 M. Zinser et al., Phys. Rev. Lett. 75 (1995) 1719. 1301 M. Zinser et al., Nucl. Phys. A 619 (1997) 151. 1311 H.G. Bohlen, W. Von Oertzen, Th. Stolla, R. Kalpakchieva, B. Gebauer, M. Wilper, Th. Wilpert, A.N. Ostrowski, S.M. Grimes and T.N. Massey, Nucl. Phys. A 616 (1997) 254c. 321 H. Kitagawa and H. Sagawa, Nucl. Phys. A 551 (1993) 16. 331 K. Kat6 and K. Ikeda, Prog. Theor. Phys. 89 (1993) 623. 341 J. Wurzer and H.M. Hofmann, Z. Phys. A 354 (1996) 135. 351 P. Descouvemont, Phys. Rev. C 52 (1995) 704. 361 D. Baye, Y. Suzuki and P. Descouvemont, Prog. Theor. Phys. 91 (1994) 271. 371 P. Descouvemont and M. Vincke, Phys. Rev. A 42 (1990) 3835. 381 P Descouvemont, Nucl. Phys. A 596 (1996) 285. 39] A.B. Volkov, Nucl. Phys. 74 (1965) 33. 140] D. Baye and N. Pecher, Bull. C1. Sc. Acad. Roy. Belg. 67 (1981) 835. 1411 D.R. Thompson, M. LeMere and Y.C. Tang, Nucl. Phys. A 286 (1977) 53. 1421 E Ajzenberg-Selove, Nucl. Phys. A 506 (1990) 1. 1431 E Ajzenberg-Selove, Nucl. Phys. A 490 (1988) 1. [441 D. Brink, Prec. Int. School "Enrico Fermi" 36, Varenna 1965 (Academic Press, New-York, 1966) p. 247. 1451 Y.C. Tang, in Topics in Nuclear Physics II, Lecture Notes in Physics (Springer, Berlin, 145, 1981) p. 572. [461 Yu.E. Penionzhkevich, Nucl. Phys. A 616 (1997) 247c. 1471 D. Baye, P. Descouvemont and N.K. Timofeyuk, Nucl. Phys. A 577 (1994) 624. [481 EC. Barker and G.T. Hickey, J. Phys. G 3 (1973) L23. [491 N.A.F.M. Poppelier, L.D. Wood and P.W.M. Glaudemans, Z. Phys. A 346 (1993) 11. 1501 P. Descouvemont, Nucl. Phys. A 615 (1997) 261. 151] EC. Barker, Phys. Rev. C 53 (1996) 1449