Nuclear Physics A362 (1981) 189 - 226 ; @ North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

HIGH-RESOLUTION STUDY OF FOUR-NUCLEON PICKUP

VIA THE (d, 6Li) REACTION AT Ed = 55 MeV ON sd SHELL NUCLEI

(I). Even-A Nuclei

J. C. VERMEULEN +, A. G. DRENTJE, H. T. FORTUNE ++, L. W. PUT, R. R. DE RUYTER VAN STEVENINCK and R. H. SIEMSSEN

Kernfysisch Versneller Instituut, Rijksuniversiteit Gronmgen, Groningen, The Netherlands

J. F. A. VAN HIENEN

Natuurkundig Laboratorium der VU, Amsterdam, The Netherlands

H. HASPER

Laboratorium voor Algemene Natuurkunde, Rijksuniversiteit Groningen, Groningen, The Netherlands

Received 1 September 1980

Abstract: The 24. 26Mg(d, ‘Li)“* “Ne and *sSi(d 6Li)24Mg reactions were studied with a broad-range , magnetic spectrograph at Ed = 55 MeV with an energy resolution of _ 40 keV. Angular distributions were obtained from elab = lo”-37” for some 100 transitions to states up to EX = 14.5-15 MeV in “Ne and “‘Mg, and to E, = 9.5 MeV in 22Ne The data were analysed with zero-range and finite- range DWBA. Experimental spectroscopic factbrs are compared with those predicted by the SU(3) model and by microscopic shell-model calculations made with the Chung-Wildenthal interaction in the full (IsOd)” space.

NUCLEAR REACTIONS 24, 26Mg, *sSi(d, 6Li), E = 55.0 MeV; measured u(E,,; 0). E “. “Ne, 24Mg deduced levels, a-spectroscopic strengths, L, J, n. Magnetic spectrograph,

enriched targets.

1. Introduction

This is the first of two papers in which we deal with a study of a-particle pickup on sd-shell nuclei. In the present paper the (d, 6Li) reaction on the even-,4 nuclei 24Mg 26Mg and ‘*Si is discussed. The results for the “Mg(d, 6Li)21Ne and 27Al(i 6Li)23Na reactions will be presented in a forthcoming paper.

A h:gh-resolution study of the (d, 6Li) reaction on (sd) shell nuclei is of interest for a variety of reasons. (i) Since level spacings are sufficiently large in sd-shell

’ Present address: CERN, Geneva, Switzerland. ” Permanent address: University of Pennsylvania, Philadelphia, USA.

189

190 J. C. Vermeulen et al. 1 (d, 6Li)

nuclei, transitions to many individual states within the same final nucleus can be studied. Furthermore much information already exists on the properties of most of these states, thus removing some of the ambiguities that would otherwise enter into the analysis of the (d, @jLi) reaction. (ii) Experimental a-spectroscopic factors, which contain info~ation about four-nucleon cluster correlations, can be readily com- pared with those of the SU(3) model, that has been successfully applied to the first half of the sd shell, and with predictions of shell-model calculations done in the full sd space. The sd shell mass region is thus ideally suited to serve as a “provingground” for the potential and possible limitations of four-nucleon pickup via the (d, 6Li) reaction for nuclear structure studies. (iii) In contrast to a-particle stripping, high- quality a-particle pickup data are scarce. Most studies on sd shell nuclei to this date were done with a relatively poor energy resolution*and thus were limited to a few states in the residual nuclei ’ -6) An exception is a recent inv~tigation of the . 24M g( 7 d 6Li)20Ne reaction at Ed = 28 MeV by Fortune eb al. ‘), in which angular distributions were measured for states up to 9 MeV excitation energy +.

In the present investigation the (d, 6Li) reaction on 24,2bMg and 2aSi was studied with a Q3D-type broad-range magnetic spectrograph with an energy resolution of approximately 40 keV. Angular distributions were obtained for about 100 transi- tions to states up to an excitation energy of 14-15 MeV in 20Ne and =Mg, and to 9.5 MeV in 22Ne. The data were analysed with zero-range and finite-range DWBA. Experimental spectroscopic factors for a-pickup were compared with those ob- tained from the SU(3) model based on the work of Ichimura et al. *), Draayer 9, and Hecht and Braunschweig lo), Theoretical spectroscopic factors were also computed from shell-model wave functions that were generated by one of us (H.H.) by diagonalization of the Chung-Wildenthal ii) interaction in the full sd shell space. The present work has been part of a thesis 12).

2. Experimentai procedure

Self-supporting and metallic targets of 24Mg, 26Mg and 28Si were bombarded with an energy-analysed beam of 55 MeV deuterons from the K-VI variable energy cyclotron with typical beam currents between 100 and 400 nA. The outgoing 6Li ions were momentum analysed and detected with a QMG/2 broad-range magnetic spectrograph. The spectrograph and focal plane detection system are described in detail elsewhere ’ 3, 14).

The targets, approximately 80 @g/cm2 thick, were produced by evaporating isotopically enriched 24Mg and 26Mg, and natural silicon (28Si abundance: 92 %), To minimize carbon and oxygen contaminations the magnesium targets were mounted in an argon atmosphere and were stored and transported either in vacuum or in argon. In spite of the precautions taken, contaminations were not as small as anticipated.

’ After the present paper was submitted for pubIication a rather extensive study of the 24S 26Mg(d, ‘Li)‘O* 22Ne reactions at Ed = 80 MeV was published by Oelert et al. 6*).

J. C. Vermeulen et al. / (d, ‘Li)

TABLE 1

Target compositions and thicknesses

Target target material 12C

Thickness (pg/cn@

160 other. constituents

=Mg 66k 15 13*3 6&-2

Z6Mg 73+15 8*3 7*3 4.3kO.7 =Mg

%4 65klO 5+2 2*1 %i 3oSi.

Target thicknesses were determined from energy-loss measurements with an a-

particle source, taking into account the energy losses from the other constituents in the targets. The stopping powers of ref. ’ 5, were used. In addition the relative cross sections for the 24*25’26Mg(d, (jLi) 203 21 9 “Ne reactions were determined by using a thick (270 pg/cm2) natural magnesium target (79 % 24Mg, 10 % “Mg and 11 % 26Mg). The thicknesses of the Mg targets as listed in table 1 were obtained by combining the result of this measurement and the results of the energy-loss measure- ments. Also listed are carbon and oxygen contents which were estimated from the yield of the (d, 6Li) reaction on these contaminants 12). (No evidence for carbon or oxygen build-up during the experiments was found.)

The horizontal opening angle of the spectrograph was 5.5”, the vertical opening angle 6O, and the solid angle had a value of 8.5 msr. In determining the average value of the reaction angle the effect of the large vertical opening angle was taken into account. For relatively strong transitions ray-tracing techniques were applied (the focal plane detecting system measured the angle of incidence of the 6Li ions on the detector) to divide the full solid angle into two bins with nearly equal horizontal widths [see for details ref. i4)].

The focal plane detector consisted of two nearly identical subsystems, placed side-by-side, with an active area of 52 x 6 cm2 each. Each subsystem contained two position-sensitive gas detectors, 10 cm apart, followed by a scintillator. The posi- tion-sensitive detectors were a combination of a (vertical) drift chamber and a resistive wire proportional detector. The horizontal position was obtained via charge division, the vertical position via the drift time. In addition to the position information a signal proportional to the differential energy loss was obtained from the gas detectors.

All data were recorded in the event-by-event mode on magnetic tape with the program SPEK and subsequently analysed (off-line) with the program PLAY 16). Particle identification was achieved by a combination of two-dimensional gates on the differential energy signal versus position and on total energy loss (from the scintillator) versus position. In this way a clear separation between a-particles, (jLi and ‘Li ions could be obtained. The total dead time (hardware plus software) was kept below 20 %. The total energy resolution of about 40 keV at forward angles

192 J. C. Vermeulen et al. / (d, ‘Li)

was mainly dete~i~~d by the target thickness, due to kinematic effects it was 60-80 keV for angles larger than 25”.

The contents of non-overlapping peaks in the position spectra were determined by summing their content after background subtraction, for partially overlapping peaks an interactive peak-fitting program employing gaussian peak shapes was used. Absolute cross sections were determined from the peak contents, the solid angle of the spectrograph, the target thicknesses and the integrated charges corrected for dead time. The estimated error in the absolute cross sections is 20 %, which is mainly due to uncertainties in the target thicknesses.

The energy versus position calibration of the spectrograph, including kinematic corrections, was deduced from a number of known transitions in the three reactions studied. The typical accuracy in the determination of excitation energies was about 7 keV [ref. “)I.

Differential cross sections were measured from Blat, = 10” to 37”. The energy range of one detector subsystem was N 8 %, which was about 3 MeV of excitation energy in the present experiment. The gap between the two detector subsystems was about 1.5 MeV wide. For 24Mg and 28Si for which differential cross sections were measured for states up to N 15 MeV excitation energy, three magnet settings were used; for 26Mg two.

3. Theoretical aspects of the (d, 6Li) reactions and calculation of ~-s~~o~opic factors

The spectroscopic amplitude for a direct a-cluster pickup reaction like the A(d, 6Li)B reaction is defined as i7)

with @Jo, (b, and Qi, the fully antisymmetrized intrinsic wave functions of the nuclei B, 01 and A, and @‘L being the wave function of the relative motion of the cc-particle and the nucleus B. Following ref. *) eq. (1) can be written as

(2)

In the derivation of this equation it is assumed that only one N-value per trans- ferred L-value contributes as is the case for harmonic oscillator shell-model wave functions when equal size parameters are used for the description of the E-particle and nuclei A and B [ref. 8)]. Furthe~ore that the wave functions Yy, and Y, are free of spurious c.m. excitation. The last factor in eq. (2), which is the “G”- factor of refs. *, ’ 7), gives the overlap of a four-nucleon cluster wave function with specified N and L with a four-nucleon shell-model wave function Yt(80) with the SU(3) classification (1~) = (SO) for the (IsOd) configuration space.

J. C. Vermeulen et al. / (d, 6Li) 193

Relative c+spectroscopic factors for positive-parity transitions have already been calculated in the conventional SU(3) shell model 9), where SU(3) harmonic oscil- lator shell-model wave functions in the (1 sod) configuration space were used in the reduced matrix element of eq. (2). Absolute values can be calculated from the relative factors and the C- and D-factors of ref. lo).

For the negative-parity states, some of which are relatively strongly excited in the (d, 6Li) reaction, two different approaches can be used for the calculation of the a-spectroscopic factor lo). (i) In the SU(3) strong coupling model a Op hole in the I60 core and the particles in the sd shell outside this core are coupled to a state of good SU(3) symmetry. (ii) In the SU(3) weak coupling model the total angular momenta and isospins of the particles and the hole are coupled, whereas the space symmetries and the SU(3) quantum numbers of the particles and hole separately are assumed to be good quantum numbers. In this study we follow approach (i). The reduced matrix element of eq. (2) was treated in a similar way as in ref. lo) and the SU(3) Wigner and Racah coefficients were calculated with the code of Akiyama and Draayer 19). Possible spurious c.m. excitations were removed by applying the techniques of Hecht 20).

For the positive-parity transitions also calculations with the j-j coupled shell- model wave functions have been performed. The values of the a-spectroscopic factors calculated with such wave functions are known to be very sensitive to the amount of configuration mixing i7) and thus probably provide a good test for the wave functions. A computer code for constructing the reduced matrix elements of eq. (2) has been written by Bennett i8). This code was modified in order to handle the large vector spaces typical for the untruncated sd configuration space. The reduced matrix element of eq. (2) can symbolically be written as:

= (illkID (3)

where (iIlk 1~) refers to the reduced matrix element between basis states and is only a geometrical coefficient, i.e. independent of the hamiltonian used.

The coefficients a, and bj have been obtained by diagonalizing the Chung- Wildenthal effective interaction ’ ‘) in a full sd shell configuration space for the target and final state, respectively. The diagonalization of the matrices was per- formed with a computer code employing the Lanczos method ‘. The matrices were constructed with the Oak Ridge-Rochester shell-model code 21). The coefficients ck in eq. (3) correspond to the representation of four-particle wave functions. of “Ne with SU(3) symmetry (80) in a j-j shell-model basis and were generated by diagonalizing a pure quadrupole-quadrupole interaction to which, in order to remove degeneracies, 0.1 y0 of the Chung-Wildenthal interaction was added. The coefficients G&O) of eq. (2) have been taken from ref. 8). Following Bennett

+ One of the authors (H. H.) will supply the wave functions upon request.

194 J. C. Vermeulen et al. / (d, 6Li)

et al. 18) the transferred cluster can also be assumed to have the configuration of the four particles outside the 160 core in “Ne. The coefficients ck are then taken to be the amplitudes of the wave functions of the 0:) 2:) 4:) 6: and 8: states of 20Ne for L-transfers of 0, 2, 4, 6 and 8, respectively. In this case the appropriate G,, is not known but can be approximated as i8):

GdZoW = (20W@W, G,,@O), (4)

where (20Nel(80)), is the overlap between a 20Ne wave function and the SU(3) wave function with @cl) = (80) for specified L. This overlap was found to have values 0.88, -0.90, -0.83, -0.81 and 0.68 for the O:, 2:, 4:, 6: and 8: states of 20Ne, respectively.

Relative cr-spectroscopic factors resulting from the various approaches mentioned here are presented and discussed in sect. 5, where also a comparison is made with the experimental results. Values of the absolute spectroscopic factors for the g.s. transi- tions are given in sect. 6. More details and more detailed results of the present calcu- lations are given in refs. 12,22).

4. DWBA analysis

Both zero-range and finite-range DWBA calculations were done under the assumption that the transferred four nucleons have the internal configuration of an a-particle in its ground state. The zero-range calculations were performed with DWUCK IV [ref. 23)], whereas DWUCK V [ref. ‘“)I was used for the finite-range calculations.

In all calculations the bound-state wave function of the a-particle in the target nucleus was generated by binding the “cr-cluster” in a Woods-Saxon potential well of which the depth was adjusted to tit the binding energy. The number of nodes N was determined from the harmonic oscillator condition : 2N+ L = cf= 1 2ni + Zi, assuming OS intrinsic motion of the four nucleons. Thus 2N+ L = 8 for (sd)4 pickup and 2N+ L = 7 for transitions to negative-parity states with (sd)“(Op)- ’ con- figurations.

The size parameters of the bound-state potential as well as the deuteron and 6Li optical-model parameters were chosen such that the experimental angular distri- butions of the ground-state transitions for the 24Mg(d, 6Li)20Ne, 26Mg(d, 6Li)22Ne and the 28Si(d, 6Li)24Mg reactions were reproduced satisfactorily. The deuteron potential of ref. 24), which is very similar to that of Hinterberger et al. 25) for 27A1, and a 6Li potential similar to the potentials of Chua et al. 26) were found to give satisfactory results. No spin-orbit term is contained in these potentials. The optical- model and bound-state parameters are listed in table 2.

In the finite-range calculations a WS potential was used to generate the 1s wave function for the a and deuteron clusters in 6Li. To obtain a good finite-range description it was not sufficient to choose a proper radius of the cr+ d potential

J. C. Vermeulen et al. / (d, 6Li) 195

TABLE 2

Optical-model potential parameters used in the DWBA analysis

(ML”) &) (2) (MT”) (M%) (k) (2) (5) d 85.5 1.05 0.80 38.8 1.30 0.82 1.3

‘Li-ZR 240.0 1.25 0.70 25.0 1.70 0.90 1.3

6Li-FR 256.5 1.05 0.80 b, 25.0 1.70 0.90 1.3

a+B 1.275 0.675 1.3

a+d R = 3.78 0.65 1.3

“) Well depth adjusted to fit binding energy. ‘) For the 26Mg(d, ‘Li)“Ne reaction a, = 0.85 has been used, which causes an increase of the absolute

cross sections of 15 %, but the shapes of the angular distributions remain nearly the same.

but it was also necessary to adjust the 6Li optical-model parameters (6Li-FR in table 2). Surprisingly, the new 6Li potential satisfies the well-matching criterion of DelVecchio 27) which states that finite-range effects will be small if the volume integral per nucleon of the real part of the optical-model potential is the same for the in- and outgoing channel. The large value of the radius of the WS potential of the cr+d system (3.78 fm) is justified by the fact that it should be about the same as the average separation of the a- and d-cluster centers for which Raphael 2*) found value of 3.75 fm. For the accuracy parameters ACC(l), . . ., ACC(6) in DWUCK V the values 1.5, 1.0, 0.6, 0.4, 1.0 and 10.0, respectively, were used. The integration step size was taken to be 0.03 fm (0.1 fm in DWUCK IV).

Absolute values of the a-spectroscopic factors extracted from DWBA analyses of the experimental data are rather sensitive to the parameters of the potentials used, especially to the radius of the potential for the 01 bound state. Reasonable variations in the parameters which do not seriously affect the quality of the fits to the three g.s.. transitions may result in a change of the absolute spectroscopic factor by a factor of 2 to 3. Relative spectroscopic factors, however, are much less sensitive to variations in the parameters and therefore we will concentrate on relative spectro- scopic factors which were determined from the relations

exp =

3 DWUCK V

(5)

(6)

with normalization constants NZR and NFR. The estimated over-all errors in the relative spectroscopic factors vary between l&20 y0 for good fits and 50 y0 or more for low-quality tits.

196 J. C. Vermeufen et al. / (d, 6Li)

5. Results and discussion

In this section we present the experimental results for the 3 target nuclei studied and compare them with results of the theoretical calculations. Spectra for the (d, 6Li) reaction on 24Mg, 26Mg and 28Si are shown in Iigs. 1, 3 and 5, respectively. The angular distributions for the observed transitions, with data points for the full solid angle, are shown in figs. 2, 4 and 6, together with results of zero-range (solid curves) and finite-range (dashed curves) DWBA calculations. The excitation energies of the observed levels in the final nuclei “Ne, 22Ne and 24Mg as obtained from the energy calibration of the spectrograph are tabulated in tables 4, 7 and 10. Many of the observed states, even up to about 12 MeV excitation energy, could be identified with known states 29,30), a survey of which is also given in the tables. Values of the differential cross section at the maximum of the angular distribution are also tabulated.

Tables 5,8 and 11 give the relative values of the spectroscopic factors, normalized to an assumed value S, ZE 1 for the g.s. transitions, as obtained from the com- parison of the data with DWBA. For some unresolved doublets of states with known spins and parities a least-squares method was used to extract spectroscopic factors for each of the constituents, for these cases the errors due to the least-squares fitting procedure are quoted. The tables also give the (relative) spectroscopic factors resulting from shell-model calculations with the Chung-Wildenthal interaction ‘r) (this work) and with the Preedom-Wildenthal interaction 31) [results from ref. 32)] and from SU(3) calculations [this work and results from refs. 7,9)]. Theoretical excitation energies from the present shell-model calculations are also given. The variation of the spectroscopic factor for the g.s. transitions for the three target nuclei studied and the absolute values of the spectroscopic factors for these transi- tions are presented and discussed in sect. 6.

Only a very limited comparison of our a-spectroscopic factors with values ob- tained from (d, 6Li) reactions at other energies [28 MeV, ref. 7), 35 MeV, ref. 4), and 54.25 MeV, ref. ‘)I and from the (31-Ie, ‘Be) reaction at 70 MeV [ref. 5)] can be made. This is because in those experiments, with the exception of the work of Fortune et al. 7), only a few transitions were observed. Furthermore the errors in those spectroscopic factors are large because of either a rather poor quality of the data “) or a poor representation of the data 5,6) by the DWBA. The values from those experiments will be quoted in the text without further discussion.

The discussion of the results will as much as possible be done per K” band. The K” band assignments from the literature for 20Ne [ref. 29)], 22Ne [refs. 31*33-3J)] and 24Mg [ref. 36)] are given in tables 3, 6 and 9, respectively.

Fig. 1. Position spectra for the **Mg(d, 6Li)zoNe reaction at Ed = 55 MeV and 8, = IO“. Q is the effective (dead-time corrected) collected charge. For E, Q 9.1 i 5 MeV qupted excitation energies, spins and parities are from ref. 2g), for higher excitation energies the values were obtained from the

present energy calibration.

i

24Mg(df?_i)20Ne

,oo Ed’ 55 MeV 8,,= IO’

50

I 1..

I

0 0 100

Q=515 PC

1 *- I -lLl

200

Q=398rC

CHANNEL NUMBER

Fig. 1.

198

to’{ i

8.694 I-

JT 10 c 8.777

E / l@ 10.648

6’ I\ L-2+3

O0

~

N&a

\

I 0’ ID.90

I.

IO’ t

12.95

i IO' 1 13.68

IO0

F

f 3-

f 1°F 13. @I

,oo ’

\ i 1

L-2

E i

LI._ u_ “-i-L-l t 0 20 40 SC

l”..LLii 0 20 40 L

B,,we) Fig. 2.

J. C. Vermeulen et al. / (d, ‘Li) 199

I-

”

mm

iu”“’

26Mg (d,%i) 22Ne

,8= IO0 Ed = 55 Me!/ 100

200

100 200

s :

0=888 pC $ - $

IO0 -

50 -

B E n Q 1

200 -

IGO-

100 20

CHANNEL NUMBER

Fig. 3. Position spectra for the 26Mg(d, 6Li)22Ne reaction at Ed = 5.5 MeV and B,_,, = IO*. Q is the effective (dead-time corrected) collected charge. For I?, d 6.817 MeV quoted excitation energies, spins and parities are from ref. 30), for higher excitation energies the values were obtained from the present

energy calibration.

Fig. 2. Angular dist~butions for the 24Mg(d, sLi)%e reaction at Ed = 55 MeV. The full (dashed) curves are the results of ZR(FR) DWBA calculations. Q = 2N+ 15, with N the number of nodes and L the orbital

angular momentum transfer.

200 J. C. Vermeulen ef al. / (d, 6Li)

I.275

10’ - 2+

~

0 0

10°.

1 3.357 4+

IO-’ 1 5.146

t 7.32

Id 1 7.397

2 L-3

IO’ - \

k

’ -0 \

L

7.623

I

I

I

I

I

I

I \

I

IO’ 8.134

! 8.452

5 8.573

16’

~ - --- i!R FR

0 20 40 60 0 20 40 60 0 20 40 60 0 20 40 60

@,,(desf

Fig. 4. Angular ~st~butions for the 26Mg(d, ‘Li)**Ne reaction at Ed = 55 MeV. The full (dashed) curves are the results of ZR(FR) DWBA calculations.

COUN

TS

PER

CHAN

NEL B

o”

8 I

I I

I I

202 126Si(d.6Lit24Mg 10q

F i

Ed=55 MeV

t 5.236

“l- IO

1 7.616

3-

i 6.113

IO0 6’

h I ‘L. \ \ ’

I \

I

I

I

j

/I 5,

I

IO’%- 10.352

IO0

L

I

IO-‘?

10.68

‘6-

II.13

L

II.20

*’

‘I

5 11.693

Il.851

i

f

Il.966

L.2

0 20 40 60 0 20 40 60 0 20 40

@C,,(deq.)

Fig. 6.

I 0 20 40 60

J. C. Vermeulen et al. / (d, ‘Li) 203

TABLE 3

Rotational band structure of *ONe “)

K”

0:

J” E, (MW

0+ g.s.

2+ 1.634 4+ 4.248 6+ 8.777 8+ 11.949

K”

0:

J” Ex WW

0+ -8.3 2+ -8.8 4+ 10.79 6+ 12.591

0- 1- 5.784

0: Of 6.724 3- 7.168 2+ 7.421 5- 10.261 4+ 9.99 7- 15.336

0: 0+ 7.191 2- 2- 4.968 2+ 7.829 3- 5.621 4+ 9.030 4- 7.004 6+ 12.134 5- 8.449

6- 10.609 7- 13.334

“) Ref. 29).

5.1. THE t4Mg(d, 6Li)20Ne REACTION

By far the strongest transitions in the 24Mg(d, 6Li)ZoNe reaction are those populating the g.s. rotational band (O+, 2+, 4+) and the first 3- state at 5.62 MeV (see fig. 1). The background at low excitation energies is due to the population of the broad 2+ state in ‘Be at E, = 2.9 MeV via the “C(d, 6Li)8Be reaction. In general the agreement between the excitation energies from the present experiment and previously known values is quite good (see table 4). Exceptions are the states found at E, = 9.300+ 0.007 MeV and 9.466* 0.007 MeV. The first state might corre- spond to the 9.318 f 0.006 MeV state of ref. 29). The second state differs too much in excitation energy to correspond to the 9.508 + 0.012 MeV state of ref. 29) ; besides, its angular distribution is not indicative for J” = 2+. Remarkably, Betts et al. 37) quote excitation.energies in good agreement with our values. By assuming that T = 1 states are not excited in the (d, 6Li) reaction, and by comparing the known and observed level widths quite a number of states above E, z 9 MeV could be identified with states with known spins and parities.

The theoretical results for the spectroscopic factors obtained by Horiuchi 38)

Fig. 5. Position spectra for the %i(d 6Li)24Mg reaction at Ed = 55 MeV and Olnb = 13“. Q is the , effective (dead-time corrected) collected charge. For Ex G 9.418 MeV quoted excitation energies, spins and parities are from ref. 30), for higher excitation energies the values were obtained from the present

energy calibration. Fig. 6. Angular distributions for the *sSi(d 6Li)24Mg reaction at I?, = 55 MeV. The full (dashed) curves ,

are the results of ZR(FR) DWBA calculations.

204 J. C. Vermeulen et al. / (d, bLi)

TABLE 4

Excitation energies and maximum differential cross sections observed in the Z4Mg(d, 6Li)20Ne reaction

Present results

EX (MeV + keV)

Previous results “)

0.0 b) 27 1.632 ‘) 9 4.248 b, 8 4.963* 7 3.2 5.619 “) 18 5.786* 7 2.2 6.715+ 10 0.9 7.004* 7 2.1

7.180& 7 4.1 7.416+ 7 2.8 7.829 b, 2.5

8.449 b, 1.8 8.704* 15 1.2 8.777 b, 2.5

8.86 +20 0.7 9.026* 7 1.5 9.100+ 15 1.8 9.300* 7 1.6 9.466_+ 7 2.5

9.943 * 15 1.0 10.04 +30 2.7

10.572+ 7

10.848k 7 2.6

10.90 +20 1.8

11.00 +20 1.8

11.22 *20 1.1

11.30 *20

2.4

1.3

0.0 1.634 4.248 4.968

5.621 5.784 6.724* 5 7.004* 4 7.168+ 5 7.191* 3 7.421 7.829 8.3 “) 8.449+ 3 8.694* 6 8.777* 3 8.8 ‘) 8.848+ 5

9.030* 5 9.115+ 4 9.318* 6

0+ 2+ 4+ 2-

3-

;z

4-

3- 0+ 2+ 2+ 0+ 5- 1- 6+ 2+ 1- 4+

3-

800

800

9.508+ 12 9.873& 4 9.92 +20 9.99 *20

10.261+ 4 10.272 d, 10.403+ 5 10.548+ 5 10.583k 6 10.609* 6 10.694k 6 10.79 $-loo 10.838 ‘) 10.84Ok 5 10.89 + 10 d,

2+

(ii) 155

5- 145 2+ 3- 80 4f

2+ 6- 4-, 3+ 4+ 350 3- 45 2+ 3+

10.97 ‘) Of 580 11.015+ 6 4+ 11.073+ 8 d, 4+ 11.23 k30 1- 11.23 flOd) 1+ 11.256+ 8 d, 1- 11.322* 7 2+ 40 11.528+ 6 3+, i-

J. C. Vermeulen et al. / (d, 6Li) 205

TABLE 4 (continued)

Present results Previous results “)

Ex (MeV + keV) (zi).., (tz) Ex (MeV f keV)

J”

11.56 +20

11.85 *20 1.5 11.92 k20 2.3

12.39 k20 1.7

12.54 k20 2.2

12.95 k20 3.2

13.34 f20’)

13.68 k20

13.91 *20 ‘)

1.2

1.5

1.3

1.7

11.552* 8 11.555* 6 ll.601+10d) 11.656 ‘) 11.866k 9 11.926+ 5 11.949+ 5

t 12.39 $0 12.412+ 5 12.49 k30 12.591+ 10 12.919* 10 13.010+ 10 13.304 13.334* 6 13.343* 6 13.64 &I5

(13.66) 13.673 13.886k 15 13.904 13.904*20

2-

2+ 4+ 8+ 3- 0+

6+ 88

(4+) 60 1+ 7- 4+ 0+

110 2- 2- 2+ 47 6+ 100

“) Ref. *‘), horizontal lines indicate that states have been omitted, errors < 2 keV have been omitted. ‘) Used for energy calibration. ‘) Error unknown. d, T= 1. ‘) Widths < 30 keV have been omitted. ‘) No evidence present for identification of this state with the state mentioned at the same line under

“previous results”.

and by Kate and Bando 3g) are very similar to the SU(3) predictions listed in table 5 and have therefore not been included in that table.

5.1.1. Ground-state band. The g.s. transition is the strongest transition observed, with a maximum cross section of 27’pb/sr. Its angular distribution is rather well described by the DWBA. In contrast, the angular distribution for the transition to the E, = 1.634 MeV, J” = 2+ first excited state has a shape different from the DWBA prediction, while also the experimental and theoretical relative spectro- scopic factors for all models disagree by a factor of 4 to 6. This may be due to effects of channel coupling. The transition to the 4.248 MeV, J” = 4+ state is predicted in all models with about the right (relative) strength, and the shape of the angular distribution is well described. Our spectroscopic factors for excitation of the lowest O+ , 2+ and 4+ states may be compared with results from previous studies: 1 .OO,

206 J. C. Vermeulen et al. / (d, 6Li)

TABLE

Experimental and theoretical values of relative a-spectroscoptc

Present Prevtous

E, (MeV) “1 J” “) s 2. ZR s *. FR s* 7

g.s.

1.634 4.248 5.621 5.784 6.724 7.191

7.421 7.829 8.449 8.694 8.777 8.848 9 030 9.115 9.99

10.583 10.838 10.840 11.015 II.23 11.322 11.866 11.926 12.39 12.412

0+ 2+ 4+

3-

;+

0+ 2+ 2+

5-

;r

:r

3- 4+ 2+ 3- 2+ > 4+

;I

2+ 4+

Of 3- 1

EE 1.00 0.77 0.82 2.95 0.24 0.04 2.19 ‘)/0.53 “) 0.18 3.07 ‘)/0.69 g, 0.96 0.13 1.37 0.07 5.37 q/o.74 8) 0.32 0.33 0.15 0.28~0.10 0.07*0.04 0.24 0.13 0.07 0.11 0.30 0.05 + 0.03 0.14+0.12

-1.00 0.83 1 .oo 3.08 0.25

0.03 3.75 ryo.88 “) 0.21

4.79 o/O.92 ‘) 1 08 0.12 1.92 0.06 8.33 ‘)/l.OO g, 0.28 0.35 0.17 0.21 kO.09 0.10~0.05 0.30 0.1 I 0.09 0.14 0.34

0.02kO.06 0.21 kO.16

= 1.00 0.31 0.85 3.09 0.42

0.67 ‘)/0.15 g, 0.67 5.93 ‘)/I.12 8) 1.98

7.04

13.83 ‘)/8.03 “)

“) Ref. 29). “) (d, 6Li), Ed = 28 MeV, ref. ‘). ‘) Shell-model, Chung-Wildenthal interaction (this work).

d, Shell-model, Preedom-Wlldenthal interaction [results from ref. 3’)]. ‘) ($) = (84) for 24Mg g.s. r) 2N+L=4. “) 2N+L=6.

0.89 and 4.2, respectively 4); 1 .OO, 0.47 and 0.74 [ref. 6)] ; 1 .OO, 2.00 and 1.46 [ref. 5)] ; 1.00,0.31 and 0.85 [ref. ‘)I. The value of S,,,, for the J” = 6+ state lies between the predictions of the SU(3) model and the shell model, whereas S,,,, agrees with the SU(3) prediction. On basis of the deviation in excitation energy and the shape of the angular distribution it is not likely that the state observed at E, = 11.92 + 0.02 MeV is the J” = 8+ member of the g.s. band which lies at E, = 11.949 MeV. Instead, we have identified this state with the known J” = 4+ state at E, = 11.926 MeV.

5.1.2. K” = 2- band. The shape of the angular distribution for the strong transi-

J. C. Vermeulen et al. / (d, “Li) 207

5

factors for the “‘Mg(d, 6Li)ZoNe reaction

Theory

s o.cw ‘)

"'Ne

0.0 F1.00 = 1.00 G1.00 F1.00 ‘) 1.747 0.20 0.19 0.12 0.13 ‘) 4.132 0.41 0.50 0.71 0.80 ‘)

2.65 ‘) 0.37 ‘)

6.240 0.09 0.04 0.14 0.26 ‘)/0.43 “) 2.39 “)/0.32 “)

7.374 0.01 0.01 0.02 0.00 ‘)/0.14”) 12.00 “)/0.07 “) 0.51 J)

8.545 0.71 0.81 0.91 1.84 ‘)

21.60 “)/0.18 “)

9.861 0.20 0.05 0.22 0.78’)/0.001m)

10.967 0.13 0.21 0.24

11.695 0.02 0.11

:;

:;

“) “) “)

(&I) = (80) [results from ref. ‘)I. (I&c, = (82)2 [results from ref. ‘)I. (&L)K, = (90)0 [results from ref. ‘)I. (,$)K, = (42)0 (this work). (,$)K, = (04)O (this work). Weak coupling “C, 0 “‘Mg [results from ref. ‘)I. 6p-2h, (@)K, = (84)0 [results from ref. ‘)I.

TABLE 6

Possible rotational band structure of “Ne, refs. 3’.33-35)

K” = O+ K” = 2+ K” = 2- K”= l+ Ku= l-

J” Ex J” 6 J” Ex J” Ex J” Ex

0+ g.s. 2+ 4.45715.365 2- 5.148 1+ 5.336 l- 7.052 2+ 1.275 3+ 5.641 5.910 6.817 3- 7.721 4+ 3.357 4+ 6.345

(F) 8.44 “)

(:I) 7.341

6+ 6.311 (5+) 7.423 (4+) 8.382

Excitation energies (in MeV) are from ref. 30) except where noted. “) Excitation energy from ref. 35).

208 J. C. Vermeulen et al. / (d, 6Li)

TABLE 7

Excitation energies and maximum differential cross sections observed in the 26Mg(d, 6Li)22Ne reaction

Present results Previous results “)

EX EX (MeV f keV) (MeV f keV)

J”

gs. b) 19 1.273 ‘) 7 3.365k 7 1.1 4.460 “) 4.0 5.142 ‘) 0.7

5.359* 7 2.3 5.520 b, 6.5 5.629f 10 0.6 5.908 b, 5.9 6.113k 10 0.9 6.233 b, 2.3

6.340* 7

6.62 k20 6.676+ 7 6.818 b)

3.1

0.4

1.3 1.5

6.886 * 10 7.08 k20 7.32 +20

7.397* 7

7.623+ 10 1.4

7.70 *20 0.4 7.920 f 10 0.6

8.134* 10 1.2

8.37 +20 8.452k 7

0.5 1.2

8.573* 10 1.2

8.724+ 10 0.7

9.042+ 7 1.5

9.229 k 10 1.4

9.49 +20 0.6

0.4 0.4 0.9

12

gs. 1.275 3.357 4.457 5.148 5.336k 5 5.365 5.523 5.641 5.910 6.115+ 6 6.237+ 5 6.311 6.345

6.636 6.691 k 4 6.817 6.854+ 8 6.904k 9 7.052* 7 7.341 1.342+ 6 7.406 7.423

(7.470? 20) 7.489* 6 7.644+ 4 7.664k 8 7.721 k 3 7.924+ 6 8.081 k4 8.131f 7 8.162+ 4 8.382& 7

8.491+ 2 8.548k 10 8.592k 4 8.73lk 7 8.861k 4 8.902* 9 8.979k 9 9.040* 9 9.097+ 3 9.170+ 4 9.223* 9

0+ 2+ 4+ 2+ 2- 1+ 2+ 4+ 3+ 3- 2+ 0+ 6+ 4+

(2,3)+

;r

1+

(0, l)+

(:-4,+ 0;

(1, 3)) (3. 5)+

;+

2- 3- 2+

2+

(:-, 4+)

2+

3-

“) Ref. “), errors < 2 keV have been omitted. b, Used for energy calibration.

TABLE 8 209

Experimental and theoretical values of relative u-spectroscopic factors for the 26Mg(d, ‘Li)**Ne reaction

J” “)

Expertment Theory

s S E 9

s .,CW 9 x.cw

a,m a.FR NV) s D, SW31 ‘1

(80) ‘*Ne

g.E.. 0+ 1.275 2+ 3.357 4+

4.457 5.365 ;:

5.523 4+ 5.910 3- 6.115 6.237 ;:

6.311 6+ 6.345 4+

6.691 6.817 ;;

7.052 1- 7.406 3- d) 7.644 2+ 7.721 3-

7.924 8.131 ‘) ;:

8.452 r) 5- 8.737 3-

= 1.00 =l.OO 0.86 0.80 0.12 0.17 0.14 0.13 0.22 0.19 0.87 0.98 1.45 1.38 0.09 0.09 0.15 0.10 1.21+0.11 0.80&0.12 0.17+0.02 0.31+0.03 0.23 0.24 0.13 0.11 0.06 0.06 2.66 2.40 0.14 0.12 0.09 0.06 0.07 0.07 0.13 0.12 0.75 0.73 0.13 0.13

0.0 G1.00 el.00 1.393 0.15 0.10 3.424 0.02 0.04 4.283 0.04 0.09 5.271 0.00 0.01 5.3% 0.69 1.16

5.909 0.10 0.05

5.768 0.08 0.10 6.259 0.03 0.03 6.225 0.18 0.17

6.686

7.499

8.342 0.00 0.03 8.808 0.02 0.02

0.01

0.01

0.00

0.02

E 1.00 0.77 0.11 1.32

O.SOh. ‘)

0.21 0.23

0.67 ‘) 0.00 k) 1.72 ‘)

0.00 ‘)

0.99 h)

“) Ref. 3”). b, Shell-model, Chung-Wildenthal interaction (this work).

“) (A&@ = (48), (2~) = (82) for positive-parity states [results, except where noted, are from ref. 9)], d, Spin assignment from this work (see text). “) The state at 8.161 MeV is possibly also excited. ‘) Present energy calibration. “) (A&K, = (84)2 (this work). ‘) (&4)K, = (65)l: S, = 0.03; (&i)K, = (92)0: Se = 0.53 (this work). j) (&)K, = (63)l (this work). ‘) (A&, = (11, 1)l (this work). ‘) ($)K~ = (65)3 (this work).

TABLE 9

Rotational band structure of 24Mg “)

K” = 0; K” = 0; 1y” = 2+ K” = o- K” = 3-

J” 4 J” EX J” E. J” 4 J ” Ex

0+ gs. 0+ 6.432 2+ 4.238 1- 7.553 3- 7.616 2+ 1.369 2+ 7.348

;: 5.236 3- 8.356 4- (9.306)

4+ 4.123 6.010 S- 10.026 s- 11.596 “) 8.113

;: (7.812) 7- 12.450 “)

(13.213) 9.528 (12.347) 14.152

Excitation energies are experimental values (in MeV) from ref: “3. “) Ref. ““). “) Ref. 56).

210 J. C. Vermeulen et al. / (d, “Li)

TABLE 10

Excitation energies and maximum differential cross sections for the %(d, 6Lr)24Mg reaction

Present results Prevtous results “)

IT, (MeV f keV) IT, (MeV f keV) J”

g.s. 18 1.368* 7 9 4.115* 7 4.1 4.231 k 7 10 5.224* 7 1.5 6.002+ 7 4.0 6.427+ 10 1.0 7.349* 7 0.9

7.616 b, 16

8.105+ 10 8.357k 7

8.439& 7

0.8 1.3

3.8

8.651+ 7 0.8 8.865? 7 0.8 9.000+ 10 0.5 9.144* 7 1.4

9.301* 7

9.437* 15

9.526+ 7 1.9

10.024k 10 0.8

10.11 +20 0.7

10.352+ 7 2.6

10.580) 7 1.0 10.68 *20 0.6

10.925* 10 0.6 11.016* 7 1.4 11.13 +20 1.3

11.20 *20

2.4

0.5

0.9

g.s. 1.369 4.123 4.238

5.236 6.010 6.432 7.348 7.553 7.616 7.747 7.812 8.113

8.358 8.437 8.438 8.653 8.863 9.002 9.148* 3 9.283 9.298 9.306* 5 9.456 9.515

1

9.521k 3 9.528 9.827 9.966

10.026 10.059 10.112* 3

C

10.161* 3 10.328+ 3 10.357* 2 10.578 10.660 10.68Ok 3 10.712 10.731 10.824* 3 10.922* 3 11.018 11.128* 3 11.161+ 3 11.181* 3

Of 2+ 4+ 2+

3+ 4+ 0+ 2+ 1- 3- 1+

6+ 3-

(F4)- 2+ 2- 2+

(y) 1+ 1+ ‘) 5-

4+ 2+

Of 1+ ‘)

2+ 2+

J. C. Vermeulen et al. / (d, 6Li)

TABLE 10 (continued)

Present results

E, (MeV k keV)

Previous results “)

E, (MeV + keV) J”

11.31 *20 0.5 11.39 &20 2.5

11.46 k20

11.51 *20 11.596+ 7

11.693* 7

1.7

1.1 2.1

0.7

11.851*10 1.2

11.986+ 10 2.2

12.14 k20 1.5

12.24 +20 0.9

13.31 f30’) 0.9

13.79 k20’) 0.7

14.10 k30’) 1.5

11.207+ 3 11.217* 3 11.293&- 3 11.318* 3 11.330 11.389

11.455+ 5 11.456* 5 11.520* 3 11.596 11.618k 3 11.6941 3 11.727 11.828* 3

{ 11.861k 11.860+ 3 5 11.930 11.964 11.986 12.000+ 3 12.014 12.049 12.116

{ 12.126* 12.159* 3 3 12.181 12.242& 3 12.257

\ 12.257 13.213k 3 13.450+20 13.750 13.840*20 14.152+ 3

(1,2)+ 4+

1-

2+ 0+ 2+ 5- ‘)

4+ 0+

1-

2+ 2+

3- 4+

3- 2-

6+ 5- 6+ 8+

“) Ref. 30), horizontal lines indicate that states have heen omitted, errors i 2 keV have heen omitted. “) Used for energy calibration. ‘) T= 1. “) Ref. s’). ‘) Ref. 56). ‘) No evidence present for identification of this state with the state mentioned at the same line under

“previous results”.

tion to the 3- state at E, = 5.621 MeV is well described both by ZR and FR calcu- lations. The relative spectroscopic factor (x 3.0) deviates from two previously reported values C1.24, ref. 4)], and 1.6, ref. 6)] but is in good agreement with the value of ref. ‘) (see table 5) and with the predictions of the SU(3) strong coupling

212 J. C. Vermeulen et al. 1 (d, 6Li)

model, assuming coupling of a (2~) = (01) hole and (1~) = (81) to (APL) = (82). This hole character is also borne out by results of (p, t) experiments 40) and indicated by the calculations of ref. 41).

The J” = 5- state of this band is excited with a strength a factor of two larger than the SU(3) prediction, but a factor of two weaker than the result of Fortune et al. ‘). The 5- state is connected to the 3- state with a large B(E2) value of 26 f 6 W.U. [ref. 29)], thus CCBA effects may be important. The J” = 7- state at 13.334 MeV has not been identified with certainty.

The J” = 2- and 4- unnatural parity states at E, = 4.968 MeV and 7.004 MeV of this K” = 2- band are observed with moderate strengths, although their excitation via direct a-pickup is forbidden. The observation of the 2- state in the (d, 6Li) reaction at Ed = 28 MeV [ref. ‘)] and 55 MeV (this work) with a strength of the order of 10-30 ‘A of that of the g.s. transition excludes in our opinion a compound reaction process. Since the 2- and 4- states are connected via large B(E2)‘s to the 3- state, excitation of these states might be possible via an cl-transfer to the 3- state followed by inelastic excitation.

5.1.3. K” = O- band. The DWBA does not reproduce the measured angular distribution of the transition to the J” = l- state at E, = 5.784 MeV. The weak excitation of this state is in agreement with the result at Ed = 35 MeV for which S, = 0.19 was obtained 4, and is an indication for the particle character of this band. This particle character also followed from results of (6Li, d) [ref. 42)] and (‘Li, t) [refs. 43’44)] reactions. No evidence was found for the excitation of other members of this band. Though the 3- member is not separated from the E, = 7.191 MeV, J” = O+ state, the corresponding transition was found to be consistent with a pure L = 0.

5.1.4. K” = 0: band. The E, = 6.724 MeV, J” = O+ state is only weakly excited. Its angular distribution is not reproduced by the DWBA. The weak excitation of this state, which has a strong (dt)3 (s+) component 36), is in agreement with shell- model predictions. The 2+ and 4+ members of this band at E, = 7.421 and 9.99 MeV should likewise be weakly excited, but instead they are seen with moderate strength. This might be due to mixing with the K” = 0; band.

5.1.5. K” = 0: band. For the transitions to the O+, 2+ and 4+ states at E, = 7.191,7.829 and 9.030 MeV of this band the values of S, and the shape of the angular distributions deviate considerably from ZR and FR calculations when an 8p-2h configuration, for which 2N+ L = 4, is assumed. For a 6p-2h configuration with 2N+ L = 6, a better agreement in shape is obtained (the ZR calculations yield about the same shape for 2N+ L = 4 and 2N+ L = 6), but a 6p-2h configuration with SU(3) symmetry (84) gives spectroscopic factors ‘) a factor of 5 to 10 too low for the 2+ and 4+ states. A 12CJ @ 24Mgo weak coupling (8p-4h) configuration on the other hand gives spectroscopic factors ‘) about three to four times too large. From the strong excitation ofmembers of this band in “‘Be” stripping reactions 45, 56) an 8p-4h character of this band was concluded, which supports the prediction of an

J. C. Vermeulen et al. 1 (d, 6Li) 213

8p-4h 0’ state at E, FZ 6 MeV by Arima et al. 47). These experiments, however, do not exclude the presence of an appreciable 6p-2h component. McGrory and Wilden- thal 48) conclude from their calculations a 6p-2h character for this band.

5.1.6. Kn = 0: band. No excitation of members of this band has been observed. This may be due to the large width of these states (see table 4) and to important (Oflp) contributions to the wave functions of K” = 0: states 41).

5.1.7. Other states. For states above E, = 10 MeV an anticorrelation with the results of the ‘9F(3He, d)‘“Ne reaction 37) at EjHe = 18 MeV is observed. States which are excited in the 24Mg(d, ‘Li) reaction are excited only weakly in the (3He,d) reaction. and vice versa.

5.2. THE 26Mg(d, 6Li)22Ne REACTION

5.2.I. Gro~d-state band. The angular distribution of the g.s. transition is repro- duced by the DWBA. For the E, = 1.275 MeV, J” = 2+ first excited state a rather flat angular distribution is observed which does not agree with the DWBA predic- tion. Also the spectroscopic factor of this state is much larger than predicted by the shell model, although it agrees with the SU(3) model. CCBA effects are probably as important for this transition as for the 24Mg(d, 6Li)20Ne reaction to the first 2+ state in ‘ONe. The deviation from the shell-model prediction may thus be due to those effects. The weak excitation of the 4+ state at E, = 3.357 MeV is in agreement with theoretical predictions. The present spectroscopic factors for the 2+ and 4+ states, relative to the g.s. value, are smaller than the values obtained from the (d, 6Li) reaction at 35 MeV [ref. 4)], namely 1.80 for the 2+ state and from the (“He, ‘Be) reaction 5, (1.30 for the 2+ state and 0.35 for the 4+ state).

The spectroscopic factor for the E, = 6.311 MeV, J” = 6+ member of the g.s. band, which forms a doublet with the E, = 6.345 MeV, J” = 4’ state, was deter- mined by a least-squares fit and was found to be considerably larger than both the shell-model and the SU(3) predictions.

5.2.2. K” = 2+ band. The SU(3) symmetry of the g.s. band, which is probably (S2), also predicts a K” = 2+ band 33). It is not yet clear whether this band starts with the E, = 4.457 MeV state or with the E, = 5.365 MeV state. The experimental angular distributions for transitions to these two states differ from each other, al- though both states should have the same angular distribution in a direct one-step reaction mechanism. The E, = 5.365 MeV state is not separated from the E, = 5.336 MeV, J” = If band head of the K” = l+ band. Because its unnatural parity the latter state is assumed to be not excited and indeed this seems to be the case in view of the good description of the angular distribution of the Ex = 5.365 MeV state with an L = 2 DWBA curve. It seems that for the 4.457 MeV state the results of the ZR calculations have to be shifted towards larger angles in order to fit the data. The same behaviour is also observed for the transition to the second 2+ state in 24Mg (see also subsects. 5.3.2 and 6.1). The shell model predicts only the E, =

214 J. C. Vermeulen et al. / (d, 6Li)

4.457 MeV, J” = 2+ state to be excited with appreciable strength. Instead both the 4.457 MeV and the 5.365 MeV states are excited with about equal strength, whereby the combined strength of both transitions by far exceeds the shell model prediction. The unnatural parity E, = 5.641 MeV, J” = 3’ state is weakly excited, in contrast to the 3+ member of the K” = 2+ band in 24Mg for which the transition strength is about a factor of four larger.

5.2.3. K” = 2- band. This band has SU(3) symmetry (84) [ref. 33)]. The un- natural parity member of this band, the J” = 2- state at E, = 5.148 MeV, is weakly excited in contrast to the J” = 2- state at E, = 4.968 MeV in 20Ne which is excited with moderate strength. The E, = 5.910 MeV, J” = 3- state is excited strongly. As this state is weakly excited in the stripping reactions (d, p), (t, p), (6Li, d) and (7Li, t) [refs. 49-51, 33)] a hole character of this state is very likely. The SU(3) model underpredicts the strength of this state by a factor of 3 when a (&+c, = (84)2 con~guration is assumed (see also the discussion at the end of this section). Broude et al. 35) suggest the E, = 8.441 kO.020 MeV state, which is not mentioned in the compilation of Endt and Van der Leun 30), to be the J” = 5- member of the K” = 2- band. In the present experiment a state is observed at E, = 8.452 -I_ 0.007 MeV with a typical L = 5 angular distribution. Its strength is in reasonable agreement with the SU(3) model prediction.

5.2.4. K” = I + band, As discussed above, the J” = 1’ band head of this band, which has (np) = (63) symmetry 33), is not excited. The J” = 2+ member at E, = 6.817 MeV is excited with a strength which lies between the strengths predicted by the shell model and the SU(3) model. The states observed at Ex = 7.32 and 8.37 MeV might be identified with the (3+) and (4+) members of this band.

5.2.5. K” = I- band. In contrast to the J” = 3- member of the K” = 2- band, the J” = 3- state of the K” = l- band is only weakly excited. In stripping reac- tions 33,49-51) the situation is just reversed and cons~uently one might surmise a predominant particle character for this state.

5.2.6. Other states. Preedom and Wildenthal 34) suggest the E, = 5.523 MeV, J” = 4+ state to be the head of a K” = 4+ band. The shell model predicts this state to be strongly excited and this is indeed seen in the present study.

The excitation of the J” = Of state, E, = 6.237 MeV state is weak while it is strongly excited in the “80(6Li, d)22Ne reaction “) This leads to the supposition . that this state probably has an excited (sd)4 configuration, similar to the J” = O’, E=, = 6.724 MeV state in 20Ne.

At E, = 7.397~0.~7 MeV a very strong excitation is observed. Endt and van der Leun 30) give as possible spins and parities: J” = l-, 3- for a state at E, = 7.406 MeV and J” = 3+, S+ for one at E, = 7.423 MeV. In our opinion such a strong excitation of an unnatural parity state is excluded and therefore only a J” value of 1 - or 3- is possible. From the shape of the angular distribution the state is concluded to have J” = 3-. The SU(3) model gives a reasonable prediction when (,Q+c,. is assumed to be (65)3. This classification is obtained by coupling a Op,

J. C. Vermeulen et al. / (d, 6Li) 21.5

hole with symmetry (01) to 23Ne with possible (2~) = (64), while the (84) sym- metry of the J” = 3- state at E, = 5.910 MeV can be obtained by coupling a (Op,) hole to 23Na with (+) = (83). In both cases several symmetries result, but following the pure SU(3) model the symmetry with maximum 12* + p2 f 3A + 2~ + 3~ has been assumed to lie lowest in excitation energy. This also shows a possible reason why two 3- states with hole character occur: the Op, hole can couple to two nuclei with different isospin (7’ = 3 for 23Na and T = 3 for 23Ne), while for *‘Ne and 24Mg the nuclei to which the hole can couple have the same isospin and therefore only one 3- state will occur,

For the J” = 2” states with E, > 6.115 MeV, the strength found experimentally is considerably larger than predicted by the shell model. It is not clear whether this is a real effect. Since the spectroscopic factors are small and the excitation energies are high, other reaction processes and/or imperfections in the DWBA may play a role.

5.3. THE ?Si(d, 6Li)24Mg REACTION

53.1. crowed-siute band. The relative spectroscopic factors for the J” = 2+, 4+ and 6+ members of the g.s. band are in reasonable agreement with the shell-model and SU(3) predictions. The only significant deviations are the predictions based on the CW interaction for the J” = 2+ state and on the PW interaction for the J” = 6+ state.

5.3.2. K” = 2+ band. The shell-model predictions for the strength of the J” = 2+, 4+ and 6+ members of this band are close to the experimentally observed strengths, whereas the SU(3) model gives consistently too large strengths. The results of ref. 4r for the Zf and 4+ state (see table 11) are in good agreement with our values. For the transition to the 2’ state one notices a deviation between the ZR calculations and the data similar to that for the 2: state in 22Ne (see subsect. 5.2.2). The J” = 6+ member of the K” = 2+ band at E, = 9.528 MeV is not separated from the E, = 9.521 MeV state for which a J” value of 2+ has been suggested 52). (The shell model predicts a J” = 2+ state at E, = 9.259 MeV, but this can probably be identified with the E, = 9.283 MeV, J = 2+ state.) Also a J” = 4+, T = 1 state at E, = 9.515 MeV is known, but this state should not be populated because of the isospin selection rule for the (d, 6Li) reaction. Therefore, a least-squares fit with L = 2 and L = 6 contributions has been made to the experimental angular distribution.

The E=, = 5.236 MeV, J” = 3+ member of the K” = 2+ band is also excited, although the transition is forbidden in direct a-transfer because of its unnatural parity, but CCBA effects might play a role. The excitation of this state in the inverse reaction 2oNe(6Li, d)24Mg at EeLi = 32 MeV [ref. 53)J was shown to be probably due to coupled channels effects.

At Ex = 14.10 MeV+ 30 keV a state is excited, which lies near the known 8+ member of the K” = 2+ band at E, = 14.152 MeV. But these states cannot be

216 J. C. Vermeulen et al. / (d. 6Li)

TABLE 11

Experimental and theoretical values of relative cc-spectroscopic factors for the Z8Si(d, 6Li)24Mg reaction

Present Previous Theory

s Ir,cw ‘) s #,ZR s 01, FR S

(80) ‘ONe ol.PW 9 S,.,“,,,‘)

g.s. 1.369 4.123 4.238

6.010 6.432 7.348 7.616 8.113 8.358 8.437 8.438 8.653

9.002 9.148

9.521 9.528

10.026 10.328 10.357 10.922 11.018 11.389 11.455 11.456 11.520 11.596

11.694 11.964 11.986

E 1.00 1 .oo

4+ 0.55 i: 0.69 0.44

0+ 0.16 2+ 0.16 3- 3.40 6+ 0.67

(& 0.49 0.29 kO.08

;: ? 0.30~0.08

0.10 2+ 0.05

(:I) 0.24

6+ i

0.11 kO.02 1.08+0.15

5- 0.59

2+

i

0.29kO.04 4+ 0.02 + 0.08

2+ 0.07 2+ 0.21 ;r 0.51

>

0.16+0.03 0+ 0.01 kO.02 2+ 0.10 5- 1.49 4+ 0.18 2+ 2+ >

0.21

= 1 .oo 1 .oo 0.71 0.36 0.54 0.06 0.11 3.43 0.59 0.30 0.61 kO.09 0.09 kO.08 0.07 0.04 0.20 0.11+0.01 0.65+0.10 0.64 0.21+0.04 0.03 + 0.08 0.05 0.14 0.34 0.11+0.03 0.02 kO.02 0.08 1.50 0.14

0.16

El.00 0.0 s1.00 2.25 1.564 0.64 1.67 4.422 0.69 0.61 4.118 0.22 0.39 5.891 0.54

7.420 0.01 7.528 0.02

8.470 0.34

8.791 0.25

El.00 s1.00 z1.00

0.54 0.88 1.20 0.74 0.38 1.07 0.15 0.42 2.10 0.49 0.93 1.69 0.00 0.01 0.00 ‘) 0.00 0.00 0.00 ‘)

1.56g) 0.48 0.06 0.83

O.OOh) 0.77

9.514 0.80 0.85 1.26 1.79

0.00 h) 10.344 0.04 0.00

1.31 ‘)

“) Ref. 30). b, (d, 6Li), Ed = 35 MeV, ref. 4). ‘) Shell-model, Chung-Wildenthal interaction (thts work). “) Shell-model, Preedom-Wildenthal interaction [results from ref. 32)].

‘) (N,,,, = (0, 12); (Au) = (84) for positive-parity states [results from ref. 9)]. ‘) (Au) = (46) (this work). r) (I& = (75)3 (this work). “) (&)K~ = (94)0 (this work).

identified with each other since the excitation energies differ too much and since the angular distribution does not display the characteristic L = 8 shape (an increase of the cross section by a factor of 10 from 8,,,, = 10” to 8,,,, = 40’).

5.3.3. K” = 0: band. The J” = O+ and 2’ members of the second K” = Of

J. C. Vermeulen ef al. / (d, 6Li) 217

band at E, = 6.427 and 7.349 MeV, respectively, are only weakly excited, in agree- ment with theoretical predictions. Since they are strongly excited in the (6Li, d) and (7Li, t) reactions 53) a predominant particle character is very likely. The intrinsic structures of the second K” = O+ bands in ‘ONe, 22Ne and 24Mg thus appear to be very similar.

5.3.4. K” = O- and K” = 3- bands. The results of the orthogonality model calculations of Kate and Bando 54) strongly suggest a predominant particle character for the K” = O- band and a hole character for the K” = 3- band. The experimental results of refs. 53,55) confirm these suggestions.

The E, = 7.616 MeV, J” = 3- state, band head of the K” = 3- band, is excited nearly as strongly as the ground state. The E, = 11.596, J” = 5- member of this band, which was recently found by Fitield et al. 56) is excited with a very charac- teristic angular distribution. The present results thus agree with a hole character of this band. The spectroscopic factor for the 3- state, as predicted by the SU(3) model, assuming strong coupling of (APL) = (01) and (1~) = (66) to (1~) = (75)3 is a factor of two too small, while the 5- state is predicted with the right strength. (2~) = (67) is also possible,.but this would yield a K” = O- or 2- band.

At an excitation energy of 9.30 MeV there are three states which are separated by less than 25 keV and whose spins and parities are not all known 30). One of these states is the possible J” = 4- member of the K” = 3- band. The angular distri- bution for the transition to the triplet has about the same shape as was observed for the transition to the E, = 7.004 MeV, J” = 4- state in 20Ne. The contributions of different L-transfers of course may also cause this behaviour, but from the strong excitation of the band head at E, = 7.616 MeV and in view of possible CCBA effectsonewould expect the J” = 4” state to beexcited. In fact, the E, = 9.301 f 0.007 MeV triplet is excited with about the same cross-section as the 4- state in 20Ne (about 3 pb/sr), while the strength for the 3- states in both nuclei are also about the same (about 15-18 pb/sr).

The E, = 7.553, 8.358, 10.026 MeV, J” = l:, 33, 5- members of the K” = O- band all are excited weakly in comparison with the members of the K” = 3- band, which is in agreement with the supposed particle character of these states.

According to Kurath and Towner 17) 3- states which are strongly excited in inelastic scattering should also be excited strongly in a-particle transfer reactions. Inelastic scattering, however, does not distinguish between a particle or hole character of the states, and indeed the J” = 3-, E, = 7.616 MeV (hole) and E, = 8.358 MeV (particle) states are both excited strongly in inelastic scattering 52).

5.3.5. Other stwes. The state observed at E, = 8.439kO.007 MeV is a doublet, consisting of the E, = 8.437, J” = (3, 4)+ and the E, = 8.438, J” = l- states. The angular distribution shows a typical L = 4 shape and therefore it is concluded that the E, = 8.437 MeV state has J” = 4+. The J” = 2- state at E, = 8.863 MeV, which might be the J” = 2- member of the K” = l- band with the E, = 8.438 MeV state as band head 36) is excited weakly in the present work. Direct a-

218 J. C. Vermeulen et al. / (d, 6Li)

particle pickup to this state is strictly forbidden because of the unnatural parity of the state.

By assuming that T = 1 states are not excited in the (d, 6Li) reaction a number of states with E, > 10 MeV could be identified with states with known spin and parity. The possible doublet at E, = 11.694 + 0.007 MeV displays a typical L = 4 shape, which is very well described by DWBA, while the excitation energy agrees with that of the known E, = 11.694 MeV, J” = 4+ state. The nearby lying E, = 11.727 MeV state with J” = O+ which also could contribute to the observed cross section therefore has been assumed to be not excited.

6. Discussion of general features

In the previous section the results for individual transitions were discussed. In the following we concentrate on the common aspects of the reactions studied.

6.1. ANGULAR DISTRIBUTIONS

Angular distributions for a number of strong transitions in the three reactions studied are compiled in fig. 7. The differential cross sections, except those for the transition to the E, = 3.357 MeV state in **Ne were obtained by dividing the full , solid angle of the spectrograph with a ray tracing technique into two parts (see sect. 2). Also included in fig. 7 are the results of the zero-range (solid lines) and the finite-range (dashed lines) DWBA calculations. The large differences in shape of the angular distributions of the transitions to the first and second 2+ states are remarkable. The angular distributions of the first 2+ states are rather flat and featureless. The transitions to the second 2+ states show a sharp rise in differential cross section towards small angles. The shape of this rise is better reproduced by the ZR calculations, but there is a systematic angle shift between the data and the DWBA calculations. A similar systematic angle shift is observed for the 4+ transi- tions between the ZR DWBA and the data except for the transition to the first 4+ state in “Ne. The 4+ states are systematically better fit by the FR calculations.

The fact that the DWBA fits some of the data, and not others, points towards other possible contributing reaction mechanisms such as inelastic excitations of the target and/or the final nucleus. The population of unnatural parity states, which cannot be excited if the four nucleons that are transferred have total spin zero, is another indication for competing reaction mechanisms. Some preliminary CCBA calculations showed that the strength of the 2: state and of the 2- state at E, = 4.963 MeV in 2oNe can be qualitatively reproduced by assuming that these states are excited via an a-transfer to the g.s. and to the 3- state, respectively, followed by inelastic excitation. If the unnatural parity states are excited via inelastic excitation the cross section at 0” should be very small 57). A 0“ measurement might thus provide a test of the reaction mechanism responsible for excitation of these states. It is

-fR

IO”

219

0 20 40 60 0 2c) 40 60 0 20 40 60

@&deg f

Fig. 7. Comparison of angular distributions of strong transitions from the t4- ‘“Mg(d, 6Li)2a* “Ne and 28Si(d, 6Li)24Mg reactions at Ed = 55 MeV. The full (dashed) curves are the results of ZR(FR) DWBA calculations. The data points have been obtained by su~ivid~ag the full spectrograph solid angle into two by applying ray-tracing techniques, data points for the E, = 3.357 MeV state in ‘*Ne are for the

full solid angle.

220 J. C. Vermeulen et al. 1 (d, ‘Li)

remarkable that CCBA effects seem to be of minor importance for the (6Li, d) reactions to “Ne, 22Ne and 24Mg [refs. 42’ ‘I, “)I. A further study of these effects is thus desirable in view of the observations made above.

Another interesting feature is the difference in the shape of the angular distri- butions for transitions to the J” = S- states in 20Ne, 22Ne and “Mg at E, = 8.449, 8.452 and 11.596 MeV, respectively (see figs. 2, 4 and 6). The shape of the experi- mental angular distribution for the transition to the 5- state in “Ne is described well by the FR calculation but not by the ZR calculation. For the transition to 24Mg the ZR calculation reproduces the data, while this is not the case for the FR curve. The transition to 22Ne is an intermediate case. This result is another example of the fact that some features of the data are reproduced better by ZR than by FR calculations, while other features are described better by FR calculations.

6.2. SPECTROSCOPIC STRENGTHS

Although the angular distributions calculated with zero-range DWBA differ from the finite-range curves in a number of cases, the extracted relative a-spectro- scopic factors are not much dependent on the type of DWBA calculation (see tables 5, 8 and 11). This enables a less ambiguous comparison of experimental and theoretical spectroscopic factors.

The shell-model calculations indicate 12,22) that the four transferred particles have a larger probability to have the internal configuration of the four-nucleon “cluster” which together with the ’ 6O core forms “Ne than the internal configuration of an alpha cluster with SU(3) symmetry (80). For most cases, however, the relative spectroscopic factors for the transfer of a four-nucleon cluster with the internal structure of an a-particle or with the “distorted” (“Ne-like) internal structure, have about the same value (see tables 5, 8 and 11).

Although the relative spectroscopic factors from ref. 32) calculated with the Preedom-Wildenthal interaction agree in a number of cases with the present results obtained with the Chung-Wildenthal interaction, the latter results are supposed to be more appropriate to compare with experiments since in the derivation of the CW interaction considerably more information on sd shell nuclei was used than in the case of the PW interaction.

In fig. 8 experimental and predicted relative spectroscopic factors for the strong transitions are shown. The experimental spectroscopic factors are those obtained with the finite-range DWBA calculations (solid bars), the shaded bars are the SU(3) predictions, and the open bars the shell-model results computed with the 20Ne transition operator. The shell model does remarkably well for the 28Si(d 6Li)24Mg reaction, except for the 2+ transitions, for which the predicted strengih is too small. This discrepancy, however, which is even more pronounced for the 24, 26Mg(d, 6Li) transitions to the 2+ members of the g.s. band might be, at least partially, a coupled-channel effect (see subsect. 6.1). The shell model does well

J. C. Vermeulen et al. / (d, 6Li) 221

3.0

2.0

1.0

30

2.0

s Q)

t er

uJ IO

30

2.0

IO

exp. m SU(3) w

model -

0’ 2’ 4+ 6* o+ 2+ 4+ 3- 5-

w 1.63 4.25 6.76 720 763 9.03 562 6.45

0+ 2* 4’ 6+

0s 1.26 3.36 6.31 4.45 5.37 5.52 6.35 5.91 7.41 6.r

0’ 2, 4* 6+ 2* 3- s-

g.+ 1.37 4.12 6.11 4.24 6.01 8.44 7.62 II.60

Fig. 8. Comparison of relative a-spectroscopic factors for pick-up to states in *“Ne, **Ne and 24Mg. The black bars are the experimentally deduced numbers, the shaded bars those predicted by the SU(3)

model, and the open bars those obtained from shell-model wave functions.

222 J. C. Vermeulen et al. / (4 6Li)

for the second 2+ state and the second and third 4+ states in ‘*Ne, and it correctly predicts the trend in the relative spectroscopic factors of the members of the g.s. band in *‘Ne. It completely fails to predict the strength of the 6+ state in **Ne. Altogether, however, the agreement between the shell-model predictions and the measured spectroscopic factors is gratifying.

The SU(3) predictions for the g.s. bands in *‘s2*Ne and 24Mg are in good agree- ment with the data except for the 2: state in *‘Ne and the 6+ state in **Ne. Because of the coupled-channels effects that may influence the transitions to the 2: states it is impossible to conclude whether or not the agreement found for the 2: states in 22Ne and 24Mg is fortuitous.

The transition strengths for the positive-parity states not belonging to the g.s. band are only poorly predicted by the SU(3) model, The negative-parity states in *‘Ne are predicted with about the right strength, whereas the 3- strength predicted for **Ne and 24Mg is too small.

TABLE 12

Comparison of experimental and theoretical ground-state transition strengths to different final nuclei

Experiment Theory

Target Final nucleus nucleus

S il. cw “) s m, ZR s

target final DI, PR S iI, SU13) Y

W) Z”Ne (h)K, (h‘)K,

24Mg 20Ne s L~O(0.57) = l.OO(O.24) = l.OO(O.103) = l.OO(O.125) = l.oO(O.081) (84)0 (80)0

=Mg **Ne 1.02 1.01 ‘) 0.74 0.77 0.49 (48)O (82)O

*sSi =Mg 1.23 1.17 0.65 0.61 1.17 (4 12)O (84)o

The spectroscopic factors have been normalized relative to the 24Mg(d 6Li)2”Ne ground-state transition. , The numbers in parentheses are absolute values of the spectroscopic factor. “) Shell-model, Chung-Wildenthal interaction (this work). ‘) Calculated from spectroscopic amplitudes given by Draayer ‘) and C- and D-factors given by Hecht

and Braunschweig lo). “) Renormalized by a factor 1.15 to account for different a, value of ‘Li potential [see footnote ‘) of table 21.

Spectroscopic factors quoted thus far have been relative numbers within one nucleus. In table 12 the strengths of the ground-state transitions for the 24*26Mg(d, 6Li)20*22Ne and **Si(d, 6Li)24Mg reactions are compared with each other as well as with theoretical predictions. The relative numbers from one nucleus to the other are accurate to about 30 %, due to uncertainties in the target thicknesses. Within these uncertainties the strength of the ground state transitions remains the same, whereas the shell model predicts a decrease from *‘Ne to 24Mg, and the SU(3) model gives about the same strength for *‘Ne and 24Mg, and half the strength for

22Ne. The relative strength of the transition to **Ne is in reasonable agreement

J. C. Vermeulen et al. 1 (d, 6Li) 223

with the result of Oelert et al. 3, who find a value of 0.84 + and with the shell-model prediction. But it is too large compared to the value 0.50 extracted from results for the (6Li, d) reaction on 20Ne and 22Ne [refs. 53*59)]. The strength for the 28Si(d, 6Li)24Mg reaction is in good agreement with the value 1.07 obtained for the 24Mg(6Li, d)28Si reaction 59,60). The values of the g.s. strength for the (d, 6Li) reaction on 26Mg and 28Si at Ed = 35 MeV, relative to the g.s. strength for the 24Mg(d, 6Li)20Ne reaction 4), 3.14 and 2.48, respectively, deviate considerably from our values and from the theoretical values.

Theabsolutevalues ofthe theoreticalspectroscopic factors for the 24Mg(d, 6Li)20Ne reaction are also given in table 12. They may be compared with experimental spectro- scopic factors obtained with the normalization constants NzR and NFR of eqs. (5) and (6) set equal to 1.0. In view of the sensitivity of the spectroscopic factor to the parameters of the DWBA the agreement is rather good.

7. Summary and conclusions

The (d, 6Li) reaction on the target nuclei 24Mg, 26Mg and 28Si was studied at a deuteron energy of 55 MeV. The good energy resolution (w 40 keV) enabled the study of transitions to many states over a large range of excitation energies in the nuclei “Ne, 22Ne and 24Mg.

A large fraction of the total observed transition strength was found to be con- tained in transitions to members of the ground state bands, to the first excited J” = 3- states in 20Ne and 24Mg, and to the lowest two J” = 3- states in 22Ne. For each of the three final nuclei also a relatively strong transition to a J” = 5- state was observed. This confirms the predominant hole character of these 3- and 5- states. Negative-parity states with a predominant particle-like structure are weakly excited, as expected. Some of the unnatural parity states are excited with a strength of at most 10 % of that of the ground-state transition. The second O+ state in these nuclei is excited weakly, which is in agreement with a probable excited (sd)” configuration of these states.

Spectroscopic factors were obtained from comparison of the experimental angular distributions with the results of zero- and finite-range DWBA calculations which were made under the assumption that the four transferred nucleons have the con- figuration of an a-particle in its ground state. For most cases the two calculations resulted in about the same value of the relative spectroscopic factor although the calculated angular distributions were sometimes different. The occurrence of systematic differences in shape between the results of the DWBA calculations and the experimental angular distributions which were observed for a number of transi- tions, and the observation of transitions to some unnatural parity states, indicate

’ Here it should be noted that the W(3) results quoted in table II of ref. “) are wrong since the C- and D-factors of ref. lo) and also the “G-factor” and the (A/B) N+L’2 factor of eq. (2) have not been included.

224 J. C. Vermeulen et al. / (d, 6Li)

that reaction mechanisms other than direct a-cluster pick-up probably play a role in the excitation of these states via the (d, 6Li) reaction.

The experimental values of the cc-spectroscopic factors were compared with the results of SU(3) shell-model calculations and, in case of transitions to positive- parity states, with the results of a shell-model calculation in which the Chung- Wildenthal effective hamiltonian was diagonalized in a full (sd)” configuration space. Both calculations predict m-strengths that are concentrated in a few transitions only, which is indeed observed experimentally. As far as the SU(3) model is con- cerned, reasonable agreement with experiment was found for many of the transi- tions observed. This might have been expected from the good SU(3) predictions in the case of the inverse (6Li, d) reaction on “Ne, 22Ne and 24Mg [refs. 53,58)]. In general there is good agreement between the j-j coupling shell-model predictions and the experimental spectroscopic factors. A notable exception is the strength of the transitions to the 2+ member of the g.s. band which is predicted too low for all nuclei. Two-step processes may be responsible for a major part of the observed strength of these transitions and require additional investigation.

Shell-model calculations with j-j coupling in which lp holes are explicitly included in the configuration space would be helpful for the investigation of transitions to negative-parity states. Extension of the present type of four-nucleon transfer ex- periments to heavier s-d shell nuclei would be of interest since p-hole admixtures are of minor importance for these nuclei, while coupled-channels effects are expected to be of less importance because of smaller deformations.

The authors wish to thank the technical staff of the KVI for their help and assistance in all phases of these experiments. We thank A. van der Woude for his help in data taking. We acknowledge the granting of special facilities for the shell- model calculations by the computer centre of the Rijksuniversiteit Groningen.

This work has partly been performed as part of the research program of the Stichting voor Fundamenteel Onderzoek der Materie (FOM) with financial support from the Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (ZWO).

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