Fusion cross sections for the 16O + 16O reaction

2.B:2.N I

Nuclear Physics A306 (1978) 259-284: (~) North-HollandPublishino Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher

FUSION CROSS SECTIONS FOR THE 160-['160 REACTION B. FERNANDEZ *, C. GAARDE, J. S. LARSEN, S. PONTOPPIDAN and F. VIDEBAEK The Niels Bohr Institute, Unit,ersi O' Of Copenhayen, DK-2100 Copenhagen O, Denmark

Received 24 February 1978 (Revised 28 April 1978) Abstract: The fusion cross section for the ~O+ ~60 reaction has been measured with a time-of-flight technique at energies Ela b = 35, 45, 52, 60, 70 and 80 MeV. Excitation functions for the total fusion

and for the individual evaporation residues have been deduced from one-angle measurements of mass yields as a function of energy by normalizing to the energies given above. The ~60+ ~2C reaction has been studied by the same method at energies of 45, 52, 70 and 80 MeV, and fusion cross sections and mass distributions are given. Small amplitude oscillations are observed in the fusion cross section for the ~60+ ~60 reaction and are explained as effects of the individual grazing angular momenta, The measured fusion cross section is compared with a TDHF calculation of this quantity. A possible low angular momentum cut for fusion as predicted by the TDHF calculation is examined. NUCLEAR REACTIONS 160, 12C(160, X), E = 35-80 MeV: measured o(E, 0), E = 35, 45, 52, 60, 70, 80 MeV, a(E, 7.5°) 35 _< E -< 70 MeV, a(E, 6°) 70 _< E _< 80 MeV: deduced fusion a( E).

I. Introduction Light heavy ion fusion has for some years received c o n s i d e r a b l e a t t e n t i o n . These lighter systems are on the o n e h a n d complex e n o u g h to show trends o f the heavier systems, b u t o n the other h a n d they exhibit some specific features which are n o t yet completely u n d e r s t o o d : (a) the m a x i m u m fusion cross section can vary a p p r e c i a b l y between systems which differ by only a few nucleons, a n d (b) oscillations have been observed in the fusion cross sections for some reactions: 12C + ~2C a n d 160-.}- 12C. However, while the reactions 12C + 12C a n d 160 + 12C are experimentally well studied a n d reported in the literature, this is n o t the case for the 160 + 160 reaction. O n l y recently have m e a s u r e m e n t s of the fusion cross section a b o v e the barrier been reported 1,2). In the latter paper a v-yield t e c h n i q u e is used with the usual p r o b l e m s of different accuracy a n d sensitivity for the different final nuclei. The technique is, however, very useful for m e a s u r e m e n t s of excitation functions, a n d results o f such m e a s u r e m e n t s are reported in the energy interval Ela b = 25 to 60 MeV. Small * Permanent address: CEN, Saclay, France. 259

260

B. FERNANDEZ

et al.

amplitude oscillations with a period of 3.5 MeV (c.m.) are observed in the 2~ channel, and this being the strongest channel also in the total fusion cross section. It is an interesting question whether the oscillations are of the same nature as the oscillations observed in the reactions involving ~2C [refs. 3, 4)]. We should in this connection point to the difference observed for these reactions below the Coulomb barrier with narrow resonances for e.g. the ~2C+ ~2C reaction 5), but not for the 160 _[_160 reaction 6). Another interesting aspect of this reaction is the recent development of realistic three-dimensional time-dependent Hartree-Fock calculations 7). The simple models one has to work with could be especially valid for this system. The fusion cross section as function of energy is an aspect of the calculation which is relatively well defined. The data for this reaction are therefore important for testing these calculations. The T D H F calculations predict a low cut in the angular momenta that leads to fusion. This "bouncing off" for the smallest impact parameters starts already at an energy of 0.83 MeV/A corresponding to Elab = 54 MeV, and at 80 MeV only I > 10 leads to fusion - an interesting prediction. We should in this connection point to the relatively high energy for this system one can reach with tandem accelerators. At e.g. Ela b ----80 MeV we see that Ec.m./B -~ 4, and the energy per nucleon available (above the Coulomb barrier) is around 1 MeV/A. In the present experiment we have measured the fusion cross section with a time-of-flight technique for the ~60+ 160 reaction from Ela b = 35 to 80 MeV. We have extracted excitation functions for individual masses of the evaporation residues in this energy interval. We believe that our cross sections are accurate to about 5 ~o. In obtaining this accuracy we have corrected for the carbon present in the target. We have therefore also measured the fusion cross section and mass distribution for the ~60 + 12C reaction. We have extended the data on the fusion cross section for this reaction up to 80 MeV 160 energy, corresponding to Ec.m. = 34.3 MeV. 2. Experimental procedure The beams of 1 6 0 in the energy region from 35 to 80 MeV were supplied by the super F N tandem accelerator at the Niels Bohr Institute. The outgoing products were measured in a time-of-flight setup. The targets consisted of self-supporting TiO 2 films obtained by the sputtering of TiO 2 onto BaC12 coated glass plates s). The TiO 2 was pressed into 1 cm 3 blocks and the sputter beam (Ar) was defocused and moved continuously over the TiO 2 surface to avoid local heating. The sputtering of a set of targets,lasted as long as 12 h and a significant part of the carbon contamination came during sputtering. The target thickness ranged from 120-180 pg/cm 2. The carbon targets used in the experiments were around 50/zg/cm 2. The TiO 2 targets were very stable, i.e. the ratio between Ti and O was found to

~0+

~60 CROSS

261

SECTIONS

be constant for a given target to a level of 1%. This was tested directly by measuring the elastic yield at a given angle and energy at the beginning and at the end of fusion cross section measurements. It was tested indirectly by the amount of Ti and O obtained from optical model fits to the elastic data with the same target at different energies. Including all targets and all energies the ratio Ti:O was found to be 1 :(1.93+_0.02). The beam is led into the scattering chamber through a 1 cm diameter tube which is part of a liquid nitrogen trap. Another cold trap is placed just above the target to reduce carbon build-up during the experiments. The slit system consisted of 3 sets of slits yielding a final beam spot of 1 × 1 mm 2. The smallness of the slits causes, however, problems in an experiment where the energy spectra studied are continuous• This problem is discussed below. The details of the time-of-flight setup are described in ref. 9). The particles to be detected pass a thin carbon foil ( ~ 5/~g/cm2). Electrons produced by this passage' are amplified in a channel plate resulting in a charge pulse 0.7 nsec wide and typically 0.2 V. This signal is eventually amplified and then used in a front trigger discriminator to get the stop signal for a time-to-amplitude converter. The flight path is 63 cm and the stop detector is a 100 mm 2 Si surface-barrier detector overbiased by 200 ~o and thermoelectrically cooled to - 10 °C to obtain optimum energy 500

A 160.160

80MeV

etob =6* 30

0¢.. t'U, v

250

N

2O

.,.o

j.

,~

I

--,2

•

x -

IJJ

.

.o~'~

o

10

256

512

E(CHANNEL NUMBER) F i g . 1. Contour plot of two-dimensional E versus E t 2 spectrum for a TiO 2 target. The lowest contour

represents 0.25 count per channel. The vertical line through the elastic peak originates from accidental coincidences giving rise to incorrect time-of-flight signals. The line going diagonally from the origin to the elastic peak is caused by pile-up in the stop detector giving incorrect energy signals. Contributions to the tail in A = 16 are discussed in sect. 3 in the text.

262

B. F E R N A N D E Z

8 0 MeV 300

150

z z <{ "lu n¢ hi n

300

et al.

OLo b = 6 °

'60 . 160

~f --

le 0

,,

12C

z o u

150

20

22

24

26

F

[

28

30

F.ig. 2. Mass distributions at 0jah = 6°. The figure shows the projections from the raw two-dimensional E-versus-M spectra onto the mass axis.

and time resolution. The signal is split into a voltage-sensitive preamplifier (timing) and a charge-sensitive preamplifier (energy). Typical time resolution was 200 psec and the energy resolution was 200-300 keV. The angle and solid angle defining aperture is a 9 m m diameter hole in front of the solid-state detector placed 72 cm from the target. This corresponds to a solid angle of 0.1 msr and the large distance also results in a well defined scattering angle, AO ~ 0.05 °. The resulting energy and time signals were digitized and processed using an R C 4000 computer. Two-dimensional mass (A oc Et z) versus energy spectra were formed on-line. Fig. ! shows a contour plot of such a spectrum and in fig. 2 is given the projection onto tile mass axis of this spectrum. The projection covers the mass range A = 18 31 and the energy range E - - 0-75 MeV (to avoid the accidental coincidences). Examples of projections on the energy axis are shown in figs. 4 and 13. The resulting mass resolution is around 0.5 mass unit when no energy restriction is imposed and the mass distribution can be obtained from such projections. At the lowest bombarding energy the different mass yields are extracted from the two-

~60+ J60 CROSS SECTIONS

263

dimensional spectra. This is because the mass lines for the heavy evaporation residues at low energies ( ~ 5 MeV) bend over because the full energy is not recorded, an effect we ascribe to a reduced electric field in the surface of the solid-state detector. The total fusion cross sections are based on measurement from 4 ° to 30 ° in varying angular steps, whereas the excitation functions are obtained from spectra at an angle 0 = 7.5 ° up to 70 MeV and at 0 = 6.0 ° from 7(L80 MeV. The yields in the angular distributions were normalized to the elastic yield measured in a monitor detector at 0 = 27 °. The absolute cross sections were obtained from optical model fits to the simultaneously measured elastic yields. As a further check of the consistency of the data the fusion cross section for the t 6 0 + 160 reaction was remeasured at bombarding energies of 60 and 80 MeV with A120 3 targets. These were prepared by anodizing an AI foil and thereafter desolving the remaining AI in a solution of 1 O//oHCI in alcohol. The thickness was around 50 l~g/cm 2. With the A1203 target the elastic data at forward angles are more difficult to extract and the absolut normalization is therefore based on elastic scattering for O and A1 calculated with the optical parameters determined with the TiO 2 target. The errors in the fusion cross section are therefore not independent of the results from the TiO 2 target but with this in mind the fusion cross section and mass distributions are reproduced within the uncertainty, as shown in table 2 and fig. 7.

3. Experimental results

Angular distributions for masses with A __> 12 were measured from 0~ab = 4 ° to 30 ° for the 1 6 0 + ~60 reaction at bombarding energies of 35, 45, 52, 60, 70 and 80 MeV. Further, the excitation function for the fusion yield at 0ja b = 7.5 ° was obtained from 35 to 70 MeV in steps of 1 MeV and at 6 ° from 70 to 80 MeV in steps of 2 MeV. F r o m the 160 + t 2C reaction we have similar angular distributions at 45, 52, 70 and 80 MeV, whereas the data at 35 MeV are only for four angles. The excitation function covers in this case 60~70 MeV at 0 = 7.5 ° and 70~80 MeV at 0 = 6°. Fig. 3 depicts the angular distributions at the highest and lowest bombarding energies for the evaporation residues in units of b/rad which means that the area under the curves represents the total fusion cross section. The total yield of the masses 21 32 is included while for A = 20 only a non-direct part contributes. The way the division is done is discussed below. Energy and angular distributions of individual masses are characteristic of the process by which the evaporation residue is formed. This is illustrated in figs. 4 and 5. Fig. 5 shows the angular distributions of individual masses at a medium bombarding energy. The distributions are quite different whether three nucleons (A = 29), a mixture of one a-particle and four nucleons (A = 28), one a-particle and one nucleon (A = 27) or two a-particles (A = 24) are emitted.

264

B. F E R N A N D E Z et al. I

r

[

Total fusion 160° =16080 MeV Elab

45 I/]- f~'~ t

3

A=2"/ E[ob =45 MeV e=20" 200

\ 0

zz

|

2

~

0

~

/~* ~ \

-Q

/

'

E lab = 35

\

/

2

8

.1u

.

¢) "o 3

~

MeV

0

0=5"

o o

100

I

"~,,L

20 30 e lab ( deg )

40

0

Fig. 3. Angular distributions for the sum of evaporation residues. The lines are drawn to guide the eye and used for calculation of the area, the total fusion cross section.

20 E (MeV)

~

/~", \

A=29

500

\,

0

z7

~ ~soo

leo ~'

2

A =

,,

1000

"1o

\

J 1000

A=28 .,~ 150 "O "1o

100

\ 50

500

\L,/'\ ~e

0

10

20

40

Fig. 4. Energy spectra for A = 27 at different angles indicating the preference of emission of the ~-particle at forward or backward angles.

160-160 Ela b =52 MeV

" E

=10"

200

l-.Z

o\.

I

10

,

nr laJ Q.

"\ 0

,

t 10 30 e|Qb (deg)

' 20

"~ 0 30

Fig. 5. Angular distributions of individual evaporation residues.

~60+ ~'O CROSS SECTIONS

0 l5

Ecm = 22.5

265

\

MeV

Ecm=35 l v l e ~ l

\

8

I

0,2

tF

1 _ ~ , ~ ~ a ,

0.5

Eem=

26 MeV ""\ i,

Ecm=40 MeV~ /

0.2

O.t r

•

0.05 .

.

.

.

i 10

20

I ..... 30

'i

.

.

.

,

.

,

,

i

10

40

I

L_

20

30

40

(deg)

ecm

Fig. 6. Elastic scattering angular distributions. At E.,,. = 26 and 35 MeV two and three independent measurements, respectively, are shown with different symbols. The full drawn curves correspond to set 1 in table 1 for the optical model parameters. The lashed curve is calculated with parameters set 2. In the dotted curve at Ec.m. = 35 and 40 MeV the imaginary term is increased for the set 1 from 3.9 to 5.0 MeV at 35 MeV and from 4.4 to 5.6 MeV at Ec.m. = 40 MeV. At E~.m. = 30 MeV, 0.... = 15° was measured as the first and repeated as the last angle at this energy. The figure shows the agreement.

TABLE 1 Optical model parameters Set

V (MeV)

rx (fro)

a R (fm)

W (MeV)

r, (fm)

a, (fm)

Ref.

12+0.25 E¢.m. 17 7.5 +0.4 Ec.m.

1.35 1.35 1.35

0.49 0.49 0.45

0.4+0.1 Ec.m. 0.8+0.2 Ec.m. 0.4+0.125 Ec.m.

1.35 1.27 1.35

0.49 0.15 0.45

to) l~) 27)

The elastic scattering and reaction cross section for sets 1 and 2 are shown in figs. 6 and 7. Set 3 is used for the absolut normalization at 0,ab = 4 ° and 5° f'or the ~60+ ~2C reaction.

266

B. FERNANDEZ

et al.

The absolute cross section scale is determined from optical model fits to the elastic scattering data. For the ~60+ 160 reaction this scale is dependent on the optical parameters chosen, because at the very forward angles the 160 peak cannot be separated from the Ti peak. The elastic angular distributions are shown in fig. 6. Also shown are calculated cross sections using different optical model parameters. The full drawn curves are based on parameters from ref. 10) (set I in table 1). At 26 and 30 MeV cross sections calculated with parameters from ref. ~1) (set 2 in table 1) are shown as the dashed curves. The effect of increasing the imaginary potential in the former potential from 3.9 to 5.0 MeV at 35 MeV and from 4.4 to 5.6 MeV at E~.m. = 40 MeV is illustrated by the dotted curves for E .... = 35 and 40 MeV. It is seen that the potential from ref. 10) modified with an increased imaginary depth accounts well for the data, whereas the potential from ref. l a) does not. Attempts were made for fixed r R and a R to vary V around the value given in set 1 with the conclusion that the energy dependence needed to fit the present data is one very close to V = 12 + 0.25 Ec.m. MeV, as of ref. 10). In establishing the absolute cross section scale the angle definition is important. This is especially the case in the present experiment where the forward angle data are used to avoid as much as possible the dependence on optical model calculations. A systematic change in scattering angle of 0.05 ° is observed to change the qualities of fits for energy spectra as well as for the elastic scattering on Ti. We conclude that the absolute cross section scale is determined from elastic scattering with an accuracy of 2 O//o.E.g., the normalization constant obtained from fitting all the 12 data points at 52 MeV is determined with a standard deviation of 1.3 ')~;;. For the 160 + ~2C reaction these problems do not arise since the elastic yields can be extracted at 0~,b = 4 ° and 5 ° where the dependence on optical parameters is small. The tail in the beam mentioned above has to be taken into account to get the absolute cross section. The energy spectra of the evaporation residues are continuous and it is therefore impossible to distinguish between evaporation residues produced by particles in the energy tail of the beam and those of full energy. The tail in the A = 16 energy spectrum as shown in fig. 1 has been analyzed and it is found to contain, apart from real inelastic events, three major contributions" (i) A bump (around channel 450 in fig. 1) due to particles from the elastic peak loosing energy in the grid of the start detector. The intensity of the bump is 3 ~/o of the elastic peak, consistent with the 97 ~o transmission of the grid. (ii) A continuous tail from particles arriving at the target with energy less than the beam energy (mainly caused by the smallness of the slits). This part of the tail contained in some cases as much as 7 },~, of the intensity in the elastic peak. (iii) A continuous tail from particles not giving full energy signals in the stop detector (1 2 ~o of the elastic intensity). The correction in the fusion cross section has to be treated individually in the three cases and the

60 + J60 CROSS SECTIONS

267

total correction ranged from (2+ 1)~/~, to (8+3)~o for 35 and 80 MeV bombarding energy respectively. For the 160 + 1 6 0 reaction the yields must be corrected for the 12C contaminant in the target. As mentioned above most of the carbon is built up during the making of the target and very little during the experiment. The correction for carbon is made in two ways. Angular distributions are extracted for masses that have large cross sections in the 160 + 12C reaction and small in the 160 + 160. This procedure is only possible with complete measurements for the 1 6 0 + 12C reaction at the same bombarding energies. The carbon correction is also determined from the elastic carbon yield, but this procedure is often less accurate. The correction from carbon is generally smaller than 5 ~o, except for 80 MeV. This energy is chosen to illustrate the way the correction is done (fig. 2). Mass 22 is seen to have a large cross section for the 160 + 12C reaction and at this energy it is assumed that all mass 22 in the 1 6 0 + 160 measurement is from the carbon contaminant. The energy spectra and angular distributions for mass 22 are consistent with this interpretation. The disadvantage in this procedure is that small amounts of M = 22 from the 160+ 160 reaction itself cannot be determined. The correction for carbon was in this case (8.5 +3)Vo. The total fusion cross sections are given in table 2. For the 160 + 160 reaction we included masses with A > 20, and for A = 20 the part of the energy spectrum corresponding to Q > - 12 MeV (Eex = 9.40 MeV). This division is quided by the shape of the spectrum for A = 20. The mutual excitation of the 2 + (4.43 MeV) state in lzC and the 4.97 MeV state in 2°Ne is seen as a peak in the spectra. In fig. 13 is shown the spectrum at 01ab = 5° and angular distributions at 80 MeV. TABLE 2 Mass distributions and total fusion cross section for the ~60+ 160 reaction Elab (MeV)

35

A = 20 23 24 25 26 27 28 29 30 31 arus (mb)

45

52

60

70

80

14_+ 7

40_+10

96_+10

140_+14

158_+15

164_+10

352_+18

20_+ 7 366_+18

5+ 5 312_+15 30_+10 36+ 5 195-+10 10-+ 2

12-2- 6 379_+20 26-+ 5 106-+10 117-+10

86_+10 382+20 22-+ 7 145-t-15 65-I- 7

63-t- 8 340+_20 2_+ 2 197_+15 243_+15 23_+ 5 138_+10 25-+ 5

140_+14 252_+14 26_+10 278_+15 128_+10 50+ 5 104_+10 7-+ 2

233_+15 168_+12 87_+10 288_+15 70+10 79+10 64+ 6 ~2+ 1

765-t-35

1010_+40

1120_+40

1120+50 1115+_50 a)

1125_+55

") Numbers obtained with A120 3 targets.

1150+60 1125_+60 a)

268

B. F E R N A N D E Z

et al.

TABLE 3 Comparison of mass distributions at E . . . . = 30 MeV Ela b = 60 MeV

Kolata ") (rob)

A = 20 22 23 24 25 26 27 28 29 30 31

93.7± 6.6

Weidinger b) (mb) 93 ± 3.3± 7 ± 252 +

36.3± 2.5 323 ± 2 3 75 ± 5 168 ± 1 3 14.4± 1.6 112 ± 8 12 ± 2 16 ± 2 851

af~ (mb) a) Ref. 2).

Present (mb)

16 1.I 10 24

9 6 + 10 63+ 8 340 ± 20 2+ 2 197_+ 15 243 -t- 15 23+ 5 138_+10 25+ 5

219 ± 2 5 197 ± 9 25 +50 -25 161 ± 19

± 60

1017± 153

1120_+50

u) Ref. 1).

i

i

i

i

~

r

~60 , 160

1500

A

x3 1000

t=

111

"

v

500

I

I

I

t

I

I

30

40

50

60

70

80

E l a b (MeV)

Fig. 7. Excitation function for the total fusion cross section. The triangles represent full angular distributions with the error bars indicating the absolute magnitude of the uncertainty. The filled circles represent one-angle measurements with error bars only .showing the relative uncertainty. The curves are optical model reaction cross sections corresponding to the curves in fig. 6.

F o r the 160--1-12C reaction the fusion cross section includes A > 19. The direct reaction contribution in e.g. A = 20, is negligible in this case. The 3~ channel is not included for this reaction as it is for the 1 6 0 + 160 reaction. In fig. 7 the fusion yield is given at 01ab = 7.5 ° in steps o f 1 MeV up to 70 MeV, and at 01ab = 6 ° from 70 to 80 MeV in steps o f 2 MeV. The fusion yields are measured relative to the elastic Ti yield, and from a calculated elastic Ti cross section (parameters from ref. 12)) the relative cross section as function of energy is deduced.

i 1.5

f

I

I

'

'

I

'

'

I

'

, -

'

269

160 .,. n c

1.0

F v

"o

05

12

I

I

16

20

,

I

I

I

24

28

32

,

,

t

Ecru (MeV) Fig. 8. Excitation function for the fusion cross section for the 160+ ~zC reaction. The filled points and the curve are taken from refs. 4. ]3). In the present experiment complete angular distributions were obtained at E~,b = 45, 52, 70 and 80 MeV (large open triangles). The small open triangles represent one-angle measurements of fusion yields (0~ab = 7.5 ° from 60 to 70 MeV and 0~,b = 6° from 70 to 80 MeV). The cross sections given correspond to masses A = 19 22, and the 30~ channel is therefore not included.

160 "160 40 -

~

.

MASS .

.

.

DISTRIBUTION .

i

. E l. a b =. 80. I ~ V

-~500 -~300

20

I

II,

,oo

, , ,

40 -

7 0 MeV -:. so0

20 L

I,

40-20 --

, .-a-

,

tlt I I[ .

I

I

,,

I

_,

I

, .

40--

5 2 MeV

I

20 40--

-,oo

60 MeV

5oo

--,oo

3OO

-~ 5oo -- 300 I00

, I . I , 4 5 MeV

.1 40--

I

IT

I

A

v

b

-

too

35 i e V - 300

2O ~-

I F

I

20

'

I

22

tO0

24

26

28

30

A

Fig. 9. Mass distributions of evaporation residues from the ~60+ ~60 reaction. See also table 2 where errors are given. The thin bars at 45 and 70 MeV are calculated mass distributions.

270

B. F E R N A N D E Z et al.

;60 ÷ ;2C T

I

MASS DISTRIBUTION I

I

T ~

40

.I

I1.=.

1oo

40 20

40

70 MeV

.I

H

n

]

2O

40

z.o -

loo

60 MeV

40 -

20

300

,I.,,

20

2O

300

Etab = 80 MeV

20

_!

[~_

__ 52 MeV

],,I

l, I I!

20

~

22

,oo

E

300

Io

1oo

45 MeV

l. l,

24

300

26

-

300

100

35 MeV

300 IOO

I

28

30

A Fig. 10. Same as fig. 9 for the ~ ' O + ~2C reaction. The data at 60 MeV are taken from ref. ~). See also table 4.

The excitation function is then normalized to the cross sections determined from complete angular distributions. Therefore the assumption is that the yield at a particular angle relative to the total fusion yield changes slowly as a function of energy. The uncertainties in the data points are statistical only. The results for 1 6 0 + 12C of refs. 4.13) are shown in fig. 8 together with results from the present experiments. Mass distributions have also been extracted from the spectra and angular distribution, as illustrated in figs. 2 and 5. The results are given in table 3 and figs. 9, 10 and I 1. For the 160+ 12C reaction the mass distribution at 35 MeV has large uncertainties because it is based on few angles, and was used to get a n estimate on the carbon contamination for the 160-]-160 reaction. The mass distribution at 60 MeV for 160+ 12C is taken from ref. 1). In fig. 11, the excitation functions are shown for the individual masses. As described above for the total fusion cross section the yields for the different masses are normalized at the energies where complete angular distributions are measured.

'~'O+ ~'O CROSS SECTIONS

271 x • o I o

400

I,..

A=20 A=23 A=24 A =25 A =26

./

300

200

e~/""

/ ¢

{{1{11{{ i+

100

.,I0

E

,

I

i

I

,

I

,~ ,

+ I

i

'0

I A =27

400 z~z~~ ~ z~

~_

~

+

A=29

~

, =30

300

200

_;~--

-

±+:

100

=

I

40

=

I

L

50

I

,

6O

I

7O

L

I

80

Elob (MeV) Fig. 11. Excitation functions of individual evaporation residues from the J60+ ]~O reaction. The full drawn line represents the excitation function for 24Mg measured by Kolata et al. 2).

T h e u n c e r t a i n t i e s are m a i n l y statistical b u t also reflect the s e p a r a t i o n o f the m a s s regions in the t w o - d i m e n s i o n a l spectra. F r o m the s a m e d a t a we h a v e also e x t r a c t e d the cross section at 01ab = 7.5 for the 160(160, 12C)2°Ne r e a c t i o n to definite states in ~2C a n d Z°Ne. T h e results are s h o w n in fig. 12 which also shows p a r t o f the fusion cross section.

272

B. FERNANDEZ et al. 1

1500

160 .16 0

J 'i',i,,}i,,I,(,~Z~'~f,i,'~')

1000

E 1sO ( 'sO, 'zC) ZONe

b {

40 ~n 20

500

2¢)Zc ¢

o

~ 40 "10

{{{}

"10

I }

1{t

G.S. * 2*ZONe

t

{t}~{}

r

"1"

"1"

40

50

60

70

Etctb (MeV) Fig. 12. Excitation function for the (~60, 12C) reaction leading to the g.s. or 2 + state in 2°Ne (collected in one curve) and to the 2 + state in 12C(4.43 MeV) at 0tab ~ 7.5°. Also shown is part of the excitation function for the total fusion cross section and the calculated reaction cross section,

4. Discussion 4.1. THE

160+160

REACTION

4.1.1. Fusion and reaction cross sections. The results for the fusion cross section are shown in fig. 7 together with the calculated total reaction cross section. The fusion cross section is seen to increase up to E c.m. a r o u n d 26 MeV and then to stay almost constant in the energy region studied. The maximal value for the fusion cross section o f a r o u n d 1125 m b is s o m e w h a t larger than expected from the general systematics observed for reactions involving nuclei in the p-shell 4). In the region up to a r o u n d Ec.m. = 26 M e V the fusion cross section accounts for a b o u t 90 ~ o f the calculated reaction cross section. This is confirmed in an experiment by Rossner et al. t , ) w h o find O'quasie I ~- 110 m b for the transfer plus inelastic cross section. This value is averaged over energy (49 to 65 MeV) and a 1/sin 0. angular distribution is assumed to calculate the cross section. They find as we do in the present experiment that only the a-transfer channels and the inelastic channels give significant contributions to the total cross section, This is understood as a conse-

1 6 0 + =60 CROSS S E C T I O N S

160÷160

273

Eta b = 80 MeV A= 20 ~,.~

"~

(MeV)

0to b =5'

25

c:D

O

O

20

40

60

o 500

•

80

EA=z0(MeVI

0 < Ee=c< 20 MeV 20 <- Eexc~ 30 MeV Eexc >20 MeV

÷

2OO .JO

E 100

c~

"10

"10

50

20

10 ~

~ 10

20

30

Otab (deg) Fig. 13. Energy spectrum at 0~.b = 5° for A = 20 and angular distributions for different sections of the energy spectra. The shapes are indicative of the separation between direct and c o m p o u n d reactions.

quence of the large negative Q-values for the one- and two-nucleon transfer reactions. At Ec.m. = 26 MeV attempts were made to account for the total reaction cross section. The mass-12 yield is used instead of the mass-20 yield to determine the ~60(160, ~2C)2°Ne cross section in order to avoid to subtract the A = 20 contribution from the 160+ 12C fusion reaction. We find a,2c ~ 45 + 8 mb where the "less than" sign comes from the not specifically measured (but small) contribution from the Ti in the target. For the inelastic reaction the published angular distribution

274

B. F E R N A N D E Z

et al.

in ref. J4) at Ec.m. = 25.75 MeV, and 0 .... > 20° is used to continue our measured inelastic cross section at E~.m. = 26 MeV. We find al,e~ = 8 0 + 2 5 mb with the large uncertainties caused by the rather p o o r background conditions we have in the A = 16 spectrum. Therefore atotaI. . . . = O'fusqt-O'12C"~-ainel = ( 1 1 2 0 _ _ _ + 4 5 ) + ( 4 5 + 8 ) + ( 8 0 _ _ _ + 2 5 ) mb= 1245+55 mb [a R = 1230__+55 mb using the value from Rossner et al. 14) for the quasielastic cross section]. The calculated reaction cross section is 1230 mb at E~.m. = 26 MeV with the optical model parameters from ref. lo). These parameters describe well the elastic scattering as shown in fig. 6. The reaction cross section corresponding to the extremely surfacetransparent potential with r I = 1.27 fro, a~ = 0.15 fm [ref. 11)] gives a R = 1140 mb. This potential, however, does not describe the elastic data well, as also shown in fig. 6. We conclude that the measured total reaction cross section in the energy region Ec.m. = 25-30 MeV is accounted for by the calculated a R consistent with the elastic scattering data. At 80 MeV bombarding energy the one- and two-nucleon transfer cross sections are also small. As demonstrated in fig. 13 there seems to be a natural division in the mass 20 spectrum at Eex ~ 20 MeV. The yield for 0 < Eex < 20 MeV corresponds to a cross section of 44 m b which is ascribed to the transfer reaction 160(160, 20Ne)~2C. This cross section added to O'fus = 1150 mb gives 1195+60 mb, whereas aR = 1450 m b ( W = 4.4 MeV) or 1486 m b ( W = 5.6 MeV). That means that the remaining 250-300 m b is going into channels with masses 12 and 16 as final products. The present experiment cannot decide on the distribution on these two channels since the Ti( 160, 12C) reactions give a contribution to the mass-12 spectra. In conclusion at 80 MeV bombarding energy there is a significant cross section into the inelastic channel eventually followed by a-emission and three-body reactions (or both). 4.1.2. Comparison with other data. Weidinger et al. ~) have measured the fusion cross section at Ec.m. = 30 MeV in a time-of-flight experiment. They find af, s = 1000__+150 mb whereas the present experiment gives 1120+ 50 mb. The mass distributions are compared in table 3. The two sets of data agree fairly well with the exception of the cross section for A = 23. Weidinger et al. correct for carbon contamination of the target by assuming that all A = 25 is from carbon, whereas we claim (from the angular distribution of A = 25) that part o f A = 25 is from ~60. A carbon correction that is too large would decrease the cross section for A = 23 (figs. 9 and 10). K o l a t a et al. 2) have measured the 7-ray yield from the 160 + 160 reaction, between 12.5 and 30.5 MeV c.m. energy. The fusion cross section is determined from the sum of yields for the different final nuclei. At E¢.m. = 30 MeV they find 8 5 0 + 6 0 mb, a significant discrepancy from our value. By comparison of the excitation function for A = 24 with that of 24Mg (shown as a solid curve in fig. 11), a striking similarity in the variations with energy is seen. At E~.m. = 30 MeV the difference in absolute value is less than 2 ~o. The mass distributions are summarized in table 3 from which is

t60+t60

CR.OSS SECTIONS

275

seen that the 7-ray fusion cross section differs especially for A = 26. The mass distributions from a time-of-flight measurement are based on spectra as shown in fig. 2, and it is difficult to make mistakes in the relative cross sections whereas a 7-ray technique is dependent on a detailed knowledge on decay modes for the different final nuclei. The angular distribution for a final product with A = 26 is very different for the 160--~-12C and the 160-1-160 reactions, making the assignment of the present cross section unambiguous. TABLE 4 Mass distributions and fusion cross sections for the 160+ 12C reaction

Elab (MeV)

35

45

52

70

80

E,.m. (MeV)

15

19.3

22.3

30.0

34.3

208_+40

312_+30

349_+28

20_+ 4 223_+25

144-+40 120_+30 232_+50 96_+24

230-+22 69_+17 17_+ 6 217+22 23_+ 4

281 _+28 55_+19 59-+19 186_+19 9_+ 5

97+_25 261 _+25 29-+ 8 133_+16 51_+ 8

44_+ 8 161 + 16 15_+ 5 199-+ 16 192_+ 16 42-+ 8 134_+12 29_+ 3

800 _+ 100

865 _+45

940 _+50

815 _+50

815 _+50

A = 19 20 21 22 23 24 25 26 27 o'fu~ (m b)

The numbers for Ela b = 35 MeV are based on only four measured angles resulting in large uncertainties.

Recently, measurements of the fusion cross section for the 160+ 160 reaction have been performed at Argonne National Laboratory in the energy range Ela b = 30-60 MeV [ref. 29)]. A AE-E telescope with a thin AE detector is used in these experiments. The fusion cross sections do agree with the present data around E,, b = 45 MeV, but both at lower and higher energies the Argonne data give smaller cross sections. The discrepancy is at present not understood. 4.1.3. The nucleus-nucleus potential. The fusion cross section as function of energy can with certain assumptions be related to the nucleus-nucleus potential. Analyses of fusion data to establish such a relation have recently been made by Glas and Mosel ,s), Bass 16) and Schr6der and Huizenga 17). Fig. 14 shows the results of analysis of the present data following the prescription given by Glas and Mosel 15). The figure illustrates the information that can be extracted from fusion data within such a model and the role of the uncertainties. It is assumed that friction can be neglected, an assumption that is questionable for distances where the nuclear potential is large. It is further assumed that fusion occurs for angular momenta from zero to lmax. In the model we discuss in the next section this is not the case.

276

B. F E R N A N D E Z et al.

160.160

potentials

10-/

.."

;~'i// i/

.." _ ."

/;/ /

I

A

5

:E 0

-5

l I

I

I

L

7

9

10

R [frn] Fig. 14. The 160+ 160 nuclear+Coulomb potentials are shown together with two sets of values V(R) extracted from the data in fig. 7 using the expression aru ~ = nR2(l - V(R)/Ecm. ). The nuclear potentials are: the real part of the optical potential, set 2 of table 1 (dotted), the proximity potential ~8) (dashed and a potential calculated by Zint and Mosel ~9) (dot-dash). Also shown (filled circle) is a point calculated from systematics of elastic data (Christensen and Winther) 2o).

A general problem in the analysis of fusion data as referred to above is the experimental definition of the cross section. In most experiments where only the evaporation residues or their 7-decay are observed, preequilibrium emission of particles followed by fusion of the rest of the system cannot be distinguished from normal fusion. 4.1.4. Time-dependent Hartree-Fock calculation of the fusion cross section. A threedimensional T D H F calculation has been performed for the 160 + 160 reaction ~). The calculation is made with a single determinant and with symmetry restrictions corresponding to a picture of the scattering of two wave functions each describing four a-particle like objects. A further symmetry through the c.m. restricts the system to stay symmetric also after the collision. The total binding energy and the surface energy are fairly well reproduced.

~ 6 0 + ~ 6 0 C R O S S SECTIONS

277

For a given bombarding energy and impact parameter the reaction is defined as leading to fusion if the system sticks together for more than 2 × 10- 2 ~ sec. The results are shown in fig. 15, and it is seen that the calculations with no adjustable parameters reproduce the overall shape and magnitude of the measured fusion cross section. An interesting aspect of this calculation is that at sufficiently high energy (Ec.m. > 27 MeV) the two ions "bounce off" for the small impact parameters against an inner barrier of dynamical origin. E.g., at Ec.m. = 40 MeV the impact parameters corresponding to 0 < l < 10 should lead to inelastic scattering (in this model). In fact, the results of the T D H F calculations are consistent with a picture with a critical distance for both the high and low/-cuts (up to E~.m. = 40 MeV). The agreement with the experimental results is therefore dependent on the existence of the low/-cut. We shall comment on this question below.

160

÷

160

1500

\

,~

present data

11 M Conjeaud et cat 1000

E

b 500

\

,

I 2

,

I 4

,

[ 6

0

, 8

' x 10-2

1]Ecm (MeV-1) Fig. 15. As a function of I / g o . r e . w e show the fusion cross section measured and calculated by a T D H F method ~). The curve is the calculated reaction cross section.

4.1.5. Oscillations in the fusion cross section. In subsect. 4.1.2, we have pointed to the striking similarities in the excitation function for 24Mg from a 7-experiment 2) and for A = 24 from the present T O F experiment, fig. 11. We therefore conclude that the variations of the fusion cross section with energy are real. Fig. 7 shows that also the calculated reaction cross section exhibits oscillations that are seen to be in phase with the measured fusion cross section.

278

B. F E R N A N D E Z el al.

The calculations show that the oscillations observed in the (calculated) cross section are due to the contribution from the different partial waves. In this light system of identical particles the spacing between the energies at which the different (even) partial waves become grazing is large. Furthermore, the absorption suggested by the elastic data is small and one therefore sees the effect of each partial wave separately. The oscillations at e.g. 38, 45 and 52 MeV are in this way related to the angular momenta l = 14, 16 and 18h becoming the grazing angular momenta at these energies. Also the excitation function for the individual masses exhibits oscillations. The A = 24 has already been mentioned. In fig. 12 is shown the excitation function for some specific transitions in the direct (160, 12C) reaction. The very pronounced resonance around 52 MeV has been observed previously 21) and is seen to coincide with the onset of the / = 18h partial wave. We have attempted to calculate the direct cross section for the transitions in question as a function of bombarding energy. We have used the code D W U C K 22) and treated the s-transfer in a zero-range and norecoil approximation. We only observed a small effect in the calculated excitation function, but cannot exclude the failure to explain the resonance as due to the simplicity of the reaction model. It is further seen from fig. 11 that there might be an effect in A = 27 around 52 MeV. The yield for A = 28 is too small relative to A = 27 and 29 to extract an excitation function for A = 28. We conclude that the pronounced oscillations seen in the total fusion cross section, in the 2~ channel (A = 24) and in the direct s-transfer channel seem to be related to particular angular momenta in the entrance channel. In the energy region where oscillations are observed the highest spins in the compound nucleus are expected to decay by 2~ emission and the highest spins correspond to grazing partial waves. 4.1.6. Decay o f the compound nucleus. The mass distributions are shown in fig. 9 together with calculated mass distributions at 45 and 70 MeV. These calculations have been made with the evaporation code G R O G I 2, ref. 23), with parameters a = A/7.5 MeV 1, j = Jrigid with r 0 = 1.20 fm and 6 = 1.2 and 2.4 MeV for odd-A and doubly even nuclei, respectively. The initial compound population was calculated from transmission coefficients from the optical model and including angular momenta up to some maximal value as to correspond to the measured fusion cross section. No attempt to fit the measured mass distributions was made, but a fair agreement is obtained as demonstrated in fig. 9. In this mass region some data do exist on the level density parameter, but the data refer to low spin states. We have also used such parameters 24), but obtained a poorer agreement with the data. In particular the yields in the 3~ channel are much smaller than the measured values. For this channel the inclusion of the low-lying levels in the final nucleus would increase the calculated yields. We have not attempted such a procedure.

160+160 CROSS SECTIONS

279

The calculations show that at 80 MeV bombarding energy the first e-particle is emitted so as to leave 288i at an average Eex of 31 MeV and with average angular m o m e n t u m of 9h. This corresponds rather well to the c o m p o u n d system 28Si obtained from 160 + 12C at 35 MeV 160 bombarding energy. The difference merely being that the energy is sharp. We see in comparing figs. 9 and 10 that except for A = 25 the two mass distributions 1 6 0 + ~60 (80 MeV) and 1 6 0 + 12C (35 MeV) do look quite similar, indicating that a c o m p o u n d picture with statistical decay seems relevant for the decay of 28Si formed by 1 6 0 + ~2C (at fairly low bombarding energy) or by a-decay o f 32S at high excitation energy. 4.1.7. Low.l-cut. Test of TDHF prediction. The T D H F calculation predicts a low /-cut o f 10h at 80 MeV corresponding to 200 mb. We have tried to test this aspect o f the calculation. In fig. 16 is shown schematically the population and decay o f the c o m p o u n d nucleus 32S as formed at 80 MeV bombarding energy. The statistical calculation shows that 5

10 I

15

20 [

25 r

160.~60 _ _ 3zS E,ob =80

._e-

\

',

M,v

Ofus = 1150 m b

_.-~J'~

\

\ 60-

/.

/..f'" .~f ~ . . . . . - . ~ ' ~ . i ...........

_ - z ~ ' -2 . . . .

_

,i

\ ..............

~. ....

L

// "

// /

i :

/

,

/

......... ! ......... . / . ".z i / '..

•, ' .."

z

//

',..

""

1....

LU

]

/

/

~"

ii

/ /i //.,." tl •

/ ii

/"

//

20

ZSSi

".. " ~,j"//// ....... , . ~1• ""

iI

I

/ ' ..,". a~p ..,

11 ..' ....-

,.,

."

0.I ,

...... ::::-

%

J

1o

20

25

i

l (h)

Fig. 16. The figure schematicallyshows the initial population and the proton (dotted) and ~ (dashed) decay of the compound nucleus a2S. The effect of a low/-cut at l = 10h is given by the dashed-dotted curve. Also shown are contour lines in the daughter nuclei 3~p and ZsSi where the population cross section is half the maximum value.

280

B. F E R N A N D E Z et al.

the high angular m o m e n t a decay by a-emission from an average spin /~(32S) of 18h ending up in 28Si at Eex ~ 31 MeV and Lssi ~ 9h. Protons are emitted from the low spin states with an average [p(32S) ~ 10h ending up in 31p at Ee, ~ 38 MeV and L,p ~ 9h. The ratio between a, p and neutron decay is calculated to be 808 : 255 : 87, corresponding to the sum afu s -- 1150 rob. The figure shows that a low cut of 10h in the spin distribution would have a major effect on the ratio between a and nucleon emission. In the experiment the masses 2% 30 and 31 must be the result of nucleon emission in the first step of the particle emission. Also A = 28 is above 60 MeV bombarding energy the result of four-nucleon emission. This is quite obvious from the angular distribution of A = 28. In fig. 17 we show the cross section for masses 28-31 as function of energy. F o r A = 28 the la emission is excluded a t E~a b = 45 and 52 MeV. This division of A = 28 will only m a k e a small contribution to the error on the cross section for the sum of the masses. This sum can be calculated as the cross section for emitting a nucleon in the first decay minus the cross section for emitting a-particles in the further decay. The calculations show, however, that this latter cross section is very small and uncertainties in this correction do not influence strongly the calculated yield of masses 28-31.

300

JD

--

standard param

(~A = 28, 29, 30

200

E

a (z$sJ)adjusted

'""-,o..

•''""-..

100

0

I

~0

,

I

50

,

I

60

,

low l- cut "''"'.....

I

70

80

E lab

Fig. 17. The measured cross section for the masses 28 31 (for A = 28 only the four-nucleon part) together with calculated cross sections with different parameters for the first step of the decay of 32S. The dotted curve is calculated with the adjusted a(28Si) value and with a low/-cut as given by the T D H F calculation of ref. 7).

160+ 160 CROSS SECTIONS

281

The results of such calculations are shown in fig. 17. The standard values as given above for a, 6 and moment of inertia lead to a calculated cross section more than 50 ~o too high. The ratio between nucleon and ~ emission was also calculated with parameters corresponding to a liquid drop model. It is assumed that at sufficient excitation energy (above the yrast line) a liquid drop description.is adequate. The effective excitation energy relevant for calculation of the level donsity is then obtained by subtracting pair and shell corrections. These corrections are taken as the difference between the experimental mass and the spherical liquid drop mass 25). Such a procedure has recently been discussed by P/ihlkofer 26). The nucleon-~ ratio calculated in this picture gives much poorer agreement with the data than the standard parameters shown in fig. 17. We do not conclude from this that the picture is not relevant as other effects could be important, e.g. a difference in deformation for the two daughter nuclei. This could have a significant effect on the level density. We have next tried to change the a-value for 28Si as to fit the experimental value for 0"28_31 at 45 MeV bombarding energy and calculate 0"28-31 a t the higher energies. The change in a for 2Ssi (from the standard value a = A/7.5) is as small as 10 % (increase) to get a 50 70 change in 0"28-31. As the last step of the calculation a cut in spin is introduced as predicted by the T D H F calculations. It should be pointed out that in this procedure we are not dependent (or at least only indirectly through the nucleon-~ correction) on details for levels in the final nuclei but only on the constancy of ratios between a~p : a2~si for the density of states 10 to 25 MeV above the yrast line. It is seen from fig. 16 that a low/-cut at E~.m. = 40 MeV of 10h decreases the nucleon emission by more than a factor of 2. This is also illustrated in fig. 17 in the decrease of the calculated cross section for masses 28-31. Very little is, however, known about the level density far from the yrast line which makes it difficult alone from the yield of masses 28 31 to make firm conclusions on the existence of a low /-cut. Another question is the importance of non-statistical processes. A pre-equilibrium emission of a nucleon (from the low non-fusing partial waves) followed by the fusion of the rest of the system would lead to a mass distribution similar to that from a normal fusion reaction. Such a nucleon emission channel is not included in the T D H F calculations of ref. 7). 4.2. THE t60+12C REACTION In fig. 8 is shown the total fusion cross section for the 160 + 12C reaction together with data from refs. 4, 13). We observe a good agreement with the measurements at 45 and 52 MeV bombarding but do not observe the decrease in cross section at 63 MeV (E~.m. ~ 27 MeV). It can be concluded that the fairly large oscillations observed at lower energy do not seem to be present at higher energies.

282

B. F E R N A N D E Z et al. TABLE 5

Cross sections for different channels for the ~ 6 0 + ~'O reaction together with calculated total reaction cross sections

Ela b

O'fus (mb)

O'( 160, 12C) (mb)

O.inel (mb)

52

1120+45

45_+8

80+25

80

1150_+60

44+5

(MeV)

O_R(caIc ) (rnb) 1230 1450 ( W = 4.4 MeV) 1486 ( W = 5.6 MeV)

The measured mass distributions are shown in fig. 10. The figure demonstrates especially when compared with the corresponding mass distribution for 160 + 1 6 0 that the emission of 3~ particles could be a strong channel. It is, however, difficult experimentally to distinguish between the 160 evaporation residues and 160 ions from other reactions. So from 60 to 80 MeV bombarding energy we do not expect to have the full fusion cross section as given in fig. 8. It has not been attempted to calculate the 3~ contribution with the rather limited success for the similar problem in the ' 6 0 + 160 reaction.

5. Summary In the present study accurate measurements of the fusion cross section and mass distributions for the ' 6 0 + 160 reaction have been made at energies 35, 45, 52, 60, 70 and 80 MeV with a time-of-flight technique. Excitation functions for the total fusion and for the individual masses have been deduced from one-angle measurements of mass yields as function of energy by normalizing at the energies mentioned above. The ' 6 0 + ~2C reaction has been studied by the same method at energies 45, 52, 70 and 80 MeV, and fusion cross sections and mass distributions were determined. For the 1 6 0 + 160 reaction we find small amplitude oscillations in the fusion cross section. These variations seem closely related to similar oscillations in the total reaction cross section. This latter quantity is calculated with optical model parameter that gives excellent fits to the measured elastic yields. In the calculated a Rwe understand the oscillations as effects of the individual (even) angular momenta. We only with certainty observe oscillations in the 2~ channel and the (160, 12C) reaction and in both cases this can be explained as an effect of the grazing angular momenta. At Ela b : 52 MeV, the total reaction cross section is accounted for within the accuracy (5 %) of the measurement. We have at two energies calculated the mass distribution in a statistical model. We have not tried to fit the measured distributions but standard parameters reproduce the general trends of the data.

160+ 160 CROSS SECTIONS

283

W e h a v e f u r t h e r tried to test the p r e d i c t i o n o f T D H F as to a p o s s i b l e l o w c u t in t h e a n g u l a r m o m e n t a l e a d i n g to f u s i o n . W e are, h o w e v e r , t o o d e p e n d e n t o n p a r a m eters in the statistical m o d e l to m a k e f i r m s t a t e m e n t s o n such a c u t in I. T h e a n a l y s i s p o i n t s to a s i g n i f i c a n t l o w e r c u t t h a n p r e d i c t e d b u t we h a v e p o i n t e d to t h e p o s s i b i l i t y o f this low c u t n o t s h o w i n g u p in the m a s s d i s t r i b u t i o n s for the h e a v y e v a p o r a t i o n residues because of preequilibrium nucleons. T h e a u t h o r s are g r a t e f u l to Dr. S. K o o n i n for s t i m u l a t i n g d i s c u s s i o n s . O n e o f us (B.F.) w a n t s to t h a n k the N i e l s B o h r I n s t i t u t e for the h o s p i t a l i t y received d u r i n g his stay a n d a c k n o w l e d g e s the s u p p o r t f r o m the D a n i s h N a t u r a l Science R e s e a r c h C o u n c i l as well as the C o m m e m o r a t i v e A s s o c i a t i o n o f the J a p a n W o r l d E x p o s i t i o n . O n e o f us (S.P.) g r a t e f u l l y a c k n o w l e d g e s the f i n a n c i a l s u p p o r t f r o m the D a n i s h

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28) M. Conjeaud, S. Harar, F. Saint-Leurent, J. M. Loiseaux, J. Menet and J. B. Viano, Proc. Int. Conf. on nuclear structure, Tokyo 1977, p. 663; S. Harar, private communication 29) J. P. Schiffer, D. F. Geesaman, W. Henning, D. G. Kovar, K. E. Rehm, S. L. Tabor, F. W. Prosser, J. V. Maher and W. Jordan, Bull. Am. Phys. Soc. 22 (1977) 1019, and to be published; J. P. Schiffer, private communication