Alpha-particle strengths from the 16O(6Li, d)20Ne reaction

I2.B: 2.N i p

I

NuclearPhysics A313 (1979) 445-466; (~) North-HollandPublishing Co., Amsterdam Not to be reproducedby photoprintor microfilmwithout writtenpermissionfrom the publisher

ALPHA-PARTICLE STRENGTHS F R O M T H E 160(6Li, d)Z°Ne R E A C T I O N N. ANANTARAMAN, H. E. GOVE, R. A. LINDGREN t, j. TOKE tt and J. P. TRENTELMAN Nuclear Structure Research Laboratory, University of Rochester, Rochester, New York 14627 tit

J. P. DRAAYER tt* Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803

and F. C. JUNDT and G. GUILLAUME LPNIN Centre de R/cherches Nucl/aires, Strasbour#, France

Received 22 May 1978 (Revised 1 August 1978) A~tract: The 160(6Li, d)2°Ne reaction has been studied at bombarding energies of 20, 32 and 38 MeV. The a-particle spectroscopic strengths have been extracted for levels up to 12.15 MeV in excitation. Nondirect processes appear to contribute significantly to all levels at 20 MeV and to high spin levels (6 + and 8 +) at 32 MeV. Strengths extracted for members of the ground st~ite band assuming (sd)4 transfer are unequal at both 32 and 38 MeV, in marked contrast to theoretical predictions. To explain this, particle-hole correlations in ~60(g.s.), inelastic channel coupling in the reaction and perhaps other effects as well, have to be considered. Strengths extracted for members of excited bands and a-decay reduced widths compare poorly with each other and with simple SU(3) predictions.

E

NUCLEAR REACTIONS 160(6Li, d), E = 20, 32, 38 MeV; measured tr(Ed, 0). 2°Ne levels deduced ct spectroscopic strengths. DWBA analysis. Natural target.

I. Introduction

T h i s p a p e r deals w i t h a s t u d y o f the 160(6Li, d ) 2 ° N e r e a c t i o n at b o m b a r d i n g energies high a b o v e the C o u l o m b b a r r i e r , w h e r e the r e a c t i o n is expected to be p r e d o m i n a n t l y direct. T h e or-particle s p e c t r o s c o p i c strengths e x t r a c t e d f r o m this s t u d y are c o m p a r e d w i t h t h e o r e t i c a l p r e d i c t i o n s . T h e s t r u c t u r e o f levels in the g r o u n d state b a n d o f 2°Ne is well k n o w n 1,2). T h e y are f o u n d to b e n e a r l y p u r e eigenstates w h e n l a b e l e d a c c o r d i n g t o the S U ( 3 ) - S U ( 4 ) t Present address: Department of Physics and Astronomy, University of Massachusetts, Amherst, MA 10002. ** Present address: Institute of Experimental Physics, Warsaw University, Warsaw, Poland. tit Supported by a grant from the National Science Foundation. 445

446

N. A N A N T A R A M A N

et al.

classification scheme. Using these wave functions, theoretical calculations of a-particle strengths for transitions populating members of the ground state band have been made 3.4). Predictions of strength based on the simplest SU(3) approximation differ but little from those made with much more complicated shell model wave functions for 2°Ne. The dominant components in the configurations of the excited bands in 2°Ne are also known 5). But the prediction of c~-particle strengths for these levels usually proves to be more of a challenge for one of two reasons: either particle-hole excitations are involved, which then require that attention be paid to the spurious c.m. problem and related questions; or seemingly small symmetry admixtures have a greater effect for excited bands than for the ground state band. For example, the simplest sd shell model calculation is inadequate to explain the a-decay width of the 6.72 MeV level of 2°Ne [ref. 6)]. Also, analyses of a-decay data give evidence for a shrinkage of the rms radius with increasing spin within each band in 2°Ne [refs. 7, 8)]. To reproduce this effect theoretically, Tomoda and Arima found it necessary to carry out an extended calculation involving excitations up to 19hco [refs. 9.1o)]. The structure of 160 has also been well studied 11). Of importance here is the ground state, in which there is strong evidence for both two-particle two-hole (2p-2h) and 4p-4h components ~2-16). These components can alter the a-particle strength predictions. This has been probed 17) and the results suggest that inclusion of the particle-hole components in 160 is important in, for example, an analysis of the a-transfer data for the 2°Ne ground state band. The present study was motivated by the expectation that a comparison of the extracted c~-particle strengths with predictions would throw additional light on some of the above aspects of the structures of Z°Ne and 160. Many experiments involving the transfer of an c~-particle onto 160 have been reported in the literature, and quantitative spectroscopic strengths have been extracted for the levels of the 2°Ne ground state band 18-21). A new feature in our analysis of the data is the inclusion of the effect of particle-hole (p-h) admixtures in the ground state of 160. Very little quantitative information exists on the c~-particle strengths to members of the excited bands. The a-decay widths for levels unbound to c~-particle emission are, however, available 7, 8). Therefore, a comparison between the decay widths and the transfer strengths'measured in the (6Li, d) reaction is also made in this paper.

2. Experimental procedure The 160(6Li, d)2°Ne reaction was studied at three different bombarding energies at the University of Rochester MP tandem accelerator using a 6Li beam of 300-500 nA. The most complete analysis was performed on data taken at an effective beam energy of 32.0 MeV. Angular distributions for most of the states populated up to 12.15 MeV excitation were obtained in the range 5°-60 °, in 2.5 ° steps between 5° and 20 ° (where L = 0 cross sections change very rapidly with angle) and

160(6Li, d)2°Ne

447

in 5° steps beyond that. The measurements at beam energies of 20.0 and 38.0 MeV were restricted to obtaining the angular distributions for the 0 +, 2 + and 4 ÷ members of the ground state band of 2°Ne. Both solid and gas targets have been used in these studies. The solid target was a self-supporting foil of Ta20 5 about 100 #g/cm 2 thick, prepared in Strasbourg by anodizing a thin evaporated layer of Ta using water as the electrolyte. In the gas cell 22), natural oxygen at a pressure of about 6.0 cm of Hg was used. The energy loss (300 keV) of 6Li ions in the entrance window of the gas cell and in the layer of gas preceding the active region was compensated for by suitably increasing the bombarding energy. The outgoing deuterons were momentum analyzed in an Enge split-pole spectrograph and detected in a focal plane detector. In the runs at 20 and 38 MeV, the detector was a sonic spark counter 23), while in the runs at 32 MeV, both the spark counter and Kodak NTB-50 emulsion plates were used. Fig. 1 shows a spectrum measured at 30° with the spark counter. Peaks were identified by comparing measured energies with tabulated energy levels 24). The resolution obtained in this particular case was about 55 keV and ranged between 50 and 75 keV in the series of runs. It was primarily target-thickness limited. Absolute cross sections were determined in the case of the solid target by normalizing the reaction data to the elastic scattering from Ta, which was pure 512

• 4+ 425

=60(6Li,d)=°Ne Esu = 3 2 MeV

d

9 = 30 ° 6"

5" !6

8.~8

7.17

~256 6+

(~

2÷

128

..

0

256

512

1°.

768

.

1.

I024

CHANNEL Fig. 1. Spectrum of deuterons from the 160(6Li, d)2°Ne reaction. The excitation energies of the levels in 2°Ne are indicated in MeV.

448

N. A N A N T A R A M A N

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TABLE 1

Differential cross sections (#b/s0 for the ground state band of 2°Ne measured in the 160(6Li, d) reaction at 20, 32 and 38 MeV ELi = 20 MeV

0lab (deg) 4 5 7.5 10 12.5 15 17.5 20 24 30 35 40 45 50 60

ELi = 38 MeV

ELi = 32 MeV

0+

2+

4+

250 211 110

312 380 422

395 400 404

80 96 110 114 56 16 16 16 9

497 485 440 322 263 358 315 202 109

419 410 455 480 520 494 378 355 380

0+

2÷

4+

554

656

248

219 128 70 51 30 30 18 17

550 569 546 433 258 146 69 82

203 246 341 405 406 478 472 275

31

82

131

9 21

90 38

66 139

0+

2+

4+

178 114 47 27 47 57 50 37 17 35 34

603 606 533 436 362 161 70 77 147 124 101 101 80

265 345 422 482 580 677 637 509 255 171 154 235 322

The error irt these absolute cross sections is + 30 %.

Rutherford scattering at forward angles at 20 MeV. This involved the assumption that the tantalum in the target was fully anodized. In the case of the gas target, the cross sections were determined from a knowledge of the pressure of gas and the thickness of the active layer. The two measurements gave values that agreed with each other to within __+5 %. Thus the accuracy claimed for the absolute cross sections, _ 30 %, is perhaps a conservative estimate. Table 1 lists the (6Li, d) differential cross sections for the transitions to the 0 ÷, 2 + and 4 + members of the ground state band of 2°Ne measured at bombarding energies of 20, 32 and 38 MeV. Recently, t60(6Li, d) excitation functions in the lab energy range from 19.8 to 32 MeV for these three levels have been measured at forward angles by House and Kemper 2s). Our 20 and 32 MeV data agree with theirs to within the quoted errors. 3. Results Angular distributions were obtained at 32 MeV bombarding energy for transitions to all but two of the 2°Ne levels labeled in fig. 1 and are displayed in figs. 2-6. Figs. 7 and 8 show angular distributions obtained at 20 and 38 MeV, respectively. They are grouped according to rotational bands in 2°Ne, the dominant shell-model configurations s) and SU(3) classifications 8) of which are given in table 2. Assuming a closed-shell structure for 160, the levels which are expected to be selectively populated are those belonging to the positive parity (sd) 4, 0,#) = (80) ground state

t 60(6Li ' d)2ONe

5o \ i

;oo

i

i

+\

i

i

O* 0.0 MeV

# IO00

449

#~

500 -('-'~'e •

('

-Q

"0

50( ~

100 I

2oo I

*

100

2oo!

8+ 11.95MeV

=ool 50 I0'

5O '

4'0

5O '

6O '

Oc.rn.(deg) Fig. 2. Angular distributions for transitions to members of the ground state band in 2°Ne at 32 MeV. The solid and dashed curves are D W B A fits.

band and to the negative parity (2p) = (90) band through its (sd)a(fp) component. Fig. 1 shows that such is indeed the case. It is evident from figs. 2-8 that the shapes of the angular distributions for the natural-parity levels are characteristic of the transferred orbital angular momentum (L) and change little with excitation energy or bombarding energy. This is indicative of a direct transfer process. Specifically, the analysis assumes the validity of an approximation wherein a cluster of mass 4 and charge 2, with quantum numbers that are determined by those of its constituent nucleons, is transferred unaltered from projectile to target. Angular momentum and parity selection rules do not permit the population of unnatural parity levels in 2°Ne by a one-step cluster transfer process. Thus the non-

450

N. A N A N T A R A M A N

et al.

i

50

160(6Li ' d)ZONe

2O

,•

•

•

• 2-

4.97• MeV

I0

I00: ..o

t •

• •

50 - ~ ~ . .

•

3-

MeV

5.62

::k

v

5(

b

~

20

4- ZOIMe~

I,,~'._+_v,,

50l

" ' °

t'~'~'t----.-., 5- 8.45 MeV

+' ~*. ',

2oi

I0

20

30

40

50

60

70

8c.m.(deg) Fig. 3. Angular distributions for transitions to m e m b e r s of the (sd)Sp - 1 band in Z°Ne at 32 MeV. The solid curves are D W B A fits for the natural-parity levels.

i

I000

i

5OO "~.,,

f

i

i

i

i

WSO(SLi,d)Z°N e

_

\

2OO

"~

I- 5.79 MeV

...t-,,-,--,,-,~,• •

3- ZI7 MeV

-

I00 5O

iooo ..Q

::k 50O

v

2OO I00

iood

•

5OO 2OO I00

--, . . . .

"\,

5O I0

20

30

40

,50

60

70

Oc.m.(deg) Fig. 4. Angular distributions for transitions to members of the (90) band in 2°Ne at 32 MeV. The solid curves are D W B A fits.

160(6Li, d)2°Ne

451

|

Ioo ,,~

iSo(6Li,d)2ONe

5O

0 + 6.72MeV

2O "C I0

:L 5 v

"X3 5O

{ --*T~""~ik~

2 + 7.42MeV

' --R~._~-- L ,b 2'0 30 4'0 5'0 6'0 8c m(deg) Fig. 5. Angular distributions for transitions to members of the excited (sd) 4 band in 2°Ne at 32 MeV. The solid curves are DWBA fits.

I

A

~- 200

v

--½ e ' ~

::k I00

~

i

i

I

i60(6Li, d

t't"~"t ~

i

ZONe'

6 + 12.15 MeV

50

I0 '

20 '

30 '

;o

50 '

ec.m.(deg) Fig. 6. Angular distribution for the transition to the 6 + (12.15 MeV) level in 2°Ne at 32 MeV. The solid curve is a DWBA fit.

vanishing cross sections measured for the excitation of the 2-(4.97 MeV) and 4-(7.01 MeV) levels (fig. 3) indicate significant contributions from mechanisms other than one-step a-particle transfer at 32 MeV. This has been discussed in an earlier paper 26). The experimental resolution was such that angular distributions for only the strongly populated levels up to about 12 MeV in excitation could be obtained. All but three of the levels, those at 7.17 MeV (3-), 8.78 MeV (6 +) and 11.95 MeV (8÷), were isolated from nearby levels. Near the 3- level is a 0 ÷ level 24), which however is expected 27) to be weakly populated in the ~-particle transfer reaction. The absence of forward peaking in the measured angular distribution indicates that this is so. Similarly, the lack of forward peaking in the 6 ÷ and 8 ÷ angular distributions suggests that the nearby 1 - (8.72 MeV) and 1 - (11.95 MeV) levels contribute little

452

N. A N A N T A R A M A N eta/.

160 (6Li,d) 2°Ne

ELi=20Me 0+ 0.0 MeV]

200

oo .~,

50

~",,

t-~

::L 2°

~{

" .

v

~ ~oo .-"'~'~....~. 200 :.~

~

•

5oo~ _ . . ' 2 " ~ . - - :

-*-,., ..-.

20

,.~-~ • •

4 4.25 Me~

oF-

I0 20 30 40 50 60 Oc.m.

Fig. 7. Angular distributions for transitions to members of the ground state band in 2°Ne at 20 MeV, The solid curves are DWBA fits and the dashed curves are the results of compound nuclear calculations

2oo~...'

' '6o/~,'i.dl~°Ne'

,o° I- ', so ~

~,!

~ooL~'~.. ] 200 I,oo

E.,-~8,.,e, ~* ,

( 7 o.o M~

~

T \ . \

5OO2oO "olo~ ~ """.

2 +,.63 M~V

.

".,

",",~ .4+425, ,, ., MeV"

IO 2o 30 40 50 60 Fig. 8. Angular distributions for transitions to members o f the ground state band in 2°Ne at 38 MeV The solid and dashed curves are DWBA fits.

160(6Li, d)Z°Ne

453

TABLE 2 Classification of levels in Z°Ne populated in the 160(6Li, d) reaction at 32 MeV E x (MeV)

J"

Configuration

(2#)K

[ 2 ( N - 1) + L]

0.0

0+

(sd)*

(80)0

8

1.63

2+

4.25 8.78 11.95

4+ 6÷ 8+

4.97 5.62 7.01

234-

(sd) Sp- 1

(82)2

... 7 ...

8.45

5-

5.79 7.17 10.26

135-

(sd)Sp- 1, (sd)3(fp)

(90)0

9

6.72 7.42

0÷ 2+

(sd)4exc

(42)0

6

12.15 "

6÷

(sd)Sp- 4

(88)0

8

7

to these two cases. There is however a possible contribution from the 4 + (11.93 MeV) level to the 8 + (11.95 MeV) angular distribution. 3.1, D W B A C A L C U L A T I O N S

A finite-range distorted wave Born approximation (DWBA) analysis of the angular distributions for the natural-parity levels was done with the code LOLA 28); the procedure was the same as that described elsewhere 29, 30) for (6Li, d) reactions on other targets. The optical model and bound-state parameters, listed in table 3, were taken from refs. 31, 32). The number of radial nodes for the bound state of the ~-particle in 2°Ne was fixed by the harmonic oscillator energy conservation relation 4

2(N-1)+L = ~ [2(n,-1)+/,]. i=1 TABLE 3 Optical model parameters used in the D W B A analysis of the reaction t60(6Li, d) at ELi = 20, 32 and 38 MeV Channel

V (MeV)

RR (fro)

aR (fm)

W (MeV)

6Li + 160 d+Z°Ne bound state in 2°Ne

72.6 93.5 a)

3.45 2.84 3.28

0.87 0.81 0.65

8.0

W' = 4W D (MeV)

RI (fm)

al (fm)

Rcoul (fro)

41.2

5.80 3.76

0.81 0.71

6.30 3.39 5.14

") Adjusted to give the correct binding energy as determined by the separation energy procedure.

454

N. A N A N T A R A M A N et al.

The values of 2 ( N - 1)+ L obtained from this relation are listed in the last column of table 2. F o r 160(g.s.) a closed core, the population of the (sd) 4 and ( s d ) 3 ( f p ) configurations in Z°Ne involve 2 ( N - 1 ) + L = 8 and 9, respectively, while the (sd) 5p- 1 configuration cannot be populated. The population of this last configuration via the (sd)2p -2 component in 160(g.s.) leads to 2 ( N - 1 ) + L = 7. The results of the DWBA calculations are shown superimposed (solid curves) on the data in figs. 2-8. The relative spectroscopic strengths, defined as (da/dfZ)exp/(dff/dfZ)owB A and normalized to unity for the transition to the ground state of 2°Ne, are given in table 4. Levels above 4.73 MeV excitation in 2°Ne are unbound to a-particle emission. The problem of calculating the "bound-state" wave functions for these levels was solved by artificially binding the a-particle at two different binding energies in 2°Ne (while using the experimental Q-value of the reaction). The cross sections thus calculated were extrapolated to the positive (unbound) energy side on a semilogarithmic plot of cross section versu~ binding energy. The third column in table 4 gives the factors by which spectroscopic strengths obtained using cross sections calculated with 0.1 MeV or-particle binding energy are to be multiplied in order to get the corrected strengths. The strengths given in the table are the corrected ones. To obtain an idea of the validity of this correction procedure, a more sophisticated procedure involving calculation of a quasi-bound state wave function 33) was used for the transitions to the 5.62 and 5.79 MeV levels. The strengths obtained thereby were respectively 4 ~ less and 9 ~ more than those given in table 4 for the two transitions. As was the case with (6Li, d) angular distributions for 4 + levels in other sd shell nuclei 30, 34), it was found that a good fit to the shape of the distribution for the TABLE 4 Relative ~t-particle spectroscopic strengths from the ~60(6Li, d)Z°Ne reaction at 32 MeV bombarding energy

E x (MeV)

J"

0.0 1.63 4.25 8.78 11.95

0+ 2+ 4+ 6+ 8+

5.62 8.45 5.79 7.17 10.26

B.E. correction factor

S(6Li, d)

Sth[SU(3)]

0.49 0.34

1.00 0.41 0.22 0.20 0.51

1.00 1.00 1.00 1.00 1.00

35-

0.82 0.55

0.06 0.04

0.00 0.00

1 3 5

0.94 0.68 0.41

0.54 0.26 0.15

1.50 1.50 1.50

6.72 7.42

0+ 2+

1.07 0.85

0.56 0.13

0.00 0.00

12.15

6+

0.25

0.05

0.00

160(6Li, d):°Ne

455

4 + (4.25 MeV) level of 2°Ne could not be obtained with the parameters that gave good fits for smaller L-values. A calculation with a bound-state radius of 2.52 fm provided an improved fit at larger angles, as the dashed curve in fig. 2 shows. But no further use of this will be made in this paper. The strengths given in table 4 are those obtained with a constant bound-state radius of 3.28 fm for all the transitions. When the shape of the calculated angular distribution disagrees with the data, there is the problem of choosing a way to normalize the two in order to extract a spectroscopic strength. This problem arises in connection with the angular distributions for the 4 ÷, 6 + and 8 + levels in fig. 2, the 4 ÷ level in figs. 7 and 8 and perhaps the 0 ÷ level in fig. 8, and there is no unique method for dealing with it. The prescription that has been followed here is to normalize the maximum in the experimental distribution to that in the calculated one (even when these two maxima are separated in angle by several degrees), provided the calculated shape gives a tolerable fit to the experimental shape over at least a limited range of angles. This leads to the normalizations shown in the figures for all the cases mentioned above except the 6 ÷ level in fig. 2 and the 4 + in fig. 7. For the 6 + level, lining up the maxima in the measured and calculated curves would have resulted in a gross misfit at all angles; so, instead, the calculated curve was normalized such that it fitted the measured one over as large an angular range as possible. For the 4 + level in fig. 7, in view of the total lack of similarity in the shapes of the (finite-range) DWBA curve and the data, no normalization was attempted and so no spectroscopic strength was extracted. The ambiguity in the fitting procedure is estimated to lead to an uncertainty of about + 25 ~ in the strengths quoted for the 4 ÷ and 8 ÷ levels at 32 MeV and for the 0 ÷ and 4 ÷ levels at 38 MeV. For the 6 ÷ level at 32 MeV, the uncertainty is about _ 50 ~ , and results in an extracted strength of 0.20 + 0.10. 3.2. RELATIVE SPECTROSCOPIC STRENGTHS IN THE GROUND STATE BAND

Table 4 compares the empirical strengths extracted from the 32 MeV data with theoretical predictions based on the simplest SU(3) model. In this limit, the ground state of 160 is a closed shell, while the 2°Ne levels have pure SU(3) wave functions with the (2#) quantum numbers given in table 2. This is a good approximation for the ground state band of 2°Ne, for which wave functions obtained from shell-model calculations in the full (sd) 4 space 1,2) have over 0.9 overlap with the SU(3) (2/~) = (80) wave functions. The ~-particle strengths predicted 3) for the 0 +, 2 + , 4 +, 6 + and 8 + levels of the ground state band using the wave functions of ref. 1) are 0.21:0.21 : 0.19 : 0.20 : 0.19, while those predicted using the wave functions of ref. 2) are 0.18 : 0.19 : 0.17 : 0.15 : 0.12. These numbers are to be compared with the constant strength of 0.23 predicted by the simple SU(3) model. In this subsection, the strengths obtained for transitions to the ground state band of 2°Ne are discussed. Table 4 shows that there is considerable disagreement in

456

N. A N A N T A R A M A N et al.

strength between experiment and the simple SU(3) theory, with theory predicting substantially stronger transfers than are experimentally observed. The disagreement is considered significant for several reasons. The experimental (6Li, d) spectra in the region of excitation of the 0 +, 2 + and 4 + levels are clean, with no unresolved or contaminant peaks. The DWBA analysis of the angular distributions of these three levels is straightforward because they are all bound states for the a-particle. The theoretical structure of levels in 2°Ne is well known and no extension of the simplest SU(3) picture there yields a-particle strengths that are unequal to any significant extent. And finally, there is good agreement with theory in the case of 2°-22Ne(6Li, d) reactions 30) and the 24Mg(6Li, d) reaction 35) at nearly the same bombarding energy. In view of this disagreement, the effect of particle-hole admixtures in the ground state of 160 (as well as other effects discussed in sect. 4) was considered. It was assumed that the 2p-2h component results from the SU(3) strong coupling of two p holes, (2~t) = (02), and two sd particles, (2#) = (40), yielding (02)× (40) ~ (42); and that the 4p-4h component results from the strong coupling of four p holes, (2#) = (04), and four sd particles, (2/~) = (80), yielding (04) × (80) ~ (84). In the (6Li, d) reaction, the members of the (2#) = (80) ground state band of 2°Ne are fed from the 0p-0h, 2p-2h and 4p-4h components in 160 via (00) × (80) ~ (80), (42) × ( 6 0 ) ~ (80) and (84)× ( 4 0 ) ~ (80), respectively. The theoretical spectroscopic amplitudes for a-particle transfer corresponding to the three routes are given in table 5. They have been calculated using a formula derived by Hecht and Braunschweig 36), who have given an analytic expression for strong coupled spectroscopic amplitudes when particles are transferred into two different major shells. The effect of the p-h admixture was taken into account in the DWBA calculation as follows. First, the DWBA amplitudes corresponding to the three routes were calculated using the appropriate number of nodes in the bound-state wave function : 2 ( N - 1) + L = 8, 6 and 4, respectively. These were then weighted by the theoretical spectroscopic amplitudes given in table 5 and by the square roots of the strengths with which the three components are present in the ~60 ground state. The products TABLE 5 Spectroscopic amplitudes for a-particle transfer from SU(3) strong coupled components in the ground state of ~60 to levels in the ground state band of 2°Ne 160(g.s.) particle-hole structure/(2#)

j.

0÷ 2+ 4+ 6+ 8+

(0p-0h)/(00)

(2p-2h)/(42)

(4p-4h)/(84)

0.4786 0.4786 0.4786 0.4786 0.4786

-0.2990 -0.2084 -0.0234 0.1753 0.0

0.3810 -0.1228 0.0291 0.0 0.0

160(6Li, d)2°Ne

457

were then coherently added and squared, to yield the calculated DWBA cross section (dtr/dfl)ca~c for the (6Li, d) reaction. [The difference between this quantity and the (dtr/dt2)I~WBA introduced in subsect. 3.1 is that the spectroscopic strength information is contained in the former but not in the latter.] If the nuclear structure and the reaction mechanism are correctly treated, (dtr/dt2)ca~¢ calculated for transitions to the various levels should be equal, except for an overall normalization, to the corresponding experimental cross sections. The ratios (da/df2)exp/(da/dt2)¢~l ~ are given in table 6. Now, the angular distribution for a given L is found to have a shape almost completely independent of the number of nodes in the bound-state wave function. Therefore, the angular distribution calculated using a superposition of tbrm factors with different nodal structures, each appropriately weighted, is quite similar in shape to that calculated using a simple (sd) 4 transfer picture. But the magnitudes are changed, as shown in table 6. The last three columns in table 6 list the ratios (dtr/dt2)exp/(dtr/dt2)¢~¢corresponding to three different assumptions regarding the strength of the p-h components in 1 6 0 ( g . s . ) . One normalization has been used for all the columns, it being the same as for the empirical strengths given in table 4. The first of these columns gives the ratios in the absence of p-h admixtures; these ratios are the same as the relative spectroscopic strengths listed in table 4. The next column gives the ratios corresponding to the admixture (85 %, 15 %, 0 %) for the (0p-0h, 2p-2h, 4p-4h) components in ~60(g.s.). The last column gives the ratios resulting from the use of strengths suggested by Brown and Green ~3): (76 %, 22 %, 2%). As remarked above, the ratios in each column should be equal if the reaction mechanism and nuclear structure have been correctly treated. It is seen from table 6 that the use of the p-h strengths suggested by Brown and Green produces an improvement in the ~-particle strengths but does not bring experiment and theory into full agreement. It is further seen that the improvement comes about both because the cross sections predicted for the ground state become stronger and because those predicted for the excited states become weaker as a result of the p-h correlations in ~60(g.s.). TABLE 6 The ratios (&r/dl2)e~p/(da/df2)cal ~ for the 160(6Li, d)2°Ne reaction at 32 MeV for three clifferent assumptions about the t60(g.s.) wave function (do'/d f~)exp/(da/df~)¢al¢ E x (MeV)

.0.0 1.63 4.25 8.78 11.95

Jn

0+ 2+ 4+ 6+ 8÷

(100, 0, 0)

(85, 15, 0)

(76, 22, 2)

1.00 0.41 0.22 0.20 0.51

0.82 0.39 0.26 0.29 0.60

0.77 0.42 0.28 0.33 0.66

458

N. ANANTARAMAN

et al.

It will be seen in sect. 4 that besides p-h admixtures in 160(g.s.), channel coupling in the (6Li, d) reaction also plays an important role in explaining the relative yields to members of the ground state band of Z°Ne. 3.3. RELATIVE SPECTROSCOPIC STRENGTHS IN EXCITED BANDS

Large and constant s-particle strength are predicted 3) for transfer to members of the (90) band. For ~60 a closed core, the transfer will proceed via the (sd)3(fp) component in the (90) band. The theoretical predictions given in table 4 are based on that assumption. The table shows that the experimental strengths are considerably smaller than the predictions and unequal to each other. It is likely that, as in the case of the ground state band, inclusion of the effect of p-h admixtures in 160(g.s.) is important. But that is not attempted here, for now the effect is much harder to calculate: it involves the transfer of particles into three different major shells, rather than two as for the ground state band. The experimental angular distribution for the 3- (5.62 MeV) level is quite different from that for the 3- (7.17 MeV) level. Moreover, the cross sections for the 3- (5.62 MeV) and 5- (8.45 MeV) levels of the (82) K = 2 [(sd)Sp - l] band are an order of magnitude smaller than those for the corresponding levels of the (90) band. This inhibition is expected, for a (sd)Sp - ~ configuration cannot be populated by direct s-particle transfer from the 0p-0h component of 160(g.s.). The smallness of the cross sections measured for the transitions to the 3- (5.62 MeV) and 5- (8.45 MeV) levels, and the unsatisfactory nature of the DWBA fits obtained for the angular distributions, indicate that multistep processes may contribute to them. On the other hand, if a one-step transfer process is assumed, the population of the natural-parity members of the (82) band is due either to mixing with the corresponding members of the (90) band, or to the presence of 2p-2h and 4p-4h components in 160(g.s.), o r to a combination of the two. If mixing between the two bands is the mechanism, the strengths given in table 4 imply an admixture of 20 ~o by intensity. This value is the same for the 3- and 5- levels. But the sum of the strengths in the two bands (for each J) is not constant, as would be expected if 160 were a closed-~hell nucleus and the lower band obtained its a-particle strength solely through mixing. The excited (sd) 4 band, whose configuration has been confirmed by strong excitation of its members in the 19F(3He, d) reaction 37), is predicted to have vanishing s-particle strength in the simplest SU(3) model. In the full sd shell calculation, the strengths predicted for the 0 ÷ (6.72 MeV) and 2 ÷ (7.42 MeV) levels are 5 ~o of that for the ground state 3). The (6Li, d) data exhibit considerably larger transfer strengths. An explanation has been offered 6) for this in terms of mixing with a band of predominantly (sd)2(fp) 2 configuration with 0 ÷ bandhead at 8.3 MeV and 2 + level at 8.6 MeV. Members of this latter band are very broad 24) and angular

160(6Li, d)2°Ne

459

distributions for them could not be extracted because of the rather large background in the (6Li, d) spectra above 8 MeV excitation (see fig. 1). An extended shell model calculation by Tomoda and Arima 9, 1o), in which the model space consists of all states of permutation symmetries [4] and 1-31] in the (sd) 4 space and ~-particle cluster states of symmetry [4] with up tO 19h~oexcitations above the sd shell, has verified that the excited ( s d ) 4 band gets its or-particle strength through mixing. The calculation predicts equal strengths for the 0 ÷ (6.72 MeV) and 2 ÷ (7.42 MeV) levels, these being half that for the ground state. The observed strengths are respectively half and one-eighth of that for the ground state. Lack of agreement between experiment and the predictions of the simplest SU(3) model has been seen also for or-decay reduced widths obtained from analyses 7, 8) of data 38)' on resonant ~-particle scattering on 160. Table 7 shows such a comparison, the reduced widths being those extracted by Arima and Yoshida s) from the data of H/iusser et al. as). The absolute values of the reduced widths depend very strongly on the channel radius used in the analysis. This dependence has been removed to a large extent in table 7 by normalizing to unity the width for the 7.17 MeV level. The same normalization has been used for the columns listing the (6Li, d) strengths and the theoretical strengths. The decay widths given in the table are those calculated at a channel radius of 3.53 fm, which corresponds to the radius parameter generally used to describe 0t-particle scattering on light nuclei. Table 7 shows that the discrepancy between the decay widths and the strengths obtained from the (6Li, d) reaction is most pronounced for levels with small decay widths. Moreover, in almost all cases the (6Li, d) strength is greater than the decay width. Both these features are explained if multistep or compound nuclear processes contribute for levels weakly excited in the reaction. Indeed, the spectroscopic TABLE 7 Comparison of relative ~t-particle strengths in 2°Ne from the 160(6Li, d) reaction and from decay widths E x (MeV)

J"

S(6Li, d)

Sa. . . . id ")

Sth[SU(3)]

8.78 11.95

6+ 8+

0.77 1.96

0.28 O.15

0.67 0.67

5.62 8.45

35-

0.23 0.15

0.04 0.00

0.00 0.00

5.79 7.17 10.26

135-

2.08 1.00 0.58

> 0.34 1.00 1.19

1.00 1.00 1.00

6,72 7.42

0+ 2+

2.15 0.50

0.40 0.09

0.00 0.00

12,15

6+

0.19

~0.00

0.00

Each column is normalized relative to unity for the strength of the 7.17 MeV level a) From ref. a).

460

N. ANANTARAMAN et al.

strengths quoted in the tables for weakly excited levels should be considered as upper limits in view of the uncertainty as to the reaction mechanism. In the case of the decay widths, the disagreement between experiment and theory has been interpreted 7,8) as indicating substantially different channel radii in different bands as well as a decrease of radius with increasing spin in each band. If the measured widths of the 6 ÷ and 8 ÷ levels of the ground state band are to agree with SU(3) predictions, channel radii of 3.78 fm for the 6 ÷ level and 3.53 fm for the 8 ÷ level are needed. Some support for this interpretation comes from results of cluster model calculations 39) and Hartree-Fock variation-after-projection techniques <3, whcih show a decrease of rms radius with increasing spin. But the relation between the channel radius, the rms radius and the radius of the potential well in which the ~t-particle moves, is not obvious. For example, it has been shown 41) that the decrease of rms radius from the 6 ÷ to the 8 ÷ level can be produced with a constant Woods-Saxon well radius as a result of the different binding energy and higher centrifugal barrier for the 8 ÷ level. Because of this uncertainty, we did not attempt (except in one case, as mentioned in the next section) to'obtain agreement between the experimental and theoretical ~-particle strengths by changing the radius of the well for each level. 3.4. ABSOLUTE STRENGTH OF THE GROUND STATE TRANSITION

The absolute value of the ~-particle strength predicted by the simple SU(3) theory for the 160(6Li, d) transition to the ground state of 2°Ne is 0.23. The experimental values extracted from the data at 20, 32 and 38 MeV by the finite-range DWBA analysis are 2.7, 10.3 and 7.4, respectively, assuming a value of 0.5 for the 6Li --* ct+ d spectroscopic strength. These values are 10 to 40 times larger than predicted, but the question of whether this is significant must be left unanswered, for the values are very sensitive to the parameters, especially the radius of the bound-state well, used in the DWBA analysis. It is however significant that the absolute strength extracted a t 20 MeV differs substantially from strengths extracted at 32 and 38 MeV, which agree with each other to within the uncertainty in the cross section determination. It is possible that nondirect processes play an important role at 20 MeV (see sect. 4). It must also be pointed out that similar large differences in the absolute magnitudes are found in the analysis of ~-decay data (which involves much less ambiguity in parameters than does a DWBA calculation). For instance, Arima and Yoshida find 8) that the experimental or-decay widths for the 6 + and 8 + levels of the 2°Ne ground state band are too large compared to calculations by factors of 8 and 32, respectively, when they use a channel radius equal to the sum of the radii of 160 and the ~-particle.

160(6Li, d)2°Ne

461

4. Further calculations

As discussed in subsect. 3.2, there are serious discrepancies between relative 0t-particle strengths extracted from the 32 MeV data and those predicted by a simple SU(3) model. It was seen there that part of the discrepancy was due to neglect of p-h correlations in 160(g.s.). But this does not fully explain the differences. So a number of other possibilities were also explored. These will now be described. In order to determine whether the difficulty was peculiar to the bombarding energy used, angular distributions were also obtained at energies of 20 and 38 MeV for the 0 +, 2 + and 4 + levels. Figs. 7 and 8 show the data and the results of finite-range DWBA (FRDWBA) calculations (solid lines) obtained with the same optical model and bound-state parameters as were used in the analysis of the 32 MeV data. The dashed lines in fig. 7 are the results of compound nuclear calculations which are discussed later. The dashed line for the 4 + angular distribution in fig. 8 is similar to the dashed line in fig. 2, viz. it is the result of a DWBA calculation with a bound-state radius of 2.52 fm. As was the case in fig. 2, here also this results in an improved fit to the ~ngular distribution. Table 8 gives the strengths extracted. The normalization used for this table is the same as for tables 4 and 6. It is seen that differences in the relative strengths between experiment and theory persist at all three bombarding energies. Moreover, there is an apparent dependence of the extracted relative strengths on the bombarding ~nergy. The error in the relative strengths is determined by the statistics of the experimental yields and by the uncertainty in normalizing the DWBA calculations to the data. It was estimated above to be about _ 25 %. So the occurrence of considerably larger differences in the strengths determined at the three energies is cause for concern. Both these problems are considered in this section. Agreement between experiment and theory for the relative strengths could not be brought about within the framework of the one-step cluster-transfer DWBA TABLE 8 The ratios (da/dtZ)cxp/(dtr/dtZ)ca,c for three levels of the ground state band of Z°Ne from the t60(6Li, d) reaction at 20, 32 and 38 MeV for two different assumptions about the composition of the t60 ground state 60(g.s.) composition (100, 0, 0)

(76, 22, 2)

(da/dtZ)exp/(da/dtZ)cal~ ELi (MeV)

0+

2+

4+

20.0 32.0 38.0

0.26 1.00 0.72

0.12 0.41 0.54

0.22 0.37

20.0 32~0 38.0

0.20 0.77 0.55

0.11 0.42 0.55

0.28 0.50

The normalization is the same as in tables 4 and 6.

462

N. ANANTARAMAN et al.

calculations. The relative strengths were stable with respect to small changes in the bound-state parameters and the use of different optical model parameters as long as the calculated angular distributions fitted the data. Further, for the 32 and 38 MeV data, very nearly the same strengths were obtained with zero-range DWBA (ZRDWBA) as with F R D W B A . For the 20 MeV data, however, the Z R D W B A calculation using the code D W U C K 42) gave the ratio 1 . 0 : 0 . 8 : 0 . 8 for the 0 ÷, 2 + and 4 + strengths. The F R D W B A calculation gave the ratio 1.0 : 0.4 for the 0 ÷ and 2 + strengths, while it could not fit the 4 ÷ angular distribution. This is the only case in all our studies of the (6Li, d) reaction on sd shell nuclei where Z R D W B A and F R D W B A calculations gave different relative strengths. The results of Z R D W B A analyses of the data obtained at 20 MeV in the present experiment and at 18 MeV in earlier measurements at the University of Pennsylvania 43) are in good agreement with each other. House and Kemper 25) found that 160(6Li, d) excitation functions measured for the 0 ÷ and 2 ÷ levels of 2°Ne had structure at bombarding energies below 26 MeV which were not explainable by a direct or-particle transfer mechanism. In the present analysis, it was seen (subsect. 3.4) that the absolute ground state strength determined at 20 MeV was substantially different from strengths determined at 32 and 38 MeV. Both these facts suggest that at 20 MeV the reaction mechanism may not be completely direct. To assess the importance of compound nuclear (CN) contributions at the various bombarding energies, Hauser-Feshbach calculations were done with the code STATIS 44). Five open channels, those for protons, neutrons, deuterons, s-particles and 6Li, were included. The details of the calculation are similar to those described in ref. 26). The calculated cross sections were multiplied by a factor of 0.1 in order to make the magnitude at forward angles for the 4+ level at 20 MeV equal the measured value. The dashed curves in fig. 7 show the calculated CN cross sections thus normalized. While the calculated shapes do not at all agree with the measured shapes, it is seen that the magnitudes for the 0 ÷, 2 ÷ and 4+ levels are 25 ~o to 50 ~o of the measured values at 20 MeV. We conclude therefore t h a t , / f all the cross section at forward angles at 20 MeV for the 4 ÷ level is compound nuclear, then this mechanism makes an appreciable contribution to the 20 MeV data. With the same normalization, the cross section calculated at 32 MeV is half the measured value at all angles for the 2- (4.97 MeV) level and 1.5 times the measured value for the 4 - (7.01 MeV) level. As regards the natural parity levels, the CN cross sections calculated at 32 MeV for the 0+, 2+ and 4 + levels of the ground state band are nine times smaller than those calculated at 20 MeV, and hence are very small compared to the measured cross sections. For the 6 ÷ and 8 ÷ levels, however, they are 30 ~o and 50 ~o, respectively, of the measured values at forward angles. (It is well known that the CN reaction model favors the population of high spin states.) Thus the strengths quoted in table 4 for the 6 + and 8 + levels assuming the direct transfer of an s-particle are upper limits.

t 60(6Li, d)2°Ne

463

Next, an explanation of the discrepancy in relative strengths observed for the 0 f, 2 ÷ and 4 + levels at bombarding energies of 32 and 38 MeV was sought by extending the one-step cluster transfer calculation in several ways: (i) by using microscopic wave functions for the bound states of the or-particle in 2°Ne, (ii) by using a different bound state radius for each level, and (iii) by doing a coupledchannels calculation. When microscopic wave functions generated by Vary et al. 45) for the 0 f, 2 ÷ and 4 ÷ levels of 2°Ne were used in the DWBA calculations, the results fitted the measured angular distributions poorly and did not significantly change the extracted relative strengths. The extended shell model calculation of Tomoda and Arima 9), as well as other calculations 39, 40), have shown that the rms radius decreases with increasing spin for the levels of the ground state band of 2°Ne. We found that a 15 ~ decrease in the radius of the Woods-Saxon well was needed in going from the 0 f to the 2 ÷ level in order to have equal extracted strengths for the two levels. This is comparable to the 9 ~ decrease that Yoshida found was necessary to get equal s-particle strengths for the same two levels in an analysis of the 12C(160, 2°Ne)aBe reaction 46). We feel, however, that decrease of bound-state radius with increasing spin is probably not the major reason for the discrepancy observed in the relative 0t-particle strengths. A preliminary study of the effect of channel coupling through strong E2 excitations in 2°Ne has been done using the zero-range coupled channels code C H U C K 47). Direct a-particle transfer from 160(g.s.) to the 0 ÷, 2 ÷ and 4 ÷ levels of 2°Ne, along with inelastic E2 excitations between those three levels, were included in the calculation. A comprehensive investigation of the effect of channel coupling in the 160(6Li, d) reaction, which simultaneously seeks to explain measured angular correlations between the outgoing deuterons and de-exciting v-rays, is being carried out by Eswaran et aL 48). Suffice it for the present to state that the effect increases the strenghts of the 2 ÷ and 4 + levels relative to that of the 0 f level by 15-20 ~ . A finiterange coupled channels analysis of our data by Koshel and Nagel is also underway 49). Cobern et aL 2o) have obtained relative strengths of 1.0 : 1.0 : 0.75 for the 0 +, 2 ÷ and 4 ÷ levels of 2°Ne from a coupled channels analysis of the 160(7Li, t) reaction at a beam energy of 38 MeV, without considering the effect of p-h admixtures in t60(g.s.). In the analysis of the present 160(6Li, d) reaction, however, it is found that a coupled channeles analysis ignoring the p-h admixtures does not give satisfactory results. It appears that the two effects have to be considered together to explain the data. The effect of the 160(g.s.) p-h components on the strengths extracted from the data at the three energies is shown in the lower half of table 8 for the ~60 wave function of Brown and Green 13). The calculations are similar to those described in subsect. 3.2 in connection with results presented in table 6. It is seen from table 8 that at 38 MeV, the ratios (dtr/df2)exv/(dcr/dt2)c,lc are nearly equal for the three levels. Thus the 38 MeV data on transitions to the 2°Ne ground state band are satisfactorily

464

N. ANANTARAMAN

et al.

explained in terms of the effect of p-h components in 160(g.s.) alone. This is in contrast to the situation with the 32 MeV data. This does not however mean that the effect of the p-h admixture is greater at 38 MeV. As table 8 shows, the admixture enhances (dcr/dO)exp/(da/dQ)c,l c for the excited levels relative to that for the ground state by about the same amounts at 38 MeV as at 32 MeV (33 ~o enhancement for the 2 + level, 70 ~o for the 4+). Nor is it true that channel coupling becomes less important at 38 MeV. The calculated channel coupling effect is nearly the same at 38 MeV as at 32 MeV. But the uncertainties (_+25 ~o) in the relative strengths extracted for the 0 +, 2 + and 4 + levels are not sufficiently small for the effect to be discernible in analysis of the data.

5. Summary and conclusions Angular distributions for 160(6Li, d) transitions populating 2°Ne levels up to 12.15 MeV excitation have been measured at 32 MeV and for the 0 +, 2 + and 4 + members of the ground state band at 20 and 38 MeV also. The a-particle spectroscopic strengths have been extracted for the natural parity levels by means of a cluster-model DWBA analysis. The DWBA calculations satisfactorily reproduce the measured angular distributions for 0 +, 1 -, 2 +, 3- and 5- levels. But they do a poor job of fitting the distributions for the higher members of the ground state band, so that the spectroscopic strengths extracted for these levels are uncertain by about +25 ~ for the 4 + and 8 + levels and by ___50~ for the 6 + level. The evidence suggests that nondirect processes contribute appreciably to the 20 MeV data. But the CN process alone is not able to reproduce the observed angular distributions. At 32 MeV, a CN calculation fits the angular distributions for the 2- (4.97 MeV) and 4- (7.01 MeV) levels but is off by a factor of 3 in predicting their relative magnitudes. It was concluded in ref. 26) that both CN and multistep processes contribute to the population of these unnatural parity levels. Calculated CN contributions are negligible for the 0 +, 2 + and 4 + levels of the ground state band but are significant for the 6 + and 8 + levels. The combined uncertainty, due to the presence of CN contributions and the poor fits to DWBA calculations, in the spectroscopic strengths extracted for the 6 + and 8 + levels is about _+50~o. A comparison has been made between (6Li, d) transfer strengths measured at 32 MeV for levels unbound to s-particle emission and their s-decay widths. The two sets of numbers compare poorly with each other and with simple SU(3) predictions. The transfer strengths are larger than the decay widths, suggesting that multistep or CN processes contribute for levels weakly excited in the reaction. Because of this uncertainty concerning the reaction mechanism, the strengths given in the tables for weakly excited levels should be regarded as upper limits. At 38 MeV, the population of the 0 +, 2 + and 4 + members of the ground state band is explained by direct processes.

160(6Li ' d)2ONe

465

A pronounced discrepancy is observed between the simplest SU(3) theoretical predictions and the relative strengths obtained at both 32 and 38 MeV for members of the 2°Ne ground state band assuming a simple (sd)4 transfer. It is found that inclusion of the effects of both the p-h correlations in 160(g.s.) and the inelastic channel couplings in the reaction is important for explaining the data. At 38 MeV, the p-h admixture by itself is able to remove the discrepancy when the wave function of Brown and Green 13) is used for 160(g.s.). The channel coupling effect may well be present at this energy, but the uncertainty in the extracted relative strengths is large enough (_+ 25 %) to obscure it. Calculations 4s) show that this effect enhances the 2 + and 4 + strengths relative to the ground state by 15-20 % at both 32 and 38 MeV. At 32 MeV, even when both effects are included, there is some residual discrepancy between experiment and theory. This may indicate the presence of some multistep contributions to the 0 ÷, 2 ÷ and 4 + levels at 32 MeV, and the absence of the same at 38 MeV. The only other sd shell case 34) where experiment and theory show significant differences for ground state band strengths is for 180(6Li, d)EENc. There p-h admixtures in the target ground state are expected to be less 50). It is worth pointing out that good agreement between experiment and theory has been found in studies of the 20, 22Ne(6Li ' d)24.26Mg and 24Mg(6Li, d)2SSi reactions without invoking the channel coupling effect, even though these are reactions where one might expect it to be more important than in the 160(6Li, d)2°Ne reaction. The effect ought to be investigated in these cases to find how it affects the extracted strengths. It is a pleasure to thank Dr. A. Arima and Dr. D. Kurath for discussions regarding the discrepancy in relative strengths for the 2°Ne ground state band, Dr. C. L. Bennett and Dr. H. T. Fortune for critical readings of an early draft of this paper, Dr. J. P. Vary for providing us with microscopic 2°Ne wave functions, and Dr. R. N. Boyd for communicating the results of the coupled channels calculations. References l) Y. Akiyama, A. Prima and T. Sebe, Nucl. Phys. A138 (1969) 273 2) E. C. Halbert, J. B. McGrory, B. H. Wildenthal and S. P. Pandya, Advances in nuclear physics, vol. 4, ed. M. Baranger and E. Vogt (Plenum Press, New York, 1972) 3) M. Ichimura, A. Arima, E. C. Halbert and T. Terasawa, Nucl. Phys. A204 (1973) 225 4) J. P. Draayer, Nucl. Phys. A237 (1975) 157 5) M. Harvey, Advances in nuclear physics, vol. 1, ed. M. Baranger and E. Vogt (Plenum Press, New York, 1968) 6) H. T. Fortune, R. Middleton and R. R. Betts, Phys. Rev. Lett. 29 (1972) 738 7) O. H~iusser, T. K. Alexander, D. L. Disdier, A. J. Ferguson, A. B. McDonald and I. S. Towner, Nucl. Phys. A216 (1973) 617 8) A. Arima and S. Yoshida, Nucl. Phys. A219 (1974) 475

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9) T. Tomoda and A. Arima, Proc. INS-IPCR Symp. on cluster structure of nuclei and transfer reactions induced by heavy ions, ed. H. Kamitsubo, I. Kohno and T. Marumori (Tokyo, 1975) p. 90 10) A. Arima, Proc. Second Int. Conf. on clustering phenomena in nuclei (University of Maryland, 1975) p. 38 11) J. M. Irvine, C. D. Latorre and V. F. E. Pucknell, Adv. in Phys. 20 (1971) 661 12) J. P. Elliott, Proc. Roy. Soc. A245 (1958) 562 13) G. E. Brown and A. M. Green, Nucl. Phys. 75 (1966) 401 ; 85 (1966) 87 14) T. Engeland and P. J. Ellis, Phys. Lett. 25B (1967) 57 15) A. P. Zucker, B. Buck and J. B. McGrory, Phys. Rev. Lett. 21 (1968) 39 16) G. Kluge and P. Manakos, Phys. Lett. 29B (1969) 277 17) C. L. Bennett, H. W. Fulbright, H. E. Gove and J. P. Draayer, Bull. Am. Phys. Soc. 22 (1977) 995 18) R. Middleton, B. Rosner, D. J. Pullen and L. Polsky, Phys. Rev. Lett. 20 (1968) 118 19) K. I. Kubo, F. Nemoto and H. Bando, Nucl. Phys. A224 (1974) 573 20) M. E, Cobern, D. J. Pisano and P. D. Parker, Phys. Rev. C14 (1976) 491 21) K. Nagatani, C. W. Towsley, K. G. Nair, R. Hanus, M. Hamm and D. Strottman, Phys. Rev. C14 (1976) 2133 22) H. W. Fulbright and J. T6ke, Nuclear Structure Research Laboratory Annual Report (University of Rochester, 1974) p. 191 23) H. W. Fulbright, R. G. Markham and W. A. Lanford, Nucl. Instr. 108 (1973) 125 24) F. Ajzenberg-Selove, Nucl. Phys. AI90 (1972) 1 ~5) L. J. House and K. W. Kemper, Phys. Rev. C17 (1978) 79 26) G. D. Gunn, R. N. Boyd, N. Anantaraman, D. Shapira, J. T6ke and H. E. Gove, Nucl. Phys. A275 (1977) 524 27) H. T. Fortune, in Symp. on heavy ion transfer reactions, vol. 1 (Argonne National Laboratory Physics Division Informal Report PHY-1973B, March 1973, unpublished) 28) R. M. DeVries, Phys. Rev. C8 (1973) 951 29) R. M. DeVries, H. W. Fulbright, R. G. Markham and U. Strohbusch, Phys. Lett. 55B (1975) 33 30) N. Anantaraman, H. E. Gove, J. T6ke and J. P. Draayer, Nucl. Phys. A279 (1977) 474 31) U. Strohbusch, C. L. Fink, B. Zeidman, R. G. Markham, H. W. Fulbright and R. N. Horoshko, Phys. Rev. C9 (1974) 965 32) E. Newman, L. C. Becker, B. M. Preedom and J. C. Hiebert, Nucl. Phys. A100 (1967) 225 33) R. M. DeVries, Comp. Phys. Comm. 11 (1976) 249 34) N. Anantaraman, H. E. Gove, J. P. Trentelman, J. P. Draayer and F. C. Jundt, Nucl. Phys. A276 (1977) 119 35) J. P. Draayer, H. E. Gove, J. P. Trentelman, N. Anantaraman and R. M. DeVries, Phys. Lett. 5311 (1974) 250 36) K. T. Hecht and D. Braunschweig, Nucl. Phys. A244 (1975) 365 37) R. R. Betts, H. T. Fortune and R. Middleton, Phys. Rev. C l l (1975) 19, and references therein 38) O. Hiiusser, A. J. Ferguson, A. B. McDonald, I. M. Sz6ghy, T. K. Alexander and D. L. Disdier, Nucl. Phys. A179 (1972) 465 39) H. Horiuchi, Ph.D. thesis (University of Tokyo, 1970, unpublished) 40) H. C. Lee and R. Y. Cusson, Phys. Rev. Lett. 29 (1972) 1525 41) C. A. Mosley and H. T. Fortune, Phys. Rev. C9 (1974) 775 42) P. D. Kunz, University of Colorado, 1969, unpublished 43) H. T. Fortune, private communication 44) R.G. Stokstad, Wright Nuclear Structure Research Laboratory internal report no. 52, Yale University (1972), unpublished 45) J. P. Vary, private communication; B. Buck, C. B. Dover and J. P. Vary, Phys. Rev. CII (1975) 1803 46) H. Yoshida, Phys. Lett. 4711 (1973) 411 47) P. D. Kunz, University of Colorado, 1973, unpublished 48) M. A. Eswaran, R. N. Boyd, E. Sugarbaker, R. Cook and H. E. Gove, private communication 49) R. D. Koshel and P. Nagel, Bull. Am. Phys. Soc. 21 (1976) 512 50) H. G. Benson and B. H. Flowers, Nucl. Phys. A126 (1969) 332