INSTITUTE OF ADVANCED STUDIES

ANU-P/9 3 9

SHELL MODEL CALCULATIONS FOR NEUTRON RICH NUCLEI WITH A~35-4'

C.L. WOODS. Department of Nuclear Physics,

Research School of Physical Sciences, \ The Australian National University,

GPO Box 4, Canberra, A.C.T. 2601, Australia,

INSTITUTE OF ADVANCED STUDIES

ANU-P/939 November 1985

Accepted for publication in Nuclear Physics A.

SHELL MODEL CALCULATIONS FOR NEUTRON RICH NUCLEI WITH A=35-41

C.L. WOODS. Department of Nuclear Physics,

Research School of Physical Sciences, The Australian National University,

GPO Box 4, Canberra, A.C.T. 2601, Australia.

Abstract

Shell model calculations are presented for neutron-rich nuclei in the mass region A=35-41 using two new interactions. The usefulness and reliability of the interactions are evaluated with particular emphasis on their predictions for neutron-rich Isotopes. The calculations are performed in a Ofiw basis space with active protons in the Id-,-, 2s,.- and ld 3 /„ orbitals and active neutrons in the If 7/? a n <* ^3/2 o r b i t a l s -

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1. Introduction

36 n >

The recent availability of highly enriched S targets"' has generated a flurry of experimental activity, aimed at determining the properties of very neutron-rich nuclei in the mass range 33-40. Such nuclei, which were difficult to study before, can now be formed with measurable cross sections in few-36 nucleon transfer and charge-exchange reactions initiated on S targets. Several experimental studies have already appeared in the literature, reporting excitation energies and spectroscopic factors for 3 7 S , 3 3 P and 3 5 P Cref. 2 , 3 )3, ^-decay data for 3 7 S Cref. 4 )3, excitation energies in 3 3 ' 3 4 S i , 3 5 ' 3 6 p and 3 7 ' 3 8 s Cref. 1, excitation energies and tentative J*-assignments in 33,34Si Z r e f ^ 2 a n d 34,35 T36 p C r e f , 7 ) ] a n d excitation energies and J^-assignments in P Cref. 1. Hitherto no shell-model interactions have been developed specifically for nuclei in this region of the mass table. However, the rapid increase in experimental data creates a need for reliable shell-model predictions with which to compare these exotic nuclei.

Several groups have published shell-model calculations of spectroscopic factors and electromagnetic decay properties as well as excitation energies for less neutron-rich nuclei having at least one nucleon in the fp-shell and it is of interest to compare their results with the present work where possible.

9) Gloeckner et al performed a detailed study of the low-lying -2 n states of argon isotopes in ir(d 3 / 2) v ( f 7 / 2 ) a n d

w ( d 3 / 2 ) ~ y,^j/2r?3/2) s P a c e s ' searching the matrix elements in each space to fit binding and excitation energies of CI, Ar, K

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and Ca Isotopes. Hsieh et al have performed calculations for negative parity states in nuclei with masses 36 to 40 in a ( d 3 / 2 s 1 / 2 ) ' n ( f 7 / 2 P 3 / 2 ) 1 space. Maripuu et al U ) have published

38 a detailed study of CI using a model space which includes active d 5 /2' &1/2' d3/2' f7/2 a n d p3/2 p a ^ 1 0 1 6 8 - Maripuu and Hokken studied negative parity states in A=35, 37 and 39 nuclei, using a ( d3/2 ) n ( f7/2 p3/2 space. SJcouras has

38 studied the importance of 2p-4h components in states of Ar and Gray et al have studied 2-particle excitations in Ar and 3. Finally, Hasper has published shell-model calculations

of the binding and excitation energies only for nuclei in this mass region, using a (0+2Hiw ( sx/2 d3/2 f7/2 p3/2 ) c o n figuration space. For neutron-rich nuclei within the sd-shell, such as 3 3' 3 4Si and 3 4 ' 3 5 P , the 'universal' sd-shell interaction of Nildenthal and its predecessors give valuable predictions. However, none of the above interactions is designed specifically for the more exotic nuclei now being studied.

In this paper the problem of finding a reliable shell-model interaction for these nuclei is addressed. Two new

17) interactions are discussed, having been selected originally 30 5) because they reproduce well the yrast band in S Cref. 1.

Details of these interactions and of the configuration space employed in the calculations are presented in section 2. This section also explains the choice of experimental data with which to assess the usefulness of the interactions. The detailed comparison is presented in section 3, which ends with a brief discussion of the merits and limitations of the new calculations relative to the previous studies mentioned above. In section 4

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predictions for neutron-rich isotopes that are currently being studied experimentally by groups in several laboratories < 3 6' 3 7P, 3 5 ' 3 6 S i , 3 8 ' 3 9 s and 4 0' 4 1C1) are presented and the work is summarised in section 5.

2. Description of the shell-model calculations

Both of the interactions discussed in this work use the 9) f_ / 2 P 3 / 2 two-body matrix elements of Gloeckner et al and the

'best fit' sd-fp cross-shell matrix elements of Schiffer *nd 18) True . One interaction (hereafter referred to as I) uses the

19) Chung-Wildenthal A=28-39 two-body matrix elements in the sd-shell, while the other (hereafter referred to as II) uses tue Wildenthal 'universal' sd-shell interaction . The sd-shell single-particle energies in I were chosen to reproduce differences between A=39 and A=40 binding energies. In both cases the f 7fo~^2/2 s* n9*l e particle energy difference was chosen to reproduce the excitation energy of the lowest 3/2" state in 41 Ca. Calculations were performed using a (dc/o3! /2(^3/2^7/2p3/2)

space, restricted to allow Oftw configurations only. For the evaluation of excitation energies in this space the energy separation between the sd- and fp- shells is irrelevant.

For most of the neutron-rich isotopes of interest, it was further necessary to restrict the number of proton holej in the l d5/2 o r b i t a l i n order to limit the dimensions of the matrices in the calculations. The effects of various restrictions for the l d 5 / 2 occupancy on the predictions for less neutron-rich isotopes were studied and are summarised in section 3. In every

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case it is possible to choose a configuration space for the neutron-rich nuclei in which the effects of this truncation are expected to be small.

The restriction to Ottw configurations was also necessary to limit the dimensions of the matricej and is perhaps the most important limitation of this work. As a consequence, the calculations do not predict states containing significant 2-particle excitation admixtures in their wavefunctions. The increasing occurrence of large 2tiu> components in nuclear states as the shell closure at Z=20 is approached has led to the choice of Z=18 as an upper limit for the nuclei with which these interactions are tested in section 3. Even for the argon isotopes many low-lying states have sizeable 2-particle

9) excitation components , necessitating care when these data are compared with the calculations. Only those nuclei having at least twenty neutrons and for which some spectroscopic factors or electromagnetic decay data are available were included in the comparison. These criteria together resulted in the selection of the nuclei 3 8" 4 1Ar, 3 7 ~ 3 9 C 1 , 3 6 ' 3 7 s and 3 5 P for the assessment of the new interactions and the available experimental data for each nucleus are compared in detail with the shell-model predictions in section 3.

The shell-model binding energies are not included in the discussion because these are insensitive to details of the wa/efunctions but do depend on the method used to correct for the Coulomb contribution and are therefore a poor criterion with which to assess the interactions. Experimental measurements of

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the binding energies of neutron-rich nuclei are of interest in determining whether they are spherical or deformed. In the mass

20 21) region considered here, various mass relations 'v'**' are applicable and, for spherical nuclei within the sd-shell, they generally give somewhat more accurate predictions than the best shell-model calculations. Comparisons of the recent mass

2-8) measurements for the neutron-rich nuclei considered here with the predictions of the mass relations show that these nuclei are not deformed. The spherical shell model should, therefore, provide an adequate description of their low-lying energy spectra and other properties. Spectroscopic factors and electromagnetic decay data have been included in the appraisal of the new interactions because these are sensitive to different properties of the wavefunctions than those tested by the excitation energies.

The data quoted in section 3 for the neutron-rich nuclei 37 35 2 3) S and P have only recently become available ' and for

several other nuclei more extensive-data have been published since the earlier shell-model studies of other groups were completed. The experimental data are taken from the compilation 22) of Endt and van der Leun unless otherwise stated. The calculations were performed using the Oxford-Buenos Aires shell-

23) model code . Experimental electromagnetic transition rates were only available for Ml and E2 transitions225. The shell-model transition rates were calculated using experimental values for the Y-ray energies. Effective charges of 1.5e for protons

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and 0.5e for neutrons were employed throughout for the E2 rates. Harmonic oscillator wavefunctions were used with oscillator

-1/3 -2/3 quantum -ttw=45A - 25A MeV. The mixing ratios are defined as CX(E2)/X(M1)31/2.

3. Comparison of shell-model predictions with established properties of nuclei.

3.1 Argon isotopes Calculations for these isotopes were performed both with

no restrictior on the number of holes in the proton-ld5/2

orbital and with this orbital closed. The results obtained with interaction II were less sensitive to the choice of basis space than those obtained with interaction I. For interaction I the

41 excitation energies changed by up to 2JO JceV in Ar, 500 keV in 39 40 38 Ar and Ar and 1 MeV in Ar, changes approximately half as

large occurring for interaction II. In all cases the full calculations gave better agreement with experiment. The larger spectroscopic factors changed by N<25% for interaction I and by N<12% for interaction II. For both spectroscopic factors and electromagnetic decay properties, better agreement with experiment was obtained without restrictions on the ld 5 / 2

occupancy and the results presented in detail below were obtained in the full configuration space.

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38 3.1.1 The nucleus Ar This nucleus has no particles in the fp-shell in the

lowest configuration and therefore a comparison with the experimental data is a test of the success of the Chung-Nildenthal and Nildenthal 'universal' sd-shell interactions.

24) ir + Betz et al identify a deformed K =0 rotational band having members J*=0+ at 3.377 MeV, 2 + at 3.937 MeV, 4 + at 5.350 MeV, 6 +

at 7.289 MeV and 8 + at 9.341 MeV. These states should not be predicted within the configuration space adopced here.

39 3 38 Spectroscopic factors for both the K(d, He) Ar and 3 7CK 3He,d) 3 8Ar reactions are available22* and new K(d, He) Ar values have been obtained recently . Branching

22) ratios for the low-lying states are also well established Fig. la shows the experimental and predicted energy levels

of 3 8Ar. As expected the deformed 0+,3.377 MeV, 2 +, 3.9-37 MeV and 4 , 5.350 MeV states are not predicted in the present model space. The second predicted 2 levels are tentatively Identified with the experimental level at 4.565 MeV. The lowest 4 state is predicted at 7.765 MeV in interaction I and at 8.521 MeV by interaction II. The lowest experimental candidate for a 1 level is that observed at 5.552 MeV. Spectroscopic factors

39 3 38 for the K(d, He) Ar reaction are shown in table 1. The calculated spectroscopic factors are in good agreement with the measured values for the lowest two states and support the identification of the 2* states calculated at 4.169 MeV (I) and 4.490 MeV (ID with the 2 + 4.565 MeV level and the 1 + states at 5.104 MeV (I) and 5.555 MeV (II) with the (1,2)* level at 5.552

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MeV. Spectroscopic factors for the Cl( He,d) Ar reaction are shown in table 2. As for the pickup case both interactions give similar predictions for the strong transitions and th^se are in good agreement with the experimental values. However, the distributions of small components of the reaction strength differ considerably and neither interaction gives quantitative agreement for the weak transitions. The experimental 0 , 4.71 MeV and 2 +, 5.16 MeV levels also contain significant 2p-4h admixtures ' and are not predicted within the present model space.

A large amount of experimental data is available on the 38 electromagnetic transitions in Ar but only the branching

ratios of low-lying, positive parity states are of relevance to this work. The 2 + 3.94 MeV level has a 94% branch to the ground state and a 6% branch to the 2 2.17 MeV state, while the 2 4.57 MeV level has a 96% branch to the 2 + 2.17 MeV level. The 2 4.17 MeV state predicted by interaction I has a 93% branch to the ground state and 7% to the lowest 2 state and therefore resembles the core-excited 2 3.94 MeV level in this respect. The 2 4.49 MeV state rredicted by interaction II, on the other hand, has a 21% branch to the ground state and a 79% branch to the lowest 2 + state, in better agreement with the experimental 4.57 MeV level. The measured mean lifetimes of the 2.17 and 4.57 MeV 2 + states are 680±30fs and 60+12fs respectively. Interaction I predicts 1680fs and 47fs for these quantities whereas interaction II gives 1140 and 22fs respectively.

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Thua interaction II gives a somewhat better description of 38 the low-lying states in Ar than interaction I. The calculated

energies of the second 2 + and lowest 1 + states agree more closely with those of the experimental levels with which they are identified and both the (d, He) spectroscopic factors and branching ratios of the second 2 + state support its identification with the experimental 2 4.57 MeV level. For interaction I, however, the second 2 state has some features in common with the 2 + 3.94 MeV level and others with the 2 4.57 MeV level.

39 3.1.2 The nucleus *Ar 39 The experimental and calculated energy levels of Ar are

shown in fig. lb and clearly both interactions predict too large 38 39 a spacing between states. The Ar(d,p) Ar spectroscopic

factors are shown in table 3. Those to the lowest two states in 39 Ar are well reproduced by both interactions. The second

calculated 3/2" states are identified with the third experimental 3/2" level on the basis of their spectroscopic strengths. The second calculated 7/2" states are tentatively identified with the experimental state at 2.481 MeV although

9) there is some evidence that this is a core-excited state . IT - 38 J =5/2 states cannot be populated from the ground state of Ar within the present model space. The single-nucleon pickup

40 spectroscopic factors from the ground state of Ar are shown in table 4. There is good agreement with the experimental values for both interactions for the 7/2" 0.0 MeV, 3/2" 1.27 MeV and 3/2" 2.63 MeV states. The 7/2" 2.481 MeV state is not seen in

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single-nucleon pickup and the calculated spectroscopic factors for this level are very small.

Experimental and theoretical branching ratios for electromagnetic decays are shown in table 5 and mixing ratios are shown in table 6. Both interactions reproduce the experimental branching ratios well for the lowest 5/2~ state and the 3/2~ state at 2.63 MeV. The measured branching ratios for the 7/2" 2.48 MeV and (5/2, 7/2)" 3.06 MeV states are similar und it is therefore not possible to use these data to assign the second theoretical 7/2 states to an experimental counterpart. Interaction II predicts a rather small mixing ratio for the 7/2l-»7/2T transition if the initial state is to be identified with the level at 2.48 MeV.

In summary, both interactions give a good description of the 7/2" 0.0 MeV, 3/2" 1.267 MeV, 5/2" 2.092 MeV and 3/2" 2.632 MeV states of 3 9Ar. The 7/2" 2.481 MeV state is less weZ1

9) reproduced and may be a cere-excited state . Five other levels 22) below 3 MeV have been assigned negative parity , but none of

38 39 them is populated significantly in the Ar(d,p) Ar stripping reaction and they are therefore probably core-excited states.

3.1.3 The nucleus 4 0A* 40 The energy level spectrum of Ar is shown in fig. 2a.

•ye. \

Recent work by Bitterwolf et al ' has established the existence of a deformed, K^'O rotational band with members J^O* at 2.121 MeV, 2 + at 2.524 MeV, 4 + at 3.515 MeV, and 6 + at 4.959 MeV. The present model cannot predict these levels but both interactions reproduce the energies of the spherical, ground state band

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containing the 0 + 0.0 MeV, 2 + 1.461 MeV, 4 + 2.893 MeV and 6 +

25) 3.464 MeV levels well. Bhat et al have recently measured 41 3 40 K(d, He) Ar spectroscopic factors and their results are

compared with the predictions of both interactions in table 7. Only the large spectroscopic factors to the lowest two states are well reproduced and both interactions fail to reproduce the character of the 2 + states between 3 and 4 KeV. Predictions for the lifetimes of the ground state band are in fair agreement with the experimental values as seen from table 8. For this nucleus the presence of significant core-excited components in the wavefunctions of low-lying states severely limits the usefulness of the present shell-'model calculations.

41 3.1.4 The nucleus Ar 41 The experimental and calculated level spectra of Ar are

shown in fig. 2b up to an excitation energy of 2.5 MeV. Above this many lev.is lack definite J^-assignments. Also, substantial i=l strength observed to 1/2" states above 2 MeV in

22) stripping reactions indicates that the 2p, / 2 orbital (which is excluded from the present model space) becomes important at these excitation energies. Both interactions predict a 5/2" ground state, inverting the ordering of the experimental 7/2" 0.0 MeV and 5/2" 0.167 MeV doublet. Single-nucleon stripping spectroscopic factors are compared wxch experimental values in table 9. Both interactions overpredict the strength to the lowest 3/2" state at 0.52 MeV. No theoretical counterpart is predicted for the 1.64 MeV 3/2~ level and its small (d,p) spectroscopic factor suggests that its wavefunction

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contains significant core-excited components. Branching ratios for the low-lying, negative parity states

are shown in table 10. Both interactions reproduce the experimental values well, except for the ratio of the lowest l/2~ strengths to the first and second 3/2~ states. All Ml

41 9) transitions in Ar are known to be severely inhibited and the transition rates are therefore sensitive to small components of the wavefunctions. The E2 transition from the 3/2~ 0.516 MeV level to the ground state, on the other hand, is strongly enhanced. These properties are reflected in the measured lifetimes of the levels, with which the shell-model predictions are compared in table 11. Both interactions seriously over­estimate the lifetime of the low-lying 5/2" state and, in fact, the Ml decay is predicted to be so strongly inhibited that the mixing ratio is approximately 2 for both interactions, although

2 the energy factor of E should favour the Ml transition strongly for a 167 keV transition. Both interactions give B(E2)-values

2 4 of approximately 20e fm for the transition and it is unreasonable to expect a larger enhancement, therefore the smaller experimental lifetime must be due to more Ml strength

3 than the model predicts. Ml transitions between pure ^7/9' 3 2

configurations or between (£7/9) a n d ^7/2 ^3/2 configurations are forbidden for identical particles and therefore the small calculated strength must come from components containing p 3 / 2 admixtures in both wavefunctions.

27) Fossan and Poletti , who originally measured the lifetime of 2 this state, calculated that a 2% intensity of (£7/9) f5/2 i n

the 5/2" wavefunction would be enough to produce the observed Ml

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transition rate. This component is sufficiently small to escape detection in (d,p) studies of the nucleus. Such an admixture is excluded a priori from these shell-model calculations, thus accounting for their over-estimation of the lifetime of the low-lying 5/2" state. For the 3/2" 0.516 MsV and 3/2" 1.354 MeV states both interactions give reasonable agreement with the experimental lifetimes.

3.2 Chlorine isotopes. Calculations were performed both with no restrictions

specifically on the ldc/2 orbital and with it closed. Generally, closing this orbital changed excitation energies by up to 500 keV, worsening the agreement with experiment. The larger spectroscopic factors changed by <10% and the smaller ones by larger amounts. Closing the ldc/? orbital also worsened

37 the agreement for electromagnetic decay properties for CI and 38 39 CI but improved it for CI, although the unrestricted

calculations still predicted the major electromagnetic decay branches correctly. In most cases the effects of different ld5/_-space truncations were larger for interaction I than for interaction II, in some cases by a factor of two. The results presented in detail below are for calculations in which no restrictions were placed specifically on the number of holes in the ld 5 /2 orbital.

3.2.1 The nucleus 3 7C1 A comparison of the predictions with experimental data for

this nucleus tests only the sd-shell part of the interactions.

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The experimental and calculated level spectra are shown in fig. 3a. Both interactions predict fewer levels than are observed above 3.5 MeV and give similar predictions for the lowest three

38 3 37 excited states. Ar(d, He) CI spectroscopic factors have been measured and are compared with the predictions in table 12. Both interactions describe the character of the lowest three states correctly but fail to reproduce higher states, seriously underestimating the amount of configuration mixing. The

3 experimental i=2 (d. He) strength is divided fairly evenly over nine states between 4 and 8 MeV, whereas the calculated strength is concentrated in the second 5/2 level for both interactions. Because of this shortcoming the -y-decays of only the lowest two excited states will be considered. The lifetimes of the l/2+

and 5/2 states are 185±30 and 45+15 fs respectively. Interaction I gives 600 and 140 fs for these quantities while interaction II gives better agreement with values of 380 and 25fs respectively. The mixing ratios for the l/2,->3/2 and 5/2^->3/2+

s transitions are 0.25±0.02 and 1.5+0.4 respectively whereas I predicts 0.31 and 9.2, the Ml 5/2"'"->3/2+ transition being severely inhibited, while II is more accurate again, giving 0.27 and 0.45 for these quantities. The 5/2^ level is observed to decay solely to the ground state and both interactions predict a 99% ground-state branch in agreement with this. 3.2.2 The nucleus 3 bCl

In a simple shell-model picture of this nucleus the low-lying 2" 0.0 MeV, 5" 0.671 MeV, 3" 0.755 MeV and 4" 1.309 MeV levels are a quadruplet having pure ir(d 3 / 2)v(f 7 / 2)

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configurations and are related to the corresponding -l 40

ir(d3/2) v(f7,„) quadruplet in K via the Pandya transformation 28)

. The accuracy of the energy levels calculated using this transformation suggested in the early days that they do have almost pure configurations. A second quadruplet having ir(d3/2)v(p3/2) configurations is expected and its members have been identified29} as the 3" 1.617 MeV, l" 1.692 MeV, 0~ 1.746 MeV and 2~ 1.981 MeV levels. However, measurements of the •37 -JO 22)

Cl(d,p) CI spectroscopic factors and the discovery of a strong Ml transition between the second 3 and lowest 4 states suggest the presence of sizeable admixtures of other configurations in the wavefunctions. These observations have invoked considerable theoretical interest and a good description of the nucleus has been obtained by Maripuu et al within the framework of the spherical shell model. 38 The experimental and predicted energy levels of CI are shown in fig. 3b. Only the observed normal parity levels below 2 MeV are shown, these including all the members of the supposed ir(d_/2)\»(f7 . 2) and ir(d_/2)\i(p3/2) quadruplets. Both interactions give similar results, predicting the correct spin sequence for the lowest multiplet but getting the 0" member of the higher multiplet out of order. In addition, the predicted spacings between members of the higher multiplet are too large. The single-nucleon stripping spectroscopic factors are compared with experiment in table 13. Both interactions give similar results again, overpredicting the strength to the 1 and 2 2

levels by a factor of 2 and underpredicting that to the 4~ level by the same amount.

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Calculated Ml transition rates are compared with experimental values quoted by Maripuu et al and Hsieh et al in table 14. The assumed correspondence between experimental and calculated levels is indicated in figure 3. Surprisingly, both interactions overpredict the strength of the enhanced Ml transition between the second 3~ and lowest 4~ states by a factor of 4. Interaction II overpredicts the ^n^l

transition by a similar factor and badly underestimates the ^2~^1 3 t r e n 9 t ^ i b u t gives satisfactory agreement with all the other experimental transition probabilities. Interaction I generally gives poorer agreement with the data, except for the 32~*31 t r a n s i t i o n which it predicts correctly. The experimental and calculated branching ratios are compared in table 15 and the meari lifetimes in table 16. Interaction II gives slightly better overall agreement-with the data than interaction I.

The second predicted 4" states and the observed 1.785 MeV level shown in fig. 3b are not members of either quadruplet. The calculated (d,p) spectroscopic factors to the 4l, 1.740 MeV (I) and 1.983 MeV (II) states are 0.58 and 0.66 respectively,

22) whereas the measured value to the experimental (2-4) , 1.785 MeV level is (2J+1)S**0.96 which corresponds to S* = 0.104 for a

n r n J=4 assignment. It is therefore impossible to identify these calculated states with the observed 1.785 MeV level.

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39 3.2.3 The nucleus CI Very few experimental data are available for this 22 ) it +

nucleus . However, the J -assignments of 3/2 for the ground state and 1/2 for the 0.396 MeV state are certain and a recent study of the 0-decay of S Cref. ;3 suggests that the 1.301 MeV level has J =5/2 . The energy level spectra are shown in fig. 4a and it is seen that both interactions predict the 1/2 level approximately 240 keV too high. The observed branching ratio '.I the 5/2+ 1.301 MeV level is 94±2% to the ground state and 6±2% to the 1/2 state and both interactions predict 100% of the decay strength to the ground state.

For both interactions 70% of the ground state wavefunction 2 intensity is the *(d3.-)v(f7.,) configuration expected from the

simple shell model, while 70% of the first 1/2 state corresponds to the excitation of one s, / 2 proton to the d-,-

2 2 orbital (i.e. the ir(s, /2^3/2^v^7/2 configuration). The remainder of the intensity is spread over many configurations in both cases. The first 5/2 states contain <10% admixtures of configurations in which a d 5 / 2 proton is promoted to a higher

2 2 2 2 orbital, the i r ( d3/2 si/2 ) v < f7/2 ) a n d i r ( d3/2 sl/2 ) v ( f7/2 )

configurations comprising 70% of their intensities.

L J Sulphur isotopes 3.3.1 The nucleus 3 6 3

The energy level spectra calculated for this nucleus are compared with the observed spectrum in fig. 4b. Only normal parity states below 5 MeV are shown because above this energy many spin assignments are ambiguous and parities are unknown. A

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J*=4+ state has, however, been established at 6.509 MeV. Both interactions predict the energies of the lowest 0 ,1 and 2 states correctly but get the second 2 and lowest 4 states at approximately 6.3 and 7.6 MeV, overpredicting their energies by 1.8 and 1.1 MeV respectively. The energy of the second 0 state is also overpredicted by 1.2 MeV. Spectroscopic factors for the 37 3 36 22) CMd, He) S reaction have been measured and are compared

with calculated values in table 17. Both interactions predict the structures of the second 2 and lowest 4 states incorrectly but predict the lowest 0 ,1 and 2 states well.

The branching ratios of the 1 + 4.52 MeV state are 75+10% + 22) to the ground state and 25+10% to the lowest 2 state

Interaction I gives values of 31% and 69% whereas interaction II gives 47% and 53% respectively. The lifetime of the lowest 2 state is 110±30fs while interaction I predicts 600fs and II predicts 570fs for this quantity. For the 1 + 4.52 MeV state the measured lifetime is 25±12fs, compared with calculated values of lOfs for I and 9fs for II. Thus neither interaction gives good

36 agreement with the -y-decay properties of S. These calculations were repeated in a truncated

configuration space, allowing a maximum of two proton-holes in the ldc/- orbita] to examine the severity of such a restriction for the sulphur isotopes. The calculated energies of all states shown in fig. 4b changed by less than 100 keV while that of the lowest 4 state increased by 400 keV. The spectroscopic factors, branching ratios and lifetimes changed by less than

-4 10%, indicating that ^dc/o* components are unimportant in the

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36 wavefunctions of low-lying states of S.

37 3.3.2 The nucleus S 37r The observed and calculated levels of S are compared in

fig. 5a, where only experimental levels populated in the 36 37 S(d,p) S reaction and having normal or unknown parity are

Included. The experimental and calculated spectroscopic factors for this reaction are compared in table 18. The presence of significant 1=1 strength to the l/2~ state at 2.638 MeV indicates the importance of the 2 p 1 / 2 orbital, which is excluded from the present configuration space, at these excitation energies. These spectroscopic factors have been used to make the identification between theoretical and experimental levels that is shown in fig. 5a. Calculations performed with a maximum of two proton holes allowed in the ld 5 / 2 orbital yielded very similar results, the energy levels changing by less than 70 keV and the spectroscopic factors by less than 10*

3r4 The nucleus £ This nucleus has recently been studied via the

3 6S(d, 3He) 3 5P reaction 2 ' 3 ' 8 ) , the (6Li,7Be) reaction7' and the 14 15 5) ( C, N) reaction ' and the energy levels that were populated

K 8) strongly enough for their J -values to be deduced are compared with the shell-model predictions in fig. 5b. Both interactions reproduce the energies of the lowest three states well but predict no other normal parity states below 7 MeV, suggesting that 2fiw admixtures are important in the wavefunctions of the 4.66 and 5.20 Mev levels. The calculated spectroscopic factors

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2) are compared with those measured by Thorn et al and Khan et 8) al In table 19 and are in qualitative agreement with them.

However, both interactions concentrate the l =2 strength in a single 5/2 state whereas it is observed to be spread over several levels.

Various configuration spaces were used for calculations of 3 the excitation energies and (d. He) spectroscopic factors.

Allowing a maximum of one proton hole in the Idc/o orbital gave excitation energies that differed by <150 JceV from the full-space results. The spectroscopic factors for the 1/2, and 5/2, states changed by <8% while that for the 3/2, state was reduced by a factor of two worsening the agreement with experiment. Allowing three proton holes in this orbital gave results that were very similar to the full calculation.

3.5 Comparison with other calculations 13) 38 The calculations of S*ouras for Ar and Maripuu et

11) 38 al for CI give better agreement with the data than that 13) obtained in the present work. Skouras includes 2p-4h

38 components in the Ar wavefunctions by projecting good angular momentum states from pure 4-hole Nilsson-model intrinsic wavefunctions and coupling these to all possible v(fp) configurations. The two-body matrix elements are calculated from a Rosenfeld-type interaction with Yukawa radial form. Single particle energies are chosen to give correct excitation

41 39 39 11) energies of states in Ca, Ca and K. Maripuu et al use two-body matrix elements calculated from the Sussex relative oscillator matrix elements with space truncation effects added.

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They also choose the single particle energies to yield excitation energies for CI, K and K in best agreement with the experimental spectra. However, both these studies make no attempt to assess the reliability of the interactions over an extended mass region and therefore a comparison of the relative merits of their interactions with those used here is impossible.

38 36 Calculations for Ar and S were performed by Gray et 14) al"' in both Ohu and 0+2ftw bases. The Oliw calculations were

performed using a Yukawa interaction with a general exchange mixture and strength parameters chosen to give acceptable

38 37 binding energies for Ar and CI and a reasonable splitting + + 38

between the 2, and 0, states in Ar. The agreement between the data and their Ottw results is qualitatively similar to that obtained in this work. 2fiu> configurations were included without modifying the Ofiw interaction, by calculating the particle-hole interaction using the matrix elements of Kuo and Brown, defining three core-excited states which contain no spurious centre-of-mass motion and then mixing the 2h and 2p-4h states. The inclusion of 2fiw configurations improves their agreement substantially but such calculations are outside the scope of

12) this work. Marlpuu and Hokken have performed calculations in 32 Ohu space, using MSDI matrix elements and assuming an inert S

core, for A*35,37 and 39 nuclei. Their results give similar agreement to that obtained here but their restricted basis space makes the Interaction unsuitable for calculations of the lighter neutron-rich nuclei considered here.

Hsleh et al have performed a detailed study of negative parity states in nuclei with A*36-40, in a Oftw space and with

-23-

the ld 5 / 2 orbital closed. Their agreement with experimental excitation energies, spectroscopic factors and electromagnetic transition rates is of a similar quality to that obtained in this work. All the nuclei discussed by Hsieh et al have at most one nucleon in the fp-shell and therefore f?-shell two-body matrix elements are not included in their interaction, nor are matrix elements involving the Id,-,, orbital. Thus the interaction is too restricted for use in the present context.

15) Hasper has also performed shell-model calculations for nuclei in the mass region being considered here, using a 0+2ftw < 2 sl/2 1' i3/2 l f7/2 2 p3/2 ) configuration space. For levels which are known to have predominantly Ofcw configurations the agreement with experimental excitation energies is of similar quality to the present calculations. However, Hasper also predicts states with large 2iiw components well and thus obtains better overall agreement with the observed energy level spectra. Unfortunately he does not report spectroscopic factors and electromagnetic transition rates for the nuclei and his interaction excludes the ld 5 /2 orbital.

A comparison of the results obtained in this work for the argon isotopes with those of Gloeckner et al is particularly interesting because the fp-shell matrix elements derived by Gloeckner et al have been used in the new interactions, with different cross-shell and sd-shell matrix elements and a larger basis space. The agreement vith experiment is qualitatively similar, the predictions of Gloeckner et al being slightly more

39 40 accurate for Ar and Ar and those of the present work being 41 slightly better for Ar. This shows that the use of different

-24-

cross-shell and sd-shell interactions and the enlargement of the basis space have not necessitated changes in the fp-shell matrix elements.

In summary, the new hybrid interactions gi/e qualitatively si illar predictions for the low-lying, Oftw states of stable and near-stable nuclei to those obtained with earlier interactions. This is particularly encouraging because many of the previous studies searched the interaction parameters or two-body matrix elements to optimise the agreement with data in this mass region whereas that approach was impractical in the present work because of the large number of matrix elements involved. The present work cannot predict states having significant 2ftu> admixtures and is inferior in this respect to previous calculations performed in a 0+2ftw space. However, the new interactions can be used for silicon and phosphorus isotopes, for which it is essential to allow holes in the ld 5 / 2 orbital whereas the previous interactions could not. The new interactions have also been tested with recent data for neutron-rich isotopes that were unavailable when the previous interactions were derived. They are found to give a good account of the low-lying Ofiw states for these nuclei as well as for the more stable Isotopes (for excitation energies at which the 2p. / 2 orbital is uniuportant). These interactions should therefore be suitable for a description of low-lying simple structure states in the neutron-rich nuclei currently being studied experimentally, and their predictions for the energy level spectra of 3 5 ' 3 6 S i , 3 6 ' 3 7 p , 3 8 ' 3 9 s and 4 0' 4 1C1 are given in the next section.

-25-

4. Predictions for neutron-rich isotopes

40 41 4.1 Chlorine isotopest CI and XC1 The energy levels predicted for the neutron-rich isotopes

CI and CI are shown in fig. 6, the calculations being performed with the lde/2 orbital closed. Both interactions

40 predict a 2 ground state for CI, as is also deduced from its B- decay to Ar Cref. 1. This nucleus has been studied recently via the Ar( Li, Be) CI charge-exchange reaction , which populated the ground state and levels at 0.64, 0.84 and 1.16 MeV strongly. As explained by Fifield et al , these levels are expected to have structures which, in the weak-

+ 37 coupling model, correspond to the 3/2 ground state of CI 43 coupled to the 7/2 ground state of Ca. The shell-model

37 43 states having large overlaps with Cl_ ft Ca_ are y # S • ^ • S •

indicated in fig. 6a and suggest a correspondence to those 40 7 7 40 observed strongly in the Ar( Li, Be) CI experiment. The

predicted ordering of the 3/2+i7/2" multiplet in 4 0C1 is 38 -

different from that in CI, the 5 state not being isomeric in 40

the "CI case. 41 No experimental studies of the excited states of CI have been published and only the limits J* = (1/2, 3/2) + have been

y • 9 •

set from studies of Its 0~-decay to Ar Cref. 3 Both interactions predict a 3/2+ ground state with a very low-lying 1/2 first excited state.

-26-

38 39 4.2 Sulphur isotopes; S and S The calculations for these isotopes were performed

allowing a maximum of two »roton holes in the Idc/o orbital. 38 The predicted and observed spectra for S are shown in fig. 7a.

As explained above, the interactions were originally selected

0 +, 2 +, 4 + and 6 + yrast states observed in the 3 6 S ( 1 8 0 , 1 6 0 Y ) 3 8 S

for more detailed study because they reproduce correctly the 0 , 2 , 4 and 6 yrast states <

5 17) experiment of Mayer et al ' 39 The energy level spectra predicted for S are shown in

fig. 7b. In a simple shell-model picture the lowest 3 configuration is v('-j/2> ' suggesting that the ground state has

J = 7/2 . It is therefore surprising that both interactions predict a 3/2" ground state. In order to shed light on this result calculations were performed with the ld5 .„ orbital closed

39 43 39 for both S and Ca. The ordering of the levels in S was unchanged while both interactions predicted the correct ordering

43 for the lowest three states in Ca, interaction I giving 7/2" 0.0 MeV, 5/2" 0.260 MeV, 3/2" 0.679 MeV, compared to the observed sequence of 7/2" 0.0. MeV, 5/2" 0.373 MeV, 3/2" 0.593 MeV and interaction II giving similar results. The lowest three 39 S states obtained in both basis spaces exhibit strong

configuration mixing. For the 7/2^ states the occupancy of the 2 3

2 p3/2 o r l > i t a l i s °'32» ^ e T ( sl/2 d3/2 7/2 configurations accounting for approximately 65% of the intensity of the wavefunction while for the 3/27 states the corresponding quantities are approximately 0.77 and 42%, the

2 2 1 f < sl/2 ) v ( f7/2 p3/2 } configuration contributing 14% to the intensity. The 5/27 states have a 2p 3 /„ occupancy of roughly

-27-

2 3 0.50, 55% of the intensity coming from the T ( si/2 d3/2 ) v ( f 7 / 2 )

conf igurati ons.

4.3 Phosphorus isotopes; 3 6 P and P These calculations were performed allowing <4(3) proton

holes In the ld 5 / 2 orbital for 3 6 P ( 3 7 P ) . The predicted and 36 observed energy levels in P are shown in fig. 8a. In a simple

shell-model picture the lowest configuration is ir(s 1 /2 , v ( f7/2 )' giving a 3~, 4~ ground state doublet. Both interactions predict

32) the 4 state as the lower of the two. Hill et al have set the experimental limits J* = (2,3,4)" for the ground state from

36 a study of its 0-decay to S. There is some uncertainty in the literature over the

36 energy of the first excited state in P. A study of the 3 6S( 7Li, 7Be) 3 6P and ^ S ^ B , 1 3 ^ ) 3 6 ? reactions recently reported a state at 0.252 MeV 7 ), while a study of the 3 6S( 1 4C, 1 4N) 3 6P reaction reported a state at 0.450 MeV . All of these are charge-exchange reactions and would therefore be expected to populate both members of the ground state doublet strongly. In a weak-coupling picture the 1~ and 2~ if<Si/2)v<P3/2) doublet might also be expected to be seen in these reactions. The 1~ state predicted at 2.37 MeV by both interactions Is approximately of this structure while the lowest two 2" states each have approximately 50% intensity of this character, most of the remainder being *(d 3 / 2)v(f 7 / 2). However, the semi-classical

33) kinematic matching conditions of Brink are much less well satisfied if the neutron Is captured into an % =1 orbital than if It is captured into an i»3 one, suggesting that the 1 and 2

-28-

states will he populated less strongly than the 3 , 4 doublet. Drumm et al ' do tentatively report a small amount of strength

36 to states in P near 2 MeV excitation energy. 37 The predicted energy levels of P are shown in fig. 8b.

No experimental data are available yet for this nucleus. In a simple ^hell-model picture the ground state is expected to have J*=l/2 , corresponding to a *(s., / 2

) v ( f 7 / 2 * configuration and the first excited state to have J*=3/2 , corresponding to a

2 1 ( d 3 / 2 > v ( f 7 / 2 ) configuration. Both interactions predict this level sequence but with considerable configuration mixing of the wavefunctions. Interaction I gives a 58% intensity of the expected configuration for the ground state while interaction II gives a value of 63%. The strengths of all other components are <11% in both cases. 42% of the first excited 3/2+ state

2 consists of the expected ir(d 3 / 2)v(f 7 / 2) configuration for Interaction I and 54% for interaction II, 22% and 15% being

2 * ( sl/2 ) v < f7/2' respectively and all other components being <8%

4.4 Silicon isotopes 1 3 53i and 3 63i These calculations were performed allowing a maximum of

four proton holes in the ld 5 / 2 orbital. The energy level 35 spectra for Si are shown in fig. 9a. The ground state of this

nucleus has recently been observed via the S( C, 0) Si 5) reaction but no excited states have been reported to date.

The ground state has a 72% (78%) intensity admixture of the ir(dg/2) v(f 7 / 2) configuration for interaction I (II) and the 3/2^ state contains 70% (interaction I) and 77% (interac- on

-29-

II) of the 'tde/o* v ( p 3 / 2 J con^iguration expected from a simple shell-model picture of this nucleus.

36 The predicted energy levels of Si are shown in fig. 9b. As yet no experimental studies of this nucleus have been published.

5. Summary and conclusions

A reliable shell-model interaction is required for neutron-rich nuclei in the mass region A=35-41, to assist in the interpretation of data now becoming available on such nuclei from reaction studies using isotopically enriched S targets. Previous interactions including both sd- and fp-shell orbitals have concentrated on the higher end of this mass region and on nuclei in the valley of stability.

Two new 'hybrid' interactions are presented in this work, 17) having been selected because they correctly reproduce the

38 yrast band of S. A detailed comparison of their predictions for less neutron-rich nuclei in the same mass region with experimental data and with the results of earlier calculations shows that they give an acceptable account of the Onw low-lying states. They also reproduce recent data for the low-lying

37 3fl 35 states of the neutron-rich nuclei S, S and P adequately. These interactions should therefore prove useful in interpreting the new data currently being obtained on neutron-rich nuclei in this mass region and their predictions for the energy levels of 3 5 ' 3 6 S i , 3 6 ' 3 7 P , 3 8 ' 3 9 S and 4 0' 4 1C1 are presented. Further information is available and will be supplied on request.

-30-

The author is grateful to Prof. R.D. Lawson for supplying the interactions and for several useful discussions concerning this work, and to Dr F.C. Barker for a critical reading of the manuscript.

-31-

References 1) P. Maier-Komor, Proc. 11th World Conf. of the

International Nuclear Target Development Society, Seattle, 1982, ed. G.M. Hinn, p.164.

2) C.E. Thorn, J.W. Olness and E.K. Warburton, Phys. Rev. C30 (1984) 1442.

3) S. Piskor, P. Franc, J. Kremenek and W. Schaferlingova, Nucl. Phys. A414 (1984) 219.

4) S. Raman, W. Ratynski, E.T. Jurney, M.E. Bunker and J.W. Starner, Phys. Rev. C30 (1984) 26.

5) W.A. Mayer, W. Henning, R. Holzwarth, H.J. Korner, G. Korschinek, W.U. Mayer, G. Rosner and H.J. Scheerer, Z. Phys. A319 (1984) 237.

6) L.K. Fifield, C.L. Woods, R.A. Bark, P.V. Drumm and M.A.C. Hotchkis, Nucl. Phys. A440 (1985) 531.

7) P.V. Drumm, L.K. Fifield, R.A. Bark, M.A.C. Hotchkis and C.L. Woods, Nucl. Phys. A441 (1985) 95.

8) S. Khan, Th. Klhra, K.T. Knopfle, G. Mairle, V. Bechtold and L. Friedrich, Phys. Lett. 156B (1985) 155.

-32-

9) D.H. Gloeckner, R.D. Lawson and F.J.D. Serduke, Phys. Rev. C2 (1973) 1913.

10) S.T. Hsieh, M.C. Wang and D.S. Chuu, Phys. Rev. C23 (1981) 521.

11) S. Maripuu, B.H. Wildenthal and A.O. Evwaraye, Phys. Lett. 43B (1973) 368.

12) S. Maripuu and G.A. Hokken, Nucl. Phys. A141 (1970) 481.

13) L.D. Skouras, Phys. Lett. 3_1B (1970) 439.

14) W.S. Gray, P.J. Ellis, T.Wei, R.M. Polichar and J. Janecke, Nucl. Phys.. A140 (1970) 494.

15) H. Hasper, Phys. Rev. C19 (1979) 1482.

16) B.H. Wildenthal, M.S. Curtin and B.A. Brown, Phys. Rev. C28 (1983) 1343.

17) R.D. Lawson, private communication

18) J.P. Schiffer and W.W. True, Rev. Mod. Phys. 48 (1976) 191.

19) W. Chung, Ph.D thesis, Michigan State University, 1976.

-33-

20) G.T. Garvey, W.J. Gerace, R.L. Jaffe, I. Talmi and I. Kelson, Rev. Mod. Phys. 41 (1969) 51.

21) N.A. Jelley, J. Cerny, D.P. Stahel and K.H. Wilcox, Phys. Rev. Cll (1975) 2049.

22) P.M. Endt and C. van der Leun, Nucl. Phys. A310 (1978) 1.

23) B.A. Brown, A. Etchegoyen, N.D.M. Rae and N.S. Godwin, OXBASH, the Oxford-Buenos Aires Shell Model Code, Internal Report, Dec. 1984, Cyclotron Laboratory, Michigan State University.

24) P. Betz, H. Ropke, F. Glatz, G. Hammel, V. Glattes and W. Brendler, Z. Phys. A271 (1974) 195.

25) CM. Bhat, M. Raja Rao and N.G. Puttaswamy, Nucl. Phys. A394 (1983) 109.

26) E. Bitterwolf et.al., Z. Phys. A313 (1983)123.

27) D.B. Fossan and A.R. Poletti, Phys. Rev. 152 (1966) 984.

28) 'Theory of the nuclear shell model' R.O. Lawson, Clarendon Press, Oxford 1980.

29) G.A.P. Engelbertink and J.W. Olness, Phys. Rev. C5 (1972) 431.

-34-

30) J.C. Hill, R.F. Petry and K.H. Wang, Phys. Rev. C21 (1980) 384.

31) L.K. Fifield, M.A.C. Hotchkis, P.V. Drumm, T.R. Ophel, G.D. Putt and D.C. Weisser, Nucl. Phys. A417 (1984) 534.

32) J.C. Hill, H.R. Koch and K. Shizuma, Phys. Rev. C25 (1982) 3104.

33) D.M. Brink, Phys. Lett. 40B (1972) 37.

TABLE 1

K(d, He) Ar spectroscopic factors

Experimental % < x ** II y r E X c s" b' • » E X s; E X s"

Wi68f Gr70 B h 8 3 C )

0 + 2 0.0 0.76 0.80 0.66 0.0 0.68 0.0 0.71 2 + 0,2 2.17 <0.05,[3.8] 0.08, 3.8 3.62 2.028 0.0, 3.69 2.013 0.09, 3.49

2 + 0,2 3.94 0.37, 0.35 0.24, 0.26 0.39 0.26 2 + 0(,2) 4.57 0.87,.<0. 5 0.74, 1.22, 4.169 1.77, 0.01 4.490 1.69, 0.22 2 + 0(#2) 5.16 0.47 <0.3 0.35, 0.56, (1,2) + 0 5.5 1.17 0.95 1.40 5 . 1 0 4 d ) 1.13 5 . 5 5 5 d ) 1.12

a* 22) Notation as in ref. for all tables unless otherwise stated.

b ) • 221 Experimental values are taken from ref. ' for all tables unless otherwise stated.

C ) Ref. 2 5> d> J = 1

TABLE 2

37 3 38 CI( He,d) Ar spectroscopic factors

Experimental _^ < , j ^ « JJ

J* £ E S+ E S+ E S+ P X P X P X P

0 + 2 0.0 2.5 0.0 2.76 0.0 2.66 2 + (0,)2 2.17 (0.018),2.6 2.028 0.001,2.47 2.013 0.007,2.43 2 + 0,2 3.94 0.008, 0.13 2 + 0(,2) 4.57 0.034,(0.002) 4.169 0.017,0.012 4.490 0.025,0.079 2 + 0(,2) 5.16 0.012,(0.012) (1,2) + 0(,2) 5.55 0.047 a ) (0.777)a)5.104a) 0.018,0.002 5.555a) 0.021,0.034

TABLE 3

38 39 Ar(d,p) Ar spectroscopic factors

Experimental -> <r -> *r II

n n n n

Ho67 Fi68b Se72

7 / 2 3 0 . 0 10 . 6 0 ] 0 . 6 0 0 . 6 6 0 . 0 0 . 8 0 0 . 0 0 . 7 7

3 / 2 " 1 1 . 2 6 7 0 .63 0 . 5 5 0 . 5 3 1 . 7 0 3 0 . 6 6 1 . 7 9 0 0 . 7 4

5 / 2 " 3 2 . 0 9 2 0 . 0 1 0 . 0 2 2 . 4 3 5 0 . 0 2 . 5 8 0 0 . 0

3 / 2 " 1 2 . 4 3 3 0 . 0 2

7 / 2 " 3 2 . 4 8 1 0 . 0 8 8 0 . 0 6 5 3 . 1 6 1 0 . 1 4 3 . 0 9 7 0 . 1 7

3 / 2 " 1 2 . 6 3 2 0 23 0 . 1 9 0 . 2 0 2 . 4 2 4 0 . 2 2 2 . 8 4 4 0 . 1 2

I - J I

TABLE 4

40 39 Ar->- Ar spectroscopic factors

< Experimental - *> < • I — b * II ^

J* I n Ex «

(p,d) s~

Ex S~ n E x s" n I n Ex «

(p,d) 3n

(d,t) (x,a) Ex S~ n E x s" n

Jo68a To77c Ja65b Fi686 Se75a Wi72a

7/2" 3 0.0 0.58 2.4 1.3 0.91 1.3 1.8 0.0 1.90 0.0 1.91 CO 00 1

3/2" 1 1.27 0.10 0.20 0.05 0.12 0.14 <0.1 1.703 0.14 1.790 0.14 5/2" 3 2.09 0.03 0.14 2.435 0.0 2.580 0.0 3/2" 1 2.63 0.02 0.04 2.424 0.02 2.844 0.01

-39-

CM \ in

o» o CN

<

+ CM \ ro

CN in

in

03

c •H

01 c 0 c «s J3 >i |Q U I >-

CM ro

CN

CM N.

t= M-(

o o

W

VO o -H m CM • +1 r- •-•

i-i co

r» co ro v v I-I

o +i

ro v

00 •

o +1 •

m

oo vo O o +1 +i •H in r* o • • • o vo CM o H at 00 V

r*» o\ oo oo fM O T VO

. . . H CM CM CN

I I * N CM CM CN X \ \ \ no in r» ro

CM

o

o o

in

n oo

CM in o o ro

o o 00 CM 00 VO in

at

o v ^ VO CM

i-t CM ro CM

I I I I CM CM CM CM \ \ \ \ ro m r» ro

ao m CM

en

iw x W

•H ro f-l ao

» r» TT . . . ro o o

vo o o • ao vo • o

o VO en

00 <-\

CO 00 O ^> r» m I-I ao r* CM ro CN

I I I I CM CN CM CM \ \ \ \ ro in r» ro

•p c a) e •H M 4) a x w

-» <- •* < T M

TABLE 6

39 Mixing ratios for transitions in Ar

Experimental I

_>

II

5/2~2.09+7/2" g. s.

0.21±0.06 0.18 0.13

7/2"2.48+7/2" B g. s. 7.1 ±5.1 3.3 0.66

3/2"2.63+5/2"2.09 0.07±0.14 0.02 0.01

7/2~2.48+5/2~2.09 0.03+0.12 0.02 0.02

I o I

I

TABLE 7

41 3 40 K(d, He) Ar spectroscopic factors

Experimental — > <& I * <: II

J* *• E v S~ a ) E S" E S" p x p ~x p x p

0 + 2 0.0 0.54 0.0 0.54 0.0 0.55 2 + 2 1.461 0.90 1.444 0.92 1.365 0.94 2 + 0,2 2.524 0.025,0.14 I

2 + 0 , 2 3 . 2 0 8 0 . 3 5 , 0 . 2 3 3 . 3 9 3 0 . 1 6 , 1 . 2 5 3 . 7 2 5 0 . 1 4 , 0 . 6 7

( 1 , 2 ) + 0 , 2 3 . 5 1 1 , 0 . 8 3 3 . 8 2 6 b ) 0 . 1 7 , 0 . 1 7 3 . 8 0 8 b > 0 : 0 6 , 0 . 4 7 ( 1 , 2 + ) 0 , 2 4 . 3 5 8 0 . 4 9

( 1 , 2 ) + 0 5 . 1 6 6 0 . 4 4

a ) R e f . 2 5 > .

b > J = 2

*»

TABLE 8

40 Mean lifetimes of the ground state band in Ar

J* E x li Units J* E x * Mean li retime • Units

Experimental I II

2 + 1.461 1.61±0.06 6.1 5.6 ps 4 + 2.893 4.3±1.5 6.6 6.1 ps 6 + 3.465 0.98+0.03 1.9 2.2 ns

I

I

TABLE 9

40 41 Ar(d,p) Ar spectroscopic factors

Experimental• -* <t- II

n n -» E n E n Fi68b Se75a

7 / 2 " 3 0 . 0 0 . 4 0 0 . 5 5 0 . 0 5 2 0 . 6 4 0 . 1 0 9 0 . 6 4

3 / 2 " 1 0 . 5 2 0 . 0 7 5 0 . 0 9 0 0 . 6 9 7 0 . 2 3 0 . 7 6 7 0 . 2 7

3 / 2 " 1 1 . 3 5 0 . 3 8 0 . 4 3 1 . 4 5 4 0 . 5 2 1 . 4 5 7 0 . 4 6

3 / 2 " 1 1 . 6 4 0 . 0 2 3

1/2" 1 2 . 4 0 0 . 2 6 0 . 3 1 2 . 3 1 8 2 . 2 5 5

I

- 4 4 -

<N \ rn

CN

U <

C •H

O ~

w ID

(0 0> C

•H A O c

>1 fl) n I

CN

\ in

CN

•H x w

• 3

m ro

CN m

o

o o

<w X W

in +l r> vo

CN > o -H •H r» vo 00 • iH CN o

PO CN CN + 1 "fl +1 N H m CN 00

CO CN O +1 -H o oo n i- i r *

r» CN m O iH m m "J • • • • O o •H CN

t i l l CN CN CN CN \ N. S \ m en m iH

m

o o

i n o

W

00 en

o o ^ vo

V fl\ (N CN m

o vo .-I o P-

o m CN © r» <» m

© o CN

I I I i CN CN CM CN

s s s \ in en cn eH

vo

r r

o

o

O

«M X w

o o

in cn

r> rn m vo

m CN CN vo CM

m

r* vo VO o r T CN • • * * o o H CN

I I I CN CN CN CN \ \ \ \ in m en ..-i

AS •p c 0) e

• H M d) 0. X w

-» <- • * 4 -

TABLE 11 41 Mean lifetimes of levels in Ar

J* E <% Mean Lifetime > Units x

Experimental I II 5/2" 0.167 0.605+0.045 14 127 ns 3/2" 0.516 475+30 890 840 ps 3/2" 1.354 640+85 770 600 fs *

I

TABLE 12

I 3 37 Ar(d, He) Cl spec t roscopic f ac to r s

Experimental-** I > « I I *

J* *P

E X s"

P E x S P E x s ;

3 / 2 + 2 0 . 0 3 . 1 0 . 0 2 . 7 6 0 . 0 2 . 6 6

V 2 + 0 1 . 7 3 1 .6 1 . 7 5 2 . 5 1 1 . 8 4 2 . 4 8

• V 2 + (2) 3 . 0 9 < 0 . 0 4 3 . 1 1 0 . 0 0 2 3 . 2 7 0 . 0 7 1

( 3 / 2 , 5/2) + 2 4 . 0 2 O i l l 4 . 0 5 a ) < 0 . 0 0 1 4 . 1 1 a ) 0 . 0 8 7

3 / 2 + 2 4 . 2 7 0 . 2 3 5 . 0 9 0 . 0 3 4 5 . 6 9 0 . 0 0 6

5 / 2 + 2 4 . 8 1 1 .9 5 . 6 5 7 . 3 5 6 . 3 7 7 . 6 5

a ) J = 3 / 2

TABLE 13

37 38 Cl(d,p) CI spectroscopic factors

<— Experimental — i <: I > < II ^

J * n E X n E X < E x • :

2~ 1.3 0.0 0.024,0.72 0.0 0.00,0.92 0.0 0.00,0.93 5~ 3 0.67 0.68 0.577 0.91 0.648 0.94 3~ 1,3 0.76 0.08,0. 54 0.986 0.03,0.85 1.092 0.08,0.80 4~ 3 1.31 0.66 1.455 0.32 1.577 0.24 3~ 1 1.62 0.29 1.683 0.32 1.779 0.25 l" 1 1.69 0.5 1.991 0.91 2.088 0.92 o" 1 1.75 0.89 1.332 1.00 1.359 1.00 2~ 1 1.98 0.48 2.526 0.83 2.681 0.82

a) Results labelled Fi74b in ret*

TABLE 14.

Ml Transition rates in CI

Transition < Ml Transition rate (s ) ->

Experimental3' I II

3~0.755-.-2~ g. s.

4~1.309H-3~0.755

3.19xl012 9.72X1011 1.34xl012 3~0.755-.-2~ g. s.

4~1.309H-3~0.755 3.34X1011 1.48xl09 1.27X1011

3~1.617H-4~1.309 2.28X1011 8.40X1011 9.09X1011

3~1.617H.3~0.755 1.28X1011 2.06X1011 6.52X1011

3~1.617H.2~ O g. s. l~l. 692-»-2~ o g. s. 2~1.981H.3~0.755

8.48xl010 1.72X1011 7.63xl010 3~1.617H.2~ O g. s. l~l. 692-»-2~ o g. s. 2~1.981H.3~0.755

4.66X1011 b) 9.84X1010 5.24X1011

3~1.617H.2~ O g. s. l~l. 692-»-2~ o g. s. 2~1.981H.3~0.755 5.61X1011 b) 9.00xl06 1.27xl09

2~1.981H.2~ „ g. s. 7.24X1011 b) 4.79X1011 1.02xl012

2~1.981H.1~1.692 2.70X1011 b) 2.35X1011 2.54X1011

2~1.981-»-3~l. 617 6.89X1011 b) 2.58X1011 3.08X1011

4~1.309H.5~0.671 1.43xl012 2.16X1011 4.52X1011

Data from ref. unless otherwise stated. b ) Ref. 1 1*

TABLE 15

-is 4*

II

5 l" 4" 3" 1" 0" 2"

38 Y-ray branchings (%) in CI 2" 5" 3" 4"

5 0 . 6 7 3 0 . 7 6 4" 1 . 3 1

E x p e r i m e n t a l 3" 1 . 6 2 1 1 . 6 9 0" 1 . 7 5 2 1 . 9 8

' /

0, 0. 1, 1. 1. 1.

58 99 46 68 99 33

2.53

EJ: 0.0 0.67 0.76 1.31 1.62 1.69

100 100 6.5i0.7 ld.8±0.8 92.6*1.4 100 43.7+0.7

76.0*0.9 3.3±0.4 <0.4 <3

17.5+0.8 28±2 7.4±1.4 <9 23.8*0.6

50±2 <4 <6 <2

<4 22.1*0.6

0.0 0.58 0.99 1.46 1.68 1.99

100 100 23 16 92 100 49

77 0.7

0.5 16 8

67

<0.1 25 24

0.0 0.65 1.09 1.58 1.78 2.09

5 0 . 6 5 3 1 . 0 9 4 1 . 5 8 3 1 . 7 8 1 2 . 0 9 0 1 . 3 6 2 2 . 6 8

100 100 11 8

98 100 64

69 0.3

20 38 2

54

<0.1 19 16

1.75

10.4*0.8 <1

1.33

<0.1

1.36

<0.1

i 4k VO I

TABLE 16 38 Mean lifetimes of states in Cl

J* E < Mean lifetime * Units

Experimental I II

3 4" 3" l" 0" 2"

0.76 320±40 1000 730 fs 1.31 530+80 3500 1900 fs 1.62 2.2+0.2 0.79 0.59 ps 1.69 1.3+0.3 7.4 1.8 ps 1.75 1.0+0.4 5.8 5.3 ps 1.98 500±80 1000 620 fs

I O I

TABLE 17

37 3 36 Cl(d, He) S Spectroscopic factors

Experimental > « I i * II j 1 F C E s^ E« SZ Ev s~

p x P x p x p Pu69 Gr70

0 + 2 0.0 1.6 1.3 0.0 1.15 2 + 0 3.291 1.5 1.1 3.420 1.40 1 + 0 4.523 ] 0.94 4.182 0.88 + J 1' 6 a) 2 0 4.575 ] 0.31 6.262 0.01,0.06a' 4 + 2 6.511 1.1 0.24 7.535 2.62 7.743 2.67 ,'.

0 . 0 1 . 1 5

3 . 4 0 6 1 . 4 1

4 . 3 5 7 0 . 8 7

6 . 4 9 1 0 . 0 1 , 0 „ 0 6 a )

7 . 7 4 3 2 . 6 7

a ) a =0, a =2 p P

U1

TABLE 18 3 6S(d,p) S spectroscopic factors

Experimental i< I =* * II J " *n E x *— <2J f+DS+—> E x (2Jf+l)S+ E j c (2J f +l) S;

Ref 9 ) Ref. 1 0 )

7 / 2 " 3 0 . 0 6 . 1 6 7 . 3 3 0 . 0 7 . 0 5 0 . 0 7 . 0 8

3 / 2 " 1 0 . 6 4 4 2 . 6 2 2 . 8 0 ' 1 . 1 0 9 3 . 2 4 1 . 0 1 0 2 . 9 8

( 3 / 2 " ) 1 1 . 9 9 3 0 . 1 5 0 . 3 0 ^

(5 /2 " ,7 /2 " ) (3) 2.024 0.08

( 5 / 2 " , 7 / 2 " ) (3) 2 . 5 1 7 0 . 1 4 0 . 2 4 3 . 3 4 4 a ) 0 . 3 2 3 . 2 6 1 a ) 0 . 3 7

1 / 2 " 1 2 . 6 3 8 1 . 5 4 1 . 6 1

1 3 . 1 7 0 0 . 1 8

3 / 2 " 1 3 . 2 6 1 0 . 6 0 0 . 5 7 3 . 0 6 5 0 . 6 2 2 . 7 1 6 0 . 9 0

a ) j " = 7 / 2 ~

to I

I

TABLE 19

S(d, He) P spectroscopic factors

Experimental -» « •»• i <_ II

ana) £-a) a) s " E S" E S" P x p x p x p

Ref. 2 ) Ref. 8 )

l/2+ 0 0.0 2.8 1.96 0.0 2.11 0.0 2.13 • on

+ w

3/2 2 2.386 0.37 2.675 0.14 2.630 0.15 ' 5/2+ 2 3.857 1.3 3.49 4.171 6.62 4.298 6.76 5/2* 2 4.665 0.49 1.27 5/2* 2 5.189 0.36 1.66 a>Ref. 8>

-54-

Figure Captions

38 Figure 1. Low-lying, normal parity levels of a) Ar and b ) 3 9Ar.

Figure 2. a)Low-lying, normal and unknown parity levels of 40 41 Ar and b) low-lyirig normal parity levels of Ar.

Figure 3. a) Low-lying, normal and unknown parity levels of 37 38 CI and b) low-lying normal parity levels of CI.

39 Figure 4. a) Low-lying states of CI and b) low-lying, normal 36 parity states of S.

37 35 Figure 5. Low-lying states of a) S and b) P.

40 Figure 6. a) Energy levels of CI below 1.5 MeV. Experimental data from 4 0Ar( 7Li, 7Be) 4 0Cl; Ref. 3 1 )

31) * Strongly populated; Ref. overlap with 3 7C1 „ M 4 3Ca r r „ b) Energy levels of 4 1C1 below 3.0 MeV.

A Large direct - see text.

38 Figure 7. a) Energy levels of S below 4.0 MeV. * Yrast band seen in 3 6S( 1 4C, 1 2C) 3 8S and 3 6 S ( 1 8 0 , 1 6 0 Y > 3 8 S ; Ref. 5 )

* Yrast states. A4 0 A r ( 1 1 B , 1 3 N ) 3 8 S ; Ref. 3 1 )

39 b) Energy levels of S below 2.8 MeV.

-55-

a) Energy levels in P below 5.0 MeV. * 3 6S( 7Li, 7Be) 3 6P and ^ ( " B , 1 ^ } 3 ^ ; Ref 7 ). A 3 6S( 1 4C, 1 4N) 3 6P; Ref 5 ). b) Energy levels in 3

below 5.0 MeV.

35 a) Energy levels in Si below 6.0 MeV. * 3 6S< 1 4C, 1 50) 3 5Si; Ref. 5 ). b) Energy levels in 3 6Si below 6.0 MeV.

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