A study of the 30Si(d,p)31Si reaction

Nuclear Physics A 662 Ž2000. 112–124

A study of the 30 Si Žd,p .31 Si reaction ˇ Piskorˇ a, J. Novak ˇ ˇ ´ a, J. Cejpek a, V. Kroha a , J. Dobesˇ a , S. ´ a, E. Simeckova P. Navratil ´ b,c a

c

ˇ ˇ Czech Republic Nuclear Physics Institute, Academy of Sciences of the Czech Republic, 250 68 Rez, b Department of Physics, UniÕersity of Arizona, Tucson, AZ 85721, USA ˇ ˇ Czech Republic Nuclear Physics Institute, Academy of Sciences of the Czech Republic, 250 68 Rez, Received 12 July 1999; received in revised form 5 October 1999; accepted 11 October 1999

Abstract The reaction 30 SiŽd,p. 31 Si is studied at the incident deuteron energy of 12.3 MeV. Proton groups are analysed using a multi-angle magnetic spectrograph with the resolving power ErD E f 1000. Angular distributions of proton groups corresponding to 14 bound states and to 8 unbound states of the 31 Si nucleus are analysed using DWBA calculations. Deduced transferred orbital angular momenta and spectroscopic factors differ for some states from those obtained previously. In particular, we attribute l s 1 transfer to the state at 5.281 MeV. Shell-model calculations are performed in a complete 0" v model space for positive-parity states and in a restricted 1" v model space including f 7r2, p3r2 and p1r2 neutron orbits for negative-parity states. The excitation energies as well as spectroscopic strengths of most of strongly excited positive- and negative-parity states are reasonably well interpreted within the shell model. q 2000 Elsevier Science B.V. All rights reserved. PACS: 27.30qt; 25.45.Hi; 21.60.Cs

Keywords: NUCLEAR REACTIONS 30 SiŽd,p. 31 Si; E s 12.3 MeV; Measured s ŽQ .; 31 Si deduced levels, l, p , spectroscopic factors; Enriched target; NUCLEAR STRUCTURE 30,31 Si; Shell model calculations

1. Introduction Not many works devoted to the structure 31 Si have been published in recent years. Most of the results were presented by Endt in the compilation of Ref. w1x. Raman et al. w2x reported on thermal neutron capture g-ray measurements. Ten low-spin levels out of the 33 known excited levels in 31 Si below the neutron separation energy w1x were populated by primary g-transitions. Accurate and precise neutron separation energy and excitation energies of nine excited states in 31 Si were deduced from the Žn,g . experiment. 0375-9474r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 Ž 9 9 . 0 0 4 2 5 - X

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High-resolution neutron transmission measurements performed on 30 Si for neutron energies from 0.2 to 1400 keV were reported in Refs. w3,4x. Single-particle orbits were ascribed to most of the 29 observed resonances above the neutron threshold in 31 Si. Direct single-nucleon transfer reactions are commonly considered to be a powerful tool in studying the structure of final nuclei. The 30 SiŽd,p. 31 Si reaction to bound as well as to unbound states has been investigated in Refs. w5–9x. The 30 SiŽt,d. 31 Si reaction to several low-lying states of the final nucleus 31 Si is discussed in Refs. w10,11x. As can be seen in the compilation by Endt w1x and as was also pointed out by Raman et al. w2x, serious discrepancies exist between spectroscopic information extracted from the Žd,p. reaction by different authors w5–8x. This served us as an incentive to inspect carefully the primary data presented in those papers and, more importantly, to repeat the Žd,p. measurements. Several shell-model studies w12,13x have dealt with the structure of states in 31 Si employing the 1 s–0 d model space. No shell-model calculation including orbits of the 0 f–1 p shell have up to now been reported. In the present paper, the levels in 31 Si are studied by means of the 30 SiŽd,p. 31 Si reaction. Experimental procedures are discussed in Section 2. The angular distributions for most of bound states and some unbound states are analysed using DWBA calculations with parameters given in Section 3. Section 4 contains results and discussion of the corresponding spectroscopic information. In Section 5, the shell-model calculations for nuclei 30 Si and 31 Si are performed in model spaces that include the 1 s–0 d and 0 f–1 p shells. Results of these calculations satisfactorily agree with the experimental data. Conclusions are given in Section 6.

2. Experimental procedure and methods of data analysis

2.1. Data acquisition The 30 SiŽd,p. 31 Si reaction was studied at an incident energy of 12.3 MeV. The cyclotron beam of deuterons was momentum separated by means of 1008 analysing magnet and focused onto the 50 m grcm2 target of SiO 2 , enriched to 95.5% in 30 Si. Carbon backing was 10 m grcm2 . The multi-angle magnetic spectrograph w14x, giving 11 spectra at different angles in one run, was used for momentum analysis of the protons. Three runs were performed with different orientations of the spectrograph with respect to the beam direction, covering the angular range 08–808. The exposures were 1500 m C in the first two runs and 500 m C in the third one. Three additional runs were made with 55 m grcm2 natural SiO 2 target under the same conditions. The proton spectra were registered by means of 700 mm long nuclear emulsion plates Ilford L4 located in the focal planes of the spectrograph channels and covered by the teflon-coated aluminium absorbers to remove the heavier products of competing reactions. The plates were scanned with the automatic plate scanner w15x and recorded as binary files in computer memory. The 66 resulting files were processed by computer code enabling to decompose the spectra in an interactive mode, part by part, determining the peak position, peak areas and corresponding error estimates. The overall achieved

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Fig. 1. Proton spectrum from the Žd,p. reaction on the 30 SiO 2 target, taken near the zero angle and at Ed s12.3 MeV. Levels of 31 Si are labelled by corresponding excitation energies in MeV. Impurity lines are marked by the final nucleus symbol only.

resolving power ErD E amounted to about 1000. A part of typical proton spectrum taken at forward reaction angles is displayed in Fig. 1. 2.2. Calibration and energy eÕaluation Though the reactions 30 SiŽd,p. 31 Si were investigated earlier w5,7,8x with rather high resolution, the accuracy was not sufficient to make unambiguous comparison with neutron transmission measurements Žsee Refs. w3,4x.. This was one of the reasons why the calibration and state identification have been made with extreme care in our study. The best estimates of the Žd,p. reaction Q-values were used for simultaneous calibration of spectra taken in all spectrograph channels and each run. The neutron binding energies for all nuclei were taken from Ref. w16x. The excitation energies of 29 – 31 Si were obtained from Refs. w1,2x, while those of 13,14 C and 15 N were from Ref. w17x and those of 17 O from Ref. w18x. Error estimates of peak positions and of Q-values were taken properly according to data processing procedures described elsewhere w19x.

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2.3. Absolute differential cross sections The relative differential cross sections have been extracted for all states at angles where the statistics was sufficient or where the impurity states did not interfere. The absolute values of cross sections are obtained by multiplying the relative ones by a conversion factor calculated as a weighted mean of values found by several independent ways: 1. Cross sections were calculated from the nominal thickness of 50 m grcm2 for the used SiO 2 target Ženriched to 95.5% in 30 Si.. 2. The ground state of 31 Si and the first excited state at 1.273 MeV in 29 Si have the q same spin and parity J p s 32 . For both states, the angular distributions are constructed in relative units from experiment made on SiO 2 target with natural silicon. Both angular distributions have very similar form w5x. Employing the absolute values of cross section for the 28 SiŽd,p. 29 Si reaction performed at 11 and 13 MeV deuteron energy by Schiffer et al. w20x and the known natural abundance of silicon nuclides, we converted the relative differential cross section for the 30 SiŽd,p. 31 Si ground state transition into the absolute one. 3. Employing the known abundance of 30 Si and 31 Si in the natural Si target the same y procedure as in b. was applied to the 3.533 MeV J p s 32 state in 31 Si and to the 7.508 MeV J p s 2y state in 30 Si. The Žd,p. reaction Q-values for these states differ only by 47 keV. The absolute value of differential cross section for the 30 Si state is taken from the work of Baxter and Hinds w21x. Though the differential cross section found by these three ways fluctuated within about 20%, the resulting absolute values of cross sections, based on the weighted mean of the conversion factor from relative to absolute cross section, are believed to be accurate to better than 10%.

3. DWBA analysis The local finite-range DWBA calculations for the bound states were performed by means of the code DWUCK3 w22x using conventional separation energy prescription. The Woods–Saxon well with r 0 s 1.25 fm, a 0 s 0.65 fm and the factor l s 25 in the Thomas spin–orbit term of the same geometry were used. The DWBA calculations for neutron stripping to fractionated single-particle resonances were done by the GDWUCK code, a modified version of the DWUCK3 in accordance with Coker w23x. The Gamow functions, i.e. the eigenstates of complex energy, were calculated by the code GAMOW w24x. The overlap integrals with Gamow 2 functions as external form factors and with the Gaussian weight factors eya r were calculated by GDWUCK for three values of a with a subsequent quadratic extrapolation to a s 0. The optical model potential parameters used in present DWBA calculation for entrance and exit channels are given in Table 1. Non-locality correction factors for

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Table 1 Optical potential parameters Ženergies in MeV, lengths in fm. 30

Siqd 31 Siqp 30 Siqn a b c

a

V

r0

a0

WD

rD

aD

Vso

rso

aso

rc

97.7bc

1.12 1.25 1.25

0.746 0.65 0.65

16.9 13.5

1.40 1.25

0.705 0.47

7.0 7.5 l s 25

1.12 1.25 1.25

0.746 0.47 0.65

1.30 1.25

Deuteron OM potentials from Ref. w9x, Ed s12.3 MeV. V s 51.1490y0.5678Q from Ref. w25x. Depth of the bound-state potential adjusted to give the correct binding energy.

deuterons and protons were taken as 0.54 and 0.85, respectively. The finite-range parameter 0.621 was used. The spectroscopic factors were extracted by the least square fit of theoretical angular distribution to points within the main maximum according to the formula Sl j s sexp Ž u .r1.53 s DWUCK Ž u .. The experimental angular distributions together with the fitted DWBA curves are given in Fig. 2.

4. Results and discussion

4.1. Energy leÕels of

31

Si

The most precise data on excitation energies of nuclei or on their neutron binding energies are obtained from Žn,g . experiments. Excitation energies of 31 Si below the neutron separation energy and their error estimates evaluated from the present experiment together with the results from previous Žd,p. measurements and Žn,g . experiment are given in Table 2. Orbital angular momenta l transferred in Žd,p. reaction are also included. Table 3 contains analogous data for states above the neutron separation energy in comparison with results from neutron transmission experiments. With the exception of Žn,g ., some Žn,n. and the present Žd,p. measurements, the excitation energies were given without quoted errors. Accuracy of excitation energies determined from reactions with charged particles does not allow us, in some cases, to conclude unambiguously about their mutual correspondence. 4.2. Spectroscopic strengths and l-assignments Table 4 contains the spectroscopic strengths Ž2 j q 1. S Ž G factors. as extracted from neutron stripping reactions by different authors w5–8,10,11x including results of the present experiment, the estimated accuracy of our spectroscopic strengths is about 30%. The l value for the 5.281 MeV level has been changed from the previous l s 0 to l s 1 for two reasons: Ži. the measured angular distribution Žsee Fig. 2. for this level is q reproduced better with l s 1 than with l s 0 and Žii. the calculated 12 states Žsee 1q Section 5. are at 815, 4493 and 5914 keV. The known 2 states at 0.752 and 4.719 MeV can be associated with the first two calculated states, but associating the 5.281

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q MeV level Žwith a 12 assignment. with the 5.914 MeV state would lead to a mismatch of 633 keV, which, for a positive-parity state in the sd shell, is very unlikely. Moreover,

Fig. 2. Angular distribution of protons from the of DWBA analysis.

30

SiŽd,p. 31 Si reaction at Ed s12.3 MeV. Curves are the result

118

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Fig. 2 Ž continued ..

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Fig. 2 Ž continued .. y the 5.281 MeV level Žwith a 12 assignment. can be matched better with the calculated y 1 state at 5.606 MeV. 2 It seems that there exist inconsistencies in previous DWBA analyses of the 30 Ž Si d,p. 31 Si reaction. We have repeated the DWBA calculations for some cases with OM parameters from Refs. w5–8x. We properly reproduced results given by Wildenthal and Glaudemans w6x where transitions to states up to 3.53 MeV are discussed. Betigeri et al. w5x did not specify the geometry of the bound state well. This fact was not the only difficulty with a proper reproduction of their DWBA calculations. We reproduced l s 2 transfers sufficiently well, but we obtained about two times smaller S-factors for l s 0 transfers. Similarly, we reproduce the calculations of Ref. w7x with the exception of the 5.281 MeV state. We were not also able to reproduce fully the DWBA calculations of Ref. w8x. For l s 1 transfers, we deduce spectroscopic factors which are smaller than those given in

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Table 2 Energy levels of 31 Si below the neutron separation energy and l-values found in Žd,p. reactions Žthis work, w5,7,8x. in comparison with energy levels from Žn,g . reaction w2x and from the present shell-model calculations Ženergies in MeV. Ref. w5x

This work

Ref. w7x

Ex

l

Ex

l

0.0 0.7520Ž4. 1.695Ž2. 2.3161Ž8. 2.7875Ž4. 3.1330Ž3. 3.5330Ž3. 3.8736Ž10. 4.2616Ž5. 4.3824Ž3. 4.6905Ž16. 4.7191Ž3. 5.2813Ž3. 5.4419Ž4. 5.604Ž3. 5.680Ž4. 5.8725Ž5. 5.9573Ž7. 5.986Ž5. 6.071Ž2. 6.106Ž1. 6.239Ž5. 6.351Ž4. 6.417Ž4. 6.461Ž1. 6.583Ž1.

2 0

0.0 0.750 1.687 2.310 2.782 3.130 3.526 3.866 4.257 4.377 4.687 4.715 5.275 5.433 5.592 5.671 5.864 5.944 5.975 6.063 6.097 6.240 6.351

2 0 2 2 2 3 1

a

2 2 3 1 2 1 0 1 3

1 1Ž2.

3

Ref. w2x

l

Ex

l

Ex

2 0 Ž2. Ž2. Ž2. 3 1 Ž4. Ž2. Ž1.

0.0 0.75223Ž3. 1.69492Ž4. 2.31694Ž10. 2.78803Ž6.

0 Ž2.

0.0 0.752 1.695 2.320 2.787 3.134 3.534 3.874 4.259 4.381 Ž4.691. 4.719 5.280 5.441 5.603 5.677 5.867 Ž5.956. 5.983 6.068 6.108 6.251 6.348 6.413 6.462 6.587

2 1 0 0

Ref. w8x

1 2 2 3 2

0 0 3

1 Ž2. Ž2. Ž2.

3.53292Ž3.

4.38237Ž4.

5.28136Ž4.

5.87315Ž7. 5.95791Ž19.

Shell model, Ref. w27x a Ex 2 Jp 0.0 0.815 1.606 2.295 2.871 3.072 3.635 3.697 3.825 5.383

3q 1q 5q 3q 5q 7y 3y 7q 3q 3y

4.493 5.606 5.458

1q 1y 7y

6.276

1y

6.182

7y

Ž3. Ž2. Ž2.

This work.

Ref. w8x. There are as well discrepancies in calculations for other states. Note that the spectroscopic factors of Ref. w8x would imply a significant excess of the sum rule limits.

5. Shell model calculations Shell-model calculations for the 30 Si and 31 Si isotopes were performed using the many-fermion-dynamics shell-model code w26x. The positive-parity-state calculations were done in a complete 0" v space using the Wildenthal’s USD interaction w27,28x scaled by the factor Ž Ar18.y0 .3. For both 30 Si and 31 Si, we used the identically scaled A s 31 interaction. The negative-parity states were calculated in a restricted 1" v space. We limited the model space to the neutron 0 f 7r2, 1 p3r2, and 1 p1r2 levels combined with the unrestricted occupation of the sd-shell levels. The effective interaction for the negativeparity calculation consists of two parts. For the sd-shell part we kept the sd-shell USD

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Table 3 Energy levels of 31 Si above the neutron separation energy and l-values found in Žd,p. reaction Žthis work, w4,7,8x. compared with energy levels, l- and 2 J-values from the Žn,n. reaction Ž w3x. Ženergies in MeV. This work Ex

l

Ref. w7x Ex l

Ref. w8x Ex

l

Ref. w4x Ex

2 Jp

6.5920

1y

6.7649

1q

6.8149 6.8803

l02 3y

7.2116 7.2693 7.3084 7.3584 7.3685 7.4044

l02 l02 3y 1y 1y 3y

7.4377 7.5353 7.7311 7.7656 7.8209 7.8470 7.8554 7.8818 7.8993 7.9259 7.9428 7.9535

l02 1y 1q l02 1y

6.600Ž1. 6.771Ž2. 6.8141Ž5.

2

6.9152Ž5.

2

7.209Ž2. 7.266Ž3. 7.311Ž2.

6.78 6.81 6.87 6.90

Ž2.

6.817 6.886 6.915

Ž1.

7.368Ž2. 7.401Ž2.

6.58956Ž5. 6.59222Ž5. 6.60205Ž6. 6.7646Ž8. 6.7718Ž4. 6.8150Ž5. 6.8802Ž5.

7.397

6.996Ž1. 7.2108

3

7.42

Ž2.

7.765Ž2.

7.9046Ž8.

7.991Ž4. 8.0175Ž12. 8.0349Ž12. 8.070Ž3. 8.1159Ž13. 8.1648Ž15.

7.432

Ž2.

7.757

3

7.88

Ž3.

3 3

7.99 8.01 8.04 8.09

3 3 3 3

7.899

l

Ž3. 1

01 Ž1.

1 Ž3. Ž3.

0 Ž2. 2

3

2

3

2

7.3107

7.4096 7.4383Ž4.

2J

Ž2.

7.205 7.265 7.305 7.356

7.29

Ref. w3x Ex

Ž1.

3y 1y 3y 1y l )1 3y

3 3

interaction used for the positive-parity-state calculations. The sd and fp cross-shell interaction we obtained from sdpf interaction described by Warburton et al. w29x. The fp-shell single-particle energies were varied to improve the fit to the experimental negative-parity energy spectrum. The following set of the single-particle energies Žin MeV. was employed: 0, 0.784, 5.594, 7.0, 7.4 and 8.2 for the 0 d5r2, 1 s1r2, 0 d3r2, 1 p3r2, 1 p1r2 and 0 f 7r2 orbits, respectively. The harmonic-oscillator energy, " v s 13 MeV, was employed for the basis states. We note, however, that the choice of " v does not greatly influence the energy and spectroscopic factor calculations. For the negative-parity state calculation, we used the standard projection method w30x to remove the spurious centre-of-mass components. In particular, one adds the term b HCM y 32 " v 4 with a large b to the shell-model Hamiltonian. In the present case we

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Table 4 Comparison of spectroscopic strengths extracted from neutron stripping reactions and values from shell-model calculations Ž2 jq1. S Žd,p. E x wMeVx

l

a

w5x

0.000 0.752 1.695 2.316 2.788 3.133 3.533 4.262 4.382 4.719 5.281

2 0 2 2 2 3 1 2 1 0 0 1 Ž2. 3 0 1 1 Ž2. 2 Ž2. 3 2 Ž3. Ž2. 3 Ž2. 2 2 2 Ž2. 3 3 3 3 3 3 3 3

2.812 0.506 ( 0.1 0.160 0.258 4.760 1.612 0.176 0.552 0.220

3.27 1.33 0.3 0.33 0.37 6.67 2.23 0.35 0.60 0.47 0.08

5.442 5.817 5.872 5.957 6.071 6.106 6.239 6.417 6.461 6.583 6.814 6.915 6.952 7.438 7.905 8.018 8.035 8.070 8.116 8.145 8.165 a

Žt,d.

Shell-model

w7x

w8x

w6x

w10x

w11x

w27x a

w12x

w13x

3.4 0.54 0.12 0.24 0.24 4.4 1.9

2.52 0.65 0.22 0.20 0.72 3.49 1.54

1.64 0.28

2.362 0.429 0.113 0.103 0.283 5.742 2.822 0.183 0.248 0.148

2.88 0.28

1.92 0.56

0.08 0.12

0.36 0.30

0.076

3.0 0.88 0.07 0.26 0.33 6.0 5.3 0.19 1.3 0.37 3.9

0.888

0.11 0.29 2.06 1.08

0.24 0.10

1.131 Ž0.268.

0.568 0.116 0.034

0.152 Ž0.092. 0.080

0.66 0.05 0.33

0.326 0.006

0.01 0.04

0.288 0.268

0.367 0.08 0.14

0.120 0.18 0.104 0.152

0.608 0.304 0.160 0.144

Ž0.200. 0.088 0.044 0.096 Ž0.020. 0.0608 0.08 0.064

0.384 0.056 0.120

This work.

set b equal 100. We note that the interaction used is not translationally invariant and also the type of model-space restriction that we employed does not guarantee the separation of internal and centre-of-mass motion. Therefore the centre-of-mass components cannot be removed completely.

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The spectroscopic factor calculation was performed using a code developed to handle the outputs from the many-fermion-dynamics shell-model code. Calculated excitation energies and spectroscopic strengths of 31 Si states are compared with the experimental values in Tables 2 and 4, respectively. Calculated positive-parity states above 4.493 MeV are not listed. The positive-parity states up to 4.5 MeV calculated with the USD interaction agree quite well as concerns the excitation energies as well as spectroscopic strengths. Some more l s 2 states are observed experimentally with excitation energy above 5 MeV which are not given by the shell model and whose wave functions might belong to configurations outside the presently used model space. The agreement of calculated negative-parity states with the experiment is not perfect but is still reasonable. For the l s 1 states, experimental excitation energies are more compressed than the calculated ones. For the l s 3 states, experimental states above 7 MeV are not present in the shell-model output. Generally we conclude that for the better description of experimental data a stronger configuration mixing than that provided by the present shell-model calculations should be considered.

6. Conclusion The 30 SiŽd,p. 31 Si reaction has been studied with high resolution using a multi-angle magnetic spectrograph. The experiments yielded new spectroscopic information on the structure of the states of the 31 Si nucleus. Transferred orbital angular momenta and spectroscopic factors as deduced by the DWBA analysis differ for some states from those obtained previously. Namely, we attribute the l s 1 transfer to the state at 5.281 MeV. Shell-model calculations have been performed in a complete 0" v model space for positive parity states and in a restricted 1" v model space including f 7r2, p3r2 and p1r2 neutron levels for negative parity states. Most of the strongly excited positive- and negative-parity states are reasonably well interpreted within the shell model as concerns the excitation energies as well as spectroscopic strengths.

Acknowledgements P.N. and J.D. acknowledge gratefully support from the NSF Grant No. PHY96-05192 and GACR Grant No.202r99r0149, respectively.

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