Inverse-kinematics proton scattering from 42,44 S, 41,43 P and the collapse of the N=28 major shell closure
L. A. Riley, D. Bazin, J. Belarge, P. C. Bender, B. A. Brown, P. D. Cottle, B. Elman, A. Gade, S. D. Gregory, E. B. Haldeman, K. W. Kemper, B. R. Klybor, M. A. Liggett, S. Lipschutz, B. Longfellow, E. Lunderberg, T. Mijatovic, J. Pereira, L. M. Skiles, R. Titus, A. Volya, D. Weisshaar, J.C. Zamora, R. G. T. Zegers
aa r X i v : . [ nu c l - e x ] A p r Inverse-kinematics proton scattering from , S, , P and the collapse of the N = 28 major shell closure L. A. Riley, D. Bazin,
2, 3
J. Belarge,
2, 3, ∗ P. C. Bender, B. A. Brown,
2, 3
P. D. Cottle, B.Elman,
2, 3
A. Gade,
2, 3
S. D. Gregory, E. B. Haldeman, K. W. Kemper, B. R. Klybor, M.A. Liggett, S. Lipschutz,
2, 3
B. Longfellow,
2, 3
E. Lunderberg,
2, 3
T. Mijatovic, J. Pereira,
2, 3
L. M. Skiles, R. Titus,
2, 3
A. Volya, D. Weisshaar, J.C. Zamora, and R. G. T. Zegers
2, 3, 5 Department of Physics and Astronomy, Ursinus College, Collegeville, PA 19426, USA National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI, 48824, USA Department of Physics and Astronomy, Michigan State University, East Lansing, MI, 48824, USA Department of Physics, Florida State University, Tallahassee, FL 32306, USA Joint Institute for Nuclear Astrophysics - Center for the Evolution of the Elements,Michigan State University, East Lansing, MI 48824, USA (Dated: April 9, 2020)Excited states of the neutron-rich isotopes , S and , P have been studied via inverse-kinematics proton scattering from a liquid hydrogen target, using the GRETINA γ -ray trackingarray to extract inelastic scattering cross sections. Deformation lengths of the 2 +1 excitations in , S have been determined and, when combined with deformation lengths determined with elec-tromagnetic probes, yield the ratio of neutron-to-proton matrix elements M n /M p for the 2 +1 exci-tations in these nuclei. The present results for , P( p, p ′ ) are used to compare two shell modelinteractions, SDPF-U and SDPF-MU. As in a recent study of Si, the present results on , Pfavor the SDPF-MU interaction.
I. INTRODUCTION
One of the highest scientific priorities for nuclear struc-ture physicists during the last few decades has been todetermine the behavior of the major neutron shell closureat N = 28 and to understand the mechanism underlyingits collapse in neutron-rich nuclei near Si, which is closeto the neutron drip line. This shell closure is strongly de-fined in the stable N = 28 isotone Ca, but appears tonarrow and then collapse as protons are removed. Theenergy of the 2 +1 state decreases from 3832 keV in Ca[1] to 1329 keV in the radioactive nucleus S [2, 3] andthen to 742 keV in Si [4]. In fact, the 2 +1 state energyin Si is lower than it is in the N = 26 isotope Si(986 keV [5]) so that the most recognizable signature ofa major shell closure – a significant increase in the en-ergy of the 2 +1 state – has disappeared entirely in the Siisotopes at N = 28.In the present work, we report results of inelastic pro-ton scattering studies of the radioactive N = 28 isotones S and P and the N = 26 isotones S and P per-formed in inverse kinematics with a liquid hydrogen tar-get to uncover several new aspects of the behavior ofnuclei in the vicinity of Si. In , S, we are able tocompare the results of the present inelastic scatteringmeasurement of the 2 +1 states to previous Coulomb ex-citation measurements of the same transitions to deter-mine whether the excitations of these states are isoscalar.In addition, we use the results of the , P( p, p ′ ) mea-surements to compare the SDPF-U and SDPF-MU shell ∗ J. Belarge is currently a MIT Lincoln Laboratory employee. NoLaboratory funding or resources were used to produce the re-sults/findings reported in this article. model interactions, as was done in a recent study of Si[6].
II. EXPERIMENTAL DETAILS
The experiment was performed at the Coupled-Cyclotron Facility of the National SuperconductingCyclotron Laboratory at Michigan State University(NSCL) [7]. The secondary beams were produced byfragmentation of a 140 MeV/nucleon Ca primary beamin a 1222 mg/cm
Be production target and separatedby the A1900 fragment separator [8]. The momentumacceptance of the A1900 was set to 2%. A 300 mg/cm aluminum achromatic wedge was used to further separatethe secondary beams by Z . The beams of interest in thepresent work were produced with two magnet settings ofthe A1900 and are summarized in Table I.Secondary beam particles were identified upstream ofthe reaction target by times of flight from the A1900 ex-tended focal plane and the object position of the S800spectrograph [9]. A scintillator in the focal plane of theS800 was used to stop both timing measurements. Thebeam then passed through the NSCL/Ursinus CollegeLiquid Hydrogen Target, based on the design of Ryuto etal. [10]. The target was installed at the target positionof the S800. Outgoing beam particles were identified byenergy loss in the S800 ionization chamber and time offlight. The reaction kinematics were such that all fourbeams, over the full range of possible projectile kineticenergies within the target, were scattered into labora-tory angles below 2 ◦ , falling entirely within the 7 ◦ × ◦ angular acceptance of the S800. The GRETINA γ -raytracking array [11, 12] was centered on the target. Eight TABLE I. Secondary beam properties and yields.Secondary Purity Mid-Target Energy Mid-Target TotalBeam (%) (MeV/nucleon) v/c
Particles S 2 62.5 0.349 9 . × P 30 57.7 0.336 1 . × S 32 70.2 0.368 5 . × P 9 64.7 0.354 1 . × modules housing four 36-fold segmented high-purity ger-manium crystals were installed on one of the GRETINAmounting hemispheres to accommodate the target. Twomodules were centered at 58 ◦ , four at 90 ◦ , and two at122 ◦ with respect to the beam axis.The liquid hydrogen was contained by a cylindrical alu-minum target cell with 125 µ m Kapton entrance and exitwindows, mounted on a cryocooler. The nominal targetthickness was 30 mm. The target cell and cryocooler weresurrounded by a 1 mm thick aluminum radiation shieldwith entrance and exit windows covered by 5 µ m alu-minized Mylar foil. The temperature and pressure of thetarget cell at 17.00(25) K and 880(10) Torr were moni-tored throughout the experiment. The variations in thetemperature and pressure of the target cell correspondedto a 0.3 % uncertainty in target density.The pressure difference across the Kapton entrance andexit windows caused them to bulge outward. The result-ing additional target thickness was determined by fitting geant4 [13] simulations of the beam particles travers-ing the target to the measured kinetic energy distribu-tion of the outgoing beam particles. Before the liquidhydrogen target was filled, the kinetic energy spectra ofthe secondary beams passing through the empty targetcell were measured. Simulations of the beams passingthrough the full target were run in which initial beamenergies were drawn from these measured empty-cell ki-netic energy distributions. The thickness of the outwardbulge of the Kapton entrance and exit windows was var-ied in the simulations of each beam, and the resultingoutgoing kinetic energy distributions were were fit by asimple scaling of the simulated spectra.Measured kinetic energy spectra of the beams aftertraversing the full target, relative to the kinetic energycorresponding to the center of the S800 momentum ac-ceptance, are shown in Fig. 1. The dashed spectra in thefour main panels of Fig. 1 are simulated spectra assum-ing the target bulge thickness giving the best fit to themeasured spectra. The insets show the figure of meritfrom the log-likelihood fitting procedure plotted vs. thesimulated window bulge thickness. This process yieldedbest-fit bulge thicknesses of 1.06 mm and 0.96 mm forthe S and P beams and 1.20 mm and 1.22 mm forthe S and P beams. The statistical uncertainties ineach of these results, corresponding to the minimum fig-ure of merit + 1, are on the order of 10 − mm. We at-tribute the discrepancies between the best-fit simulations and the measured spectra, as well as the larger ≈ . .The mid-target beam energies and average beam veloc-ities given in Table I were also determined using thesesimulations. III. ANALYSIS AND RESULTS
Projectile-frame γ -ray spectra measured via inverse-kinematics proton scattering from , S and , P ap-pear in Figs. 2-5. The average projectile velocities inTable I were used in the Doppler reconstruction of the γ rays emitted in flight. The solid curves in the figures arefits consisting of a linear combination of geant4 simu-lations of the response of GRETINA to the observed γ rays with a prompt background, shaded in grey, consist-ing of two exponential functions. The contribution of thenon-prompt room background to the fits was negligible.The γ -ray energies and intensities extracted from the fitsare listed in Table II. The γ -ray energies reported to theright in Table II were determined by varying the simu-lated energies of the emitted γ rays to optimize the fitsof the response functions to the measured spectra. Theerror ranges correspond to 95% confidence intervals.The position of the liquid hydrogen target relative tothe focus of GRETINA along the beam axis strongly im-pacts the energies of γ rays in Doppler reconstruction.We found the target offset to be 11.1(4) mm by fixingthe energy of the 1329 keV 2 +1 → +g . s . transition in Sand varying the target offset in simulations to obtain abest fit to the measured spectrum. We chose this tran-sition, because its energy was determined to a precisionof 1 keV in a measurement of S nuclei at rest in thelaboratory [14]. The best fit to the 1329 keV peak alongwith a plot of the figure of merit from the log-likelihoodfit vs. the offset of the target along the beam axis appearin Fig. 6. We accounted for the mean lifetime of the 2 +1 state of S, deduced from the B ( E
2; 0 + g.s → + ) valuemeasured via Coulomb excitation [15], of 3.5(10) ps inthe simulations. However, we found that the impact ofthe lifetime on the resulting best-fit target offset was be-low 0.1 mm, a statistically insignificant effect relative tothe 0.4 mm uncertainty in the result.The mean lifetimes of the 2 +1 state of S of 20.6(15) psand the 4 + state of S at 2457 keV of 76(24) ps re-ported by Parker et al. [3] are long enough to impact theDoppler-corrected γ -ray line shapes and were includedin the simulations of the 903 keV and 1150 keV γ raysde-exciting these states. These transitions were not ob- S P -8 -6 -4 -2 0 2 4 6 802×10 S P F O M ( a r b . un i t s ) F O M ( a r b . un i t s ) F O M ( a r b . un i t s ) F O M ( a r b . un i t s ) Relative Kinetic Energy (%) C o un t s / ( . % ) FIG. 1. (Color online) Relative kinetic energy spectra of the beams measured downstream of the target in the S800. Thedashed spectra are the geant4 fits described in the text. The insets are plots of the figure of merit from log-likelihood fits ofthe simulated beam particles vs. the thickness of the outward bulge of the Kapton entrance and exit windows of the target.
Energy (k eV)
500 1000 1500 2000 2500 3000 C o un t s / ( k e V ) + ⟶ + g . s .
903 1820 29902190 v/c = 0.349
FIG. 2. (Color online) Projectile-frame spectrum of S mea-sured via inverse-kinematics proton scattering. served with sufficient statistics in the present work toperform independent mean lifetime measurements. Theline shapes corresponding to the γ rays de-exciting the173 keV and 188 keV first excited states of P and Pshow low-energy tails consistent with lifetimes on the or-der tens to hundreds of picoseconds. The low-energy re-gions of the projectile-frame γ -ray spectra of P and Pare shown in Figs. 7 and 8. We varied both the energiesand mean lifetimes of the states to determine best-fit val- ues of τ (3 / + ) = 550(70) ps in P and τ / + = 130(40) psin P. In both cases, the best-fit transition energy variesby less than 1 keV over the full uncertainty range of themean lifetimes. Plots of the figure of merit from the log-likelihood fits vs. mean lifetime appear in the insets inFigs. 7 and 8. In addition to the statistical uncertainties,we have included the contribution of the 0.4 mm uncer-tainty in the position of the target along the beam axisin the error ranges. This is an 11% effect in P and a20% effect in P.Partial level schemes of , S and , P including thelevels populated in the present work are shown in Fig. 9.We observe several known γ rays, and have identified twonew transitions in S at 1570 keV and 2190 keV, whichwe are unable to place in the level scheme. In S, weobserve two new γ rays at 2696 keV and 3076 keV. Weplace the 2696 keV transition feeding the 2 +1 state due tothe fact that it is seen in the spectrum of γ rays measuredin coincidence with the 1329 keV 2 +1 → +g . s . transitionshown in Fig. 3(b). We are unable to place the 3076 keVtransition in the level scheme. In P, we place a new733 keV transition in the level scheme on the basis ofits observation in coincidence with the 1415 keV-gatedspectrum in Fig. 4(b), where we also see a possible weak γ ray at ≈
930 keV, which we are unable to confirm or placein the level scheme. The 420 keV transition observed byBastin et al. [17] to de-excite the state at 1588 keV alongwith the 1415 keV γ ray, and which appears in Fig. 9 as a TABLE II. Level energies, spins and parities, and γ -ray energies from Refs. [2, 16–18] and γ -ray energies, relative intensities,and cross sections from the present work. E level [keV] J π [¯ h ] E γ [keV] E γ [keV] I γ [%] σ [mb] S Ref. [16]902 2 + + ) 1820(4) 1820(30) 8(4) 2.4(12)3002 (2 + ) 3002(4) 2990(30) 20(5) 6.2(15)2100(4) <
1— 1570(30) 9(4)— 2190(30) 8(4) S Ref. [2]1329 2 + + ) 949(5) 954(4) 17(3) 4.5(8)2479(11) (4 + ) 1128(6) 1150(11) 11(3) 2.7(8)3264(6) (2 + ) 1891(10) 1899(6) 13(2) 3.7(7)1929(7) 1955(25) 2(2)4027(13) — 2698(13) 8(2) 2.1(5)— 3076(10) 8(2) P Ref. [17]173(1) (3 / + ) 172(12) 173(1) 100(2) 6(2)1150(3) 964(22) 972(1) 43(2) 13.5(6)1146(28) 1148(2) 24(2)1589(4) 1408(19) 1415(3) 23(2) 4.6(4)420(22) < P Ref. [18]188(1) 3 / + / + ) 845(4) 846(11) 15(7) 4.9(14)661(4) 656(6) 15(5)1015(4) (5 / + ) 825(5) 827(4) 38(8) 6.2(12)1104(5) (5 / + ) 911(6) 916(5) 22(5) 3.5(8)2039 (5 / + ) 1851(11) 1851 7(4) 1.8(8)1018(6) <
8— 283(6) 7(3)— 352(13) 11(4) dashed arrow, was below our detection threshold. We arealso unable to place a new 1729 keV transition. In P, weincluded in the fit the 1018 keV and 1851 keV transitionsde-exciting the excited state at 2035 keV observed in one-proton knockout from S [18]. We were only able toplace an upper limit on the intensity of the 1018 keVtransition. We were unable to place new 283 keV and352 keV transitions.The cross sections for inelastic proton scattering to ex-cited states of , S and , P listed in Table II weredetermined from the measured γ -ray yields, corrected forfeeding by transitions from higher-lying states, based onthe partial level schemes in Fig. 9. In the case of the 2 +1 state of S, the 2100 keV γ ray observed to de-excitethe 3002 keV (2 + ) state by Lunderberg et al. [16] andshown as a dashed arrow in Fig. 9 was below our detec-tion threshold. We included the 2100 keV γ ray in the fit to place an upper limit on its intensity, and included thatupper limit in the feeding correction. We observed γ rays,at 1570 keV and 2190 keV in S and at 3076 keV in S,that we could not place in the respective level schemes.We have included possible feeding of the 2 +1 states bythese γ rays in the error ranges of the measured crosssections. We find cross sections for populating the 2 +1 excitations via proton scattering of 23(6) mb in S and15(3) mb in S.We used the coupled-channels code
ECIS97 [21] andthe global optical potentials of Ref. [22] and [23] to deter-mine deformation lengths from our measured cross sec-tions for inelastic scattering to the 2 +1 states of δ =1 . S and 1 . S. The errorranges include both the uncertainties in measured crosssections and any discrepancy due to the two global opti-cal potential sets and the use of vibrational and rotational
Energy (k eV)
500 1000 1500 2000 2500 3000 C o un t s / ( k e V ) C o un t s / ( k e V ) (a)(b) + ⟶ + g . s . G(cid:0)(cid:1)(cid:2)
FIG. 3. (Color online) (a) Projectile-frame spectrum of S measured via inverse-kinematics proton scattering. (b)Proton-scattering spectrum gated on the 1329 keV γ ray. models. The impact of the choice of optical potential pa-rameter set was 2% for S and 7% in the case of S.The variation in the deformation lengths determined us-ing the vibrational and rotational models for the excita-tions was at the 3% level. Proton-scattering deformationlengths of the 2 + states of even-even neutron-rich sulfurisotopes from the present work and Ref. [19] are plottedalong with electromagnetic deformation lengths from theevaluation of Ref. [20] in Fig. 10.We did not collect sufficient statistics to perform γ -ray angular distribution measurements. However, signif-icant alignment of the residual nucleus can be expectedin direct reactions with fast beams [24–26]. We usedthe ECIS calculations described above to evaluate thepotential impact of γ -ray angular distributions on ourmeasured γ -ray yields. We integrated the angular distri-butions from ECIS of the components t and t of thepolarization tensor of the , S nuclei after excitationto their 2 +1 states via ( p, p ′ ) to determine their expec-tation values, which correspond to the orientation pa-rameters B and B in the usual notation. We foundroughly 30% oblate alignment for S and 20% oblatealignment for S, with very similar γ -ray angular distri-butions predicted by the vibrational and rotational mod-els. Following the formalism outlined in Refs. [24, 25], wecalculated the corresponding angular distribution coeffi- Energy (k eV) C o un t s / ( k e V )
500 1000 1500 2000 / + ⟶ / +
733 972 1148 17291415 C o un t s / ( k e V ) (a)(b) (cid:3)(cid:4)(cid:5)(cid:6) FIG. 4. (Color online) (a) Projectile-frame spectrum of P measured via inverse-kinematics proton scattering. (b)Proton-scattering spectrum gated on the 1416 keV γ ray. Energy (k eV) C o un t s / ( k e V ) / + ⟶ / +
283 352 656 827 /
846 916 1018 1851
200 400 600 800 1000 1200 1400 1600 1800 2000
FIG. 5. (Color online) Projectile-frame spectrum of P mea-sured via inverse-kinematics proton scattering. cients and performed simulations including the resulting γ -ray angular distributions and compared the resulting γ -ray yields with those obtained assuming isotropic γ -rayemission in the projectile frame. The predicted angulardistribution affected the yield of the 903 keV γ ray in Sat the 3% level and that of the 1329 keV γ ray in Sat the 1% level. These effects fall well within the sta-tistical uncertainties in the γ -ray yields. It is important C o un t s / ( k e V ) E(cid:7)(cid:8)(cid:9)g(cid:10) (cid:11)(cid:12) eV)
Ta(cid:13)(cid:14)(cid:15)(cid:16) (cid:17) (cid:18) ff s(cid:19)(cid:20) (cid:21)(cid:22)(cid:23) ) F.(cid:24)(cid:25)(cid:26)(cid:27)(cid:28)(cid:29)(cid:30)(cid:31) !n"
FIG. 6. (Color online) The region of the projectile-frame spec-trum of S surrounding the photopeak of the 1329 keV γ rayde-exciting the 2 +1 state. The smooth curve is the geant4 fitcorresponding to the best-fit value of the target offset fromthe focus of GRETINA along the beam axis of 11.1 mm. Theinset shows the figure of merit from the fit vs. the simulatedtarget offset. The dashed line corresponds to the 95% confi-dence interval of 0.4 mm.
100 120 140 160 180 200 220 240 260 280 300
Energy (k eV) C o un t s / ( k e V ) &’()*+, a r b . un i t s )
450 500 600 650
Mean Lifetime (ps)
FIG. 7. (Color online) The low-energy region of the projectile-frame spectrum of P measured via inverse-kinematics pro-ton scattering. The smooth curve is the geant4 fit corre-sponding to a mean lifetime of the J π = (3 / + ) first excitedstate of 550 ps. The inset shows the figure of merit from the fitvs. the simulated mean lifetime. The dashed line correspondsto the 95% confidence interval of ±
35 ps. to note that the significant feeding of the 2 +1 states byde-excitations of higher-lying states leads to a reduceddegree of alignment relative to these estimates. IV. DISCUSSION
Given the importance of Si and neighboring isotopesfor building our understanding of nuclear structure closeto the neutron dripline, a range of observables in thesenuclei should be measured and understood. In this sec-
160 180 200 220
260 280 30020 M9:; <=>?@ABC DHI ]
80 100 120
JKL
NOPQRSUVWXYZ n i t s ) [\]^_‘bcdkefh ijlmop qrtuv FIG. 8. (Color online) The low-energy region of the projectile-frame spectrum of P measured via inverse-kinematics pro-ton scattering. The smooth curve is the geant4 fit corre-sponding to a mean lifetime of the J π = 3 / + first excitedstate of 130 ps. The inset shows the figure of merit from the fitvs. the simulated mean lifetime. The dashed line correspondsto the 95% confidence interval of ±
30 ps. tion, we use the present , S( p, p ′ ) results to extract M n /M p , the ratio of the neutron and proton transitionmatrix elements for the 0 + gs → +1 excitations, whichprovide insights about the presence or absence of closedshells. In addition, we use the , P( p, p ′ ) results toadd to a comparison of the SDPF-U and SDPF-MU shellmodel interactions recently begun by Gade et al. in astudy of Si [6].The comparison of the present , S( p, p ′ ) results onthe 0 + gs → +1 excitations with the previous Coulomb ex-citation measurements of the same transitions [15, 27]allows us to determine M n /M p . Coulomb excitationmeasures the proton transition matrix element exclu-sively, while proton scattering involves both the protonand neutron transition matrix elements. If the excita-tion is isoscalar, then the ratio M n /M p of the neutronand proton transition matrix elements is equal to the ra-tio N/Z of the neutron and proton numbers, and hence( M n /M p ) / ( N/Z ) is equal to 1. This ratio is determinedfrom the proton inelastic scattering deformation length δ ( p,p ′ ) and the proton deformation length δ p using theequation, [28] M n M p = b p b n (cid:18) δ ( p,p ′ ) δ p (cid:18) b n b p NZ (cid:19) − (cid:19) , (1)where b n /b p is the ratio of the sensitivities of the protonscattering reaction to the neutron and proton contribu-tions to the excitation. The ratio b n /b p is approximately3 at proton energies below 50 MeV, and approximately1 at 1 GeV. However, there is considerable uncertaintyabout the value of b n /b p at the energy of this exper-iment – 60-70 MeV in the center of mass frame. Sofor the purposes of the present analysis, we assume that b n /b p = 2 ±
1, which despite the large uncertainty al- wxyz{ | }~ (cid:127)(cid:128) (cid:129)(cid:130)(cid:131)(cid:132) S (2 ) (cid:133) (cid:134)(cid:135)(cid:136)(cid:137)(cid:138) (cid:139) (cid:140)(cid:141) (cid:142)(cid:143) (cid:144)(cid:145)(cid:146)(cid:147) S (cid:148) (cid:149) (cid:150)(cid:151)(cid:152)(cid:153) (2 ) (cid:154) (cid:155)(cid:156)(cid:157) (cid:158)(cid:159)(cid:160)¡ ¢£⁄¥ P ƒ§¤'“«‹› fifl(cid:176)–†‡ ·(cid:181)¶ •‚„”»… ‰(cid:190)¿(cid:192)` ) ´ˆ˜¯˘ ) 0 ˙¨(cid:201) P ˚¸(cid:204) ˝˛ˇ + —(cid:209)(cid:210) + (cid:211)(cid:212)(cid:213)(cid:214) + ) (cid:215)(cid:216)(cid:217)(cid:218) + ) (cid:219)(cid:220)(cid:221)(cid:222)(cid:223)(cid:224)Æ (cid:226)ª(cid:228)(cid:229) + ) (cid:230)(cid:231)ŁØ + ) FIG. 9. Partial level schemes of , S and , P showing levels populated in the present work. Arrow widths are proportionalto the measured γ -ray intensities. Œº(cid:236)(cid:237)(cid:238)(cid:239)(cid:240)æ(cid:242)(cid:243)(cid:244)ı(cid:246)(cid:247)łøœß(cid:252) ( f m ) (cid:253)
20 22 (cid:254)(cid:255)
26 28
E(cid:0)((cid:1)(cid:2)(cid:3)(cid:4)(cid:5)
FIG. 10. (Color online) Proton-scattering and electromag-netic deformation lengths δ for the 0 + g.s. → +1 excitationsof even-even neutron-rich sulfur isotopes. Proton-scatteringdeformation lengths are from Ref. [19] (open circles) and thepresent work (filled circles), and electromagnetic deformationlengths (open squares) are from Ref. [20]. lows us to reach important conclusions about the presentmeasurements.Figure 11 illustrates the values of ( M n /M p ) / ( N/Z ) forthe 0 + gs → +1 excitations in the N = 20 −
30 even-evenisotopes of Si, S, Ar and Ca. We have combined proton-scattering deformation lengths from the present work andRefs. [19, 29–33] with the proton deformation lengths of ( M n /(cid:12) p )(cid:13)(cid:14)(cid:15)(cid:16)Z(cid:17) Ca A(cid:18)
SSi N
20 22 (cid:19)(cid:20)
26 28 30
FIG. 11. (Color online) Ratios of neutron to proton transitionmatrix elements M n /M p expressed relative to N/Z for even-even neutron-rich calcium, argon, sulfur, and silicon isotopesfrom Refs. [19, 29–33] (open symbols) and the present work(filled circles).
Ref. [20]. Of the isotopes shown in the plot, only two varysubstantially from the value of ( M n /M p ) / ( N/Z ) = 1 ex-pected for isoscalar transitions — the N = 28 and 30 iso-topes of Ca, which have a closed major proton shell ( Z =20). The ( M n /M p ) / ( N/Z ) values for these two isotopesreflect the fact that while there are valence neutrons tocontribute to the 0 + gs → +1 excitation, there are no va-lence protons. Therefore, the only proton contributions (cid:21)(cid:22)(cid:23) (cid:24)(cid:25)(cid:26) (cid:27)(cid:28)(cid:29) 4(cid:30) Si (cid:31) S !" Ca ( M n ’* p +,-.356 789:S;<=>?@BCDFGH FIG. 12. (Color online) Ratios of neutron to proton transitionmatrix elements M n /M p expressed relative to N/Z for even-even N = 28 isotones from Ref. [30] and the present workcompared with shell model predictions, described in the text. must result from the mechanism of core polarization.The results from the present work, ( M n /M p ) / ( N/Z ) =0 . S and ( M n /M p ) / ( N/Z ) = 0 . Sare both statistically consistent with 1.0.In Fig. 12, the measured values for the N = 28 isotonesare compared with shell model calculations performedwith the SDPF-U [34] and SDPF-MU [35] effective in-teractions. For both sets of shell model calculations, M p and M n are calculated using the “bare” transition matrixelements A p and A n and a parameter d that reflects corepolarization in the transitions: M p = A p (1 + d ) + A n ( d ) M n = A p ( d ) + A n (1 + d ) (2)We adopt d = 0 .
5, which gives effective charges of e p = 1 . e n = 0 . M n /M p values,with the largest discrepancy of 6% for Ar. The calcula-tions predict that for Si, S and Ar M n /M p < N/Z ,signaling that protons play a disproportionately large rolein the 0 + gs → +1 excitations. The present S( p, p ′ ) dataare not sufficient to distinguish between the isoscalar sit-uation ( M n /M p = N/Z ) and the shell model predictionsof M n /M p = 0 . M n/M p in Si, Sand Ar are provocative. An M n /M p value of less than N/Z generally indicates a closed neutron shell. Giventhe collapse of the N = 28 shell closure in Si and S,the shell model predictions for M n /M p in those nucleiare interesting and important to test experimentally.To determine whether the M n /M p values for Si, Sand Ar are less than
N/Z and consistent with the shellmodel predictions with statistical confidence, three issueswill have to be addressed. First, the uncertainty in the( p, p ′ ) data for these nuclei will have to be reduced by in-creasing the numbers of counts in the experimental spec- I ( m b ) JKLMOPQRTUV ) WXY[g\ ]^_‘abcd efh ijk lmn opqrstu P SDPF-USDPF-MU / + / + / + / + / + / + / + / + / + / + FIG. 13. Measured cross sections for populating excited statesin P (top panel) compared with shell-model predictions ofthe proton-scattering transition strength B ( p, p ′ ) calculatedusing the SPDF-U [34] (middle panel) and SPDF-MU [35](bottom panel) effective interactions. In the bottom panels,the filled bars correspond to b n /b p = 1 and the open bars to b n /b p = 3. tra significantly – and the new Facility for Rare IsotopeBeams (FRIB) will have that capability. Second, a pre-cise Coulomb excitation measurement of Si is needed,and the uncertainty in the Coulomb excitation result for S will need to be improved, once again through theimprovement in statistics possible at FRIB.Third, we must address the uncertainty in the ratio b n /b p , for which we have used the value 2 ±
1. Re-markably, this issue will be addressed at FRIB as well.The beams in the present work were at energies of 70MeV/nucleon and lower - energies for which there is con-siderable uncertainty regarding the value of b n /b p forinverse kinematics ( p, p ′ ) reactions. At FRIB, intensebeams ( > particles per second) of Si, S and Ar will be available at energies much greater than 100MeV/nucleon. It has been known for more than thirtyyears that inelastic hadron scattering at energies over 100MeV is approximately isoscalar; that is, it has b n /b p ≈ p, p ′ ) reactions at FRIB will nearlyeliminate the uncertainty in the value of b n /b p .The present , P( p, p ′ ) results provide an opportu-nity to expand upon the recently reported comparison ofthe SDPF-U [34] and SDPF-MU [35] shell model inter-actions using a measurement of the level scheme of Sivia the one-proton knockout reaction [6]. The authorsof Ref. [6] demonstrated that the SDPF-U interactionpredicted a number of states at low excitation energy vwxyz{ |}~(cid:127)(cid:128)(cid:129)(cid:130)(cid:131) (cid:132)(cid:133)(cid:134) (cid:135)(cid:136)(cid:137) (cid:138)(cid:139)(cid:140) (cid:141)(cid:142)(cid:143) (cid:144) (cid:145) ( m b ) (cid:146)(cid:147)(cid:148)(cid:149)(cid:150)(cid:151)(cid:152)(cid:153)(cid:154)(cid:155)(cid:156) ) (cid:157)(cid:158)(cid:159)(cid:160) P SDPF-USDPF-MU / + / + / + / + / + / + / + / + / + / + / + FIG. 14. Measured cross sections for populating excited statesin P (top panel) compared with shell-model predictions ofthe proton-scattering transition strength B ( p, p ′ ) calculatedusing the SPDF-U [34] (middle panel) and SPDF-MU [35](bottom panel) effective interactions. In the bottom panels,the filled bars correspond to b n /b p = 1 and the open bars to b n /b p = 3. (below 4 MeV) that was significantly larger than whatwas observed in the experiment. In contrast, SDPF-MUpredicted a smaller number of states in the same range ofexcitation energy that more accurately reflected the ob-served spectrum. The authors of the Si study thereforeconcluded that SDPF-MU is a more useful interaction forinvestigating the effects of weak binding in Mg. Here,we find that the SDPF-MU is also better able to describethe more deeply bound P isotopes in the neighborhoodof Si.In Fig. 13, we compare the cross sections of thestates observed here in P( p, p ′ ) with the distributionof ( p, p ′ ) strength, B ( p, p ′ ), predicted using the SPDF-Uand SPDF-MU interactions. The strength B ( p, p ′ ) is cal-culated for each state using the proton ( M p ) and neutron( M n ) transition matrix elements for the decay from thestate to the ground state using the equation, B ( p, p ′ ) = 1(2 J i + 1) ( C p M p + C n M n ) (3)where J i = 1 /
2, since the ground states of both , Phave J = 1 / M n and M p are calculated for , P in thesame way that the corresponding values are calculatedfor the even-even isotopes (as described in the discussionabove). The normalized coefficients C p and C n accountfor the sensitivity of ( p, p ′ ) to protons and neutrons andtake on the values C p = C n = 0 . b n /b p = 1 and C p = 0 . C n = 0 .
75 for b n /b p = 3. In the bottom two panels of Fig. 13 the filled bars correspond to b n /b p = 1,and the open bars correspond to b n /b p = 3. The valuesbetween the top of the filled bar and the top of the openbar correspond to the range b n /b p = 2 ± P,which has J π = 3 / + . SDPF-U gives two strong 5 / + states at about 1.1 MeV, while SDPF-MU gives only one.The experiment shows only one strong state at that en-ergy, so that observation favors the SDPF-MU interac-tion. The experiment shows a state near 1.6 MeV andanother near 2.4 MeV. Both SDPF-U and SDPF-MU givesuch states – with a 3 / + state near 1.6 MeV and a 5 / + state near 2.4 MeV.We conclude that the comparison of the theoreticalcalculations with the data on P favors SDPF-MU.Figure 14 illustrates the situation in P. As in P, theexperiment shows that the lowest excited state is stronglyexcited, and that excitation is reproduced by both theSDPF-U and SDPF-MU interactions. In the experiment,there is a cluster of three strongly populated states near1 MeV. While SDPF-MU predicts two strongly popu-lated states near 1 MeV (one having J π = 3 / + and theother J π = 5 / + ), the SDPF-U interaction gives a clusterof five states distributed from 1.0 to 1.5 MeV, with thestrongest being a 5 / + state near 1.4 MeV. We concludethat the SDPF-MU interaction gives a better accountingof the situation in P than SDPF-U does.In short, the present results on , P( p, p ′ ) and therecently reported Si results all favor the SDPF-MU in-teraction in this neutron-rich region.
V. CONCLUSIONS
Comparison of the proton inelastic scattering measure-ment of the 0 + gs → +1 excitation in S reported herewith previous Coulomb excitation measurements of thesame excitation gives an M n /M p value that is, becauseof experimental uncertainties, consistent with both theisoscalar value of N/Z and the shell model predictionthat M n /M p = 0 . N/Z ), which would indicate that pro-ton excitations play a disproportionately large role in thisexcitation and that there is a residual N = 28 shell clo-sure effect. However, the higher beam rates and higherbeam energies available at FRIB will provide an oppor-tunity to resolve this issue not only in S but also in Si. The , P( p, p ′ ) measurements reported here pro-vide a means for expanding upon the comparison of theSDPF-U and SDPF-MU shell model interactions begunin a recent study of Si [6]. As in the case of Si, thepresent results favor the SDPF-MU interaction.0
ACKNOWLEDGMENTS
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