Isoscalar and Isovector spin response in sd shell nuclei

Transcription

1 Isoscalar and Isovector spin response in sd shell nuclei H. Sagawa,), T. Suzuki,), ) RIKEN, Nishina Center, Wako, 5-98, Japan ) Center for Mathematics and Physics, University of Aizu, Aizu-Wakamatsu, Fukushima 95-85, Japan ) Department of Physics, College of Humanities and Science, Nihon Univerity, Sakurajosui, Setagaya-ku, Tokyo 5-855, Japan ) National Astronomical Observatory of Japan, Mitaka, Tokyo , Japan arxiv:8.9v [nucl-th] 9 Jan 8 The spin magnetic dipole transitions and the neutron-proton spin-spin correlations in sd shell even-even nuclei with N = Z are investigated using shell model wave functions taking into accout enhanced isoscalar (IS) spin-triplet pairing as well as the effective spin operators. It was shown that the IS pairing and the effective spin operators gives a large quenching effect on the IV spin transitions to be consistent with observed data by (p,p ) experiments. On the other hand, the observed IS spin strength show much smaller quenching effect than expected by the calculated results. The IS pairing gives a substantial quenching effect on the spin magnetic dipole transitions, especially on the isovector (IV) ones. Consequently, an enhanced isoscalar spin-triplet pairing interaction enlarges the proton-neutron spin-spin correlation deduced from the difference between the isoscalar (IS) and the IV sum rule strengths. The beta-decay rates and the IS magnetic moments of sd shell are also examined in terms of the IS pairing as well as the effective spin operators. PACS numbers:..jz,.5.ef,..cz,..gd I. INTRODUCTION The spin-isospin response is a fundamental process in nuclear physics and astrophysics. The Gamow-Teller (GT) transition, which is a well-known allowed charge exchange transition, involves the transfer of one unit of the total angular momentum induced by σt ± []. In a no-charge-exchange channel, magnetic dipole (M) transitions are extensively observed in a broad region of the mass table. Both the spin and the angular momentum operators induce M transitions [], and depending on whether the isospin operator is included also induce the isovector (IV) and the isoscalar (IS) modes. Compared to the relevant theoretical predictions by shell model and random phase approximations(rpa) [ 7], the experimental rates of these spin-isospin responses are quenched. A similar quenching effect also occurs in the observed magnetic moments of almost all nuclei compared to the single-particle unit (i.e., the Schmidt value) [, 8, 9]. The quenching effect of spin-isospin excitations influences many astrophysical processes such as the mean free path of neutrinos in dense neutron matter, the dynamics and nucleosynthesis in core-collapse supernovae explosions [], and the cooling of prototypeneutron stars []. Furthermore, the exhaustion of the GT sum rule is directly related to the spin susceptibility of asymmetric nuclear matter [] and the spin-response to strong magnetic fields in magnetars []. Although the quenching phenomena of magnetic moments and spin responses have been extensively studied, previous research has focused mainly on the mixings of higher particle-hole (p-h) configurations [8, 9, ] and the coupling to the resonances [5, ]. In particular, the measured strength of the GT transitions up to the GT giant resonance is strongly quenched compared to the non-energy weighted sum rule, (N Z) []. This observation has raised a serious question about standard nuclear models because the sum rule is independent of the details of the nuclear model, implying a strong coupling to. After a long debate [7], experimental investigations by charge-exchange (p, n) and (n, p) reactions on 9 Zr using multipole decomposition (MD) techniques have revealed about 9% of the GT sum rule strength in the energy region below E x =5 MeV [, 8], demonstrating the significance of the p h configuration mixings due to the central and tensor forces [], although the coupling to is not completely excluded. IV spin M transitions induced by σt z can be regarded as analogous to GT transitions between the same combination of the isospin multiplets. Therefore, they should show the same quenching effect as GT transitions. On the other hand, the IS spin M transitions are free from the coupling to and their strength quenching should be due to higher particle-hole configurations. Various theoretical studies have pointed out that the quenching of IS spin operators is similar to that of IV ones [9]. However, recent high-resolution proton inelastic scatteringmeasurementsate p =95MeVhaverevealedthatthe IS quenching is substantially smaller than the IV quenching for several N =Z sd shell nuclei []. These empirical findings give rise to a positive value for the proton-

2 neutron spin-spin correlations in the ground state of N=Z nuclei. Recently, it has been reported that the isoscalar (IS) spin-triplet pairing correlations play an important role in enhancing the GT strength near the ground states of daughter nuclei with mass N Z [ ]. At the same time, the total sum rule of the GT strength is quenched by ground state correlations due to the IS pairing [5]. In this paper, we study the IS and IV spin M responses based on modern shell model effective interactions for the same set of N=Z nuclei asthose in Ref. []. We introduce effective spin and spin-isospin operators to make quantitative study of the accumulalted strength in the responses. We consider that simultaneous calculations of these responses within the same nuclear model may be advantageous to distinguish the effect of the higher order configurations from the hole coupling due to the fact that the IS spin M transition is independent of the hole coupling strength. We discuss also the effect of IS spin-triplet pairing interaction on the spin responses and the proton-neutron spin-spin correlations in the ground states of N=Z nuclei. Thespin Moperatorsareintroducedin SectionII and their sum rules are also defined. Section III is devoted to the shell model calculations of N=Z even-even nuclei in comparisons with available experimental data by (p,p ) reactions. The accumulated sum rule values of IS and IV spin transitions are extracted in Section. The protonneutron spin-spin correlations are also discussed in terms of the IS spin-triplet pairing correlations. The beta-decay rates and IS magnetic moments in sd shell are studied in the same context of the shell model calculations in Section V. The summary is given in Section VI. II. SPIN M OPERATORS AND SUM RULES We consider the IS and IV spin M operators, which are given as Ô IS = i Ô IV = i σ(i), () σ(i)τ z (i), () as well as the GT charge exchange excitation operators, which is expressed as Ô GT = i σ(i)t ± (i). () The sum rule values for the M spin transitions are defined by S( σ) = f S( στ z ) = f J i + J f ÔIS J i, () J i + J f ÔIV J i. (5) For the GT transition, the sum rule value is defined by S( σt ± ) = f J i + J f ÔGT J i, () and satisfies the model independent sum rule, S( σt ) S( σt + ) = (N Z). (7) According to Ref. [], the proton-neutron spin-spin correlation is defined as = f f spin = (S( σ) S( στ z)) J i i J i i σ n (i)+ σ p (i) J f J f i σ n (i) σ p (i) J f J f i σ n (i)+ σ p (i) J i σ n (i) σ p (i) J i = J i S p S n J i, (8) where S p = i p s p(i) and S n = i n s n(i). The correlation value is.5 and.75 for a proton-neutron pair with a pure spin triplet and a singlet, respectively. The former corresponds to the ferromagnet limit of the spin alignment, while the latter is the anti-ferromagnetic one. III. SHELL MODEL CALCULATIONS WITH EFFECTIVE OPERATORS AND IS PAIRING CORRELATIONS The shell model calculations are performed in full sd shell model space with the interaction []. Among the effective interactions of the USD family, USD [7], USDA [], and [], the results of spin excitations with J π = + are quite similar to each other both in excitation energies and transition strengths for collective states with large transition strengths. Hereafter, we present results based on interaction. To take into account the effects of higher order configuration mixings as well as meson exchange currents (MEC) and isobar effect, effective operators are commonly adopted in the study of magnetic moments, GT transitions and spin and spin-isospin dependent β decays. IV magnetic transitions in sd shell nuclei and GT transitions have been studied extensively in experiments and theories. On the other hand, experimental evidence of IS magnetic transitions are not well known so far except recent experimental data by Matsubara et al.. In the literature [8, 9, 8], the effective operators have been introduced to mimic the effects of higher-order configuration mixings, meson-exchange currents, isobar coupling and the relativistic corrections. For the spin operators, the effective operators read for the IS operator Ô eff IS = fis s σ +fis l l+f IS 8π[Y σ] (λ=) (9) and also for the IV spin operator, Ô eff IV = fiv s στ z +fl IV lτz +fp IV 8π[Y σ] (λ=) τ z () p

3 where f IS(IV) i (i = s,l,p) are the effective coefficients of IS(IV) spin, orbital an spin-tensor operators. The summation of index i in Eq. () is discardeed in the effective operators. The effective coefficients for the IS spin operator obtained by Towner are fs IS =.75, fl IS =.5 and fp IS =-.57. For the IV part, Towner obtained the corrections for the spin, orbital and the spin-tensor operators of GT transitions of d orbit as Ô eff GT = (+δg s) σt ± +δg l lt± +δg p 8π[Y σ] (λ=) t ± () with δg s =.9, δg l =., δg p =.8 () due to the various higher order effects. In the shell model calculations with USD interaction, the IV spin and charge exchange GT excitations are the same features since no isospin breaking interaction such as Coulomb interaction and charge symmetry breaking forces is not included. We adopt the GT effective operators for IV spin transitions. For the isoscalar part, the quenching factor for spin operator is introduced to check the sensitivity of transition strength on the effective operator. The effective operators for IS orbital and spin-tensor are not introduced in the present study. For the IS case, we take different versions of calculations: : the original interaction with the bare spin operator : the IS spin-triplet pairing matrix is enhanced multiplying a factor. on the relevant matrix elements of interaction. The bare spin operator is adopted. :the IS spin-triplet pairing matrix is enhanced multiplying a factor. on the relevant matrix elements of interaction. The IS spin operator is % quenched: fs IS =.9. :the IS spin-triplet pairing matrix is enhanced multiplying a factor. on the relevant matrix elements of interaction. The IS spin operator is % quenched: fs IS =.9. For the IV case, we take different cases of calculations: : the original interaction with the bare spin operator q:the IS spin-triplet pairing matrix is enhanced multiplying a factor. on the relevant matrix elements of interaction. The effective IV spin operator is adopted. q:the IS spin-triplet pairing matrix is enhanced multiplying a factor. on the relevant matrix elements of interaction. The effective IV spin operator is adopted. A. C In C, IS and IV + states are observed at Ex=.7 and 5.MeV, respectively. The B(M) values are extracted from (e,e ) scattering experiments to be B(M)=. and.79 in terms of nuclear magneton (e /mc) [9]. The shell model calculations with CK- POT interaction give B(M)=. and. in the unit of nuclear magneton at Ex=.5 and 5.9MeV, respectively, with the bare magnetic transition operators. A recent p sd shell Hamiltonian, SFO [], gives B(M) =. and.55 µ N for the IS and IV transitions, respectively. The model space of SFO is p sd shell and the excitations from p shell to sd shell are included up to ω. It is noticed that the experimental value is about times larger than the calculated value for the IS + state at Ex=.7MeV, while the calculated value for the IV state is close to the experimental value. The (p,p ) data was reported for the two + states to be B(σ) =.7±.8 at Ex=.7MeV and B(στ) =.99 ±.9 at Ex=5.MeV, respectively. The shell model results with SFO are B(σ)=.5 and B(στ)=.97, respectively. The proton inelastic scattering data of the state at Ex=.7MeVshow also a factor larger value than the shell model results. The isospin mixingbetweenthetwo + statehasbeendiscussedasan origin of the enhancement of IS spin matrix element. A large isospin mixing was claimed to enhance IS magnetic transition observed by the electron scattering. The same effect is expected for B(σ). When the +, T=,.7 MeV and +, T=, 5. MeV state are mixed by isospin-mixing, +,.7MeV >= a +,T = > +a +,T = > +,5.MeV >= a +,T = > a +,T = () >, we get an enhancement of B(σ) as well as a reduction of B(στ). B(σ) is enhanced from.5to.7while B(στ) is reduced from.97 to.75 and the mixing amplitude a =.5 []. The difference spin (Eq. (8)) is found to be enhanced by.. Though the isospin-mixing gives rise to favorable effects, it is still not enough to reproduce the experimental value of B(σ). B. Ne Figures (a) and (a) show the energy spectra of the IS spin excitations and their accumulative sums, respectively, in Ne. The IS spin-triplet matrix elements are enhanced by a factor. for and, and by a factor. for case. Together with the enhancement of IS pairing, the quenched spin operator is introduced in the cases and with q=.9. The calculated results are smoothed by a Lorentzian weighting factor with the width of.5 MeV to guide the eye. The shell model results with give + states at Ex=. and.98 MeV with B(σ) =.

4 and.59, respectively. The results with the enhanced pairing and the quenched spin operator are also shown in the same figure. The lowest state in the case of is found at Ex=.55MeV with a smaller strength B(σ) =.78, which is a half of one. The higher energy strength is fragmented into three peaks at Ex.5 and. MeV with the summed strength B(σ)=.9. The accumulated values are shown in Fig. for four cases,, and. In Ne, the accumulated sum increases up to Ex MeV. The enhanced IS pairing in gives about % quenching compared with results, while more enhanced IS pairing in gives further quenching compared with. In the (p,p ) data, the IS strengths are not found so far. The results of IV spin response are shown in Figures and. The original gives IV strength at Ex=. and.9mev with B(στ)=. and.8, respectively, below Ex=5MeV. The excitation energies ofthesetwopeaksareshiftedtohigherenergies.and.9mev in the case of enhanced IS pairing q. Comparing with the results of q, the q gives essentially the same excitation energies for + spectra, while the B(στ) are decreased from.9 to. for the first peak and from. to.7 for the second peak. The calculated results give large strength also in the energy region above Ex=5MeV. The accumulated sums are shown in Figure. Up to Ex=MeV, the accumulated sums are.577,. and.9 for, q and q, respectively. Due to the strong IS pairing and the effective IS spin operator, the accumulated strength decrease by % for q and 5% for q. The (p,p ) experiments found two IV strength at Ex =. and.mev with B(στ)=.9 and.8, respectively. The summed strength.87 is comparable with the result of q. C. Mg Figures and show the energy spectra of the spin excitations and their accumulative sums, respectively, in Mg. For the IS case, the experimental data give the strong spin M strength at Ex=9.88MeV with B(σ) =.88 ±.. The shell model results with give IS + state at Ex=9.88MeV with B(σ) =.78. In the (p,p ) data, other IS strengths are also found at 7.78MeV with B(σ)=.58 and at Ex MeV with B(σ).. The calculated results reproduce strong M states at very similar energies Ex=7.8 and.7mev with B(σ) =. and.59, respectively. The calculations with show also the same amount of B(σ) value as the experimental data around Ex=MeV. The summed strength up to Ex=MeV is B exp (σ:ex MeV)=5.±., while the calculated sum is B cal (σ:ex MeV)=.5. The calculated results of USB changes only slightly the excitation energies of + state by about -kev, while db(σ)/dex db(στ)/dex (a) Ne IS 5 5 Ex Ne IV q q 5 5 Ex FIG.. (Color online) IS (top) and IV spin-m (bottom) transition strengths in Ne. Shell model calculations are performed in the full sd shell model space with an effective interaction. For the IS case, the and has the % enhanced IS spin-triplet interaction, while has % enhanced ones. The quenching factor for the IS spin operator fs IS =.9 is introduced for and calculations. For the results of the IV spin-m transitions, an effective IV spin operator () is adopted in q and q cases. The IS spin-triplet interaction is enhanced by multiplying the relevant matrix elements by factors. and. in the cases of q and q, respectively, together with the effective operaror. Calculated results are smoothed by taking a Lorentzian weighting factor with the width of.5mev, while the experimental data are shown in the units of B(σ) for the IS excitations and B(στ) for the IV excitations. Experimental data are from ref. []. the summed B(σ) value is decreased by %. The experimental analysis show a strong IV spin strength at Ex=.7MeV with B(στ)=.7. The calculation gives at Ex=.7MeV with B(στ)=.85. Experimental data show also substantial strength around Ex=.8MeV with B(σ) and at Ex=9.98 and.mev with B(στ)=.8 and.9, respectively. The calculated results give large strengths at Ex=9.99MeV and.75mev with B(στ)=.8 and

5 5 Accumulated B(σ).8... (a) Ne IS db(σ)/dex (a) Mg IS 5 5 Ex 5 5 Ex Accumulated B(στ).8... Ne IV q q db(στ)/de x Mg IV q q 5 5 Ex FIG.. (Color online) Accumulative sum of the IS spin-m strength (top) and the IV spin-m strength (bottom) as a function of the excitation energy in Ne. The calculated energy spectra are also shown for in the IS channel and for q in the IV channel. Calculated results are smoothed in the same manner as Fig.. Dot with a vertical error bar denotes the experimental accumulated sum of the strengths. See the text and the caption to Fig, for details. 5 5 Ex FIG.. (Color online) IS (top) and IV spin-m (bottom) transition strengths in Mg. See the text and the caption to Fig, and the text for details. effects are favorable and consistent with the experimental data though their magnitudes are not significant..5, respectively. The experimental summed strength is B exp (στ:ex MeV)=.8±., while the calculated value is B cal (στ:ex MeV)=.855. There are about % quenching in the empirical IV spin sum rule strength below Ex=MeV compared with results with the bare spin operator. The q and q results with the effective spin operator show about and % quenching of the accumulated strength up to Ex=MeV, respectively. We study the effects of the isospin-mixing in Mg. The +, T= states at E x =7.77 MeV and 9.87 MeV are mixed with the +, T= state at E x = 9.9 MeV []. Using the mixing amplitudes obtained by < T = V C D T = > / E with < T = V C D T = > = 9 kev [], an enhancement of S( σ) from.5 to.9, a reduction of S( στ z ) from.85 to.5 and an enhancement of spin by. are obtained for. These D. 8 Si The calculated results of IS response are shown in Figs. 5(a) and (a). The calculated results give a IS + state at Ex=9.MeV with interaction, which reproduces well the experimental IS + state with a strong spin transition at E x =9.58MeV exhausting about 7% of the total IS strength. Another strong state is observed at Ex=.57MeV with B(σ)=.75±., while the calculation shows no sign of strong B(σ) transition above Ex=MeV. Other IS spin transitions are found experimentally at Ex MeV with B(σ).7. The calculations show also the IS spin strength of B(σ). at Ex=(.-.)MeV. The summed empirical IS strength below Ex=MeV is B exp (σ:ex MeV)=.89±., while the calculated results are B cal (σ:ex MeV)=.8,., 5.55 and

6 Accumulated B(σ) 7 5 (a) Mg IS db(σ)/dex 5 (a) 8 Si IS Accumulated B(στ) 7 5 Mg IV q q 5 5 FIG.. (Color online) Accumulative sum of the IS spin-m strength (top) and the IV spin-m strength (bottom) as a function of the excitation energy in Mg. See the text and the caption to Fig, and the text for details..5 for,, and, respectively. The B cal (σ:ex MeV) values show, and % quenching for, and interactions, respectively, compared with results. The IV spin response is shown in Figs. 5 and. Gross structure of empirical IV spin response is well reproduced by the calculations based on interaction. Empirical IV spin strength is rather fragmented, while two IV + states with strong spin strengths of B(στ)=.5 and.9 are reported at E x =.5 and.mev, respectively. Calculated results show also largely fragmented IV spin strength and two strong strength are found at Ex=.8 and.mev with B(στ)=.5 and.77, respectively, with interaction. The empirical IV summed strength is B exp (στ:ex MeV)=.59±., while the calculated one is B cal (στ:ex MeV)=7., 5. and. for, q and q cases, respectively. In the spin IV sum rule value, we see a largequenching spin factorq IV s (eff) B exp (στ)/b cal (στ : )=.79,which db(στ)/dex 8 Si IV q q 5 5 FIG. 5. (Color online) IS (top) and IV spin-m (bottom) transition strengths in 8 Si. See captions to Fig. and the text for details. is close to the ratio of summed values of q to. It is pointed out also in ref. [] that the enhanced IS pairing multiplying a factor. on the IS pairing matrices reduces the IV spin transition strength, corresponding to the renormalization factor of fs IV =.87 for the accumulated IV spin strength. On the other hand, the same enhanced IS pairing gives the IS quenching factor fs IS =.9. This difference between IS and IV spin response induces a positive value for the proton-neutron spin-spin correlations in the ground state. This point will be discussed more in Section IV. E. S For the IS case shown in Fig. 7(a), the experimental data show a strong state at Ex=9.95MeV with B(σ) =.8 ±.8. The corresponding state is found in the calculated results at Ex=9.MeV with B(σ) =.. Another strong IS transition was found at Ex=9.97MeV with B(σ) =. ±., while

7 7 Accumulated B(σ) 8 (a) 8 Si IS db(σ)/dex 5 (a) S IS Ex(MeV) Accumulated B(στ) 8 8 Si IV q q db(σt)/dex S IV q q 5 5 FIG.. (Color online) Accumulative sum of the IS spin-m strength (top) and the IV spin-m strength (bottom) as a function of the excitation energy in 8 Si. See the captions to Fig. and the text for details. the calculations show a state at Ex=9.5MeV with B(σ) =.9MeV. There are two IS states observed below Ex=7.MeV. The calculations found also two states at the same energy region with almost the same B(σ) values as the observed ones. The observed IS sum rule strength is B exp (σ:ex MeV)=.±.7,while theoreticallyb cal (σ:ex MeV)=7.. We can see a small quenching effect corresponding to fs IS (eff)=.9 for the sum rule strength below Ex=MeV. The IV response in S is shown Fig. 7. The IV spin strength is concentrated at Ex.MeV having 8% of the total strength below Ex=MeV. The calculated results show also very large fraction of the total strength of about 87% of the total strength. Another strong state is found experimentally at Ex=8.5MeV with B(στ) =.7±., while the calculations show a state at Ex=7.959MeV with B(στ)=.7. The agreement between theory and experiment is quite satisfactory as far as the gross feature of IV spin response is concerned. The summed strength of IV transitions is 5 5 Ex(MeV) FIG. 7. (Color online) IS (top) and IV spin-m (bottom) transition strengths in S. See captions to Fig. for details. B exp (στ:ex MeV)=.±.7,while the calculated results are B exp (στ:ex MeV)=7.99. We see a large quenching for IV case with fs IV (ef f)=.7. The results q with the enhanced IS pairing and the effective IV spin operator give good account of the accumulated strength. F. Ar The IS spin response in Ar is given in Fig. 9(a). The experiments are found two states at Ex=8.985 and.8mev with B(σ)=.7±.5 and.87±., respectively. The shell model results of show a strong IS strength at Ex=8.55MeV with B(σ)=.558. Above Ex=MeV, the calculated IS strength is rather fragmented with the summed B(σ) in the energy region Ex=(-5)MeV. The experimental summed strength in Fig. (a) is B exp (σ:ex MeV)=.9±.9, while the calculated value is B cal (σ:ex MeV)=.75. We see a small

8 8 Accumulated B(σ) 8 (a) S IS db(σ)/dex (a) Ar IS Accumulated B(στ) 8 S IV q q db(στ)/dex.5.5 Ar IV q q 5 5 FIG. 8. (Color online) Accumulative sum of the IS spin-m strength (top) and the IV spin-m strength (bottom) as a function of the excitation energy in 8 Si. See the captions to Fig. for details. 5 5 FIG. 9. (Color online) IS (top) and IV spin-m (bottom) transition strengths in Ar. See captions tofig. for details. IV. ACCUMULATED STRENGTH OF IS AND IV SPIN M EXCITATIONS quenching with the factor f IS s (eff)=.88. The IV spin strength is shown in Fig. 9. Experimental data show two large IV strength at Ex=8 and MeV having % and 55% of the observed total strength below Ex=MeV. The calculated results show also two strong spin strengths at Ex=8.MeV and Ex=9.8MeV with B(στ)=.8 and.788, respectively. The calculations show another strong IV transition strength with B(στ). at Ex MeV, while experimental data observed a small strength with B(στ). around Ex=MeV. The observed summed strength in Fig. is B exp (στ:ex MeV)=.98±., while the calculated one is B cal (στ:ex MeV)=.5. The quenching factor is rather large with fs IV (eff)=.9 for the IV case so that the results of q in Fig. give the closest value to the empirical one. FigureshowsthesumrulevaluesofS( σ)ands( στ z ) for the variations of interactions, respectively. The % enhanced IS pairing in give a small quenching effect on the accumulated IS sum rule value about 5%, at most 7% in S and Ar. With the quenching factor fs IS =.9 in, the IS accumulated strength is further decreased by -5% compared with the original value by interaction. The decrease of the IS summed value is going down further by the % enhanced IS pairing to be 9-% in the case of USB results. Compared with calculations, the empirical accumulated IS values are % enhanced in Mg and gradually quenched from A=8,.95,.88 and.77 in 8 Si, S and Ar, respectively. In general, the quenching effect is rather small and at most % of the original calculations with the bare spin operator. The IV accumulated sum rule values up to Ex=MeV are shown in Fig.. The IS pairing interactions are

9 9 accumulated B(σ) 5 (a) Ar IS S(IS) A Accumulated B(στ) 5 Ar IV q q S(IV) 8 q q 5 5 FIG.. (Color online) Accumulative sum of the IS spin-m strength (top) and the IV spin-m strength (bottom) as a function of the excitation energy in 8 Si. See the captions to Fig. for details. enhances by factors of. for q and. for US- DBq, respectively, with the effective operators from ref. [9]. The q results give -% quenching sum rule values, while the stronger IS pairing in q gives additional quenching of the strength, i.e., -5% quenching of the summed strength. The empirical values show also large quenching from % in Ne, 5% in Mg, 7% in 8 Si, 7%in S and5%in Ar, respectively, compared with the summed value of calculations. Figure shows the experimental and the calculated proton-neutron spin-spin correlations (8). Although the experimental data still have large error bars, the calculated results with the interaction show poor agreement with the experimental data. This is also the case for the other USD interactions such as USD and USDA. The results with an enhanced IS spin-triplet pairing improve the agreement appreciably. For the IS channel, adopts the bare spin operator, while the effective spin operator fs IS =.9 is used for case. The effective operator gives a smaller spin-spin correla- 8 A FIG.. (Color online) Accumulated spin-m transition strengths of IS channel (a) and IV channel. Experimental and theoretical data are summed up to Ex = MeV. Shell model calculations are performed with the effective interaction: (a) In the results of and, the IS spin-triplet pairing interaction is enhanced by multiplying the relevant matrix elements by a factor of. compared to the original interactions with the quenching factor q=. and.9 for IS spin operator, respectively. For, the IS pairing interaction is enhanced by a factor of. with a quenching factor q=.9 for the IS spin operator. Experimental data are from ref. []. Long thin error bars indicate the total experimental uncertainty, while the short thick error bars denote the partial uncertainty from the spin assignment. The effective IV operators are adopted for spin, orbital and spin-tensor operators. The effective operators are taken from ref. [9]. For the results of q and q, the IS pairing interaction is enhanced by a factor of. and., respectively, with the effective operators. tion spin than the case of bare IS spin operator. The positive value of the correlation indicates that the population of spin triplet pairs in the ground state is larger than that of the spin singlet pairs. We should remind that the spin and the spin-isospin M strength may exist in the energy region above Ex=MeV. In the analysis of Figure, these higher energy contributions are assumed to be the same for both the IS and IV channels. In the

10 S... +q +q for proton and neutron excitations. The indices (i,,, ) stand for the quantum numbers of the singleparticlestatei = (n i,l i,j i ). InEq. (), theperturbative coefficient is given by α(,,, ) = ( π π )J,T ;( ν ν )J,T : J = T = H p E (5). 8 A FIG.. (Color online) Experimental and calculated protonneutron spin-spin correlation spin. Spin M transition strengths are summed up to E x=mev. Shell model calculations are performed with an effective interaction. In the results of and for the IS channel, the IS spin-triplet interaction is enhanced multiplying the relevant matrix elements by a factor of. compared to the original, and the IS quenching factor is q=. and.9, respectively. The effective spin operators are used for the q for the IV transitions. Experimental data are taken from ref. []. See the caption of Fig. for a description of the experimental error bars. shell model calculations in the full sd shell, the spin M strength above MeV is small in the N=Z nuclei except Ne and 8 Mg. In Ne, % and 5% of the total strength exist in the energy region Ex=(-)MeV for the IS and IV channels, respectively. In the case of 8 Mg, the strengths above Ex=MeVare 9.9% and % of the total strengths for IS and IV channels, respectively. In the other nuclei, the spin strengths above Ex=MeV are rather small around few % of the accumulated strength below Ex=MeV and the strengths are more or less the sameinbothisandivchannels. Thehypothesisofequal IS and IV strengths in the higher energy region is valid in the present theoretical calculations in nuclei A > 8, which should be checked experimentally in future. To clarify the physical mechanism of the IS spin-triplet interaction, we make a perturbative treatment of the - particle -hole (p-h) ground state correlations on the spin-spin matrix element. We express the wave function for the ground state with proton-neutron correlations for even-even N=Z nuclei,, as = + α(,,, ) i=,,, ( π π )J,T ;( ν ν )J,T : J = T =. () Here the first term on the right hand side,, is the wave function with seniority ν = (i.e., without the spin-triplet correlations). The second term represents the states of particle hole pairs ( π π ) and ( ν ν ) where H p is the IS spin-triplet two-body pairing interaction and E = E E( ; ). The p-h states are the seniority ν= states in Eq. (). The two-body matrix element in Eq. (5) is rewritten as ( π π )J,T ;( ν ν )J,T : J = T = H p { jπ j = π J }{ } / / T j ν j ν J / / T ĴĴ ˆT ˆT ( ) j π +jν+j+j ++T+T ( π ν )J,T ;( π ν )J,T : J = T = H p whereĵ J +andj symbolsareused. Thematrix element is further expressed as ( π ν )J,T ;( π ν )J,T : J = T = H p = J + ( π ν )J,T H p ( π ν )J,T () Since the pairing interaction H p is attractive and the energy denominator E is negative, the perturbative coefficient α(,,, ) should be positive. The effect of the ground state correlations on the proton-neutron spin-spin matrix is then evaluated as S p S n = α(,,, ),,, S p S n ( π π )J,T ;( ν ν )J,T : J = T = (7) where the angular momenta and the isospins are selected to be J = J = and T = T = by the nature of the neutron-proton spin-spin matrix. The matrix element in Eq. (7) is further expressed as a reduced matrix element in the spin space, S p S n ( π π )J ;( π π )J : J = = δ J,J δ J, ( ) j+j j j j s p j j s n j. (8) In Eq.(8), the coupled angular momentum J is taken as J = due to the selection rule of the spin matrix element. The isospin quantum number is discarded since it gives a trivial constant in Eq. (8). We can obtain the effect of the IS spin-triplet pairing correlations on the proton-neutron spin-spin correlation matrix element taking a p h configuration j = j = j < = d /

11 TABLE I. Beta-decay rate between mirror nuclei with T = / and T z = ±/. The shell model calculations are performed by using and * [] interactions with the bare spin-g factor as well as by q and q interactions with the Towner s effective GT operator. A J * q q Exp. 9 /.7.8.7,9.5 / / / / / / / / / σ rms and j = j = j > = d 5/. Taking the spin-orbit splitting between d 5/ and d / as 5MeV and the spintriplet pairing matrix element as -MeV, the α coefficient in Eq. (5) is evaluated to be.5. The effect on the neutron-proton spin-spin correlation becomes a large positive value, spin =.7. Thus, the positive value obtained by numericalresults shown in Fig. can be qualitatively understood by using these formulae for p-h configuration mixing due to the IS spin-triplet pairing. V. BETA-DECAY AND IS MAGNETIC MOMENTS The effects of IS pairing and the effective spin operators are examined for the beta-decay rates between mirror nuclei with T = / and T z = ±/. The results are given in Table I for, * [], q and q interactions. The IS spin-triplet pairing matrix is enhanced by a factor. in the relevant matrix elements and bare spin operator is used for *. The r.m.s. deviations σ rms of the calculations and experiments are.7,.9,.5 and. for USB, *, q and q, respectively. We do not see any significant difference in the results between and * interactions. Let us make the optimal χ square fits with experimental data by using the quenching spin q factor for the bare GT operator. The r.m.s. deviation σ rms of q GT (eff) B(GT; ) from B(GT; ) becomes the minimum value. at q GT (eff)=.78, while that of q GT (eff) B(GT; *) from B(GT; ) becomes the minimum value.9 at q GT (eff)=.77. We obtain more or less the same quenching factor for and *. Thisis because odd nucleonin these T=/ nuclei masks the pairing effect by the single-particle nature of the transition. The r.m.s deviations σ rms get reduced for the cases with the Towner s effective operator, q and q, though they are not as small as those for the χ square fitted cases with the quenching factor q GT (eff). The traditional source of information for the IS spin operator is from isoscalar magnetic moments, i.e., the average of the magnetic moments of mirror nuclei. The IS magnetic moments are also calculated for, * and aswellasforthe casewith Towner seffective operator (Eq. (9)) and tabulated in Table II. In case for the Towner s operator which will be denoted as Towner, the IS spin-triplet pairing matrix is not enhanced. The case for using the effective operator (Eq. (9)) where the coefficients are obtained by χ square fitting the experimental isoscalar spin expectation value of sd shell nuclei with T =/, and (TABLE II of Ref. [8]). The coefficientsobtainedarefs IS =.,fl IS =.5andfp IS =. and the r.m.s. deviation σ rms is.. The results using enhanced IS spin-triplet pairing matrix by a factor. is also tabulated denoted as * eff. The σ rms of the calculations and experiments shown in Table II are.,.,.,. and.9 for USB, *,, Towner and * eff, respectively. The results of Towner and * eff are close to each other with smaller r.m.s deviations and quite satisfactory. We do not see any appreciable difference among the results of, * and, whose r.m.s deviations are largerbut only.5times thoseoftownerand * eff. The IS magnetic moment is expressed by using the effective operators as [8] [µ IS J/]/(g IS s g IS l ) = J s z J +g IS s /(gis where s g IS l )δs (9) δs = δ s J s z J +δ l J l z J +δ p 8π J [Y s] () J. () The effective factors δ s, δ l and δ p are essentially equivalent to the effective factors in Eq. (9) as δ s = fs IS, δ l = fl IS / and δ p = fp IS. Since the effects of spin and orbital contributions in Eq. () to the IS magnetic moment cancel largely because of different signs of δ s and δ l so that the net effect of the effective operators is rather small. The summed IS and IV spin M transitions are tabulated in Table III for N=Z s d shell nuclei and C. The calculated values are obtained by the and * interactions with the bare spin operator as well as with the Towner s effective operators both for IS and IV transitions. The values of C are for the IS state at Ex=.78MeV and for the IV one at Ex=5.MeV, respectively. For the summed strength, the enhanced IS interaction * gives -8% quenching for the IS spin transitions and 8-% quenching for the IV ones except Ne (8% for IS and % for IV). The different values between Ne and the other nuclei is due to the smaller summed values below Ex=MeV for Ne, which exhaust only 8% and half of the total strength

12 TABLE II. IS magnetic moments extracted from the average of magnetic moments in mirror nuclei. The shell model calculations are performed by using, * and interactions as well as for the case with the Towner s effective operator (denoted as Towner). The case with the effective operator (Eq. (9)) where the coefficients are obtained by chisquare fitting procedure (denoted as * eff ) is also given. The values are given in a unit of nuclear magneton µ N. A J * Towner * eff Exp. 9 / / / / / / / / / / σ rms of full model space for IS and IV channels, respectively, while more than 9% of the total strength exists below Ex=MeV in other nuclei (except IV channel of 8 Mg; 8% ). The Towner s effective operators give large quenched values, about 7-8% quenching for all nuclei in the IS channel and 7-8% quenching (% for Ne) for the IV one. The experimental summed values of the IS strengths show rather minor quenching or even enhanced in Mg compared with results, while the IV data show a large quenching consistent with those obtained by Towner s effective operators. More quantitatively, we need a combined effect of the enhanced IS paring and the effective operators to get better agreement with the experimental values of IV channel for A 8 as seen in Fig.. Although the IS moments are very well described by Towner and * eff, the calculated S(IS) values prove to be much suppressed compared with the experimental data. The values of accumulated strengths S(IS) are.85(.8),.(.5),.579 (.5),.8 (.9) and.975 (.8) for Ne, Mg, 8 Si, S and Ar, respectively, for Towner s operators (* eff ). The difference between the IS magnetic moment and the spin M transition will be clarified by the following way. The IS spin M transition is expressed as J Ô(σ) = (fis s fis l ) J σ +f g IS 8π J [Y σ] () () where the non-diagonal matrix element of the operator J vanishes, and the effective operators f s and f l have different signs so that the net effect gives a large quenching effect. This is rather different from the IS magnetic moment in Eq. (9) where the effects of effective spin and orbital operators cancel largely. VI. SUMMARY In summary, we studied the IS and IV spin M transitions in even-even N=Z sd shell nuclei using shell model calculations with interactions in full sd shell model space. We introduced the effective operators for the spin and spin-isospin M operators in Eqs. (9) and () as well as the enhanced IS spin-triplet pairing. In general, the calculated results show good agreement with the experimental energy spectra in N=Z nuclei as far as the excitation energies are concerned. Compared with the experimental M results, the accumulated IS spin strengths up to MeV show small quenching effect, corresponding to the effective quenching operator q IS (eff).9, while a large quenching q IV (eff).7 is extracted for the IV channel. The similar quenching on the IS spin M transitions is obtained by the % enhanced IS spin-triplet pairing correlations with the bare spin operator. The enhanced IS pairing does not change much the excitation energy spectra themselves. Positive contributions for the spin-spin correlations are found by an enhanced isoscalar spin-triplet pairing interaction in these sd shell nuclei. The effects of the effective spin operators and enhanced IS pairing is also examined on the beta decay rate and the IS magnetic moments in sd shell nuclei. The r.m.s. deviation between the calculations and experiments of beta decay rate are improved by the effective operator, while the IS pairing has only a minor effect on these observables since the unpaired particle masks the effect of pairing correlations on the matrix elements. The Towner s effective spin operators works well to reproduce the accumulated experimental IV spin strength, while the quenching of the effective operators is much larger than the observed one in the IS spin channel. In the past, a large quenching of IS magnetic transition strength was suggested in the literatures. However, the (p,p ) data in ref. [] do not show any sign of the large quenching effect on the IS spin transitions. This point should be studied further experimentally by possible IS probes such as (d,d ) reactions [] together with comprehensive theoretical calculations. ACKNOWLEDGMENTS We would like to thank H. Matsubara for providing the experimental data. We would also like to thank M. Ichimura, A. Tamii, M. Sasano and T. Uesaka for the useful discussions. This work was supported in part by JSPS KAKENHI Grant Numbers JPK57 and JP5K59.

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