Deformation-induced splitting of the monopole giant resonance in 24Mg

The strong deformation splitting of the isoscalar giant monopole resonance (ISGMR), recently observed in ($\alpha,\alpha'$) reaction in prolate $^{24}$Mg, is analyzed in the framework of the Skyrme quasiparticle random-phase-approximation (QRPA) approach with the Skyrme forces SkM*, SVbas and SkP$^{\delta}$. The calculations with these forces give close results and confirm that the low-energy E0-peak is caused by the deformation-induced coupling of ISGMR with the K=0 branch of the isoscalar giant quadrupole resonance.


I. INTRODUCTION
The isoscalar giant monopole resonance (ISGMR) represents the main source of information on the nuclear incompressibility [1] and is a subject of intense studies during last decades, see the book [2] and recent reviews [3,4]. In deformed nuclei, ISGMR is coupled with the K π = 0 + branch of the isoscalar giant quadrupole resonance (ISGQR), which leads to a double-bump structure (splitting) of the ISGMR [2]. The energy interval between the bumps is larger than the ISGMR and ISGQR widths [2]. Besides, in well deformed nuclei, both bumps carry a significant monopole strength. These two factors favor an experimental observation of the ISGMR splitting.
The first experimental data of this kind have been recently obtained. Namely, the ISGMR splitting in 24 Mg was observed in (α, α ′ ) reaction to forward angles [11]. The experiment was performed in the Research Center for Nuclear Physics (RCNP) in Osaka University. The nucleus 24 Mg has a huge prolate quadrupole axial deformation with β=0.605±0.008 [10] and thus promises a strong E0-E2 coupling. However fist attempts to observe a discernible ISGMR splitting in this nucleus using inelastic scattering of α-particles and 6 Li have failed [6,[12][13][14]. Only in [11], the experimentalists have managed to reliably discriminate the splitting. It turned out to be huge, with a strong narrow peak at E 1 ∼ 16 MeV and a broad structure at E 2 ∼ 24 MeV, see Fig. 1.
The calculations in the framework of the quasiparticle random-phase-approximation (QRPA) with the Skyrme force SkM* [15], presented in Ref. [11], have confirmed the E0-E2 origin of the ISGMR splitting observed in 24 Mg. However it is known that QRPA description of giant resonances can noticeably depend on the applied Skyrme parametrization [16][17][18][19]. Such dependence can take place for ISGMR in deformed nuclei as well [20]. In this connection, it is worth to check the Skyrme QRPA results obtained with SkM* in Ref. [11] by using other Skyrme parameterizations. This is just the aim of the present study.
Here we explore the ISGMR splitting in 24 Mg within Skyrme QRPA approach with the forces SkM* [15], SVbas [19] and SkP δ [21]. The force SkM* (K ∞ =217 MeV) is used for comparison with the previous calculations [11]. Two other forces are chosen as representatives of essentially different nuclear incompressibilities: K ∞ =234 MeV for SVbas and 202 MeV for SkP δ . As shown below, all three forces give qualitatively close results and confirm that the ISGMR splitting arises just because of the E0-E2 coupling.

II. MODEL
The calculations are performed with the twodimensional (2D) QRPA code [22]. As compared to our previous calculations for ISGMR [20], the present code does not use any separable ansatz. The method is fully self-consistent because: i) both the mean field and residual interaction are obtained from the same Skyrme functional, ii) the residual interaction includes all terms of the functional, including Coulomb (direct and exchange) terms. Both time-even and time-odd densities are involved. The code exploits a mesh in cylindrical coordinates with a mesh size of d=0.7 fm and a calculation box of about three nuclear radii.
The experimental value of the deformation β=0.605 [10] is used for all three forces. The δ-force volume pairing is treated at the BCS level [23]. The pairing particleparticle channel in the residual interaction is taken into account.
The ISGMR strength function reads where |0 is the ground state wave function, |ν and E ν are QRPA states and energies,M ISGMR (E0) = , and SkP δ (c). In the panel a), the SkM* strength function from [11] is also given (blue dotted curve). In all the panels, the TAMU [11,24] (yellow stars) and RCNP [11] (red squares) experimental data are exhibited. The calculated energy centroids of the ISGQR(K=0)-branch are indicated by the red dash arrows. The estimation EISGMR = 78A −1/3 MeV [2] for the ISGMR energy is marked by the black bold arrow.
A i (r 2 Y 00 ) i is the isoscalar (T=0) transition operator, ] is the Lorentz smoothing with the averaging parameter ∆. The Lorentz function approximately simulates smoothing effects beyond QRPA and makes convenient comparison of the calculated and experimental strengths. The calculations [11]used the smearing 3 MeV. Here the averaging ∆= 2.5 MeV is found optimal.
Our calculations use sufficiently large basis. In SVbas the single particle (s-p) spectrum includes 750 proton and 920 neutron levels in the energy intervals (-32, +61) MeV and (-37, +85) MeV, respectively. The two-quasiparticle basis involves about 3700 states with the energies up to ∼200 MeV. The energy-weighted sum rule for the isoscalar monopole excitations is exhausted by 94%. The similar basis is used for SkM* and SkP δ .

III. RESULTS AND DISCUSSION
Results of the calculations are presented in Fig. 1. They are compared with the experimental data from RCNP [11] and TAMU (Texas A&M University) [11,24]. Following the statement in Ref. [11], RCNP data do exhibit the ISGMR splitting while TAMU data do not.
As seen from Fig. 1, the calculations with SkM*, SVbas and SkP δ give a qualitatively similar picture and justify origin of the narrow peak at E ∼ 16 MeV as a result of the coupling between ISGMR and K=0 branch of ISGQR. Indeed the position of this peak well coincides with the energy of ISGQR(K=0) branch, marked by the dash arrows in the figure. The second evidence is that this peak is almost disappears in the spherical limit (β=0) when the E0-E2 coupling is absent. The agreement with the experimental energy E ∼ 16 MeV is very nice for SkM* and SVbas and somewhat worse for SkP δ . The latter is explained by a low incompressibility K ∞ =202 MeV in SkP δ .
Our and previous [11] SkM* calculations give about the same description of the peak at E ∼ 16 MeV (compare the black bold and blue dotted curves in the panel (a)). Both results are also similar for E0 strength above the peak. A noticeable difference between two calculations takes place only in a low energy region with E<13 MeV, which is beyond of our interest. Perhaps this difference is partly caused by using variant pairing schemes: (HF+BCS in our case and HFB in [11]).
In Figure 1, the RCNP experimental data give a highenergy distribution peaked at ∼24 MeV. This distribution constitutes the familiar ISGMR. The broad IS-GMR structure seems to continue to the low-energy region and form a massive background of the narrow peak at E ∼ 16 MeV. This is confirmed by the distribution of E0 strength calculated in the spherical limit. Note that the energy E ∼ 24 MeV gives E ISGMR = 69A −1/3 MeV which is closer to the empirical estimation for IS-GQR (E ISGQR = 64A −1/3 MeV) rather than for ISGMR (E ISGMR = 78A −1/3 MeV) [2]. The latter gives for IS-GMR in 24 Mg the energy E ISGMR ∼27 MeV.
As seen from Fig. 1, our calculations in general reproduce the ISGMR at ∼24 MeV. The forces SkM* and SVbas demonstrate a better performance than SkP δ with its very low incompressibility. Note also that present SkM*, SVbas and SkP δ calculations, as well as the SkM* results [11], underestimate the observed E0 strength at E> 24 MeV. Thus the computed ISGMR looks somewhat downshifted as compared to the experimental data. Most probably the QRPA is not enough to describe details of present experimental distribution and we need here the coupling with complex configurations.

IV. SUMMARY
The analysis of the recent experimental data for the deformation-induced splitting of the isoscalar gi-ant monopole resonance (ISGMR) in strongly deformed 24 Mg [11] has been done within the Skyrme quasiparticle random-phase approximation (QRPA) approach. The Skyrme forces SkM* [15], SVbas [19], and SkP δ with different values of nuclear incompressibility were applied. The calculation generally well reproduce the experimental data and confirmed the origin of the peak at E ∼ 16 MeV as a result of the E0-E2 coupling. For a more precise description of the experimental E0 distribution, inclusion of the coupling with complex configurations is desirable.