Compression-mode resonances in the calcium isotopes and implications for the asymmetry term in nuclear incompressibility

Citation for published version (APA): Howard, K. B., Garg, U., Itoh, M., Akimune, H., Bagchi, S., Doi, T., Fujikawa, Y., Fujiwara, M., Furuno, T., Harakeh, M. N., Hijikata, Y., Inaba, K., Ishida, S., Kalantar-Nayestanaki, N., Kawabata, T., Kawashima, S., Kitamura, K., Kobayashi, N., Matsuda, Y., ... Yang, Z. (2020). Compression-mode resonances in the calcium isotopes and implications for the asymmetry term in nuclear incompressibility. Physics Letters B, 801, [135185]. https://doi.org/10.1016/j.physletb.2019.135185

The isoscalar giant monopole resonance (ISGMR) has been wellestablished as the most direct means by which one can constrain the incompressibility of nuclear matter. The incompressibility of a nucleus is extracted from the resonance energy, E ISGMR , such that: where K A is the incompressibility of the nucleus of mass A undergoing the excitation, m is the free-nucleon mass, and r 2 0 is the mean-square radius of the ground-state density. Within different * Corresponding author.
E-mail address: garg@nd.edu (U. Garg). model frameworks, the value of E ISGMR is associated with different moment ratios of the ISGMR strength distribution [1,2], the extraction of which is the primary goal of many experiments venturing to constrain the incompressibility, K ∞ , and, thence, the Equation of State (EoS) of infinite nuclear matter [3].
In a system with a proton-neutron imbalance, the EoS depends additionally on the asymmetry parameter η = (N − Z )/A, and the symmetry energy, S(ρ). In the same way that the ISGMR provides a direct measurement of K ∞ , the curvature of the EoS of symmetric nuclear matter, the trend of measurements of nuclei with varying values of η yields a direct constraint of the curvature of S(ρ). For a more complete discussion of the means by which properties of the giant resonances provide constraints on the EoS of such asymmetric nuclear matter, we refer the reader to Refs. [3,4]. The microscopic formalism for extracting properties of infinite nuclear matter from measurements on finite nuclei is detailed in Ref. [5]. The macroscopic leptodermous expansion of K A in terms of properties of infinite nuclear matter gives: Equation (2) is useful in determining the value of K τ for finite nuclei, owing in part to the isolated dependence on η within the expression, as well as the fairly minimal changes in the surface term within an isotopic chain. The general prescription for the same is detailed in Refs. [6,7], and involves quadratically fitting the dependence of K A − K Coul Z 2 /A 4/3 on η with a model function of the form K τ η 2 + c, with c being a constant. The K τ values so extracted are consistent with one another: K τ = −550 ± 100 MeV and K τ = −555 ± 75 MeV, for the even-A 112−124 Sn and 106−116 Cd isotopes, respectively [6][7][8].
The corresponding definition of K ∞ τ in terms of properties of the EoS for infinite nuclear matter is 1 [9]: within which L and K sym are, respectively, the slope and curvature of S(ρ), and Q ∞ /K ∞ is the skewness parameter for the EoS of symmetric nuclear matter. The implications of this are that experimental constraints on K τ arising from measurements of K A on finite nuclei are helpful in determining the density dependence of the symmetry energy; this argument is predicated on the smoothness with which the values of K A vary across the nuclear chart. Indeed, as has been argued in Ref. [10], any nuclear structure effects on properties of the giant resonances which arise in a narrow region of the chart of nuclides would dramatically affect our understanding of the collective model upon which has rested the understanding of these resonances.
In light of all this, the results reported recently for 40,44,48 Ca by the TAMU group [11][12][13] were very surprising: the moment ratios for the ISGMR and, therefore, the K A values for 40,44,48 Ca increased with increasing mass number. The most immediate consequence of this, considering Eq. (2), is that K τ is a positive quantity, and it was shown in Ref. [12] that a large positive value of K τ models the data well. In a test of hundreds of energy-density functionals currently in use in the literature, the values of K τ extracted were consistently between −800 MeV ≤ K τ ≤ −100 MeV [14].
Examination of Eq. (3) also directly suggests that the symmetry energy would need to be extremely soft in order to accommodate K τ > 0 [15]. Moreover, the hydrodynamical model predicts E ISGMR ∼ A −1/3 , while the results of Refs. [11][12][13] indicated exactly the opposite: the ISGMR energies increasing with mass number over the isotopic chain.
These results clearly demanded an independent verification before significant theoretical efforts were expended in understanding, and explaining, this unusual and unexpected phenomenon. This Letter presents the results of such an experiment; it is found that the ISGMRs in the measured calcium isotopes ( A = 40, 42, 44, 48) follow the "normal" pattern of E ISGMR decreasing with increasing mass, ruling out a positive value for K τ .
The measurements were carried out at the Research Center for Nuclear Physics (RCNP) at Osaka University, utilizing a beam of 1 It should be noted that K ∞ τ is not equal to the value of K τ extracted from finite nuclei utilizing the methodology of Eq. (2), just as K ∞ = K A . However, through the same self-consistent mechanisms by which measurements of K A serve to constrain K ∞ as described by Blaizot [5], determining values of K τ from finite nuclei can constrain the EoS for asymmetric infinite nuclear matter. 386-MeV α-particles which was, for all practical purposes, "halofree". This beam impinged onto enriched 40,42,44,48 Ca target foils of areal densities ∼1-3 mg/cm 2 . The scattered α-particles were momentum analyzed in the high-resolution magnetic spectrometer, Grand Raiden [16]. The focal-plane detection system was comprised of a pair of vertical and horizontal position-sensitive multiwire drift chambers in coincidence with plastic scintillators which provided the particle identification signal [17,18]. A recent and comprehensive description of the procedure employed in the offline data analysis has been presented in Ref. [10]. Here, we briefly revisit the most salient points: in addition to the lateral dispersion of the spectrometer allowing for the scattered particles to be distributed across the horizontal focal plane according to their momentum, the unique optical properties of Grand Raiden allow for α-particles whose momentum transfers occurred only at the target to be coherently focused onto the median of the vertical focal plane distribution. On the other hand, particles that undergo scattering processes before or after the target are, correspondingly, over-or under-focused along the vertical axis. The latter events constitute the instrumental background, which in the present methodology can be eliminated from the inelastic spectra prior to further analysis. This removes any, and all, ambiguities associated with modeling the instrumental background in the subsequent extraction of the ISGMR strength distributions. In contrast, the measurements reported in Refs. [11][12][13] employed a phenomenological modeling of the instrumental background, thus introducing an additional uncertainty in the analysis [19]; this process was suggested as the most likely reason behind differences in the extracted ISGMR strength distributions noted recently for the A ∼ 90 nuclei [10,[20][21][22].
Inelastic scattering data were obtained over a broad angular range 0 • ≤ θ Lab ≤ 10 • , and the acceptance of the spectrome-   [24] relative to this data with the empirical density distributions reported in [25]. The extracted OMPs are presented in Table 1; further details of this procedure have been provided elsewhere [10]. Because of unavailability of elastic scattering data on 40 Ca, OMPs extracted for 42 Ca were employed for that nucleus. The use of OMPs from a nearby nucleus has been shown to have negligible effect in the results of the giant resonance strength extraction [7], which is further evidenced by the minimal variation in the OMPs themselves as seen in Table 1 for 42 Ca and 44 Ca. We further note that Refs. [12] and [13] has also employed the same OMP-set in the analyses of the 44 Ca and 48 Ca giant resonance data which originally motivated this work. The Multipole-Decomposition Analysis (MDA) of the inelastic spectra was carried out employing the now "standard" procedure, described, for example, in Refs. [10,22,26,27]. The experimental double-differential cross sections over the E x = 10 −31 MeV region were decomposed into a linear combination of the DWBA angular distributions for pure angular momentum transfers: The A λ (E x ) coefficients correspond to the fraction of the energyweighted sum rule (EWSR) for the multipolarity λ exhausted within a particular energy bin [2]. DWBA cross sections for isoscalar modes were included in the MDA up to λ max = 8, and the contribution from the isovector giant dipole resonance (IVGDR) was accounted for using the Goldhaber-Teller model and the available photoneutron data for the calcium isotopes [28,29]. Typical results of the MDA are presented in Fig. 2.
From the A λ coefficients, the strength distributions for the IS-GMR were calculated using the corresponding EWSR relationships [2]. Shown in Fig. 3 are the extracted ISGMR strength distributions for each of the calcium nuclei investigated in this work. From these extracted strength distributions, S(E x ), the moments of the strength distribution were extracted in the usual way: The moment ratios √ m 1 /m −1 , m 1 /m 0 , and √ m 3 /m 1 that are customarily used in characterizing the excitation energy of the IS-GMR [30] are presented in Table 2. The quoted uncertainties have been estimated using a Monte Carlo sampling from the probability distributions of the individual A λ (E x ) and constitute a 68% confidence interval. The pattern of moment ratios observed in the calcium isotopic chain (decreasing with A, as expected from the A −1/3 rule) is contrary to that reported in Ref. [12] viz. increase in the moment ratios with increasing A.  In addition to the moment ratios exhibiting the expected behavior over the isotopic chain, the demonstrated trend for the extracted finite incompressibilities, K A , is even more illustrative (see Fig. 4): The agreement of the extracted K A values with the behavior modeled by the leptodermous expansion of Eq. (2) using the accepted values for K τ and K ∞ is rather good, and stands in stark contrast to the results from Ref. [12]. While the extracted K A for 44 Ca is consistent with that which was measured in Ref. [12], the K A for the extrema of 40 Ca and 48 Ca follow precisely opposite trends between the two analyses. However, the presence of an see Table 2). The expected trend for these values utilizing the previously documented central value for K τ = −550 MeV, and K ∞ = 220 MeV as input to Eq. (2) is presented (blue dashed line), along with the same calculation but with the value K τ = +582 MeV reported in Ref. [12] (red dotted line). A fit to the data leads to a curve that is nearly identical to that shown above (blue dashed line) and leads to a value of K τ = −510 ± 115 MeV. For comparison purposes, the data from Ref. [12] are shown (red circles), as well as the K A values calculated from the ISGMR responses predicted by the relativistic FSUGarnet interaction (green squares) [15,31]. The solid lines through the data points are merely to guide the eye.
additional data point for 42 Ca -which was absent in the TAMU analysis -that follows the same general trend as the other three isotopes found in the present work inspires greater confidence in our results. These data, thus, conclusively exclude the possibility of a positive K τ value for the calcium nuclei. We also note that the fit presented for the TAMU data in Ref. [12] corresponded to K ∞ = 200 MeV, which is significantly lower than the currently accepted value of 240±20 MeV for this quantity [3,32]. Also presented in Fig. 4 [14]. These results, of course, beg the question as to the origin of the differences in the extracted ISGMR responses from those obtained by the TAMU group. The most obvious difference between the experimental techniques lies in the accounting of the instrumental background and physical continuum. Whereas in the present work the former is almost completely eliminated and the physical continuum is included within the extracted ISGMR strength, the TAMU group subtracts both by approximating a smooth background underlying the inelastic spectra. As stated earlier, this has resulted in similar discrepancies in the extracted ISGMR strengths, especially at the higher excitation energies (E x > 20 MeV), in the A ∼ 90 nuclei [10,[20][21][22].
In summary, motivated by the great cause for concern that would arise were K τ > 0 a reality, we have carried out a systematic measurement of the ISGMR response of 40,42,44,48 Ca and extracted the nuclear incompressibilities, K A , therefrom. In contrast to prior results [11][12][13], the ISGMR strength distributions, and the metrics that are generally used to characterize the excitation energy of the response, obey expected trends. It may be concluded, therefore, that there are no local structure effects on the ISGMR strength distribution in the calcium region of the nuclear chart and that a positive value for the asymmetry term of nuclear incompressibility, K τ , is ruled out.