Why is Tin so soft?
The distribution of isoscalar monopole strength in the neutron-even Sn-isotopes has been computed using a relativistic random-phase-approximation approach. The accurately-calibrated model used here (“FSUGold”) has been successful in reproducing both ground-state observables as well as collective excitations — including the giant monopole resonance (GMR) in Zr, Sm, and Pb. Yet this same model significantly overestimates the GMR energies in the Sn isotopes. It is argued that the question of “Why is Tin so soft?” becomes an important challenge to the field and one that should be answered without sacrificing the success already achieved by several theoretical models.
The compression modulus of nuclear matter (also known as the nuclear incompressibility) is a fundamental parameter of the equation of state that controls small density fluctuations around the saturation point. While existing ground-state observables have accurately constrained the binding energy per nucleon ( MeV) and the baryon density () of symmetric nuclear matter at saturation, the extraction of the compression modulus () requires to probe the response of the nuclear system to small density fluctuations. It is generally agreed that the nuclear compressional modes — particularly the isoscalar giant monopole resonance (GMR) — provide the optimal route to the determination of the nuclear incompressibility Blaizot (1980). Moreover, the field has attained a level of maturity and sophistication that demands strict standards in doing so. It is now demanded that the same microscopic model that predicts a particular value for the compression modulus of infinite nuclear matter (an experimentally inaccessible quantity) be able to accurately reproduce the experimental distribution of monopole strength.
Earlier attempts at extracting the compression modulus of symmetric nuclear matter relied primarily on the distribution of isoscalar monopole strength in Pb — a heavy nucleus with a well developed giant resonance peak Youngblood et al. (1977, 1981). However, as was pointed out recently in Refs. Piekarewicz (2002, 2004) — and confirmed since then by several other groups Vretenar et al. (2003); Agrawal et al. (2003); Colò et al. (2004) — the GMR in Pb does not provide a clean determination of the compression modulus of symmetric nuclear matter. Rather, it constraints the nuclear incompressibility of neutron-rich matter at the particular value of the neutron excess found in Pb, namely, . As such, the GMR in Pb is sensitive to the density dependence of the symmetry energy. The symmetry energy represents a penalty levied on the system as it departs from the symmetric limit of equal number of neutrons and protons. As the infinite nuclear system becomes neutron rich, the saturation density moves to lower densities, the binding energy weakens, and the nuclear incompressibility softens Piekarewicz (2007). Thus, the compression modulus of a neutron rich system having the same neutron excess as Pb is lower than the compression modulus of symmetric nuclear matter. We note in passing that the symmetry energy is to an excellent approximation equal to the difference between the energy of pure neutron matter (with ) and that of symmetric nuclear matter (with ).
The alluded sensitivity of the distribution of isoscalar monopole strength to the density dependence of the symmetry energy proved instrumental in resolving a puzzle involving : how can accurately calibrated models that reproduce ground state data as well as the distribution of monopole strength in Pb, predict values for that differ by as much as 25%? This discrepancy is now attributed to the poorly determined density dependence of the symmetry energy Piekarewicz (2002). Indeed, models that predict a stiffer symmetry energy (one that increases faster with density) consistently predict higher compression moduli than those with a softer symmetry energy. Thus, the success of some models in reproducing the GMR in Pb was accidental, as it resulted from a combination of both a stiff equation of state for symmetric nuclear matter and a stiff symmetry energy Piekarewicz (2004). Since then, the large differences in the predicted value of have been reconciled and a “consensus” has been reached that places the value of the incompressibility coefficient of symmetric nuclear matter at MeV Agrawal et al. (2003); Colò et al. (2004); Todd-Rutel and Piekarewicz (2005); Garg (2006).
An example of how this consensus was reached is depicted in Fig. 1 where the distribution of isoscalar monopole strength in Zr, Sn, Sm, and Pb at the small momentum transfer of MeV (or ) is displayed for the relativistic FSUGold model of Ref. Todd-Rutel and Piekarewicz (2005) — a model that predicts an incompressibility coefficient for symmetric nuclear matter of MeV. Note that the distribution of strength was obtained from a relativistic random-phase-approximation (RPA) approach as described in detail in Ref. Piekarewicz (2001). Further, the inset on Fig. 1 shows a comparison of the theoretical predictions against the experimental centroid energies reported in Ref. Youngblood et al. (1999). Finally, the solid line in the inset provides a fit to the mass dependence of the theoretical predictions that yields .
The isoscalar monopole strength displayed in Fig. 1 is extracted from the low momentum transfer behavior of the longitudinal response defined as follows:
Here is the exact nuclear ground state, is an excited state with excitation energy , and is the Fourier transform of the isoscalar baryon density. That is,
where is an isodoublet nucleon field, is the timelike (or zeroth) component of the Dirac gamma matrices, and is the identity matrix in isospin space.
The important realization that the distribution of monopole strength in heavy nuclei is sensitive to the density dependence of the symmetry energy has motivated a recent experimental study of the GMR along the isotopic chain in Tin. Indeed, the distribution of isoscalar monopole strength in the neutron-even Sn-isotopes has been measured at the Research Center for Nuclear Physics (RCNP) in Osaka, Japan Garg (2006); Li et al. (2007). This important experiment probes the incompressibility of asymmetric nuclear matter by measuring the distribution of isoscalar strength in a chain of isotopes with a neutron excess ranging from (in Sn) to (in Sn). The experiment represents a hadronic compliment to the purely electroweak Parity Radius Experiment (PREX) at the Jefferson Laboratory that aims to measure the neutron radius of Pb accurately and model independently via parity-violating electron scattering Horowitz et al. (2001); Michaels et al. (2005). Such an accurate determination will have far-reaching implications in areas as diverse as nuclear structure Todd and Piekarewicz (2003), heavy-ion collisions Danielewicz et al. (2002); Tsang et al. (2004); Chen et al. (2005); Li and Steiner (2005); Shetty et al. (2007); Horowitz (2006), atomic parity violation Todd and Piekarewicz (2003); Sil et al. (2005); Piekarewicz (2006) and nuclear astrophysics Horowitz and Piekarewicz (2001); Buras et al. (2003); Lattimer and Prakash (2004); Steiner et al. (2005).
In Fig. 2 the experimental distribution of isoscalar monopole strength measured at the RCNP Garg (2006); Li et al. (2007) is compared against the predictions of the highly successful NL3 Lalazissis et al. (1997, 1999) and FSUGold Todd-Rutel and Piekarewicz (2005) models. As one is only interested in comparing the shape of the distribution and a particular ratio of its moments, the maximum of the theoretical curves — computed from the longitudinal response as described in the text — has been normalized to the experimental data. The -dependence of the corresponding centroid energies is also displayed in Fig. 3 and compiled in Table 1. Note that the centroid energy is computed from the ratio of the moment to that of the moment. That is,
where, consistent with the experimental analysis Garg (2006); Li et al. (2007), the limits of integration have been chosen to be MeV and MeV. Further, to mimic the forward-angle experiment, the longitudinal response was evaluated at the “small” momentum transfer of .
A subtle telltale problem with Tin barely discernible in the inset on Fig. 1, becomes magnified in Fig. 2 as one compares the experimentally extracted distribution of monopole strength against the theoretical predictions. While mean-field plus RPA calculations are typically unable to describe the experimental width — which is in general composed of both an escape (particle-hole) and a spreading (multiparticle-multihole) width — such is not the case for the description of the centroid energies. Indeed, accurately calibrated models, both non-relativistic Colò et al. (2004) and relativistic (see Fig. 1), provide an adequate description of the GMR centroid energies in both Zr (with ) and Pb (with ) — nuclei with a neutron excess similar to those at the two extremes of the isotopic chain considered here. Why is then that both non-relativistic Garg (2006); Li et al. (2007); Colò (2007) and relativistic models consistently overestimate the centroid energies in the Sn-isotopes? Or more colloquially, why is Tin so soft? And why is that the discrepancy between theory and experiment continues to grow as the neutron excess increases? A stiff symmetry energy leads to a rapid softening of the nuclear incompressibility Piekarewicz (2007). This is the main reason behind the slightly larger (negative) slope displayed by NL3 relative to FSUGold in Fig. 3. The even larger (by more than 50%) slope displayed by the experimental data is unlikely to be solely related to the stiffness of the symmetry energy, as NL3 already predicts a neutron skin thickness in Pb that appears overly large Todd-Rutel and Piekarewicz (2005).
So why is Tin so soft and why does it become even softer with an increase in the neutron excess? Could there be a systematic error in the experimental extraction? While possible, this is unlikely as an earlier independent measurement on Sn Youngblood et al. (1999) appears to confirm the present (RCNP) result (see Fig. 3). Could the GMR in Tin probe physics that has not been already constrained by nuclear observables? This also appears unlikely as existing density functionals are successful at describing a host of ground-state observables as well as collective excitations — including the GMR in Zr, Sm, and Pb (see Fig. 1 and Ref. Todd-Rutel and Piekarewicz (2005)). Could Tin be sensitive to pairing correlations and more complicated multiparticle-multihole excitations? The answer appears to be negative Colò (2007), but even if it would be positive, why should Tin be sensitive to these effects but not Zr, Sm, and Pb? Clearly, the distribution of isoscalar monopole strength in the Sn-isotopes poses a serious theoretical challenge, perhaps suitable for the new Universal Nuclear Energy Density Functional (UNEDF) initiative. Whatever the theoretical approach, however, one must remember that the challenge is not solely to describe the distribution of monopole strength along the isotopic chain in Tin, but rather to do so without sacrificing the enormous success already achieved in reproducing a host of ground-state properties and collective modes.
Acknowledgements.The author is grateful to Professors G. Colò and U. Garg for many fruitful discussions. The author also wishes to thank Prof. Garg and his collaborators for sharing the experimental data prior to publication. This work was supported in part by DOE grant DE-FD05-92ER40750.
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