Emergence of lowenergy monopole strength
in the neutronrich calcium isotopes
Abstract
 Background

The isoscalar monopole response of neutronrich nuclei is sensitive to both the incompressibility coefficient of symmetric nuclear matter and the density dependence of the symmetry energy. For exotic nuclei with a large neutron excess, a lowenergy component emerges that is driven by transitions into the continuum.
 Purpose

While understanding the scaling of the giant monopole resonance with mass number is central to this work, the main goal of this paper is to explore the emergence, evolution, and origin of low energy monopole strength along the eveneven calcium isotopes: from Ca to Ca.
 Methods

The distribution of isoscalar monopole strength is computed in a relativistic random phase approximation (RPA) using three effective interactions that have been calibrated to the properties of finite nuclei and neutron stars. A nonspectral approach is adopted that allows for an exact treatment of the continuum without any reliance on discretization. This is particularly critical in the case of weaklybound nuclei with singleparticle orbits near the continuum. The discretization of the continuum is neither required nor admitted.
 Results

For the stable calcium isotopes, no evidence of lowenergy monopole strength is observed, even as the 1f neutron orbital is being filled and the neutronskin thickness progressively grows. Further, in contrast to experimental findings, a mild softening of the monopole response with increasing mass number is predicted. Beyond Ca, a significant amount of lowenergy monopole strength emerges as soon as the weakbinding neutron orbitals (2p and 1f) become populated. The emergence and evolution of lowenergy strength is identified with transitions from these weaklybound states into the continuum—which is treated exactly in the RPA approach. Moreover, given that models with a soft symmetry energy tend to reach the neutrondrip line earlier than their stiffer counterparts, we identify an inverse correlation between the neutronskin thickness and the inverse energy weighted sum.
 Conclusions

Despite experimental claims to the contrary, we found a mild softening of the giant monopole resonance in going from Ca to Ca. Measurements for other stable calcium isotopes may be critical in elucidating the nature of the discrepancy. Moreover, given the early success in measuring the distribution of isoscalar monopole strength in the unstable Ni nucleus, we advocate new measurements along the unstable neutronrich calcium isotopes to explore the critical role of the continuum in the development of a soft monopole mode.
pacs:
21.60.Jz, 24.10.Jv, 24.30.CzI Introduction
What are the relatively few combinations of neutrons and protons that form a bound atomic nucleus is a question of critical importance to both nuclear structure and astrophysics Lon (); NuP (). Probing the limits of nuclear existence lies at the heart of the commissioning of state of the art rare isotope facilities throughout the world. However, mapping the precise boundaries of the nuclear landscape is enormously challenging. Given that the Coulomb interaction severely limits the number of neutrons that may be removed from a given isotope, the proton drip lines remains relatively close to the valley of stability. Indeed, the proton drip line has been determined experimentally up to protactinium, an isotope with atomic number . In stark contrast, the neutron drip line has been mapped only up to oxygen Thoennessen (2004). Thus, the neutronrich landscape constitutes a fertile ground for research in nuclear structure and—due to its relevance to the rprocess and to the composition of the neutronstar crust—also in astrophysics Utama and Piekarewicz (2017).
Exceptional experimental advances have led to a paradigm shift in fundamental core concepts that have endured the test of time, such as the traditional nuclear magic numbers. Understanding the impact of such a paradigm shift in the development of novel modes of excitation in exotic nuclei has become a fruitful area of research Paar et al. (2007). Given the richness of these excitations, they offer a unique window into the nuclear dynamics that is often closed to other means Harakeh and van der Woude (2001). In particular, the isovector dipole resonance has been shown to be highly sensitive to the equation of state of neutronrich matter, specifically to the density dependence of the symmetry energy Reinhard and Nazarewicz (2010); Piekarewicz (2011); Piekarewicz et al. (2012); RocaMaza et al. (2013). Indeed, measurements of the electric dipole polarizability in a variety of stable neutronrich nuclei Tamii et al. (2011); Poltoratska et al. (2012); Hashimoto et al. (2015); Birkhan et al. (2016) and in the unstable Ni isotope Wieland et al. (2009); Rossi et al. (2013), provide an attractive alternative to parityviolating experiments to determine the neutronskin thickness of Pb Abrahamyan et al. (2012); Horowitz et al. (2012) as well as the density dependence of the symmetry energy.
Of specific interest in this paper is the emergence and evolution of isosocalar monopole strength along the calcium isotopes. There are several reasons for undertaking such a study. First, the compressibility of neutronrich matter is sensitive to both the incompressibility coefficient of symmetric nuclear matter and the density dependence of the symmetry energy Piekarewicz and Centelles (2009). Particularly interesting is to examine this sensitivity along an isotopic chain having several stable isotopes, such as calcium and tin. Note that although a decade has already passed since the “fluffiness” of tin was first identified Garg et al. (2007); Li et al. (2007, ), a theoretical explanation continues to elude us Piekarewicz (2007); Sagawa et al. (2007); Avdeenkov et al. (2009); Li et al. (2008); Piekarewicz (2010); Khan (2009a, b); Khan et al. (2010); Cao et al. (2012); Vesely et al. (2012); Piekarewicz (2013); Colò et al. (2014); Chen et al. (2013, 2014). That is, theoretical models that account for Giant Monopole Resonance (GMR) energies in Zr, Sm, and Pb, overestimate the GMR energies in all stable tin—and cadmium () Patel et al. (2012)—isotopes. The imminent report of the GMR energy of the unstable Sn nucleus may prove vital in elucidating the softness of tin Garg (2017). In turn, an exploration of the evolution of the GMR energies along the isotopic chain in calcium above and beyond the already measured Ca and Ca isotopes Youngblood et al. (2001); Lui et al. (2011); Anders et al. (2013) may be highly illuminating. This is particularly relevant given that existing experimental data seem to suggest an increase in the GMR energies with increasing mass, a fact that is difficult to reconcile with theoretical expectations Anders et al. (2013). Second, theoretical and computational advances have evolved to such an extent that ab initio calculations in the calcium region are now possible. Ab initio calculations now exist that can predict the electric dipole polarizability as well as the charge and weak form factors of both Ca and Ca Hagen et al. (2015). Indeed, a primary motivation for the Calcium Radius EXperiment (“CREX”) at Jefferson Lab is to bridge ab initio approaches with nuclear density functional theory Mammei et al. (2013). Finally, extending GMR calculations beyond Ca highlights the role of the continuum in the emergence of lowenergy monopole strength. We expect that transitions out of the weakly bound neutron orbitals dominate the development of a soft monopole mode.
Although the compressibility of neutronrich matter is primarily sensitive to the incompressibility coefficient of symmetric nuclear matter, a sensitivity to the symmetry energy develops in heavy nuclei with a significant neutron excess. Unfortunately, this sensitivity is often hindered by the relatively small neutron excess of the stable nuclei investigated up to date; note that the contribution from the symmetry energy scales as the square of the neutronproton asymmetry Piekarewicz and Centelles (2009). Hence, measuring the distribution of isoscalar monopole strength in unstable nuclei with a large neutron excess is highly appealing. Despite the relatively modest value of (and ) the measurement of the isoscalar monopole response of Ni by Vandebrouck and collaborators represents an important first step in the right direction Vandebrouck et al. (2014). Although this pioneering experiment established the great potential of inelastic alpha scattering in inverse kinematics to probe the distribution of isoscalar monopole strength in unstable neutronrich nuclei, a controversy ensued on the interpretation of the observed lowenergy structure. Based on RPA predictions that used a discretized continuum, it was suggested that the observed lowenergy strength consisted of individual peaks that were well separated from the main giant resonance Capelli et al. (2009); Khan et al. (2011). Shortly after, Hamamoto and Sagawa concluded based on SkyrmeRPA calculations that did not rely on a discretization of the continuum, that the development of isoscalar monopole peaks in the lowenergy region was very unlikely Hamamoto and Sagawa (2014). Subsequently, relativistic RPA calculations confirmed that while significant amount of lowenergy strength in Ni is indeed generated, such lowenergy structure lacks any distinct features Piekarewicz (2015). These findings suggest that a proper treatment of the continuum is critical. As we shall see below, the continuum plays a fundamental role in the development of a soft monopole mode in all unstable neutronrich calcium isotopes.
The paper has been organized as follows. In Sec. II a brief summary of the relativistic RPA formalism is presented highlighting the role of the continuum. In Sec. III we display both uncorrelated and RPA predictions for the distribution of isoscalar monopole strength along the isotopic chain in calcium for all eveneven isotopes: from Ca up to Ca. In particular, we pay special attention to the role of the continuum in shaping the distribution of lowenergy strength. Finally, we provide a summary of the most important findings and suggestions for the future in Sec. IV.
Ii Formalism
The main goal of this section is to provide the technical material necessary to follow the discussion presented in Sec. III. The relativistic RPA formalism as implemented here has been reviewed extensively in earlier publications; see for example Refs. Piekarewicz (2013, 2015) and references contained therein. In particular, the critical role that the continuum plays on the emergence of lowenergy isoscalar monopole strength has been extensively addressed in Ref. Piekarewicz (2015).
ii.1 Isoscalar Monopole Response
In the relativistic meanfield (RMF) approach pioneered by Serot and Walecka Walecka (1974); Serot and Walecka (1986) the basic constituents of the effective theory are protons and neutrons interacting via the exchange of the photon and various mesons of distinct Lorentz and isospin character. Besides the conventional Yukawa couplings of the mesons to the relevant nuclear currents, the model is supplemented by a host of nonlinear meson coupling that have been found essential to improve the standing of the model Boguta and Bodmer (1977); Mueller and Serot (1996); Horowitz and Piekarewicz (2001). In the widely used meanfield approximation, the nucleons satisfy a Dirac equation in the presence of strong scalar and vector potentials that are the hallmark of the relativistic approach. In turn, the photon and the mesons satisfy classical KleinGordon equations with the relevant nuclear densities acting as source term. As in most meanfield approaches, this close interdependence demands that the equations be solved selfconsistently. In particular, the selfconsistent procedure culminates with a few observables that fully characterize the meanfield ground state.
With these observables in place, one may proceed to compute the linear response of the meanfield ground state to an external perturbation. All dynamical information relevant to the excitation spectrum of the system is encoded in the polarization propagator which is a function of both the energy and momentum transfer to the nucleus Fetter and Walecka (1971); Dickhoff and Van Neck (2005). The first step in the calculation of the nuclear response is the construction of the uncorrelated (or meanfield) polarization propagator. That is,
(1) 
where and are singleparticle energies and Dirac wavefunctions obtained from the selfconsistent determination of the meanfield ground state, contains information on the Lorentz and isospin structure of the operator responsible for the transition, and is the “Feynman” propagator. Note that the sum is restricted to positiveenergy states below the Fermi level. In turn, may also be expressed as a sum over states by relying on its spectral decomposition:
(2) 
where now and represent singleparticle energies and Dirac wavefunctions associated with the negativeenergy part of the spectrum; recall that in the relativistic formalism the positive energy part of the spectrum by itself is not complete. However, note that now the spectral sum is unrestricted, as it involves bound and continuum states of positive and negative energy. In practice, one avoids carrying out this infinite sum by invoking a nonspectral representation of the Feynman propagator. This involves solving the following inhomogeneous differential equation for the Green’s function in the meanfield approximation. That is,
(3) 
where and are Dirac gamma matrices, and is the same exact meanfield potential obtained from the selfconsistent solution of the groundstate problem Piekarewicz (2013).
The uncorrelated meanfield polarization together with the residual interaction constitute the two main building blocks of the RPA polarization depicted in Fig.1. By iterating the uncorrelated polarization to all orders, coherence is build through the mixing of all particlehole excitations of the same spin and parity. If many particlehole pairs get mixed, then the resulting RPA response displays strong collective behavior that manifests itself in the appearance of one “giant resonance” that often exhausts the classical sum rule Harakeh and van der Woude (2001). If the residual interaction is attractive in the channel of interest (as in the case of the isoscalar monopole response) then the RPA distribution of strength is softened and enhanced relative to the uncorrelated response. If instead the residual interaction is repulsive (as in the case of the isovector dipole response) then the RPA response is hardened and quenched. For further details see Refs. Piekarewicz (2013, 2015).
Given that the distribution of isoscalar monopole strength is the main focus of this paper, we now briefly describe how to extract the response from the RPA polarization. To excite the appropriate channel, it is sufficient to use for the transition operator given in Eq. (1) the timelike component of the Dirac gamma matrices, namely, . That is,
(4) 
The above expression is still a function of both the energy and momentum transfer to the nucleus. In particular, “peaks” in the energy transfer are associated to the nuclear excitations and the area under the peaks are proportional to the transition form factor at the given momentum transfer. Since inelastic scattering at forward angles is the experimental method of choice in isolating the isoscalar monopole strength from the overall cross section, we compute in the long wavelength limit. That is,
(5) 
Often one is interested in computing moments of the strength distribution to extract centroid energies of the GMR. The moments of the distribution are defined as suitable energy weighted sums:
(6) 
Here we concentrate on the energy weighted , energy unweighted , and inverse energy weighted sums Harakeh and van der Woude (2001).
A few comments that highlight the power of the nonspectral approach are in order. First, concerning the uncorrelated meanfield polarization depicted by the thin blue “bubble” in Fig.1, the nonspectral framework adopted here is immune to most of the features that hinder the spectral approach, such as an energy cutoff and the discretization of the continuum. Indeed, some of these artifacts were responsible for the faulty interpretation of the structure of the lowenergy monopole strength observed in Ni Vandebrouck et al. (2014). Second, besides avoiding any reliance on artificial cutoffs and truncations, the nonspectral approach has the added benefit that both the positive and negativeenergy continua are treated exactly. Finally, in the context of the RPA response, it is important to underscore that in the interest of consistency, both the meanfield potential appearing in Eq. (3) as well as the residual particle hole interaction depicted by the red wavy line in Fig.1, must be fully consistent with the interaction used to generate the meanfield ground state. It is only in this manner that one can guarantee both the conservation of the vector current and the decoupling of the centerofmass motion from the physical response Thouless (1961); Dawson and Furnstahl (1990); Ring and Schuck (2004).
We close this section by briefly addressing the emergence of low energy strength in neutronrich nuclei and its connections to fundamental parameters of the equation of state (EOS). Evidence of a soft dipole mode in exotic nuclei has generated considerable excitement as both a novel mode of excitation and as a possible constraint on the EOS Tsoneva et al. (2004); Piekarewicz (2006); Tsoneva and Lenske (2008); Klimkiewicz et al. (2007); Carbone et al. (2010); Papakonstantinou et al. (2014); Savran et al. (2013). One of the goals of the present study is to investigate how does the distribution of isoscalar monopole strength in the calcium isotopes—particularly the appearance of a soft monopole mode—scales with mass number. To quantify the sensitivity of the bulk parameters of infinite nuclear matter to the the isoscalar monopole response we introduce the energy per particle of asymmetric nuclear matter at zero temperature:
(7) 
where is the nuclear density, is the energy per particle of symmetric nuclear matter, is the symmetry energy, and the neutronproton asymmetry. Now expanding the energy per particle around saturation density () one obtains
(8) 
where quantifies the deviation of the density from its value at saturation—and , , and denote the binding energy per nucleon, curvature (i.e., incompressibility), and skewness parameter of symmetric nuclear matter. In turn, , , and represent the corresponding quantities for the symmetry energy. However, unlike symmetric nuclear matter which saturates, the symmetry pressure—or equivalently the slope of the symmetry energy —does not vanish. In particular, induces changes in the saturation density and incompressibility coefficient of asymmetric neutronrich matter Piekarewicz and Centelles (2009) that are given by
(9a)  
(9b) 
The above expression for suggests that measurements of the isotopic dependence of the giant monopole resonance in exotic nuclei with a very large neutron excess—such as the calcium isotopes—may place important constraints on the density dependence of the symmetry energy Garg et al. (2007); Piekarewicz (2007). Important first steps along this direction have been already taken by Garg and collaborators Li et al. (2007, ); Patel et al. (2012).
Iii Results
We start this section by computing the distribution of isoscalar monopole strength for three relativistic meanfield models: (a) NL3 Lalazissis et al. (1997, 1999), FSUGold ToddRutel and Piekarewicz (2005), and FSUGarnet Chen and Piekarewicz (2015). Among these three, FSUGarnet is a recently calibrated parametrization that has been fitted to the groundstate properties of magic and semimagic nuclei, a few giant monopole energies, and well established properties of neutron stars Chen and Piekarewicz (2014). In particular, FSUGarnet predicts that the isotopic chain in oxygen can be made to terminate at O, as it has been observed experimentally Thoennessen (2004). The parameters for the three relativistic models adopted in this work are listed in Table 1 in terms of the Lagrangian density defined in Ref. Chen and Piekarewicz (2014).
Model  

NL3  508.194  782.501  763.000  104.3871  165.5854  79.6000  3.8599  0.015905  0.000000  0.000000 
FSUGold  491.500  782.500  763.000  112.1996  204.5469  138.4701  1.4203  0.023762  0.060000  0.030000 
FSUGarnet  496.939  782.500  763.000  110.3492  187.6947  192.9274  3.2602  0.003551  0.023500  0.043377 
Using the model parameters listed in Table 1, one can then proceed to make predictions for a variety of bulk properties of infinite nuclear matter as defined in Eqs. (8) and (9). We display these set of bulk properties in Table 2 alongside quantities denoted as , , and that represent the incompressibility coefficient of asymmetric matter having the same neutron excess as Ca(), Ca(), and Ca(), respectively. Although NL3 predicts an incompressibility coefficient that is considerably larger than the other two models, NL3 becomes the softest of the three models for infinite nuclear matter with a neutron excess equal to that of Ca. This rapid softening of the incompressibility coefficient is attributed to NL3’s stiff symmetry energy ( MeV) which in turn is responsible for generating a large and negative value for . Hence, exploring the isotopic dependence of the ISGMR over an isotopic chain that includes exotic nuclei with a large neutron excess may become instrumental in constraining the density dependence of the symmetry energy.
Model  

NL3  0.148  16.24  271.5  209.5  37.29  118.2  100.9  699.4  271.5  252.1  193.8 
FSUGold  0.148  16.30  230.0  522.7  32.59  60.5  51.3  276.9  230.0  222.3  199.3 
FSUGarnet  0.153  16.23  229.5  4.5  30.92  51.0  59.5  247.3  229.5  222.7  202.1 
The distribution of isoscalar monopole strength for all stable eveneven calcium isotopes from Ca to Ca is displayed in Fig. 2 as predicted by the new FSUGarnet parametrization with the centroid energies shown in parenthesis; see Eq. (6) and Table 3. Although an enhancement of the response is evident as the f neutron orbital is progressively being filled (see inset in the figure) no appreciable softening is observed. This is somehow surprising given that as the f orbital is being filled and the neutronskin thickness steadily increases, the appearance of a soft monopole mode involving neutronskin oscillations may have been natural. Yet, no lowenergy monopole strength is detected. Values for the neutronskin thickness of the calcium isotopes alongside other ground state properties are listed in Table 4.
NL3Lalazissis et al. (1997, 1999)  FSUGoldToddRutel and Piekarewicz (2005)  FSUGarnetChen and Piekarewicz (2015)  ExperimentYoungblood et al. (2001); Lui et al. (2011)  

Isotope  
Ca  22.29  21.52  21.53  20.76  20.81  20.09  
Ca  22.04  21.28  21.39  20.74  20.64  20.02  
Ca  21.81  21.10  21.16  20.47  20.44  19.81  
Ca  21.56  20.89  21.00  20.34  20.31  19.69  
Ca  21.27  20.58  20.85  20.20  20.22  19.57 
The predicted trend displayed in Fig. 2, however, is consistent with the expectation of a softening of the response with increasing mass number Harakeh and van der Woude (2001). Hence, experimental evidence in favor of a centroid energy that is actually lower in Ca than it is in Ca Youngblood et al. (2001); Lui et al. (2011) has baffled theoretical explanations; see Ref. Anders et al. (2013) and references contained therein. As illustrated in Table 3, we too have no explanation for this apparent anomaly. Mapping the experimental distribution of strength in the intermediate region, from Ca to Ca, is encouraged as it may ultimately proved vital in solving the puzzle.
NL3Lalazissis et al. (1997, 1999)  FSUGoldToddRutel and Piekarewicz (2005)  FSUGarnetChen and Piekarewicz (2015)  

Isotope  
Ca  8.543  3.329  3.377  0.048  8.539  3.287  3.338  0.051  8.531  3.293  3.345  0.052 
Ca  8.552  3.412  3.374  0.038  8.538  3.372  3.342  0.030  8.539  3.369  3.347  0.022 
Ca  8.572  3.484  3.373  0.111  8.546  3.445  3.348  0.097  8.554  3.430  3.350  0.081 
Ca  8.603  3.548  3.375  0.172  8.561  3.508  3.356  0.152  8.578  3.482  3.353  0.128 
Ca  8.641  3.605  3.379  0.226  8.585  3.563  3.366  0.197  8.613  3.524  3.357  0.167 
Ca  8.489  3.732  3.398  0.334  8.421  3.693  3.388  0.305  8.448  3.651  3.384  0.267 
Ca  8.367  3.839  3.416  0.423  8.284  3.802  3.411  0.391  8.313  3.755  3.412  0.343 
Ca  8.204  3.962  3.438  0.524  8.115  3.921  3.435  0.486  8.135  3.874  3.440  0.434 
Ca  8.038  4.035  3.476  0.559  7.937  3.982  3.471  0.512  7.923  3.943  3.482  0.461 
Ca  7.903  4.100  3.514  0.586  7.789  4.036  3.506  0.530  7.746  4.002  3.523  0.478 
Ca  7.793  4.159  3.551  0.608  7.667  4.084  3.541  0.543  7.600  4.051  3.563  0.488 
While no evidence of lowenergy monopole strength was found in the stable calcium isotopes as the 1f neutron orbital was being filled, the situation changes dramatically with the occupation of the weaklybound 2p and 1forbitals. In an effort to elucidate the nature of the lowenergy monopole strength observed in the unstable calcium isotopes [see Fig.5(a)] it is convenient to start by displaying in Fig. 3 the singleparticle spectrum of Ca as predicted by FSUGarnet; the arrows are used to indicate three prominent discrete proton excitations. Given that information about the meanfield excitation spectrum is fully contained in the uncorrelated response, the three discrete proton excitations in the MeV region are clearly discernible in Fig. 4(a) (the d and p excitations can not be individually resolved in the figure). Besides these discrete proton excitations, two additional peaks are prominent in the 2528 MeV region that can not be inferred from the singleparticle spectrum, as these involve transitions into the continuum. To illuminate the structure of these additional transitions, we artificially remove the two valence 2s and 1d proton orbitals, both having a binding energy of about 25 MeV (see Fig. 3). In particular, the curve denoted by (red line) was obtained by removing the 2s orbital. The curve clearly shows how all the strength below MeV practically disappears. This effect is further accentuated by removing both (2s and 1d) proton orbitals, resulting in the blue curve denoted by . Essentially, no monopole strength remains below 30 MeV.
If the role of the continuum is important in elucidating the character of the proton excitations, it becomes even more critical in understanding the nature of the neutron excitations—given that they all involve transitions into the continuum. Indeed, in Fig. 4(b) we observe a significant amount of low energy monopole strength resulting from the excitation into the continuum of the weakly bound 2p and 1f states; see black line. As in the proton case, we underscore the role of these lowenergy excitations by suppressing either three (red line) or four (blue line) transitions out of the valence 2p1f orbitals. By doing so, essentially no monopole strength remains below MeV. Moreover, it is interesting to note the two emerging discrete transitions out of the 1p states at MeV. The fact that these two Pauliblocked transitions are completely suppressed from the complete calculation (black line) further validates the nonspectral treatment of the nuclear response. Before addressing the fully correlated RPA response, it is important to underscore the virtues of the nonspectral approach, especially in the context of weaklybound nuclei. The nonspectral Green’s function approach adopted here allows for an exact treatment of the continuum without any reliance on artificial parameters. Indeed, introducing an energy cutoff or the discretization of the continuum is neither required nor admitted.
NL3Lalazissis et al. (1997, 1999)  FSUGoldToddRutel and Piekarewicz (2005)  FSUGarnetChen and Piekarewicz (2015)  

Isotope  Pygmy  Giant  Total  Pygmy  Giant  Total  Pygmy  Giant  Total  
Ca  0.334  23.78  96.55  120.33  0.305  35.50  100.61  136.11  0.267  28.27  102.99  131.26 
Ca  0.423  36.30  105.06  141.36  0.391  47.66  109.01  156.67  0.343  46.76  110.45  157.21 
Ca  0.524  75.42  114.24  189.66  0.486  104.15  117.62  221.77  0.434  98.61  118.15  216.75 
Ca  0.559  83.30  126.32  209.62  0.512  102.10  128.28  230.38  0.461  117.39  129.09  246.48 
Ca  0.586  68.16  138.88  207.05  0.530  98.52  139.36  237.88  0.478  113.47  140.13  253.60 
Ca  0.608  76.73  152.31  229.04  0.543  99.29  150.83  250.12  0.488  122.86  151.37  274.22 
In Fig. 1 the diagrammatic representation of the RPA equations indicates how the correlated RPA response (solid black line) is obtained from the meanfield response (thin blue line) and the residual particlehole interaction (wavy red line). Given that the RPA equations involve an iterative procedure to all orders, the singularity structure of the polarization is often changed dramatically. In the particular case of an attractive residual interaction, mixing a large number of particlehole excitations results in the development of a single collective peak that exhausts most of the energy weighted sum Harakeh and van der Woude (2001), as is clearly evident in Fig. 2. However, for the unstable neutronrich isotopes a significant amount of lowenergy strength is expected to emerge as the result of transitions from weaklybound states into the continuum. This notion is confirmed in Fig. 5(a) that displays the distribution of isoscalar monopole strength for all unstable eveneven calcium isotopes from Ca to Ca; the singlepeak response of Ca is also shown for reference. As in the case of Fig. 2, a giant resonance peak in the MeV region is noticeable. Yet, a significant amount of lowenergy monopole strength that progressively increases with mass number is also clearly evident. This can be quantified by plotting the running sum in the inset of Fig. 5(a). In contrast to the corresponding inset displayed in Fig. 2, a considerable amount of lowenergy strength appears below 10 MeV. By somehow artificially dividing the giantmonopole region from the lowenergy “Pygmy” region at an excitation energy of MeV, we display in Fig. 5(b) both the pygmy and giant contributions to the total unweighted energy sum as predicted by the FSUGold and FSUGarnet parametrizations. The figure clearly indicates the important contribution from the lowenergy region to the overall moment. As expected, the pygmy contribution can be correspondingly enhanced or quenched by computing the inverse energy weighted sum or energy weighted sum as shown in Figs. 6(a) and (b), respectively. In particular, we observe nearly equal contributions from the pygmy and giant regions to the overall moment for Ca. We recall that the electric dipole polarizability, which is proportional to the moment of the isovector dipole response, was identified as a strong isovector indicator that is strongly correlated to both the neutronskin thickness of neutronrich nuclei and the slope of the symmetry energy Reinhard and Nazarewicz (2010). As indicated in Table 5 and illustrated in Fig. 7, we too find a correlation between the neutronskin thickness of the neutronrich calcium isotopes and the total moment of the isoscalar monopole response. As already alluded in earlier references, see for example Refs. Chen and Piekarewicz (2015) and Todd and Piekarewicz (2003), the softer the symmetry energy the earlier the neutrondrip line is reached. In essence, models with thin neutron skins predict weak binding energies for the valence neutron orbitals in neutronrich nuclei. In the present case, FSUGarnet having the softest symmetry energy of the models employed here (see Table 2) predicts the weakest binding for the valence 2p1f neutron orbitals in the neutronrich calcium isotopes and, as a consequence, the largest amount of lowenergy monopole strength. In turn, NL3 having the stiffest symmetry energy displays the opposite trend: whereas it predicts the thickest neutron skins, its displays the least amount of lowenergy monopole strength.
We conclude this section by displaying in Fig. 8 centroid energies for all three models considered in the text. Energies have been computed as both (black circles) and (red squares). Note that values also appear between parenthesis in both Fig. 2 and Fig. 5(a). Besides these two choices, we include for reference centroid energies computed by assuming a BreitWigner (or Lorentzian) fit to only the giant monopole resonance (purple triangles). As the strength distribution for all stable calcium isotopes displayed in Fig. 2 is dominated by a single collective peak, fairly consistent results are obtained for all three choices. However, with the appearance of low energy monopole strength in Ca—and progressively more strength with increasing mass number—significant distortions to the simple Lorentzian shape emerge. This is reflected in a large dispersion among the three adopted definitions. Naturally, the Lorentzian fit to the GMR peak predicts the largest energy value, followed by , and predicting the lowest value, because of the large enhancement in . Finally, although the last panel in the figure collects predictions from all three models, the notion of a centroid energy for a distribution of strength that is no longer dominated by a single collective peak loses most of its appeal.
Iv Conclusions
The fascinating dynamics of exotic nuclei has lead to a paradigm shift in nuclear structure. “How does subatomic matter organize itself and what phenomena emerge?” and “What combinations of neutrons and protons can form a bound atomic nucleus?” are only a few of the most interesting questions energizing nuclear physics today. The main goal of this contribution was to explore the nature of the isoscalar monopole response in exotic calcium isotopes with extreme combinations of neutrons and protons.
At the most basic level, one would like to understand the scaling of the giant monopole resonance with increasing mass number, as it encodes information on the incompressibility of neutronrich matter which, in turn, is sensitive to both the incompressibility coefficient of symmetric nuclear matter and the density dependence of the symmetry energy. Perhaps the biggest challenge in uncovering the sensitivity of the monopole resonance to the symmetry energy stems from the fact that its contribution scales as the square of the neutronproton asymmetry. Thus, one most probe the dynamics of exotic nuclei far away from the valley of stability where the neutronproton asymmetry is large. Important first steps along this direction have been taken along the isotopic chains in tin and cadmium. However, given that these measurements have been limited so far to stable isotopes, the neutron excess, although appreciable, is not yet sufficiently large. Yet, one is confident that in the new era of rare isotope facilities experimental studies of this kind will be extended well beyond stability.
An important motivation behind the current work was to confront the experimental observation that the distribution of isoscalar monopole strength in Ca appears softer than in Ca, in stark contrast with theoretical expectations. To do so, we have used three relativistic meanfield models that are known to provide a good description of groundstate properties throughout the nuclear chart. The disagreement with the experimental trend was confirmed for all three models considered here. Indeed, in examining the distribution of monopole strength in all stable eveneven isotopes from Ca to Ca, we found a gradual, albeit small, softening of the monopole response with increasing mass number. This in spite that one of the models adopted here (“FSUGarnet”) was fitted to GMR energies of several magic and semimagic nuclei. Hence, measurements of the isoscalar monopole response of other stable calcium isotopes is strongly encouraged as it may play a vital role in resolving this discrepancy.
Yet the central goal of the present work was to study the emergence evolution, and origin of lowenergy monopole strength as a function of neutron excess. Somehow surprising, we found no evidence of a soft monopole mode in the stable calcium isotopes despite a steady increase in the thickness of the neutron skin. However, the situation changed dramatically with the occupation of the weaklybound neutron orbitals. Indeed, starting with Ca and ending with Ca, a well developed soft monopole mode of progressively increasing strength was clearly identified. The origin of the lowenergy monopole strength was attributed to neutron excitations from the weaklybound orbitals into the continuum. As such, the RPA formalism employed here, based on a nonspectral approach that treats the continuum on the same footing as the bound states, is particularly advantageous.
Due to the prominent role played by the weakly bound neutron orbitals in generating lowenergy monopole strength, we searched for a correlation to the density dependence of the symmetry energy. This is motivated by the realization that models with a soft symmetry energy reach the neutrondrip line before their stiffer counterparts. In our particular case, this is a reflected in the fact that the 2p1f neutron orbitals are more weakly bound in FSUGarnet (the softest of the models employed here) than in NL3 (the model with the stiffest symmetry energy). As a result, FSUGarnet predicted a larger amount of low energy monopole strength than NL3. On the other hand, models with a stiff symmetry energy generate thicker neutron skins. Hence, we uncovered the following inverse correlation: the thiner the neutron skin the larger the moment of the monopole distribution. Given that the isoscalar monopole response of the unstable Ni isotope has already been measured in a pioneer experiment using inelastic alpha scattering in inverse kinematics, we are confident that such techniques may also be used to explore the monopole response of the exotic calcium isotopes. Such experiments will provide critical insights into the role of the continuum in understanding the physics of weaklybound systems and on the development of novel modes of excitation in exotic nuclei.
Acknowledgements.
The author wishes to express his gratitude to Professor Umesh Garg for many stimulating conversations. This material is based upon work supported by the U.S. Department of Energy Office of Science, Office of Nuclear Physics under Award Number DEFD0592ER40750.References
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