An approximately linear correlation between r skin and L was demonstrated in Hartee–Fock calculations of ²⁰⁸Pb with relativistic [31] and Skyrme [32] density functionals, as illustrated in Figure 3A. A comprehensive investigation of correlations between IV experimental observables and the bulk parameters of DFT models [33] demonstrates that these two quantities are also correlated with α D in heavy nuclei. Figure 3B shows, as an example, the correlations with the neutron form factor of ²⁰⁸Pb, which can be derived from a parity-violating elastic electron scattering experiment.

While this type of correlation is observed for all interactions, absolute values show large differences. In general, the magnitude of IV quantities like α D is not well constrained in DFT models because the model parameters are typically fitted to binding energies and charge radii of selected nuclei, which show little sensitivity to the IV parts of the nuclear interaction. A study of the relation between α D and the neutron skin in ²⁰⁸Pb for a large number of interactions illustrates the problem [34]. Figure 4A shows that the predictions for the neutron skin vary from 0.12 fm to 0.32 fm, and for a given value of r s k i n , predictions for α D scatter wildly. However, the product of α D × J plotted versus r s k i n (or L ) shows a linear dependence with a high correlation coefficient [34], cf. Figure 4B. This relation can be understood within the droplet model [35] and provides a correlated range of J , L values, as indicated for the case of ²⁰⁸Pb [36] in Figure 1C.

Ab initio calculations based on interactions derived from χ EFT play an important role in the attempt to systematically describe the EOS of neutron-rich matter at all densities [2]. Figure 5A displays examples of next-to-next-to-next-to-leading order predictions of the density behavior in the nuclear regime [37]. The upper and lower parts present the neutron and symmetric matter results, respectively, for two different families of interactions with somewhat different symmetry energy parameters shown in the left and right columns. The gray boxes indicate the value of the saturation density. The colored curves correspond to different cutoff parameters of the model space; for details, see [37].

Predictions of α D and correlations with proton and neutron radii based on χ EFT interactions have been obtained from calculations based on a coupled-cluster expansion of the wave functions [21] combined with the Lorentz-integral-transform approach to extract the E 1 strength [41]. An example of such calculations for ⁴⁸Ca [40] is presented in Figure 5B, where the correlation of r p with r n , r skin and α D is displayed together with representative examples of DFT predictions. While the DFT results predict neutron skin values ranging from 0.16 fm to 0.22 fm, the ab initio results based on a set of interactions from [38, 39] consistently favor rather small values varying from 0.12 fm to 0.15 fm.

A major difference between the two theoretical approaches lies in the predicted relation between the proton and neutron radii. The DFT predictions of r p are approximately constant, most likely because the charge radius of ⁴⁸Ca is in all cases part of the data set used to fix the model parameters. The ab initio calculations, on the other hand, predict a linear correlation, leading to the approximate constancy of the neutron skin. The absolute value of α D in the ab initio models shows a larger variation than the DFT calculations but can be well described by a linear correlation similar to r n . As discussed in the following, these correlations allow extracting constraints on the range of symmetry energy parameters based on the successful description of experimentally measured polarizabilities and charge radii. This type of calculation has been limited so far to closed-(sub)shell nuclei. For recent attempts of an extension to open-shell nuclei, see [42, 43].

A general problem of all experimental methods discussed above is the removal of magnetic contributions to the photoabsorption cross sections and the derived DP. Overall, contributions of M 1 strength to the DP are small except for very light nuclei [71]. However, they become relevant in the excitation energy region of the spinflip M 1 resonance [72]. Therefore, techniques to decompose E 1 and M 1 contributions are important.

No E 1 / M 1 decomposition can be performed for the photoneutron and total photoabsorption experiments. In the excitation energy regime relevant to determining the DP, they can be distinguished in Compton scattering by combining measurements at forward and backward angles. The multipolarity can also be determined in NRF experiments using transversely polarized photons [73]. Measurements of the response relative to the polarization plane permit a unique assignment of the electric or magnetic character of the emitted radiation. A polarized beam can be extracted from off-axis bremsstrahlung or LCBS. The latter method is particularly efficient because the polarization of the laser light is fully transferred to the photon beam [70].

In relativistic Coulomb excitation, the virtual photon spectrum in the forward direction is dominated by E 1 . However, in the proton scattering experiments close to 0 ° , one must consider contributions to the cross sections due to the nuclear excitation of the spinflip M 1 strength. Two independent methods have been applied to separate E 1 and M 1 cross section parts based either on the total spin transfer derived from the combined information of polarization-transfer observables or from a multipole decomposition analysis (MDA) of the cross section angular distributions [24]. Figure 7 presents some illustrative examples. The lower panel of Figure 7A displays the total spin transfer at 0 ° for the nucleus ¹²⁰Sn [30] derived from measurements of the polarization-transfer observables D L L and D S S [74]. The upper panel presents the differential cross sections at 0 ° and their decomposition in non-spinflip ( E 1 from Coulomb excitation) and spinflip ( M 1 from nuclear excitation) parts.

An example of the MDA analysis is presented in Figure 7B for ⁴⁰Ca [29]. Spectra at different scattering angles are displayed in the upper panel, demonstrating strongly forward-peaked cross sections in the energy region of the IVGDR expected for Coulomb excitation. The lower panel shows the partial contributions to the cross sections at the most forward angle measured resulting from the MDA: E 1 (red), multipoles L > 1 (orange), and nuclear background from quasifree scattering (green). Note that M 1 strength was neglected for ⁴⁰Ca because it is concentrated in a single state [75]. A comparison of the two independent methods was made in studies of ⁹⁶Mo [76], ¹²⁰Sn [30], and ²⁰⁸Pb [36], and good correspondence of the resulting E 1 and M 1 cross sections was found. The results shown in Figure 7A have also been confirmed in an MDA analysis [77]. Because the polarization-transfer measurements require secondary scattering, statistics are limited. Thus, in most ( p , p ′ ) experiments, the E 1 / M 1 decomposition was restricted to MDA.

The largest contribution to the DP stems from the IVGDR, whose energy centroids lie well above S n . It is experimentally accessible with different techniques, and the comparison of results for the same nucleus provides an estimate of the typical accuracy of the DP. One can also average over results obtained with independent methods, thereby reducing the error bars. Some illustrative examples are presented in Figures 8A,B for ⁴⁸Ca and ¹¹⁶Sn, respectively.

The B ( E 1 ) strength distributions in ⁴⁸Ca obtained from ( p , p ′ ) [78] (blue circles) and ( e , e ′ n ) [81] (green triangles) agree well. Results derived from the bremsstrahlung-induced activity of ⁴⁷Ca [82] agree on the low-energy wing of the resonance but are significantly larger than the other data on the high-energy side. This can probably be traced back to the problems discussed in Sec. 3.1.4. The second example (B) compares photoabsorption cross sections for ¹¹⁶Sn from relativistic Coulomb excitation [79] (blue squares) with ( γ , x n ) data [83, 84] (green left arrows and red right arrows). Reasonable agreement is observed in the maximum region of the IVGDR, but one finds significant differences on the low-energy flank. Such deviations are systematically observed in the stable Sn isotope chain, and for some isotopes also at the high-energy flank [77].

In general, studies of the ( γ , n ) and ( γ , x n ) reactions with LCB beams at NewSUBARU agree well with the ( p , p ′ ) results from the RCNP; see, for example, [85] for a study of Sn isotopes (black upward arrows in Figure 8B) or for ²⁰⁸Pb [53] in Figure 6. However, a puzzling result reported for ²⁰⁹Bi is shown in Figure 8C. Although it differs from ²⁰⁸Pb by only one extra neutron, additional strength is seen on the high-energy side of the IVGDR, leading to a difference in α D not predicted by any model. This particular case certainly needs further investigation.

All particle-emission coincidence experiments accessing the E 1 strength are limited to the excitation region above the lowest particle separation threshold. Experimental evidence has accumulated that in nuclei with significant neutron excess E 1 strength—often concentrated in a resonance-like structure commonly termed pygmy dipole resonance (PDR)—can be found below [86, 87]. Low-energy E 1 strength is also found in lighter nuclei with N ≈ Z . Its contribution to the DP can be significant because of the inverse energy weighting, cf. Equation 4. As examples, they amount to approximately 10% in ⁵⁸Ni [43] and 8%–13% in the stable Sn isotopes [79].

Most data on low-energy E 1 strength stem from ( γ , γ ′ ) experiments [70]. They suffer from the problem that branching ratios to excited states are typically unknown, and the extracted strength based on the g.s. transitions represents a lower limit only. Taking ¹²⁰Sn as an example, the resulting B ( E 1 ) strength distribution [88] reasonably agrees with a ( p , p ′ ) experiment [89] measuring the total excitation strength up to approximately 6.5 MeV but totally underestimates the strength at higher excitation energies, cf. Figure 9A. Attempts have been made to model the inelastic contributions assuming statistical decay (see, e.g., [90]) but tend to overestimate contributions at low excitation energies. However, progress has been made recently by analyzing the cumulative population of the first 2 + state in even-even nuclei [70]. For the quoted example ¹²⁰Sn, good agreement between the two experimental methods is achieved [88].

The origin of the low-energy E 1 strength in nuclei with neutron excess is a topic of current debate. It has been suggested to arise from an oscillation of the excess neutrons forming a skin against the (approximately) isospin-saturated core [92, 93]. If true, its strength should be related to the neutron skin thickness and, in turn, to the parameters of the symmetry energy [94–96]. However, a recent study of the Sn isotope chain for mass numbers 111–124 casts doubts on such a picture [91]. The correlation between neutron excess and neutron skin thickness in Sn isotopes has been experimentally demonstrated with different methods [97], but based on combined data from Oslo [98–100] and ( p , p ′ ) experiments [77], only a minor fraction of the photoabsorption cross section (expressed as fraction of the Thomas-Reiche-Kuhn (TRK) sum rule) can be related to the PDR [101]. A decomposition into the tail of the IVGDR and two resonance-like structures is shown in Figure 9B [91]. The contribution interpreted as PDR is much smaller than those of the IVGDR and the prominent structure approximately 8 MeV. These findings rather point to an interpretation of the PDR as a low-energy part of a toroidal E 1 mode [102, 103]. At present, understanding the nature of the PDR remains an open problem. It is clear, however, that DFT predictions restricted to 1p-1h excitations cannot reliably estimate the low-energy E 1 strength distribution for cases where data are unavailable [91].

At excitation energies beyond the giant resonance region, photonuclear cross sections typically contribute a few percent only to the DP. However, for precision results, they must be considered. Data up to the pion threshold have been measured for a few cases, viz., ^natCa [58], ^natSn [104], and ²⁰⁸Pb [61, 51]. They show approximately constant cross sections as a function of excitation energy and were considered for the extraction of the DP from ( p , p ′ ) experiments [30, 36], neglecting an isotopic dependence. The dominant excitation mechanism in this energy regime is the quasi-deuteron effect [105]. It has been pointed out by Roca-Maza et al. [106] that these contributions are not included in model calculations based on DFT and should thus be removed compared to theoretical predictions. For heavy nuclei, they can be estimated using [107], while in light nuclei, they are negligible in the energy range covered by the models [29, 43, 78].

The ratio of Coulomb excitation to quasifree cross sections in the ( p , p ′ ) experiments [24] drops with decreasing mass number limiting, in some cases, the excitation energy range accessible with an MDA for the extraction of E 1 cross sections. In such cases, model-dependent corrections must be applied. In the study of the Sn isotopic chain [79], these were based on quasiparticle random phase approximation (QRPA) calculations folded with a Lorentzian to reproduce the experimentally measured width of the IVGDR. A particularly promising approach is discussed in [43] for the example of ⁵⁸Ni. An extension of the QRPA calculations to include quasiparticle vibration coupling has been successful in describing the width of the ISGMR and curing a longstanding discrepancy between the compressibility values extracted from ²⁰⁸Pb and lighter nuclei [108, 109]. The application to ⁵⁸Ni demonstrates that the predicted high-energy tail of the IVGDR is largely independent of the chosen interaction [43]. This can be understood to result from the dominance of stochastic coupling [110]; that is, the strength distribution is mainly determined by the density of states and an average coupling matrix element between the 1p-1h and more complex states.

The DPs of ⁴⁰Ca and ⁴⁸Ca have been studied in [29, 78], respectively. Figure 10A depicts their correlation and a comparison to selected DFT results. The four functionals are representative of widely used forms: non-relativistic Skyrme functionals SV [111] and RD [112] with different forms of density dependence, and relativistic functionals DD [113] with finite-range meson-exchange coupling and PC [114] with point coupling. All four have been calibrated to the same set of ground-state data to determine the model parameters.

The predictions are displayed as filled ellipses that represent the 1 σ error as defined in [116]. The DD functional performs rather well. The other models tend to slightly overestimate the experimental mean values of both ⁴⁰Ca and ⁴⁸Ca, but their 1 σ error ellipses do overlap with the experimental bands, except for PC. In all cases, the α D values for both nuclei are highly correlated. The dashed ellipses show the effect of additionally including the experimental α D value of ²⁰⁸Pb [36] in the fit, yielding functionals denoted “-alpha.” This improves the agreement with the experiment and shrinks the error ellipsoids. The models incorporate a span of symmetry energy parameters J = 30 − 35 ( 30 − 32 ) MeV and L = 32 − 82 ( 35 − 52 ) MeV for the calculations excluding (including) the ²⁰⁸Pb data point.

The B ( E 1 ) strength distribution of the unstable neutron-rich nucleus ⁶⁸Ni determined in an experiment measuring Coulomb excitation in inverse kinematics [26] is displayed in Figure 10B. The DP was extracted from a comparison to the model of [115]. The model results show a sensitivity to the assumed neutron skin thickness, as illustrated by the colored curves. A value of 0.17 (2) fm was extracted for the neutron skin thickness from the correlation between the two quantities.

A study of the DP in a long isotopic chain is particularly suited to investigate the connection with the neutron skin thickness. This can be best done in the Sn isotopes with neutron numbers between 50 and 82, where the proton shell closure stabilizes the g.s. deformation. There are many stable isotopes, and a study of the systematics of the DP was presented in [79]. The results are summarized in Figure 10C, which shows the evolution of α D between mass numbers 112 and 124. All DFT calculations predict an approximately linear increase as a function of neutron excess with roughly the same slope. The experimental results indicate a saturation between mass numbers 120 and 124, but the uncertainties (blue band) do not exclude a mass dependence similar to the theoretical results. The rightmost part of Figure 10C shows the predictions of the different models for the ²⁰⁸Pb DP after subtraction of the quasi-deuteron part (see the next paragraph). The models closest in absolute magnitude to the data tend to underpredict α D (²⁰⁸Pb), while those reproducing it overshoot the absolute values in the Sn chain, indicating that the functionals cannot yet fully describe the mass dependence of the DP. We note that E 1 strength distributions have also been measured for the unstable neutron-rich isotopes ^130,132Sn [25] but the extracted values of α D cannot be compared directly to the results of [79] because the experiment only provided data above neutron threshold.

Roca-Maza et al. [106] combined the experimental DP data for ⁶⁸Ni [26], ¹²⁰Sn [30], and ²⁰⁸Pb [36] to test a large variety of density functionals. Because the DFT calculations do not include contributions from the quasi-deuteron process dominating the photoabsorption cross sections above the energy region of the IVGDR, these had to be removed for a comparison [106]. Figure 10D presents correlation plots between the experimental results and theoretical predictions from a wide range of DFT interactions. Only a handful (marked in red) are capable of simultaneously describing all three data points. Based on this reduced set, systematic predictions of α D for other masses, r skin , and the symmetry energy parameters could be derived. The experimental results for ^40,48Ca discussed above are fairly well described by these predictions.

An experimental study of the DP in ⁴⁸Ca [78] is of particular interest because it is accessible for both DFT and ab initio calculations, and a measurement of the neutron skin with parity-violating electron scattering is available [19]. The comparison is summarized in Figure 11A, where the blue band describes the experimental uncertainty. Ab initio results for the set of interactions from [38, 39] are displayed as green triangles, and a prediction from [40] based on a normalization to the ⁴⁸Ca charge radius is displayed as a green bar. Results from the set of density functionals described in [40] are shown as red squares with some representative error bars, and the prediction from the analysis of [106] discussed above is shown as a black bar.

The DFT results tend to be somewhat high compared to the experiment. The ab initio results show a significant dependence on the chosen interaction, but it can be well approximated by a linear dependence. In principle, this allows for the derivation of boundaries on the neutron skin thickness and the symmetry energy. However, while the ab initio results shown were truncated in the coupled-cluster expansion at the 2p-2h level, subsequent work [118] demonstrated that inclusion of 3p-3h correlations lowers the α D values by 10 − 20 %. The refined results in ⁴⁸Ca are plotted in Figure 11C against corresponding calculations for ⁴⁰Ca [29]. A high correlation similar to the DFT results shown in Figure 10A is observed. The purple uncertainty band from the ab initio results overlaps with the crossing of the experimental 1 σ error bands. In particular, the NNLO_sat interaction [38] accurately describing binding energies and radii of nuclei up to ⁴⁰Ca and the saturation point of symmetric nuclear matter now reproduces both DP values. The importance of including 3p-3h correlations has also been demonstrated in a recent measurement of the ⁶⁸Ni charge radius [117]. Figure 11B illustrates the improvement in reproducing the correlation between the charge radius and α D [26] when going from the 2p-2h level (light blue band) to the inclusion of 3p-3h correlations (dark blue band).

As noted in Section 2.2, independent of the chosen interaction, a neutron skin thickness of approximately 0.14 fm is predicted for ⁴⁸Ca, consistent with the value deduced from the measurement of the weak form factor [19]. The simultaneous description of the data in ^40,48Ca and ⁶⁸Ni implies that the underlying symmetry energy parameters are correct. A conservative estimate is provided by taking the full range of values from the set of ab initio interactions, viz., J = 27 − 33 MeV and L = 41 − 49 MeV.

Recent work has, for the first time, been able to extend the range of ab initio DP calculations based on χ EFT interactions to ²⁰⁸Pb [119]. A different technique was used to construct the interactions by history matching [120] using selected experimental observables in light nuclei. Moreover, low-energy nucleon–nucleon scattering phase shifts were additionally considered. The latter are responsible for tight constraints to rather small values of the resulting neutron skin thickness ( 0.14 − 0.20 fm for ²⁰⁸Pb). The variation of the density dependence of the symmetry energy in these calculations is L = 38 − 69 MeV.

While in ⁴⁸Ca there is fair agreement between the neutron skin thickness and symmetry energy properties derived from the different experiments, the parity-violating elastic electron scattering experiment on ²⁰⁸Pb [18] finds a much larger neutron skin r skin = 0.28 ( 7 ) fm than most other work. Accordingly, an extraction of symmetry energy parameters based on the correlations established in DFT (see Section 2.1) leads to large symmetry energy values of J = 38 ( 5 ) MeV and L = 106 ( 37 ) MeV in contradiction to limits derived from astrophysical observations of neutron star radii and masses as well as the tidal deformability of neutron star mergers [7]. All astrophysical constraints point toward a softer EOS. This has led to speculations about a phase transition at intermediate densities [121].

Because of the strong correlation between r s k i n and α D for a given nucleus and r s k i n values of different nuclei in DFT models, Reinhard et al. [116, 122] investigated whether it is possible to construct a DFT interaction capable of simultaneously describing the data for ⁴⁸Ca and ²⁰⁸Pb. The analysis was based on representative families of non-relativistic and relativistic functionals. The isovector properties of EDFs are typically not well constrained by the input data used to fit the model parameters. As illustrated in Figure 12A for the case of ²⁰⁸Pb, it is possible to vary the symmetry energy parameters—and thereby the predicted r skin and α D —over a fairly large range maintaining comparable description of ground-state properties [116]. Figure 12B [122] demonstrates that the polarizabilities and the neutron skin thickness of ⁴⁸Ca could be consistently described, but it was impossible to construct an EDF simultaneously accounting for the neutron skin thickness of ²⁰⁸Pb extracted from the PREX experiment [19]. Similar conclusions were drawn in [123, 124]. In another recent attempt [125], a DFT interaction reasonably accounting for the measured parity-violating asymmetries in both the PREX and CREX experiments was constructed, but at the expense of unusual properties of the symmetry energy curvature and a very strong isovector coupling leading to density fluctuations in the nuclear interior.

Another way of extracting properties of the symmetry energy is a study of the mass dependence of the DP. A simple power law σ − 2 ∝ A 5 / 3 based on a model of two interpenetrating fluids has been given by Migdal, where σ − 2 denotes the second inverse moment of the photoabsorption cross sections and σ 2 ≃ α D in units of mb/MeV ([126] and Refs. therein). A proportionality constant 2.4 × 1 0 − 3 has been determined by Orce [127] from a fit to ( γ , x n ) data over a wide mass range. Figure 13 [128] shows a comparison with a combined data set of α D measurements in light nuclei [58] with the then-available (2016) data from relativistic Coulomb excitation for heavier nuclei as a green short-dashed line. Note that results for A < 12 from [58] are neglected because the hydrodynamical picture is highly questionable and corrections due to the magnetic polarizability are large [71] for these very light nuclei. Results are severely underestimated in lighter nuclei where charged-particle decay dominates. The mass dependence is reasonably described for larger masses, but the proportionality coefficient of [127] is too low because additional contributions from the strength below the neutron threshold, as discussed in Sec. 3.3, must be considered.

For masses A ≤ 40 , surface contributions must be considered, modifying the volume term of the symmetry energy dominating for heavy nuclei. These can be parameterized as [128] σ − 2 = 0.0518 A 2 S v A 1 / 3 − κ m b / M e V . (5)

Here κ = S s / S v , and S s and S v denote the surface and volume coefficients of the symmetry energy, respectively. The numerical coefficient in Equation 5 is obtained from Migdal’s approach. A fit with S s , S v parameters from binding energies of isobaric nuclei [130] shown in Figure 13 as a long-dashed blue line still underestimates the lower-mass data. Parameters of the study of [129] provide a better description (dotted red line). Results of a free fit of Equation 5 crucially depend on the inclusion (dotted-dashed black line) or exclusion (solid black line) of the ¹²C data point. The latter provides a better fit with S v = 25.6 ( 8 ) MeV, κ = 1.66 ( 5 ) [128] close to [129]. S v can be interpreted as J , but measured at about 2 / 3 of the saturation density [34, 131].