Enhanced symmetry energy bears universality of the r-process

url]https://nuclear.uwc.ac.za

The binding energy of a nucleus with $Z$ protons and $N$ neutrons can be described by the well-known Bethe-Weizsäcker semi-empirical mass formula (SEMF) [1, 2],

where $A=Z+N$ is the mass number and $a_{v},a_{s},a_{c},a_{sym}$ and $a_{p}$ are the volume, surface, Coulomb, symmetry energy and pairing coefficients, respectively. The symmetry energy, $a_{sym}(A)(N-Z)^{2}/A$, reduces the total binding energy $B(Z,A)$ of a nucleus as the neutron-proton asymmetry becomes larger, i.e. for $N\gg Z$, and yields the typical negative slope of the binding energy curve[5] for $A>62$. It is divided by A to reduce its importance for heavy nuclei, and it depends on the mass dependency of $a_{sym}(A)$. Its convergence for heavy nuclei establishes the frontiers of the neutron dripline for particle-unbound nuclei and eventually leads to the disappearance of protons at extreme nuclear densities[6].

Furthermore, $a_{sym}(A)$ is relevant to understanding neutron skins[7], the effect of three-nucleon forces[8] and – through the equation of state (EoS) – supernovae cores, neutron stars and binary mergers[9, 10, 11]. The latter are the first known astrophysical site where heavy elements are created through the rapid neutron-capture or $r$-process[12, 13]. The identification of heavy elements in neutron star mergers is supported by the short duration gamma-ray bursts via their infrared afterglow[14] – only understood by the opacities of heavy nuclei – as well as blueshifted Sr II absorption lines[15], following the expansion speed of the ejecta gas at $v=0.1-0.3~{}c$. Mergers are expected to be the only source for the creation of elements above lead and bismuth, as inferred from the very scarce abundance of actinides in the solar system[16].

The universality of the $r$-process for the heaviest elements with $56<Z<90$ is further inferred from the similar abundance patterns observed in both extremely metal-poor stars and the Sun[17, 18]. Other potential sources of heavy elements involve different types of supernova (e.g. collapsars [4] – the supernova-triggering collapse of rapidly rotating massive stars – and type-II supernova[12]), which need to be considered in order to explain all neutron-capture abundances[19, 20].

It is the motivation of this work to understanding the limits of the neutron dripline and heavy-element production by investigating $a_{sym}(A)$ at different temperatures $T$ using available data of potential interest to the $r$-process; namely, data from photoabsorption cross sections, binding energies and giant dipole resonances.

Generally, $a_{sym}(A)$ is parametrized using the leptodermous approximation of Myers and Swiatecki, where $A^{-1/3}\ll 1$[21],

which considers the modification of the volume symmetry energy, $S_{v}$, by the surface symmetry energy $S_{s}$. This particular leptodermous parametrization was chosen on the account of its better fit to the masses of isobaric nuclei [22]. Constraints on these parameters have been investigated using experimental and theoretical information concerning properties of ground states, i.e. at $T=0$ MeV[23, 24].

The giant dipole resonance (GDR) represents the main contribution to the absorption and emission of electromagnetic radiation (photons) in nuclei[25]. The dynamics of this quantum collective excitation is characterized by the inter-penetrating motion of proton and neutron fluids out of phase[26], which results from the density-dependent symmetry energy, $a_{sym}(A)(\rho_{{}_{N}}-\rho_{{}_{Z}})^{2}/\rho$, acting as a restoring force[25]; where $\rho_{{}_{N}}$, $\rho_{{}_{Z}}$ and $\rho=\rho_{{}_{N}}+\rho_{{}_{Z}}$ are the neutron, proton and total density, respectively, which spread uniformly throughout the nucleus.

The ratio of the induced dipole moment to an applied constant electric field yields the static nuclear polarizability, $\alpha$. Using the hydrodynamic model and assuming inter-penetrating proton and neutron fluids with a well-defined nuclear surface of radius $R=r_{{}_{0}}A^{1/3}$ fm and $\rho_{{}_{Z}}$ as the potential energy of the liquid drop, Migdal[26] obtains the following relation between the static nuclear polarizability, $\alpha$, and $a_{sym}$,

where $r_{{}_{0}}=1.2$ fm, $e^{2}=1.44$ MeV fm in the c.g.s. system, and a constant value of $a_{sym}=23$ MeV was utilized.

Alternatively, $\alpha$ can be calculated for the ground states of nuclei using second-order perturbation theory[27] following the sum rule,

where $E_{{}_{\gamma}}$ is the $\gamma$-ray energy corresponding to a transition connecting the ground state $|i\rangle$ and an excited state $|n\rangle$, $M$ the nucleon mass, $f_{in}$ the dimensionless oscillator strength for $E1$ transitions [27] and $\sigma_{{}_{-2}}$ the second moment of the total electric-dipole photo-absorption cross section,

where $\sigma_{{}_{total}}(E_{{}_{\gamma}})$ is the total photo-absorption cross section, which generally includes $(\gamma,n)+(\gamma,pn)+(\gamma,2n)+(\gamma,3n)$ photoneutron and available photoproton and photofission cross sections [28], in competition in the GDR region [29, 30]. By comparing Eqs. 3 and 6, a mass-dependent symmetry energy, $a_{sym}$(A), is extracted in units of MeV,

Empirical evaluations reveal that $\sigma_{{}_{-2}}$ can also be approximated by $\sigma_{{}_{-2}}=2.4\kappa A^{5/3}$, where the dipole polarizability parameter $\kappa$ measures GDR deviations between experimental and hydrodynamic model predictions[31].

Figure 1 shows the distribution of $a_{sym}(A)$ for the ground state of stable isotopes, along the nuclear landscape, determined from empirical $\sigma_{{}_{-2}}$ values. Data include all available emission channels. The contribution of $(\gamma,p)$ cross sections are evident in light nuclei, which significantly reduces the symmetry energy. For heavy nuclei, $(\gamma,n)$ cross sections are dominant because of the higher Coulomb barrier. A fit to the data using Eq. 2 (solid line) yields $a_{sym}(A)=31.2(12)\left(1-1.12(10)A^{-1/3}\right)$ MeV, with an RMS deviation of 22%[34]. Unfortunately, $(\gamma,p)$ cross-section data are very scarce, which directly affects the $a_{sym}(A)$ trend for $A\lessapprox 70$ in Fig. 1.

In addition, Tian and co-workers determined $a_{sym}(A)=28.32\left(1-1.27A^{-1/3}\right)$ MeV from a global fit to the binding energies of isobaric nuclei with mass number $A\geq 10$[22] – extracted from the 2012 atomic mass evaluation[35] – with $S_{v}\approx 28.32$ MeV being the bulk symmetry energy coefficient and $\frac{S_{s}}{S_{v}}\approx 1.27$ the surface-to-volume ratio. Similar coefficients are calculated in Refs.[9, 36]. Within this approach, the extraction of $a_{sym}(A)$ only depends on the Coulomb energy term in the SEMF and shell effects[37] – which are both included [22] – and $a_{sym}(A)$ presents a maximum energy around 23 MeV. This description of $a_{sym}(A)$ has been used to explain the enhanced $\sigma_{{}_{-2}}$ values observed for low-mass nuclei[34].

The symmetry energy $a_{sym}(A)$ is the fundamental parameter that characterizes the energy of the GDR, $E_{{}_{{}_{GDR}}}$, within the Steinwedel-Jensen (SJ) model of proton and neutron compressible fluids moving within the rigid surface of the nucleus[38]. Danos improved the SJ model by including the GDR width, $\Gamma_{{}_{GDR}}$[39, 25] in the second-sound hydrodynamic model[39, 25], where $E_{{}_{{}_{GDR}}}$ and $\Gamma_{{}_{GDR}}$ are related to $a_{sym}(A)$ as[30],

where $K$ is the real eigenvalue of $\bigtriangledown^{2}\rho_{{}_{Z}}+K^{2}\rho_{{}_{Z}}=0$, with the boundary condition $(\mathbf{\hat{n}}\bigtriangledown\rho_{{}_{Z}})_{{}_{surface}}=0$, and has a value of $KR=2.082$ for a spherical nucleus[40]. For quadrupole deformed nuclei with an eccentricity of $a^{2}-b^{2}=\epsilon R^{2}$, where $a$ and $b$ are the half axes and $\epsilon$ the deformation parameter, the GDR lineshape splits into two peaks with similar values of $Ka$ and $Kb\approx 2.08$ [39]. For deformed nuclei, we estimate a similar equation to Eq. 9, but using the average centroid energy and the FWHM of the total Lorentzian (see e.g.[41]). Uncertainties in the quoted values arise from the error propagation of Eq. 9.

The GDR cross-section data for each nucleus were obtained from the EXFOR and ENDF databases [32, 33] and fitted with one or two Lorentzian curves to extract $E_{{}_{{}_{GDR}}}$ and $\Gamma_{{}_{GDR}}$, as shown e.g. in Fig. 2 for ²⁰⁸Pb. The data set for each nucleus was selected based on the number of data points, experimental method and energy range. In this work, the maximum integrated $\gamma$-ray energy, E${}_{\gamma}^{max}$, was in the range 20–50 MeV, therefore excluding contributions resulting from high energy effects such as pion exchange and other meson resonances. The resulting distribution of $a_{sym}(A)$ is shown in the left panel of Fig. 3, which converges at approximately 27 MeV for heavy nuclei. It is reassuring that the two methods based on photoabsorption cross-section data — namely $a_{sym}(A)$ extracted from $\sigma_{{}_{-2}}$ values and parameters of GDRs built on ground states – present similar trends.

Data obtained from GDR parameters at $T=0$ can also be fitted to Eq. 2, which yields $a_{sym}(A)=35.3(7)\left(1-1.58(5)A^{-1/3}\right)$ MeV (red solid band in Fig. 3), with an RMS deviation of 15%. Larger values of $S_{v}=42.8$ and $S_{s}=89.9$ were determined by Berman using Eq. 9 for 29 nuclei ranging from $A=75$ to 209 [42]. Furthermore, Berman argued that assuming a surface binding energy coefficient of $a_{{}_{S}}=20$ MeV in the SEMF, the large symmetry to surface energy ratio, $S_{s}/a_{{}_{S}}=4.5$, favors – as a result of a steeper slope of the binding energy curve for heavy nuclei – a close-in neutron dripline for heavy elements; hence, constraining the reaction network that produces heavy elements by the $r$-process in neutron mergers and supernovae. Using our value of $S_{s}=46$ and $a_{{}_{S}}=20$ MeV, a more reasonable ratio of $S_{s}/a_{{}_{S}}=2.3$ is determined. Slightly smaller values of $a_{{}_{S}}\approx 17$ MeV are also found in the literature[5, 36], yielding $S_{s}/a_{{}_{S}}=2.7$.

Furthermore, it is interesting to investigate the behavior of $a_{sym}(A)$ using the available information on GDRs built on excited states, below the critical temperatures and spins where the GDR width starts broadening; i.e. for moderate average temperatures of $T\lessapprox T_{c}=0.7+37.5/A$ MeV and spins $J$ below the critical angular momentum $J\lessapprox J_{c}=0.6A^{5/6}$. In fact, similar centroid energies, $E^{exc}_{{}_{GDR}}$, and resonance strengths, $S^{exc}_{{}_{GDR}}$ – relative to the Thomas–Reiche–Kuhn $E1$ sum rule [27] – to those found for the ground-state counterparts [29, 43] indicate a common physical origin for all GDRs, in concordance with the Brink–Axel hypothesis that assumes that a GDR can be built on every state in a nucleus [44, 45].

Applying again Eq. 9, the right panel of Fig. 3 shows $a_{sym}(A)$ values for GDRs built on excited states in slightly-deformed nuclei ³¹P [46], ⁶³Cu [47], ⁹⁷Tc [48], ¹²⁰Sn [49] and ²⁰¹Tl [50], as well as for well-deformed nuclei in the $A\approx 160-180$ mass region (^158,160,166Er, ¹⁶⁹Tm and ¹⁸⁵Re [41, 51, 52]). With an average temperature between $T\approx 0.7$ and 1.0 MeV and below $J_{c}$, these nuclei were selected to investigate the symmetry energy at temperatures relevant to the r-process nucleosynthesis. Surprisingly, $a_{sym}(A)$ values for heavy nuclei are relatively larger than previously observed at $T=0$ MeV. A fit to the data using Eq. 2 (green solid line in Fig. 3) yields $a_{sym}(A)=46.2(2.4)\left(1-2.22(14)A^{-1/3}\right)$ MeV, with an RMS deviation of 6%. A value of $a_{sym}(A)=30$ MeV for heavy nuclei yields larger values of $S_{v}=44.44$, $S_{s}=97.32$ and $S_{s}/a_{{}_{S}}=4.87$ (again for $a_{{}_{S}}=20$ MeV). Two bands showing the loci limits of the two fitting curves at $T=0$ and $T\approx 0.7-1$ MeV are shown for comparison. Such a distinct behaviour could clearly affect nucleosynthesis of heavy elements via the $r$-process during the cooling down of the ejecta.

Lighter or heavier seed nuclei are generally produced depending on the density and temperature of the ejecta gas. Assuming nuclear-statistical equilibrium – when forward and reverse reactions are balanced – abundances follow a Maxwell-Boltzmann distribution where lighter seed nuclei are favoured at very high temperatures ($\propto kT^{-3/2(A-1)}$) and heavier nuclei are favoured at very high densities ($\propto\rho^{A-1}$), as those found in the ejecta of neutron-star mergers [3]. At temperatures below $T=1$ MeV (or $1.2\times 10^{10}$ K), seed nuclei are produced before charge reactions freeze out – impeded by the Coulomb barrier – at about $T\approx 0.5$ MeV (or $5\times 10^{9}$ K). Thereafter, heavy nuclei are produced through subsequent neutron capture until neutron reactions freeze out – as neutrons are finally consumed – at a few 10⁸ K.

Our work may not be sensitive to the lower temperatures occurring during neutron capture in neutron-star mergers, which likely range from $T\approx~{}0.5\times 10^{8}$ K [53] to $T\approx 5\times 10^{9}$ K [54] (i.e. in the range from $T\approx 0.04$ to 0.43 MeV, respectively). Nevertheless, Fig. 4 shows that the symmetry energy does not change with temperature in the [0.74,1.3] MeV range, which suggests that this relation may still hold at lower temperatures.

Such an increase in the symmetry energy results from the change in the effective mass of the nucleon, which decreases as $T$ increases in the temperature interval $0<T<1$ MeV [55]. The temperature dependence of the symmetry energy has been studied within the liquid-drop and Fermi gas models [56], where an effective nucleon mass – the so-called ‘$w$’ mass – is introduced to account for the non-locality of the Hartree-Fock potential. This leads to an increase in the centroid energy of the GDR and, hence, the symmetry energy of medium and heavy mass nuclei also increases by approximately 8% at $T\approx 1$ MeV [55]. In the current work, we notice a slight increase of   3-5% in the centroid energy at $T\approx 0.7-1$ MeV as compared with the ground-state values for nearly-spherical ¹²⁰Sn [57], ²⁰⁸Pb [58] and ²⁰¹Tl [50] nuclei as well as for the deformed nuclei in the $A=160-180$ mass region [41, 51, 52]. Although such an increase is within the experimental errors, it leads to a distinct systematic behaviour, as shown in the right panel of Fig. 3.

Finally, the effects from the larger symmetry energy at $T\approx 0.7-1$ MeV are illustrated in Fig. 5, which shows the corresponding nuclear charts (top) and binding energy curves (bottom) using $a_{sym}=23.7$ MeV[60] (left) and $30$ MeV (right), respectively. The nuclear chart determined using $a_{sym}=30$ MeV illustrates a substantial close-in of the neutron dripline, as a result of the decreasing binding energy per nucleon in neutron-rich nuclei. For instance, the dripline closes in from ²⁵⁴Pt to ²²⁰Pt for $a_{sym}=23.7$ and $30$ MeV, respectively. Figure 6 shows the respective neutron driplines and clearly illustrates the dramatic effect of an enhanced symmetry energy in the production of heavy elements, which constrains exotic $r$-process paths and plausibly explains the universality of $r$-process abundances inferred from the observation of extremely metal-poor stars and our Sun.

Consequently, such an increase in the symmetry energy leads to the reduction of radiative neutron capture rates as neutron-rich nuclei become less bound. The corresponding change in the capture cross section has been calculated using TALYS [61] and EMPIRE [62] codes by changing only the mass excess with standard input parameters. Both codes yield similar results with a reduction of the neutron-capture cross section by a factor of the order of 10² in the $A\approx 200$ mass region relevant to the $r$-process. More detailed calculations will be presented in a separate manuscript. These findings support the rapid drop of the neutron capture rates at increasing neutron excesses inferred from Goriely’s microscopic calculations at $T=1.5\times 10^{9}$ K [12].

More experimental data regarding GDRs built on excited states below $T\approx 0.7$ MeV are crucially needed in order to elucidate the nature of the symmetry energy as a function of temperature. Modern high-efficient spectrometers such as GAMKA in South Africa [63] – with up to 30 HPGe clover and LaBr₃ detectors – may provide such data.