Origin of the heaviest elements: The rapid neutron-capture process

The production of about half of the heavy elements found in nature is assigned to a specific astrophysical nucleosynthesis process: the rapid neutron-capture process ($r$ process). Although this idea was postulated more than six decades ago, the full understanding faces two types of uncertainties or open questions: (a) The nucleosynthesis path in the nuclear chart runs close to the neutron-drip line, where presently only limited experimental information is available, and one has to rely strongly on theoretical predictions for nuclear properties. (b) While for many years the occurrence of the $r$ process has been associated with supernovae, where the innermost ejecta close to the central neutron star were supposed to be neutron rich, more recent studies have cast substantial doubts on this environment. Possibly only a weak $r$ process, with no or negligible production of the third $r$-process peak, can be accounted for, while much more neutron-rich conditions, including an $r$-process path with fission cycling, are likely responsible for the majority of the heavy $r$-process elements. Such conditions could result during the ejection of initially highly neutron-rich matter, as found in neutron stars, or during the fast ejection of matter that has previously experienced strong electron captures at high densities. Possible scenarios are the mergers of neutron stars, neutron-star–black hole mergers, but also include rare classes of supernovae as well as hypernovae or collapsars with polar jet ejecta, and possibly also accretion disk outflows related to the collapse of fast rotating massive stars. The composition of the ejecta from each event determines the temporal evolution of the $r$-process abundances during the “chemical” evolution of the Galaxy. Stellar $r$-process abundance observations have provided insight into and constraints on the frequency of and conditions in the responsible stellar production sites. One of them, neutron-star mergers, was just identified thanks to the observation of the $r$-process kilonova electromagnetic transient following the gravitational wave event GW170817. These observations, which are becoming increasingly precise due to improved experimental atomic data and high-resolution observations, have been particularly important in defining the heavy element abundance patterns of the old halo stars, and thus in determining the extent and nature of the earliest nucleosynthesis in the Galaxy. Combining new results and important breakthroughs in the related nuclear, atomic, and astronomical fields of science, this review attempts to answer the question “How were the elements from iron to uranium made?”

Figure 1

Abundances $Y_i$ of elements and their isotopes in the Solar System as a function of mass number $A_i=Z_i+N_i$. $A_i{Y}_i$ is equal to the mass fraction of isotope $i$, and the sum of mass fractions amounts to 1, ${\sum }_i{A}_i{Y}_i=1$. A scaling, leading for historical reasons to an abundance of ${10}^6$ for the element Si is utilized. Element ratios are obtained from solar spectra, the isotopic ratios from primitive meteorites and terrestrial values ([42]; [504]). These values represent a snapshot in time of the abundances within the gas that formed the Solar System.

Figure 2

Solar $r$-process abundances as determined by [154] and [263]. The largest uncertainties are clearly visible for $A\lesssim 100$ (weak $s$-process region) and around lead.

Figure 3

Top panel: neutron-capture abundances in 13 $r$-II stars (points) and the scaled Solar System $r$-process-only abundances of [811], adapted mostly from [808]. The stellar and Solar System distributions were normalized to agree for the element Eu ($Z=63$), and vertical shifts were then applied in each case for plotting clarity. The stellar abundance sets are (a) CS 22892-052 ([821]), (b) HD 115444 ([928]), (c) $\mathrm{BD}+17$ 3248 ([155]), (d) CS 31082-001 ([811]), (e) HD 221170 ([363]), (f) HD $1523+0157$ ([225]), (g) CS 29491-069 ([311]), (h) HD 1219-0312 ([311]), (i) CS 22953-003 ([224]), (j) HD 2252-4225 ([535]), (k) LAMOST $\mathrm{J}110901.22+075441.8$ ([488]), (l) RAVE J203843.2-002333 ([679]), and (m) 2MASS $\mathrm{J}09544277+5246414$ ([334]). Bottom panel: mean abundance differences for the 13 stars with respect to the Solar System $r$-process values.

Figure 4

Differences between stellar and $r$-process-only Solar System (s.s.) abundances for four VMP stars with $r$-process abundance mixes, modeled after Fig. 5 given by [337] and Fig. 11 given by [742]. The “s.s., $r$-only” abundances are those given by [811], mostly from [808]. The stellar abundance sets are CS31082-001 ([811]), HD 88609 ([337]), HD 122563 ([338]), and HD 221170 ([363]).

Figure 5

Comparisons of laboratory data on Sm ii from University of Western Ontario (UWO) and University of Wisconsin (UW) groups. (a) Histogram of differences in lifetimes ($\tau$ divided by their uncertainties added in quadrature), along with a dashed line representing a 1 standard deviation Gaussian. (b) Similar histogram and Gaussian representation for transition probabilities ($A$ values). Adapted from [481].

Figure 6

Abundances of $[\mathrm{Sr}/\mathrm{Fe}]$ vs $[\mathrm{Ba}/\mathrm{Fe}]$ in a large number of galactic and extragalactic stars. From [734].

Figure 7

Abundances as a function of metallicity for (a) $[\mathrm{Mg}/\mathrm{Fe}]$and (b) $[\mathrm{Eu}/\mathrm{Fe}]$. Red solid lines are approximate fits to the averages of halo, thick disk, and thin disk stars. Black dashed lines in (b) highlight the growing star-to-star scatter in $[\mathrm{Eu}/\mathrm{Fe}]$ with decreasing metallicity. Individual data points were taken from [244], [320], [714], [127], [138], [808], [57], [713], [224], [81], [741], and [61]. Adapted from [815].

Figure 8

Synthesis and derived abundance for U in the star 2MASS $\mathrm{J}09544277+5246414$. From [334].

Figure 9

$\mathrm{Th}/\mathrm{Eu}$ ratios for stars with detected thorium abundances. One can see that at low metallicities around $[\mathrm{Fe}/\mathrm{H}]\approx -3$ a number of so-called actinide-boost stars can be found. If utilizing initial $r$-process production ratios that fit solar $r$ abundances ([768]), unreasonable, and even negative, ages of these stars are obtained, which is not at all consistent with their metallicity, which points to the formation of these stars in the early Galaxy. From [334].

Figure 10

Left panel: bolometric light curve of AT 2017gfo, the kilonova associated with GW170817. The filled black triangles are from [814]. Uncertainties derived from the range of values given in the literature ([156]; [814]; [921]) are shown as a gray band. Also shown are lower limits on the late-time luminosity as inferred from the Ks band with VLT/HAWK-I ([858]) (empty circles) and the $4.5\mu \mathrm{m}$ detections by the Spitzer Space Telescope from [899] (empty triangles) and [405] (empty squares). Adapted from [949]. Right panel: evolution of the kilonova flux spectrum during the first 10 d. Each spectrum is labeled by the observation epoch. The shaded areas mark the wavelength ranges with low atmospheric transmission. From [673].

Figure 11

Plane of maximum temperatures and densities ([937]), indicating for explosive Si burning the boundaries of conditions after freeze-out of charged-particle reactions with an adiabatic expansion for an electron fraction $Y_e=0.498$. High densities, permitting three-body reactions to build up C and heavier nuclei, lead to a normal freeze-out NSE composition. For lower densities unburned $\alpha$ particles remain; i.e., the final outcome is a so-called $\alpha$-rich freeze-out (see the lines of remaining He mass fractions $X_{\mathrm{He}}$). Also displayed are typical conditions experienced in Si-burning mass zones of SNe Ia and CCSNe, determining the nucleosynthesis outcome of such explosions.. Adapted from [871], [873], and [875].

Figure 12

The figure shows the line of stability (black solid squares), the proton and neutron magic numbers (blue horizontal and vertical lines), and an $r$-process path. The conditions here are taken from a neutron-star merger environment that is further discussed later. The position of the path follows from a chemical equilibrium between neutron captures and photodisintegrations in each isotopic chain [$(n,\gamma )\rightleftarrows (\gamma ,n)$ equilibrium]. However, the calculation was performed with a complete nuclear network, containing more than 3000 nuclei. The colors along the path indicate how well the full network calculation follows such an $(n,\gamma )\rightleftarrows (\gamma ,n)$ equilibrium. It can be seen that such full calculations agree with this equilibrium approach within a factor of 2 along the $r$-process path, which continues to the heaviest nuclei. From [184].

Figure 13

Color-coded time derivatives of nuclear abundances $Y$ during an $r$-process simulation describing the destruction via (top panel) neutron-induced fission and (center panel) $\beta$-delayed fission ([656]) and (bottom panel) the production of fission fragments ([411]). The largest destruction rates occur at or close to the neutron closure $N=184$ due to the smallest fission barriers encountered at these locations. Fission fragments are produced in a broad distribution, ranging in mass number $A$ from 115 to 155. From [184].

Figure 14

Evolution of temperature, density, and nuclear energy generation for different trajectories corresponding to material ejected dynamically in neutron-star mergers. Gray (darker) [brown (lighter)] lines correspond to the “slow (fast) ejecta” discussed in Sec. 6b1. For both gray and brown lines, light (dark) colors correspond to hotter (colder) conditions. Adapted from [547].

Figure 15

Upper panel: evolution of the abundances of neutrons, alpha particles, and heavy nuclei during the $r$ process. Bottom panel: evolution of the neutron-to-seed ratio and average timescales for neutron captures, photodissociation, and $\beta$ decays. The absolute value of the neutron lifetime, defined in the text, is also shown. Notice that for $t\lesssim 1\mathrm{s}$ neutron-capture and photodissociation timescales coincide.

Figure 16

$r$-process elemental abundances at freeze-out compared to the effective $\beta$-decay half-life of an isotopic chain.

Figure 17

Abundances of neutrons $Y_n$, ${}^4\mathrm{He}$ ($\alpha$ particles), $Y_{\alpha }$, and so-called seed nuclei $Y_{\mathrm{seed}}$ (in the mass range $50\le A\le 100$), resulting after the charged-particle freeze-out of explosive burning (for $Y_e=0.45$), as a function of entropy in the explosively expanding plasma. One can realize that the ratio of neutrons to seed nuclei ($n_s=Y_n/Y_{\mathrm{seed}}$) increases with entropy. The number of neutrons per seed nucleus determines whether the heaviest elements (actinides) can be produced in a strong $r$ process, requiring $A_{\mathrm{seed}}+n_s\gtrsim 230$. From [198].

Figure 18

Evolution of the mass fractions of neutrons, alpha particles, and heavy nuclei vs $Y_e$ (left $y$-axis scale) and neutron-to-seed ratio (right $y$-axis scale) at a temperature of 3 GK for an adiabatic expansion with an entropy per nucleon of $s=20k_B$.

Figure 19

Nuclear chart with stable nuclei indicated by black squares, the limit of known masses from the 2016 Atomic Mass Evaluation (blue line), the future reach of radioactive ion-beam (RIB) facilities like FRIB or FAIR, and abundance results (color coded) at neutron freeze-out from an $r$-process calculation, utilizing the FRDM masses, combined with Thomas-Fermi fission barriers. Adapted from [255].

Figure 20

$\beta$-decay half-lives (solid circles) for a number of isotopic chains as a function of neutron number compared to the 2012 NUBASE evaluation (open triangles) ([44]) and the predictions of the models: finite-range droplet model with quasirandom phase approximation (FRDM-QRPA) (dark blue lines) ([580]), gross theory KTUY-GT2 (light green lines) ([845]; [438]), and Fayans density functional with continuum quasirandom phase approximation (DF3-CQRPA) (magenta dashed lines) ([95]) when available. From [507].

Figure 21

Summary of recent efforts undertaken at experimental facilities worldwide to attain precise (a) nuclear masses and (b) $\beta$-decay properties like half-lives $T_{1/2}$ and delayed neutron probabilities $P_n$. The individual results at TRIUMF, JYFLTRAP, GSI, CERN, ANL, NSCL, and RIKEN (discussed earlier in this section) are indicated via color coding. Also shown is the region in the nuclear chart in reach of the RIB facilities. Adapted from [342].

Figure 22

Differences, in MeV, between experimental energies taken from the 2012 version of the atomic mass evaluation AME12 ([918]) and theoretical binding energies. The following mass models are shown: (a) FRDM-1992 ([579]), (b) HFB-21 ([269]), (c) DZ31 ([174]), (d) WS3 ([501]). From [548].

Figure 23

Final mass-integrated $r$-process abundances obtained in a neutron-star merger simulation using four different mass models. Adapted from [547].

Figure 24

Comparison of experimental ([672]; [172]; [210]; [507]) and shell-model half-lives ([966]) for $N=82$ $r$-process nuclei.

Figure 25

Left panels: evolution of the luminosities and average energies of neutrinos emitted during the protoneutron star cooling phase following a core-collapse supernova explosion. Adapted from [531]. Right panels: luminosities and average energies of neutrinos emitted after a NS-NS merger that forms a hypermassive neutron star surrounded by an accretion disk. Adapted from [666].

Figure 26

Results of an $r$-process calculation assuming an initial $Y_e=0.45$, adiabatic expansion of matter in a so-called neutrino wind with a given expansion speed $v_{\exp }$ of ejected mass shells, and a superposition of entropies $S$ between $120k_B/\mathrm{baryon}$ and $280k_B/\mathrm{baryon}$ with equal amounts of matter ejected per entropy interval. Changes due to the utilization of an improved nuclear-mass model are indicated ([577]; [583]). From [446].

Figure 27

In a MHD-jet supernova the winding up of magnetic field lines causes the “squeezing out” of polar jets along the rotation axis ([932]). This environment leads to low entropies, much lower than those discussed in Fig. 17. But in opposition to the $Y_e$ values utilized in Fig. 17, the collapse to high densities resulted in large amounts of electron captures and $Y_e$ values close to 0.1–0.15; see the top of the panel as well as Fig. 18. Such low $Y_e$’s, as under neutron-star merger conditions (where even values as low as 0.03–0.05 can be attained; see Sec. 6a6), lead to a strong $r$ process and abundance predictions displayed in the bottom of the panel [shown for the two fission fragment distributions utilized ([411]; [657])]. As $Y_e$ is only moderately low, the effect of late neutron capture by fission neutrons is also moderate, thus avoiding a final shift of the third $r$-process peak, as indicated in Figs. 32 and 33. From [867].

Figure 28

Abundances from nucleosynthesis calculations with varying ratios of magnetic field strength vs the neutrino heating of regular core-collapse SNe, increasing for the models $h$, $\mathit{i}-$, $i$, $\mathit{i}+$, and $m$. For comparison abundances from MP stars with a weak $r$ process are also shown, i.e., HD122563 (black squares) ([338]), and solar-type $r$-process observations from CS22892-052 (blue circles) ([817]). Abundances are normalized for $Z=40$ of HD122563. Observations of low-metallicity stars with strong $r$-process contributions vary for abundances below $Z=50$ ([815]). From [621].

Figure 29

Nucleosynthesis features of rotating core-collapse SN models ($h$, $\mathit{i}-$, $i$, $\mathit{i}+$, $m$) with varying ratios of neutrino luminosity and magnetic field strengths, as in Fig. 28. Model $m$ represents a strong MHD-jet supernova. One can see the transition from a regular core-collapse SN pattern, dominated by ${}^{56}\mathrm{Ni}$, total Fe (after decay), and Zn, to a strong $r$-process pattern with a high Eu abundance. From [621].

Figure 30

Radial distribution of $Y_e$ (thick solid line) and proton fraction (dashed line) in a disk, with $\alpha =0.1$ and $M_{\mathrm{BH}}=3{\hspace{0.17em}\hspace{0.17em}\mathrm{M}}_{\odot }$, for different accretion rates. $Y_e$ indicates neutron-rich conditions deep inside the disk but develops aymptotically to values of 0.5 in the outer layers above $100r_g$ from where the nucleosynthesis outflow will occur. From [365].

Figure 31

Ejection channels in compact binary mergers including estimates based on simulations of the ejecta mass, $Y_e$, and velocity during the different ejection phases. The NS-NS merger system shown in the upper part includes two possible outcomes: a long-lived massive neutron star and a hypermassive neutron star that collapses to a black hole on a timescale shorter than the disk lifetime. The BH-NS merger is shown in the lower part. Adapted from [749].

Figure 32

Resulting $r$-process abundances for dynamic tidal ejecta [compared to solar values (black dots)] from neutron-star merger simulations ([184]), making use of $\beta$-decay half-lives from [580] (red line) and recent $\beta$-decay half-life predictions (black line) ([522]) together with the fragment distributions from fissioning nuclei of [411]. From [867].

Figure 33

$r$-process abundances after a decay time of 1 Gyr for all trajectories shown in Fig. 14. The dark gray (light brown) curves correspond to the abundances of the trajectories of the slow (fast) ejecta. The mass-averaged abundances for all trajectories (red solid curves), the slow ejecta (green dotted curves), and the fast ejecta (blue dashed curves) are also shown. The abundances for the slow and fast trajectories and their averages have been scaled by the value of their fractional contribution to the total ejecta. Adapted from [547].

Figure 34

Neutrino-wind contribution to neutron-star merger ejecta, dependent on the delay time between the merger and BH formation. For comparison the dynamic, tidal ejecta given by [434] are also shown. The neutrino wind, ejected dominantly in polar regions, contributes nuclei with $A<130$ due to the effect of the neutrinos on $Y_e$. From [527].

Figure 35

Resulting $r$-process abundances [compared to solar values (black dots)] from black hole accretion disk simulations, making use of a black hole mass of $3{\hspace{0.17em}\hspace{0.17em}\mathrm{M}}_{\odot }$, a disk mass of $0.03{\hspace{0.17em}\hspace{0.17em}\mathrm{M}}_{\odot }$, an initial $Y_e$ of 0.1, entropy per baryon of $8k_b$, an alpha parameter of the viscous disk of 0.03, and a vanishing black hole spin. The impact of two different mass models (FRDM and DZ31) in the final abundances is illustrated. From [945].

Figure 36

Specific energy generation rate $\dot{Q}$ in $r$-process ejecta (black line). For comparison the energy production from the decay chain ${}^{56}\mathrm{Ni}\to {}^{56}\mathrm{Co}\to {}^{56}\mathrm{Fe}$ (red line) and the analytical estimate $\dot{Q}\sim t^{-1.3}$ are also shown. Adapted from [559].

Figure 37

Bolometric light curve of the optical-infrared counterpart AT 2017gfo of GW170817 (red circles) from multiband photometry ([156]) compared to the fiducial model of [559]. A line with the approximate power-law decay $\dot{Q}\sim t^{-1.3}$ for $r$-process heating is given for comparison; see Fig. 36.

Figure 38

Derived $[\mathrm{Eu}/\mathrm{Fe}]$ abundances as a function of metallicity: $r$-I stars (green triangles), $r$-II stars (blue squares), limited-$r$ observations (red stars), and non-$r$-process-enhanced stars (black dots); see classifications defined in Sec. 2. Upper limits are indicated by black arrows. Gray light dots refer to an earlier overview ([744]). From [304].

Figure 39

Required $r$-process ejecta masses as a function of the occurrence frequency of the production site. As shown, on a typical SN frequency of ${10}^{-2}{\mathrm{yr}}^{-1}$ about ${10}^{-4}$ to ${10}^{-5}{\hspace{0.17em}\hspace{0.17em}\mathrm{M}}_{\odot }$ of $r$-process matter would need to be produced, for binary merger ejecta with about ${10}^{-2}{\hspace{0.17em}\hspace{0.17em}\mathrm{M}}_{\odot }$ the frequency must be rarer by a factor of 100 to 1000, and if $1{\hspace{0.17em}\hspace{0.17em}\mathrm{M}}_{\odot }$ of $r$-process matter is ejected in specific events, the frequency must again be lower by another factor of 100. From [749].

Figure 40

Results from chemodynamical inhomogeneous evolution models predicting median values (lines) and distributions of $[\mathit{r}/\mathrm{Fe}]$ ratios in newly born stars of the Milky Way for three different neutron-star merger rates, different indices of the coalescence delay-time distributions $t^{\text{index}}$, and an additional admixture of $r$-process events occurring with rates proportional to CCSNe. The combination of additional $r$-process events proportional to the CCSN rate, which were of great importance at low metallicities early in the evolution of the Galaxy, and neutron-star mergers permits an $[\mathit{r}/\mathrm{Fe}]$ dependence on $[\mathrm{Fe}/\mathrm{H}]$ that does not decline toward low metallicities. From [890].

Figure 41

Evolution of $[\mathrm{Eu}/\mathrm{Fe}]$ in a stochastic inhomogeneous galactic chemical evolution model, including both neutron-star and neutron-star–black hole mergers as $r$-process sites, under the assumption that all NS-BH mergers eject $r$-process matter. Magenta crosses represent observations, whereas different choices for black hole formation are utilized at low metallicities. Red (green, blue) squares represent models where all stars $\ge 20{\hspace{0.17em}\hspace{0.17em}\mathrm{M}}_{\odot }$ ($\ge 25{\hspace{0.17em}\hspace{0.17em}\mathrm{M}}_{\odot }$, $\ge 30{\hspace{0.17em}\hspace{0.17em}\mathrm{M}}_{\odot }$) at metallicites $\mathit{Z}\le {10}^{-2}{\mathit{Z}}_{\odot }$ lead to failed SNe and black holes at the end of their life. The combination of these two $r$-process sources also permits a good fit with observations at low metallicities. From [924].

Figure 42

Utilizing the Duflo-Zuker mass model ([174]) and trajectories from [58] permits large variations in actinide production, even at low $Y_e$. The highest actinide production is found at $Y_e=0.125$. From [947].