Dependence of $(n,\gamma )\text{−}(\gamma ,n)$ equilibrium $r$-process abundances on nuclear physics properties

In most $r$-process expansions, the dominant nuclear evolution occurs in an $(n,\gamma \left)\text{−}\right(\gamma ,n)$ equilibrium in which nuclei rapidly exchange neutrons but change charge much more slowly by beta decay. Freeze-out from this equilibrium shapes the final abundances but does not significantly alter the overall global abundance pattern; therefore, it is important to understand the details of $(n,\gamma \left)\text{−}\right(\gamma ,n)$ equilibrium both because it is the main evolution phase that determines the final abundance pattern and because it is the starting point for the freeze-out. Through use of a simple but realistic phenomenological nuclear physics model, we show that isotopic abundances versus neutron number in $(n,\gamma \left)\text{−}\right(\gamma ,n)$ equilibrium are well approximated as Gaussians. Nuclear pairing causes isotopic abundances to alternate between two Gaussians, and shell effects cause the isotopic abundances to shift from one Gaussian to another when the neutron number crosses a magic number. More complex neutron-separation energy curves versus mass number can be generated by adding a spike function to a linearly declining curve. In such a case, the equilibrium abundance curve jumps from one Gaussian to another for each added spike. Insights from our model can help shed light on how detailed theoretical or experimental nuclear data affect $r$-process nucleosynthesis during the $(n,\gamma )\text{−}(\gamma ,n)$ equilibrium phase.

Figure 1

Chemical-potential offset differences for the reference r-process calculation at the time step $t=0.243$ s, $T_9=4.44$, and mass density $\rho =8.78\times {10}^7$ g/cc. The chemical-potential offset difference is the chemical-potential offset relative to that for the maximum abundance species within the scope of the diagram (nuclides from Cr, $Z=24$, to Ru, $Z=44$); thus, the quantity shown is $\tilde{\mu }(Z,A)/kT-\tilde{\mu }(Z_{\max },A_{\max })/kT$, where $Z_{\max }$ and $A_{\max }$ are the atomic number and mass number of the maximum abundance species, respectively. The maximum abundance species within the scope of the diagram is indicated by the triple-outline octagon, while the maximum abundance species for other elements is indicated by the double-outlined circle. The color indicates the value chemical-potential offset difference, and the value for any species may be inferred from Fig. 2. The chemical-potential offset difference is only shown for species with abundance greater than ${10}^{25}$ per nucleon.

Figure 2

The color bar for the chemical-potential offset differences in Figs. 1, 3, and 4. The number inside each box gives the offset difference divided by $kT$ for the corresponding color.

Figure 3

The same as Fig. 1 but for the time step 0.332 s, $T_9=3.30$, and $\rho =3.60\times {10}^7$ g/cc.

Figure 4

The same as Fig. 1 but for the time step 0.432 s, $T_9=2.369$, and $\rho =1.33\times {10}^7$ g/cc.

Figure 5

The abundances as a function of nucleon number for the last moment of $(n,\gamma )\text{−}(\gamma ,n)$ equilibrium and the final frozen-out abundances.

Figure 6

Neutron-separation energies as a function of neutron number $N=A-Z$ for $Z=50$ from our model (orange line) and from those derived from the REACLIB V2.2 database [35] (blue line). The separation energies in our model were computed from Eq. (73) with $[S_0,S_1,S_2,{\delta }_n,C_s]=[17,0.25,0.001,1.2,3.5]$.

Figure 7

The upper panel shows the ratio values of $G(Z,A)m{(Z,A)}^{3/2}$. The middle and lower panels show the ratio values of masses and partition functions, respectively, in an isotopic chain ($Z=50$). Data are from the REACLIB V2.2 database.

Figure 8

The upper panel shows the intersection of $S_n$ and $-{\mu }_n^{\prime }$. The lower panel presents the isotopic abundances within an isotope chain. The intersection position in the top panel is the abundance peak position in the lower panel.

Figure 9

This figure shows the abundance distribution including both the linear and pairing term of $S_n$. The upper panel shows the intersection of $S_n$ and $-{\mu }_n^{\prime }$, while the solid line in the lower panel illustrates the abundance distribution. The dashed and dash-dotted lines are abundances curves, including purely even and odd $N$, respectively.

Figure 10

The upper panel shows the intersection of $S_n$ and $-{\mu }_n^{\prime }$ at two different values of $-{\mu }_n^{\prime }$. The intersections each occur at neutron numbers between two magic numbers. The lower panel presents the resulting equilibrium abundance distributions, which appear as full Gaussians since they do not cross any closed shell. This figure uses $A_0-Z=40$.

Figure 11

Results for (a) the $N=50$ shell crossing and (b) the $N=82$ shell crossing. The upper panel shows the intersection of relevant energies. The solid curve in the lower panel represents the full abundance distribution. The dash-dotted and dotted lines are Gaussian curves before and after a closed shell, respectively.

Figure 12

The upper panel shows the intersection of $S_n$ and $-{\mu }_n^{\prime }$ for the case of the linear and quadratic $S_n$. The lower panel shows the resulting equilibrium abundance distributions. The distribution for the quadratic $S_n$ is normalized, and the distribution for the linear $S_n$ is computed from the same values of $S_0$ and $S_1$ as in the quadratic case. The quadratic term in $S_n$ shifts the intersection to higher neutron number relative to that for the linear case. It also gives the resulting equilibrium abundance distribution a positive skew.

Figure 13

Each panel shows the intersection of $S_n$ and ${\mu }_n^{\prime }$ in the upper part and the abundance distribution considering pairing and shell as well as quadratic terms in the lower part. From left to right, and top to bottom, we used $-{\mu }_n^{\prime }=15$, 10, 5, 3 in each panel to demonstrate the abundance pattern.

Figure 14

(a) The neutron-separation energy $S_n$ with a single spike (upper panel) and the resulting equilibrium abundance curve. (b) The neutron-separation energy $S_n$ with two successive negative spikes and the resulting equilibrium abundance curve.

Figure 15

Neutron-separation for a single-shell transition. The shell effect may be considered a sequence of negative spikes added to the linearly declining separation energy. The lower panel shows the full abundance distribution. The abundance point after the neutron number corresponding to the first spike is on its own Gaussian distribution labeled in the legend. The resulting abundance distribution is a new Gaussian after the magic number.

Figure 16

Neutron-separation for two shell transitions. As in Fig. 15, the shell effect may be considered a sequence of negative spikes added to the linearly declining separation energy. The second shell is then a second sequence of spikes added to the first. The lower panel shows the full abundance distribution and illustrates the transformation from one Gaussian to a second Gaussian after the first shell and then to a third Gaussian after the second shell.

Figure 17

The upper panel shows the fractional change in $S_0$, $S_1$, and $S_2$ with the given variation in $a_S$, the surface-term coefficient in the liquid-drop model of Eq. (15). The lower panel shows the normalized equilibrium abundance curves resulting from the indicated variations in the surface term coefficient away from the reference value $a_S$ for a fixed $-{\mu }_n^{\prime }$.

Figure 18

The same as Fig. 17 but for variation in the Coulomb term coefficient $a_C$.

Figure 19

The same as Fig. 17 but for variation in the asymmetry term coefficient $a_A$.

Figure 20

Results for $Z=70$. The upper panel shows the intersection of $S_n$ and the given $-{\mu }_n^{\prime }$, and the lower panel presents the abundance distribution.

Figure 21

Results for $Z=70$. The upper panel shows the intersection of $S_n$ and the given $-{\mu }_n^{\prime }$, and the lower panel presents the abundance distribution.

Figure 22

Results for $Z=55$. The upper panel shows the intersections of energies. The lower panel presents the abundance distribution when the pairing effect in $S_n$ is distinctive in each shell.

Figure 23

Results for $Z=68$. The upper panel shows the intersections of energies. The lower panel demonstrates the abundance pattern when the pairing effect varies inside a single shell.

Figure 24

The upper panel shows the intersection of $S_n$ and $-{\mu }_n^{\prime }$ (3 and 9 MeV). The lower panel presents the abundances distribution with different temperatures.

Figure 25

Chemical potential offset differences for the reference $\nu p$-process calculation at the time step $t=0.914$ s, $T_9=2.18$, and mass density $\rho =1.73\times {10}^5$ g/cc. The color indicates the value chemical-potential offset difference, and the value for any species may be inferred from the color bar.

Figure 26

Relevant nuclear energies and abundances for the $N=24$ isotones in the $\nu p$ process at the time step shown in Fig. 25. The left panel shows the intersection of proton-separation energy ($S_p$) and proton chemical potential less the proton rest mass energy ($-{\mu }_p^{\prime }$). The right panel presents the isotonic network abundances which overlap the $(p,\gamma )\text{−}(\gamma ,p)$ equilibrium abundances at this time step.