OPPORTUNITIES TO CONSTRAIN ASTROPHYSICAL REACTION RATES FOR THE s-PROCESS VIA DETERMINATION OF THE GROUND-STATE CROSS-SECTIONS

Nuclear reactions occurring in astrophysical plasma are described by astrophysical reaction rates; e.g., for neutron capture, the rate is defined as (Fowler 1974; Rauscher 2011)

$$\begin{equation} r^*=n_\mathrm{n} \, n_\mathrm{target} \, \langle \sigma v \rangle ^* = n_\mathrm{n} \, n_\mathrm{target} \, R^*, \end{equation} \tag{ 1 }$$

where n_n and n_target are number densities of the neutrons and a species of target nuclei, respectively, and the stellar rate factor (also called reactivity) R* = 〈σv〉*. The relative velocities v between the neutrons and the target nuclei are given by the Maxwell–Boltzmann (MB) distribution Φ_MB(v, T) at temperature T. It can also be expressed as a function of the relative interaction energy Φ(E, T).

In stellar plasma, nuclei are in thermal equilibrium with the plasma. This also implies that internal states can be excited and the nuclei do not necessarily have to be in the ground state (g.s.), depending on the temperature. In most cases, the excited states within a nucleus are also in thermal equilibrium because the equilibration timescale is short (Gintautas et al. 2009). There are a few cases with isomeric or long-lived states that may not be thermalized at s-process temperatures, such as ¹⁷⁶Lu (Heil et al. 2008; Mohr et al. 2009; Gintautas et al. 2009) or ¹⁸⁰Ta (Belic et al. 1999; Mohr et al. 2007). Assuming thermal equilibrium, the total stellar reactivity R* is (see, e.g., Ward & Fowler 1980)

$$\begin{equation} R^*(T) = w_0 R_0 + w_1 R_1 + w_2 R_2 + \cdots,\ \quad \end{equation} \tag{ 2 }$$

with the reactivities of the ith state in the target

$$\begin{equation} R_i(T) = \int _0^\infty \sigma _i(E_i) \, \Phi (E_i,T) \, dE_i. \end{equation} \tag{ 3 }$$

The cross-section σ_i is the total neutron-capture cross-section of the ith state, i.e., summed over all final states j in the residual (A + 1) nucleus: σ_i = ∑_jσ^{i → j}. The relative energy E_i between projectile and target is always measured in the system of the respective target state (i.e., the g.s. or excited state), and therefore the integration always starts at zero (Fowler 1974; Rauscher 2011). Thus, the full MB energy-distributed neutrons are also acting on nuclei that are in excited states, not only on nuclei in the g.s., and the reaction rate in Equation (1) has to be defined appropriately by the sum in Equation (2).

The normalized weights w_i of the excited-state contributions in Equation (2) are given by

$$\begin{equation} w_i=\frac{P_i}{\sum _i P_i}=\frac{P_i}{G}, \end{equation} \tag{ 4 }$$

with the partition function G = ∑_iP_i or, normalized to the g.s., G₀ = G/(2J₀ + 1). The population factor P_i of the excited state i with excitation energy E_i and spin J_i is given by the usual Boltzmann factor:

$$\begin{equation} P_i=(2J_i+1)\exp \left(-\frac{E_i}{kT}\right). \end{equation} \tag{ 5 }$$

Normalized partition functions G₀(T) are usually given along with reaction rate predictions and can be found, e.g., in Holmes et al. (1976), Woosley et al. (1976), and Rauscher & Thielemann (2000).

Often, the integral in Equation (3) is divided by the mean thermal velocity v_{i, T}, leading to the so-called Maxwellian-averaged cross-section (MACS). The MACS for the g.s. (i = 0) can be derived from a measurement of the neutron-capture cross-sections over a properly chosen energy interval. The MACS for the g.s. and its uncertainty are provided in the compilations by Bao et al. (2000) and Dillmann et al. (2006, 2009).

Laboratory experiments are performed on nuclei in the g.s. (with the exception of ¹⁸⁰Ta, which is the only naturally occurring isomer and has a short-lived g.s.). Thus, experiments determine only the neutron-capture cross-section of the g.s., i.e., σ₀. Therefore, only the g.s. reactivity R₀ (or g.s. MACS) can be derived, instead of the desired astrophysical reactivity R*. To assess how much the g.s. rate differs from the astrophysical rate, the so-called stellar enhancement factor (SEF),

$$\begin{equation} f_\mathrm{SEF} = \frac{R^*(T)}{R_0}, \end{equation} \tag{ 6 }$$

usually is used. It is found that the SEF remains relatively close to unity, within about 20%–30%, for most nuclei in the s-process path up to temperatures of kT ≈ 100 keV (Bao et al. 2000; Dillmann et al. 2006, 2009). As shown below, however, this does not necessarily imply a small contribution of transitions from excited states or a small remaining uncertainty once R₀ has been well determined.

It can be shown that the weighted sum of integrals in Equation (2) can be transformed to an integral over a single MB distribution (Fowler 1974; Holmes et al. 1976; Rauscher 2011),

$$\begin{equation} R^*= \frac{1}{G_0(T)} \int _0^\infty \sigma ^\mathrm{eff}(E_0) \, \Phi (E_0,T) \, dE_0, \end{equation} \tag{ 7 }$$

thereby avoiding the different energy scales found in Equation (2). Following Fowler (1974) and Holmes et al. (1976), σ^eff is the so-called effective cross-section, summing over all final states and initial excited states up to the interaction energy E:

$$\begin{equation} \sigma ^\mathrm{eff}(E)=\sum _i \sum _j \frac{2J_i+1}{2J_0+1} \frac{E-E_i}{E} \sigma ^{i \rightarrow j}(E-E_i). \end{equation} \tag{ 8 }$$

This transformation proves convenient in determining the actual weights with which transitions from excited states i are contributing to stellar rates. These are not the P_i as defined in Equation (5) because of the different MB distributions appearing in Equation (2). They are rather (Rauscher 2011)

$$\begin{equation} \mathcal {W}_i=(2J_i+1)\frac{E-E_i}{E}=(2J_i+1)\left(1-\frac{E_i}{E}\right) \end{equation} \tag{ 9 }$$

and are therefore linearly declining with increasing excitation energy E_i. The energies E appearing in Equation (9) are given by the range of energies significantly contributing to the integral in Equation (7). Therefore, the maximum reference energy E that appears is the upper end of the relevant energy window (Gamow window). The relevant energy window for neutron captures is located close to the peak of the MB distribution, and its width is comparable to the width of the MB distribution (Iliadis 2007; Rauscher 2010). As can be seen in Iliadis (2007) and Rauscher (2010), the upper edge of the effective energy window for neutrons is located not far above the energy E = kT, leading to $\mathcal {W}_i \approx (2J_i+1)\left(1-E_i/(kT)\right)$.

The above discussion shows that for the s-process temperature of kT = 30 keV, excited states up to excitation energies of ≈30–40 keV can be important in the target nucleus due to the linearly decreasing weight; for a temperature of kT = 80 keV, excited states even up to ≈80–100 keV have to be considered. Obviously, σ^{i → j} contains information on how strong the contribution from the excited state is. However, $\mathcal {W}$ allows us to estimate whether we have to bother with calculating σ^{i → j}.

Even though many nuclei in the s-process path show excited states below 100 keV, especially the heavier nuclides, the SEFs nevertheless remain close to unity in most cases. Clearly, this implies that the (predicted) stellar reactivity is close to that for the g.s., because either the w_i ($\mathcal {W}_i$) or the σ^{i → j} is negligible for i > 0 or they conspire to yield a value close to that obtained for the g.s. transitions alone. The SEF does not allow us to distinguish those cases. For illustration, we examine a nucleus with only two levels characterized by excitation energies E₀ = 0 and E₁, spins J₀ and J₁, and thermal occupation weights w₀ and w₁. Let us consider two different cases in this simple model. For case A, we assume that 90% of the target nuclei are in the g.s. (w₀ = 0.9) and 10% are in the excited state (w₁ = 0.1). Let us further assume that the cross-section of the excited state σ₁ is proportional to σ₀ but larger by a factor of two. This yields

$$\begin{equation} f_\mathrm{SEF}^{\mathrm{A}} = \frac{0.9 R_0 + 0.1 R_1}{R_0} = 0.9 + 0.1 \times 2 = 1.1. \end{equation} \tag{ 10 }$$

For case B, we assume that two-thirds of the target nuclei are in the first excited state and only one-third remain in the g.s. This can be easily achieved for a small excitation energy E₁ ≪ kT and J₀ ≪ J₁. Furthermore, we assume that the cross-section σ₁ is proportional to σ₀ but larger by 15%. This results in

$$\begin{equation} f_\mathrm{SEF}^{\mathrm{B}} = \frac{R_0/3 + 2R_1/3}{R_0} = 1/3 + 2/3 \times 1.15 = 1.1. \end{equation} \tag{ 11 }$$

Although the SEF is identical in both cases and close to unity, the essential difference is in the contribution of the excited state, which is small in case A but dominant in case B.

Instead of using the SEF, the importance of excited states in the stellar rate factor R* can be better assessed by the g.s. contribution X to R*, which is given by

$$\begin{equation} X = \frac{w_0 R_0}{R^*} = \frac{w_0}{f_\mathrm{SEF}} \end{equation} \tag{ 12 }$$

for the two simple examples above. The quantity X is positive and assumes its maximum value of unity when only the target g.s. is contributing to R*. It follows that in case A the g.s. contribution to the stellar rate factor is X = 0.9/1.1 ≈ 0.82, i.e., the stellar rate factor is essentially defined by the dominating g.s. contribution. On the other hand, in case B, we find a g.s. contribution of only X = (1/3)/1.1 ≈ 0.30, i.e., about 70% of the stellar rate stems from the excited state. This shows that the excited-state contribution may significantly exceed the g.s. contribution, even when f_SEF ≈ 1.

The above result for the g.s. contribution can easily be generalized for many contributing thermally excited states. Equation (7) clearly shows that f_SEF is not a constant of order unity without the contribution of excited-state transitions; rather, f_SEF = 1/G₀ in this case (Rauscher 2011). Therefore, the g.s. contribution X ⩽ 1 in the general case is given by

$$\begin{equation} X(T) = \frac{1}{f_\mathrm{SEF}(T) \, G_0(T)} = \frac{R_0}{R^* G_0}. \end{equation} \tag{ 13 }$$

The quantity X will be included in future versions of the KADoNiS (Karlsruhe Astrophysics Database of Nucleosynthesis in Stars) compilation (Dillmann et al. 2009). Even when it is not directly given in a compilation, however, it can be easily computed from G₀ and either f_SEF or R₀, as shown above. The SEF itself is still interesting in the sense that it specifies whether the stellar rate, albeit incidentally, is similar to the g.s. rate.

Note, however, that X itself is a theoretical quantity (like the SEF). The weighted uncertainties in the rates or cross-sections of the ground and excited states determine the total uncertainty in X and f_SEF. Entering this total uncertainty, however, are the uncertainties in the ratios of the rates of excited states and the g.s., as well as the uncertainty in the ratios of the weights, i.e., of G₀. This can be seen easily when looking at the reciprocal of X (which must have the same relative uncertainty),

$$\begin{equation} \frac{1}{X}=1+\frac{w_1}{w_0} \frac{R_1}{R_0}+\frac{w_2}{w_0} \frac{R_2}{R_0}+\frac{w_3}{w_0} \frac{R_3}{R_0}+\cdots. \end{equation} \tag{ 14 }$$

Similar considerations apply to the SEF. For example, using the simple two-level system from Section 2.2, we may assign some uncertainty, say a factor of two, to the term w₁R₁/(w₀R₀). This translates into uncertainties f^A_SEF = 1.1^+0.20_−0.10, X^A = 0.82^+0.08_−0.13 and f^B_SEF = 1.1^+0.77_−0.38, X^B = 0.30^+0.16_−0.12.

In the following, we study the influence of an uncertainty factor u on the calculations of the resulting g.s. contribution X and its inverse 1/X. Only nuclei at or close to stability participate in the s-process. These nuclides generally have well-determined level schemes at low excitation energies, and therefore G₀ and the weights w_i are well determined. Thus, the uncertainties in X are limited to the uncertainties u_i arising from the predicted ratios R_i/R₀,

$$\begin{equation} u_{1/X}=1+\frac{w_1}{w_0} u_{1}\frac{R_1}{R_0} +\frac{w_2}{w_0} u_2\frac{R_2}{R_0} +\frac{w_3}{w_0} u_3 \frac{R_3}{R_0} +\cdots. \end{equation} \tag{ 15 }$$

For the two-level system above, we assumed an uncertainty of a factor of two, i.e., u₁ = 2 and u₁ = 1/2, to arrive at the given total uncertainty in X.

Another convenient property of X is that its uncertainty scales with X itself. This can be seen when assuming $\overline{u}=u_1=u_2=u_3=\cdots.$ For realistic cases in the s-process, this is a good approximation of the actual errors in the ratios because only the first few terms in the sum will contribute to X, and the weighted average of the uncertainties $\overline{u}$ will be close to the values of u₁ and/or u₂. The applicability of the assumption can be checked through the normalized partition function. The closer G₀ is to unity, the better the above assumption holds.

Under the assumption above, we obtain

$$\begin{eqnarray} u_{1/X}&=&1+\overline{u}\left\lbrace \frac{w_1}{w_0}\frac{R_1}{R_0} +\frac{w_2}{w_0}\frac{R_2}{R_0} +\frac{w_3}{w_0}\frac{R_3}{R_0} +\cdots \ \right\rbrace \nonumber \\ &=&1+\overline{u}\left\lbrace \frac{1}{X}-1\right\rbrace. \end{eqnarray} \tag{ 16 }$$

Then, the factor X'/X, by which X changes when assuming an averaged uncertainty factor $\overline{u}$ in the rate ratios, is

$$\begin{equation} u_X=\frac{X^{\prime }}{X}=\frac{1}{\overline{u}\left(1-X\right) + X}. \end{equation} \tag{ 17 }$$

This is an important result because it shows that the uncertainties will be negligible for X ≈ 1. Uncertainties will be larger for X ≪ 1 but will still yield X' ≪ 1 when they are included. In other words, large X values always remain large and very small X values remain small. Therefore, X is a robust measure of the g.s. contribution and can be used to determine the direct impact of measurements of g.s. cross-sections on the astrophysical reaction rate. This is another advantage of using X instead of the SEF. Figure 1 shows the connection between the uncertainty and X.

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The uncertainty factor will reach its maximum value, $\max \left(\overline{u},1/\overline{u}\right)$, only for very small values of X. Note that $\overline{u}$ gives the uncertainty in the ratio R_i/R₀ and not in the rates or cross-sections. Global reaction rate predictions typically find average deviations from measurements of 30% for (n,γ) MACS at 30 keV, with local deviations of up to factors of 2–3 (Rauscher et al. 1997; Hoffman et al. 1999; Bao et al. 2000). When using locally adjusted parameters, these uncertainties can be reduced further (at the expense of predictive power for unknown nuclides and their properties). It is commonly assumed, however, that the cross-section ratios σ_i/σ₀, and thus the reactivity ratios R_i/R₀, are predicted with much smaller uncertainty. This is especially true for the s-process. Unknown spins and parities of low-lying states would give rise to the largest part of the uncertainty in the prediction of σ_i/σ₀, but these are known for nuclei in s-process conditions. The remaining uncertainty is dominated by the uncertainty in the energy dependence of the γ widths (Rauscher 2010). However, note that neutron captures in the s-process generally have large reaction Q values, and therefore the relative change in energy for the contributing transitions (E − E_i + Q)/Q (Rauscher 2008, 2011), when varying E_i, is small. This leads to an uncertainty much smaller than those in specific cross-sections, which are probably lower than 10%.

For a more conservative estimate, the uncertainties in X in Table 1 and Figures 2 and 3 are given for uncertainty factors $\overline{u}=1.3$, i.e., 30%.

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The g.s. contribution X to stellar rates, as derived in Section 2.3, is not just of theoretical interest. It gives directly the maximum reduction in a rate's uncertainty that can possibly be achieved by completely determining the g.s. rate factor, e.g., by completely measuring the g.s. cross-sections σ₀ within the relevant energy window. Let us define an uncertainty factor $\mathcal {U}$ of the stellar rate R*, implying that the "true" rate R*_true is in the range $R^*/\mathcal {U}\le R^*_\mathrm{true}\le \mathcal {U}R^*$ for $\mathcal {U}\ge 1$. For instance, a 50% uncertainty in the rate is then expressed as $\mathcal {U}=1.5,$ and a rate known to be without uncertainty is characterized by $\mathcal {U}=1$. This overall uncertainty factor $\mathcal {U}$ of the rate can be split into a contribution to the uncertainty stemming from the g.s. uncertainty $\mathcal {U}_\mathrm{gs}$ and the combined uncertainty of the (usually only theoretically) predicted excited-state contributions $\mathcal {U}_\mathrm{exc}$, leading to

$$\begin{equation} \mathcal {U}=\mathcal {U}_\mathrm{gs}X+\mathcal {U}_\mathrm{exc}\left(1-X \right). \end{equation} \tag{ 18 }$$

(It is important to note that the uncertainty factors $\mathcal {U}$, $\mathcal {U}_\mathrm{gs}$, and $\mathcal {U}_\mathrm{exc}$ defined here are different from, but related to, the factors $\overline{u}=\mathcal {U}_\mathrm{exc}/\mathcal {U}_\mathrm{gs}$, u₁, u₂,..., used in Section 2.4.)

The uncertainty $\mathcal {U}$ is determined by $\mathcal {U}_\mathrm{gs}$ for X ≈ 1 and by $\mathcal {U}_\mathrm{exc}$ for X ≪ 1. In both cases, it is safe to assume that the uncertainty of a theoretically predicted rate is $\mathcal {U}=\mathcal {U}_\mathrm{gs}=\mathcal {U}_\mathrm{exc}$ for simplicity. If the g.s. cross-section σ₀ can be experimentally determined without error, i.e., $\mathcal {U}_\mathrm{gs}^\mathrm{exp}=1$, within the full energy range required to compute the reaction rate integral, then the maximum possible percentage reduction of $\mathcal {U}$ will be simply 100X. It is immediately seen that the reduction is negligible for X ≪ 1. On the other hand, if the experimentally determined g.s. rate R₀ has a non-negligible uncertainty $\mathcal {U}_\mathrm{gs}^\mathrm{exp}>1$, then the new, reduced uncertainty factor can be computed by replacing $\mathcal {U}_\mathrm{gs}$ in Equation (18) with $\mathcal {U}_\mathrm{gs}^\mathrm{exp}$. Again, this will only have appreciable impact on $\mathcal {U}$ when X ≈ 1.

This maximum possible reduction in rate uncertainty is especially important for the s-process. Astrophysical modeling and cross-section measurements have entered a high-precision era, with σ₀ for many targets determined to a few percentage points of precision and with stellar models demanding very small uncertainties in stellar rates for a detailed understanding of the production of s nuclei and for constraining the conditions within a star. It is important to realize, however, that the stellar rate is constrained by the experimental uncertainty only when X = 1. In the extreme case when X ≪ 1, a measurement of g.s. cross-sections or MACS will not be able to constrain the rate in any way, even when performed with the highest precision. In this case, R* is essentially given by the theoretical prediction of the cross-sections of excited states, and further experiments are required to constrain theoretical predictions. For example, (n, n') reactions populating excited target states (see, e.g., Mosconi et al. 2010b) or (γ, n) reactions on the residual nucleus (e.g., Sonnabend et al. 2003a) have been suggested to study specific transitions to excited states in the target.

It should be noted that even when X < 1, the knowledge of σ₀ can be used to test and improve a model's prediction of the reaction cross-section, even though the measurement will not directly constrain R*. In many cases, deficiencies in the description of the cross-sections of the ground and excited states are correlated, and the prediction of a stellar rate may also be improved in such instances by a renormalization of the rate. Whether this is possible depends on reaction details and must be investigated separately for each target nucleus.

For completeness, it must be mentioned here that the uncertainties in the SEFs f_SEF and g.s. contribution X are not the only nuclear uncertainties playing a role in s-process nucleosynthesis calculations. In particular, for interesting branching nuclei, often only theoretical predictions of the capture cross-sections are available. The feasibility of a considerable reduction in theoretical uncertainties via measurements can be checked by inspecting X for such nuclei. However, branching may be additionally affected by a temperature-dependent β-decay half-life (Takahashi & Yokoi 1987). Also, for decay rates (as for any other type of rate), the g.s. contribution can be determined by applying Equation (13). In this work, however, we focus on neutron captures that are close to stability.

We calculated SEFs f_SEF and g.s. contributions X for a set of nuclides important to s-process studies using the reaction rate code SMARAGD, version 0.8.1s (Rauscher 2011). This includes the excited states from the recent version of Evaluated Nuclear Structure Data File (ENSDF 2010). Nuclides present in KADoNiS v0.3 (Dillmann et al. 2009) were supplemented with several additional nuclei under consideration for inclusion in future KADoNiS versions, yielding a set of 412 nuclei. Table 1 shows the nuclei considered with their G₀, X, and f_SEF values at kT = 30 keV and kT = 80 keV (results for further temperatures will be included in future versions of KADoNiS). Also given is the relative uncertainty factor u_X, which defines the lower limit X_lower = X/u_X and upper limit X_upper = u_XX of X, as discussed in Section 2.4. The last column of Table 1 provides information on the current experimental status of the (n,γ) reaction on the given target nucleus and identifies the nuclei marked in Figure 4 (marked by an asterisk in the comment column). When a reaction is measured (marked by "e") and X is close to unity, the uncertainty in stellar rate is directly connected to the experimental uncertainty. If X is small, the theoretical uncertainty was not reduced, even when the rate was measured. For several reactions, only a 30 keV MACS was measured (marked by "30"), and the rates at other temperatures were determined by a renormalized rate prediction. Also listed are purely theoretical rates (marked by "t") and rates not appearing in KADoNiS v0.3 (marked by "n").

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The table is useful for both astrophysicists and experimentalists and is meant to be interpreted as follows. The maximum possible reduction in theoretical uncertainty can be obtained by inserting the value of X into Equation (18) for a given reaction. When striving to better constrain reactions experimentally, those with values of X close to unity should be chosen. If reactions were measured previously, it is possible to better constrain them by measuring with higher precision across the full relevant energy range (see Rauscher 2010). An unmeasured rate will only be constrained by measuring σ₀ if X ≈ 1. At the other extreme, some measured reactions show low X values (marked by the asterisk). It would be incorrect to assume for these cases that the uncertainty in stellar rates is similar to that of the measurements. Thus, this allows one to assess which reactions presently are well constrained in s-process calculations.

For a number of reactions, only kT = 30 keV MACS are directly obtained from experiments whereas the energy dependence is taken from theory. Obviously, this will only constrain stellar rates at 30 keV when X ≈ 1. Although it is usually preferable to obtain a rate from a measurement within the full range of contributing energies, one has to realize that X may decrease with higher energy, even when it is close to unity at kT = 30 keV. In such a case, additional measurements above 30 keV may only partially reduce the uncertainty in the rates at higher temperatures. To see how X changes with energy, values for kT = 80 keV are also shown in Table 1.

For a better overview, further comparisons of g.s. contributions X to (n,γ) rates at different temperatures are shown in Figures 2 and 3 for the nuclei included in Table 1. The error bars on the X values were computed as described in Section 2.4. We chose the reference temperature kT = 30 keV for the s-process and a high temperature of T = 2.5 GK, which is typical for explosive nucleosynthesis. As can be seen, the values of X are smallest in the region of deformed nuclei, where their deviation from the SEF is also the largest. This is because the level density in such nuclei is high, i.e., there are already many excited states at low excitation energies. At kT = 30 keV, we find X ≈ 0.16–0.3 for several nuclei (with the lowest values of X = 0.04, 0.16, and 0.2 for ¹⁶⁶Ho, ¹⁹³Pt, and ¹⁴²Pr, respectively) and X ≈ 0.5–0.8 for most nuclei in the mass range 150 ⩽ A ⩽ 190. Although experimental data with small uncertainties exist for most stable nuclei in this mass range (Bao et al. 2000; Dillmann et al. 2006, 2009), the stellar reactivity R* is at least partially based on the calculated contribution of excited states. Consequently, the uncertainties in the stellar reactivities R* may be much larger than the uncertainties in the g.s. MACS, which are provided in the compilations by Bao et al. (2000) and Dillmann et al. (2006, 2009).

At a glance, Figure 4 provides a guide to which targets are and are not well suited for experimentally constraining the theoretical uncertainties in stellar rates. The filled squares mark nuclei with X < 0.8 at 30 keV (using the lower limit of X from the errors discussed in Section 2.4). For neutron captures on these targets, even a complete experimental determination of g.s. rates will not be able to reduce the error from theory by more than 80%, and for even smaller X values it will not provide an improved rate. The open squares mark nuclei in our set that have X ⩾ 0.8 and therefore show cases for which the rate uncertainty mainly stems from experimental uncertainties. If not already measured, they may provide good opportunities for future experiments.

The limit of 80% error reduction was chosen because a significant reduction in the uncertainty from theory is required to achieve the precision desired for s-process nucleosynthesis. For example, assuming an 80% error reduction in an uncertainty of 30% in the rate predictions (Rauscher et al. 1997), the remaining error stemming from theory will then be on the order of 6%. On the other hand, an 80% reduction of the uncertainty factors of 2–3 in global reaction rate predictions would lead to a theoretical uncertainty in the stellar rate of about 40–60%, coming from uncertainties in the excited-state contributions. In addition, the experimental uncertainty for σ₀ also has to be included. This leads to uncertainties in stellar rates that already may be too large for high-precision s-process nucleosynthesis models.

Neutron captures and their inverse photodisintegration reactions at high temperatures are also important in the γ-process production of p nuclei (Woosley & Howard 1978; Rauscher 2006). For that reason, they are also included in KADoNiS (Dillmann et al. 2009). It is apparent from Figure 3 that X can assume very low values at high temperatures. This indicates that one must be very careful in extrapolating experimental rates from s-process temperatures (T = 0.384 GK is E = 30 keV), e.g., to a typical γ-process temperature of 2.5 GK. This implies an increasing theoretical contribution that is only constrained by experiments if, as pointed out above, the predictions of the ground- and excited-state cross-sections are correlated. Furthermore, although the partition function G₀ is well determined under s-process conditions by experimentally known level schemes, higher temperatures (and/or unstable nuclei) imply an increasing uncertainty in G₀.

Finally, let us discuss some specific cases in more detail to further illustrate the difference between f_SEF and X: neutron-capture reactions on ⁷⁹Se, ⁹⁵Zr, ¹²¹Sn, ¹⁸⁷Os, and ¹⁹³Pt. We focus here on the stellar temperatures kT = 8 keV and kT = 30 keV. A temperature of kT = 30 keV is often taken as the reference temperature of the s-process (see, e.g., Dillmann et al. 2009). It is close to the typical burning temperature of the ²²Ne(α, n)²⁵Mg reaction, which is one of the two neutron sources for the s-process in thermally pulsing AGB stars. The main neutron source in AGB stars is ¹³C(α, n)¹⁶O, which operates at lower temperatures around kT = 8 keV. At these relatively low temperatures, the g.s. contribution X to the stellar reaction rate factor R* is already far below unity in some cases. The weak s-process in massive stars operates at higher temperatures, and thus the g.s. contribution X is even further reduced.

The reaction ⁷⁹Se(n,γ)⁸⁰Se is an example of a case where f_SEF < 1 due to the weak influence of excited states. The nucleus ⁷⁹Se has several low-lying levels (J^π = 1/2⁻, 95.8 keV; 1/2⁻, 128 keV(?); 9/2⁺, 137.0 keV) above its 7/2⁺ g.s. (Singh 2002; ENSDF 2010). Because J₀ = 7/2 and J₁ = 1/2 in the first excited state, its population remains below about 1% up to kT = 30 keV, and a similar value is found for the large J = 9/2 state at 137.0 keV. This leads to G₀ = 1.027, f_SEF = 0.99, and X = 0.98 at kT = 30 keV. Moreover, ⁷⁹Se is considered to be an important branching nucleus with a highly temperature-dependent half-life (Takahashi & Yokoi 1987) in the weak s-process. For kT = 100 keV, we find f_SEF = 0.85 and X = 0.78. This means that not only are the excited states more populated (G₀ = 1.51), but the captures in these states also impact the resulting stellar rates. Were this not the case, i.e., were the capture cross-sections of ⁷⁹Se negligible in the excited states, then X = 1 and f_SEF = 1/G₀ = 0.66.

The reaction ⁹⁵Zr(n,γ)⁹⁶Zr is an example of both f_SEF = 1 and X = 1 because the excited states are not populated. The first excited state of ⁹⁵Zr is located at E = 954 keV, with J^π = 1/2⁺. The g.s. has J^π = 5/2⁺ (ENSDF 2010). The SEF and X do not deviate from unity over the entire temperature range of the s-process because G₀(T) = 1. Note that there was an earlier claim for a low-lying, first excited state with J^π = (3/2, 5/2)⁺ at 23 keV, but it was only observed in one particular experiment (Frota-Pessôa & Joffily 1986). The existence of such a level was excluded in a later high-resolution experiment (Sonnabend et al. 2003b). There is no evidence of another low-lying state "with quite different angular momentum," as recently discussed by Huther et al. (2010). Consequently, the state at 23 keV was removed from the adopted levels of ⁹⁵Zr (Basu et al. 2010; ENSDF 2010). An experimental determination of the ⁹⁵Zr(n,γ)⁹⁶Zr capture cross-section would be extremely valuable because the experimental neutron-capture data completely define the stellar reaction rate. These data are a prerequisite for the understanding of isotopic patterns of zirconium and molybdenum isotopes in meteorites that may be of s-process origin.

The reaction ¹²¹Sn(n,γ)¹²²Sn is close to case B in Section 2.2. The g.s. of ¹²¹Sn has a low spin of J^π = 3/2⁺. There is a very low-lying intruder state at E = 6.3 keV, with J^π = 11/2⁻, and a further low-J state at E = 60.3 keV, with J^π = 1/2⁺. Then, there is a gap in the level scheme up to the next state at E = 663.6 keV, which is too high to play a significant role in the s-process (Ohya 2010; ENSDF 2010). At kT = 8 keV, the population of the first excited state already exceeds the g.s., and we find a g.s. contribution to the stellar rate of X = 0.36. At the reference temperature of kT = 30 keV, the rate is further reduced to X = 0.27. The SEFs remain close to unity, with f_SEF = 1.19 and 1.07 at 8 keV and 30 keV. Because of the dominating contribution from the first excited state, experimental data for the ¹²¹Sn(n,γ)¹²²Sn reaction can only provide a minor one-third contribution to the stellar rate.

The reaction ¹⁸⁷Os(n,γ)¹⁸⁸Os is important for the Re/Os cosmochronometer (Mosconi et al. 2010a, 2010b). The nucleus ¹⁸⁷Os exhibits a very low-lying 3/2⁻ state at 9.8 keV above the 1/2⁻ g.s. The next levels appear at 74.4 keV, 75.0 keV, and 100.5 keV (3/2⁻, 5/2⁻, and 7/2⁻, respectively). Above the level at 117 keV and without spin assignment, the next levels are found at 187.4 keV and 190.6 keV, which are too high for the s-process (Basunia 2009; ENSDF 2010). At kT = 8 keV, the SEF is f_SEF = 1.0. The first excited state, however, is already populated, and we find a g.s. contribution of X = 0.63. At higher energies, the g.s. contribution reduces further to X = 0.28 for kT = 30 keV, although f_SEF = 1.19 remains close to unity. Because of the relevance of this reaction to the Re/Os cosmochronometer, neutron-capture experiments on ¹⁸⁷Os have been performed (see, e.g., Mosconi et al. 2010a; Fujii et al. 2010, and references therein). The importance of the low-lying, first excited state was noticed in that work (Mosconi et al. 2010b; Fujii et al. 2010), and additional experiments on the inelastic neutron scattering of ¹⁸⁷Os and the photodisintegration of ¹⁸⁸Os have been performed (see, e.g., Shizuma et al. 2005; Mosconi et al. 2010b) to study transitions proceeding on this state.

Another example of the importance of including excited-state transitions is ¹⁹³Pt(n,γ)¹⁹⁴Pt. An extremely low-lying 3/2⁻ state at E = 1.6 keV is found above the 1/2⁻ g.s. of ¹⁹³Pt, and a further low-lying 5/2⁻ state is located at E = 14.3 keV. Then, there is a small gap in the level scheme up to the next 3/2⁻ state at E = 114.2 keV (Achterberg et al. 2006; ENSDF 2010). At kT = 8 keV, the first excited state already dominates the reaction rate factor R*, and the g.s. contribution is only X = 0.26. At kT = 30 keV, the g.s. contribution drops to X = 0.16, and the reactivity R* is essentially determined by comparable contributions of the states at 1.6 keV and 14.3 keV. Again, as in the previous examples, the SEF remains relatively close to unity, with f_SEF = 1.22 at kT = 8 keV and f_SEF = 1.31 at kT = 30 keV. Thus, an experimental determination of the ¹⁹³Pt(n,γ)¹⁹⁴Pt cross-section can be used only to restrict theoretical predictions of the g.s. cross-section. Similar to the above examples of ¹²¹Sn and ¹⁸⁷Os, it is not possible to directly derive the stellar reaction rate from experimental neutron-capture data.