Astromers: Nuclear Isomers in Astrophysics^*

We express the effective transition rate Λ_AB from A to B as

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{{AB}}={\lambda }_{{AB}}+\displaystyle \sum _{i}{\lambda }_{{Ai}}{P}_{{iB}}.\end{eqnarray} \tag{ 6 }$$

The rate λ_AB is for transitions directly from A to B; this quantity may be negligible, but in some cases it is not, so we include it. The sum is over intermediate states i, and P_iB is the probability the system follows a chain of internal transitions from intermediate state i to the endpoint state B without passing through A. We require that the system not pass through A as that would be like starting over, amounting to a failed transition. In effect, Equation (6) sums the transition rate from A to each other state times that state's probability of going to B.

We now must find the P_iB. These can be computed by considering the first step of all allowed routes from i to B. A step from i to state s occurs with probability b_is. Now, P_iB can be identified as the sum over the probabilities of the possible first steps from i folded with the probabilities P_sB that the new state eventually reaches B. That is,

$$\begin{eqnarray}&&{P}_{{iB}}=\displaystyle \sum _{s}{b}_{{is}}{P}_{{sB}},\end{eqnarray} \tag{ 7 }$$

The sum is over all states s. If s = A, the transition has failed, and if s = B, the transition is complete; therefore, P_AB ≡ 0 and P_BB ≡ 1. States do not transition to themselves, so b_ss ≡ 0 ∀ s. These features allow a slight rewrite:

$$\begin{eqnarray}&&{P}_{{iB}}={b}_{{iB}}+\displaystyle \sum _{j}{b}_{{ij}}{P}_{{jB}}.\end{eqnarray} \tag{ 8 }$$

The summation in the second term is now only over intermediate states. From here, we may express the problem in matrix form,

$$\begin{eqnarray}\begin{array}{rcl}{\overrightarrow{P}}_{{IB}} & = & {\overrightarrow{b}}_{{IB}}+{{\boldsymbol{b}}}_{{II}}{\overrightarrow{P}}_{{IB}}\\ \to \left({\mathbb{1}}-{{\boldsymbol{b}}}_{{II}}\right){\overrightarrow{P}}_{{IB}} & = & {\overrightarrow{b}}_{{IB}}.\end{array}\end{eqnarray} \tag{ 9 }$$

In this expression, the subscript I reminds us that the vector and matrix indices run over the intermediate states. So, ${\overrightarrow{P}}_{{IB}}$ is the vector with components P_iB, ${\overrightarrow{b}}_{{IB}}$ is the vector with components b_iB, and b_II is the matrix with elements b_ij; in each of these, i and j are intermediate states. Equations (7) and (9) represent a system of N linear equations in N variables, where N is the number of intermediate states included in the calculation. Thus, ${\overrightarrow{P}}_{{IB}}$ can be found with a linear equation solver, and inserting ${\overrightarrow{P}}_{{IB}}$ into Equation (6) yields the effective transition rate from A to B.

In the case of a system with more than two endpoint states, Equation (6) holds for all pairs of initial and final endpoint states. The method is straightforwardly extended by redefining P_iE as the probability of eventually reaching endpoint state E from intermediate state i without passing through any other endpoints, and all of the same arguments apply.

By the invertible matrix theorem, if 1 − b_II is invertible, then a unique solution to Equation (9) exists:

$$\begin{eqnarray}&&{\overrightarrow{P}}_{{IE}}={\left({\mathbb{1}}-{{\boldsymbol{b}}}_{{II}}\right)}^{-1}{\overrightarrow{b}}_{{IE}}.\end{eqnarray} \tag{ 10 }$$

To show that that 1 − b_II is invertible, we will use the logic of Gupta & Meyer (2001).

We note that the full transition matrix b with elements b_st (which includes endpoint and intermediate states) is a stochastic matrix: its rows sum to unity. Obviously, b is nonnegative (all entries are real and positive or zero). Furthermore, because every state is reachable via some path from every other state, b is irreducible, meaning the rows and columns cannot be permuted to produce a block diagonal form. If it were reducible, there would be at least two sets of states that communicate only with members of their set, and the graph of the system would not be connected.

Stochastic matrices have an eigenvector with eigenvalue 1; to see this, observe that ${\boldsymbol{b}}\overrightarrow{1}=\overrightarrow{1}$, where $\overrightarrow{1}$ is the column vector with all entries equal to 1. From the Gershgorin circle theorem on upper bounds of spectral radii, the largest absolute value among eigenvalues (the spectral radius) of b is 1. Because b is nonnegative and irreducible, the Perron–Frobenius theorem guarantees that there is exactly one eigenvector corresponding to an eigenvalue equal to the spectral radius, i.e., 1.

We now rely on a corollary to the Perron–Frobenius theorem: any principal submatrix of an irreducible nonnegative matrix has a spectral radius that is strictly less than the full matrix. A principal submatrix is obtained by deleting rows and columns from a matrix, with the row and column indices being equal. The matrix b_II is a principal submatrix of b obtained by deleting the rows containing transitions from endpoints and the columns containing transitions to endpoints. Therefore, it has a spectral radius less than 1, and all of its eigenvalues have absolute value (modulus) less than 1.

Finally, let M and N be square matrices with N = M + 1. For every eigenvalue ^Mλ_i of M, there will be an eigenvalue ^Nλ_i = ^Mλ_i + 1 of N. To see this, let $\overrightarrow{x}$ be an eigenvector of M with eigenvalue λ. We then have

$$\begin{eqnarray}\begin{array}{rcl}{\boldsymbol{N}}\overrightarrow{x} & = & \left({\boldsymbol{M}}+{\mathbb{1}}\right)\overrightarrow{x}\\ & = & {\boldsymbol{M}}\overrightarrow{x}+\overrightarrow{x}\\ & = & \lambda \overrightarrow{x}+\overrightarrow{x}\\ & = & (\lambda +1)\overrightarrow{x}.\end{array}\end{eqnarray} \tag{ 11 }$$

Taking M = − b_II and remembering that all eigenvalues of b_II have modulus less than 1, we deduce that all eigenvalues of 1 − b_II have a positive real component and are therefore nonzero, and the matrix is invertible by the invertible matrix theorem.

²⁶Al is produced by massive stars in substantial quantities. Its ground-state β-decay lifetime of ∼1 million years means that it lives long enough to be observed after a massive star dies, but not so long as to become uncorrelated with where that star died. Because massive stars have short lifetimes, we infer that if one died recently in a location, that location is probably a region of active star formation. Hence, ²⁶Al is an important tracer of star formation (Mahoney et al. 1982; Diehl et al. 1995), and we are interested in accurately computing its abundance in nucleosynthesis networks. ²⁶Al production is sensitive not only to the rates in which it is directly involved, but also to many other reaction rates in nearby nuclei (Iliadis et al. 2011). However, its treatment is complicated by a long-lived isomer at 228.305 keV that can greatly impact ²⁶Al destruction rates (Ward & Fowler 1980; Coc et al. 1999; Gupta & Meyer 2001; Runkle et al. 2001; Banerjee et al. 2018; Reifarth et al. 2018).

To compute ²⁶Al transition and β-decay rates, we included 67 total states: the two endpoint states and 65 intermediate states. In this nucleus and all others in this section, we used available experimental data for the spontaneous transition and β-decay rates (Basunia & Hurst 2016). In ²⁶Al, we supplemented the data with transition and β-decay strengths computed using the shell model codes NuShellX (Brown & Rae 2014) and OXBASH (Brown et al. 2004) along with the USDB interaction (Brown & Richter 2006). When a transition rate is unmeasured and a shell model calculation is impractical, we use the Weisskopf approximation (Weisskopf & Wigner 1930). For the β-decay rates, we used the measured ft values for ground-state and isomer β decays, and we computed matrix elements with the shell model for all other weak transitions.

Once we have the b_st, obtained from Equations (1)–(5), we use Equation (9) to compute the P_IE. Figure 2 shows P_IE for the first three intermediate states as functions of temperature; in this figure and Figure 3, the index g indicates the ground state, and m is the isomer.

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While most of the P_IE are nearly constant, the jump in P_3m at around 30 keV portends a behavioral change in the nucleus at this temperature. Indeed, this manifests as an increase in the rate for the ground state to transition to state 3 and from there ultimately to the isomer. Figure 3 illustrates this jump; it shows rates for endpoint states to transition to specific intermediate states, and from there to eventually transition to the other endpoint state. That is, Figure 3 shows the terms in Equation (6). Whereas most intermediate states' contributions increase smoothly with temperature, the third state going from g to m (black solid) has a kink at ∼30 keV.

It is no mere coincidence that the kink occurs at the same temperature where λ_m3P_3g (black dashed) crosses λ_m4P_4g (red dashed). As the temperature rises, state 4 becomes more accessible from the isomer, driving the rise of λ_m4P_4g. Once it dominates λ_m3P_3g, the most probable isomer-to-ground transition pathways change.⁹ Figure 4 shows the three most probable paths from ground to the isomer at T = 25 keV and 35 keV; by symmetry (Section A.3), it also shows the isomer-to-ground paths.

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At T = 25 keV, the dominant pathway between 1 (ground) and 2 (isomer) is simply through state 3. But at T = 35 keV, state 4 is thermally accessible from 2, and because it is much more strongly coupled to the isomer than is state 3 (P_4m ≫ P_3m in Figure 2),¹⁰ 2 preferentially transitions to 4. State 4 is only weakly coupled to ground (P_4g in Figure 2), and the preferred route from 4 to 1 is via 3. Therefore, when the isomer preferentially transitions to state 4 (above T = 30 keV), the most probable isomer-to-ground path is 2 → 4 → 3 → 1. This "opens up" the 1 → 3 → 4 → 2 path and creates the kinks. Figure 5 further illustrates this point by showing the probabilities for state 3 to go directly to the isomer (b_3m), to go to 4, and then eventually to the isomer (b₃₄P_4m, and to go to 5 and then eventually to the isomer (b₃₅P_5m).

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Finally, we use Equation (6) to compute the effective transition rates Λ₁₂ and Λ₂₁, shown in Figure 6. From here on, 1 should be understood as the ensemble corresponding to the ground state, and 2 is the ensemble corresponding to the isomer; the ensembles are constructed as described at the beginning of this section in Equations (12)–(15). The figure also shows various β-decay rates: the ensemble decay rates Λ_1β and Λ_2β;  the thermal decay rate Λ_{β therm}, computed using a thermal distribution of level occupations; the steady-state decay rate Λ_{β SS};  the steady-state decay rates ${{\rm{\Lambda }}}_{\beta \,{SS}}^{P1}$ and ${{\rm{\Lambda }}}_{\beta \,{SS}}^{P2}$ when the nucleus is produced exclusively in either the ground state (P1) or isomer (P2) ensemble; and—for comparison—the effective decay rate found by Coc et al. (1999). We computed the steady-state rates with the formalism of Misch et al. (2018) assuming a two-level system comprised of 1 and 2. At low temperatures, we compute a slightly higher β-decay rate than Coc et al. (1999); this is because they used laboratory rates, whereas our electron capture rates are enhanced by the dense environment. We are otherwise in good agreement.

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Let us first understand Λ₁₂, Λ₂₁, Λ_1β, and Λ_2β. Below T ≈ 7 keV, excited states are inaccessible. The isomer transitions directly to ground at a slow trickle, the ground state cannot reach the isomer, and each β decays at its laboratory rate. Above T ≈ 7 keV, state 3 becomes accessible, the path 1 ↔ 3 ↔ 2 opens up, and the transition rates begin to rise. At T ≈ 13 keV, Λ₁₂ surpasses Λ_1β, and the dominant "destruction" channel for ensemble 1 is transitions to ensemble 2. At T ≈ 17 keV, the weight w₃₁ of state 3 in ensemble 1 grows great enough that it contributes significantly to the ensemble β-decay rate. State 3 has allowed decays to ²⁶Mg, and Λ_1β rises accordingly. At T ≈ 30 keV, the path 1 ↔ 3 ↔ 4 ↔ 2 opens up, and the transition rates rise faster. At T ≈ 35 keV, Λ₂₁ surpasses Λ_β2, and the dominant destruction channel for ensemble 2 is transitions to ensemble 1.

Now let us examine Λ_{β therm} and Λ_{β SS}, which are the thermal-equilibrium β-decay rate and the steady-state (without any ²⁶Al production) β-decay rate, respectively. The thermal decay rate is straightforward to understand. If ²⁶Al had no isomer, the nuclear level occupation probabilities would simply follow a Boltzmann distribution. At T ≈ 8 keV, the first excited state would be sufficiently populated to contribute to Λ_{β therm}, and the decay rate would rise concomitantly. However, ²⁶Al does have an isomer that suppresses thermally driven transitions up from ground. Because there is no thermally accessible path to populate the isomer, it decays away rapidly, is not replenished, and has a less-than-thermal population in steady state; this suppresses Λ_{β SS}. Once Λ₁₂ exceeds Λ_1β at T ≈ 13 keV, the ground state begins to appreciably feed the isomer. But up until T ≈ 35 keV, the isomer essentially always β decays. Furthermore, Λ_2β ≫ Λ_1β, so ²⁶Al decays as fast as the isomer is fed, and the steady-state β-decay rate is approximately equal to Λ₁₂. At T ≈ 35 keV, when Λ₂₁ exceeds Λ_2β, transitions between the ensembles dominate the ensemble decay rates, and the system at last reaches the thermal equilibrium found by other authors (Coc et al. 1999; Iliadis et al. 2011; Banerjee et al. 2018). This defines the thermalization temperature below which the ²⁶Al isomer is an astromer.

This leaves us to understand ${{\rm{\Lambda }}}_{\beta \,{SS}}^{P1}$ and ${{\rm{\Lambda }}}_{\beta \,{SS}}^{P2}$, which are the steady-state decay rates when ²⁶Al is produced solely in the ground-state or isomer ensembles, respectively. Observe that ${{\rm{\Lambda }}}_{\beta \,{SS}}^{P1}$ tracks Λ_{β SS} at all temperatures. This is because in steady state without production, essentially all of the population is in the ground state anyway, so it makes no difference if ²⁶Al is produced in the ground-state ensemble.

In contrast, when ²⁶Al is produced in the isomer ensemble at low temperature, the ground state is fed exceedingly slowly. Gradually, a small equilibrium amount of material will build up in the ground state that brings down the total decay rate, though this may take many years (see Misch et al. 2018; Figure 5), and it will never be enough to bring the decay rate to near the ground-state rate. At T ≈ 7 keV, Λ₂₁ increases dramatically, so the ground state is more rapidly fed. The increased feeding yields a larger quantity of ground-state material, and the steady-state decay rate consequently falls. Once Λ₁₂ overtakes Λ_1β at T ≈ 13 keV, the steady-state decay rate goes to the thermal-equilibrium rate. This follows from Equation (18) in Misch et al. (2018) using our two-ensemble system. That equation is expressed in matrix notation, but we may analyze one of the embedded scalar equations by selecting a single vector component. Taking the first element of the vectors, we have

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{21}{n}_{2}-{{\rm{\Lambda }}}_{12}{n}_{1}-{{\rm{\Lambda }}}_{1\beta }{n}_{1}+\left({{\rm{\Lambda }}}_{1\beta }{n}_{1}+{{\rm{\Lambda }}}_{2\beta }{n}_{2}\right){p}_{1}=0.\end{eqnarray} \tag{ 16 }$$

Here, n₁ and n₂ are the occupation fractions of the two ensembles, and p₁ is the fraction of ²⁶Al production that goes into the ground-state ensemble. By assumption, p₁ = 0, and we are considering a regime where Λ₁₂ dominates Λ_1β;  the latter point implies that Λ₁₂n₁ + Λ_1βn₁ ≈ Λ₁₂n₁. Using these facts along with Equation (A13), we find

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\Lambda }}}_{21}{n}_{2}-{{\rm{\Lambda }}}_{12}{n}_{1}\approx 0\\ \quad \to \displaystyle \frac{{n}_{2}}{{n}_{1}}\approx \displaystyle \frac{{{\rm{\Lambda }}}_{12}}{{{\rm{\Lambda }}}_{21}}=\displaystyle \frac{{g}_{2}}{{g}_{1}}{e}^{\tfrac{{E}_{1}-{E}_{2}}{T}},\end{array}\end{eqnarray} \tag{ 17 }$$

which is precisely a thermal distribution between the ground state and isomer. We conclude that the nuclear levels are thermally distributed when Λ₁₂ ≫Λ_1β, leading to the observed agreement between ${{\rm{\Lambda }}}_{\beta \,{SS}}^{P2}$ and Λ_{β therm}. Note that this is not necessarily an excuse to just use the thermal decay rate, because it may take many thousands of years to reach steady state.

Up to now, we have used exact values for the λ_st. However, uncertainties in the individual rates lead to uncertainty in the effective transition rates. We therefore performed a sensitivity study by varying the λ_st. Figure 7 shows the results of adjusting measured rates all up or all down by one or two standard deviations and shell model rates all up or all down by a factor of 3 or 30. Shell model rates using the USDB Hamiltonian tend to be good to within a factor of a couple with occasional large errors relative to experiment of one to two orders of magnitude (see Banerjee et al. 2018), so this choice of variations should reasonably probe the range of the Λ_AB. Dark bands show the smaller variations, and light bands show the larger. Because the λ_st were adjusted up and down together, the bands represent reasonable bounds on the effective rates. Furthermore, our analysis finds that the thermalization temperature of ²⁶Al is uncertain within the range between 30 and 40 keV; this is at odds with Coc et al. (1999), who concluded that the thermalization temperature is insensitive to remaining nuclear uncertainties.

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A more detailed picture emerges when the λ_st are varied separately. We identified which individual rates to change by examining the dominant transition paths; if a transition existed in paths carrying a total of at least 1% of the effective rate and that transition's uncertainty was at least 10%, we varied it. We adjusted measured rates up and down by one standard deviation and shell model rates up and down by a factor of 3. Table 1 shows the sensitivity of the effective transition rate to variations in the individual rates.

Our results generally agree quite well with Gupta & Meyer (2001). The greatest uncertainty in the effective transition rates arise from λ₃₂ at lower temperatures and λ₄₃ at higher temperatures.

³⁴Cl may be observable immediately after a nova (Coc et al. 1999), which means that accurately computing its abundance could be interesting. However, it has a long-lived isomer at 146.36 keV, bringing with it the familiar difficulty. But unlike ²⁶Al and the other nuclei in this paper, the ³⁴Cl isomer is more β-stable than the ground state; the ground-state β-decay half-life is 1.5266 s, while the isomer's total half-life is 31.99 minutes (Nica & Singh 2012).

We computed the ³⁴Cl rates using 30 states. As with ²⁶Al, we supplemented unmeasured transition rates and β-decay ft values with calculations from NuShellX, OXBASH, and the USDB interaction. Experimental data for the key 3 → 2 transition consist only of an upper bound, so we used our shell model result. Our computed effective transition rates and β-decay rates are shown in Figure 8 along with the rates computed by Coc et al. (1999). As with ²⁶Al, we are in good agreement with the earlier work.

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Below T ≈ 10 keV, no paths apart from the direct transition contribute meaningfully to the transition rate, but above this temperature, the path 2 ↔ 3 ↔ 1 opens up and Λ₂₁ begins to rise. Because 2 is the more β-stable state, ${{\rm{\Lambda }}}_{\beta \,{SS}}^{P2}$ mostly tracks Λ_{β SS}. Once Λ₂₁ dominates Λ_2β, Λ_{β SS} follows until it joins Λ_{β therm}, while ${{\rm{\Lambda }}}_{\beta \,{SS}}^{P1}$ is delayed slightly; ³⁴Cl thermalizes at T ≈ 20 keV. Below this temperature, the isomer is an astromer. With ground being the less stable state, Λ_{β therm} and ${{\rm{\Lambda }}}_{\beta \,{SS}}^{P1}$ both track Λ_1β until T ≈ 30 keV, at which point the isomer ensemble is thermally populated at a sufficient level to pull the thermal and steady-state rates down a bit.

Figure 9 shows the uncertainty bands in the ³⁴Cl Λ_AB. The uncertainty is driven almost entirely by the 3 → 2 transition. Even up to T = 50 keV, all dominant paths travel through states no higher than state 4, and apart from λ₃₂, the λ_st are measured with small experimental uncertainties. Our dark uncertainty bands (smaller variations) agree well with the findings of Coc et al. (1999). The light bands (larger variations) exhibit some asymmetry: the lower bands are narrower than the upper bands at higher temperatures. As λ₃₂ is increased, the nucleus flows more freely through the 1 → 3 → 2 path, and 3 → 2 is the bottleneck. But when λ₃₂ is sufficiently decreased, the 1 → 4 → 2 path dominates at higher temperatures, and the effective transition rate becomes insensitive to further decrease in λ₃₂.

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In slow neutron capture (s-process) nucleosynthesis, neutron captures are generally slow relative to β decays, so few nuclei are produced, which are more than one or two steps from stability. In most cases, this makes it easy to trace out the s-process path in the chart of nuclides. Starting with a seed nucleus, add neutrons until you make a β-unstable nucleus. Allow that unstable nucleus to β decay until it is stable. Resume adding neutrons and repeat until you have lead, bismuth, or no more neutron source.

⁸⁵Kr lies along the s-process nucleosynthesis path, but the simple procedure above is derailed by the ground state's long, but not too long, β-decay half-life of 10.739 yr. This creates competition between β decay and neutron capture, which influences the production of nearby nuclei (Abia et al. 2001). Because the s-process path can go in two different directions at ⁸⁵Kr, it is known as a branch-point nucleus and serves as an interesting diagnostic of the s-process environment. However, its current usefulness as such is greatly diminished by two complications: an uncertain neutron capture cross section (Dillmann et al. 2010) and an isomer at 304.871 keV with a total half-life of 4.480 hours (Singh & Chen 2014).

We will not address the neutron capture cross section, but we did apply our method to study the isomer's effect on β decay. We used experimental data on the lowest 30 levels of ⁸⁵Kr. We supplemented unmeasured transition rates with the Weisskopf approximation; ⁸⁵Kr (N = 49) is near a closed neutron shell (N = 50)—which diminishes the effectiveness of shell model calculations—so we made no further data supplements. Figure 10 shows our results.

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For the most part, ⁸⁵Kr does not behave in any remarkable way; the thermal and steady-state decay rates track each other, including when ⁸⁵Kr is produced in the ground state, so the isomer is not an astromer. This is because in contrast with ²⁶Al, Λ₂₁ exceeds Λ_2β (the green lines cross) at a lower temperature than Λ₁₂ exceeds Λ_1β (the red lines cross). Therefore, by the time the ground-state ensemble can appreciably feed it, the isomer ensemble preferentially transitions back to ground rather than β decays, and the thermal β-decay rate is appropriate for all situations except when ⁸⁵Kr is produced in the isomer ensemble.

The latter case, however, can give rise to interesting behavior. Below T ≈ 21 keV, material produced in the isomer ensemble has an  ∼ 80% chance to β decay rather than transition to ground; this drives up the decay rate of the species and favors β decay over neutron capture, rendering the ⁸⁵Kr isomer an astromer. Above T ≈ 25 keV, the nuclear levels thermalize, and the thermal β-decay rate is always correct. The weak component of the s-process occurs in massive stars with temperatures between 30 and 90 keV, well above the ⁸⁵Kr thermalization temperature. But interestingly, the main s-process occurs in pulsing asymptotic giant branch (AGB) stars of mass 1 − 3 M_⊙;  short pulses reach temperature T ∼ 30 keV, while longer interpulse periods have a temperature of T ∼ 8 keV. The entire cycle has a period of ∼50,000 yr (Busso et al. 1999). This means that the ⁸⁵Kr isomer alternates being an astromer in the main s-process.

The uncertainties in the ⁸⁵Kr effective transition rates are somewhat different from the ²⁶Al and ³⁴Cl cases in two ways. First, the experimental uncertainties are asymmetric (see table 2). Second, the unmeasured rates are estimated with the Weisskopf approximation rather than the shell model; the Weisskopf approximation tends to only be precise to one or two orders of magnitude.

Figure 11 shows the uncertainties in the ⁸⁵Kr Λ_AB. The upper light band is much narrower than the lower light band due to a bottlenecking effect similar to ³⁴Cl. As the λ_st are turned up together, the experimental λ₄₁ becomes a bottleneck; because its uncertainty is much less than the Weisskopf rates, it is turned up less than the Weisskopf rates and eventually controls the Λ_AB. However, as the rates are turned down, the opposite happens: λ₄₁ is reduced much more slowly, the Weisskopf rates become the bottleneck, and the lower band is wider than the upper band. If we decrease the Weisskopf rates much more, then as with ³⁴Cl, the 1 → 2 direct transition will be favored up to higher temperatures, and the Λ_AB will be insensitive to further decreases.

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We show the sensitivity of the Λ_AB to selected λ_st in Table 2; we used the same selection criteria as with ²⁶Al. Similarly, we only vary by one "unit" of uncertainty, which will avoid the bottleneck effect. We examine the fairly narrow range of temperatures from 21–25 keV because that is the range of thermalization temperatures found from Figure 11. At all temperatures, the unmeasured λ₄₃ is the largest contributor to the effective transition-rate uncertainty. At T = 23 keV, the unmeasured λ₅₄ plays a significant role, and at T = 25 keV, the unmeasured λ₆₅ and the measured (but uncertain) λ₄₁ are also important sources of uncertainty in the Λ_AB.

To reiterate from Section 3.3, the main s-process occurs in thermally pulsing AGB stars. The total pulsation period is ∼50 thousand years, with a pulse temperature of T ∼ 30 keV and an interpulse temperature of T ∼ 8 keV (Busso et al. 1999).

All rates in this section were computed using all measured levels in each nucleus (up to 30 levels) with unmeasured γ-decay rates computed from the Weisskopf approximation. We do not provide sensitivity studies for these nuclei because nearly all of the intermediate-level lifetimes (and hence γ rates) along dominant transition paths are unknown, and the uncertainties in the Λ_AB are driven by the Weisskopf approximation.

The 6.31 keV isomer of ¹²¹Sn (T_1/2 = 43.9 y) is more stable than the ground state (T_1/2 = 27.03 h). Unlike other isomers considered in this work, this one decays primarily to the ground state (77.6%) rather than β decay, so Λ₂₁ always exceeds Λ_2β. However, Figure 12 shows that Λ₁₂ does not catch up to Λ_1β until temperature T ∼ 20 keV, making this an astromer in the interpulse periods of the main s-process. Moreover, the long half-life of the astromer provides ample time for it to capture a neutron rather than deexcite or β decay. This would reduce the s-process production of ¹²¹Sb, ¹²²Te, and ¹²³Te.

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Takahashi & Yokoi (1987) took advantage of the 122.845 keV isomer to compute a thermal β-decay rate for ¹⁷⁶Lu. We also computed the transition and β-decay rates in this isotope; the results are in Figure 13. Takahashi & Yokoi (1987) performed a very careful calculation, but two of their temperature points (0.05 and 0.1 GK) fall in the range where the two states should be treated as separate species. Furthermore, ¹⁷⁶Lu is made in the main s-process, and the ∼10 keV thermalization temperature implies its isomer could be an astromer during the main s-process interpulse periods.

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All rates in this section were computed using all measured levels in each nucleus (up to 30 levels) with unmeasured γ-decay rates computed from the Weisskopf approximation. We do not provide sensitivity studies for these nuclei because nearly all of the intermediate-level lifetimes (and hence γ rates) are unknown, and the uncertainties in the Λ_AB are driven entirely by the Weisskopf approximation.

Isomers may play an important role in the rapid neutron capture process (r-process). Fujimoto & Hashimoto (2020) identified nine nuclei with isomers that could affect the light curve of a post-neutron-star-merger kilonova, four of which were particularly impactful: ^123,125,127Sn with isomers at 24.6, 27.50, and 5.07 keV respectively, and ¹²⁸Sb with an isomer of unknown energy. Figures 14, 15, and 16 show the transition and β-decay rates for the Sn isotopes. Indeed, each of these thermalizes at T ≈ 25 keV, well above the expected ambient temperature when the isotopes are populated in such environments. The disparity between the ground-state and isomer β-decay rates could have a marked impact on the radioactive heating as nuclei decay back to stability, so further investigation is in order.

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The fourth isotope identified by Fujimoto & Hashimoto (2020), ¹²⁸Sb, suffers from incomplete data. The rates are shown in Figure 17, but it does not thermalize within the computed temperature range. This is almost certainly due to missing data in the excited states. The ground state has spin and parity J^π = 8⁻, the isomer has J^π = 5⁺, and all other measured states have J ≤ 4. Furthermore, the isomer has unknown energy (we take the likely upper bound of 20 keV), and all other excited states have energies measured with respect to that unknown energy (Elekes & Timar 2015). Because this is such a potentially important nucleus, closer investigation is warranted. If the isomer is indeed influential, this would motivate experiments to measure the excited-state properties.

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¹⁸²Hf is a cosmochronometer that may be produced in the s-process (Lugaro et al. 2014) or the r-process (Wu et al. 2019). It has an isomer at 1172.87 keV; Figure 18 shows the rates for this isotope. Although ¹⁸²Hf thermalizes at a fairly low T ≈ 6 keV and is therefore not an s-process astromer, this is nevertheless higher than the expected temperature at which it would be produced in some r-process events; the two-minute β-decay half-life of ¹⁸²Lu likely implies that a rapid neutron capture event would have adequate time to cool. However, the ¹⁸²Lu β decay exclusively feeds the ¹⁸²Hf ground-state ensemble (Kirchner et al. 1982). With this information, we conclude that the ¹⁸²Hf isomer does not affect the γ-ray signal predicted by Wu et al. (2019).

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Finally, we raise the interesting case of ¹⁷⁰Ho. This nuclide has only two measured levels: the ground state with J^π = 6⁺ and a β-decay half-life of 2.76 minutes, and a 120 keV isomer with J^π = 1⁺ and a β-decay half-life of 43 s. We anticipate that this difference in decay rates could affect energy generation in an r-process event with observable effects in the light curve of a kilonova. ¹⁷⁰Ho is produced by β decay of ¹⁷⁰Dy, but the parent β-decay intensities are unmeasured. However, the parent ground state has J^π = 0⁺, so it likely decays predominantly to the ¹⁷⁰Ho isomer. This fact—coupled with the lack of data on ¹⁷⁰Ho excited states (which greatly hinders estimating the Λ_AB), the computed abundance of A = 170 nuclei (Sprouse et al. 2020; Vassh et al. 2020), and the potential kilonova implications—motivates experimental examination of this pair of nuclei.

We show here two examples of isomers that look promising as astromers but for which we are unaware of any environments where they may behave as such. We computed unmeasured γ-decay rates using the Weisskopf approximation.

⁵⁸Mn is an important antineutrino source in presupernova stellar cores (Patton et al. 2017), and it has an isomer at 71.77 keV. We computed effective transition and β-decay rates using the lowest 30 measured levels. As shown in Figure 19, ⁵⁸Mn thermalizes at T < 5 keV, far below the presupernova core temperatures of 300−900 keV, so it is not an astromer in this environment. Nevertheless, there is a point to be made about this isomer. Because ⁵⁸Mn is such a prodigious antineutrino source (with consequences for presupernova detection), we naturally desire precise weak-interaction rates. In most cases, we must rely entirely on theory for the weak-interaction strengths of excited states, but the existence of a low-lying isomer gives precise experimental data that reduces uncertainties in the high-temperature weak-interaction rates of ⁵⁸Mn.

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¹¹³Cd has a 14.1 yr isomer at 263.54 keV that nearly always β decays (Blachot 2010). Because it lies on the s-process path and the isomer β-decay rate could make it a branch point, it is worth taking a closer look at. We computed effective transition rates and β-decay rates in ¹¹³Cd using the 30 lowest measured energy levels; our results are shown in Figure 20.

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The ¹¹³Cd rates are qualitatively similar to ⁸⁵Kr in that Λ₂₁ overtakes Λ_2β at a lower temperature than Λ₁₂ exceeds Λ_1β, with similar consequences for the β-decay rates. The major difference is that the thermalization temperature for ¹¹³Cd is only ∼5 keV. This is below any s-process temperatures, so this isomer is likely never an s-process astromer.

Of course, as with ⁵⁸Mn, the existence of the isomer means that we have a precise measurement of the β-decay rate for the first excited state in this nucleus, providing a more precise value of the thermal β-decay rate for this species. Also, if there exists an environment where the isomer is produced at a cooler temperature, ¹¹³Cd that is produced in the ground state will remain nearly stable, but that produced in the isomer will decay to ¹¹³In. However, we are unaware of any specific astrophysical environments where this would play a role, as ¹¹³Ag—the ¹¹³Cd β decay parent produced in the r-process—essentially always decays to the ground-state ensemble.

The probability of following a path is the product of the weights b_ij for each step in the path. We will show here precisely what that means. A probability quantifies how frequently an outcome will occur from a set of possibilities, and the probabilities must sum to unity. In discussing path probabilities, we must define the possibilities under consideration.

The first set of possibilities we discuss is not directly applicable to calculations of effective transition rates, but rather is useful in interpreting the P_iE. Consider the set of all paths of length N which begin at state i and end anywhere. Naturally, the probabilities of following these paths must sum to unity because the system must end up somewhere. For paths of length 1, we can easily see that the probability of following each path to each final state f is b_if. Those are the individual transition probabilities, and we know that they sum over f to unity. Now consider paths of length 2: i → j → f. We know the probabilities of the individual transitions, so we may simply multiply them then sum the products over all values of j and f to get a normalization factor K for the total probability:

$$\begin{eqnarray}&&K=\displaystyle \sum _{j}{b}_{{ij}}\displaystyle \sum _{f}{b}_{{jf}}.\end{eqnarray} \tag{ A1 }$$

The inner sum is unity, which implies that the total sum is unity, and the normalization factor is simply 1. Considering paths of greater length simply extends the number of sums, and we see that paths of length N form a well-defined set of possibilities with probabilities computed from direct multiplication of the individual transition probabilities.

Our graph represents an irreducible Markov chain (every state is reachable from every other state by at least one path), so all states are members of a single communicating class (Gagniuc 2017). Consequently, each intermediate state can reach A via at least one path with a finite number of steps N and that path has a positive probability with respect to other paths of length N. This implies that with repeated transitions, the probability of not having followed a path that reaches A will decay toward zero. Thus, A is recurrent, meaning the system will eventually reach A with unit probability. Recurrence is a class property (all members of a communicating class are either recurrent or not recurrent), so a system will eventually reach A with probability 1 and it will eventually reach every other endpoint state with probability 1.

The P_iE from Section 2 quantify the probability of starting from intermediate state i and reaching endpoint E before reaching any other endpoint. The system definitely reaches every endpoint eventually, and it must reach one of them first; the probabilities P_iE therefore sum to 1:

$$\begin{eqnarray}&&\displaystyle \sum _{E}{P}_{{iE}}=1\end{eqnarray} \tag{ A2 }$$

Now we describe a more directly applicable set of possibilities. We are interested in starting at A, leaving to some state s (which in principle could be B), and finding the subsequent probability of successfully transitioning to endpoint state B. Again, for every state s, the sum over t of b_st is unity, which combines with Equation (A2) to give a set of possibilities whose probabilities sum to unity:

$$\begin{eqnarray}&&\displaystyle \sum _{s}\left({b}_{{As}}\displaystyle \sum _{E}{P}_{{sE}}\right)=\displaystyle \sum _{s}{b}_{{As}}=1.\end{eqnarray} \tag{ A3 }$$

In words, the set of possibilities for which we compute probabilities is paths that begin at A, terminate at an endpoint state, and do not have any endpoint states in between. When the system leaves A, it must eventually reach an endpoint state (Equation (A2)), so the probabilities of all possibilities sum to 1 (Equation (A3)). The possibilities are also mutually exclusive (no two paths are alike), giving a well-defined set of possibilities and associated probabilities.

Iterated expansion of the P_jB in Equation (8) reveals that P_iB is the sum over all paths of any length from i to B of the compounded individual transition probabilities:

$$\begin{eqnarray}&&{P}_{{iB}}=\displaystyle \sum _{\mathop{\mathrm{paths}}\limits_{i\to j\to ...\to k\to B}}{b}_{{ij}}...{b}_{{kB}}.\end{eqnarray} \tag{ A4 }$$

Note that this includes short paths that might not have a j or k. Multiplying Equation (8) by b_Ai gives an unnormalized probability that when the system leaves A, it goes to i, and from there via some path that eventually leads to B, with the contribution of each path to that probability explicitly computable. Likewise, we may compute the unnormalized probabilities of paths that begin at A and go to any endpoint E (including returning to A) by using the P_iE. But comparison with Equation (A3) shows that the sum of the compounded individual transition probabilities for all paths that start at A and end at any endpoint is unity. In other words, the compounded individual transition probabilities for these paths directly answer the question "when the system leaves A, what is its probability of following a particular path to a particular endpoint state given that the path terminates at an endpoint state?" Naturally, we may replace A with any endpoint state.

The total effective transition rate Λ_AB is ultimately the product of the rate λ_A at which systems leave A and the sum of the probabilities of paths from A to B (see Equation (6) in the context of the arguments in this section). Each path A → ... → B contributes to the total effective transition rate in proportion to its probability, and we may now ask which paths contribute the most to transition rates.

To find the most probable paths through intermediate states, we use a version of the A^* pathfinding algorithm (Hart et al. 1968). In its original form, A^* finds the single path of least cost between two vertices that does not revisit intermediate vertices on a weighted graph. The edge costs are also typically nonnegative and additive. For our purposes, each of the emphasized points requires generalizing.

The first generalization is well known: finding the k shortest paths, rather than the single shortest (Eppstein 1998). Although k is the standard symbol for finding multiple shortest paths, we will for the remainder of this section use N to avoid confusion with a state labeled k.

The second generalization requires a notion of "legality." Consider a graph that represents a geographic map, and we are looking for routes from A to B. We would not be interested in paths that revisit vertices as it would effectively be backtracking. We would therefore declare any paths that include revisiting to be illegal and not consider such paths as candidates. In general, at each iteration of the algorithm described below, we would determine whether each next increment in the path is legal according to our specific needs, then as appropriate either add it to or exclude it from the list of candidate paths.

In the class of cases under consideration here, a physical system may meander between intermediate states—even tracing loops in the graph—before reaching B. If we exclude paths that revisit intermediate states, we will miss the contribution of these meanderings to the total effective transition rate. Indeed, some paths with loops may contribute more than some direct paths. This leads to our statement of path legality: we allow any path that starts at A, ends at B, and does not have A or B as intermediates.

The final generalization addresses how we compute cost. In the graphs considered here, the edge weights are multiplicative probabilities. These may be converted to nonnegative additive weights by taking the negative logarithm:

$$\begin{eqnarray}\begin{array}{rcl}{w}_{{ij}} & = & -\mathrm{log}({b}_{{ij}})\\ \to {w}_{{ij}}+{w}_{{jk}}+... & = & -\mathrm{log}({b}_{{ij}}\cdot {b}_{{jk}}\cdot ...).\end{array}\end{eqnarray} \tag{ A5 }$$

But this is an unnecessary step if we change the optimization criterion from "least cost" to "highest probability." We may just as easily compute the cumulative effect of each incremental addition to a path; we simply multiply instead of add, and we prefer paths with higher cost (probability) instead of lower.

A* requires a heuristic function h(v) that estimates the cost from each vertex v to the goal (here, the goal is B). Candidate paths are then selected according to the criterion of optimizing the computed actual cost to get to the current vertex combined with the estimated cost to get the rest of the way to the goal. This heuristic must be "admissible," meaning it never overestimates the actual cost. With our multiplicative weights, our choice of heuristic must therefore never underestimate the probability. For simplicity, we take h(v) = 1∀v. In principle, we could achieve better performance with a more intelligent choice of h(v), but we will not worry about that here.

Finally, we describe our algorithm in detail. To find the N most probable paths from A to B, follow these steps.

• 1.  
  Start an empty list of candidate paths.
• 2.  
  Add to this list all possible paths of length 1 (one increment) that start at A. Using the graph in Figure 1 as an example, the list will now consist of the paths A → i, A → j, and A → k.
• 3.  
  Select the highest probability path from the candidate list and delete it from the list.
• 4.  
  If the selected path does not end at B, add to the candidate list all legal paths that are one increment farther along and return to step 3. If, for example, the most probable candidate path is A → i, delete it from the list and append to the list the candidates A → i → B, A → i → j, and A → i → k.
• 5.  
  If the selected path ends at B, record it as the nth most probable path, where n is the number of most probable paths found so far. If N most probable paths have been found, you are done. Otherwise, return to step 3.

It is worth mentioning that it is in principle possible to get caught in a loop of transitions where probability decays slowly with each iteration, and a path may traverse this loop many times before it fails to be selected in step 3. This can be avoided if in step 4 we augment the legality condition to say that a path may not include more than N − 1 total loops. Because every path increment at best does not increase a path's probability, we may be assured that a path P that visits a state n times will be at least as probable as an otherwise identical path that includes a loop to visit the state n + 1 times. Therefore, there will be at least n − 1 paths at least as probable as P, and we may safely reject all paths with more than N − 1 loops.

In a general weighted digraph, the optimal path from A to B is not necessarily the optimal path from B to A. But when transitions are mediated by a thermal bath, the most probable path between two states—and hence the greatest contributor to the transition rate—is the same in both directions. In fact, all paths maintain their relative probability when traversed in either direction. For example, the second most probable path from A to B is also the second most probable path from B to A, and so on. Figure 21 illustrates this for the five most probable paths through ²⁶Al at temperature T = 500 keV. This is an unrealistically high temperature for ²⁶Al production, but it nicely illustrates the symmetry. The states are labeled in increasing order of energy starting with ground = 1 and isomer = 2. In the figure, the most probable path (rank = 1) from ground to the isomer is 1 → 36 → 47 → 21 → 8 → 4 → 2, while from the isomer to ground it is 2 → 4 → 8 → 21 → 47 → 36 → 1. All other ranks exhibit the same symmetry. We detail the inputs to our calculations in Section 3.1.

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A stochastic process for which all paths obey the symmetry described above is known as a reversible Markov chain (RMC), and there are at least three ways to show that nuclear energy levels with thermally mediated transitions comprise such.

First, and most simply, a system that meets the detailed balance condition is an RMC. The detailed balance condition is that there exists a configuration of occupation probabilities for which the transition rates between every pair of states is the same forward and backward. That is, if we denote the occupation probability of state i as n_i, there must be some set of occupations such that n_iλ_ij = n_jλ_ji for all i and j (Durrett 1999). The thermal-equilibrium distribution ${n}_{i}\propto (2{J}_{i}+1){e}^{-{E}_{i}/T}$ satisfies this equation (see Equation (5)), guaranteeing reversibility. Importantly, the system need not be in this configuration; the configuration must only exist.

Second, the specific symmetry here is directly provable, which we will do presently. Third, by a trivial extension of the direct proof, we can show that thermally mediated transitions satisfy Kolmogorov's criterion, which states that a Markov chain is reversible if for every closed loop, the probability of following the loop is the same in both directions (Kelly 2011).

To directly prove path reversal symmetry, we begin with the probability p_Ai...jB that a transition out of state A follows the path A → i → ... → j → B:

$$\begin{eqnarray}&&{p}_{{Ai}...{jB}}={b}_{{Ai}}\cdot {b}_{i...}\cdot ...\cdot {b}_{...j}\cdot {b}_{{jB}}.\end{eqnarray} \tag{ A6 }$$

Rewrite the b's terms of the λ's:

$$\begin{eqnarray}&&{p}_{{Ai}...{jB}}=\displaystyle \frac{{\lambda }_{{Ai}}}{{\lambda }_{A}}\cdot \displaystyle \frac{{\lambda }_{i...}}{{\lambda }_{i}}\cdot ...\cdot \displaystyle \frac{{\lambda }_{...j}}{{\lambda }_{...}}\cdot \displaystyle \frac{{\lambda }_{{jB}}}{{\lambda }_{j}}.\end{eqnarray} \tag{ A7 }$$

We use Equation (5) to reverse the indices of the numerators, using g_s ≡ 2J_s + 1 for brevity:

$$\begin{eqnarray}\begin{array}{rcl}{p}_{{Ai}...{jB}} & = & \left(\displaystyle \frac{{g}_{i}}{{g}_{A}}{e}^{\tfrac{{E}_{A}-{E}_{i}}{T}}\displaystyle \frac{{\lambda }_{{iA}}}{{\lambda }_{A}}\right)\left(\displaystyle \frac{{g}_{...}}{{g}_{i}}{e}^{\tfrac{{E}_{i}-{E}_{...}}{T}}\displaystyle \frac{{\lambda }_{...i}}{{\lambda }_{i}}\right)\\ & & \times ...\times \left(\displaystyle \frac{{g}_{j}}{{g}_{...}}{e}^{\tfrac{{E}_{...}-{E}_{j}}{T}}\displaystyle \frac{{\lambda }_{j...}}{{\lambda }_{...}}\right)\left(\displaystyle \frac{{g}_{B}}{{g}_{j}}{e}^{\tfrac{{E}_{j}-{E}_{B}}{T}}\displaystyle \frac{{\lambda }_{{Bj}}}{{\lambda }_{j}}\right).\end{array}\end{eqnarray} \tag{ A8 }$$

The exponents contain a collapsing sum, and the g_s≠A,B all cancel, yielding

$$\begin{eqnarray}&&{p}_{{Ai}...{jB}}=\displaystyle \frac{{g}_{B}}{{g}_{A}}{e}^{\tfrac{{E}_{A}-{E}_{B}}{T}}\displaystyle \frac{{\lambda }_{{iA}}}{{\lambda }_{A}}\cdot \displaystyle \frac{{\lambda }_{...i}}{{\lambda }_{i}}\cdot ...\cdot \displaystyle \frac{{\lambda }_{j...}}{{\lambda }_{...}}\cdot \displaystyle \frac{{\lambda }_{{Bj}}}{{\lambda }_{j}}.\end{eqnarray} \tag{ A9 }$$

We now multiply by ${\lambda }_{B}/{\lambda }_{B}$ and shift all of the numerators to the right:

$$\begin{eqnarray}&&{p}_{{Ai}...{jB}}=\displaystyle \frac{{g}_{B}}{{g}_{A}}{e}^{\tfrac{{E}_{A}-{E}_{B}}{T}}\times \displaystyle \frac{1}{{\lambda }_{A}}\cdot \displaystyle \frac{{\lambda }_{{iA}}}{{\lambda }_{i}}\cdot \displaystyle \frac{{\lambda }_{...i}}{{\lambda }_{...}}\cdot ...\cdot \displaystyle \frac{{\lambda }_{j...}}{{\lambda }_{j}}\cdot \displaystyle \frac{{\lambda }_{{Bj}}}{{\lambda }_{B}}\cdot {\lambda }_{B}.\end{eqnarray} \tag{ A10 }$$

Rewrite the λ's in terms of the b's:

$$\begin{eqnarray}&&{p}_{{Ai}...{jB}}=\displaystyle \frac{{g}_{B}}{{g}_{A}}{e}^{\tfrac{{E}_{A}-{E}_{B}}{T}}\displaystyle \frac{{\lambda }_{B}}{{\lambda }_{A}}{b}_{{iA}}\cdot {b}_{...i}\cdot ...\cdot {b}_{j...}\cdot {b}_{{Bj}}.\end{eqnarray} \tag{ A11 }$$

Finally, we recognize that this product of b's is the reverse path probability:

$$\begin{eqnarray}&&{p}_{{Ai}...{jB}}=\displaystyle \frac{{g}_{B}}{{g}_{A}}{e}^{\tfrac{{E}_{A}-{E}_{B}}{T}}\displaystyle \frac{{\lambda }_{B}}{{\lambda }_{A}}{p}_{{Bj}...{iA}}.\end{eqnarray} \tag{ A12 }$$

The factor relating the forward and reverse path probabilities is path independent, and depends only on temperature and the properties of the endpoint states. Therefore, the fractional contribution of each path to transition rates from endpoint to endpoint is the same forward and backward; if a path contributes 10% of the rate from ground to isomer, then it contributes 10% of the rate from isomer to ground. Reversal factors that are not equal to unity imply that one endpoint state or the other is more likely to fail to transition, returning to its starting point.

To demonstrate satisfaction of Kolmogorov's criterion, we need only set A = B in the direct proof. Because the proof does not rely on any special properties of A and B (e.g., longevity), A and B may be taken to be arbitrary and equal. Then, per Equation (A12), the reverse probability is identical to the forward probability for all closed loops.

As a final point, the rate to follow a particular path starting at endpoint E is the product of the rate to leave E and the probability of following that path. That is, λ_path = λ_E p_path. Note from Equation (A12) that the rates to follow each path in the forward and reverse directions obey the same relationship as the direct transition rates in Equation (5). Summing over all paths from A to B leads us to conclude that effective transition rates between endpoint states exhibit the thermal relationship

$$\begin{eqnarray}&&{{\rm{\Lambda }}}_{{AB}}=\displaystyle \frac{{g}_{B}}{{g}_{A}}{e}^{\tfrac{{E}_{A}-{E}_{B}}{T}}{{\rm{\Lambda }}}_{{BA}}.\end{eqnarray} \tag{ A13 }$$

This result is not surprising given an intuition of detailed balance, but it is nevertheless useful to have it made explicit and to observe that it does not require the nuclear levels to be in a thermal-equilibrium distribution. Indeed, relying on methods that do not include path reversal symmetry can lead to erroneous results. Reifarth et al. (2018) made the seemingly reasonable assumption that all dominant paths consist of one up transition from a long-lived state followed by a cascade of down transitions through intermediate states; this is justified by the fact that down transitions are much faster than up transitions and therefore dominate the deexcitation of intermediate states. However, from the present analysis and Figure 4, we see that at temperature T = 35 keV, this misses the single greatest contributing path (1 → 3 → 4 → 2) from the ground state to the isomer in ²⁶Al; the consequence is a drastic underestimate of the effective transition rate in an important temperature range.