Alpha decay of thermally excited nuclei

One of the prominent decay modes of heavy nuclei which are produced in astrophysical environments at temperatures of the order of $10^9$ K is the $\alpha$ ($^4$He) decay. Thermally enhanced $\alpha$ decay rates are evaluated within the standard scheme of a tunneling decay where the $\alpha$ particle tunnels through the potential barrier formed by its interaction with the daughter nucleus. Following the observation that there exist several excited levels with the possibility of an $\alpha$ decay when the daughter nucleus is at a shell closure, we focus in particular on decays producing daughter nuclei with the neutron number, N = 126. Within a statistical approach we find that the half-lives, $t_{1/2}(T)$, for temperatures ranging from $T$ = 0 to 2.4 GK can decrease by 1 - 2 orders of magnitude with the exception of the decay of $^{212}$Po which decays to the doubly magic daughter $^{208}$Pb, where $t_{1/2}(T)$ decreases by 5 orders of magnitude. The effect of these thermally enhanced $\alpha$ decays on the $r$-process nucleosynthesis can be significant in view of the mass build up at the waiting point nuclei with closed neutron shells.

neutronized atmosphere above the proto-neutron star in a Type II supernova, (b) neutronrich jets from supernovae or neutron star mergers, (c) inhomogeneous Big Bang, (d) He/C zones in Type II supernovae, (e) red giant He flash, (f) spallation neutrons in He zone (g) neutrino driven wind from freshly born neutron stars and (h) outflows from black hole accretion discs originated in compact object mergers or collapsars (for recent reviews of the possible r-process sites see Cowan et al [9] and Côté et al [10]). The abundance of elements is found through a network [11,12] of coupled differential equations involving nuclear reaction rates at elevated temperatures. Theory and models play an important role in determining the latter for neutron-rich nuclei which cannot be measured in terrestrial laboratories.
The r-process nucleosynthesis path is along highly unstable, exotic, and neutron-rich nuclei that in principle does not involve alpha emitters. However, once heavy neutron rich nuclei in the region with Z > 82 are formed, and with the depletion of further neutron captures (i.e. after r-process freeze out), those nuclei decay by different modes (e.g. beta, alpha, fission). Some decay modes would compete. This process can even lead to the formation of nuclei in the actinide region. The nuclei studied in this work are part of the mass region (A > 208) where several alpha emitters are found [13,14]. Thus, it is not only the photo-dissociation and neutron capture cross sections but also fission (spontaneous and induced) and the decay rates which are important for the abundance evolution. The explosive conditions in supernovae and neutron star mergers [15][16][17] leading to considerably high temperatures could result in nuclei existing in excited states. Though the possible influence of these nuclear thermal excitations is taken into account in the production reactions as well as in their reverse reactions, with libraries publicly available for the scientific community (e.g. [18][19][20]), the same is not true in the case of α decay. These decay rates, entering as an input to the network calculations, are taken to be the ground state (or terrestrial) half-lives [11]. However, one must note that for high ambient temperatures, the population factor for the excited energy levels of a nucleus is large. These are thermal excitations and one must take into account the possibility of α decay of thermally excited nuclei. This is in particular quite important for the r-process nucleosynthesis where the closed neutron shells present waiting points due to the fact that it takes a long time for the successive β decays, which are slow at the shell closures, to occur and allow progression through higher N nuclei.
In section IV, we will see that the α decays of parent nuclei producing daughter nuclei at the shell closures display a stronger temperature dependence with increased decay rates at higher temperatures.
Apart from the paper of Perrone and Clayton [21], published in 1970, there is indeed no estimate of the possible effects of temperature on the α decay half-lives. However, given the fact that about 50 years ago, the data on excited levels of nuclei was scarce, calculations were performed assuming a continuum of states described by the available density of states.
The latter assumption as we will see leads to a very large overestimate of the enhancement in the decay rate due to temperature.
In the present work, we investigate the temperature dependence of the α decay rates relevant for the r-process nucleosynthesis. The calculations are performed within two different approaches: (i) a statistical model which makes use of the experimentally measured excited levels and an empirical decay law (fitted to data) in the absence of available data on the halflives and (ii) a theoretical model which treats the α decay as a semiclassical tunneling of the α particle through the barrier created by the interaction of the α and the daughter nucleus which exist inside the parent in the form of a preformed cluster. The latter calculation is performed using a density dependent folding model which has been reasonably successful in reproducing the measured α decay half-lives [22,23]. The formalism is presented in sections II, and III. In section IV we connect to shell closures, and in section V we discuss the results.
Finally, we summarize our findings in section VI.

II. ALPHA DECAY FORMALISM
One of the most successful achievements of the quantum theory is the explanation of the α-decay of radioactive nuclei as a tunneling problem. This approach was developed independently by Gamow [24] and by Gurney and Condon [25] in the late twenties. Though α-decay has been studied since then within different quantum mechanical approaches [26], semiclassical approaches based on the tunneling of an α particle through the potential barrier created by its interaction with the daughter nucleus produced in the decay are some of the most popular and widely used methods for calculating half-lives. The interaction potential between the α ( 4 He nucleus) and the daughter nucleus, and the Q-value, which is usually taken to be the energy of the tunneling α, play the main role in determining the tunneling probability and hence the half-life. Using the JWKB approximation [27], different semiclassical approaches lead to the same expression for the α-decay width [22] with the so-called wave number κ(r) = 2µ 2 |V (r) − Q| and µ the reduced mass of the daughter-α system. The classical turning points r 1 , r 2 and r 3 are obtained by solving the equation V (r) = Q where Q is the energy of the tunneling α-particle. The factor in front of the exponential arises due to the normalization of the bound state wave function in the region between r 1 and r 2 . The exponential factor is the penetration probability. The α-decay half-life of an isotope is evaluated as Since the tunneling decay assumes the existence of a preformed cluster of the 4 He and daughter nucleus inside the decaying parent nucleus, one must include a preformation probability P α in the expression for half-life. This factor, in principle, can be expressed as an overlap between the wave functions of the parent nucleus and the decaying-state wave function describing the α-particle coupled to the daughter nucleus. Such a microscopic undertaking is still considered a difficult task and a phenomenological way to determine P α is simply taking the ratio of the theoretical and experimental half-lives, such that, P α = t theory 1/2 /t exp 1/2 , where, t theory 1/2 is evaluated using Γ from Eq. (1) but with P α = 1.
The total potential between the α and the daughter nucleus is typically written as a function of the distance between their centers of mass as, where V n (r) and V C (r) are the nuclear and Coulomb potentials, respectively. The last term in equation 3 represents the Langer modified centrifugal potential [28] which must be used while using the JWKB approximation. Some of the calculations presented in this work will be performed within the density dependent double folding model (DFM) which is based on realistic nucleon-nucleon interactions and has been reasonably successful in reproducing the experimental half-lives. The details of this potential can be found in [23,29]. Here we describe it briefly. In the DFM, the nucleus-nucleus interaction is related to the NN interaction by folding an effective NN interaction over the density distribution of the two nuclei. The folded nuclear potential is written as where ρ i (i = d, α) are the densities of the alpha and the daughter nucleus in a decay, and v N (|s|) is the nucleon-nucleon (NN) interaction (see [29] for the figure with details). The matter density distribution of the heavy daughter is calculated as where ρ 0 is obtained by normalizing ρ(r) to the mass number, ρ(r) dr = A, and the constants are given as R = 1.07A 1/3 fm and a = 0.54 fm. The alpha or 4 He density distribution is given using a standard Gaussian form [30], namely, We use the popular choice of the effective NN interaction which is based on the M3Y- where |s| = |r + r 2 − r 1 | is the distance between a nucleon in the daughter nucleus and a nucleon in the alpha. The above NN interaction consists of a short-ranged repulsive part and a long-ranged attractive one, in addition to the zero-range contribution J 00 δ(s) with J 00 = −276(1 − 0.005 E/A c ). The latter is the so-called knock-on exchange term which takes into account the antisymmetrization of identical nucleons in the alpha and the daughter nucleus. It represents a kind of nonlocality in the DFM potential and in order to avoid double counting, is usually not included in the calculation if one uses nonlocal nuclear potentials [29]. The strength of the nuclear potential, λ, is deduced by requiring the Bohr-Sommerfeld quantization condition to be satisfied [23]. The Coulomb potential, V C (r), is obtained in a similar way with the matter densities of the alpha and the daughter replaced by their charge densities (which have the same form as above but are normalized to the number of protons).
The angular momentum, l, carried by the alpha particle must satisfy the following spinparity selection rules, where (J p , π p ) and (J d , π d ) are the (spin, parity) of the parent and daughter nuclei, respectively.
In Table I we present the half-lives for some nuclei using the DFM. We examine transitions for which the alpha particle has the minimum angular momentum value, l min , satisfying equations (8). For the decays considered in Table I, l min = 0. The experimental half-lives and the corresponding preformation factors are also listed in Table I  The double folding model calculations can in principle be improved with the inclusion of deformation and nonlocalities in the interaction potential [23,29]. However, the objective of the present work is to perform a comparative study of approaches for half-lives measured on earth and in a hot astrophysical environment and hence it suffices to perform calculations within a model which can reproduce alpha decay half-lives reasonably well. Indeed, we shall also use an empirical formula (a universal decay law (UDL) for α and cluster decay, obtained from fits to extensive data) for the half-lives calculated within the statistical approach (to be explained in the next section) since (i) the UDL gives the right order of magnitude estimate of half-lives and (ii) it would be quite a tedious undertaking to evaluate the half-lives of several excited states within the double folding model (DFM) without any significant advantage.
Such universal decay laws are usually obtained by starting with an analytical expression [47] which is based on the assumption of a rectangular well for the nuclear potential and a point-like Coulomb potential between the decay products of the radioactive nucleus. The constants appearing in such an expression are then assumed to be free parameters and fitted to an extensive set of data. The latter compensates for the simplistic assumptions made in the derivation of the empirical formula and provides a useful expression depending on the number of nucleons and the Q-value of the decay. We use the following UDL obtained in [32,33]. In the next section we shall also describe the effective Q-value approach to evaluate the temperature dependent half-lives using the DFM and the UDL. Comparing it with the statistical approach gives us an idea of the usefulness of this approach in the context of α decay half-lives in an astrophysical environment. We shall also discuss one of the earliest attempts to evaluate the thermally enhanced α-decay rates in connection with the s-process nucleosynthesis [21]. The authors in [21] predicted a decrease in the half-lives by about 20 -60 orders of magnitude depending on the nucleus for a temperature around 2 GK. We do not find such spectacular effects of temperature in our calculations. The reason for this difference will become evident in the next sections.

III. TEMPERATURE DEPENDENT HALF-LIVES
Nucleosynthesis in the later stages of stellar evolution, especially through the r-process is considered to take place at considerably high temperatures of the order of 10 9 K. The calculation of the abundance of heavy elements depends on a precise determination of the nuclear reaction rates of the processes which produce the elements as well as the processes which destroy the newly formed nuclei. Though the neutron capture cross sections and their reverse reaction rates at elevated temperatures are carefully taken into account, the network codes usually rely on the laboratory values of the half-lives of α decays from the ground states of nuclei. Perrone and Clayton [21] investigated the effect of thermally excited states in the alpha decay of some nuclei and its application in the s-process nucleosynthesis.
However, such effects are not included in the r-process simulations.
In what follows, we present a statistical calculation of thermally enhanced α-decay rates that includes experimentally observed excited levels for some nuclei. We also propose a model that uses an effective Q-value approach and which makes use of an average excitation energy at a given temperature. Finally, we discuss the approach of Perrone and Clayton briefly for completeness.

A. Statistical calculation
The temperature-dependent half-life, t 1/2 (T ) = ln 2/λ(T ), can be evaluated within the standard statistical approach [34] by defining the temperature dependent decay constant as follows: Here the sums over i and j are over the parent and daughter states respectively. Thus, λ ij is the decay constant for the decay of the i th level in the parent to the j th level in the daughter such that The population probability, p i , is given with a Boltzmann factor as [35] where J i and E i are the spin and the excitation energy of the state i, respectively. Inserting (12) in (10) λ(T ) = ln(2) where (BR) ij is the branching fraction for the decay from the i th level of the parent nucleus to the j th level in the daughter nucleus. The detailed decay schemes and the percentage decay to a particular channel, i.e., I = [λ ij /λ tot ] * 100% can be found at the web-site in [36].
The branching fraction, (BR) ij = λ ij /λ tot , can thus be obtained from the data tables. To evaluate the temperature dependent half-life in the statistical approach, we shall use Eq.
(13) with the input half-lives, t i 1/2 and (BR) ij taken from experiment. If the experimental half-life of a level is not known, it is calculated using the UDL at an effective Q-value given by Q + E i , where E i is the energy of the excited level. In such cases, even if the experimental branching ratio is known, it is not used but taken to be 100% since the UDL per definition is formulated only for the alpha decay channel.

B. Effective Q-value model
Alpha decay half-lives are very sensitive to the penetrability factor which at the same time means that they are very sensitive to the Q-values too. For a tunneling decay of an α particle taking place in a very hot surrounding, one can model the effect of temperature by an increase in the energy, Q, with which the α tunnels through the Coulomb barrier. A higher Q-value would clearly reduce the area under the integral in the exponential term in Eq. (1) and hence lead to an increase in the decay probability (and therefore a decrease in the half-life). This increase in the Q-value can be modelled by adding an average excitation energy of the nucleus at a given temperature. Such an effective Q-value of an α-decay can be expressed as where¯ (A, Z, T ) is given by the standard definition of the average excitation energy [37] in statistical physics as,¯ with the canonical partition function Z given by where, β = 1/(k B T ), g i = 2J i + 1 and J i is the spin of the i th level and D(E) is the nuclear level density for which we choose the following form [38]: The level density parameter, a, is taken to be A/9 where A is the mass number of the parent nucleus. If we consider the discrete levels as well as the continuum, the average excitation energy, using the above equations can be expressed as The temperature, T , appearing in the above expression (through β = 1/(k B T )), is in principle the nuclear temperature. Modification of nuclear properties at finite temperatures is relevant both for applications in astrophysics [39][40][41] and for models of finite nuclei and nuclear matter at high excitation energy [42,43].  TABLE II. Average excitation energies evaluated using equation (18) in conjunction with (17).
The unique attempt in literature for the evaluation of the thermally enhanced α decay half-lives of nuclei in stellar environments can be found in [21] by Perrone and Clayton. They evaluated the α decay rates of several nuclei formed in the s-process nucleosynthesis to find that the half-lives are greatly enhanced if one considered the contributions from thermally excited nuclei at elevated temperatures. The temperature-dependent half-life in [21] was given by where t 1/2 (Z, A, E, J), the temperature-independent half-life for the decay of the parent nucleus to the daughter particle ground state is weighed by the nuclear density of states D(Z, A, E, J) and the occupation probability F (E, J, T ), which for an excited nucleus with energy E i and spin J i is given by with J 0 the spin of the ground state and k B the Boltzmann constant. The last approximation was justified by mentioning that the nuclear ground state dominates the sum over states for temperatures below 2 GK. The nuclear level density used was taken from [45]. The It is important to note that although the nuclear energy levels are discrete, the half-life t P C 1/2 (Z, A, T ) defined by (19) in [21] treats the nucleus as having a continuum of excited states with the density of states given by D(Z, A, E, J). The authors mention the need for explicitly taking the discrete levels into account if one wished to use the calculations in context with astrophysics. However, given the meagre data available in 1970, the authors found their approach appropriate for an initial survey of the problem. The present work takes into account this omission made by Perrone and Clayton due to the lack of data and finds that even if the decrease in half-lives is not as spectacular as that in [21], it is surely significant and possibly relevant for nucleosynthesis calculations.

IV. EXCITED NUCLEAR LEVELS, Q VALUES AND SHELL CLOSURES
A naive expectation for the decay of thermally excited nuclei would be that for a nucleus which decays by α decay in its ground state, there must exist some excited levels which decay by emitting an α too. However, experimental results show that this is, in general, not true. A careful examination of the nuclear data tables reveals that the α decay occurs in heavy nuclei mostly in the ground state. An excited parent nucleus often decays by emitting a photon (γ-decay). In fact, the excited nucleus undergoes several successive γ-decays before reaching its ground state. If this were true for all nuclei decaying by α decay, an undertaking as in the present work would not make much sense. However, based on a conjecture in an earlier work [46], we did find exceptions.
In [46], Ac had several experimentally observed α decays from excited levels. These nuclei are formed in the r-process nucleosynthesis and will be studied in the present work.
The above phenomenon of a larger number of excited levels decaying by α decay should in principle happen at the other N as well as Z shell closures too. Inspecting the parent nuclei near N or Z = 84 we find that they do display some such excited states, but the effect is either not so pronounced or the data is scarce. In the range of the medium heavy nuclei, with daughters near the shell closure of Z = 50, there are hardly any nuclei decaying by α decay and near N = 50, none. We have an interesting case however at the lowest magic number of 2. 8 Be decays to two α's, i.e., 4 He nuclei and hence both the daughters in the decay have N as well as Z = 2. The number of excited levels which decay by α decay are 13 and one would expect a strong effect in the thermally enhanced decay rates. However, one does not observe a big change in the decay rate due to temperature since the spacing between levels is much larger than those in the heavy nuclei. For example, the first excited state in 8 Be is around 3 MeV and 212 Po has about 50 excited levels between 0 and 3 MeV.
The high density of excited states in heavy nuclei is expected and was explained a long time ago by Bethe [48].  From a potential barrier perspective, their probability of transmission is too small due to the fact that the Q-value is small. One could consider the possibility of a γ delayed α decay of a nucleus due to the thermally excited levels. However, the γ decay half-lives of the excited levels are much smaller than the ground state α decay half-lives and the delay would be negligible. In the case of the statistical approach involving a sum over all excited states, we perform the calculation using the information provided in Table III. In the absence of data on the half-lives, we use a universal decay law (UDL) at a shifted Q-value, namely, where E i is the energy of the i th excited state of the parent nucleus. For the UDL calculation, we use the minimum allowed value of the orbital angular momentum quantum number l for each excited level decay and have listed it in Table III. Higher values of l could be included, however, the nature of the UDL is such that even if larger values of l were taken into account, the final result would not change much. Note that the difference between the b and d parameters of the second and third term in Eq. (9), respectively, is of two orders of magnitude (the third term being the smallest). The ratio of these two terms grows roughly linearly as a function of angular momentum 1/10 × l(l + 1) ∼ l/10, therefore only very high values of angular momentum would make the third term comparable to the second.
Those large momenta would happen for large values of J p and in such cases l min is also large providing a good approximation to the half-life. For the half-life using the UDL, we do not include any preformation factor, i.e., we assume it to be unity. The factor is usually calculated phenomenologically from the ratio of the theoretical and experimental half-lives.
Performing such a calculation for each individual excited level is a formidable task and out of the scope of the present work. Besides, since the contribution of this factor will not vary exponentially, it appeared to us a reasonable assumption to take it to be a constant. We are interested in the relative decrease in half-lives at elevated temperatures as compared to the terrestrial ones and expect that the error introduced due to this omission is not large.
We provide two sets of results for the half-lives of nuclei with neutron number N = 128 evaluated using two prescriptions. In the first set,  TABLE III. Energy level, spin, parity, branching ratio and measured half-lives of levels which decay by alpha emission. If the experimental half-life of a level is not known, it is calculated using the UDL at an effective Q-value given by Q + E i , where E i is the energy of the excited level. In such cases, even if the experimental branching ratio is known, it is not used but taken to be 100% since the UDL per definition is formulated only for the alpha decay channel. l min is the minimum value of the orbital angular momentum quantum number, allowed by the selection rules.
(9). We take Q → Q ef f , with the Q ef f approach of section III B, in which the extra energy is taken as in Eq. (18). In our second set, Table V, we use the statistical method where the decay constant and, in turn the half-life, are given by Eq. (13). We compare two approaches in Table V: one in which all available listed levels are included and another one in which only levels experimentally found to decay by alpha emission are used. For those levels (in both approaches) with unknown experimental half-lives, the UDL (Eq. (9)) is invoked to estimate t 1/2 of those levels. The results in Table IV ensure that the estimate obtained from Eq. (9) and used as an input in the statistical approach (for the experimentally unknown half-lives) is reasonable. In Table IV From Table IV, we see that the increase in temperature in general decreases the half-lives with the decrease being at the most an order of magnitude from T = 0 to 2.4 GK. Though the half-lives evaluated using the universal decay law are not exactly the same as those in the more realistic double folding model, the percentage decrease in both cases is roughly the same. The percentage decrease is calculated as This fact allows us with some reliability to use the UDL given by (9) in the statistical approach for the calculation of the missing half-lives in the available data for excited levels.
The temperature-dependent half-lives for several isotopes using the statistical approach are displayed in Table V. As the temperature increases the half-lives are reduced and the reduction is larger than found in the effective Q-value approach. The calculations are done using the experimentally listed half-lives. For these particular isotopes, several excited levels have been observed to decay by alpha decay, however, the half-lives of many of these excited states have not been measured. For the cases with no experimental information, we use the UDL to evaluate the half-lives to be used in (10   A small note on the comparison of the two approaches, namely, the effective Q-value approach and the statistical approach is in order here. With the aim of providing temperature dependent half-lives for nucleosynthesis applications, we began by formulating an approach where the effective Q-value would enter in the UDL with the advantage of avoiding the task of performing calculations of half-lives for many individual levels populated at very high temperatures, the use of a density level model; thus, making the proposed effective Q-value model feasible in a network calculation. However, this advantage comes at the expense of missing information about the variation in the half-lives of different levels as well as the population probability of the excited states. Performing an average over the excitation energies and calculating for just one effective Q-value is equivalent to considering decay from one excited level which occurs at an effective excitation energy. A more realistic description is provided in the statistical approach, which would be model-independent as long as the half-lives of the excited levels and the branching ratios for alpha decay are known. Our introduction of the UDL for the unknown half-lives gives a pathway to extend the calculations to a larger number of alpha emitters. In this work, we use the known UDL from literature and evaluate the half-life of an unknown level with energy E i , by replacing the Q-value in the UDL by Q + E i . However, formulating a UDL for excited levels using all available data on the alpha decay of excited parent and daughter nuclei is a task which we plan for the future. Finally, in passing we mention that there also exists the possibility that the system can transit from the excited state to the lower state and then decay by alpha. However, given the exponential nature of the population probability, namely, p i ∝ exp (−E i /kT ) (with E i being the energy of the excited state), at a given temperature, the likelihood of a higher level being populated and decaying from a lower level to which it transits will surely be smaller than the lower level itself getting populated and decaying by emitting an alpha. Furthermore, in the cases where experimental information exists, the branching fraction, BR ij , accounts for the effect.
Alpha decay plays a role, in competition with beta decay and fission, in powering and shaping the light curves of kilonovae [53][54][55]. It is customary in nucleosynthesis calculations to consider alpha emission only from the ground state, i.e at zero temperature. In view of the reduction of the α-decay half-lives in hot environments found above, it seems appropriate to replace these inputs by temperature dependent ones. Such a detailed calculation, though necessary is out of scope of the present work.

VI. SUMMARY
In this work we have explored the role of temperature in alpha emission from nuclear excited states. We used a statistical approach and proposed a model that can potentially be extended for several nuclei. We particularly focused on nuclei that can be produced in r-process, motivated by the impact that alpha decay has on the heating of light curves of kilonovae. Thermally enhanced alpha decay rates were calculated for nuclei with the neutron number N = 128 decaying to a daughter nucleus at the shell closure with N = 126. The latter choice was made due to the occurrence of more excited levels decaying by α emission as compared to other nuclei. The calculation performed within the statistical approach is in principle model independent. It requires the experimental input of the energies, spins and half-lives (as well as their branching fractions for α decay) of the excited levels. However, sometimes the experimental data are incomplete (e.g. even if it is known that an excited level decays via α decay, its half-life is not known). In such a case we supply the missing information by calculating the half-life using a universal decay law (UDL) for the half-life.
The latter introduces some uncertainty in the results, however, we do not expect the results to change drastically since the temperature dependence of the half-lives using the UDL is in good agreement with the predictions of the more elaborate double folding model for tunneling decay.
We found that temperatures of the order of GK can increase the half-lives of the nuclides studied here by at least a factor of ten. Particularly, for the case of 212 Po and depending on the model, the change can be by orders of magnitude.