Role of dynamic pairing correlations in fission dynamics

TL;DR

This paper investigates how dynamic pairing correlations influence fission dynamics, revealing that they significantly reduce collective inertias and are affected by Coulomb antipairing, thus impacting traditional fission models.

Contribution

It introduces a method to incorporate dynamic pairing correlations via particle number projected energy in fission calculations, highlighting their effect on potential energy surfaces and inertias.

Findings

Restricted variation after particle number projection has minimal impact on potential energy surface.

Collective inertias decrease substantially due to dynamic pairing correlations.

Coulomb antipairing mitigates the quenching effect on inertias.

Abstract

We study the role of dynamic pairing correlations in fission dynamics by considering intrinsic Hartree-Fock-Bogoliubov wave functions that are obtained by minimizing the particle number projected energy. For the restricted variational space, the set of self-consistent wave functions with different values of proton and neutron number particle fluctuations are considered. The particle number projected energy is used to define potential energy surface for fission whereas collective inertias are computed within the traditional formulas for the intrinsic states. The results show that the effect of the restricted variation after particle number projection in the potential energy surface is small while collective inertias substantially decrease. On the other hand, we show that this quenching is strongly mitigated when Coulomb antipairing is considered. In the light of these beyond mean-field calculations, the validity of traditional fission calculations is discussed.

Tables

Table 1. Table 1: On the left hand side, spontaneous fission half-lives (in seconds) computed with two different sets of collective inertias (ATDHFB and GCM) and for the four different sets of calculations for the nuclei considered. On the right hand side, the values of the fission barrier heights B I subscript 𝐵 𝐼 B_{I} , B I ​ I subscript 𝐵 𝐼 𝐼 B_{II} and fission isomer excitation energy E I ​ I subscript 𝐸 𝐼 𝐼 E_{II} (in MeV) are algo given.

		tSF
		ATDHFB	GCM	B${}_{\text{I}}$	E${}_{\text{II}}$	B${}_{\text{II}}$
		(s)	(s)	(MeV)	(MeV)	(MeV)
236U	HFBt	$3.0\times {10}^{43}$	$2.4\times {10}^{32}$	9.07	4.05	10.22
	HFBCep	$3.1\times {10}^{63}$	$1.2\times {10}^{46}$	9.82	4.88	10.97
	HFBRVAP	$8.3\times {10}^{41}$	$1.1\times {10}^{32}$	9.64	4.77	10.74
	PNP	$1.0\times {10}^{42}$	$1.4\times {10}^{32}$	9.74	4.69	10.72
240Pu	HFBt	$7.4\times {10}^{38}$	$7.5\times {10}^{29}$	10.23	4.39	10.20
	HFBCep	$2.0\times {10}^{54}$	$2.4\times {10}^{39}$	10.91	4.94	10.75
	HFBRVAP	$3.0\times {10}^{37}$	$9.5\times {10}^{28}$	10.74	4.74	10.57
	PNP	$2.8\times {10}^{37}$	$1.2\times {10}^{29}$	10.83	4.79	10.63
252Cf	HFBt	$2.3\times {10}^{22}$	$1.7\times {10}^{18}$	11.18	3.71	7.77
	HFBCep	$7.6\times {10}^{24}$	$2.9\times {10}^{18}$	11.60	3.45	6.86
	HFBRVAP	$7.8\times {10}^{19}$	$2.5\times {10}^{15}$	11.19	3.40	7.09
	PNP	$1.9\times {10}^{21}$	$6.2\times {10}^{16}$	11.22	3.71	7.49

Equations

$$E^{Z,N}(f_{\pi },f_{\nu })=\frac{⟨\Phi (f_{\pi },f_{\nu })|\hat{H}{P}^Z{P}^N|\Phi (f_{\pi },f_{\nu })⟩}{⟨\Phi (f_{\pi },f_{\nu })|\Phi (f_{\pi },f_{\nu })⟩},$$

Full text

I Introduction

Undoubtedly, pairing correlations represent a key ingredient in the description of the dynamics of the fission phenomenon experienced by heavy atomic nuclei Moretto and Babinet (1974); Schunck and Robledo (2016). For instance, the amount of pairing correlations has a strong impact on quantities such as spontaneous fission lifetimes Urin and Zaretsky (1966); Staszczak et al. (1985); Lazarev (1987); Staszczak et al. (1989); Łojewski and Staszczak (1999); Pomorski (2007); Mirea and Bobulescu (2010); Giuliani and Robledo (2013); Rodríguez-Guzmán and Robledo (2014); Giuliani et al. (2014); Zhao et al. (2016), the shape of the barriers separating the ground state from scission Stepień and Szymański (1968); Rutz et al. (1999); Samyn et al. (2005); Schunck et al. (2014); Abusara et al. (2010); Karatzikos et al. (2010); Sadhukhan et al. (2014) and fission fragments distributions Goutte et al. (2005); Sadhukhan et al. (2016); Tao et al. (2017); Matheson et al. (2018). At the mean-field level, pairing is traditionally described using the Hartree-Fock-Bogoliubv (HFB) theory, which is a reasonable approximate scheme when pairing correlations are strong Ring and Schuck (1980). In nuclear physics, however, the pairing strength is not strong enough and, as a consequence, many mean field configurations show little or no pairing correlations Dean and Hjorth-Jensen (2003); Brink and Broglia (2005). In this case the mean-field description of the nucleus breaks down, and the inclusion of dynamic pairing correlations stemming from beyond mean-field effects becomes necessary. The evolution of the nucleus through the different shapes involved in fission affects the level density around the Fermi energy, with a large impact on pairing correlations. This effect is reflected by the intricate behavior shown along the fission path, including many regions of very weak static pairing which points out the possible crucial role of dynamic pairing correlations in the studies of fission.

In order to account for such effects, beyond mean field calculations involving the restoration of the particle number of the nuclear wave function are required. Unfortunately the computational cost of beyond mean field calculations limited so far their application to fission studies, keeping the impact of dynamic pairing correlations unexplored. Moreover, to properly address the role of dynamic pairing correlation one should account for all those effects that may mitigate the effective pairing strength and that are usually neglected for the sake of computational time, like for instance Coulomb antiparing Anguiano et al. (2001a), which is the name given to the destructive effect of the repulsive Coulomb interaction in proton’s pairing correlations. If proton and neutron pairing strengths are independently adjusted to experimental data in the region of interest Bertsch et al. (2009), Coulomb anti-pairing is taken into account in an effective way by the fitted pairing strengths. Conversely, in forces like Gogny Dechargé and Gogny (1980) the neutron pairing strength is fitted to experimental data (for instance in the tin isotopic chain) and the proton pairing strength comes from isospin invariance. In those cases, Coulomb anti-pairing must be explicitly taken into account to avoid the self-energy problem and the breaking of the Pauli principle in particle number projected calculations. The Coulomb anti-pairing effect can reduce proton’s pairing gap by a $20-30$% Lesinski et al. (2009); Nakada and Yamagami (2011), with a strong impact on observables such as moments of inertia Anguiano et al. (2001a); Afanasjev (2014), but their effect is usually neglected due to the enormous computational cost associated to the evaluation of Coulomb’s pairing field Anguiano et al. (2001a).

In the light of this discussion, it is possible to conclude that the inclusion of dynamic pairing will have a twofold effect: On the one hand, collective inertias driving fission dynamics, with their inverse dependence on the square of the pairing gap Brack et al. (1972); Moretto and Babinet (1974); Bertsch and Flocard (1991); Giuliani et al. (2014), are expected to increase when the Coulomb anti-pairing effect is considered, increasing the collective action and leading to longer fission lifetimes $t_{\mathrm{SF}}$. On the other hand, dynamic pairing correlations are expected to increase the pairing gap reducing thereby the collective inertias. The outcome of these competing effects is uncertain and it is the purpose of this paper to clarify this situation and establish a step forward in the study of beyond mean-field effects in fission calculations.

In previous studies, angular momentum projection Bender et al. (2004) has been used to compute fission barrier heights. However, the results are almost indistinguishable from the ones obtained with the rotational correction Schunck and Robledo (2016); Rodríguez-Guzmán et al. (2000). Parity projection has also been considered in the reflection asymmetric section of the fission path Hao et al. (2012); Samyn et al. (2005) with little or no impact at all. Finally, the impact of particle number projection on fission barrier heights has been considered in Samyn et al. (2005). A change of at most $\pm$ 0.5 MeV is obtained in all the cases.

In this paper we are considering the contribution to dynamic pairing correlation coming from a restricted variation after projection for particle number projection. The evolution of the pairing properties of the nucleus as it evolves towards fission, will be studied as a function of the axial quadrupole moment $q=\langle Q_{20}\rangle$. We will analyze the impact of dynamic pairing correlations in the potential energy surface, computed with the particle number projected wave function $|\Psi^{N}(q)\rangle=\hat{P}^{N}|\varphi(q)\rangle$, and in the collective inertia computed with the intrinsic state $|\varphi(q)\rangle$ associated to the former.

II Methodology

Dynamic pairing correlations require a beyond mean field framework involving the restoration of the particle quantum number of the nuclear wave function. In order to gain more correlations, the intrinsic mean field wave function has to be determined by minimizing the projected energy in the so called variation after projection (VAP) method. In this paper we use the restricted variation after projection (RVAP) Rodríguez et al. (2005) particle number projection (PNP) method Anguiano et al. (2001b). The RVAP-PNP has been shown to be superior to other alternatives like the Lipkin-Nogami method Rodríguez et al. (2005) commonly used in the literature. In the RVAP-PNP method the variational subspace is formed by projecting onto good particle number (protons and neutrons separately) intrinsic wave functions obtained from a HFB calculation constraining on the particle number fluctuation for protons and neutrons $|\Phi(\langle\Delta N^{2}\rangle_{\pi},\langle\Delta N^{2}\rangle_{\nu})\rangle$ 111The constraint on $\langle\Delta N^{2}\rangle$ involves the two-body operator $\Delta\hat{N}^{2}$. The implementation of this constraint is straightforward when using the gradient method to solve the HFB equation Robledo and Bertsch (2011).. Henceforth, we will denote by $f_{\pi}$ and $f_{n}$ the value of the particle number fluctuation for protons and neutrons, respectively. The RVAP intrinsic state $|\Phi(f_{\pi},f_{\nu})\rangle$ corresponds to the minimum of the projected energy

[TABLE]

as a function of the $f_{\pi}$ and $f_{\nu}$ variables. The minimum of the two dimensional function $E^{Z,\,N}(f_{\pi},f_{\nu})$ is determined by a simple gradient method in two dimensions. The potential energy surface for fission is obtained by introducing an additional constraint on the quadrupole moment $Q_{20}$ of the axially symmetric intrinsic state and is given by the projected energy of the RVAP for each $Q_{20}$ value. We could also introduce easily additional constrains like the octupole moment or the necking operator to form multidimensional potential energy surfaces (PES) so popular in fission studies, but this is not the purpose of the present work. An example of both the HFB and PNP potential energy surfaces obtained as a function of $\langle\Delta N^{2}\rangle$ and $\langle\Delta Z^{2}\rangle$ is given in Fig. 1 where those energies, computed with the Gogny D1M parametrization Goriely et al. (2009), are plotted for the nucleus 240Pu and three different values of the quadrupole moment (see caption for details). The chosen quadrupole moments correspond to the ground state, first fission barrier and fission isomer. Both the HFB and PNP energies show a parabolic behaviour as a function of $(f_{\pi},f_{\nu})$ that is slightly distorted in both cases. In the figure, it is clearly observed how the minimum of the PNP energy is shifted to higher $\langle\Delta N^{2}\rangle$ and $\langle\Delta Z^{2}\rangle$ values as compared to the HFB ones. This is in agreement with the expectation that the RVAP method provides intrinsic states with more pairing correlations than those intrinsic states obtained by the HFB method. This has important consequences for fission dynamics as the collective inertia strongly depend upon the amount of pairing correlations.

The other quantity required to study the dynamics of spontaneous fission is the collective inertia associated to the collective variables used to drive the nucleus from its ground state to fission. The collective inertia plays a crucial role in several fission observables, such as the spontaneous fission lifetimes $t_{\mathrm{SF}}$ obtained within the Wenzel-Kramers-Brillouin (WKB) formula and the fission fragments distributions obtained in both time dependent frameworks Goutte et al. (2005); Regnier et al. (2016) and stochastic Langevin approaches Sadhukhan et al. (2016); Matheson et al. (2018). For instance, the $t_{\mathrm{SF}}$ has an exponential dependence on the collective inertia than can amount to changes of several orders of magnitude in this quantity Giuliani and Robledo (2013); Giuliani et al. (2014); Rodríguez-Guzmán and Robledo (2014). As mentioned before, the magnitude of the collective inertia depends on the amount of pairing correlations in a way that can be quantified as an inverse dependence on the square of the pairing gap. This dependence on the amount of pairing correlations implies that the larger the pairing correlations are the smaller the collective inertia (and therefore $t_{\mathrm{SF}}$) is. Therefore, we expect a strong dependence of the collective inertia on the combined action of both the Coulomb antipairing effect and the PNP.

There are two types of collective inertias: the one coming from adiabatic time dependent Hartree-Fock-Bogoliubov (ATDHFB) theory and the one coming for the Gaussian overlap approximation (GOA) to the Generator Coordinate Method (GCM) Schunck and Robledo (2016). Unfortunately, so far none of these schemes has been generalized to the case of non-HFB states like the PNP ones considered in this paper. In these respect, the GCM-GOA framework is more promising since its formalism is not intimately rooted to the HFB method. However, the perturbative cranking approximation (where the linear response matrix is approximated by its diagonal both in the expressions of the inertia and in the definition of the collective momentum Giuliani and Robledo (2018)) required to alleviate the computational cost of the collective inertias is not easy to implement in the PNP case. Therefore we take a pragmatic approach and use for the PNP case the perturbative cranking inertias computed with the intrinsic state $|\Phi\rangle$ obtained in the RVAP. Work to obtain a sound and easy way to compute the inertia for PNP wave functions is underway and will be reported elsewhere.

To avoid the appearance of divergences in the calculation of the PNP energy with the Gogny force, we computed the exchange, direct and pairing channels for each of the terms of the interaction Anguiano et al. (2001b). The required Hamiltonian and norm overlap between the HFB state $|\Phi\rangle$ and its rotated in gauge space $e^{i\Phi_{p}\hat{Z}}e^{i\Phi_{n}\hat{N}}|\Phi\rangle$ are computed using the methodology of the generalized Wick theorem as developed in Robledo (2009); Bertsch and Robledo (2012). For the density dependent part of the interaction we use the so-called ”PNP projected density prescription” that is commonly used for particle number projection Anguiano et al. (2001b); Robledo (2007) (be aware, however, of the fundamental difficulties encountered when using the projected density prescriptions in the context of spatial symmetries restoration Robledo (2010)).

III Results

We have considered three nuclei as prototipical examples of application of the issues discussed in the previous section. The first nucleus studied is the light actinide 236U, characterized by a double humped potential energy surface (PES) with high and wide barriers. Reflection symmetry is broken right after the fission isomer and therefore asymmetric fragment mass distribution is expected for this nucleus. In Fig. 2 we show the most relevant quantities for a theoretical understanding of fission. In panel a) potential energy surfaces (to be discussed below) are shown as a function of the quadrupole moment. The corresponding particle-particle correlation energies $\frac{1}{2}\mathrm{Tr}(\Delta_{\tau}\kappa_{\tau})$ (with $\tau=p,n$) are shown in panels b) and c). In panel d) the self-consistent octupole and hexadecapole moments are also shown along with the neck parameter given by the mean value of the neck operator $Q_{N}=\exp[-(z-z_{0})^{2}/a_{0}^{2}]$ with $z_{0}=0$ and $a_{0}=1.0$ fm. Finally, in panel e) the collective inertia computed in the traditional perturbative ATDHFB scheme is displayed.

Panel a) shows the potential energy surfaces for four different calculations. The black solid line (HFBt) corresponds to the traditional HFB calculation where Coulomb exchange is evaluated in the Slater approximation and Coulomb and spin-orbit anti-pairing are neglected. The dashed red line (HFBCep) corresponds to a HFB calculation where both Coulomb exchange and anti-paring are fully considered. Comparing the predicted isomer energies ($E_{II}$) and inner ($B_{I}$) and outer ($B_{II}$) fission barrier heights (see Table 1) we notice that HFBCep predicts values that are $0.75-0.83$ MeV larger. This increase is an expected behavior when pairing correlations get reduced Samyn et al. (2005); Rodríguez-Guzmán and Robledo (2014). Also, more pronounced structures are observed in HFBCep, particularly at large quadrupole deformations, which can be traced back to the reduced pairing correlations Bertsch et al. (2018) associated to the presence of Coulomb anti-pairing. These changes in the potential energy surface are partially washed out in the HFB calculation obtained with intrinsic RVAP states (HFBRVAP, blue dashed line). The HFBRVAP barriers heights and isomer excitation energy are $0.52-0.55$ larger than the HFBt, and the potential energy surfaces at large deformations are also similar. This result suggests that pairing correlations induced by the RVAP partially cancel out the effect of the Coulomb anti-pairing quenching (see below). Finally, the blue full curve with symbols corresponds to the RVAP projected energy ($E_{\mathrm{PNP}}$). This energy is around two MeV deeper than the intrinsic energies, being the fission parameters $0.50-0.64$ MeV larger than the HFBt results.

In order to better understand the impact of dynamic correlations on fission, it is worth to analyze the changes in the other quantities depicted in Figure 2. Proton particle-particle correlation energies are shown in panel b) for the HFBt, HFBCep and HFBRVAP intrinsic states (this quantity is meaningless in the PNP case). Coulomb antipairing quenches the particle-particle proton correlation energy, but the quenching is softened by the effect of the PNP-RVAP, being the latter results closer to the HFBt ones. In the neutron case, shown in panel c), no significant differences are observed between the HFBt and HFBCep cases as expected. The effect of PNP-RVAP is to increase neutron pairing correlations bringing the particle-particle correlation energy of the intrinsic state above the other two curves. The quadrupole, octupole and necking shape parameters are shown in panel d). For each of the three parameters, the results obtained with the three different types of intrinsic states lie each on top of the other. The impact of different types of pairing correlation regimes on the deformation parameters is negligible. Finally, in panel e) the ATDHFB perturbative collective inertia for the three intrinsic states are shown. As compared to the HFBt reference calculation, the HFBCep inertia is larger as a consequence of the quenched pairing. Overall, the HFBCep inertia is around two times larger than the HFBt one. It also shows more pronounced structures in the form of high peaks. On the other hand, the increase of pairing correlations associated to PNP-RVAP brings the HFBRVAP intrinsic inertia back to the range of the HFBt curve. It is worth mentioning that the HFBRVAP inertia looks a bit smoother than the HFBt one. From this comparison we conclude that the HFBt inertia (i.e. without Coulomb exchange, and what is more important, without Coulomb anti-pairing) represents a good approximation, in the case of the Gogny force, to the inertia obtained from the PNP-RVAP intrinsic states. It is worth mentioning that this cancellation is typical of the Gogny forces and is not expected in calculations where the strength of the pairing interaction is fitted separately for protons and neutrons to experimental data Baran et al. (2015). In this case, the effect of Coulomb anti-pairing is taken into account by the fitted pairing strength and therefore a reduction of a factor of two in the inertias has to be expected.

Finally, we have computed the spontaneous fission half-live $t_{\mathrm{SF}}$ using the traditional WKB formula (see Refs Schunck and Robledo (2016); Rodríguez-Guzmán and Robledo (2014) for details and applications) with a $E_{0}$ parameter of 1 MeV. The results for the HFBt and HFBCep cases are computed with the corresponding PES and collective inertias, whereas the PNP-RVAP is computed with the PNP PES but using the collective inertia of the HFBRVAP intrinsic state. The results are summarized in Table 1 along with the values of $E_{II}$, $B_{I}$ and $B_{II}$ discussed above. The effect of Coulomb antipairing in the inertia increases $t_{\mathrm{SF}}$ by 20 (14) orders of magnitude in the ATDHFB (GCM) cases, but this huge impact is cancelled out by the dynamic pairing effect associated to RVAP-PNP. The final RVAP-PNP $t_{\mathrm{SF}}$ values are very close to the HFBt ones. It is important to emphasize that the RVAP-PNP $t_{\mathrm{SF}}$ values are lower than the HFBt ones in spite of the larger fission barrier heights. This is due to the smaller values of the inertias in the projected case.

The results obtained for the nucleus 240Pu look qualitatively the same as those obtained for 236U, being the small differences observed mostly due to shell effects associated with the different proton and neutron numbers. The values of $E_{II}$, $B_{I}$ and $B_{II}$ are given in Table 1. The most notorious difference is in the larger values of $B_{I}$ which are around 1 MeV higher than in the 236 U case. The impact of Coulomb antipairing in $t_{\mathrm{SF}}$ is 16 (10) orders of magnitude the ATDHFB (GCM) inertias and, as in the uranium case, the inclusion of dynamical pairing correlations reduce substantially $t_{\mathrm{SF}}$ and brings it closer to the traditional HFBt value. As in the previous case, we conclude that the dynamic pairing compensates the Coulomb antipairing effect and the $t_{\mathrm{SF}}$ values obtained in the traditional HFB approach are very similar to the ones obtained in the RVAP-PNP context.

We have also carried out calculations for the heavier 252Cf isotope. The potential energy surfaces, particle particle energy correlations, deformation parameters and ATDHFB collective inertias are shown in Fig. 3. In all the cases, the PES show a rather high inner barrier (see Table 1 for the values of the different parameters). The reflection symmetric fission isomer lies at around 3.7 MeV excitation energy, whereas the slightly reflection asymmetric outer barrier is around 7 MeV high. In this particular nucleus the impact of the different theoretical schemes used in the calculation of the outer barrier is stronger with changes in its height of more than 1 MeV. It turns out that in the region of the outer barrier the HFBt PES is very flat with several coexisting minima but one of them is clearly favored when Coulomb antipairing is considered. The particle-particle correlation energy for protons looks similar to the one of 236U for the HFBCep and HFBRVAP cases but differs significantly in the HFBt case around the outer barrier region. The reason for this behavior is the same that explain the discrepancies in the PESs in that region. The $E_{pp}$ for neutrons follows the same pattern as in the uranium case and only small differences are noticed in the outer barrier region. The same observation is valid for the deformation parameters of panel d). The behavior of the ATDHFB inertia in panel e) is qualitatively similar to the one of 236U.

Concerning $t_{\mathrm{SF}}$, we observe longer values when Coulomb antipairing is considered but the difference amounts to 2 (0) orders of magnitude in the ATDHFB (GCM) case. This is in strong contrast with the 236U and 240Pu cases. A possible explanation is the reduction of the outer barrier height of more than 1 MeV seen in this particular case. Considering dynamical pairing lowers $t_{\mathrm{SF}}$ by 5 (3) orders of magnitude in the ATDHFB (GCM) cases as compared to the HFBCep result. The net effect of this opposite trends is to yield final values for the RVAP-PNP calculation which are, again, pretty close to the HFBt ones.

IV Conclusions

In this paper we studied the impact of dynamical pairing correlations in the theoretical estimation of fission properties. We found that particle number projection in the restricted variation after projection framework (using $\langle\Delta N^{2}\rangle$ for protons and neutrons as variational parameters) has a profound impact on some of the quantities related to fission such as spontaneous fission half-lives. The parameters defining the potential energy surface, like the fission barrier heights and fission isomer location are little affected by particle number projection in the three examples analyzed. On the other hand, the increase in pairing correlations due to particle number restoration leads to a quenching of the collective inertia by a factor of the order of two. The consequences for the spontaneous fission half-life depend on the nucleus but it is quantified to be large and can reach a reduction of 20 orders of magnitude. This reduction is compensated by the Coulomb anti-pairing effect, which is often neglected (or described with the Slater approximation) in mean field calculations but is required in particle number projection to avoid the self-energy and self-pairing problems. The reduction of pairing correlations associated with Coulomb anti-pairing increases the collective inertias by a factor of around two in the examples considered and can increase the calculated $t_{\mathrm{SF}}$ up to 20 orders of magnitude. On the other hand, the consequences of an exact treatment of the Coulomb potential in the potential energy surface are relatively small and have a relatively less important impact on $t_{\mathrm{SF}}$. The two opposite effects, Coulomb anti-paring and dynamical pairing correlations tend to suppress each other and the final outcome turns out to be similar to the results obtained omitting both of them. This result is relevant for calculations with nuclear forces (Gogny among them), where the nuclear pairing interaction is isospin invariant and Coulomb anti-pairing has to be considered. The effect of dynamical pairing correlations alone is relevant for other interactions where the pairing strength for protons and neutrons is fitted separately to experimental data.

For future work, the evaluation of the collective inertias with particle number projected wave functions is the next step to consider. Also, the consequences of particle number projection on induced fission half-lives and properties of the fission fragments could be an interesting subject of research.

Acknowledgements.

SAG acknowledge support from the U.S. Department of Energy under Award Number DOE-DE-NA0002847 (NNSA, the Stewardship Science Academic Alliances program). The work of LMR has been supported in part by Spanish grant Nos FIS2015-63770 MINECO and FPA2015-65929 MINECO.