Cluster properties of heavy nuclei predicted with the Barcelona-Catania-Paris-Madrid energy density functional

We study the cluster emission properties of 224Ra and 238Pu employing the Barcelona-Catania-Paris-Madrid (BCPM) energy density functional (EDF). Starting from two-dimensional potential energy surfaces, coexisting fission paths are identified. A fission valley located at large octupole deformations, corresponding to a highly-asymmetric mass distribution, is found in both nuclei. As the corresponding fragments are dominated by the presence of 208Pb, we can relate this fission path to the emergence of cluster emission. Using the octupole moment as collective degree of freedom, we compute the cluster decay half-lives and study the impact of collective inertias, pairing strength and collective zero-point energy. The agreement with experimental data resembles the results obtained for spontaneous fission half-lives, indicating the capability of BCPM to consistently describe a large variety of fission phenomena, including cluster emission.


Introduction
The phenomenon of cluster decay, or cluster radioactivity, consists in the spontaneous emission of fragments with an intermediate mass between fission and α decay.Originally postulated by Sandulescu et al. in 1980 [1] and experimentally observed by Rose and Jones four years later [2], this decay is driven by the emission of one fragment in the neighborhoods of the doubly magic nucleus 208 Pb.From a theoretical point of view, cluster radioactivity can be described as a highly-asymmetric fission mode, where the fragments emerge as the consequence of the quantum mechanical tunneling through the potential barrier of the compound nucleus.Given the extreme mass asymmetries characterizing this decay, the odd-parity octupole moment can be used as driving coordinate.Therefore, cluster emission studies explore octupole deformation not only in the traditional ground state regime, but also in the extreme elongations required for the formation of a well defined neck that defines scission point and fragment emission properties.Due to this, cluster decay has been the subject of several experimental measurements [3][4][5][6][7][8][9][10] and microscopic studies based on effective nucleon-nucleon interactions [11][12][13][14][15], with a renewed interest in the latest years arising from its predicted presence in the super-heavy landscape [16][17][18].
The Barcelona Catania Paris Madrid (BCPM) [19] functional was originally developed to describe with good accuracy binding energies and radii of finite nuclei.Large scale mean-field Hartree-Fock-Bogoliubov (HFB) calculations revealed that deformation properties of the functional are at the level of other interactions such as Gogny D1S or D1M.This property opens up the door to the use of BCPM in fission studies, and more specifically to describe spontaneous fission lifetimes [20][21][22].One of the most useful features of BCPM is its relatively low computational cost, specially when compared with finite range interactions like Gogny.This feature makes possible large-scale fission calculations like the ones required in stellar nucleosynthesis studies, including fission recycling of super-heavy neutron rich nuclei [23].
Given the success of BCPM in traditional fission studies, it seems pertinent to explore the ability of the functional to describe very asymmetric fission modes like the one taking place in cluster emission in actinides.This decay provides an optimal case study for assessing the capability of microscopic models to properly reproduce shell structure properties of parent and daughters nuclei at large deformations, as well as the collective inertias required in the calculation of cluster decay half-lives.In this work, we present the results obtained for two different nuclei: 224 Ra and 238 Pu.The former is one of the first cluster emitters detected in laboratory [4], and represents a canonical standard for studies concerning cluster emission.The latter is a well known case of cluster emission observed in the transactinide region [6], and allows the exploration of cluster phenomena in heavier systems.In this work, the cluster path is identified by computing the evolution of the energy as a function of quadrupole and octupole deformation.Cluster emission lifetimes are thus obtained using the octupole moment as collective degree of freedom.Despite the sensitivity of the lifetimes with respect to the different quantities entering the collective action integral, a good agreement with experimental data is found.
The paper is structured as follows.In Section 2, the theoretical framework employed in the calculation of the cluster decay half-lives is outlined.In Section 3, fission pathways and cluster decay half-lives predicted by BCPM are discussed.Conclusions are summarized in Section 4, together with an outlook on future work.

Methods
The self-consistent description of highly asymmetric fission is based on the HFB method with constraining operators [24,25].In the HFB method, the many-body wavefunction |Ψ⟩ is defined as the vacuum to all the annihilation quasiparticle operators β µ .Nuclear states are thus obtained by minimizing the Routhian with constraints on the neutron N and proton Z particle number, and the mass multipole moments Q µν being H the HFB Hamiltonian.The independent λ i quantities are determined by the condition that the gradient of the Routhian has to be orthogonal to the gradient of the constraints.In this work, we impose constraints on the axially symmetric quadrupole (Q 20 ), and octupole (Q 30 ) moments, defined as Higher multipolarities are self-consistently adjusted in order to minimize the total energy, e.g. the hexadecapole (Q 40 ) moment related to the development of a fission neck: The dipole moment Q 10 is set to zero to avoid spuriousness associated with the centre-of-mass motion.
The HFB equations are solved using the HFBaxial computer code, which employs an approximate second-order gradient method to determine the amplitudes of the Bogoliubov transformation to quasiparticles [26].The quasiparticle operators are expanded in a large deformed axially-symmetric harmonic oscillator (HO) basis, containing states with J z up to 35/2 and 26 quanta along the z-direction.The HO quantum numbers are restricted by the condition n z /q + 2n ⊥ + |m| ≤ N 0 , with N 0 = 17 and q = 1.5.This configuration is well suited for describing elongated shapes along the z direction, as those routinely found in fission.The two oscillator lengths characterising the HO bases are chosen as to minimize the energy for each constrained configuration considered.
One of the goals of the present paper is to evaluate the half-lives of spontaneous cluster emission, which can be obtained by means of the Wenzel-Kramers-Brillouin (WKB) method: being S the collective action computed along the fission path In this work, collective inertias M(Q 30 ) are computed using the perturbative cranking approximation within the adiabatic time-dependent HFB (ATDHFB) [28] and the Gaussian Overlap Approximation to the Generator Coordinate Method (GOA-GCM) [29] formalisms: being M −n the energy-weighted moments obtained from the one-quasiparticle energies E µ and the two-quasiparticle zero-hole component of the octupole operator (Q 20  30 ) µν (see Appendix E in Ref. [29]): The effective potential V(Q 30 ) is obtained by subracting the rotational and zero-point energy corrections from the HFB energy In this expression, E rot represents the energy gained by restoring rotational symmetry, which can be effectively computed employing the recipe from Ref. [30].E zpe is the energy correction associated to quantum fluctuations in the collective degree of freedom Q 30 , which is computed consistently with the collective inertia scheme [29,31]: being the overlaps Finally, the action integral defined in Eq. ( 5) is computed between the classical turning points a and b determined by the condition V = E 0 , where E 0 represents the ground-state value of V(Q 30 ) plus the zero-point energy associated to the quantum motion of the nucleus along the collective degrees of freedom.This quantity is usually treated as a free parameter in order to reproduce spontaneous fission lifetimes.In this work, we will study the impact of E 0 in the predicted t 1/2 by varying its value between 1.0 and 1.5 MeV.
To compute the effective potential energy V(Q 30 ) and collective inertias M(Q 30 ) entering in Eq. ( 5), we employ the BCPM EDF.For a comprehensive discussion regarding the details and applications of this interaction, we refer to the recent review from Baldo et al. [32].Here, we just recap the basic ingredients of the functional: • A bulk or volume term is given by two fifth order polynomials of the proton and neutron densities.One of the polynomial is fitted to reproduce the binding energy per nucleon in symmetric nuclear matter, while the other uses the pure neutron matter equation of state.For intermediate situations, a quadratic term proportional to β = ρ n −ρ p is used to connect both polynomials.
• A surface term, obtained from the Hartree potential of a finite range Gaussian.
• A spin-orbit potential adopting the same form as the ones of Gogny or Skyrme interactions, with a standard spin-orbit strength adapted to the effective mass of BCPM being equal to the bare mass.The kinetic energy is computed using the standard quantum mechanic expression.

Results
In order to assess the capability of the BCPM interaction to reproduce the main features of cluster decay in heavy nuclei, we follow the strategy of Ref. [13] and study the properties of two representative nuclei: 224 Ra and 238 Pu.

Potential energy surfaces and fission paths
Fig. 1 shows the potential energy surface (PES) as a function of the quadrupole Q 20 and octupole Q 30 mass moments predicted by BCPM for 224 Ra and 238 Pu.This plot, representing the evolution of the HFB energy (including the rotational correction) as a function of elongation and mass asymmetry, allows for the identification of the multiple fission paths coexisting in a particular nucleus.Calculations are performed in a grid with a spacing of 2 b in the Q 20 direction and 2 b 3/2 in the Q 30 direction.The harmonic oscillator lengths have been optimized for each configuration.For a better visualization, the predicted ground-state energy has been subtracted from the PES.We observe that for both nuclei, BCPM predicts a well deformed ground-state configuration: Q gs 20 = 8 b for 224 Ra, and Q gs 20 = 14 b for 238 Pu.In the case of 224 Ra the ground-state is also predicted to break reflection symmetry as Q gs 30 = 4 b 3/2 .Starting from these ground-state configuration, fission paths can be determined by seeking for the maximal decrease in energy.Hence, we find that BCPM predicts the existence of two distinct fission paths.The first one initially proceeds along configurations with small Q 30 values.that the elongated fission path of 224 Ra is particularly wide and has a fission barrier height of 10.76 MeV, indicating that BCPM predicts 224 Ra to be stable against fission.
Besides this elongated asymmetric fission path, both PESs in Fig. 1 show the existence of a second fission valley located at very large octupole deformations: (Q 20 , Q 30 = 34 b, 48 b 3/2 ) for 224 Ra; and (Q 20 , Q 30 ) = (66 b, 76 b 3/2 ) for 238 Pu.The octupole moment along the fission path connecting this valley to the ground-state configuration quickly grows with increasing elongation, suggesting the emergence of a highly-asymmetric fission mode.The one dimensional projection of this fission path for both nuclei is plotted in Fig. 2 as a dashed red line.We notice that a saddle point is reached at Q 20 = 26 b for 224 Ra, and Q 20 = 52 for 238 Pu.At these deformations, the nucleus starts to split into two fragments.For both 224 Ra and 238 Pu the heavy partner is a spherical nucleus in the vicinity of 208 Pb, clearly indicating that the highly-asymmetric fission path found in these nuclei corresponds to a cluster emission.We notice that the energy barrier of cluster decay is above 26 MeV, indicating that the probability of the nucleus to undergo this decay is much smaller than the one corresponding to the traditional fission decay.
In order to visualize the emerging fragments, Fig. 3 shows the mass distributions of 224 Ra and 238 Pu at the saddle point of the cluster path.The cluster decay predicted by BCPM for these nuclei are 224 Ra → 210 Pb + 14 C and 238 Pu → 208 Pb+ 30 Mg, in agreement with experimental observations [9].In both nuclei, fragments' deformation are determined by their ground-state deformation: spherical for 14 C, 208 Pb and 210 Pb, and oblate for 30 Mg (β 2 = −0.17).After overcoming the saddle point of the cluster barrier, i.e., after the splitting of the nucleus into two fragments, the energy of the nucleus starts to decrease with elongation in a parabolic fashion.This is the result of the diminishing Coulomb repulsion between the fragments as they move further apart.The outer turning point of the cluster decay can thus be obtained by following the Coulomb barrier until it falls below the ground-state energy.However, as already discussed by Warda and Robledo [13], we find that this two-fragments configuration quickly saturates the HO basis for large values of the elongation, suggesting that a larger basis should be employed in order to compute the full cluster decay path.Unfortunately, the number of HO shells is limited by numerical accuracy, and cannot be increased without the emergence of numerical instabilities in the estimation of the matrix elements.This limitation will be addressed in the next section.

Cluster decay path as a function of Q 30
As the cluster path overcomes the energy barrier, it proceeds towards larger deformations by following the bottom of the highly-asymmetric valley described in Sec.3.1.As shown in Fig. 1, this results in a cluster path placed along a diagonal line in the (Q 20 , Q 30 ) plane.Hence, a correspondence can be established between increasing elongation and increasing mass asymmetry for these cluster decays.As already proposed in Ref. [13], this proportionality can be exploited in order to parametrize the cluster path as a function of the octupole moment Q 30 .This change of collective variable is particularly convient as the PES for a fixed Q 30 is stiffer than for a fixed Q 20 , which simplifies the identification of the cluster decay path located at large Q 30 values.The one-dimensional cluster path of 224 Ra and 238 Pu as a function of Q 30 is plotted in Fig. 4. As in the Q 20 parametrization of Fig. 2, we notice a fast increase of the energy with increasing Q 30 .A saddle point is reached at Q 30 = 26 b 3/2 for 224 Ra and Q 30 = 52 b 3/2 for 238 Pu.At these configurations, the nucleus is already split in two fragments, corresponding to the cluster decay described in the previous section.Hence, increasing the octupole moment beyond this point corresponds to a separation of the centre of mass of the fragments, with the consequent diminishment of the total energy due to the decreasing Coulomb repulsion.By approximating the cluster fragments as two homogeneous spheres with charges Z 1 and Z 2 , the Coulomb repulsion energy as a function of the octupole moment can be written as being R the distance between the centers of mass of the fragments, while the decay Q-value can be extracted from the AME2020 database [33].
The relationship between R and the octupole moment can be obtained by approximating the two fragments as spheres with masses A 1 and A 2 : with where A is the mass number of the compound nucleus.The Coulomb repulsion energy from Eq. ( 11) is plotted in Fig. 4 as a solid red line.At the saddle point, the Coulomb repulsion and the HFB calculations differ by roughly 5 MeV.This discrepancy is slightly larger than the one found in Ref. [13], and can be related to the excitation of the lighter fragment in the presence of the Coulomb field produced and to the shape of the emerging fragments.As a consequence of the saturation of the HO basis discussed in Sec.3.1, we notice a departure of the HFB calculations from the Coulomb parabola with increasing Q 30 .
As this numerical limitation imposed by the finite size of the basis do not allow a proper calculation of configurations with large values of the octupole moment, in the next section the cluster decay halflives will be estimated employing the Coulomb barrier from Eq. (11).For this purpose, one shall rewrite the classical collective inertia for two fragments separated by a distance R in terms of the octupole moment [13]: being µ = m n A 1 A 2 /(A 1 + A 2 ) the reduced mass and m n the nucleon mass.The upper panel of Fig. 4 shows the evolution of the collective inertias from Eqs. (6a), (6b) and ( 14).The microscopic ATDHFB and GOA-GCM inertias present large fluctuations due to the crossing of single particle levels, which are obviously absent in the classical collective inertias.However, once the parental nucleus splits into two fragments, both microscopic inertias agree to the same asymptotic value.

Cluster decay half-lives
In order to estimate the spontaneous fission half-lives of cluster decay, we employ the formalism described in Sec.The cluster-decay half-lives, computed from Eq. ( 5), are plotted in Fig. 5 for two different values of the collective zero-point energy E 0 .The comparison with experimental data [9] shows a good agreement when the GOA-GCM collective inertias are employed, while with the ATDHFB scheme the half-lives are overestimated (particularly for 238 Pu).In general, we notice a large scatter in the predicted results depending on the collective inertia scheme and E 0 value employed for the estimation of t 1/2 .This is not a surprising result, as both nuclei have a high fission barrier (which in turn implies a small cluster emission probability and long half-life), which makes the theoretical calculations of t 1/2 extremely sensitive to the details concerning the estimation of the Results obtained with ATDHFB collective inertias (6a) and GOA-GCM (6b) are plotted as orange circles and purple diamonds, respectively.Empty and full symbols represents the half-lives computed with E 0 = 1.0 and 1.5, respectively.The horizontal gray band shows the experimental data [9].
quantities entering into Eq.( 5).Regarding the absolute value of theoretical half-lives, it should be mentioned that the inclusion of pairing as collective degree of freedom can reduce the predicted t 1/2 of long-lived nuclei by several orders of magnitude [21,[34][35][36].This is because collective inertias decrease as the square of the inverse pairing gap, leading to a reduction of the t 1/2 if the action integral in Eq. ( 5) is computed along the least-action path.While the inclusion of paring as dynamical variable goes beyond the scope of this work, its effect on the collective inertias and t 1/2 can be mimicked by increasing the pairing strength η [20,37,38].Fig. 5 shows the predicted halflives for cluster emission when pairing strength is increased by 5% (η = 1.05) and 10% (η = 1.10).From this plot it is possible to conclude that for larger η values not only the t 1/2 are quenched, as expected, but also the spread in theoretical predictions due to variations in E 0 and the collective inertia scheme is reduced, in agreement with the results obtained when pairing is included as collective degree of freedom [21,35,36].Evidently, the choice of the collective inertia scheme is the main source of uncertainty, due to the large values of the collective action integral of these decays.While a better agreement with experimental data is obtained with the GOA-GCM scheme, as these are systematically smaller than the ATD-HFB ones, an increase of the pairing strength by a 10% brings the ATDHFB predictions on top of experimental data, similarly to what already found in BCPM fission studies in transactinides nuclei [20].However, we stress out again that theoretical t 1/2 are extremely sensitive to the details of the action integral calculation, and therefore the comparison with experimental data should be always taken with a grain of salt.Nevertheless, we conclude that BCPM provides a good reproduction of cluster decay half-lives within the simplified scheme presented in this work.

Conclusions
The cluster-decay properties of 224 Ra and 238 Pu have been computed using the BCPM EDF.By studying the evolution of the potential energy surface as a function of the mass quadrupole moment (elongation) and octupole moment (mass asymmetry) we identified the coexisting fission paths in these nuclei.Besides the traditional elongated fission mode, which proceeds through low values of the octupole moment, BCPM predicts the existence of a highly-asymmetric fission valley located at large Q 30 , in agreement with previous EDFs studies [13].The large value of the cluster barrier indicates that this mode is substantially suppressed compared to the traditional fission mode, as observed in laboratory.
The cluster decay half-lives are computed using the WKB formalism, by parametrizing the fission path as a function of Q 30 .Due to the large values of the fission barriers, we find a large impact of collective inertias and ground-state zero-point energy on the estimated t 1/2 .By increasing the pairing strength, the sensitivity of the estimated half-lives on the different quantities defining the collective action diminishes.Overall, a good agreement with experimental data is found when the GOA-GCM scheme is employed.Conversely, for the ATDHFB inertias an increase in the pairing strength by a 10% is required in order to reproduce the observed t 1/2 .
In a future work, we plan to expand and improve the formalism employed for the estimation of the half-lives.First and foremost, the impact of including pairing as a dynamical variable should be assesed, since it could reduce the predicted t 1/2 by several orders of magnitude.Furthermore, the estimation of the collective inertias can be improved by employing the non-perturbative ATDHFB and GOA-GCM schemes [39][40][41].Work along these lines is in progress and will be the subject of a future publication.
• A Coulomb term, with two contributions, one the traditional Hartree term derived from the Coulomb potential and the other the Slater local approximation to Coulomb exchange.

Fig. 1 Fig. 2
Fig.1Evolution of the HFB energy (including rotational correction) in MeV for 224 Ra (upper panel) and 238 Pu (lower panel), as a function of elongation (Q 20 ) and mass asymmetry (Q 30 ).The ground state configuration of each nucleus is marked with a • symbol.The red solid line represents the traditional elongated fission path.The cluster path is marked with an orange dash-dotted line.

Fig. 3
Fig. 3 Spatial mass distribution of 224 Ra (left panel) and 238 Pu (right panel).The configurations correspond to the ■ symbols plotted in Fig. 1.

2 .
The fission path L(Q 30 ) employed in the estimation action integral is obtained by splicing the cluster path (parametrized as a function of Q 30 ) with the Coulomb repulsion barrier obtained in Sec.3.2.The transition between the two regimes occur at the saddle point (Q 30 = 22 b 3/2 for 224 Ra, and Q 30 = 46 b 3/2 for 238 Pu), keeping the collective inertias consistent with the employed barrier.

Fig. 5
Fig. 5 Spontaneous fission half-lives for the cluster decay of 224 Ra (upper panel) and 238 Pu (lower panel) as a function of different values of the pairing strength η.Results obtained with ATDHFB collective inertias (6a) and GOA-GCM (6b) are plotted as orange circles and purple diamonds, respectively.Empty and full symbols represents the half-lives computed with E 0 = 1.0 and 1.5, respectively.The horizontal gray band shows the experimental data[9].