Microscopic description of fission in nobelium isotopes with the Gogny-D1M energy density functional

Microscopic description of fission in nobelium isotopes with the Gogny-D1M energy density functional

R. Rodríguez-Guzmán Physics Department, Kuwait University, Kuwait 13060, Kuwait    L.M. Robledo Departamento de Física Teórica, Universidad Autónoma de Madrid, 28049-Madrid, Spain
July 17, 2019
Abstract

Constrained mean-field calculations, based on the Gogny-D1M energy density functional, have been carried out to describe fission in the isotopes No. The even-even isotopes have been considered within the standard Hartree-Fock-Bogoliobov (HFB) framework while for the odd-mass ones the Equal Filling Approximation (HFB-EFA) has been employed. Ground state quantum numbers and deformations, pairing energies, one-neutron separation energies, inner and outer barrier heights as well as fission isomer excitation energies are given. Fission paths, collective masses and zero-point quantum vibrational and rotational corrections are used to compute the systematic of the spontaneous fission half-lives t both for even-even and odd-mass nuclei. Though there exists a strong variance of the predicted fission rates with respect to the details involved in their computation, it is shown that both the specialization energy and the pairing quenching effects, taken into account within the self-consistent HFB-EFA blocking procedure, lead to larger t values in odd-mass nuclei as compared with their even-even neighbors. Alpha decay lifetimes have also been computed using a parametrization of the Viola-Seaborg formula. The high quality of the Gogny-D1M functional regarding nuclear masses leads to a very good reproduction of values and consequently of lifetimes.

pacs:
24.75.+i, 25.85.Ca, 21.60.Jz, 27.90.+b, 21.10.Pc

I Introduction

The theoretical description of odd-mass nuclei still remains as a major challenge in nuclear structure as compared to the treatment of even-even nuclei. Nevertheless, odd-mass nuclei provide key information regarding not only odd-even effects associated to pairing correlations but also the “time-odd” section of the nuclear energy density functional (EDF) which is of paramount relevance in the dynamical behavior of the nucleus. In addition, odd-mass nuclei could also be useful to constrain the parameters of energy density functionals (EDFs) with the aim to reach a reasonable spectroscopic quality. Moreover, the energies, spins and parities of their low-lying one-quasiparticle spectra provide insight into the underlying shell effects. Thus, the properties of odd-mass nuclei and the computational challenge in their evaluation have received renewed attention in recent years (see, for example, Hamamoto-dd (); EFA-jsut (); EFA-Bonneau (); duguet-odd (); EFA-Rayner-1 (); EFA-Rayner-2 (); EFA-Rayner-3 (); EFA-Rayner-4 (); reorient-1 (); reorient-2 (); No-REGION () and references therein).

In the context of spontaneous fission Bjor (); Specht (); Holden-paper (), it is experimentally observed that the half-lives t of odd-mass nuclei are systematically larger than the ones of their even-even neighbors. Two complementary mechanisms have been proposed to explain this feature: One is the specialization energy special-Fong (), that modifies the collective potential energy landscape felt by the odd-mass nucleus in its evolution from the ground state to scission as compared to the one of the neighboring even-even companions. The origin of part of this energy is the assumption that the K quantum number associated with the projection of the angular momentum onto the intrinsic symmetry axis characteristic of any odd-mass quantum state has to be conserved along the fission process. The reason behind this assumption is that the time scale for fission (not to be confused with the t value) is of the order of the one for the strong interaction and, therefore, much shorter than the typical time scales for the electromagnetic processes responsible for the change of the K quantum number. The second mechanism is the characteristic quenching of pairing correlations caused by the unpaired nucleon that weakens the pairing field strengths, as compared with the fully paired situation, enhancing the collective inertia that strongly depends on the pairing gap. This enhancement in turn, increases the collective action and, therefore, the spontaneous fission half-life t. One should also keep in mind, that fission observables are quite sensitive to pairing correlations owing to the strong dependence of the collective inertias on the inverse of the square of the pairing gap Robledo-Giulliani (); proportional-1 (); proportional-2 (); Rayner-UPRC-2014 (); Rayner-UEPJA-2014 (); Rayner-RaEPJA-2016 ().

Within the EDF framework, it is customary to describe the transition from a single even-even system to its fragments in terms of several deformation parameters Bjor (); Specht (); Krappe (); Baran-Kowal-others-review2015 (); Schunck-Robledo-review (). Such microscopic (constrained) EDF calculations already represent a highly demanding computational task. They are usually carried out using mean-field approximations based on nonrelativistic Gogny Rayner-UPRC-2014 (); Rayner-UEPJA-2014 (); Rayner-RaEPJA-2016 (); gogny-d1s (); Delaroche-2006 (); Dubray (); Younes-fission (); Warda-Egido-Robledo-Pomorski-2002 (); Warda-Egido-2012 (); Action-Rayner (), Skyrme Baran-Kowal-others-review2015 (); Schunck-Robledo-review (); UNEDF1 (); Mcdonell-2 (); Erler2012 (); Baran-SF-2012 (); Baran-1981 () and more recently Barcelona-Catania-Paris-Madrid (BCPM) BCPM (); Robledo-Giulliani () as well as relativistic Abusara-2010 (); Abu-2012-bheights (); RMF-LU-2012 (); Kara-RMF (); Afa-arxiv () EDFs. What further complicates the EDF description of fission in odd-mass nuclei is the need to consider, for each intrinsic configuration along the fission path, blocked one-quasiparticle wave functions rs (). The blocked wave functions break time-reversal invariance which requires the evaluation of time-odd fields. Typically, this requirement represents a factor of two more computing time in the solution of the Hartree-Fock-Bogoliubov (HFB) equations as compared to the even-even case where the time-odd fields are zero by construction. On the other hand, due to the (nonlinear) self-consistent character of the HFB equations, there is no guarantee to obtain the lowest energy solution by blocking the quasiparticle with the lowest one-quasiparticle excitation energy. Therefore, the blocking procedure has to be repeated for several low-lying one-quasiparticle states.

From what has already been mentioned above, it is clear that an approximation is required in order to afford the computational cost of the microscopic EDF description of fission in odd-mass nuclei. Within this context, the Equal Filling Approximation (EFA) represents a reasonable starting point still preserving time-reversal invariance EFA-jsut (); EFA-Bonneau (); EFA-Rayner-1 (); EFA-Rayner-2 (); EFA-Rayner-3 (); EFA-Rayner-4 (); EFA-Decharge (). The EFA has been proven to be a fully variational method by recasting it in terms of the language of quantum statistical mechanics and the introduction of a statistical ensemble where one-quasiparticle configurations and their time-reversed companions are present with equal probability EFA-jsut (). One advantage of this point of view is that the results of finite temperature HFB and its extensions to compute collective inertias in the framework of the Adiabatic Time Dependent (ATD) method ATDHFB-T-1 (); ATDHFB-T-2 (); ATDHFB-T-3 () can be used to compute the collective inertias needed to obtain the t values for odd-mass nuclei.

Nuclei in the No (Z=102) region have attracted considerable attention No-REGION () thanks to the progress made in the production of super-heavy elements (see, for example, JULIN-SHE () and references therein) but also because of the rich spectroscopic data obtained for them SPEC-HERZ (). In this paper, we mainly focus on the HFB-EFA description of the fission properties in odd-mass nobelium isotopes. To the best of our knowledge our EDF calculations are the first of their kind reported in the literature. For the sake of completeness, we also discuss results for even-even No nuclei. We have studied the sample of nuclei No. The odd-mass isotopes are considered within the (constrained) HFB-EFA EFA-jsut (); EFA-Bonneau (); EFA-Rayner-1 (); EFA-Rayner-2 (); EFA-Rayner-3 (); EFA-Rayner-4 (); EFA-Decharge () while for the even-even ones calculations have been carried out using the standard (constrained) HFB framework Rayner-UPRC-2014 (); Rayner-UEPJA-2014 (); Rayner-RaEPJA-2016 (); rs ().

Figure 1: The HFB plus the zero-point rotational energies, are plotted in panel (a) as functions of the quadrupole moment for the nucleus No. The octupole and hexadecupole moments are given in panel (b). The pairing interaction energy is depicted in panel (c) for protons (dashed lines) and neutrons (full lines). The collective GCM (dashed lines) and ATD (full lines) masses are plotted in panel (d). For more details, see the main text.

Our HFB and/or HFB-EFA calculations are based on the parametrization D1M gogny-d1m () of the Gogny-EDF gogny (). In previous studies Rayner-UPRC-2014 (); Rayner-UEPJA-2014 (); Rayner-RaEPJA-2016 (), we have performed fission calculations for neutron-rich Ra, U and Pu nuclei as well as for a sample of heavy and super-heavy nuclei for which experimental data exist. The comparison between the D1S gogny-d1s (), D1N gogny-d1n () and D1M gogny-d1m () Gogny-EDFs, with available experimental data and other theoretical results, reveals that the parameter set D1M represents a reasonable starting point to describe fission in heavy and super-heavy nuclei. This is quite satisfying as the Gogny-D1M EDF does a much better job to reproduce nuclear masses gogny-d1m () and, at the same time, seems to keep essentially the same predictive power of the well tested D1S parametrization gogny-d1m (); PRCQ2Q3-2012 (); Robledo-Rayner-JPG-2012 (); PTpaper-Rayner (); Rayner-Robledo-JPG-2009 (); EFA-Rayner-3 (); EFA-Rayner-4 () to reproduce low-energy nuclear structure data. However, more work is needed in order to substantiate this conclusion, especially in the case of fission where the D1M parametrization has been scarcely used. In particular, in this study we explore, for the first time, the ability of the Gogny-D1M EDF framework to capture basic fission properties along the No isotopic chain including not only even-even but also odd-mass nuclei.

The paper is organized as follows. In Sec. II, we briefly outline the HFB-EFA EFA-jsut (); EFA-Rayner-1 (); EFA-Rayner-2 (); EFA-Rayner-3 (); EFA-Rayner-4 (); EFA-Decharge (). For details on the HFB framework for even-even nuclei the reader is referred to Rayner-UPRC-2014 (); Rayner-UEPJA-2014 (); Rayner-RaEPJA-2016 (); rs (). The results of our calculations are presented in Sec. III where we discuss the fission paths, ground state quantum numbers and deformations, first and second barrier heights, fission isomers, spontaneous fission and -decay half-lives for No and compare, whenever possible, with the available experimental data EXP-NO (); Mass-Table-W (). In particular, we illustrate our methodology for both No and No in Secs. III.1 and III.2 while the systematic of the fission paths and spontaneous fission half-lives in No is presented in Secs. III.3 and III.4. Let us stress that in this study, we implicitly assume that the properties of the considered No isotopes are determined by general features of the Gogny-D1M EDF and therefore, no fine tuning of its parameters has been done. Finally, Sec. IV is devoted to the concluding remarks and work perspectives.

Figure 2: (Color online) The K = 9/2 HFB-EFA plus the zero-point rotational energies, are plotted in panel (a) as functions of the quadrupole moment for the nucleus No. The octupole and hexadecupole moments are given in panel (b). The pairing interaction energies are depicted in panel (c) for protons (dashed lines) and neutrons (full lines). The collective GCM (dashed lines) and ATD (full lines) masses are plotted in panel (d). Results corresponding to ”average” (AV) HFB calculations for No have also been included in each panel (red). For more details, see the main text.

Ii Theoretical framework

In this section, we briefly outline the (constrained) HFB-EFA EFA-jsut (); EFA-Rayner-1 (); EFA-Rayner-2 (); EFA-Rayner-3 (); EFA-Rayner-4 (); EFA-Decharge () used to compute the fission paths for No. Calculations for No have been carried out within the (constrained) HFB approach and, for details, the reader is referred to Rayner-UPRC-2014 (); Rayner-UEPJA-2014 (); Rayner-RaEPJA-2016 (); rs ().

The description of odd-mass systems requires, blocked (product) one-quasiparticle wave functions , which are given in terms of a fully paired reference vacuum and the quasiparticle creator rs () for the state to be blocked. The states , as well as their density matrix and pairing tensor break the time-reversal invariance and so do the Hartree-Fock and pairing fields rs () associated with them. Furthermore, the density matrix and pairing tensor are no longer invariant under unitary transformations of the Bogoliubov amplitudes U and V rs () applied to the right and therefore, reorientation effects reorient-1 (); reorient-2 () should also be taken into account in the solution of the mean-field equations. Therefore, in order to alleviate the numerical effort, we have resorted to the HFB-EFA within which one considers the following density matrix

Figure 3: The ground state fission paths obtained for the isotopes No are plotted as functions of the quadrupole moment . Starting from the nucleus No (No ) in panel (a) [panel (b)], the curves have been shifted by 15 MeV in order to accommodate them in a single plot. For No, the corresponding K values are included in the plot. The ”average” (AV) fission paths for those odd-mass nuclei are also depicted with dashed lines. For more details, see the main text.
(1)

and pairing tensor

(2)

Intuitively, and correspond to an occupancy of 1/2 for the quasiparticle state and its Kramers’ partner , thus preserving time-reversal invariance. This intuition can be substantiated by proving the existence of a statistical density matrix operator EFA-jsut () that produces exactly the same density matrix and pairing tensor. The help of Gaudin’s theorem Gaudin-the (); Gaudin-the1 () is required to compute all the basic contractions in Eqs. (1) and (2), as well as the mean values of any kind of operator within the HFB-EFA. The total HFB-EFA energy, which is the statistical average of the EDF with the special statistical density described above, can then be written in the usual form in terms of and rs (). The Ritz-variational principle applied to the HFB-EFA energy leads to the HFB-EFA equations that can be solved using the standard gradient method Robledo-Bertsch2OGM (), with the subsequent simplification in the treatment of several constrains EFA-jsut (); EFA-Rayner-1 (); EFA-Rayner-2 (); EFA-Rayner-3 (); EFA-Rayner-4 ().

Figure 4: The K = 7/2 and 9/2 fission paths obtained for the isotopes No and No are plotted as functions of the quadrupole moment in panels (a) and (b), respectively. The ”average” (AV) fission paths are also depicted with dashed lines. For more details, see the main text.

Let us now turn our attention to the practical aspects of the methodology employed in this work to obtain the ground state fission path in the case of No (see, Sec. III.2) taken, as an illustrative example. The same methodology has been used for all the studied odd-mass No isotopes.

Step 1) We have computed the fission path for No (see, Sec. III.1) using the HFB approach with constrains on the axially symmetric quadrupole and octupole operators PRCQ2Q3-2012 (); Robledo-Rayner-JPG-2012 (). Here, as well as in Steps 2, 3 and 4 below, we have imposed an additional constrain on the operator , to avoid spurious center-of-mass effects. We have used a deformed axially symmetric harmonic oscillator (HO) basis consisting of states with J quantum numbers up to 35/2 and up to 26 quanta in the -direction Rayner-UPRC-2014 (). The oscillator lengths and for each value of have been optimized as to minimize the energy. Zero-point quantum rotational and vibrational energies have been added a posteriori to the HFB energies Rayner-UPRC-2014 (); Rayner-UEPJA-2014 (); Rayner-RaEPJA-2016 ().

Step 2) Using the HFB states (Step 1) as initial wave functions in the iterative mean-field procedure, we have computed an ”average” (AV) fission path for No (see, Fig. 2). In order to obtain the AV HFB states, calculations have been carried out as for an even-even nucleus but with a constrain on the mean value of the neutron number operator to be =151. We have used the single-particle basis with the (optimized) HO lengths and resulting from Step 1. Zero-point quantum corrections have also been added to the mean-field energies Rayner-UPRC-2014 (); Rayner-UEPJA-2014 (); Rayner-RaEPJA-2016 ().

Step 3) Using the HFB wave function corresponding to the absolute (normal deformed) minimum of the AV path (Step 2) in No as starting state, we have performed a set of HFB-EFA blocking calculations to identify the K=K value corresponding to the ground state. To this end, we have repeated the blocking procedure several times, using the same single-particle basis as in Step 2. We have obtained three different solutions of the HFB-EFA equations for each of the K values from K=1/2 up to 11/2. Larger K values have not been taken into account as the neutron single-particle levels corresponding to them are too far from the Fermi surfaces. In addition to K, the ground states obtained for all the odd-mass isotopes studied in this work are characterized by their parity . We will refer to them hereafter, as K configurations.

Step 4) Using the AV HFB states (Step 2) as starting wave functions, we have computed the K (i.e., ground state) fission path for No. Note that we are assuming that for an odd-mass nucleus with the ground state quantum numbers K, the spontaneous fission will take place in a configuration with the same K (parity can be broken in the fission process). For each (Q, Q)-configuration, we have considered the blocking of several quasiparticle states so as to obtain three different K-solutions of the HFB-EFA equations. From these three K-solutions, the one with the lowest energy is used to build the ground state fission path for No. In this step we use the same single-particle basis as in Step 2 and with the same oscillator lengths.

Let us mention, that in our Gogny-D1M mean-field calculations (Steps 1 to 4) the two-body kinetic energy correction has been fully taken into account in the Ritz-variational procedure while for the Coulomb exchange term we have considered the Slater approximation. The spin-orbit contribution to the pairing field has been neglected.

The computational cost to produce a plain potential energy surface requiring 75 constrained HFB calculations is of the order of five hours in a modern personal computer. If the oscillator lengths have to be optimized at each value then the cost is a factor of 5 higher (Step 1). The computational cost of the EFA is in principle similar to the one of HFB as time odd-fields are not considered. However, in the EFA case several initial blocked configurations (three in our case) have to be considered. In addition, the number of iteration required to solve the EFA problem is typically two or three times larger than the one to solve HFB and therefore the typical cost of an EFA calculation is ten times higher than the one of a HFB calculation.

For the rotational correction to the HFB-EFA energies, we have taken the expression . However, both and the Yocooz moment of inertia have been computed using the formulas for even-even nuclei RCORR-1 (); RCORR-2 (); RCORR-3 (). The reason is that an approximate angular momentum projection has not yet been carried out within the HFB-EFA framework. Work along these lines is in progress and will be reported elsewhere. On the other hand, the Adiabatic Time Dependent HFB (ATDHFB) framework has already been developed in the realm of quantum statistical mechanics ATDHFB-T-1 (); ATDHFB-T-2 (); ATDHFB-T-3 () allowing its verbatim extension to the HFB-EFA via its statistical density matrix operator EFA-jsut (). In particular, the ATD vibrational energy correction reads

(3)

For the evaluation of the collective mass in Eq. (3), we have resorted to the cranking approximation crankingAPPROX (); Giannoni-1 (); Giannoni-2 (); Libert-1999 (). In this case

(4)

with moments

(5)

and

(6)

In Eq. (6), Q and Q are the 11 and 20-components of the considered operator in the quasiparticle representation rs (). The matrix takes the form

(7)

where the f’s are the quasiparticle HFB-EFA occupation factors (1/2 for the and levels, respectively) while

(8)

where E represents the quasiparticle energy.

Figure 5: (Color online) The one-neutron S separation energies are plotted as functions of the neutron number. Experimental values are taken from Mass-Table-W ().

In the case of even-even nuclei, one usually considers the GCM collective masses in addition to the ATD ones Rayner-UPRC-2014 (); Rayner-UEPJA-2014 (); Rayner-RaEPJA-2016 (). Though the use of the former in the case of odd-mass systems lacks the theoretical justification available for the latter within the HFB-EFA, we have also considered their phenomenology in the present study in order to examine the impact of such a prescription on the computed t values. To this end, for the collective GCM masses in odd-mass No isotopes we have adopted the expression

(9)

and E is given by Eq. (3) but replacing the ATD mass with the GCM one. The moments are given by Eq. (5). For a thorough comparison of different forms of the collective masses in the framework of the Skyrme-EDF approximation, the reader is referred to Baran-TSF-2 ().

As a consequence of the cranking approximation, which essentially amounts to neglecting the residual interaction in the evaluation of collective masses, divergences might appear in the computation of the moments Eq. (5) due to level crossings in the denominator. A phenomenological way to deal with the problem is to replace those denominators by , with being a regularization factor introduced to simulate the effect of the residual interaction Egido-reg ().

With all the required quantities at hand, we have computed the spontaneous fission half-life t for the studied odd-mass nuclei using both the ATD and GCM schemes for the collective masses and zero-point vibrational corrections. For details regarding even-even isotopes the reader is referred to Rayner-UPRC-2014 (); Rayner-UEPJA-2014 (); Rayner-RaEPJA-2016 (). Within the Wentzel-Kramers-Brillouin (WKB) formalism Baran-TSF-1 (); Baran-TSF-2 () the t value (in seconds) is given by

(10)

where the action S along the (minimal energy, one-dimensional projected) fission path reads

(11)

In this expression, the integration limits a and b are the classical turning points proportional-1 () below the barrier corresponding to the energy E+E. The collective potential V(Q) is given by the HFB-EFA energy corrected by the zero-point rotational ) and vibrational energies. The parameter accounts for the true ground state energy once the zero-point quadrupole fluctuations are considered. Although it is not difficult to estimate its value using the curvature of the energy around the ground state minimum and the values of the collective inertias Baran-SF-2012 () we have followed the usual recipe Robledo-Giulliani (); Warda-Egido-Robledo-Pomorski-2002 () of considering it as a free parameter that takes four different values (i.e., =0.5, 1.0, 1.5 and 2.0 MeV). In this way we can estimate its impact on the predicted spontaneous fission half-lives. Note, that different values provide different classical turning points and in Eq. (11) and therefore modify the final value of the action integral. As in previous studies Rayner-UPRC-2014 (); Rayner-UEPJA-2014 (); Rayner-RaEPJA-2016 (), we have overlooked the E-dependence of the pre-factor in front of the exponential of the action in Eq. (10) due to the large uncertainties in the estimation of the spontaneous fission half-lives arising from other sources.

Finally, we have computed the -decay half-lives t using the Viola-Seaborg formula Viola-Seaborg ()

(12)

with the parameters A, B, C, D and h given in TDong2005 (). The value is obtained from the calculated binding energies for No and Fm nuclei.

Iii Discussion of the results

In this section we discuss the results of our Gogny-D1M calculations. First, in Secs. III.1 and III.2, we consider the nuclei No and No as illustrative outcomes of our calculations. A similar analysis, as described below, has been carried out for all the studied even-even and odd-mass No isotopes. The systematic of the fission paths and spontaneous fission half-lives in No is presented in Secs. III.3 and III.4, respectively.

iii.1 The nucleus No

The HFB plus the zero-point rotational energies, are plotted in panel (a) of Fig. 1 as functions of the quadrupole moment for No. The zero point vibrational energies are not included in the plot, as they are rather constant as functions of Q. However, they are always included in the computation of the spontaneous fission and -decay half-lives. The octupole and hexadecupole moments are given in panel (b). The ground state for No is located at Q = 16 b and is reflection symmetric. The fission isomer at Q = 52 b lies 1.38 MeV above the ground state from which, it is separated by the inner barrier (Q = 30 b) with the height of 10.75 MeV. As in the ground state case, the fission isomer is also reflection symmetric, a property that is shared for all the configurations belonging to the first barrier. However, as deduced from panel (b), octupole correlations play a role for quadrupole deformations Q 62 b. Those correlations significantly affect the outer barrier (Q = 70 b) whose height is 4.23 MeV.

Figure 6: ( Color online) The proton and neutron pairing interaction energies E corresponding to the ground states (GS) of No are plotted as functions of the neutron number N. The ”average” (AV) E values obtained for odd-mass isotopes are also included in the plot.

The proton and neutron pairing interaction energies E are depicted in panel (c). The neutron E values exhibit minima (maxima) at Q = 16, 40 and 88b (30, 68 and 120 b) while for protons those minima (maxima) correspond to Q = 18, 50 and 68b (30, 60 and 88 b). The minima of E are related to the minima of the potential energy surface (PES), a fact that can be understood in terms of level densities: minima of the PES correspond to low level density regions (Jahn-Teller effect) and the amount of pairing correlations is proportional to the level density around the Fermi level. On the other hand, maxima of the PES are associated to high level density regions implying stronger pairing correlations.

The collective GCM and ATD masses are plotted in panel (d). Their behavior is well correlated with the one of the pairing correlations in panel (c) and the inverse dependence of the collective mass with the square of the pairing gap proportional-1 (); proportional-2 (). We also observe that the ATD masses are systematically larger than the GCM ones. For example, for Q = 20 b we have obtained the ratio B/B = 1.77. Such differences can have a strong impact on the predicted fission half-lives Robledo-Giulliani (); Rayner-UPRC-2014 (); Rayner-UEPJA-2014 (); Rayner-RaEPJA-2016 () and are the reason why both kinds of collective masses have been considered in the computation of the t values. For example, for No and E = 1.0 MeV we have obtained t = 8.7919 s within the ATD scheme while t = 6.9928 s within the GCM one. Another relevant quantity is the parameter E, because increasing it reduces the integration interval in the action and this leads to smaller t values in either the ATD or GCM scheme. For example, once more in the case of No but with E = 1.5 MeV, we have obtained t = 5.7252 s and t = 4.7378 s within the ATD and GCM approaches, respectively. In the computation of the fission half-lives, the wiggles in the collective masses have been softened by means of a three point filter Rayner-UPRC-2014 (); Rayner-UEPJA-2014 (); Rayner-RaEPJA-2016 ().

iii.2 The nucleus No

In Fig. 2, we have plotted similar curves as in Fig. 1 but for the fission path of No corresponding to the K value of its ground state, i.e., K. Results corresponding to AV HFB calculations (see, Sec. II) for this odd-mass nucleus have also been included in each panel for the sake of comparison. As can be seen from panel (a) the K = 9/2 energy, computed as the HFB-EFA plus the rotational energy, displays as a function of the quadrupole moment, a behavior reminiscent of the one found for No. The absolute minimum of the K = 9/2 path appears at Q = 16 b and corresponds to a parity-conserving configuration [see, panel (b)] with = -1, which agrees well with the experimental EXP-NO () K = 9/2 ground state for this nucleus. The 9/2 fission isomer at Q = 52 b lies 2.22 MeV above the ground state from which, it is separated by the inner barrier (Q = 30 b) with the height of 11.67 MeV. The left-right symmetry is broken for Q 62 b and the height of the outer barrier (Q = 70 b) turns out to be 5.42 MeV. All these values are obtained with the K=9/2 configuration corresponding to the lowest energy for each value of the fission path. In this way, the number just given for the fission isomer might or might not correspond to the lowest energy configuration with the deformation of the fission isomer. In fact, in the case of No, the lowest energy configuration at Q = 52 b corresponds to K=11/2 and lies at an excitation energy of 2.04 MeV.

On the other hand, the comparison between the EFA and AV paths in panel (a) of Fig. 2, reveals that the former is always higher than the latter and the difference is not constant with the quadrupole moment. The EFA inner and outer barriers (11.67 and 5.42 MeV) are higher than the AV ones (10.91 and 4.62 MeV). The same holds for the EFA (2.22 MeV) and AV (1.63 MeV) excitation energy of the fission isomer. This is a clear manifestation of the specialization energy effect partly due to following the lowest energy configuration with a fixed K = 9/2 value.

Note, that the octupole and hexadecupole moments [panel (b)] corresponding to the AV and EFA fission paths follow a rather similar pattern revealing the minute impact of the blocked configuration in the mass moments defining the nuclear shape.

As can be seen from panel (c) of Fig. 2, the HFB-EFA (black dashed line) and AV (red dashed line) proton E values can hardly be distinguished. Both the HFB-EFA and AV neutron pairing energies also display a similar pattern. However, the former are much lower than the latter as a consequence of the quenching of pairing correlations due to the unpaired neutron. A direct consequence of this quenching also reflects in the enhancement of the HFB-EFA masses as compared to the AV ones regardless of the ATD and/or GCM scheme used [see, panel (d)]. From Fig. 2, one concludes that the effect of the unpaired neutron is to increase the fission barrier heights and an enhancement of the HFB-EFA collective masses with respect to the AV ones. Both effects go in the direction of increasing the collective action Eq. (11) and therefore, the t values Eq. (10) increase in odd-mass nuclei as compared to their even-even neighbors.

Once more, one sees that the ATD masses are larger than the GCM ones leading, for example, to the values t = 24.5590 s and t = 15.3318 s within the ATD and GCM schemes for No and E = 1.0 MeV. On the other hand, for E = 1.5 MeV we have obtained the values t = 20.8810 s and t = 12.9325 s within the ATD and GCM approaches, respectively. Due to these differences, and even when there is a lack of detailed theoretical justification for the use of GCM-like masses in the case of odd-mass systems, we have also considered both schemes in the computation of their spontaneous fission half-lives.

iii.3 Systematic of the fission paths in No

In Fig. 3, we have summarized the ground state fission paths obtained for No as functions of the quadrupole moment Q. The paths show two minima, the ground state one at Q = 16 b and the fission isomer around Q = 52 b. For all the studied nuclei, our Gogny-D1M calculations predict the super-deformed minimum to be above the normal deformed one though its excitation energy decreases with increasing neutron number [see, panel (b) of Fig. 7]. On the other hand, the top of the inner and outer barriers is located at Q 30 b and Q 70 b, respectively. For all the paths shown in the figure, the outer barriers belong to a parity-breaking sector that decreases their heights as compared with a reflection-symmetric situation. The specialization energy effect becomes apparent from the comparison between the ground state and AV fission paths in the case of odd-mass nuclei.

Figure 7: The inner barrier height E, excitation energy of the fission isomer E and outer barrier height E corresponding to the ground state (GS) fission paths shown in Fig. 3 are plotted in panels (a), (b) and (c) as functions of the neutron number. The ”average” (AV) values obtained for odd-mass isotopes are also included in the plot.

In good agreement with the experimental data EXP-NO (), our calculations predict 9/2, 1/2 and 3/2 ground states for No, No and No. This is not the case for the nuclei No and No where we have obtained 5/2 and 7/2 ground states while the experimental ones correspond to 7/2 and 9/2, respectively. However, for those nuclei we have found 7/2 and 9/2 one-quasiparticle configurations located inside the ground state well whose excitation energies are only 65.55 and 149.62 keV, respectively. Therefore, we have also explored the K = 7/2 and 9/2 fission paths in the case of No and No. They are plotted in Fig. 4 as functions of Q. As can be seen, the structure of those paths and the associated specialization energy effect resemble the ones in Fig. 3 (see also, the discussion below).

Figure 8: (Color online) The spontaneous fission half-lives t, predicted within the GCM [panel (a)] and ATD [panel (b)] schemes, for the isotopes No are depicted as functions of the neutron number N. For each isotope, calculations have been carried out with E = 0.5, 1.0, 1.5 and 2.0 MeV. The experimental t values for even-even isotopes are taken from Holden-paper (). For more details, see the main text.

Using the ground state energies obtained for No, we have computed the one-neutron S separation energies shown in Fig. 5. Though our calculations capture the experimental Mass-Table-W () odd-even staggering, the predicted amplitudes are larger than the experimental ones, especially for the heavier isotopes. This could be a consequence of an inappropriate pairing strength in the regions under study or the neglect of many-body effects like particle-vibration coupling or the restoration of the broken U(1) particle number symmetry. Another source of uncertainty is the neglect of time-odd fields in the EFA used in the present calculation that can contribute up to a few hundred keV to the binding energy of odd nuclei Rob.12 ().

The proton and neutron pairing interaction energies E corresponding to the ground states of No are plotted in Fig. 6 as functions of the neutron number N. The AV E values obtained for No are also included in the plot. For protons the pairing energies exhibit a smooth increase with increasing neutron number. As expected, they are rather similar to the AV ones for No. However, in the case of neutrons, the quenching of the pairing correlations induced by blocking is very severe. We have obtained the ratios E/E = 0.40, 0.21, 0.26, 0.56 and 0.62 for No. An alternative measure of how effective is the quenching of the neutron pairing correlations is provided by the ratio / = 0.528, 0.08, 0.332, 0.79 and 0.821 for the same nuclei. For the K = 7/2 (9/2) normal deformed minimum in No (No) we have obtained the ratios E/E = 0.29 (0.63) and / = 0.422 (0.824). These results indicate that for the neutron pairing correlations the quenching induced by the self-consistent blocking depends on the considered one-quasiparticle configuration. They also show that, at least for some of the studied odd-mass isotopes (for example, No) one is dealing with a weak pairing regime. Therefore, a more realistic account of the fission process in those systems would require a dynamical description of the neutron pairing correlations as well as their coupling to the relevant deformation parameters Action-Rayner ().

Nucleus K Scheme E
0.5 MeV 1.0 MeV 1.5 MeV 2.0 MeV
No 7/2 GCM 11.3558 8.7824 6.2380 3.5589
No 7/2 ATD 19.9848 16.0185 11.3774 7.0114
No 5/2 GCM 11.7274 8.8949 6.1555 3.4046
No 5/2 ATD 20.3827 16.2834 11.6087 7.3890
No 9/2 GCM 13.2903 10.6305 7.9765 5.3722
No 9/2 ATD 22.9660 18.8139 14.2703 8.0629
No 7/2 GCM 14.1852 11.3998 8.5118 5.1894
No 7/2 ATD 23.7159 19.1671 13.8127 8.0677
Table 1: The values of log t (in s) obtained for the K=7/2 and 5/2 fission paths in No are given as functions of the parameter E (in MeV). The same quantities are also given for the K=9/2 and 7/2 fission paths in No. For details, see the main text.

The inner barrier heights E, excitation energies of the fission isomers E and outer barrier heights E corresponding to the ground state fission paths shown in Fig. 3 are displayed in panels (a), (b) and (c) of Fig. 7 as functions of the neutron number N. The corresponding AV quantities for No are also depicted. The three quantities exhibit pronounced odd-even as well as specialization energy effects. In our calculations the largest E, E and E values are obtained for the nucleus No (N = 151). Moreover, for the K = 7/2 (9/2) fission path in No (No) we have obtained E = 10.88 (10.54) MeV, E = 1.28 (1.86) MeV and E = 3.83 (2.46) MeV. These values should be compared with the ones [i.e., 10.86 (10.65) MeV, 1.29 (1.36) MeV and 4.16 (2.42) MeV] shown in the figure which correspond to the ground state paths predicted for those nuclei. Let us mention, that we are aware of the reduction, by a few MeV, of the inner barrier heights once triaxiality is taken into account Rayner-UPRC-2014 (). In the case of odd-mass nuclei, the polarization effects induced by the unpaired nucleon might also lead to triaxial solutions. Though we have kept axial symmetry along the fission paths of the studied No isotopes, we have also corroborated the already mentioned reduction in the case of the even-even nuclei No and No. However, as such a reduction comes together with an increase in the collective inertia Baran-1981 (); Bender-1998 () that tends to compensate in the final value of the action, the role of triaxiality is very limited and has not been taken into account in the computation of the corresponding t values.

iii.4 Systematic of the spontaneous fission half-lives in No

The ground state spontaneous fission half-lives t, obtained within the GCM and ATD schemes, for the isotopes No are depicted, as functions of the neutron number N, in panels (a) and (b) of Fig. 8, respectively. In the panels we have included the available experimental data for even-even No isotopes Holden-paper (). Those data are more scarce for odd-mass nuclei for which only some recommended lower bound values (t 10 s for No, t 28 min for No and t 10 h for No) can be found in the literature Holden-paper (). However, in spite of being only lower bounds, those data already reveal an increase in the spontaneous fission half-lives of the odd-mass isotopes. In our calculations, for each of the studied isotopes, we have considered four values of E (i.e., E = 0.5, 1.0, 1.5 and 2.0 MeV). As can be seen from the figure, increasing this parameter leads to a decrease in the predicted spontaneous fission half-lives as well as to an improvement (exception made of No) of the agreement with the experiment in the case of even-even nuclei . The predicted ATD t values tend to be larger than the GCM ones, with the differences being more pronounced in the case of odd-mass nuclei. For example, for No (No) and E = 1.0 MeV we have obtained t = 9.1373 s ( t = 17.0551 s) within the GCM scheme while the corresponding value within the ATD approach is t = 11.4739 s ( t = 28.1227 s) [see also, Table 1]. Though there exists a strong variance of the theoretical fission rates with respect to the details involved in their computation, the same trend is always observed, i.e., regardless of the employed scheme and/or E value the fission half-lives exhibit a pronounced odd-even staggering. The amplitude of the staggering (i.e., the difference between the spontaneous fission lifetime of an odd-mass isotope and the one obtained by averaging the values of the two neighboring isotopes) is rather insensitive to the E value considered but it depends strongly both on neutron number and on the kind of collective inertia considered. The amplitude varies from the four orders of magnitude observed for the N=149 and N=157 isotopes in the calculation with the GCM inertias to the 16 orders of magnitude for the No isotope with N=155 in the ATD calculation. The amplitude of the staggering is larger in the ATD than in the GCM case (up to a factor 1.6) as can be expected from the values of the corresponding inertias. The dependence with neutron number shows a seesaw pattern with large values at N=151 and N=155. On the other hand, the amplitude of the staggering seems to be an interesting quantity to be compared with experimental data given its insensitivity to the E values. Similar features to the ones just discussed also emerge by comparing the spontaneous fission half-lives, given in Table 1, for the 5/2 and 7/2 fission paths in No as well as the 9/2 and 7/2 ones in No (see, Figs. 3 and 4).

Figure 9: (Color online) The theoretical and experimental Mass-Table-W () -decay half-lives are plotted as functions of the neutron number N. They are compared in panel (a) [panel (b)] with the spontaneous fission half-lives obtained within the GCM (ATD) scheme with E = 2.0 MeV. For more details, see the main text.

In Fig. 9, we have plotted the -decay half-lives computed with a parametrization TDong2005 () of the Viola-Seaborg formula Viola-Seaborg (). To this end, we have used the binding energies obtained for the corresponding nobelium and fermium isotopes. The -decay half-lives obtained in our Gogny-D1M calculations compare well with the experimental ones Mass-Table-W () as could be expected from the good performance of the Gogny-D1M EDF in reproducing experimental binding energies and therefore in reproducing the values entering the Viola-Seaborg formula. In panel (a), we have also included the spontaneous fission half-lives predicted within the GCM scheme (E = 2.0 MeV). From the comparison between the t and GCM t values we conclude that -decay is the dominant decay channel for the odd-mass No isotopes considered even for No that lies in between two even-even nuclei where spontaneous fission dominates. The same conclusion can be obtained from the comparison with the corresponding ATD fission lifetimes in panel (b).

Iv Conclusions

In this paper, we have considered the fission properties of a set of nobelium isotopes including even-even and odd-mass nuclei. The selected set comprises No as a representative sample. We have resorted to the constrained Gogny-D1M mean-field approximation as our main theoretical tool. In particular, the even-even systems have been studied within the standard HFB framework while for the odd-mass isotopes we have used the HFB-EFA in order to reduce the already substantial computational effort. Besides the proton and neutron number operators, we have employed constrains on the axially symmetric quadrupole , octupole and operators. We have presented a detailed description of the (blocking) methodology used to obtain the (least energy) fission paths in the studied odd-mass isotopes while for the even-even ones calculations have been carried out along the lines discussed in previous fission studies Rayner-UPRC-2014 (); Rayner-UEPJA-2014 (); Rayner-RaEPJA-2016 (). Zero-point rotational and vibrational corrections have always been added to the HFB and/or HFB-EFA energies a posteriori. The rotational correction has been found in terms of the Yoccoz moment of inertia while both the GCM and ATD schemes have been used to compute the collective masses and vibrational corrections. All the required mean-field building blocks have then been used to compute the GCM and/or ATD ground state spontaneous fission half-lives t for No within the WKB formalism. The -decay half-lives t have been found, using the binding energies obtained for the corresponding No and Fm nuclei, with the help of a parametrization of the Viola-Seaborg formula.

For both even-even and odd-mass No isotopes our Gogny-D1M calculations provide ground state fission paths with normal deformed and isomeric minima as well as inner and outer barriers whose quadrupole deformations, remain almost constant as functions of the neutron number. In all cases, the outer barriers belong to a parity-breaking sector that decreases their heights as compared to a reflection-symmetric situation. The inner and outer barrier heights as well as the isomer excitation energies display pronounced odd-even effects as functions of the neutron number with maxima at N=151. As a result of following configurations with a fixed K quantum number, the ground state fission paths obtained for the odd-mass nuclei are always higher than the AV ones. This specialization energy effect together with the pronounced quenching of neutron pairing correlations (leading to an enhancement of the GCM and/or ATD collective masses as compared to the AV ones) are fully taken into account via the Ritz-variational minimization of the HFB-EFA energies.

The present calculations represent a first step towards a better understanding of the physics of spontaneous fission in odd-mass nuclei and there is still room for further improvements. For example, the evaluation of the collective inertias relies on approximations that are not fully understood in the finite temperature case which is used to justify the formulas within our EFA methodology. This lack of full understanding leads to phenomenological prescriptions like the one used here to avoid potential divergences when two-quasiparticle energies cross. Another relevant aspect is the kind of treatment used to find the fission path: a dynamical treatment favors stronger pairing correlations than the least energy one employed in the present study and this could substantially change not only the specialization energy but also the collective inertias. Finally, the lowering of the first barrier due to triaxiality and its impact on the spontaneous fission life-time in the case of odd-nuclei has still to be elucidated. There is no doubt that all these, still unresolved, aspects of the spontaneous fission in odd-mass nuclei deserve further studies in the future. Work along these lines is in progress and will be reported elsewhere.

Acknowledgements.
The work of LMR was supported by the Spanish grants FIS2012-34479-P MINECO, FPA2015-65929-P MINECO and FIS2015-63770-P MINECO.

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