A. Staszczak, A. Baran, J. Dobaczewski, and W. Nazarewicz
April 18, 2009
Seventy years ago Joliot Curie and Savitch  discovered that the exposure of uranium to neutrons leads to the existence of lanthanum. Following this finding, Hahn and Strassmann  proved definitively that bombarding uranium with neutrons produces alkali earth elements, ushering in what has come to be known as the atomic age. The term nuclear fission was coined one year later by Meitner and Frisch , who explained experimental results in terms of the division of a heavy nucleus into two lighter nuclei. In 1939, Bohr and Wheeler  developed a theory of fission based on a liquid drop model. Interestingly, their work also contained an estimate of a lifetime for fission in the ground state. Soon afterwards, Petrzhak and Flerov  presented the first experimental evidence for such spontaneous fission (SF).
While early descriptions of fission were based on a purely geometrical framework of the nuclear liquid drop model  (i.e., shape-dependent competition between Coulomb and surface energy), it was soon realized  that the single-particle motion of protons and neutrons moving in a self-deforming mean field is crucial for the understanding of a range of phenomena such as fission half-lives, mass and energy distributions of yields, cross sections, and fission isomers [7,8]. In the macroscopic-microscopic method (MMM) proposed by Swiatecki  and Strutinsky et al. [10,11], quantum shell effects are added atop the average (or macroscopic) behavior described by the liquid drop, and this approach turned out to be very successful in explaining many features of SF [12,13,14].
Quantum mechanically, fission represents a time-dependent solution of the many-body Schrödinger equation where all particles move collectively. To fully solve such a time-dependent problem for more than 200 particles is neither possible nor sensible because the essence of the process is in its coherence. Consequently, most of the essential physics should be contained in underlying mean fields. This determines the choice of a microscopic tool to be used: the nuclear density functional theory (DFT) . The advantage of DFT is that, while treating the nucleus as a many-body system of fermions, it provides an avenue for identifying the essential collective degrees of freedom.
Because the commonly used nuclear density functionals are usually adjusted to nuclear ground-state properties and infinite nuclear matter, and most applications are symmetry-restricted to speed up computations, self-consistent theories typically are not as quantitative as MMM when it comes to SF, except for some cases . It is only recently that an effort has been made to systematically optimize the effective forces by considering experimental data relevant to large deformations .
Many observed fission characteristics can be traced back to topologies of fission pathways in multidimensional collective space . For that reason, allowing for arbitrary shapes on the way to fission is the key. The main goal of this study is to provide a microscopic description of multi-mode fission based on the nuclear DFT. To this end, we search for the optimum collective trajectory in a multidimensional space. The barrier penetration probability, or a fission half-life, is computed by integrating the action along this optimum path. In practice, this is done by constraining the nuclear collective coordinates associated with shape deformations to have prescribed values of the lowest multipole moments, by which we explore the main degrees of freedom related to elongation , reflection-asymmetry , and necking . The effects due to triaxiality are known to be important around the top of the first fission barrier . We confirm this finding by studying the stability of axial shapes by including the triaxial quadrupole moment .
At each point, fully self-consistent DFT equations are solved, whereupon the total energy of the system is always minimized with respect to all remaining (i.e., unconstrained) shape parameters. The optimum path is then localized in the form of multipole moments, , , and , usually becoming functions of the driving moment, . The calculations were carried out using a symmetry-unrestricted DFT program based on the Hartree Fock-Bogoliubov solver HFODD [19,20] capable of treating simultaneously all the possible collective degrees of freedom that might appear on the way to fission. Based on this DFT framework, we calculated the collective inertia (collective masses) and zero point energy (ZPE) corrections to account for quantum fluctuations.
In the particle-hole channel, we use the SkM energy density functional  that has been optimized at large deformations; hence it is often used for fission barrier predictions. In the pairing channel, we adopted a seniority pairing force with the strength parameters fitted to reproduce the experimental gaps in Fm . Because the nuclei considered are all well bound, pairing could be treated within the BCS approximation. The single-particle basis consisted of the lowest 1,140 stretched states originating from the lowest 31 major oscillator shells.
In the analysis of fission pathways, we explored multidimensional collective space. (For examples of 1D energy surfaces obtained in our DFT model, we refer the reader to the previous study .) To separate fission pathways, we computed energy surfaces in the deformation spaces and . The calculations were not limited to axial shapes; triaxial deformations appear if energetically favorable (e.g., within the inner barrier). The vibrational and rotational ZPE corrections and the cranking quadrupole mass parameter were calculated as described in Ref. . The spontaneous fission half-lives were estimated from the WKB expression for the double-humped potential barrier [24,25] assuming a 1D tunneling path along .
To demonstrate the validity and generality of our method, we chose a case where several fission pathways were known to coexist and all intrinsic symmetries of the nuclear mean field were broken. In this respect, a phenomenon known as bimodal fission, observed in several fermium and transfermium nuclei [26,27,28,29], is a perfect testing ground. It manifests itself, for example, in a sharp transition from an asymmetric mass division in Fm and No to a symmetric mass split in Fm and No. Furthermore, the total kinetic energy distributions of the fission fragments appear to have two peaks centered around 200 and 233 MeV. It has been suggested [28,30,31,32,33,34] that the higher energy fission mode corresponds to a scission configuration associated with two touching, nearly spherical, fragments with the maximal Coulomb repulsion, whereas the lower-energy mode can be associated with more elongated fragments. Before this work, bimodal fission was studied within the MMM [30,31,32,33,13] and nuclear DFT [35,36,37,38,39]. All those studies were symmetry-restricted (i.e., they did not consider simultaneous inclusion of elongation, triaxiality, and reflection-asymmetry).
The triaxial deformations are important around the first (inner) fission barrier, and they reduce the fission barrier height by several MeV. Beyond the first barrier, at b, a reflection-asymmetric path corresponding to asymmetric elongated fragments (aEFs) branches away from the symmetric valley, see Fig. 1a. At b, a reflection-symmetric path splits into two branches: one corresponding to nearly spherical symmetric compact fragments (sCFs) and one associated with symmetric elongated fragments (sEFs). This bifurcation is clearly seen in Fig. 1b. Such three fission pathways were predicted in early work based on MMM [30,32] and also found recently within a DFT framework [37,38], except that the axial-symmetry was enforced in all these studies. A pattern of similarly competing fission valleys was found for all investigated isotopes.
It should be emphasized that the pathways correspond to different regions of the collective space and this is apparent when studying them in more than one dimension. Indeed, aEF is well separated from sCF and sEF in (the apparent crossing between aEF and sEF in Fig. 1b is an artifact of the 2D projection) while the symmetric trajectories sCF and sEF strongly differ in the values of higher multipole moments =4, 6, and 8 (see the inset in Fig. 1). We wish to stress that it is only through 2D surface analysis one can confirm that the pathological behavior discussed in Refs. [12,40,41] does not happen.
The energy curves corresponding to individual pathways in Fm were discussed in our earlier work  where the diagrams illustrating competition between different fission valleys can be found. While the asymmetric pathway aEF is favored in Fm and in the lighter Fm isotopes, both symmetric paths are open for Fm, due to the disappearance of the outer fission barrier in sCF and sEF. It is to be noted that the symmetric pathways sCF and sEF in Fm are predicted to bifurcate away well outside the first barrier (see also recent work  based on MMM). In the case of Fm, we find that there is no outer potential barrier along the sCF trajectory, and the sEF and aEF paths lie significantly higher in the outer region.
To assess the SF half-lives theoretically, we calculated the collective inertia parameter along and performed WKB barrier penetration calculations for even-even fermium isotopes with 242A260. We assumed two values of the ground state energy counted from the ground state potential energy minimum: =0.3MeV and the commonly used value  of 0.5MeV. The resulting SF half-lives are shown in Fig. 3. In spite of a fairly simple 1D penetration picture, it is satisfying to see a quantitative agreement between experiment [44,45] and theory (for Gogny-DFT results, see ). The existence of a small outer barrier in Fm is significant as it increases the fission half-life in this nucleus by more than four orders of magnitude compared to that of Fm, thus explaining the rapid change in experimental SF half-lives between these nuclei.
To map out the competition between different fission pathways in the heaviest elements, we carried out systematic calculations for even-even nuclei with 98108 and 154160. A transition from the usual asymmetric fission channel seen in the actinides to compact symmetric fission is seen when moving towards Fm. In the intermediate region of bimodal fission, two symmetric channels coexist. Around Sg (=106, =154), our calculations predict trimodal fission, i.e., competition between the asymmetric fission valley and two symmetric ones. (The term ``multimodal fission" has been previously used by M.G. Itkis et al.  in the context of fusion-fission and quasi-fission of hot superheavy nuclei produced in heavy ion reactions.)
The representative fission pathways for Cf (asymmetric fission), Fm (symmetric compact fission), No (bimodal fission) and Hs (trimodal fission) are displayed in Fig. 4. The inclusion of triaxiality significantly reduces the inner barrier in Cf, Fm, and No while the effect in Hs is much weaker.
In summary, the symmetry-unrestricted nuclear DFT framework has been applied to the problem of SF. As an example, we studied the challenging case of static SF pathways in Fm, Fm, and Fm and in a number of neighboring nuclei. We found competition between symmetric-compact, symmetric-elongated, and asymmetric elongated fission valleys that is consistent with the observed distribution of fission yields. The saddle points obtained in constrained 1D calculations were confirmed through an analysis of 2D energy surfaces. From the calculated collective potential and collective mass, we estimated SF half-lives, and good agreement with experimental data was found. Finally, we predicted trimodal fission for several rutherfordium, seaborgium, and hassium isotopes.
It is worth noting that calculations of self-consistent energy 2D surfaces are computer intensive. Because a single HFODD run with all self-consistent symmetries broken takes about 60 minutes of CPU time, it takes about 3 CPU-years to carry out the full fission pathway analysis for 24 nuclei; hence, massively parallel computer platforms had to be used.
In the near term, we intend to improve the theory of SF half-lives by considering multidimensional inertia tensors and by performing the direct minimization of the collective action in a multidimensional collective space . In the long term, the theory will be extended to account for nonadiabatic effects (e.g., along the lines of Refs. [48,49]). In addition, quality microscopic input for fission calculations is needed. Of particular importance is the development of the nuclear energy density functional better reproducing both bulk nuclear properties and spectroscopic data.
This work was supported in part by the National Nuclear Security Administration under the Stewardship Science Academic Alliances program through U.S. Department of Energy Research Grant DE-FG03-03NA00083; by the U.S. Department of Energy under Contract Nos. DE-FG02-96ER40963 (University of Tennessee), DE-AC05-00OR22725 with UT-Battelle, LLC (Oak Ridge National Laboratory), and DE-FC02-07ER41457 (UNEDF SciDAC Collaboration); by the Polish Ministry of Science and Higher Education under Contract Nos. N202 179 31/3920 and N N 202 328234; and by the Academy of Finland and University of Jyväskylä within the FIDIPRO program. Computational resources were provided by the National Center for Computational Sciences at Oak Ridge National Laboratory.