Thermal effects on the Fission Barrier of neutron-rich nuclei
Futoshi Minato, Kouichi Hagino
备注：
4 pages, 6 figures. Uses aipxfm.sty. A talk given at the Nuclear Physics and Astrophysics : From Stable Beams to Exotic Nuclei, June 25-30, 2008, Cappadocia, Turkey
We discuss the fission barrier height of neutron-rich nuclei in a r-process site at highly excited state, which is resulted from the beta-decay or the neutron-capture processes. We particularly investigate the sensitivity of the fission barrier height to the temperature, including the effect of pairing phase transition from superfluid to normal fluid phases. To this end, we use the finite-temperature Skyrme-Hartree-Fock-Bogolubov method with a zero-range pairing interaction. We also discuss the temperature dependence of the fission decay rate.
The fission processes of thermal excited nuclei are conventionally studied by statistical models which rely on inputs of phenomenological level densities and potential barriers. Therefore the microscopic descriptions of spontaneous fission and induced fission are very desirable for a unified understanding of various fission processes. We propose to study the fission rates, at both low and high temperatures, with microscopically calculated temperature-dependent fission barriers and collective mass parameters. The fission barriers are calculated by the finite-temperature Skyrme-Hartree-Fock+BCS …
A systematic analysis of the ground state and fission properties of actinides and superheavy nuclei important for the $r$ process modeling has been performed within the framework of covariant density functional theory for the first time in Ref. [1]. A brief review of the results related to the heights of primary fission barriers and systematic uncertainties in their prediction is presented. In addition, new results on the potential impact of the isospin dependence of pairing on fission barriers in fission cycling regions is provided for the first time.
We evaluate the performance of modern nuclear energy density functionals for predicting inner and outer fission barrier heights and energies of fission isomers of even-even actinides. For isomer energies and outer barrier heights, we find that the self-consistent theory at the HFB level is capable of providing quantitative agreement with empirical data.
Systematic calculations of fission barriers allowing for triaxial deformation are performed for even-even superheavy nuclei with charge number $Z=112-120$ using three classes of covariant density functional models. The softness of nuclei in the triaxial plane leads to an emergence of several competing fission pathes in the region of the inner fission barrier in some of these nuclei. The outer fission barriers are considerably affected by triaxiality and octupole deformation. General trends of the evolution of the inner and the outer fission barrier heights are discussed as a function of the pa…
We discuss the sensitivity of fission barrier for heavy neutron-rich nuclei to fission paths in the two dimensional neutron-proton quadrupole plane. To this end, we use the constrained Skyrme-Hartree-Fock + BCS method, and examine the difference of fission barriers obtained with three constraining operators, that is, the neutron, proton, and mass quadrupole operators. We investigate $^{220}$U, $^{236}$U, and $^{266}$U, %from proton-rich to neutron-rich uranium isotopes, that is relevant to r-process nucleosynthesis. We find that the fission barrier heights are almost the same among the three c…
Relativistic mean field theory allowing for triaxial deformations is applied for a systematic study of fission barriers in the actinide region. Different pairing schemes are studied in details and it is shown that covariant density functional theory is able to describe fission barriers on a level of accuracy comparable with non-relativistic calculations, even with the best phenomenological macroscopic+microscopic approaches. Triaxiality in the region of the first saddle plays a crucial role in achieving that.
Two systematic sources of error in most current microscopic evaluations of fission-barrier heights are studied. They are concerned with an approximate treatment of the Coulomb exchange terms (known as the Slater approximation) in the self-consistent mean fields and the projection on good parity states (e.g., of positive parity for the spontaneous fission of an even-even nucleus) of left-right reflection asymmetric intrinsic solutions (e.g., around the second barrier). Approximate or unprojected solutions are shown to lead each to an underestimation of the barrier heights by a few hundred keV.
A simple phenomenological model, based on the Świ{\c a}tecki idea for evaluation of the spontaneous fission half-lives is proposed. The model contains only one adjustable parameter fixed to the data for even-even nuclei and two additional hindrance factors to the life-times, which give the effect of odd particles. A good agreement with the experimental data for all fissioning nuclei is achieved.
We present fission-barrier-height calculations for nuclei throught the Periodic Table based on a realistic macroscopic-microscopic model. Compared to other calculations: (1) we use a deformation space of sufficiently high dimension, sampled densely enough to describe the relevant topography of the fission potential, (2) we unambiguously find the physically relevant saddle points in this space, and (3) we formulate our model so that we obtain continuity of the potential energy at the division point between a single system and separated fission fragments or colliding nuclei, allowing us to (4) d…
The restoration of the parity symmetry has been performed in the framework of the Highly Truncated Diagonalization Approach suited to treat correlations in an explicitly particle-number conserving microscopic approach. To do so we have assumed axial symmetry and used a generalized Wick's theorem due to Löwdin in a projection-after-variation scheme. We have chosen the Skyrme SkM$^*$ energy-density functional for the particle-hole channel and a density-independent delta force for the residual interaction. We have applied this approach in the region of the outer fission barrier of the $^{240}$Pu …
With a help of the selfconsistent Hartree-Fock-Bogoliubov (HFB) approach with the D1S effective Gogny interaction and the Generator Coordinate Method (GCM) we incorporate the transverse collective vibrations to the one-dimensional model of the fission-barrier penetrability based on the traditional WKB method. The average fission barrier corresponding to the least-energy path in the two-dimensional potential energy landscape as function of quadrupole and octupole degrees of freedom is modified by the influence of the transverse collective vibrations along the nuclear path to fission. The set of…
Static fission barriers for 95 even-even transuranium nuclei with charge number $Z=94-118$ have been systematically investigated by means of pairing self-consistent Woods-Saxon-Strutinsky calculations using the potential energy surface approach in multidimensional ($β_2$, $γ$, $β_4$) deformation space. Taking the heavier $^{252}$Cf nucleus (with the available fission barrier from experiment) as an example, the formation of the fission barrier and the influence of macroscopic, shell and pairing correction energies on it are analyzed. The results of the present calculated $β_2$ values and barrie…
In this contribution, we present the main relations of the Langevin approach to the description of fission or fusion-fission reactions. The results of Langevin calculations are shown for the mass distributions of fission fragments of super-heavy elements and used for the investigation of memory effects in nuclear fission.
This is a short review of methods and results of calculations of fission barriers and fission half-lives of even-even superheavy nuclei. An approvable agreement of the following approaches is shown and discussed: The macroscopic-microscopic approach based on the stratagem of the shell correction to the liquid drop model and a vantage point of microscopic energy density functionals of Skyrme and Gogny type selfconsistently calculated within Hartree-Fock-Bogoliubov method. Mass parameters are calculated in the Hartree-Fock-Bogoliubov cranking approximation. A short part of the paper is devoted t…
We investigate the systematics of fission barriers in superheavy elements in the range Z = 108-120 and N = 166-182. Results from two self-consistent models for nuclear structure, the relativistic mean-field (RMF) model as well as the non-relativistic Skyrme-Hartree-Fock approach are compared and discussed. We restrict ourselves to axially symmetric shapes, which provides an upper bound on static fission barriers. We benchmark the predictive power of the models examining the barriers and fission isomers of selected heavy actinide nuclei for which data are available. For both actinides and super…
A systematic study of fission-barrier dependence on excitation energy has been performed using the self-consistent finite-temperature Hartree-Fock+BCS (FT-HF+BCS) formalism with the SkM* Skyrme energy density functional. The calculations have been carried out for even-even superheavy nuclei with Z ranging between 110 and 124. For an accurate description of fission pathways, the effects of triaxial and reflection-asymmetric degrees of freedom have been fully incorporated. Our survey demonstrates that the dependence of isentropic fission barriers on excitation energy changes rapidly with particl…
The fission barriers of the nuclei 254Fm, 256Fm, 258Fm, 258No and 260Rf are investigated in a fully microscopic way up to the scission point. The analysis is based on the constrained Hartree-Fock-Bogoliubov theory and Gogny's D1S force. The quadrupole, octupole and hexadecapole moments as well as the number of nucleons in the neck region are used as constraints. Two fission paths, corresponding to the bimodal fission, are found. The decrease with isotope mass of the half-life times of heavy Fm isotopes is also explained.
A configuration-interaction model is presented for the barrier region of induced fission. The configuration space is composed of seniority-zero configurations constructed from self-consistent mean-field wave functions. The Hamiltonian matrix elements between configurations include diabatic and pairing interactions between particles. Other aspects of the Hamiltonian are treated statistically, guided by phenomenological input of compound-nucleus transmission coefficients. In this exploratory study the configuration space is restricted to neutron excitations only. A key observable calculated in t…
A comparative study of fission of actinides specially $^{238}$U, by proton and bremsstrahlung photon is performed. Relative mass distribution of $^{238}$U fission fragments have been explored theoretically for both proton and photon induced fission. The integrated yield along with charge distribution of the products are calculated to find out the neutron richness in comparison to the nuclei produced by r-process in nucleosynthesis. Some r-process nuclei in intermediate mass range for symmetric fission mode are found to be produced almost two order of magnitude more for proton induced fission t…
The current status of the application of covariant density functional theory to the description of fission barriers in actinides and superheavy nuclei is reviewed. The achievements and open problems are discussed.
In the frame of proposed statistical method the calculation of fission-fragment yields mass and charge spectra for pre- actinide nuclei on example of natPb isotopes had been conducted.
The concept of fission barrier - a parameter which enters in quantitative estimates of various observables related to nuclear fission - is presented from the point of view of theory based on the picture of nuclear deformation and energy dependent on it. We describe the macroscopic-microscopic method of calculating energy landscapes which is simpler than the selfconsistent mean field approach, and, due to its two-component nature, seems to be easier to adjust to experimental data. We present some models and methods used to find the fission saddles. For the purpose of illustration, we present re…
In order to describe heavy-ion fusion reactions around the Coulomb barrier with an actinide target nucleus, we propose a model which combines the coupled-channels approach and a fluctuation-dissipation model for dynamical calculations. This model takes into account couplings to the collective states of the interacting nuclei in the penetration of the Coulomb barrier and the subsequent dynamical evolution of a nuclear shape from the contact configuration. In the fluctuation-dissipation model with a Langevin equation, the effect of nuclear orientation at the initial impact on the prolately defor…
Background: Fission barriers of actinide nuclei have been mostly and for long been microscopi- cally calculated for even-even fissioning systems. Calculations in the case of odd nuclei have been performed merely within a so-called equal-filling approximation (EFA) as opposed to an approach taking explicitly into account the time reversal breaking properties at the mean field level- and for only one single-particle configuration. Purpose: We study the dependence of the fission barriers on various relevant configurations (e.g. to evaluate the so-called specialization energy). Besides, we want to…
After shortly analyzing data relevant to fission hindrance of odd-A nuclei and high-$K$ isomers in superheavy (SH) region we point out the inconsistency of current fission theory and propose an approach based on the instanton formalism. A few results of this method, simplified by replacing selfconsistency by elements of the macro-micro model, are given to illustrate its features.
The double-humped structure of many actinide fission barriers is well established both experimentally and theoretically. There is also evidence, both experimental and theoretical, that some actinide nuclei have barriers with a third minimum, outside the second, fission-isomeric minimum. We perform a large-scale, systematic calculation of actinide fission barriers to identify which actinide nuclei exhibit third minima. We find that only a relatively few nuclei accessible to experiment exhibit third minima in their barriers, approximately nuclei with proton number $Z$ in the range $88 \leq Z \le…
Theoretical uncertainties in the predictions of inner fission barrier heights in superheavy elements have been investigated in a systematic way for a set of state-of-the-art covariant energy density functionals which represent major classes of the functionals used in covariant density functional theory. They differ in basic model assumptions and fitting protocols. Both systematic and statistical uncertainties have been quantified where the former turn out to be larger. Systematic uncertainties are substantial in superheavy elements and their behavior as a function of proton and neutron numbers…
We present a systematics of fission barriers and fission lifetimes for the whole landscape of super-heavy elements (SHE), i.e. nuclei with Z>100. The fission lifetimes are also compared with the alpha-decay half-lives. The survey is based on a self-consistent description in terms of the Skyrme-Hartree-Fock (SHF) approach. Results for various different SHF parameterizations are compared to explore the robustness of the predictions. The fission path is computed by quadrupole constrained SHF. The computation of fission lifetimes takes care of the crucial ingredients of the large-amplitude collect…
Fission dynamics of excited nuclei is studied in the framework of Langevin equation. The one body wall-and-window friction is used as the dissipative force in the Langevin equation. In addition to the usual wall formula friction, the chaos weighted wall formula developed earlier to account for nonintegrability of single-particle motion within the nuclear volume is also considered here. The fission rate calculated with the chaos weighted wall formula is found to be faster by about a factor of two than that obtained with the usual wall friction. The systematic dependence of fission width on temp…
Spontaneous fission and alpha decay are the main decay modes for superheavy nuclei. The superheavy nuclei which have small alpha decay half-life compared to spontaneous fission half-life will survive fission and can be detected in the laboratory through alpha decay. We have studied the alpha decay half-life and spontaneous half-life of some superheavy elements in the atomic range Z = 100-130. Spontaneous fission half-lives of superheavy nuclei have been calculated using the phenomenological formula and the alpha decay half-lives using Viola-Seaborg-Sobiczewski formula (Sobiczewski et al. 1989)…
The dependence of fission barriers on the excitation energy of the compound nucleus impacts the survival probability of superheavy nuclei synthesized in heavy-ion fusion reactions. In this work, we investigate the isentropic fission barriers by means of the self-consistent nuclear density functional theory. The relationship between isothermal and isentropic descriptions is demonstrated. Calculations have been carried out for $^{264}$Fm, $^{272}$Ds, $^{278}$112, $^{292}$114, and $^{312}$124. For nuclei around $^{278}$112 produced in "cold fusion" reactions, we predict a more rapid decrease of f…
The neutron and proton density distributions obtained in constrained Hartree-Fock-Bogolyubov calculations with the Gogny force along the fission paths of 232Th, 236U, 238U and 240Pu are analyzed. Significant differences in the multipole deformations of neutron and proton densities are found. The effect on potential energy surfaces and on barrier heights of an additional constraint imposing similar spatial distributions to neutrons and protons, as assumed in macroscopic-microscopic models, is studied.
We systematically calculate the spontaneous fission half-lives for heavy and superheavy nuclei between U and Fl isotopes. The spontaneous fission process is studied within the semi-empirical WKB approximation. The potential barrier is obtained using a generalized liquid drop model, taking into account the nuclear proximity, the mass asymmetry, the phenomenological pairing correction, and the microscopic shell correction. Macroscopic inertial-mass function has been employed for the calculation of the fission half-life. The results reproduce rather well the experimental data. Relatively long hal…
The impact of pairing correlations on the fission barriers is investigated in Relativistic Hartree Bogoliubov (RHB) theory and Relativistic Mean Field (RMF)+BCS calculations. It is concluded that the constant gap approximation in the usual RMF+BCS calculations does not provide an adequate description of the barriers. The RHB calculations show that there is a substantial difference in the predicted barrier heights between zero-range and finite range pairing forces even in the case when the pairing strengths of these two forces are adjusted to the same value of the pairing gap at the ground stat…
We discuss the effect of kinetic energy of the relative motion becoming spurious for separate fragments on the selfconsistent mean-field fission barriers. The treatment of the relative motion in the cluster model is contrasted with the necessity of a simpler and approximate approach in the mean-field theory. A scheme of the energy correction to the Hartree-Fock is proposed. The results obtained with the effective Skyrme interaction SLy6 show that the correction, previously estimated as $\sim$ 8 MeV in $A=70-100$ nuclei, amounts to 4 MeV in the medium heavy nucleus $^{198}$Hg and to null in $^{…
Spontaneous fission barriers, quadrupole inertia, and zero-point quadrupole-energy corrections are calculated for 252,256,258Fm in the framework of the self-consistent Skyrme-Hartree-Fock+BCS theory. Two ways of computing dynamical inertia are employed: the Gaussian Overlap Approximation to the Generator Coordinate Method and cranking ansatz. The Skyrme results are compared with those of the Gogny-Hartree-Fock-Bogolyubov model.
Recent progress in the theory of fission of hot nuclei is reported. We discuss in particular the properties of the friction formfactor as function of the deformation (and possibly of the temperature) which are necessary to reproduce data concerning fission of hot nuclei and its accompanying light particle and $γ$-ray emission. Recent theoretical work gives support to a phenomenological friction form factor (proposed some time ago), which is weak for compact shapes and increases on the way to scission.
We report here the signature of bi-modal fission, one asymmetric and the other symmetric, in Uranium nuclei in the mass range A = 230 to 236. The finding is unexpected and striking and is based on a model independent analysis of experimental mass distributions (cumulative yields) at various excitations from about 23 to 66 MeV in the alpha induced fission of 232Th. It has been found that the observed asymmetry in the mass distributions and the unusually narrow peak in the symmetry region, can both be explained in a consistent manner if one assumes: a) multi-chance fission, b) bi-modal fission a…
The deformation barriers standing in the quasi-molecular shape path have been determined in the actinide region within a macroscopic-microscopic energy derived from a generalized liquid drop model, the algebraic droplet model shell corrections and analytic expressions for the pairing energies. Double and triple-humped fission barriers appear. The second barrier corresponds to the transition from one-body shapes to two touching ellipsoids. The third minimum and third peak, when they exist, come from shell rearrangements in the deformed fragment. The shape of the other almost magic one is close …
We present a comparison of our model treating fission dynamics in conjunction with light-particle (n, p, alpha) evaporation with the available experimental data for the nuclei 126Ba, 188Pt and three isotopes of the element Z=110. The dynamics of the symmetric fission process is described through the solution of a classical Langevin equation for a single collective variable characterizing the nuclear deformation along the fission path. A microscopic approach is used to evaluate the emission rates for pre-fission light particles. Entrance-channel effects are taken into account by generating an i…
We found that a true ternary fission with formation of a heavy third fragment (a new type of radioactivity) is quite possible for superheavy nuclei due to the strong shell effects leading to a three-body clusterization with the two doubly magic tin-like cores. The simplest way to discover this phenomenon in the decay of excited superheavy nuclei is a detection of two tin-like clusters with appropriate kinematics in low-energy collisions of medium mass nuclei with actinide targets. The three-body quasi-fission process could be even more pronounced for giant nuclear systems formed in collisions …
Fission properties of the actinide nuclei are deduced from theoretical analysis. We investigate potential energy surfaces and fission barriers and predict the fission fragment mass-yields of actinide isotopes. The results are compared with experimental data where available. The calculations were performed in the macroscopic-microscopic approximation with the Lublin-Strasbourg Drop (LSD) for the macroscopic part and the microscopic energy corrections were evaluated in the Yukawa-folded potential. The Fourier nuclear shape parametrization is used to describe the nuclear shape, including the non-…
Fission-fragment mass and total-kinetic-energy (TKE) distributions following fission of even-even nuclides in the region $74 \leq Z \leq 126$ and $92 \leq N \leq 230$, comprising 896 nuclides have been calculated using the Brownian shape-motion method. The emphasis is the region of superheavy nuclei. To show compatibility with earlier results the calculations are extended to include earlier studied regions. An island of asymmetric fission is obtained in the superheavy region, $106\leq Z\leq114$ and $162\leq N\leq 176$, where the heavy fragment is found to be close to $^{208}$Pb and the light f…
Using the idea of the instanton approach to quantum tunneling we try to obtain a method of calculating spontaneous fission rates for nuclei with the odd number of neutrons or protons. This problem has its origin in the failure of the adiabatic cranking approximation which serves as the basis in calculations of fission probabilities. Selfconsistent instanton equations, with and without pairing, are reviewed and then simplified to non-selfconsistent versions with phenomenological single-particle potential and seniority pairing interaction. Solutions of instanton-like equations without pairing an…
Nuclear fission of heavy (actinide) nuclei results predominantly in asymmetric mass-splits. Without quantum shells, which can give extra binding energy to these mass-asymmetric shapes, the nuclei would fission symmetrically. The strongest shell effects are in spherical nuclei, so naturally the spherical "doubly-magic" ${^{132}}$Sn nucleus (${Z=50}$ protons), was expected to play a major role. However, a systematic study of fission has shown that the heavy fragments are distributed around ${Z=52}$ to 56, indicating that ${^{132}}$Sn is not the only driver. Reconciling the strong spherical shell…
It is shown, that the Gamow-like model with only one adjustable parameter - radius constant is able to reproduce well the alpha-decay half-lives for all even-even nuclei with the proton number larger than 50. The systematics for odd-A and odd-odd isotopes can be also well described when ones introduces an additional hindrance factor. Similar model based on the W. J. Swiatecki idea from 1955 is developed to reproduce the spontaneous fission half-lives of transactinide nuclei. The achieved accuracy of reproduction of the data is better than that obtained in more advanced theories.
A systematic study of 160 heavy and super-heavy nuclei is performed in the Hartree-Fock-Bogoliubov approach with the finite range and density dependent Gogny force with the D1S parameter set. We show calculations in several approximations: with axially symmetric and reflexion symmetric wave functions, with axially symmetric and non-reflexion symmetric wave functions and finally some representative examples with triaxial wave functions are also discussed. Relevant properties of the ground state and along the fission path are thoroughly analyzed. Fission barriers, Q$_α$-factors and lifetimes wit…
The fission of highly excited compound nuclei formed in heavy ion induced fusion reactions has emerged as a topic of considerable interest in the recent years. Dissipative dynamical models based on the Langevin equation were developed and were applied successfully for fission dynamics of highly excited heavy nuclei. However, Wall Friction(WF), the standard version of nuclear friction when incorporated in the Langevin dynamical model was not able to reproduce simultaneously experimental data for both prescission neutron multiplicity and fission probability. Consequently, an empirical reduction …
We evaluate the impact of using sets of fission yields given by the GEF code for spontaneous (sf), neutron-induced ((n,f)), and beta-delayed (betadf) fission processes which take into account the approximate initial excitation energy of the fissioning compound nucleus. We further explore energy-dependent fission dynamics in the r process by considering the sensitivity of our results to the treatment of the energy sharing and de-excitation of the fission fragments using the FREYA code. We show that the asymmetric-to-symmetric yield trends predicted by GEF can reproduce the high-mass edge of the…
The potential energy of a deformed nucleus has been determined within a Generalized Liquid Drop Model taking into account the proximity energy, the microscopic corrections and compact and necked shapes. Multiple-humped potential barriers appear. A third minimum and third maximum exist in specific exit channels where one fragment is close to a magic spherical nucleus while the other one varies from oblate to prolate shapes. The heights of the fission barriers and half-lives of actinides are in agreement with the experimental results.
Energy dependence of fission observables is a key issue for wide nuclear applications. We studied real-time fission dynamics from low-energy to high excitations in the compound nucleus $^{240}$Pu with the time-dependent Hartree-Fock+BCS approach. It is shown that the evolution time of the later phase of fission towards scission is considerably lengthened at finite temperature. As the role of dynamical pairing is vanishing at high excitations, the random transition between single-particle levels around the Fermi surface to mimic thermal fluctuations is indispensable to drive fission. The obtain…
We present a large scale survey of life-times for spontaneous fission in the regime of super-heavy elements (SHE), i.e. nuclei with Z=104-122. This is done on the basis of the Skyrme-Hartree-Fock model. The axially symmetric fission path is computed using a quadrupole constraint. Self-consistent cranking is used for the collective masses and associated quantum corrections. The actual tunneling probability is estimated by the WKB approximation. Three typical Skyrme forces are used to explore the sensitivity of the results. Benchmarks in the regime Z=104-108 show an acceptable agreement. The gen…
Spontaneous-fission half-lives of the heaviest even-even nuclei are evaluated and compared with their alpha-decay mode. Calculations of spontaneous-fission half-lives are performed in the dynamical way with potential energy obtained by the macroscopic-microscopic method and the inertia tensor obtained by the cranking approximation. The alpha-decay half-lives are calculated for the same region of nuclei by use of the Viola and Seaborg formula. The ground-state properties such as mean square radii and electric quadrupole moments are also studied. From the analysis of spontaneous-fission half-lif…
