Gaussian Overlap Approximation Research Articles

Generator coordinate method for transition-state dynamics in nuclear fission

Since its beginnings, fission theory has assumed that low-energy induced fission takes place through transition-state channels at the barrier tops. Nevertheless, up to now there is no microscopic theory applicable to those conditions. We suggest that modern reaction theory is suitable for this purpose, and propose a methodology based on a configuration-interaction framework using the generator coordinate method (GCM). Simple reaction-theoretic models are constructed with the Gaussian overlap approximation to parametrize both the dynamics within the channels and their incoherent couplings to states outside the barrier. The physical characteristics of the channels examined here are their effective bandwidths and the quality of the coupling to compound-nucleus states as measured by the transmission factor $T$. We also investigate the spacing of GCM states with respect to their degree of overlap. We find that a rather coarse mesh provides an acceptable accuracy for estimating the bandwidths and transmission factors. The common numerical stability problem in using the GCM is avoided due to the choice of meshes and the finite bandwidths of the channels. The bandwidths of the channels are largely controlled by the zero-point energy with respect to the collective coordinate in the GCM configurations.

Microscopic self-consistent description of induced fission: Dynamical pairing degree of freedom

The role of dynamical pairing in induced fission dynamics is investigated using the time-dependent generator coordinate method in the Gaussian overlap approximation, based on the microscopic framework of nuclear energy density functionals. A calculation of fragment charge yields for induced fission of $^{228}$Th is performed in a three-dimensional space of collective coordinates that, in addition to the axial quadrupole and octupole intrinsic deformations of the nuclear density, also includes an isoscalar pairing degree of freedom. It is shown that the inclusion of dynamical pairing has a pronounced effect on the collective inertia, the collective flux through the scission hyper-surface, and the resulting fission yields, reducing the asymmetric peaks and enhancing the contribution of symmetric fission, in better agreement with the empirical trend.

The Time-Dependent Generator Coordinate Method in Nuclear Physics

The emergence of collective behaviors and the existence of large amplitude motions are both central features in the fields of nuclear structure and reactions. From a theoretical point of view, describing such phenomena requires increasing the complexity of the many-body wavefunction of the system to account for long-range correlations. One of the challenges, when going in this direction, is to keep the approach tractable within our current computational resources while gaining a maximum of predictive power for the phenomenon under study. In the Generator Coordinate Method (GCM), the many-body wave function is a linear superposition of (generally non-orthogonal) many-body states (the generator states) labeled by a few collective coordinates. Such a method has been widely used in structure studies to restore the symmetries broken by single-reference approaches. In the domain of reactions, its time-dependent version (TDGCM) has been developed and applied to predict the dynamics of heavy-ion collisions or fission where the collective fluctuations play an essential role. In this review, we present the recent developments and applications of the TDGCM in nuclear reactions. We recall the formal derivations of the TDGCM and its most common approximate treatment, the Gaussian Overlap Approximation. We also emphasize the Schr\"odinger Collective-Intrinsic Model (SCIM) variant focused on the inclusion of quasiparticle excitations into the description. Finally, we highlight several exploratory studies related to a TDGCM built on time-dependent generator states.

Microscopic description of induced fission dynamics with nuclear energy density functionals

Static and dynamic aspects of the fission process are analyzed in a self-consistent framework based on energy density functionals. Multidimensionally constrained mean-field calculations in the collective space determine the potential energy surface of the fissioning nucleus, the scission line, the single-nucleon wave functions, energies, and occupation probabilities. Induced fission dynamics is described using the time-dependent generator coordinate method in the Gaussian overlap approximation. The position of the scission line is analyzed as a function of the strength of the pairing interaction, as well as the effect of static pairing correlations on charge yields and total kinetic energy of fission fragments.

An Accurate Quantum-Based Approach to Explicit Solvent Effects: Interfacing the General Effective Fragment Potential Method with Ab Initio Electronic Structure Theory.

An interface between ab initio quantum mechanics (QM) methods and the general effective fragment potential (EFP2) method, QM-EFP2, is implemented in which the intermolecular interactions between a QM molecule and EFP fragments consist of Coulomb, polarization, exchange repulsion (exrep), and dispersion components. In order to ensure accuracy in the QM-EFP2 exrep interaction energy, the EFP2-EFP2 spherical Gaussian overlap (SGO) approximation is abandoned and replaced with the exact electron repulsion integrals (ERI) that are evaluated with a direct method to reduce disk usage. A Gaussian damping function for the QM-EFP2 Coulomb component damps both the EFP nuclear and electronic charges. A new overlap damping function has been implemented for the QM-EFP2 dispersion component. The current QM-EFP2 implementation has been benchmarked with the S22 and S66 data sets and demonstrates excellent agreement with symmetry-adapted perturbation theory (SAPT) for component energies and with coupled cluster theory [CCSD(T)] for the total interaction energies. Water clusters of various sizes (up to 256 water molecules) have been tested; it is shown that the QM-EFP2 method has an accuracy that is comparable to that of EFP2-EFP2. It has been shown previously that the accuracy of EFP2-EFP2 intermolecular interactions is comparable to that of second-order perturbation theory (MP2) or better. The implementation of the distributed data interface (DDI) parallelization scheme significantly improves the efficiency of QM-EFP2 calculations. The time to form the QM-EFP2 Fock operator per SCF iteration for water clusters scales linearly with the number EFP basis functions.

Time-dependent generator-coordinate-method study of mass-asymmetric fission of actinides

Low-energy positive and negative parity collective states in the equilibrium minimum, and the dynamics of induced fission of actinide nuclei are investigated in a unified theoretical framework based on the generator coordinate method (GCM) with the Gaussian overlap approximation (GOA). The collective potential and inertia tensor, both at zero and finite temperature, are computed using the self-consistent multidimensionally constrained relativistic mean field model, based on the energy density functional DD-PC1. Pairing correlations are treated in the Bardeen-Cooper-Schrieffer approximation with a separable pairing force of finite range. A collective quadrupole-octupole Hamiltonian characterized by zero-temperature axially symmetric deformation energy surface and perturbative cranking inertia tensor, is used to model the low-lying excitation spectrum. The fission fragment charge distributions are obtained by propagating the initial collective states in time with the time-dependent GCM+GOA that uses the same quadrupole-octupole Hamiltonian, but with the collective potential and inertia tensor computed at finite temperature. The illustrative charge yields of $^{228}\mathrm{Th}$, $^{234}\mathrm{U}$, $^{240}\mathrm{Pu}$, $^{244}\mathrm{Cm}$, and $^{250}\mathrm{Cf}$ are in very good agreement with experiment, and the predicted mass asymmetry is consistent with the result of a recent microscopic study that has attributed the distribution (peak) of the heavier-fragment nuclei to shell-stabilized octupole deformations.

From asymmetric to symmetric fission in the fermium isotopes within the time-dependent generator-coordinate-method formalism

Predicting the properties of neutron-rich nuclei far from the valley of stability is one of the major challenges of modern nuclear theory. In heavy and superheavy nuclei, a difference of only a few neutrons is sufficient to change the dominant fission mode. A theoretical approach capable of predicting such rapid transitions for neutron-rich systems would be a valuable tool to better understand r-process nucleosynthesis or the decay of super-heavy elements. In this work, we investigate for the first time the transition from asymmetric to symmetric fission through the calculation of primary fission yields with the time-dependent generator coordinate method (TDGCM). We choose here the transition in neutron-rich Fermium isotopes, which was the first to be observed experimentally in the late seventies and is often used as a benchmark for theoretical studies. We compute the primary fission fragment mass and charge yields for 254 Fm, 256 Fm and 258 Fm from the TDGCM under the Gaussian overlap approximation. The static part of the calculation (generation of a potential energy surface) consists in a series of constrained Hartree-Fock-Bogoliubov calculations based on the D1S, D1M or D1N parameterizations of the Gogny effective interaction in a two-center harmonic oscillator basis. The 2-dimensional dynamics in the collective space spanned by the quadrupole and octupole moments is then computed with the finite element solver FELIX-2.0. The available experimental data and the TDGCM post-dictions are consistent and agree especially on the position in the Fermium isotopic chain at which the transition occurs. The main limitation of the method lies in the presence of discontinuities in the 2-dimensional manifold of generator states.

Microscopic self-consistent description of induced fission dynamics: Finite-temperature effects

The dynamics of induced fission of $^{226}$Th is investigated in a theoretical framework based on the finite-temperature time-dependent generator coordinate method (TDGCM) in the Gaussian overlap approximation (GOA). The thermodynamical collective potential and inertia tensor at temperatures in the interval $T=0 - 1.25$ MeV are calculated using the self-consistent multidimensionally constrained relativistic mean field (MDC-RMF) model, based on the energy density functional DD-PC1. Pairing correlations are treated in the BCS approximation with a separable pairing force of finite range. Constrained RMF+BCS calculations are carried out in the collective space of axially symmetric quadrupole and octupole deformations for the asymmetric fissioning nucleus $^{226}$Th. The collective Hamiltonian is determined by the temperature-dependent free energy surface and perturbative cranking inertia tensor, and the TDGCM+GOA is used to propagate the initial collective state in time. The resulting charge and mass fragment distributions are analyzed as functions of the internal excitation energy. The model can qualitatively reproduce the empirical triple-humped structure of the fission charge and mass distributions already at $T=0$, but the precise experimental position of the asymmetric peaks and the symmetric-fission yield can only be accurately reproduced when the potential and inertia tensor of the collective Hamiltonian are determined at finite temperature, in this particular case between $T=0.75$ MeV and $T=1$ MeV.

Non-perturbative collective inertias for fission: A comparative study

The non-perturbative method to compute Adiabatic Time Dependent Hartree Fock Bogoliubov (ATDHFB) collective inertias is extended to the Generator Coordinate Method (GCM) in the Gaussian overlap approximation (GOA) including the case of density dependent forces. The two inertias schemes are computed along the fission path of the 234U and compared with the perturbative results. We find that the non-perturbative schemes predict very similar collective inertias with a much richer structure than the one predicted by perturbative calculations. Moreover, the non-perturbative inertias show an extraordinary similitude with the exact GCM inertias computed numerically from the energy overlap. These results indicate that the non-perturbative inertias provide the right structure as a function of the collective variable and only a phenomenological factor is required to mock up the exact GCM inertia, bringing new soundness to the microscopic description of fission.

FELIX-2.0: New version of the finite element solver for the time dependent generator coordinate method with the Gaussian overlap approximation

The time-dependent generator coordinate method (TDGCM) is a powerful method to study the large amplitude collective motion of quantum many-body systems such as atomic nuclei. Under the Gaussian Overlap Approximation (GOA), the TDGCM leads to a local, time-dependent Schrödinger equation in a multi-dimensional collective space. In this paper, we present the version 2.0 of the code FELIX that solves the collective Schrödinger equation in a finite element basis. This new version features: (i) the ability to solve a generalized TDGCM+GOA equation with a metric term in the collective Hamiltonian, (ii) support for new kinds of finite elements and different types of quadrature to compute the discretized Hamiltonian and overlap matrices, (iii) the possibility to leverage the spectral element scheme, (iv) an explicit Krylov approximation of the time propagator for time integration instead of the implicit Crank–Nicolson method implemented in the first version, (v) an entirely redesigned workflow. We benchmark this release on an analytic problem as well as on realistic two-dimensional calculations of the low-energy fission of 240Pu and 256Fm. Low to moderate numerical precision calculations are most efficiently performed with simplex elements with a degree 2 polynomial basis. Higher precision calculations should instead use the spectral element method with a degree 4 polynomial basis. We emphasize that in a realistic calculation of fission mass distributions of 240Pu, FELIX-2.0 is about 20 times faster than its previous release (within a numerical precision of a few percents). Program summaryProgram Title: FELIX-2.0Program Files doi:http://dx.doi.org/10.17632/t8b4h9g88r.1Licensing provisions: GPLv3Programming language: C++Journal reference of previous version: Computer Physics Communications 200, 350–363 (2016)Does the new version supersede the previous version?: YesReasons for the new version: FELIX-2.0 extends the physics capabilities of the previous version, since it includes the option to have a metric term in the collective Schrödinger equation. The computational efficiency of the code is considerably increased thanks to the implementation of several new numerical methods. This version also provides a more flexible and robust workflow to perform calculation of fission fragment distributions more efficiently.Summary of revisions: The new version includes the ability to use a metric term in the TDGCM+GOA equation. Numerical integration methods (both spatial and time integration) have been updated to more efficient schemes: matrix elements are now computed with Gauss–Legendre or Gauss–Legendre–Lobatto quadratures, the time propagation of the wave packet is computed with the Krylov approximation of the propagator instead of the previous Crank–Nicolson scheme. A new type of finite element based on n-dimensional orthotope is implemented, which enables a spectral element scheme for spatial discretization. Finally, the workflow and inputs/outputs of the code were entirely redesigned to provide more flexibility and be more user-friendly.Nature of problem: The Gaussian overlap approximation to the time-dependent generator coordinate method [1,2] yields a local, time-dependent Schrödinger equation in a small, multi-dimensional collective space. Its solution provides the time-evolution of the collective probability amplitude. For applications to nuclear fission, scission configurations are defined by a hyper-surface in this collective space referred to as the frontier. Distributions of nuclear observables such as charge or mass distributions of the fission fragments are extracted from the probability for the system to pass through any given section of the frontier. This probability is computed by first solving the evolution equation up to times greater than 10−20s and then integrating the flux of probability through the frontier over the entire timerange.Solution method: FELIX-2.0 solves the time-dependent Schrödinger equation by first discretizing the N-dimensional collective space with the continuous Galerkin Finite Element Method. This produces a large set of coupled, time-dependent Schrödinger equations characterized by the sparse overlap and Hamiltonian matrices. The solution is evolved in small time steps by applying an explicit and unitary propagator built as a Krylov approximation [3] of the exponential of the Hamiltonian.Additional comments including Restrictions and Unusual features: Although the implementation of the program gives it the ability to solve the TDGCM+GOA equation in a generic N-dimensional collective space, it has only been tested on 1-, 2- and 3-dimensional meshes.[1] J.J. Griffin, J. A. Wheeler, Collective Motions in Nuclei by the Method of Generator Coordinates, Phys. Rev. C 108, 311-327 (1975)[2] P. G. Reinhard, K. Goeke, The Generator-Coordinate Method and Quantized Collective Motion in Nuclear Systems, Rep. Prog. Phys. 50, 1-64 (1987)[3] Y. Saad, Iterative Methods for Sparse Linear Systems: Second Edition, SIAM, 2003

Microscopic study of induced fission dynamics of Th226 with covariant energy density functionals

Static and dynamic aspects of the fission process of $^{226}\mathrm{Th}$ are analyzed in a self-consistent framework based on relativistic energy density functionals. Constrained relativistic mean-field calculations in the collective space of axially symmetric quadrupole and octupole deformations, based on the energy density functional PC-PK1 and a $\ensuremath{\delta}$-force pairing, are performed to determine the potential energy surface of the fissioning nucleus, the scission line, the single-nucleon wave functions, energies, and occupation probabilities, as functions of deformation parameters. Induced fission dynamics is described using the time-dependent generator coordinate method in the Gaussian overlap approximation. A collective Schr\"odinger equation, determined entirely by the microscopic single-nucleon degrees of freedom, propagates adiabatically in time the initial wave packet built by boosting the ground-state solution of the collective Hamiltonian for $^{226}\mathrm{Th}$. The position of the scission line and the microscopic input for the collective Hamiltonian are analyzed as functions of the strength of the pairing interaction. The effect of static pairing correlations on the preneutron emission charge yields and total kinetic energy of fission fragments is examined in comparison with available data, and the distribution of fission fragments is analyzed for different values of the initial excitation energy.

Fission dynamics of Cf252

The time-dependent generator coordinate method with the gaussian overlap approximation (TDGCM+GOA) formalism is applied to describe the fission of $^{252}\mathrm{Cf}$. We perform analysis of fission from the initial states laying in the energetic range from the ground state to the state located 4 MeV above the fission barrier. The fission fragment mass distributions, obtained for different parity, energy of levels, and types of mixed states, are calculated and compared with experimental data. The impact of the total time of wave packet propagation on the final results is studied as well. The weak dependence of obtained mass yields on the initial conditions is shown.

Spontaneous fission, cluster emission and alpha decay of [formula omitted] in a unified description

Three disintegration modes of the parent nucleus 222Ra, namely the fission, the cluster emission and the alpha decay, are treated in a unified manner. The half-lives are calculated by taking into account the penetration probabilities through the potential barrier along disintegration trajectories in the deformation space appropriate to each kind of decay. These trajectories start from the same parent ground state configuration. The disintegration paths are obtained from the least action principle. The deformation energy is calculated within the macroscopic–microscopic procedure based on the Woods–Saxon two center shell model. The effective mass is determined microscopically with the cranking method and the Gaussian Overlap Approximation. The nuclear shape parametrization is characterized by the most important five degrees of freedom encountered in fission, corresponding to the elongation, the mass asymmetry, the necking and the two fragment deformations. Partial half-life results for alpha-decay and cluster emission processes treated as superasymmetric fission agree with the experimental data. The spontaneous fission half-life is predicted. Molecular states are evidenced for the superasymmetric fission process as a result of the large shell effects in the nascent fragments.

Microscopic description of fission dynamics: Toward a 3D computation of the time dependent GCM equation

Accurate knowledge of fission fragment yields is an essential ingredient of numerous applications ranging from the formation of elements in the r-process to fuel cycle optimization in nuclear energy. The need for a predictive theory applicable where no data is available, together with the variety of potential applications, is an incentive to develop a fully microscopic approach to fission dynamics. One of the most promising theoretical frameworks is the time dependent generator coordinate method (TDGCM) applied under the Gaussian overlap approximation (GOA). However, the computational cost of this method makes it difficult to perform calculations with more than two collective degree of freedom. Meanwhile, it is well-known from both semi-phenomenological and fully microscopic approaches that at least four or five dimensions may play a role in the dynamics of fission. To overcome this limitation, we develop the code FELIX aiming to solve the TDGCM+GOA equation for an arbitrary number of collective variables. In this talk, we report the recent progress toward this enriched description of fission dynamics. We will briefly present the numerical methods adopted as well as the status of the latest version of FELIX. Finally, we will discuss fragments yields obtained within this approach for the low energy fission of major actinides.

Fission description: First steps towards a full resolution of the time-dependent Hill-Wheeler equation

Dynamical description of low energy fission is, in our full microscopic approach, decomposed in two steps. In the first step we generate the Potential Energy Surface (PES) of the compound system we want to describe with the Hartree-Fock-Bogoliubov (HFB) method with a Gogny interaction. The second step uses the Time Dependent Generator Coordinate Method (TDGCM) with the Gaussian Overlap Approximation (GOA). The GOA holds in two assumptions: the overlap matrix between HFB states has a gaussian shape (with respect to the difference between coordinates of states in deformation space); and the expectation value of the collective hamiltonian between these states can be expanded up to order two, leading in this case to a Schrodinger-like equation. In this work we replace TDGCM+GOA in the second step of our approach by an exact treatment of the TDGCM. The main equation of this method is the time-dependent Hill-Wheeler equation and involves two objects: the overlap matrix and the collective hamiltonian. We first calculate these matrices on a PES. Then, we build an “exact TDGCM” solver using a finite element method and a Crank-Nicolson scheme. In this talk, we will present the time-dependent Hill-Wheeler equation and discretization schemes (in time and deformation space). The analytic calculation of overlap matrix and collective hamiltonian will be detailed. Finally, first results with an exact treatment of the TDGCM will be discussed.

Fission fragment charge and mass distributions inPu239(n,f)in the adiabatic nuclear energy density functional theory

Background: Accurate knowledge of fission fragment yields is an essential ingredient of numerous applications ranging from the formation of elements in the $r$ process to fuel cycle optimization for nuclear energy. The need for a predictive theory applicable where no data are available, together with the variety of potential applications, is an incentive to develop a fully microscopic approach to fission dynamics.Purpose: In this work, we calculate the pre-neutron emission charge and mass distributions of the fission fragments formed in the neutron-induced fission of $^{239}\mathrm{Pu}$ using a microscopic method based on nuclear density functional theory (DFT).Methods: Our theoretical framework is the nuclear energy density functional (EDF) method, where large-amplitude collective motion is treated adiabatically by using the time-dependent generator coordinate method (TDGCM) under the Gaussian overlap approximation (GOA). In practice, the TDGCM is implemented in two steps. First, a series of constrained EDF calculations map the configuration and potential-energy landscape of the fissioning system for a small set of collective variables (in this work, the axial quadrupole and octupole moments of the nucleus). Then, nuclear dynamics is modeled by propagating a collective wave packet on the potential-energy surface. Fission fragment distributions are extracted from the flux of the collective wave packet through the scission line.Results: We find that the main characteristics of the fission charge and mass distributions can be well reproduced by existing energy functionals even in two-dimensional collective spaces. Theory and experiment agree typically within two mass units for the position of the asymmetric peak. As expected, calculations are sensitive to the structure of the initial state and the prescription for the collective inertia. We emphasize that results are also sensitive to the continuity of the collective landscape near scission.Conclusions: Our analysis confirms that the adiabatic approximation provides an effective scheme to compute fission fragment yields. It also suggests that, at least in the framework of nuclear DFT, three-dimensional collective spaces may be a prerequisite to reach 10% accuracy in predicting pre-neutron emission fission fragment yields.

Microscopic predictions of fission yields based on the time dependent GCM formalism

Accurate knowledge of fission fragment yields is an essential ingredient of numerous applications ranging from the formation of elements in the r-process to fuel cycle optimization in nuclear energy. The need for a predictive theory applicable where no data is available, together with the variety of potential applications, is an incentive to develop a fully microscopic approach to fission dynamics. One of the most promising theoretical frameworks is the time-dependent generator coordinate method (TDGCM) applied under the Gaussian overlap approximation (GOA). Previous studies reported promising results by numerically solving the TDGCM+GOA equation with a finite difference technique. However, the computational cost of this method makes it difficult to properly control numerical errors. In addition, it prevents one from performing calculations with more than two collective variables. To overcome these limitations, we developed the new code FELIX-1.0 that solves the TDGCM+GOA equation based on the Galerkin finite element method. In this article, we briefly illustrate the capabilities of the solver FELIX-1.0, in particular its validation for n+ 239 Pu low energy induced fission. This work is the result of a collaboration between CEA,DAM,DIF and LLNL on nuclear fission theory.

FELIX-1.0: A finite element solver for the time dependent generator coordinate method with the Gaussian overlap approximation

We describe the software package FELIX that solves the equations of the time-dependent generator coordinate method (TDGCM) in N-dimensions (N≥1) under the Gaussian overlap approximation. The numerical resolution is based on the Galerkin finite element discretization of the collective space and the Crank–Nicolson scheme for time integration. The TDGCM solver is implemented entirely in C++. Several additional tools written in C++, Python or bash scripting language are also included for convenience. In this paper, the solver is tested with a series of benchmarks calculations. We also demonstrate the ability of our code to handle a realistic calculation of fission dynamics. Program summaryProgram title: FELIX-1.0Catalogue identifier: AEYZ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEYZ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, Version 2No. of lines in distributed program, including test data, etc.: 1192325No. of bytes in distributed program, including test data, etc.: 10214787Distribution format: tar.gzProgramming language: C++.Computer: Intel Xeon, Intel Core.Operating system: LINUX.RAM: Memory usage depends on the number of nodes in the calculation mesh as well as on the degree of the interpolation polynomials. For a 1D calculation with linear polynomials on a mesh with 600 nodes, memory usage is approximately 3.3 MB; in a realistic simulation of fission on a 2D mesh with quadratic polynomials and 1.3 105 nodes, it reaches 1.5 GiB.Classification: 17.23.External routines: The solver itself requires the BLAS and LAPACK libraries, and a Fortran compiler with OpenMP support. Building the documentation requires DoxyGen-1.8.6 or higher. Building the full set of tools also requires GSL, PETSc, SLEPc and Boost. In particular, environment variables PETSC_DIR, PETSC_ARCH, SLEPC_DIR and SLEPC_ARCH must be set.Nature of problem: Nuclear fission is a relatively slow process compared to the typical timescale of the intrinsic motion of the nucleons. In the adiabatic approximation, it can be described as a large amplitude collective motion driven by only a few collective degrees of freedom. In the time-dependent generator coordinate method (TDGCM), the nuclear wave-function is thus described as a time-dependent, linear superposition of basis functions in this collective space. Further assuming a Gaussian overlap approximation (GOA) for the basis functions, the time-dependent Schrödinger equation can be reduced into a local, time-dependent, Schrödinger-like equation in collective space. This is the TDGCM+GOA equation. Scission configurations are defined as a hyper-surface in the N-dimensional collective space. Fission fragment distributions are then computed by integrating over time the flux of the collective wave-packet across the scission hyper-surface. This microscopic approach to fission fragment distributions is fully quantum-mechanical.Solution method: FELIX solves the TDGCM+GOA equation by using the Galerkin finite element method to discretize the N-dimensional collective space, and the Crank–Nicolson scheme to solve for the time evolution. At each time step, this procedure requires solving a linear system of equation involving sparse, complex, symmetric matrices. FELIX employs an iterative QMR algorithm to perform matrix inversion.Restrictions: Although the program can operate in an arbitrary number of dimensions N, it has only been tested in practice on 1, 2 and 3 dimensional meshes.Additional comments: The code has checkpointing capabilities: the collective wave-function, norm a and energy kernels are stored on disk every n iterations, ensuring that the program can resume where it stops.Running time: Running time grows linearly with the number of time-steps requested by the user. It is also highly dependent on the number of nodes in the space mesh. Two periods of a 1D harmonic oscillator (600 nodes, 800 time steps) are typically computed in a few seconds on one thread of a Intel(R) Core(TM) i5 CPU. A 2-dimensional realistic case of fission (105 nodes, 105 time steps) requires roughly 10 h on 10 threads of an Intel Xeon EP X5660 processor.

On the Gaussian overlap approximation for the collective excitations of odd nuclei

The multi-dimensional generator coordinate method is applied to describe the quadrupole collective states of odd nuclei. The laboratory Cartesian components of the quadrupole deformation tensor are used as the generator coordinates. The two mean-field states of an odd nucleus constituting a Kramers’ doublet are taken as the two intrinsic generating states which can rotate in the laboratory reference frame. The time-signature invariance of these intrinsic states is assumed. A version of the Gaussian overlap approximation (GOA) for the two generating states is constructed. A system of two coupled differential equations for the collective excitations of odd nuclei is derived from the variational principle using the GOA. The system forms a generalized eigenvalue differential equation in the two-dimensional ‘alispin’ space. Breaking the time-reversal symmetry and alignment of the angular momentum in the mean-field states modifies the rotational part of the Hamiltonian compared to that of the standard Bohr Hamiltonian, whereas the quadrupole vibrations are modified through the coupling of the Kramers states.

The full five-dimensional Bohr Hamiltonian from the generator coordinate method

The five-dimensional generator coordinate method with the five generator coordinates being the components of the quadrupole tensor is used to generate the nuclear collective quadrupole excitations by the vibrations and rotations of the intrinsic deformed ground state of the mean-field Hamiltonian. The Gaussian overlap approximation leads to a differential eigenvalue equation with the five-dimensional Bohr Hamiltonian. The resulting collective Hamiltonian is presented. It can be calculated only approximately. However, there is a clear algorithm how to improve the approximation. Formulae for the inverse vibrational inertial functions, the moments of inertia and the potential with the vibrational and rotational zero-point energy corrections are given in the lowest-order approximation.