Adiabatic Time-dependent Hartree-Fock Research Articles

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.

Self-consistent collective coordinate for reaction path and inertial mass

We propose a numerical method to determine the optimal collective reaction path for the nucleus-nucleus collision, based on the adiabatic self-consistent collective coordinate (ASCC) method. We use an iterative method combining the imaginary-time evolution and the finite amplitude method, for the solution of the ASCC coupled equations. It is applied to the simplest case, the $\alpha-\alpha$ scattering. We determine the collective path, the potential, and the inertial mass. The results are compared with other methods, such as the constrained Hartree-Fock method, the Inglis's cranking formula, and the adiabatic time-dependent Hartree-Fock (ATDHF) method.

Multidimensional Skyrme-density-functional Study of the Spontaneous Fission of $^{238}$U

We determined the spontaneous fission lifetime of 238U by a minimization of the action integral in a three-dimensional space of collective variables. Apart from the mass-distribution multipole moments Q20 (elongation) and Q30 (left–right asymmetry), we also considered the pairing-fluctuation parameter λ2 as a collective coordinate. The collective potential was obtained self-consistently using the Skyrme energy density functional SkM*. The inertia tensor was obtained within the nonperturbative cranking approximation to the adiabatic time-dependent Hartree–Fock–Bogoliubov approach. As a result, the pairing-fluctuation parameter λ2 allowed us to control the pairing gap along the fission path, which significantly changed the spontaneous fission lifetime.

TETRAHEDRAL SYMMETRY IN NUCLEI: NEW PREDICTIONS BASED ON THE COLLECTIVE MODEL

We apply the standard collective nuclear Hamiltonian derived in the literature with the help of the Adiabatic Time Dependent Hartree-Fock (ATDHF) theory. We use a schematic two-dimensional collective space spanned by the axial-quadrupole (α20) vs. tetrahedral (α32) deformations together with the potential energy surfaces calculated microscopically. We illustrate and discuss the collective-model solutions suggesting that the tetrahedral-symmetry nuclei may generate sizeable quadrupole transitions: this result is in contrast to the previous expectations and/or predictions.

Modeling the doubly excited state with time-dependent Hartree–Fock and density functional theories

Multielectron excited states have become a hot topic in many cutting-edge research fields, such as the photophysics of polyenes and in the possibility of multiexciton generation in quantum dots for the purpose of increasing solar cell efficiency. However, obtaining multielectron excited states has been a major obstacle as it is often done with multiconfigurational methods, which involve formidable computational cost for large systems. Although they are computationally much cheaper than multiconfigurational wave function based methods, linear response adiabatic time-dependent Hartree-Fock (TDHF) and density functional theory (TDDFT) are generally considered incapable of obtaining multielectron excited states. We have developed a real-time TDHF and adiabatic TDDFT approach that is beyond the perturbative regime. We show that TDHF/TDDFT is able to simultaneously excite two electrons from the ground state to the doubly excited state and that the real-time TDHF/TDDFT implicitly includes double excitation within a superposition state. We also present a multireference linear response theory to show that the real-time electron density response corresponds to a superposition of perturbative linear responses of the S(0) and S(2) states. As a result, the energy of the two-electron doubly excited state can be obtained with several different approaches. This is done within the adiabatic approximation of TDDFT, a realm in which the doubly excited state has been deemed missing. We report results on simple two-electron systems, including the energies and dipole moments for the two-electron excited states of H(2) and HeH(+). These results are compared to those obtained with the full configuration interaction method.

Nuclear fission with mean-field instantons

We present a description of nuclear spontaneous fission, and generally of quantum tunneling, in terms of instantons, that is, periodic imaginary-time solutions to time-dependent mean-field equations. This description allows comparisons to be made with the more familiar generator coordinate (GCM) and adiabatic time-dependent Hartree-Fock (ATDHF) methods. It is shown that the action functional whose value for the instanton is the quasiclassical estimate of the decay exponent fulfills the minimum principle when additional constraints are imposed on trial fission paths. In analogy with mechanics, these are conditions of energy conservation and the velocity-momentum relations. In the adiabatic limit, the instanton method reduces to the time-odd ATDHF equation, with collective mass including the time-odd Thouless-Valatin term, while the GCM mass completely ignores velocity-momentum relations. This implies that GCM inertia generally overestimates the instanton-related decay rate. The very existence of the minimum principle offers hope for a variational search for instantons. After the inclusion of pairing, the instanton equations and the variational principle can be expressed in terms of the imaginary-time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory. The adiabatic limit of this theory reproduces ATDHFB inertia.

Boost-invariant mean field approximation and the nuclear Landau-Zener effect

We investigate the relation between time-dependent Hartree-Fock (TDHF) states and the adiabatic eigenstates by constructing a boost-invariant single-particle Hamiltonian. The method is numerically realized within a full three-dimensional TDHF which includes all the terms of the Skyrme energy functional and without any symmetry restrictions. The study of free translational motion of a nucleus demonstrates the validity of the concept of boost-invariant and adiabatic TDHF states. The interpretation is further corroborated by the test case of fusion of 16 O + 16 O. As a first application, we present a study of the nuclear Landau-Zener effect on a collision of 4 He + 16 O.

Small-amplitude limit of the nuclear Born-Oppenheimer method.

We examine here how the nuclear Born-Oppenheimer (NBO) method describes the collective dynamics of nuclei undergoing small-amplitude oscillations around the equilibrium state. After specifying the NBO trial wave function, and assuming that the intrinsic state is not very different from the Hartree-Fock (HF) ground state, we show that the NBO method yields the random phase approximation (RPA) equations. We then derive an expression for the ground state energy. This expression, which contains zero-point energy correction terms, is smaller than the static HF energy. Next, we derive the correlated ground state energy and then show that it is identical with the corresponding expressions obtained from the generator-coordinate method, from the properly quantized adiabatic time-dependent Hartree-Fock approach, and from the RPA.

A systematic study of the octupole correlations in the lanthanides with realistic forces

We have performed a systematic study of the octupole degree of freedom in the nuclei 140Ba, 142–150Ce, 144–152Nd and 146–154Sm. The static properties (ground state deformations, energy gaps, dipole moments, etc.) have been analyzed within the Hartree-Fock plus BCS approximation (HFBCS); for the dynamical ones (energy splittings, transition probabilities, etc.) the adiabatic time-dependent Hartree-Fock plus zero point energy in the cranking approximation (ATDHF+ZPE) has been applied. In both approximations the realistic density-dependent Gogny force has been used. In our parameter-free calculations we are able to describe very well the whole experimental systematic of energy splittings and B(E1), among others. The flatness of the potential energy of some nuclei makes the mean field approach unreliable for such nuclei.

Do anticommutators give additional conditions on adiabatic time-dependent Hartree-Fock path?

Utilizing one-particle one-hole variations of ATDHF states by invoking commutators obtained from generators of complete set of coordinates and of momenta, the ATDHF equations were shown in an earlier work to yield the valley condition. In principle, variations of ATDHF states should also be exploited by using anticommutators involving the said generators. It is demonstrated here that owing to the canonicity condition relating the generators of coordinates and of momenta, use of anticommutators gives rise to exactly the same valley condition on the ATDHF path.

Adiabatic time-dependent Hartree-Fock theory with the Skyrme interaction.

A derivation for reducing the adiabatic time-dependent Hartree-Fock theory with the consistency condition to an efficacious form, in which the evolution of the single-particle states is given by a time-dependent Hartree-Fock-like equation while the consistency condition is given by a one-body equation, is discussed. The density-dependent Skyrme interaction is included in the derivation.

Technique for evaluating the optimal path in adiabatic time-dependent Hartree-Fock theory.

A method of achieving the consistent solution of the adiabatic time-dependent Hartree-Fock theory as the optimal path is discussed. The method is applied to evaluate the valley path in the soluble three-level model.

Canonicity condition in adiabatic time-dependent Hartree-Fock theory.

We show how the incorporation of the canonicity condition, which relates the generator of coordinate and momentum, gives the correct Lagrange multiplier needed to complete the proof that the solution of the adiabatic time-dependent Hartree-Fock theory with the consistency condition fulfills the constrained variation principle designating the valley and thus the canonicity condition is, in principle, a normalization requirement rather than a restriction on the adiabatic time-dependent Hartree-Fock path.

Comparison of quantised ATDHF and GCM theory with application to the 12C+20Ne system

A conceptional and numerical comparison of the one parameter generator coordinate method (GCM) and the quantised adiabatic time-dependent Hartree-Fock (ATDHF) theory is performed by applying both theories to the 12C+20Ne and 16O+16O system. Different parametrisations of the Skyrme interaction are used. The single-particle wavefunctions and the operators are represented on a three-dimensional grid in coordinate and momentum space. The collective path is evaluated in the gradient method, corresponding to GDM, and by solving the numerically more involved ATDHF equations.

Adiabatic time-dependent Hartree-Fock theory in the generalized valley approximation

A classical theory of large amplitude, adiabatic collective motion, developed and applied previously to test examples, is transcribed into the language of time-dependent Hartree-Fock theory in order to initiate the application of the new method to problems in nuclear physics. The formulation which emerges can be characterized as a generalized cranking theory, in the sense that the cranking operator cannot be chosen arbitrarily, as in the conventional formulation of this method, but is instead fully constrained by the formalism. A procedure for obtaining approximate solutions to the new equations is described and illustrated with several simplified many-body models. It is inferred that traditional cranking calculations can serve as starting points for more realistic models. This paper also includes additional discussion concerning the calculation of collective masses and the problem of local stability of collective motion, not covered in our previous work.

Microscopic study of the octupole degree of freedom in the radium and thorium isotopes with gogny forces

The octupole degree of freedom of the nuclei 218–230Ra and 222–230Th is investigated in a microscopic way. Our analysis is based on the constrained Hartree-Fock plus BCS theory as well as on the adiabatic time-dependent Hartree-Fock in the cranking approximation (and generator coordinate method plus mean field). In the numerical applications we use the Gogny forces. From the mean field calculations we show octupole barrier heights, dipole moments as well as the values of β 2, β 4,β 5, β 6, and β 7 along the constrained path. From the symmetry conserving calculations we display the 0 +–1 − splitting, wave functions as well as the El and E3 transition probabilities. The overall agreement with the available experimental data is very good.

Uniqueness of the collective submanifold in the adiabatic theory of large amplitude collective motion

It is emphasized that the conditions for the existence of a collective submanifold which follow from adiabatic time-dependent Hartree-Fock theory are precisely the conditions for the existence of a manifold of solutions of Hamilton's equations confined to a surface of reduced dimensionality. A constructive procedure, valid in any number of dimensions and involving the concept of the multidimensional valley, is developed to determine whether a given system admits such a manifold. It is extended to include the idea of the approximate manifold, and an application to a generalized landscape model is described.

Quantized ATDHF and angular momentum projection: Three-dimensional applications to heavy ion scattering

The quantized adiabatic time-dependent Hartree-Fock (qATDHF) theory is extended to the calculation of observables in nuclear phenomena using general many-body techniques including angular momentum projection. All calculations are performed using three-dimensional coordinate and momentum-grid techniques. The Bonche-Koonin-Negele interaction as well as several Skyrme-type forces has been used in the simple Hartree-Fock (HF) calculations of 12C and 20Ne nuclei. Further, the maximally decoupled collective path is evaluated within qaTDHF, and the angular momentum projected kernels are inserted into the GCM formalism. As a test case, this formalism is applied to the phase shifts of elastic α-α scattering because they can be compared to experimental values. The same techniques are applied to the α- 16O system and the results are compared with those obtained by solving the collective Schrödinger equation in a Gaussian Overlap Approximation, as in a previous publication. The GCM, extended to complex energies, is used to extract the widths of the resonance levels of the 0 1 +, 0 4 +, 0 bands in α- 16O scattering. A model calculation, in which the structure of the nuclei is kept fixed to their HF ground state, is performed for the same α- 16O system. We get quite different results with this “sudden” approximation than with the adiabatic calculation. The present calculations show that it is indeed possible to connect general and symmetry conserving many-body techniques to the qATDHF theory and to obtain in this way a purely microscopic framework which can be handled numerically, thus allowing evaluation of observables which are accessible to experiments.

A measure of adiabaticity for nuclear collective motion

We map the time-dependent Hartree-Fock (TDHF) motion onto a classical Hamilton dynamics and perform an adiabatic expansion of the hamiltonian up to second order in the collective momentum. Requiring decoupling of intrinsic from collective motion in the various orders of this expansion, we obtain the adiabatic time-dependent Hartree-Fock (ATDHF) equations for the optimal collective degree-of-freedom and a measure of the validity of the description in terms of collective motion. We evaluate this validity condition for the system 16O + 16O → 32S and find that it is well fulfilled.

Deformation effects in the12C-12C system

The quantized adiabatic time dependent Hartree-Fock (quantized ATDHF) theory is applied to the12C-12C-system. Especially the intrinsic oblate ground state deformation of this nucleus is considered. For four relevant mutual orientations of the two nuclei the optimal, i.e. maximally decoupled collective paths, associated to relative motion, are evaluated by solving a coupled set of nonlinear differential equations on three dimensional grids in coordinate and momentum space. In a second step, the constituents of the corresponding collective Hamiltonian are calculated. Besides the collective potentials this includes the evaluation of the appropriate mass parameters, of the quantum corrections with regard to rotation, translation and collective relative motion and of the centrifugal potentials, all of which are shown to be important. By means of generalized WKB methods, subbarrier fusion cross sections are evaluated for different assumptions on the reaction process and compared with experimental data. Finally, the collective Hamiltonian is calculated for a head-on-collision of two12C-chains.