Adiabatic Time-dependent Hartree-Fock Research Articles

Low energy fusion of 12C + 12C and 16O + 16O systems

The adiabatic time-dependent Hartree-Fock method (ATDHF) is applied to the calculation of low energy fusion of 12C + 12C and 16O + 16O systems. The energy dependence of the results is in good agreement with experiment, while the order of magnitude is not correct. It is shown that the dynamical effects included in ATDHF are very important and cannot be neglected at the energies of astrophysical interest.

Adiabatic Time-Dependent Hartree-Fock Calculations of the Optimal Path, the Potential, and the Mass Parameter for Large-Amplitude Collective Motion

The adiabatic time-dependent Hartree-Fock theory is reformulated in order to yield a simple differential equation for the collective path with accompanying simple expressions for the collective mass and the potential. With use of three-dimensional coordinate- and momentum-space techniques and density-dependent interactions, the new adiabatic time-dependent Hartree-Fock formalism is applied to $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$ scattering and correspondingly to the fission mode of $^{8}\mathrm{Be}$. In the overlapping region the resulting collective mass deviates strongly from the reduced mass.

Isovector giant monopole resonances: A sum-rule approach

Several useful sum rules associated with isovector giant monopole resonances are calculated for doubly closed shell nuclei. The calculation is based on techniques known from constrained and adiabatic time-dependent Hartree-Fock theories and assume various Skyrme interactions. The results obtained form, together with the compiled literature, the basis for a quantitative description of the RPA strength distribution in terms of energy-weighted moments. These, together with strength distribution properties, are determined by a hierarchy of determinantal relations between moments. The isovector giant monopole resonance turns out to be a rather broad resonance centered at E = 46A −1 10 MeV with an extended width of more than 16 MeV. The consequences regarding isospin impurities in the nuclear ground state are discussed.

The generator-coordinate-method with conjugate parameters and the unification of microscopic theories for large amplitude collective motion

A dynamic theory of large amplitude collective motion of many particle systems is presented which is relevant, for example, to nuclear fission. The theory is microscopic and makes use of a collective path, i.e. a suitably constructed set of distorted nonequilibrium Slater determinants. The approach is a generalization of the generator coordinate method (GCM) and improves its dynamic aspects by extending it to a pair of conjugate generator parameters q and p (DGCM). The problems connected with redundancy and superfluous degrees of freedom are solved by prediagonalizing the local oscillations about each point of the dynamic collective basis | q, p ∼. For adiabatic large amplitude collective motion a Schrödinger equation is derived which appears to be nearly identical to the one obtained by a consistent quantization of semiclassical approaches as e.g. the adiabatic time dependent Hartree-Fock theory (ATDHF). In turn a collective path constructed by ATDHF proves to be particularly suited for being used in the present DGCM formalism. Altogether the formalism unifies two classes of microscopic approaches to collective motion, viz. the quantum mechanical GCM and the classical theories like cranking and ATDHF.

Collective subspaces for large amplitude motion and the generator coordinate method

The collection path $|\ensuremath{\varphi}(q)〉$ to be used in a microscopic description of large amplitude collective motion is determined by means of the generator coordinate method. By varying the total energy with respect to $|\ensuremath{\varphi}(q)〉$ and performing an adiabatic expansion a hierarchy of equations is obtained which determines uniquely a hierarchy of collective paths with increasing complexity. To zeroth order the $|\ensuremath{\varphi}(q)〉$ are Slater determinants, to first order they include 2p-2h correlations. In both cases simple noninterative prescriptions for an explicit construction of the path are derived. For a correlated path their solutions agree at the Hartree-Fock minimum with random phase approximation eigenmodes. The resulting equations for the path are compared with the outcome of related theories, particularly of semiclassical nature. It is remarkable that both sorts of approaches, on one hand the generator coordinate method with correlated states and on the other the quantized adiabatic time dependent Hartree-Fock theory, are virtually identical in the results, although they are of different conceptual origin and use different techniques. It is shown that the use of a correlated path does not cause numerical complications.NUCLEAR STRUCTURE Collective path derived by GCM, inclusion of RPA correlations, relation to adiabatic TDHF.

Calculation of giant monopole resonances in the adiabatic time-dependent Hartree-Fock theory

Giant monopole resonances in doubly magic nuclei are investigated by the adiabatic time-dependent Hartree-Fock method. The collective paths are constructed using constrained Hartree-Fock techniques with various Skyrme interactions. The calculation of the mass parameter is reduced to the solution of a finite set of linear equations which contain the usual cranking expression as an approximation. The resulting masses and energies allow the extraction of various energy moments of the random phase approximation monopole strength distribution. Predictions of the resonance widths are also presented.NUCLEAR STRUCTURE Giant monopole resonances: Time-dependent Hartree-Fock method, Skyrme's interactions.

The concept of a collective path and its range of validity

The space of Slater determinants is described in terms of certain non-cartesian coordinates in which the time-dependent Hartree-Fock theory (TDHF) appears to be fully equivalent to a set of classical canonical equations. In this framework the concept of a collective path is developed. This consists in selecting a subspace of Slater determinants in which the collective channel is decoupled from the non-collective excitations. The adiabatic time-dependent Hartree-Fock theory (ATDHF) and the local harmonic approach (LHA) are discussed, which both provide equations to determine the collective path. The main task is to study the limits of validity of either method and the conditions under which the decoupling of collective and non-collective channels is achieved. There the curvatures of the determinantal space appear to be crucial quantities. The LHA aiming at a dynamical path | q, p〉 has the stronger restriction than the ATDHF, being merely concerned with a static path | q, p = 0〉. The condition for the validity of ATDHF consists in requiring the collective kinetic energy per particle to be small compared with an average single-particle energy divided by a simple matrix element representing the coupling between collective and non-collective degrees of freedom. Finally, this condition is cast into a tractable form involving only RPA operations.

The generation coordinate method with conjugate parameters and its relation to adiabatic time-dependent Hartree-Fock

With the concept of redundant spaces a reduction scheme is proposed which makes the generator coordinate method with two conjugate parameters tractable. By means of that, a collective Schrodinger equation is derived which proves to be identical to the one obtained by consistent quantisation of the classical adiabatic time-dependent Hartree-Fock Hamiltonian.

An adiabatic time-dependent Hartree-Fock theory of collective motion in finite systems

We show how to derive the parameters of a phenomenological collective model from a microscopic theory. The microscopic theory is Hartree-Fock, and we start from the time-dependent Hartree-Fock equation. To this we add the adiabatic approximation, which results in a collective kinetic energy quadratic in the velocities, with coefficients depending on the coordinates, as in the phenomenological models. The crucial step is the decom-position of the single-particle density matrix ϱ in the form exp( iχ) ϱ 0 exp(− iχ), where ϱ O represents a time-even Slater determinant and plays the role of coordinate. Then χ plays the role of momentum, and the adiabatic assumption is that χ is small. The energy is expanded in powers of χ, the zeroth-order being the collective potential energy. The analogy with classical mechanics is stressed and studied. The same adiabatic equations of motion are derived in three different ways (directly, from the Lagrangian, from the Hamiltonian), thus proving the consistency of the theory. One equation of motion (Eq. I) expresses the relation between velocity and momentum; the other (Eq. II) is the dynamical equation. Equation II is not necessary for writing the energy or for the subsequent quantization which leads to a Schrödinger equation, but it must be used if one wants to check the validity of various approximation schemes, particularly when one attempts to reduce the problem to a few degrees of freedom. The role of the adiabatic hypothesis, its definition, and range of validity, are analyzed in great detail. It assumes slow motion, but not small amplitude, and is therefore suitable for large-amplitude collective motion. The RPA is obtained as the limiting case where the amplitude is also small. The translational mass is correctly given, and the moment of inertia under rotation is that of Thouless and Valatin. For a quadrupole two-body force, the Baranger-Kumar formalism is recovered. The self-consistency brings additional terms to the Inglis cranking formula. Comparison is also made with generator coordinate methods.

A consistent microscopic theory of collective motion in the framework of an ATDHF approach

Based on merely two assumptions, namely the existence of a collective Hamiltonian and that the collective motion evolves along Slater determinants, we first derive a set of adiabatic time-dependent Hartree-Fock equations (ATDHF) which determine the collective path, the mass and the potential, second give a unique procedure for quantizing the resulting classical collective Hamiltonian, and third explain how to use the collective wavefunctions, which are eigenstates of the quantized Hamiltonian.

Reality conditions for classical motions along collective paths

The equations of motion along a large amplitude collective path $|q(t),p(t)〉$ are shown to require, in general, complex parameters $q$ and $p$. It is shown, however, that real values are sufficient if $|q,p〉$ is constructed by Hartree-Fock with coordinate and momentum constraint. In addition, the motion along the particular paths suggested by the adiabatic time-dependent Hartree-Fock theory and by the local harmonic approach is proved to be stable against small imaginary displacements.NUCLEAR STRUCTURE Classical motion, collective path, constrained Hartree-Fock, adiabatic time-dependent Hartree-Fock, local harmonic approach, large amplitude collective motion, complex generator coordinates.

Exposure of relations between collective theories through their energy-weighted moments

A systematic family of one-body collective operators is generated from a given one-body operator, e.g. the standard multipole operator. In the small amplitude limit any of these operators can be used as a constraint in constrained Hartree-Fock theory and the corresponding CH F solutions can be used to describe a collective path in the cranking model, in the double projection method, in the adiabatic time dependent Hartree-Fock theory or in the generator coordinate method. In this formalism one can reproduce virtually all previous collective theories by suitable choice of constraint, thereby unifying these theories. Key quantities met constantly in the analysis are RPA energy moments, in terms of which collective masses, potentials and excitation energies are expressed. Remarkable theorems about these moments and about the collective theories have counterparts at large amplitudes.

Adiabatic time-dependent Hartree-Fock theory in nuclear physics

The adiabatic limit of time-dependent Hartree-Fock theory (ATDHF) is developed in a systematic way so as to provide a basis for the description of large amplitude, slow, collective motion in nuclei. A variational principle is used to find the best approximation to a solution of the time-dependent Hartree-Fock equation in the form of a parametrized wave packet φ( p, q), p, q being time-dependent c-numbers. The mean energy for the wave packet φ( p, q) defines a classical collective Hamiltonian H(p, q) . The evolution of the parameters p and q is described by the canonical equations of motion associated with H. A set of self-consistency conditions, of a formally stationary type, determines possible optimal wave packets. An iteration scheme for the solution of the self-consistency conditions is proposed. The quantal interpretation of the classical description of collective motion embodied in the Hamiltonian H(p, q) is discussed. Finally, two simple, well understood illustrative examples are presented to illustrate the application of the general formalism developed in this paper.

Theories for large amplitude collective motion and constrained Hartree-Fock

For a large amplitude collective motion the adiabatic time dependent Hartree-Fock theory (ATDHF), the infinitesimal generator approach (IG), the constrained random phase approximation (CRPA) and the cranking model (CR) are compared, the only assumption made being that the system moves along a collective path generated by constrained Hartree-Fock (CHF). Without restricting to a certain model it is shown, that CRPA and IG yield identical expressions for the collective mass, which depend explicitly on the constraining operator and are therefore not unique. ATDHF is shown to depend only on the collective path rather than on the constraining operator as well. The nuclear monopole vibrations in 16O are taken as an example to demonstrate the magnitude of the effect. The masses are calculated in the different theories taking the Skyrme III interaction and r 2 as constraining operator. The theories are unified by using a constraining operator, which is projected on its particle-hole subspace at each value of the time dependent collective coordinate. This choice is justified and compared with various local prescriptions for the constraining operator.

Mass parameters for quadrupole vibrations in the adiabatic approximation

Nuclear quadrupole vibrations are investigated in the framework of the adiabatic time-dependent Hartree-Fock equations. Using a continuity condition satisfied by these equations, we show that a simple analytic expression can be derived for the mass parameter in the case where the collective variable α is a scaling variable i.e. changes the cartesian coordinates x, y, z into x/ α, y/ α, α 2 z. This result depends only on the existence of the continuity equation and thus holds for any nucleon-nucleon interaction. Applications are considered in the case of small amplitude motion in oxygen-16 and calcium-40.

A generator co-ordinate approach to the breathing mode

In the harmonic approximation of the generator co-ordinate method (GCM) with the Skyrme interaction we show that the frequency of the breathing mode is essentially the same as the semiclassical result. The accuracy of the harmonic approximation is tested in 16O. Also, a comparison is made between the adiabatic time-dependent Hartree-Fock (ATDHF) theory and an anharmonic version of the GCM.

Non axial vibrations in the adiabatic approximation

The adiabatic time-dependent Hartree-Fock method, together with a scaling assumption is used for the calculations of the mass parameters associated with non-axial quadrupole (γ) collective vibrations, and for the study of the splitting of the giant quadrupole resonance in deformed nuclei.

Four approaches to the function of inertia in a solvable model

Four different methods of calculating the mass function for large-amplitude collective motion are applied to the Lipkin model: adiabatic time-dependent Hartree-Fock (ATDHF), infinitesimal generators (IG), constrained RPA and the generator coordinate method (GC). ATDHF and GC give identical results. IG and RPA lead to the same result only in the limit of vanishing constraint. The corresponding collective Hamiltonian gives good results for the low-lying excitation energies while the cranking approximation seems to be rather poor for this case.