Derivation of collective potential and mass from the generator coordinate method

The Hill-Wheeler equation of the generator coordinate method is approximated by a local collective Schrödinger equation. General expressions for the potential and the mass parameter are obtained by a symmetrized moment expansion. The validity of the approximation is tested for several examples where the exact solution is known. These include the Gaussian overlap with harmonic and anharmonic interaction, the Lipkin model, and monopole resonances of spherical light nuclei. In all cases, surprisingly close agreement with the exact solution is found. Other possible applications of the formalism are indicated.     

Derivation of collective potentials and mass parameters from the generator coordinate method

By expanding the overlap and the hamiltonian kervels in the Hill-Wheeler equation into a series of hermitian differential operators, expressions for the collective potential and mass parameter are derived.     

Collective modes in the generator coordinate method

Using the harmonic version of the generator coordinate method, and Skyrme interaction, the frequencies of the isoscalar breathing and quadrupole modes are related to the relevant incompressibility coefficients. The possibility of extending this to spin modes is also examined. It is found that a spin incompressibility coefficient is negative for a particular set of Skyrme parameter for⁴He. Other sets produce low positive values and these in turn could imply a relatively low lyingS=2,T=1 state. The replacement of the three-body term by the density-dependent one, suggested by Chang provides a cure for this pathology.     

Effective nucleus-nucleus potentials derived from the generator coordinate method

The equivalence of the generator coordinate method (GCM) and the resonating group method (RGM) and the formal equivalence of the RGM and the orthogonality condition model (OCM) lead to a relation connecting the effective nucleus-nucleus potentials of the OCM with matrix elements of the GCM. This relation may be used to derive effective nucleus-nucleus potentials directly from GCM matrix elements without explicit reference to the potentials of the RGM. In a first application local and l-independent effective potentials are derived from diagonal GCM matrix elements which represent the energy surfaces of a two-centre shell model. Using these potentials the OCM can reproduce the results of a full RGM calculation very well for the elastic scattering of two α-particles and fairly well for elastic ¹⁶O-¹⁶O scattering.     

Calculation of nuclear reaction parameters with the generator coordinate method and their interpretation

Collisions between complex nuclei are described variationally in terms of the GCM with the aim of providing evidence that it is a manageable calculational procedure. The variational principle of Kohn and Kato is used to derive the expression for the K-matrix. The space of scattering states is spanned entirely by antisymmetrized products of shell-model wave functions describing separate clusters; the generator coordinate is the separation between the two shell-model potentials. Scattering boundary conditions are enforced by solving an integral equation for the channel GC amplitude in each open channel separately. The main part of evaluation of collision parameters is performed by calculating double integrals of a form factor between channel GC amplitudes. A theorem about a property of the form factors is proved which allows one to reduce the amount of work needed to calculate double integrals.Application of the method to elastic ³H to ⁴He scattering has shown the feasibility of the calculation.It is shown how an analysis of calculated scattering parameters and corresponding scattering states in terms of quasibound states enables one to make a consistent comparison with experiment and to extract some knowledge of the reaction mechanism.Finally a comparative list of the calculational procedures of the GCM and RGM for reactions is made.     

Pairing vibrational states and the generator coordinate method

The pairing vibrational states and the two-neutron transfer cross sections between these states are calculated in Ni, Sn and Pb isotopes by the generator coordinate method (GCM). The particle number fluctuation of the BCS functions is handled by projecting in a good approximation on sharp particle numbers. The results agree quite well with the experimental data.     

Gamov states in the generator coordinate method

Starting from the generator coordinate theory, a method is developed for calculating-decay widths within a microscopic dynamical theory. Antisymmetrization is taken into account exactly between all nucleons of the decaying system. For illustration, the method is applied to the-decay of⁸Be and²⁰Ne.     

Gamov states in the generator coordinate method

Starting from the generator coordinate theory, a method is developed for calculatingα-decay widths within a microscopic dynamical theory. Antisymmetrization is taken into account exactly between all nucleons of the decaying system. For illustration, the method is applied to theα-decay of⁸Be and²⁰Ne.     

Towards a macroscopic generator coordinate method

Collective quantities are defined as macroscopic statistical averages over many level crossing points where microscopic densities are redistributed. Accordingly, the generator coordinate method (GCM) is reconsidered. It is concluded that, contrary to earlier arguments, the macroscopically defined inertia parameter which appears in the GCM Hamiltonian has a finite value close to that obtained using traditional theories assuming the existence of the adiabatic BCS ground state.     

The generator coordinate method and the theory of metals

An application is made of the Generator Coordinate Method to a system of permions interacting with bosons. The problem treated in the present work is the collective dynamics of a metal regarded as an electron system interacting dynamically with the phonons of the lattice. A generalization of the Bohn-Staver formula for the sound velocity in metals is obtained.     

The connection between the generator coordinate method and Bose expansions

It is shown that the generator coordinate method leads to a Bose expansion which in lowest order agrees with the RPA, and which for infinite order is equivalent to the Marumori expansion.     

Variational principles for scattering in the generator coordinate method

The generator coordinate method (GCM) wave function is used as a trial function in a Kohn type variational principle for scattering phase shifts. It is shown that a GCM trial function is a solution of the variational equations if the Hill-Wheeler integral equation is satisfied subject to an appropriate boundary condition. A new method for introducing the scattering boundary condition is presented. There is a uniqueness theorem for the phase shift.     

A Schrödinger equation for collective motion from the generator coordinate method

It is shown that if the collective wave function occuring in the Griffin-Hill-Wheeler integral equation is slowly varying, the equation can be transformed into a Schrödinger equation. The conditions under which this can be done, are studied in detail.     

Collective mass parameters in the nuclear fission process

The theory of nuclear fission process is considered. The effective mass parameters are calculated as a function of the following collective coordinates: the separation between the centres of harmonic oscillators, the mass asymmetry parameter and the necking parameter. Numerical calculations are carried out for the fission of²³⁶U and²³⁸U nuclei. Symmetric and asymmetric cases are considered. From the present calculation we see the importance of taking into account the necking parameter as a dynamical collective coordinate.     

Maximum overlap,atomic coherent states and the generator coordinate method

A version of the maximum overlap approach is presented, based on the generator coordinate method. Recourse is made to techniques developed with reference to atomic coherent states. Shape instabilities in quasi-spin two-level models are studied.     

A particle number projection study with the generator coordinate method

The generator coordinate method (GCM) is used in order to describe pairing vibrations. The generating wave functions are particle-number-projected BCS-wave functions. With the conventional pairing Hamiltonian the method is tested in some simple models and applied also to the even Nickel isotopes. The results obtained can be regarded as encouraging ones.     

On calculating scattering phase shifts by the generator coordinate method

The generator coordinate equations for scattering phase shifts are solved for α-α scattering. Calculations are made in the coordinate representation and the results agree well with those obtained by a more laborious procedure which is formulated in the momentum representation. The results disagree with another calculation in the coordinate representation, where the scattering boundary condition is introduced less accurately.     

Going beyond the self-consistent mean-field with the symmetry-restored generator coordinate method

Starting from successful self-consistent mean-field models, this paper discusses why and how to go beyond the mean field approximation. To include long-range correlations from fluctuations in collective degrees of freedom, one has to consider symmetry restoration and configuration mixing, which give access to ground-state correlations and spectroscopy.