Generalized cranking model for collective nuclear motion

The Inglis cranking model is generalized to take into account effects of any velocity dependence present in the single-particle potential and the reaction of the pairing field to the collective motion. The generalized model is applied to translations, rotations and some special types of vibrations. Some of our results and our numerical calculations are obtained with a harmonic-oscillator single-particle potential. Unlike the inertia calculated with the Inglis cranking model, the inertia calculated with the generalized cranking model is independent of the effective mass and approaches the irrotational value in the limit of large pairing.     

Calculation of the nuclear inertia in a generalized cranking model

By use of a time-dependent formalism, we generalize the Inglis cranking model to include the velocity dependence of the single-particle potential and the reaction of the pairing field to the collective motion. This generalized cranking model is used to calculate the inertia for spheroidal deformations of²⁴⁰Pu in both a pure harmonicoscillator potential and a harmonic-oscillator potential with spin-orbit interaction. For comparisons with irrotational values, we transform to the inertia with respect to a matter coordinate that specifies the distance between the centers of mass of the two halves of the nucleus. We study in particular the dependence of the inertia upon nuclear temperature in the absence of pairing, and its dependence upon pairing strength for zero temperature.     

Energy surfaces,collective potentials and collective inertia parameters

The Hill-Wheeler equation is used to derive a collective Hamiltonian from an energy surface. Two simple ansatz are proposed for the collective Hamiltonian, which is then calculated explicitly in a hierarchy of approximations. The consistency of the method is verified in an illustrative example.     

Analytical relationship for the cranking inertia

The wave functions of a spheroidal harmonic oscillator without spin-orbit interaction are expressed in terms of associated Laguerre and Hermite polynomials. The pairing gap and Fermi energy are found by solving the BCS system of two equations. Analytical relationships for the matrix elements of inertia are obtained as a function of the main quantum numbers and potential derivative. They may be used to test complex computer codes developed in a realistic approach of the fission dynamics. Results given for the ²⁴⁰Pu nucleus are compared with a hydrodynamical model. The importance of taking into account the correction term due to the variation of the occupation number is stressed.     

Theory of non-adiabatic cranking and nuclear collective motion

The cranking model is extended to the case of a general non-adiabatic motion. The time-dependent many-body Schrödinger equation is solved, where the time dependence of the collective motion is determined by the classical Lagrange equations of motion. The Lagrangian is obtained from the expectation value of the energy. In the case of one collective degree of freedom the condition that the expectation value of the energy is constant in time is sufficient to determine the collective motion. An iteration procedure is applied, of which the zeroth order is shown to be the common cranking formula. In an alternative approach the energy conservation is expressed in differential form. This leads in the case of one collective degree of freedom to a set of coupled, non-linear first-order differential equations in time for the expansion coefficients of the many-body wave function and for the collective variable. As an illustrative example we solve the case of two coupled linear harmonic oscillators.     

Semiclassical inertia of nuclear collective dynamics

For the low-lying collective excitations in nuclei, the transport coefficients, such as the stiffness, the inertia, and the friction, are derived within the periodic-orbit theory in the lowest orders of semiclassical expansion corresponding to the extended Thomas—Fermi approach. The multipole vibrations near the spherical shape are described in the mean-field approximation through the infinitely deep square-well potential and Strutinsky averaging of the transport coefficients. Owing to the consistency condition, the collective inertia for sufficiently increased particle numbers and temperatures is substantially larger than that of irrotational flow. The average energies of collective vibrations, reduced friction, and effective damping coefficients are in better agreement with experimental data than those found from the hydrodynamic model. The text was submitted by the authors in English.     

Solution of the cranking model to all orders

For a system with an energy gap a solution to the time dependent Schrödinger equation is obtained in form of a systematic expansion in powers of an inverse, collective time scale. The close connection to a model with a changing self consistent field is discussed as well as the applicability to nuclear fission. The fourth order correction to the almost adiabatic solution to the cranking model is given explicitly.     

Angular momentum fluctuation energy in the cranking model

Angular momentum is approximately projected from Hartree-Fock-Bogoliubov cranked (HFBC) wave functions. At each J the projected energy is $\text{E}{}_{\mathrm{<mtext>proj</mtext>}}\text{≈ E}{}_{\mathrm{<mtext>HFBC</mtext>}}\text{? (Δ}\text{J}\text{)}{}^2\text{/2}\text{J}{}_{\mathrm{<mtext>HFBC</mtext>}}$. The spin-dependent fluctuation $\text{Δ}\text{J}$ includes contributions from J_y and J_z as well as J_x. There are no correlations in the three angular momentum components. Projected energies are calculated for ^{168, 170}Yb and ¹⁷⁴Hf. When compared to experimental energies, the projected spectra are less compressed than the HFBC spectra. At low spins the projected and experimental energies are in good agreement.     

Connection between nuclear moments of inertia and collective variables

Several frequently used formulae for calculating nuclear moments of inertia are shown to be associated with specific choices of collective variables.     

Towards CRAMOLA,the cranking model for large amplitudes

Arguments are presented showing that the traditional cranking model derivations applied to cases of finite velocity face serious problem due to violation of the consistency between the density distribution and the assumed external potential. A dynamical zero-approximation function determined with the moving adiabatic basis states is suggested which may help to overcome this difficulty and be used in an expansion exploiting the incoherency of the strong-coupling matrix elements at the level crossings.     

On the validity of the cranking model for 2+-vibrations

The relation of the cranking model to more fundamental methods of determining the generator of a collective state is investigated for the quadrupole case. We are able to show that the heuristic procedure starting with an adiabatic change in the nuclear deformation and splitting the cranking operator according to main quantum number selection rules approximates the true collective states very well. However, for the low-lying collective state the analytic form of the cranking term is completely different from any type of scaling ansatz.     

Investigation of the cranking model wave function with respect to angular momentum

Projection of angular momentum on cranking model wave functions is performed for some simple cases. An extensive analysis has been possible since an algebraic projection technique is employed and a detailed numerical example is presented. The distribution of angular momentum as a function of rotational frequency and signature is analysed, and special attention is focused on the fact that the signature does not give any strict selection of angular momenta contained in the cranking wave function.     

Application of a fusion model using collective variables and microscopically related mass parameters

A calculation of nucleus-nucleus collisions is presented, using a model which starts from a TDHF equation and leads to classical equations of motion for a set of four collective variables. Restricting to axial symmetry and assuming the liquid drop mass formula to hold, a differential equation is derived, which describes nuclear deformations and energies and is used to construct a potential energy surface for the collective variables. The nuclear deformations are obtained without the need of shape parameters. The equations of motion for the collective variables are solved numerically.     

Variable collective inertia and the transition from spherical to deformed shapes in the Hg-isotopes

The observed sudden increase of the mean square nuclear charge radius between¹⁸⁷Hg and¹⁸⁵Hg is explained as an increase of the nuclear deformation. The zero point vibration has been taken into account within the cranking approach.     

A collective model description of the low lying and giant dipole resonant properties of40,42,44,46Ca

Zeitschrift für Physik A Hadrons and nuclei - The low lying and giant dipole resonant properties of the even-even calcium isotopes are calculated within the framework of the Gneuss-Greiner...     

Dependence of calculated collective nuclear properties on the form of the potential energy and inertia tensor

A systematic study of the dependence of the collective quantities (energies and matrix elements of the E2, M1 and E0 moments) on the form of approximations to the potential energy V and the inertia tensor B is performed. Various approximations used up to now are tested. Macroscopic-microscopic values for V and cranking results for B are taken as a reference. The collective quantities are calculated by solving the Schrödinger equation corresponding to the collective Bohr hamiltonian. The contribution of all nucleons is explicitly taken into account; no renormalization factors are used. Spherical, transitional and deformed even-even nuclei are considered. The quality of various approximations for V and B used in the boson-expansion method is discussed. Large effects of the microstructure of the inertia tensor B are obtained and commented on.