Time-dependent Pairing Equations Research Articles

Microscopic description of α-decay as super-asymmetric fission

The fine structure of α-decay is treated with fission-like models. The single particle levels are calculated along a least action path connecting the ground state of the parent nucleus and the configuration of two spherical tangent nuclei. The probabilities to find different seniority-1 configurations are obtaining by solving the time-dependent pairing equations generalized by including the Landau-Zener effect and the Coriolis coupling. The theoretical results for the α-decay of 211Po and 211Bi are compared with experimental data showing a good agreement.

Fine structure of $$\alpha $$ decay from the time-dependent pairing equations

The $$\alpha $$ decay half-lives and the fine structure phenomenon are investigated with fission-like models. A superasymmetric fission path in a configuration space spanned by five degrees of freedom is determined in accordance with the least action principle. The deformation energy is evaluated within the macroscopic–microscopic approach while the inertia is obtained in the framework of the cranking model. The single particle levels schemes are calculated connecting the ground state of the parent nucleus and the asymptotic configuration of two separated nuclei. The probabilities to find different seniority configurations are obtained by solving time-dependent pairing equations generalized by including the Landau–Zener effect and the Coriolis coupling. The microscopic equations of motion for even numbers of particles are deduced, those for odd-nuclear systems being obtained in previous works. The models used in the calculations are reviewed within a detailed description. The microscopic equations of motion are solved by starting from the ground state configuration and arriving at the scission point. A description of all the possible configurations at scission together with their realization probabilities is given. By fitting the inter-nuclear velocity, the best agreement between experimental and theoretical hindrance factors is retained. The theoretical results for the $$\alpha $$ decay half-lives for $$^{211,212}$$ Po and $$^{211}$$ Bi are compared with experimental data showing discrepancies ranging over three orders of magnitude. The accuracy of the model concerning the calculations of the half-lives for different channels is discussed. The connections between the classical theories concerning the preformation of the $$\alpha $$ particle and the fission-like descriptions are highlighted.

Cranking inertia of odd nuclei from time-dependent pairing equations: Application to Th cold fission

A new formalism for the nonadiabatic collective inertia from time-dependent pairing equations for odd numbers of nucleons is described in detail. The effective masses and the moments of inertia result from the occurrence of matrix elements of the time derivatives and of the angular couplings, respectively, between states of different seniority configurations. For low collective velocities, the formulas for the inertia reduce to the known cranking expressions available for quasitationary states. The effective mass and the perpendicular moment of inertia are evaluated for the $^{230--232}\mathrm{Th}$ parent nuclei along the fission path. The inertia for the even-odd system is larger than those of the even-even ones and exhibits fluctuations due to the intrinsic nuclear structure. The ground state theoretical value of the moment of inertia for $^{231}\mathrm{Th}$ agrees well with the evaluated experimental data if the blocking effect is neglected.

Fine structure of 211Bi α-decay from Coriolis coupling

The α-decay is considered as a superasymmetric fission process. A fragmentation path is obtained by mean of the least action principle in a configuration space spanned by five degrees of freedom. The potential barrier is obtained within the macroscopic-microscopic model while the effective mass and the momentum of inertia within the cranking approach. The single-particle wave functions and the single-particle energies are supplied by the Woods-Saxon two-center shell model. The fine structure of 211Bi α-decay is treated within a set of generalized time-dependent pairing equations that takes into account the Landau-Zener effect and the Coriolis coupling. A low value of the internuclear collective velocity was used. The theoretical results showed a good agreement with the experimental data. Essentially, in the framework of the formalism, the fine structure for the 211Bi is due to the occurrence of the Coriolis coupling.

Fine structure of α decay from the variational principle

Starting from the variational principle, the time-dependent pairing equations are generalized by including the Landau-Zener effect and the Coriolis coupling. A system of microscopic equations of motion for configuration mixing is deduced, allowing the determination of quantities that have the same meaning as the preformation factors of the $\ensuremath{\alpha}$ particle. These equations are solved in order to reproduce the hindrance factors of the $\ensuremath{\alpha}$ decay of an odd-$A$ mass nucleus. The $\ensuremath{\alpha}$ decay of $^{211}\mathrm{Po}$ is treated as a superasymmetric fission process, by following the rearrangement of the nuclear orbitals from the parent ground state up to the scission configuration. The probabilities of finding the excited states of the daughter at scission are obtained from the microscopic equations of motion. The intensities of the transitions to the excited states of the daughter were evaluated theoretically. The experimental data were compared with the theoretical findings. A very good agreement was obtained. A mean value of the tunneling velocity of about $2\ifmmode\times\else\texttimes\fi{}{10}^{4}$ fm/fs was extracted.

Nuclear inertia from the time dependent pairing equations

In a dynamical system, the momenta of inertia and the effective masses are not adiabatic quantities, but are dynamical ones that depend on the dissipated energy accumulated during motion. However, these parameters are calculated for adiabatic nuclear systems, leaving no room for dissipated energy. In this work, a formalism is elaborated in order to derive simultaneously the nuclear momenta of inertia and the effective masses by taking into account the appearance of dissipated energy for large amplitude motion of the nuclear system. The expressions that define the inertia are obtained from the variational principle. The same principle manages the time dependent pairing equations, offering estimations of the averaged dissipation energy for large amplitude motions. The model is applied to 232Th fission. The fission barrier was calculated along the least action trajectory. The dissipation energy, effective mass and moment of inertia are determined for different values of the collective velocities. The dissipation increases with the internuclear velocity in binary disintegration processes and modifies the effective mass parameters. We observed that the inertia decreases as long as the collective velocity increases to some moderate values and begins to grow for larger collective velocities. So, a dependence between the cranking mass parameters and the intrinsic excitation energy is evidenced. In order to investigate the overall effect, the half-lives are predicted for adiabatic and dynamics simulations.

Inversion of the Odd-Even Effect in Cold Fission from the Time-Dependent Pairing Equations

A peculiar phenomenon was observed experimentally in cold fission: the odd partition yields are favored over the even ones for excitations energies of the fragments smaller than 4 MeV. In this contribution, a microscopic model is proposed for the explanation of this odd-even effect in cold fission. This explanation is based on a mixing configuration mechanism that is produced during the fission process. This configuration mixing mechanism is obtained dynamically by solving a the generalized system of time-dependent pairing equations, which include a pair-breaking effect. The time dependent equations of motion for the pair breaking effect were corroborated with a condition that fixes dynamically the number of particles on the two fission fragment. The single particle level scheme was calculated with the Woods-Saxon superasymmetric two center shell model, providing a continuous variation of the single particle energies and of the wave functions from one nucleus up to two separated fragments. A first rule can be extracted from this model. The even-even fission products cannot be obtained at zero excitation energies because of the existence of dynamical excitations produced in the avoided- level-crossing regions when the nuclear system deforms slowly.

Dissipation for Z-channels in fission, alpha-decay and cluster emission

The dissipated energy in fission, cluster and alpha decay was estimated microscopically by using the time dependent pairing equations. The single particle level schemes were determined in the framework of the superasymmetric two center shell model. A strong dependence of the dissipated energy as function of the mass asymmetry is evidenced. The dissipated energy is much lower for alpha decay and cluster emission than for fission.

Intrinsic energy partition in fission

The intrinsic energy partition between two complementary fission fragments is investigated microscopically. The intrinsic excitation energy of fission fragments is dynamically evaluated in terms of the time-dependent pairing equations. These equations are corroborated with two conditions. One of them fixes the number of particles and the other separates the pairing active spaces associated to the two fragments in the vicinity of the scission configuration. The excitation energy in a wide distribution of fission fragments is calculated for the 234 U parent nucleus.

Energy Partition in Fission within a Microscopic Approach

Abstract The intrinsic excitation energy of fission fragments is dynamically evaluated in terms of the time-dependent pairing equations. These equations are corroborated with two conditions. One of them fixes the number of particles and the other separates the pairing active spaces associated to the two fragments in the vicinity of the scission configuration. The excitation energy in a wide distribution of fission fragments is calculated for the 234 U parent nucleus.

Microscopic description of energy partition in fission fragments

The energy sorting at scission is treated fully microscopically in terms of the time dependent pairing equations. These equations for the intrinsic motion are mixed with a dynamical particle number projection condition on the primary fragments. The shape sequences during the disintegration resort from the minimal action principle. For this purpose, the potential energy is computed within the macroscopic–microscopic approach and the inertia within semi-adiabatic cranking approximation. The disentanglement of the wave function in two semi-spaces is realized within the Woods–Saxon two-center shell model. That allows a separation of the pairing field onto two sub-spaces. It is shown that the excitation energies of the fission fragments depend on a delicate interplay between their structure at and their deformations at scission.

CONFIGURATION MIXING AND DISSIPATION FROM THE TIME-DEPENDENT PAIRING EQUATIONS

The time-dependent pairing equations are developed to include the Landau–Zener effect and a pair breaking mechanism. The new formalism provide information about the configuration mixing and the dissipation for a dynamical evolution of a nuclear system. A formula for a nonadiabatic cranking inertia is also deduced. Results concerning cluster decay and fission processes are presented.

Strong dissipation regime in nuclear disintegrations

The coupling of collective degrees of freedom with the microscopic ones causes dissipation and a modification of the adiabatic potential. The term dissipation usually refers to exchange of energy (either linear or angular momentum) by all kind of damping from collective motion to intrinsic heat. A measure of the dissipated energy can be obtained solving the time dependent pairing equations. In this presentation, a generalization of the time dependent pairing equations is presented by including the Landau-Zener effect in the superfluid model. These new equations allows a mixing of seniority one configurations that allows us to obtain a ground state at the end of the process. An application concerning the 14C emission is offered and its fine structure is explained. These new equations are furthermore used to evidence a dynamical pair breaking effect that could explain the fine odd-even effect in cold fission. Finally, the time dependent pairing equations are used to deduce a model for non-adiabatic cranking inertia.

Dissipative regime in low energy fission

Abstract In this presentation, a generalization of the time dependent pairing equations is presented by including the Landau-Zener effect in the superfluid model. These new equations allows a mixing of seniority one configurations that allows us to obtain a ground state at the end of the process. An application concerning the C-14 emission is offered and its fine structure is explained. These new equations are furthermore used to evidence a dynamical pair breaking effect that could explain the fine odd-even effect in cold fission. The partition of the dissipated energy between the two fission fragments will be also modeled. Finally, the time dependent pairing equations are used to deduce a model for non-adiabatic cranking inertia.

Energy partition in low energy fission

The intrinsic excitation energy of fission fragments is dynamically evaluated in terms of the time dependent pairing equations. These equations are corroborated with two conditions. One of them fixes the number of particles and the another separates the pairing active spaces associated to the two fragments in the vicinity of the scission configuration. The fission path is obtained in the frame of the macroscopic-microscopic model. The single particle level schemes are obtained within the two center Woods-Saxon shell model. It is shown that the available intrinsic dissipated energy is not shared proportionally to the masses of the two fission fragments. If the heavy fragment possesses nucleon numbers close to the magic ones, the accumulated intrinsic excitation energy is lower than that of the light fragment.

Cranking mass parameters for fission

A formalism for semi-adiabatic cranking mass parameters is presented. For the fission process of 234U, the time-dependent pairing equations of motion are used to calculate the excitation energy and to extract values of the cranking inertia. A fission barrier is determined by minimizing the action trajectory in a five-dimensional configuration space spanned by elongation, necking, deformations of fragments and mass asymmetry. The deformation energy is computed in the frame of the microscopic–macroscopic model. The two-center shell model with Woods–Saxon potentials is used in this context. Values of the inertia for excited fissioning systems are reported. A dependence between the cranking mass parameters and the intrinsic excitation energy is evidenced.

Time-dependent pairing equations for seniority-one nuclear systems

When the time-dependent Hartree-Fock-Bogoliubov intrinsic equations of motion are solved in the case of seniority-one nuclear systems, the unpaired nucleon remains on the same orbital. The blocking effect hinders the possibility to skip from one orbital to another. This unpleasant feature is by-passed with a new set of pairing time-dependent equations that allows the possibility that the unpaired nucleon changes its single-particle level. These equations generalize the time-dependent Hartree-Fock-Bogoliubov equations of motion by including the Landau-Zener effect. The derivation of these new equations is presented in detail. These equations are applied to the case of a superasymmetric fission process, that is, to explain the fine structure the $^{14}\mathrm{C}$ emission from $^{233}\mathrm{Ra}$. In this context, a new version of the Woods-Saxon model extended for two-center potentials is used.