Time-odd Mean Fields Research Articles

Measurements of binding energies and electromagnetic moments of silver isotopes – A complementary benchmark of density functional theory

We report on a set of high-precision measurements of nuclear binding and excitation energies, as well as nuclear spins, magnetic dipole and electric quadrupole moments of neutron-rich silver isotopes, 113−123Ag. The measurements were performed using the JYFLTRAP mass spectrometer and the collinear laser spectroscopy beamline at the Ion Guide Isotope Separator On-Line (IGISOL) facility. For the first time, we can firmly establish the ordering of the long-lived Iπ=1/2−,7/2+ states in these isotopes, and pin down the inversion of these two levels at either A=121(N=74) or A=123(N=76). We compare these findings to calculations performed with density functional theory (DFT), from which we establish the crucial role that the spin-orbit strength and time-odd mean fields play in the simultaneous description of electromagnetic moments and nuclear binding.

Implementation of nuclear time-dependent density-functional theory and its application to the nuclear isovector electric dipole resonance

Following a previous paper [Y. Shi, Phys. Rev. C 98, 014329(2018)], we present an extension of the density-functional theory to allow for dynamic calculations based on the obtained static Hartree-Fock results. We perform extensive benchmark calculations, by comparing the calculated results with that of an existing code Sky3D. To perform linear-response calculations using the TDDFT method, comparisons have been made with the finite-amplitude quasiparticle random-phase approximation (FAM-QRPA) method. We plan to apply the TDDFT method to a systematic description of the IVD resonances in the Zr, Mo, and Ru isotopes. The strengths of IVD resonances are calculated using two complementary methods: TDDFT and FAM-QRPA methods. For the TDDFT results, additional benchmark calculations have been performed using the well-tested code Sky3D. In these three models, the important ingredients which have major influence on the results, such as time-odd potentials, boundary conditions, smoothing procedures, spurious peaks etc., have been carefully examined. The current TDDFT and the Sky3D codes yield almost identical response functions once both codes use the same time-odd mean fields and absorbing boundary conditions. The strengths of the IVD resonances calculated using the TDDFT and FAM-QRPA methods agree reasonably well with the same position of the giant dipole resonance. Upon seeing a reasonable accuracy offered by the implemented code, we perform systematic TDDFT calculations for spherical Zr and Mo isotopes near $N=50$, where experimental data exist. For neutron-rich Zr, Mo, and Ru isotopes where shape evolution exist we predict the photoabsorption cross sections based on oblate and triaxial minima.

Implementation of the variation-after-projection approach in calculations with a time-odd Hartree-Fock mean field

We have implemented a new variation-after-projection (VAP) calculation based on a time-odd Hartree-Fock (HF) mean field. The exact Hessian matrix of the projected energy has been successively evaluated for the first time, which makes the VAP calculation much more stable. With the time-odd mean field, the present VAP can be applied to the yrast states not only in the even-even nuclei, but also in the odd-$A$ and odd-odd ones. The VAP energies throughout all spins for some calculated $sd$-shell nuclei are very close to the corresponding shell model ones. Our calculations clearly show that the spin projection is very important in achieving a good approximation to the full shell model.

Covariant density functional theory for antimagnetic rotation

Following the previous letter on the first microscopic description of the antimagnetic rotation (AMR) in 105Cd, a systematic investigation and detailed analysis for the AMR band in the frame-work of tilted axis cranking (TAC) model based on covariant density functional theory are carried out. After performing the microscopic and self-consistentTAC calculations with an given density functional, the configuration for the observed AMR band in 105Cd is obtained from the single-particle Routhians. With the configuration thus obtained, the tilt angle for a given rotational frequency is determined self-consistently by minimizing the total Routhian with respect to the tilt angle. In such a way, the energy spectrum, total angular momenta, kinetic and dynamic moments of inertia, and the B(E2) values for the AMR band in 105Cd are calculated. Good agreement with the data is found. By investigating microscopically the contributions from neutrons and protons to the total angular momentum, the "two-shears-like" mechanism in the AMR band is clearly illus-trated. Finally, the currents leading to time-odd mean fields in the Dirac equation are presented and discussed in detail. It is found that they are essentially determined by the valence particles and/or holes. Their spatial distribution and size depend onthe specific single-particle orbitals and the rotational frequency.

Effect of time-odd mean fields on inertial parameters of the quadrupole collective Hamiltonian

Most current implementations of the quadrupole collective Hamiltonian model do not include the contributions of time-odd mean fields to the moments of inertia and mass parameters (Thouless-Valatin dynamical rearrangement contributions). A hybrid model is introduced that allows a quantitative estimate of these contributions to the inertial functions of a five-dimensional collective Hamiltonian based on microscopic energy density functionals. Fully self-consistent constrained relativistic Hartree-Bogoliubov calculations of triaxial energy surfaces in the $\ensuremath{\beta}\ensuremath{-}\ensuremath{\gamma}$ plane are used to determine the parameters of an equivalent pairing-plus-quadrupole ($P+Q$) Hamiltonian. This Hamiltonian is employed in constrained Hartree-Fock-Bogoliubov (CHFB) plus local quasiparticle random-phase approximation (LQRPA) calculation of the deformation-dependent corrections to the Inglis-Belyaev moments of inertia and cranking mass parameters. This hybrid model is used to evaluate the influence of time-odd mean fields on vibrational and rotational collective masses and investigate their effect on the low-energy collective excitation spectra and transition rates of $\ensuremath{\gamma}$-soft Xe and Ba nuclei.

Dynamics of nuclear single-particle structure in covariant theory of particle-vibration coupling: From light to superheavy nuclei

The impact of particle-vibration coupling and polarization effects due to deformation and time-odd mean fields on single-particle spectra is studied systematically in doubly magic nuclei from low-mass $^{56}\mathrm{Ni}$ up to superheavy ones. Particle-vibration coupling is treated fully self-consistently within the framework of the relativistic particle-vibration coupling model. Polarization effects due to deformation and time-odd mean field induced by odd particle are computed within covariant density functional theory. It has been found that among these contributions the coupling to vibrations makes a major impact on the single-particle structure. The impact of particle-vibration coupling and polarization effects on calculated single-particle spectra, the size of the shell gaps, the spin-orbit splittings and the energy splittings in pseudospin doublets is discussed in detail; these physical observables are compared with experiment. Particle-vibration coupling has to be taken into account when model calculations are compared with experiment since this coupling is responsible for observed fragmentation of experimental levels; experimental spectroscopic factors are reasonably well described in model calculations.

Properties of odd nuclei and the impact of time-odd mean fields: A systematic Skyrme-Hartree-Fock analysis

We present a systematic analysis of the description of odd nuclei by the Skyrme-Hartree-Fock approach augmented with pairing in BCS approximation and blocking of the odd nucleon. Current and spin densities in the Skyrme functional produce time-odd mean fields (TOMF) for odd nuclei. Their effect on basic properties (binding energies, odd-even staggering, separation energies and spectra) is investigated for the three Skyrme parameterizations SkI3, SLy6, and SV-bas. About 1300 spherical and axially-deformed odd nuclei with 16 < Z < 92 are considered. The calculations demonstrate that the TOMF effect is generally small, although not fully negligible. The influence of the Skyrme parameterization and the consistency of the calculations are much more important. With a proper choice of the parameterization, a good description of binding energies and their differences is obtained, comparable to that for even nuclei. The description of low-energy excitation spectra of odd nuclei is of varying quality depending on the nucleus.

Time-odd mean fields in covariant density functional theory: Rotating systems

Time-odd mean fields (nuclear magnetism) and their impact on physical observables in rotating nuclei are studied in the framework of covariant density functional theory (CDFT). It is shown that they have profound effect on the dynamic and kinematic moments of inertia. Particle number, configuration and rotational frequency dependences of their impact on the moments of inertia have been analysed in a systematic way. Nuclear magnetism can also considerably modify the band crossing features such as crossing frequencies and the properties of the kinematic and dynamic moments of inertia in the band crossing region. The impact of time-odd mean fields on the moments of inertia in the regions away from band crossing only weakly depends on the relativistic mean field parametrization, reflecting good localization of the properties of time-odd mean fields in CDFT. The moments of inertia of normal-deformed nuclei considerably deviate from the rigid body value. On the contrary, superdeformed and hyperdeformed nuclei have the moments of inertia which are close to rigid body value. The structure of the currents in rotating frame, their microscopic origin and the relations to the moments of inertia have been systematically analysed. The phenomenon of signature separation in odd-odd nuclei, induced by time-odd mean fields, has been analysed in detail.

One-quasiparticle states in the nuclear energy density functional theory

We study one-quasiproton excitations in the rare-earth region in the framework of the nuclear density functional theory in the Skyrme-Hartree-Fock-Bogoliubov variant. The blocking prescription is implemented exactly, with the time-odd mean field fully taken into account. The equal filling approximation is compared with the exact blocking procedure. We show that both procedures are strictly equivalent when the time-odd channel is neglected and discuss how nuclear alignment properties affect the time-odd fields. The impact of time-odd fields on calculated one-quasiproton band-head energies is found to be rather small, of the order of $100--200$ keV; hence, the equal filling approximation is sufficiently precise for most practical applications. The triaxial polarization of the core induced by the odd particle is studied. We also briefly discuss the occurrence of finite-size spin instabilities that are present in calculations for odd-mass nuclei when certain Skyrme functionals are employed.

Time-odd mean fields in covariant density functional theory: Nonrotating systems

Time-odd mean fields (nuclear magnetism) are analyzed in the framework of covariant density functional theory (CDFT). It is shown that they always provide additional binding to the binding energies of odd-mass nuclei. This additional binding only weakly depends on the RMF parametrization reflecting good localization of the properties of time-odd mean fields in CDFT. The underlying microscopic mechanism is discussed in detail. Time-odd mean fields affect odd-even mass differences. However, our analysis suggests that the modifications of the strength of pairing correlations required to compensate for their effects are modest. In contrast, time-odd mean fields have profound effect on the properties of odd-proton nuclei in the vicinity of proton-drip line. Their presence can modify the half-lives of proton-emitters (by many orders of magnitude in light nuclei) and affect considerably the possibilities of their experimental observation.

Band terminations in density functional theory

The analysis of the terminating bands has been performed in the relativistic mean field framework. It was shown that nuclear magnetism provides an additional binding to the energies of the specific configuration and this additional binding increases with spin and has its {\it maximum} exactly at the terminating state. This suggests that the terminating states can be an interesting probe of the time-odd mean fields {\it provided that other effects can be reliably isolated.} Unfortunately, a reliable isolation of these effects is not that simple: many terms of the density functional theories contribute into the energies of the terminating states and the deficiencies in the description of those terms affect the result. The recent suggestion \cite{ZSW.05} that the relative energies of the terminating states in the $N \neq Z, A\sim 44$ mass region given by $\Delta E$ {\it provide unique and reliable constraints on time-odd mean fields and the strength of spin-orbit interaction} in density functional theories has been reanalyzed. The current investigation shows that the $\Delta E$ value is affected also by the relative placement of the states with different orbital angular momentum ${\it l}$, namely, the placement of the $d$ (${\it l}=2$) and $f$ (${\it l}=3$) states. This indicates the dependence of the $\Delta E$ value on the properties of the central potential.

Time-odd mean fields in the rotating frame: Microscopic nature of nuclear magnetism

The microscopic role of nuclear magnetism in rotating frame is investigated for the first time in the framework of the cranked relativistic mean field theory. It is shown that nuclear magnetism modifies the expectation values of single-particle spin, orbital, and total angular momenta along the rotational axis effectively creating additional angular momentum. This effect leads to the increase of kinematic and dynamic moments of inertia at given rotational frequency and has an impact on effective alignments.