Single-particle Wave Functions Research Articles

Canonical states in continuum Skyrme Hartree-Fock-Bogoliubov theory with Green's function method

The canonical single-particle energies and wave functions are obtained by diagonalizing the density matrix on a spatial mesh in continuum Skyrme Hartree-Fock-Bogoliubov theory with the Green's function method. Taking the exotic nucleus $^{66}\mathrm{Ca}$ as an illustrative example, the canonical energies and wave functions are compared with those obtained by the box-discretized method. Most of the results are consistent except for ${s}_{1/2}$ orbitals with positive energies, which have large spatial extension. By examining the canonical energies and wave functions with different box sizes, it is shown that the Green's function method can obtain the convergent canonical energy within a box size smaller than that of the box-discretized method, due to the correct asymptotic behaviors of the canonical wave functions.

Inelastic longitudinal electron scattering C42 form factors in42Ti nucleus

Inelastic longitudinal electron scattering form factors for secondexcited state C42 in 42Ti nucleus have been calculated using shellmodel theory. Fp shell model space with configuration (1f7/2 2p3/21f5/2 2p1/2) has been adopted in order to distribute the valenceparticles (protons and neutrons) outside an inert core 40Ca. Modernmodel space effective interactions like FPD6 and GXPF1 have beenused to generate model space vectors and harmonic oscillator wavefunction as a single particle wave function. Discarder space (coreorbits + higher orbits) has been included in (core polarization effect)as a first order correction in microscopic theory to measure theinterested multipole form factors via the model space.Gogny and Michigan sum of three-range Yukawapotential (M3Y-p2) have been utilized as a residual interaction tocouple the (particle-hole) pair across the model space active particlesand the excitation energy of the pair is (2ħω) and four options for theused effective and residual interactions were determined for thetransitions from (+0) to (+01,2,3), (+21,2,3) and (+41,2,3).

Study of nuclear structure of <SUP align=right>58,62</SUP>Ni isotopes using the F5PVH effective interaction

The nuclear structure of 58,62Ni nuclei has been calculated using F5PVH as effective interaction for 1f5/2 2p3/2 2p1/2 shell model wave functions. Skyrme-Hartree Fock (SKX), Harmonic Oscillator (HO) and Woods-Saxon (WS) potentials were used to calculate the single particle wave function. The core-polarisation effects have been adopted using the shape of Tassie model. The level schemes are compared with the experimental data up to 3.42 MeV and 4.018 MeV for 58Ni and 62Ni, respectively. In this study, very good agreements are obtained for all nuclei. Results from C2 form factor and Charge Density Distribution (CDD) calculations give good agreement with the experimental data with no adjustable parameters.

Study of nuclear structure of 58,62Ni isotopes using the F5PVH effective interaction

The nuclear structure of 58,62Ni nuclei has been calculated using F5PVH as effective interaction for 1f5/2 2p3/2 2p1/2 shell model wave functions. Skyrme-Hartree Fock (SKX), Harmonic Oscillator (HO) and Woods-Saxon (WS) potentials were used to calculate the single particle wave function. The core-polarisation effects have been adopted using the shape of Tassie model. The level schemes are compared with the experimental data up to 3.42 MeV and 4.018 MeV for 58Ni and 62Ni, respectively. In this study, very good agreements are obtained for all nuclei. Results from C2 form factor and Charge Density Distribution (CDD) calculations give good agreement with the experimental data with no adjustable parameters.

Nascent fragment shell effects on the nuclear fission processes in semiclassical periodic orbit theory

Making use of the semiclassical periodic orbit theory (POT), we propose, for the first time, a method to exclusively evaluate the shell effects associated with each of the nascent fragments (prefragments) generated by the neck formation in nuclear fission processes. In spite of the strong indication of such shell effects in asymmetric fragment mass distributions, they could not have been accessed by any previous theoretical approach since most of the single-particle wave functions are delocalized. In the POT, we have found that the prefragment shell effects can be naturally and unambiguously identified as the ontributions of the classical periodic orbits localized in each of the prefragments. For a numerical test, simple cavity potential models are employed with the shape described by the three-quadratic-surface shape parametrization. Deformed shell energies are studied with the trace formula for degenerate orbits in a runcated spherical cavity which was recently derived [K. Arita, Phys. Rev. C 98, 064310 (2018)]. In this simple model, it is shown that the prefragment shell effect dominates the total shell energy shortly after the neck formation, and the magicity of the heavier prefragment plays a significant role in establishing the fission saddle with asymmetric shape which leads to an asymmetric scission.

Nuclear mass parameters and moments of inertia in a folded-Yukawa mean-field approach

Collective moments of inertia as well as quadrupole and octupole mass parameters are calculated in the perturbative cranking approximation using the folded-Yukawa mean-field potential written in Cartesian coordinates. The resulting single-particle Hamiltonian is required, as a minimal symmetry, to be invariant under z-signature and time-reversal transformations. The deformation of the nucleus is defined through a shape parametrization in cylindrical coordinates, with the so-called Funny–Hills and Trentalange–Koonin–Sierk shapes as typical and performant examples. To take pairing correlations into account, the standard set of BCS equations is solved with an approximation of constant pairing strength. The numerical program determining the quadrupole–octupole mass tensor and the moments of inertia, written in Fortran, is constructed according to the here presented study as an extension and application of the “yukawa” code for the diagonalization of the folded-Yukawa mean-field potential in the basis of a harmonic-oscillator in Cartesian coordinates, published in this journal in 2016. Program summaryProgram Title: yukmomsProgram Files doi:http://dx.doi.org/10.17632/9ygk4bgkfr.1Licensing provisions: GPLv3Programming language: FortranNature of problem: The moments of inertia and the quadrupole and octupole mass parameters are calculated within the perturbative cranking approximation. As an input, the folded-Yukawa mean-field is diagonalized in the harmonic-oscillator deformed basis to generate the single-particle energies and wave functions. Nuclear shapes are given by means of well known Funny–Hills or Trentalange–Koonin–Sierk parametrizations able to describe the elongation, mass asymmetry, non-axiality and neck of the deformed nucleus.Solution method: Having diagonalized the matrix corresponding to the folded-Yukawa single-nucleon Hamiltonian expressed in the basis of the anisotropic harmonic oscillator basis, one then calculates the matrix elements of the one-body angular-momentum as well as the quadrupole and octupole-moments operators. These matrix elements together with the solutions of the coupled BCS equations are essential input to calculate the desired collective masses and moments of inertia of atomic nuclei.

Two-Particle Collisional Coordinate Shifts and Hydrodynamic Anomalous Hall Effect in Systems without Lorentz Invariance.

We show that electrons undergoing a two-particle collision in a crystal experience a coordinate shift that depends on their single-particle Bloch wave functions and derive a gauge-invariant expression for such a shift, valid for arbitrary band structures and arbitrary two-particle interaction potentials. As an application of the theory, we consider two-particle coordinate shifts for Weyl fermions in space of three spatial dimensions. We demonstrate that such shifts in general contribute to the anomalous Hall conductivity of a clean electron liquid.

Superintegrable relativistic systems in scalar background fields

We consider a relativistic charged particle in a background scalar field depending on both space and time. Poincaré, dilation and special conformal symmetries of the field generate conserved quantities in the charge motion, and we exploit this to generate examples of superintegrable relativistic systems. We also show that the corresponding single-particle wavefunctions needed for the quantum scattering problem can be found exactly, by solving the Klein–Gordon equation.

The quantum steeplechase

Quantum steeplechase is the study of a Luttinger liquid (LL) in one dimension in the presence of a finite number of barriers and wells clustered around an origin. The powerful nonchiral bosonization technique (NCBT) is introduced to write down closed formulas for the two-point functions in the sense of the random phase approximation (RPA). Unlike g-ology-based methods that are tied to the translationally invariant, free particle basis, the NCBT explicitly makes use of the translationally noninvariant single particle wave functions. The present method, that provides the most singular part of the asymptotically exact Green function in a closed form, is in contrast to competing methods that require a combination of renormalization group and/or numerical methods in addition to the bosonization techniques.

Quantum dynamics of Bose-Fermi mixtures via the stochastic-wave-function approach

We present a reformulation of the nonrelativistic many-body quantum dynamics of Bose-Fermi mixtures via stochastic wave functions. We show that, within a wide range of two-particle interactions and a certain class of greater-than-two-particle interactions, the quantum dynamics can be mapped exactly onto a set of single-particle wave functions, with their evolution governed by a stochastic process.

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.

Atomization of correlated molecular-hydrogen chain: A fully microscopic variational Monte Carlo solution

We discuss electronic properties and their evolution for the linear chain of $H_2$ molecules in the presence of a uniform external force $f$ acting along the chain. The system is described by an extended Hubbard model within a fully microscopic approach. Explicitly, the microscopic parameters describing the intra- and inter-site Coulomb interactions are determined together with the hopping integrals by optimizing the system ground state energy and the single-particle wave functions in the correlated state. The many-body wave function is taken in the Jastrow form and the Variational Monte-Carlo (VMC) method is used in combination with an ab initio approach to determine the energy. Both the effective Bohr radii of the renormalized single-particle wave functions and the many-body wave function parameters are determined for each $f$. Hence, the evolution of the system can be analyzed in detail as a function of the equilibrium intermolecular distance, which in turn is determined for each $f$ value. The transition to the atomic state, including the Peierls distortion stability, can thus be studied in a systematic manner, particularly near the threshold of the dissociation of the molecular into atomic chain. The computational reliability of VMC approach is also estimated.

Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries

We present a fully many-body formulation of topological invariants for various topological phases of fermions protected by antiunitary symmetry, which does not refer to single particle wave functions. For example, we construct the many-body $\mathbb{Z}_2$ topological invariant for time-reversal symmetric topological insulators in two spatial dimensions, which is a many-body counterpart of the Kane-Mele $\mathbb{Z}_2$ invariant written in terms of single-particle Bloch wave functions. We show that an important ingredient for the construction of the many-body topological invariants is a fermionic partial transpose which is basically the standard partial transpose equipped with a sign structure to account for anti-commuting property of fermion operators. We also report some basic results on various kinds of pin structures -- a key concept behind our strategy for constructing many-body topological invariants -- such as the obstructions, isomorphism classes, and Dirac quantization conditions.

Nuclear properties for astrophysical and radioactive-ion-beam applications (II)

We tabulate the ground-state odd-proton and odd-neutron spins and parities, proton and neutron pairing gaps, one- and two-neutron separation energies, quantities related to β-delayed one- and two-neutron emission probabilities, average energy and average number of emitted neutrons, β-decay energy release and half-life with respect to Gamow–Teller decay with a phenomenological treatment of first-forbidden decays, one- and two-proton separation energies, and α-decay energy release and half-life for 9318 nuclei ranging from 16O to 339136 and extending from the proton drip line to the neutron drip line. This paper is a new and improved version of Atomic Data And Nuclear Data Tables [66 131 (1997)]. The starting point of our present work is the new study (FRDM(2012)) of nuclear ground-state masses and deformations based on the finite-range droplet model and folded-Yukawa single-particle potential published in a previous issue of Atomic Data And Nuclear Data Tables [109–110, 1 (2016)]. The β-delayed neutron-emission probabilities and Gamow–Teller β-decay rates are obtained from a quasi-particle random-phase approximation with single-particle levels and wave functions at the calculated nuclear ground-state shapes as input quantities. A development since 1997 is we now use a Hauser–Feshbach approach to account for (n, γ) competition and treat first-forbidden decay in a phenomenological approach.

Analysis with SDHO and RMF density distributions of elastic scattering cross-sections of oxygen isotopes (16–18O) by various target nuclei

In the present study, two different density distributions of oxygen isotopes ([Formula: see text]O) that consist of the harmonic oscillator single-particle wave functions (SDHO) and the relativistic mean-field (RMF) approaches are investigated for the availability of elastic scattering cross-sections. For this purpose, the elastic scattering angular distributions of nuclear reactions with 13 target nuclei, four target nuclei and nine target nuclei are calculated for [Formula: see text]O, [Formula: see text]O and [Formula: see text]O, respectively. For these calculations, the double folding model based on the optical model is used. The optical potential parameters, volume integrals and cross-sections for all the nuclear reactions are given in this study. The comparison of theoretical results and experimental data shows very good agreement. The imaginary potential depth expressions, which will be new and more practical terms to explain the nuclear interactions of [Formula: see text]O, [Formula: see text]O and [Formula: see text]O with different nuclei, for each oxygen isotope are proposed.

Manifestly covariant canonical quantization of the scalar field and particle localization

Particle localization within quantum field theory is revisited. Canonical quantization of a free scalar field theory is performed in a manifestly Lorentz covariant way with respect to an arbitrary 3-surface [Formula: see text], which is the simultaneity surface associated with the observer, whose proper time direction is orthogonal to [Formula: see text]. Position on [Formula: see text] is determined by a 4-vector [Formula: see text]. The corresponding quantum position operator, formed in terms of the operators [Formula: see text], [Formula: see text], that create/annihilate particles on [Formula: see text], has thus well-behaved properties under Lorentz transformations. A generic state is a superposition of the states, created with [Formula: see text], the superposition coefficients forming multiparticle wave packet profiles — wave functions, including a single-particle wave function that satisfies the covariant generalization of the Foldy equation. The covariant center-of-mass operator is introduced and its expectation values in a generic multiparticle state calculated.

Theoretical Study of Density Distributions and Size Radii of 8B and 17Ne

The proton, neutron and matter density distributions, the corresponding size radii and elastic electron scattering form factors of one-proton 8 B and two-proton 17 Ne halo nuclei are calculated. The theoretical technique used to fulfill calculations is by assuming that both nuclei under study are composed of two main parts; the first is the compact core and the second is the unstable halo part. The single-particle radial wavefunctions of harmonic-oscillator (HO) and Woods-Saxon (WS) potentials are used to study core and halo parts, respectively. And other approach is studied by using HO potential for both core and halo parts, but using two HO size parameters for both supposed parts. The long tail behavior which is the main characteristic of halo nuclei are well produced for both 8 B and 17 Ne. The calculated size radii are in general in good agreement with the available experimental data. The electron scattering form factors of the C0+C2 and C0 components are also calculated for 8 B and 17 Ne, respectively and compared with corresponding stable 10 B and 20 Ne nuclei. For 8 B calculations, the core-polarization (CP) effects are taken into account by using Tassie and Bohr-Mottelson models. The contribution from model-space (MS) part C2 component is taken through pwt interaction. The results of the calculated charge form factors are left for the planned electron-radioactive ion beam colliders where the study of skin or halo on the charge form factors are going to be studied.

Probing Sizes and Shapes of Nobelium Isotopes by Laser Spectroscopy.

Until recently, ground-state nuclear moments of the heaviest nuclei could only be inferred from nuclear spectroscopy, where model assumptions are required. Laser spectroscopy in combination with modern atomic structure calculations is now able to probe these moments directly, in a comprehensive and nuclear-model-independent way. Here we report on unique access to the differential mean-square charge radii of ^{252,253,254}No, and therefore to changes in nuclear size and shape. State-of-the-art nuclear density functional calculations describe well the changes in nuclear charge radii in the region of the heavy actinides, indicating an appreciable central depression in the deformed proton density distribution in ^{252,254}No isotopes. Finally, the hyperfine splitting of ^{253}No was evaluated, enabling a complementary measure of its (quadrupole) deformation, as well as an insight into the neutron single-particle wave function via the nuclear spin and magnetic moment.

Combination of complex momentum representation and Green's function methods in relativistic mean-field theory

We have combined the complex momentum representation method with the Green's function method in the relativistic mean-field framework to establish the RMF-CMR-GF approach. This new approach is applied to study the halo structure of $^{74}$Ca. All the continuum level density of concerned resonant states are calculated accurately without introducing any unphysical parameters, and they are independent of the choice of integral contour. The important single-particle wave functions and densities for the halo phenomenon in $^{74}$Ca are discussed in detail.

Quantal diffusion description of multinucleon transfers in heavy-ion collisions

Employing the stochastic mean-field (SMF) approach, we develop a quantal diffusion description of the multi-nucleon transfer in heavy-ion collisions at finite impact parameters. The quantal transport coefficients are determined by the occupied single-particle wave functions of the time-dependent Hartree-Fock equations. As a result, the primary fragment mass and charge distribution functions are determined entirely in terms of the mean-field properties. This powerful description does not involve any adjustable parameter, includes the effects of shell structure and is consistent with the fluctuation-dissipation theorem of the non-equilibrium statistical mechanics. As a first application of the approach, we analyze the fragment mass distribution in $^{48}\mathrm{Ca}+{}^{238}\mathrm{U}$ collisions at the bombarding energy $E_{\text{c.m.}}=193$ MeV and compare the calculations with the experimental data.