E2 Transition Rates Research Articles

Energies and E1, M1, E2 transition rates for states of the [formula omitted] and [formula omitted] configurations in fluorine-like ions between Si VI and W LXVI

Energies and E1, M1, E2 transition rates from relativistic configuration interaction calculations are reported for the states of the (1s2)2s22p5 and 2s2p6 configurations in all fluorine-like ions between Si VI and W LXVI. Valence, core–valence, and core–core correlation effects were accounted for through single and double excitation expansions to increasing sets of active orbitals.

Analyses of β-Bands of 230,232Th and 232,234U by the Projected Shell Model

The ground bands and β-bands of four nuclei 230,232Th and 232,234U in the actinide region are investigated by introducing a collective D0 pair into the projected shell model. We discuss the collectivity of the D0 pair. The calculated energy schemes agree well with experimental data, and so do the E2 transition rates.

Energies, E1, M1, and E2 transition rates, hyperfine structures, and Landé [formula omitted] factors for states of the 2s 22p 2, 2s2p 3, and 2p 4 configurations in carbon-like ions between F IV and Ni XXIII

Energies, electric dipole, magnetic dipole, and electric quadrupole transition rates, hyperfine structures, and Landé g J factors from relativistic configuration interaction calculations are reported for the states of the (1s 2)2s 22p 2, 2s2p 3, and 2p 4 configurations in all carbon-like ions between F IV and Ni XXIII. Valence, core–valence, and core–core correlation effects were accounted for through single/double-excitation-multireference expansions to increasing sets of active orbitals. The calculated energy levels generally agree within a few hundred cm −1 with the experimentally compiled results, and the Babushkin (length), and Coulomb (velocity) forms of transition rates agree within less than 1% for a majority of the allowed transitions.

Excitation energies, E1, M1, and E2 transition rates, and lifetimes in Ca+, Sr+, Cd+, Ba+, and Hg+ 1This article is part of a Special Issue on the 10th International Colloquium on Atomic Spectra and Oscillator Strengths for Astrophysical and Laboratory Plasmas.

Excitation energies of ns1/2, npj, and ndj states in Cd+ (n = 5), Hg+ (n = 5) and ns1/2, npj, and (n – 1)dj states in Ca+ (n = 4), Sr+ (n = 5), and Ba+ (n = 6) are evaluated using the linearized coupled-cluster (all-order) method. Reduced matrix elements, oscillator strengths, and transition rates are determined for the ns–npj–ndj (or ns–npj–(n – 1)dj) possible electric dipole transitions in Ca+, Sr+, Ba+, Cd+, and Hg+. Electric quadrupole matrix elements are evaluated to obtain ns1/2–(n – 1)dj transition rates in Ca+ (n = 5), Sr+ (n = 5), and Ba+ (n = 6). The matrix elements are calculated using both relativistic many-body perturbation theory, complete through third order, and the relativistic all-order method restricted to single and double (SD) excitations. The SD lifetime results for the np and nd states in Ca+, Sr+, Ba+, Cd+, and Hg+, are compared with the latest available experimental measurements. The contribution of the magnetic dipole nd3/2–nd5/2 transition to the lifetimes of the lowest nd5/2 level ln Ca+, Sr+, and Ba+ ions is discussed. These calculations provide a theoretical benchmark for comparison with experiment and theory as well as data needed for various applications.

Low-lying states of valence-hole nuclei in the 208Pb region

Systematic calculations of low-lying states for Ir, Pt, Au, Hg and Tl isotopes with neutron numbers between 120 and 125 have been performed within the framework of the SDG-pair approximation of the shell model. We employ a monopole and quadrupole pairing plus quadrupole–quadrupole-type interaction with optimized parameters, which are assumed to be constants for nuclei with the same proton number or neutron number. We calculate binding energies of the ground states, low energy level schemes, electric quadrupole and magnetic dipole moments, and E2 transition rates. Our results are reasonably consistent with the available experimental data as well as previous theoretical studies, in particular, for low-lying yrast states. We also demonstrate that low-lying states of nuclei studied here are usually well represented by very simple configurations in collective nucleon-pair basis.

The E(2) symmetry and quantum phase transition in the two-dimensional limit of the vibron model

We study in detail the relation between the two-dimensional Euclidean dynamical E(2) symmetry and the quantum phase transition in the two-dimensional limit of the vibron model, called the U(3) vibron model. Both geometric and algebraic descriptions of the U(3) vibron model show that structures of low-lying states at the critical point of the model with a quartic potential as its classical limit can be approximately described by the E(2) symmetry. We also fit the finite-size scaling exponent of the energy levels and E1 transition rates in the F(2) model, which is exactly the E(2) model but with truncation in its Hilbert subspace, as well as those at the critical point in the U(3) vibron model. The N-scaling power law around the critical point shows that the E(2) symmetry is well preserved even for cases with finite number of bosons. In addition, two kinds of experimentally accessible effective order parameters, such as the energy ratios and E1 transition ratios , , are proposed to identify the second-order phase transition in such systems. Possible empirical examples exhibiting approximate E(2) symmetry are also presented.

NUCLEAR STRUCTURE OF 150Sm IN THE DPPQ MODEL AND IBM

The decay pattern of the low lying levels in the shape transitional nucleus 150 Sm is analyzed in terms of the quasi-vibrational and quasi-rotational collective model. The Dynamic Pairing plus quadrupole Model in the microscopic theory of the collective model is employed to predict extensive structure characteristics. The potential energy surface (PES), the spectroscopic factors and the E2, E0 transition rates from the DPPQ model are illustrated. Comparison is made with the predictions in the Interacting Boson Model (IBM-1). A correspondence is demonstrated of the effect of the control parameter in the two models on the calculated nuclear structure. The study of the wave functions of the two spin I = 2 excited states in IBM is carried out. The alternative view of an anharmonic vibrator and a soft rotor is discussed in terms of the E2 transition rates and other structural characteristics.

Bottomonium Spectrum with Screened Potential

As a sister work of [Phys. Rev. D 79 (2009) 094004], we incorporate the color-screening effect due to light quark pair creation into the heavy quark-antiquark potential, and investigate the effects of screened potential on the spectrum of bottomonium. We calculate the masses, electromagnetic decays, and E1 transitions of bottomonium states. We find that the fine splittings between χbJ (J = 0, 1, 2) states are in good agreement with experimental data, and the E1 transition rates of γ(2S) → γχbJ (1P) and γ(3S) → γχbJ (2P) (J = 0, 1, 2) all agree with data within experimental errors. In particular, the mass of γ(6S) is lowered down to match that of the γ(11020), which is smaller than the predictions of the linear potential models by more than 100 MeV. Comparison between charmonium and bottomonium in some related problems is also discussed.

Particle number scale invariant feature of the states around the critical point of the first order nuclear shape phase transition

We study systematically the evolutive behaviors of some energy ratios, E2 transition rate ratios and isomer shift in the nuclear shape phase transitions. We find that the quantities sensitive to the phase transition and independent of free parameter(s) are approximately particle number N scale invariant around the critical point of the first order phase transition, similar to that in the second order phase transition.

The study of reduced transition probabilities for E2 transitions in the decays of 192Os and 192Pt nuclei

The ratios of E2 transition rates, i.e., B( E2; I i − I f )/ B( E2; I i − I f ) were computed for gamma to ground state band transitions in 192Os and 192Pt. The reduced transition probabilities for the E2 transitions evaluated on the basis of Interacting Boson Approximation (IBA-I) model and those measured experimentally are compared. The SU(3) symmetry of IBA-I is not strictly obeyed by the nuclei in which T( E2; I i − I f ) transitions are observed, i.e., the symmetry SU(3) is broken. The percentage sum coincidence corrections are applied to marginalise contributions from crossover transitions and the intensities of affected transitions are found to agree fairly well with earlier sum peak method applications to the decay of 192Ir.

Low-lying states of heavy nuclei within the nucleon pair approximation

In this article we perform systematic calculations on low-lying states of 33 nuclei with A=202-212, using the nucleon pair approximation of the shell model. We use a phenomenological shell-model Hamiltonian that includes single-particle energies, monopole and quadrupole pairing interactions, and quadrupole-quadrupole interactions. The building blocks of our model space include one J=4 valence neutron pair, and one J=4,6,8 valence proton pair, in addition to the usual S and D pairs. We calculate binding energies, excitation energies, electric quadrupole and magnetic dipole moments of low-lying states, and E2 transition rates between low-lying states. Our calculated results are reasonably consistent with available experimental data. The calculated quadrupole moments and magnetic moments, many of which have not yet been measured for these nuclei, are useful for future experimental measurements.

Low-lying spectra and E2 transition rates in 160–170Er isotopes in the interaction boson model

Spectra and E2 transition rates for the 160–170Er isotopes are studied in the framework of the interaction boson model. A schematic Hamiltonian able to describe their spectra and B(E2) transition is used. It is found that the 160–170Er isotopes are in the transition from the vibrational limit to rotational limit.

QQ-onia package: a numerical solution to the Schrödinger radial equation for heavy quarkonium

This paper presents the basics of the QQ-onia package, a software based upon the Numerov O ( h 6 ) method which can be used to solve the Schrödinger radial equation using a suitable potential V ( r ) for the heavy quarkonium system. This package also allows the analysis of relevant properties of those resonances such as the square of the wave functions at the origin, their corresponding derivatives for l ≠ 0 states, typical heavy-quark velocities, and mean square radii. Besides, it includes a tool to analyze the spin dependent contributions to the heavy quarkonia spectrum, providing the splitting of n 3 S 1 − n 1 S 0 , as well as the n 3 P J − n 1 P 1 energy levels. Finally a simple software implemented in QQ-onia to compute E1 transition rates is presented. Program summary Program title: QQ-onia package Catalogue identifier: AECQ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AECQ_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 17 706 No. of bytes in distributed program, including test data, etc.: 2 334 506 Distribution format: tar.gz Programming language: PAW, Physics Analysis Workstation ( http://wwwasd.web.cern.ch/wwwasd/paw/) Computer: PC/Workstation Operating system: Windows-XX and Unix (Linux) Classification: 11.1, 11.6 Nature of problem: Software to solve the Schrödinger radial equation using a suitable potential V ( r ) for the heavy quarkonium system, allowing to perform spectroscopy. It also allows the analysis of relevant quantities of those resonances such as the square of the wave functions at the origin, their corresponding derivatives for l ≠ 0 states, typical heavy-quark velocities, and mean square radii. The package is a (userfriendly) multipurpose tool for dealing with different heavy quarkonium systems, providing a way to study the influence of a given potential on a series of relevant physical quantities, by either varying parameterized values of a well-known potential form, or by including new terms. Solution method: Based upon the Numerov O ( h 6 ) method, we perform a matching procedure to the reduced wave function at the cut point. We also perform a normalization technique for these wave functions taking into account the different domains when we use a Numerov backward–forward technique. In the case of l ⩾ 2 we present a way to find the corresponding derivatives at the origin by only calculating the reduced radial wave function and first derivative. When estimating the heavy quark velocity, we introduce an additional way to compute this quantity from the virial theorem. The calculated reduced wave functions and radial wave functions at the origin are later used to obtain the heavy quarkonia nL splitting and E1 transition rates. Additional comments: Using Windows, to optimize the edition of the files, please, open it with MFC-WORDPAD. Running time: It depends on the choice of the r range, and the number of energy steps.

Relativistic many-body Møller–Plesset perturbation theory calculations of the energy levels and transition rates in Na-like to P-like Xe ions

Relativistic multireference many-body perturbation theory calculations have been performed for Xe 43+ to Xe 39+ ions, resulting in energy levels, electric dipole transition rates, and level lifetimes. The second-order many-body perturbation theory calculation of energy levels included mass shifts, the frequency-dependent Breit correction, and Lamb shifts. The calculated transition energies and E1 transition rates are used to present synthetic spectra in the extreme ultraviolet range for some of the Xe ions.

On the unusual properties of the 282 keV state in 135Sb

Recently the first excited state in 135Sb has been observed at the unexpectedly low excitation energy of only 282keV and interpreted as mainly d 5/2 proton coupled to the 134Sn core. Based on theoretical considerations it was suggested that its low excitation energy is related to a relative shift of the proton d 5/2 and g 7/2 orbits induced by the neutron excess. We have measured the lifetime of the 282keV state by the advanced time-delayed βγγ(t) method. The measured half-life, T 1/2 = 6.1(4)ns, yields exceptionally low limits of B(M1;5/21 +→7/21 +)≤3.0×10-4 μ 2 N and B(E2;5/21 +→7/21 +)≤54e 2 fm 4. These strongly hindered M1 and slow E2 transition rates are similar to those for the transition de-populating the first excited state at 405keV in 211Bi. Results of shell model calculations with realistic interactions are presented. The M1 decay rate was found to be extremely sensistive both to the wave function and to the M1 effective operator.

Some observables as an indication of atomic nuclear shape phase transition

Using interacting boson model, shape phase transitions from U(5) to O(6) and from U(5) to SU(3) are studied in the space of two control parameters. Some simple shape phase diagrams in terms of the order parameter and one of the control parameters are presented. Spectra and E2 transition rates are used to construct observables. It is shown that these observables exhibit the behavior of the shape phase transition, namely a peak structure at and after the critical point.

Solution of the Bohr Hamiltonian for a periodic potential with minimum at [formula omitted

We present an analytic solvable γ-periodic potential of the form μ / sin 2 ( 3 γ ) in the Bohr–Mottelson collective model. The choice allows for a separation in both β, γ and θ ι if the γ-variable in the moments of inertia is approximated by its expectation value γ 0 = π / 6 . Energy spectra, E2 transition rates and subsequent collective bands with their properties are presented. We compare the approximation of γ = γ 0 = π / 6 with a full diagonalisation of the rotational part. These results clearly point out that the approximative analytical treatment describes the full numerical results very well.

Excitation energies, hyperfine constants, E1 transition rates and lifetimes of 4s2nl states in neutral gallium

Energies of the 4s2npj (n = 4–8), 4s2ns1/2 (n = 5–8), 4s2ndj (n = 4–7), 4s2nfj (n = 4–5) and 4s25gj states in neutral gallium are obtained using a relativistic many-body perturbation theory (RMBPT) method. First-, second-, third- and all-order Coulomb energies and first-order Breit and second-order Coulomb–Breit corrections are calculated. Reduced matrix elements, oscillator strengths, transition rates and lifetimes are determined for the 130 possible electric-dipole transitions. Hyperfine constants A are determined for 28 4s2nlj states in 69Ga I and 71Ga I isotopes. All above-mentioned properties are obtained in the relativistic single–double (SD) approximation, where single and double excitations of Dirac–Fock wavefunctions are included to all orders of perturbation theory. Using SD wavefunctions, accurate values are obtained for energies of the lowest states and for the possible electric-dipole matrix elements between these states. With the aid of the SD wavefunctions, we also determine transition rates, oscillator strengths and lifetimes. These calculations provide a theoretical benchmark for comparison with experiment and theory.

Relativistic many-body calculations of energies, E2, and M1 transition rates of 4s 24p states in Ga-like ions

Ionization energies, fine-structure splitting of 4s 24p states, and magnetic-dipole and electric-quadrupole transition rates for the 4s 24p 2P 1/2–4s 24p 2P 3/2 transition are calculated for Ga-like ions with nuclear charge Z ranging from 31 to 100. First-, second-, third-order, and all-order Coulomb energies and first- and second-order Coulomb–Breit energies are calculated. Electric-quadrupole and magnetic-dipole matrix elements are evaluated to calculate the lifetime of the 4s 24p 2P 3/2 state. The matrix elements are calculated using both relativistic many-body perturbation theory, complete through third order, and the relativistic all-order method restricted to single and double excitations. These calculations provide a theoretical benchmark for comparison with experiment and theory.

Description of nucleus 44 Ti in new symmetry limit in IBM-3

Analysis of the spectra and E2 transition rates shows that the empirical scheme of 44Ti is in good agreement with the predictions of the [Ud(5)⊗UTd(3)]⊕UTs(3) symmetry limit.