E2 Matrix Elements Research Articles

Approaching rotational collectivity in odd-odd NZ nuclei in pf-shell

The analysis of shell model wave functions and new experimental data for odd-odd NZ nuclei 46V and 50Mn is presented in terms of collective rotor-plus-quasideuteron model. The low-lying low-spin states exhibiting collective properties appropriate for the K=0, K=3, and K=5 bands are identified. The strong isovector M1 transitions of quasideuteron origin in 46V and 50Mn are shown to be important indicators of collective features and to give more complete information on the structure of the intrinsic states than E2 matrix elements.

Shell model description of “mixed-symmetry” states in [formula omitted]Mo

Shell model calculations have been performed for the nucleus 94Mo. The calculated excitation energies and electromagnetic properties of low-lying states are in good agreement with the data, which include states with mixed-symmetry (MS) assignments in previous interacting boson model studies. In the shell model large isoscalar E2 matrix elements are found between states with MS assignments indicating that they form a class of states with similar proton-neutron symmetry.

Excitation energies and transition rates in magnesiumlike ions

Excitation energies, transition probabilities, and lifetimes are calculated for 3 l 13l 2-3l 33l 4 electric dipole transitions in Mg-like ions with nuclear charges Z ranging from 13 to 100. Relativistic many-body perturbation theory ~MBPT!, including the Breit interaction, is used to evaluate retarded E1 matrix elements in length and velocity forms. The calculations start from a 1 s 2 2s 2 2p 6 Dirac-Fock potential. First-order perturbation theory is used to obtain intermediate coupling coefficients, and the second-order MBPT is used to determine the matrix elements. The contributions from negative-energy states are included in the second-order E1 matrix elements to achieve agreement between length-form and velocity-form amplitudes. The resulting transition energies and lifetimes are compared with experimental values and with results from other recent calculations.

Relativistic Many-Body Calculations of TransitionProbabilities for the2l12l2[LSJ]-2l32l4[L'S'J']Lines in Be-like Ions

Reduced matrix elements, oscillator strengths, and transition ratesare calculated for all allowed and forbidden 2s–2p electricdipole transitions in berylliumlike ions with nuclear charges rangingfrom Z = 4 to 100. Many-body perturbation theory (MBPT),including the Breit interaction, is used to evaluate retarded E1matrix elements in length and velocity forms. The calculations startwith a 1s2 Dirac–Fock potential and include allpossible n = 2 configurations, leading to 4 odd-parity and 6even-parity states. First-order perturbation theory is used to obtainintermediate coupling coefficents. Second-order MBPT is used todetermine the matrix elements, which are evaluated for the 16 possibleE1 transitions. The transition energies used in the calculation ofoscillator strengths and transition rates are evaluated usingsecond-order MBPT. The importance of virtual electron–positronpair (negative energy) contributions to the transition amplitudes isdiscussed.

A computer program for relativistic multiple Coulomb and nuclear excitation

A computer program is presented by which one may calculate the multiple electric dipole, electric quadrupole and magnetic dipole Coulomb excitation with relativistic heavy ions. The program applies to an arbitrary nucleus, specified by the spins and energies of the levels and by all El, E2 and M1 matrix elements. Nuclear excitation is calculated optionally for monopole, dipole and quadrupole excitations and needs inputs of optical potentials. For given bombarding conditions, the differential cross sections and statistical tensors (useful to calculate γ-ray angular distribution functions) are computed.

New features in the systematics of low-spin states in170–178W

Excited states in the range of even tungsten isotopes, from170W to178W, populated in the β-decay of170–178Re parents have been identified using γ-ray singles and coincidence techniques, utilising the high efficiency of a Compton-suppressed array, and conversion electron systems. Many new states at low excitation energies have been identified, complementing the level schemes previously established from in-beam studies. The new states include excited O+ states in170W,172W,174W,176W and178W. A large body of data on decay properties, spins and parities, and relative E2 and E0 matrix elements has been obtained. The properties of theKπ=0+ (quasi-β) bands are compared with similar bands observed in light osmium, platinum and mercury isotopes, which have been attributed to shape coexistence.

Quadrupole and octupole collectivity in 148Nd

The role of quadrupole and octupole collectivity in the shape-transitional nucleus 148Nd has been studied by Coulomb excitation using beams of 58Ni and 92Mo, and a beam of 148Nd (using a 208Pb target). The extracted E1, E2 and E3 matrix elements involving states up to 12 + ·in the ground band and 13 − in the negative-parity band are presented, and compared to calculations that assume a vibrational and rotational octupole nature for the negative-parity band. The positive-parity ground-band states are well described in terms of a prolate deformed shape with Q 20 ≈ 400 e fm 2 ( β 2 rms ≈ +0.18). The present results suggest a vibrational octupole nature for the low-spin negative-parity states, with an intrinsic moment Q 30 ≈ 1500 e fm 3 ( β 3 rms ≈ 0.12). The E2 and E3 matrix elements connecting these bands to the β- and γ-vibrational bands (and within these bands) are also presented, and compared to calculations incorporating the coupling between the rotational and vibrational modes. These calculations describe reasonably well the E2 matrix elements involving the gamma band, but do not reproduce the measured E2 matrix elements for the beta band, implying a complicated intrinsic structure for the beta band. The strong enhancement of the measured E3 matrix elements connecting the negative-parity band to the beta band could be indicative of a significant component of the two-phonon octupole vibration in the wavefunction of the so-called beta band.

Triaxiality in quadrupole deformed nuclei.

The intrinsic E2 matrix elements 〈K=2|E2|K=0〉 for 25 deformed nuclei, covering from neodymium to uranium, have been deduced from measured interband E2 matrix elements between the ground band and \ensuremath{\gamma} band after correcting for the first-order angular momentum dependence of the coupling between the rotation and intrinsic motion. Fairly precise centroids for the triaxiality of the intrinsic E2 moments are obtained, and these correlate well with the triaxiality implied by the excitation energies. The strong correlation of the triaxiality derived from the E2 properties and level energies provides a quantitative measure of triaxial quadrupole deformation of the nuclear shape for these states. \textcopyright{} 1996 The American Physical Society.

Quadrupole collectivity and shapes of OsPt nuclei

Abstract E2 collective properties for the low-lying states in 186,188,190,192 Os and 194 Pt have been investigated experimentally by means of Coulomb excitation using 3.3–4.8 MeV/nucleon 40 Ca, 58 Ni, 136 Xe and 208 Pb beams. The deexcitation γ rays following Coulomb excitation were detected in coincidence with the scattered particles. Levels with excitation energies up to 3–4 MeV of the ground-state, γ and 4 + collective bands as well as excited 0 + states were populated in each nucleus studied. A semiclassical Coulomb-excitation least-squares search code GOSIA was used to extract E2 matrix elements from the measured γ-ray yields. For each nucleus studied, a unique and almost complete set of E2 matrix elements for the low-lying states has been determined, which includes both the magnitudes and signs of the transitional and diagonal matrix elements. The completeness of the set of measured E2 matrix elements makes it possible to determine the intrinsic quadrupole deformation for the low-lying states in these nuclei via a model-independent method. The results indicate clearly that the E2 properties for the low-lying states in these nuclei are correlated well using only the quadrupole collective degrees of freedom. The extracted E2 matrix elements are compared with the prediction of various collective models such as the asymmetric rigid rotor model, the γ-soft model of Leander, and the IBA-2 model. These particular models do not reproduce the data satisfactorily, however the general trends of the data are consistent with the descriptions of γ-soft type collective models through a prolate to oblate shape-transition region. That the enhanced B (E2) values between the I ,quasi- K π = 4,4 + state and members of the I ,quasi- K π = 2,2 + band are well reproduced by the γ-soft model is consistent with the interpretation of the I ,quasi- K π = 4,4 + state being a two-phonon γ-vibration excitation.

Fragmentation of two-phonon γ vibration strength in deformed nuclei

The intrinsic E2 matrix elements between ΔK = 2 bands for 186,188,190,192Os have been deduced from previously measured interband E2 transition matrix elements after correcting for the coupling between the rotational and intrinsic motion. The ratio of the intrinsic E2 matrix element between the K = 4 and 2 bands to that between the K = 2 and 0 bands, together with the ratio of their excitation energies, provide strong evidence for the existence of rather pure harmonic double-phonon γ-vibrational excitation. A similar result is obtained for 232Th, whereas significant fragmentation of the two-phonon strength is deduced for 156Gd and 160Dy.

Effects of E2 transitions in the Coulomb dissociation of 8B

We examine the Coulomb dissociation of aB in a model that has both E1 and E2 matrix elements. We find that the interference between El and E2 amplitudes produces large asymmetries in the angular and momentum distributions of the emitted protons and 7Be fragments. By measuring these asymmetries one may be able to put constraints on the E2 component and thereby improve the accuracy of the E1 strength that can be extracted from Coulomb dissociation experiments. We also investigate the effect of higher-order dynamical processes and find that they reduce the asymmetries in reactions on high-Z targets. They also reduee the effect of E2 transitions on the dissociation probability and on the peak height of the decay energy spectrum compared to predictions of first-order perturbation theory.

Coulomb excitation of 76,80,82Se

Abstract The collective structure of the even-even nuclei 76,80,82 Se has been studied using Coulomb excitation with 16 O, 48 Ti and 208 Pb ions. About 20 E2 matrix elements were determined in each isotope. The measured matrix elements are compared with the predictions of simple geometric and algebraic collective models. The E2 properties and energy level spectra of the 76,80,82 Se nuclei exhibit features characteristic of anharmonic vibrational collective motion. Although the simplest collective vibrational models exhibit the general features of the data, they do not reproduce the fine details. The influence of a number of physical effects on the data was investigated. These include Coulomb-nuclear interference, mutual excitation and the effect due to the semiclassical approximation of the Coulomb-excitation formalism.

A relation between E2 matrix elements connecting collective states of deformed nuclei

A relation is presented stating that a sum of certain products of E2 matrix elements connecting the 0 + and 2 + levels in the ground, β-, and γ-bands in the limit of strong quadrupole deformation vanishes. As a consequence β → g E2 matrix elements are much smaller than β → γ matrix elements, contrary to the naive expectation based on the vibrational picture of independent β and γ excitations. This relation is supported by experimental data on deformed nuclei, and suggests a re-examination of the traditional understanding of these basic collective modes and their inter-relationships.

Multiphonon vibrational states in 106,108Pd

Abstract The nuclei 106 Pd and 108 Pd have been Coulomb excited using beams of 16 O, 58 Ni and 208 Pb. The data determined 25 and 31 E2 matrix elements in 106 Pd and 108 Pd, respectively. The experimental data are qualitatively in agreement with the spherical-harmonic quadrupole vibrational model, but quantitatively there are large discrepancies, even for the states generally accepted to be the two-phonon vibrational states. A sum-rule method was applied to the data, the result of which implies that the vibrational quadrupole strengths of the 0 + and 4 + two-phonon states are fragmented and shared between several E2 matrix elements. The E2 matrix elements imply appreciable quadrupole triaxiality with a γ-centroid of about 20°. A systematic comparison is made with neighbouring Pd-isotopes and with 114 Cd.

Particle-hole wave functions and residual interaction for 208Pb derived from experimental data

The level scheme of 208Pb is now nearly complete up to 5 MeV excitation energy from measurements of 209Bi(t, αγ)208Pb, 207Pb(d, pγ)208Pb, and γ-spectroscopy with inelastic scattering of 64Ni and 82Se. Also a complete set of M1- and E2-matrix elements is available from experiments or calculations, that allows us to evaluate γ-branching ratios quantitatively. This information together with the spectroscopic factors is sufficient to derive wave functions for many states strictly from the experimental data. The Ansatz for the wave functions is, that they are orthonormal superpositions of the lowest one-particle one-hole states. Redundant data are often well reproduced. Matrix elements of the residual interaction are calculated from the wave functions; they are compared with realistic Kuo-Brown matrix elements and the surface-delta interaction.

Experimental investigation of excited states in atomic dysprosium.

A pair of closely lying states of opposite parity and the same total electronic angular momentum (J=10) at 19797.96 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ in dysprosium is experimentally investigated with the goal of evaluating parameters relevant to parity and parity--time-reversal violation experiments. An atomic beam apparatus is used. The states of interest are populated with a sequence of two laser pulses, and a third laser pulse and fluorescence detection are used to probe the state population. Their lifetimes are determined from the dependence of the fluorescence signal on the time delay between the pump and probe pulses. The lifetime of the even-parity state was also measured by a different method involving detection of cascade fluorescence. Spectroscopy of radio-frequency E1 transitions between the opposite-parity states is performed when one of the states is populated and the population of the other is probed. This determines precise energy separations between various isotope and hyperfine structure components, and allows extraction of the corresponding isotope shift and hyperfine structure parameters. Electric polarizability is determined from radio-frequency line shapes in the presence of a dc electric field. As part of the preparation of a parity violation experiment, level-crossing signals which occur when various Zeeman components of the closest hyperfine components (isotope 163, F=10.5; initial separation 3.1 MHz) are brought together in the presence of collinear electric and magnetic fields were studied. These measurements provide an independent determination of the E1 matrix element between the opposite-parity states. In addition, lifetimes, isotope, and hyperfine structure of other dysprosium states are obtained and a number of narrow autoionization resonances (\ensuremath{\Gamma}10 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$) are observed.

Low-spin non-yrast states and collective excitations in 174Os, 176Os, 178Os, 180Os, 182Os and 184Os

Excited states in the range of isotopes 174, 176, 178, 180, 182, 184Os, populated in the β-decay of 174, 176, 178, 180, 182, 184Ir parent activities have been identified using γ-ray singles and coincidence techniques, utilising the high efficiency of a Compton-suppressed array, and conversion electron techniques. Many new states at low excitation energies have been identified, complementing the level schemes previously established from in-beam studies. The new states include excited 0 + states in 174Os, 176Os 178Os and 180Os. A large body of data on decay properties, spins and parities, and relative E2 and E0 matrix elements has been obtained. The systematics of the quasi-β and quasi-γ bands is discussed. Some detailed analyses in terms of the schematic band-mixing model are presented, incorporating reproduction of the yrare states in the fits. In particular, the proposed interpretation of the anomalous low-frequency alignment gains in the yrast positive-parity states, as a consequence of mixing with a low-lying intruder band, is confronted. The B(E0)/ B(E2) ratios are over-estimated by this model although their form (spin and isotope dependence) is reproduced. Better agreement is obtained within the IBM, which uses however reduced effective charges to match experiment. Absolute measurements of the B(E0) and B(E2) values may be necessary to distinguish between the models and test conjectures of shape differences.

Shell-model calculations for the A approximately 100 region: application to the even-Z N=52 isotones 92Zr-100Cd

Shell-model calculations for the Z=38-50, N>50 region, with a space of protons in the 2p1/2, 1g9/2 orbits and neutrons in the 2d5/2, 3s1/2, 2d3/2, 1g7/2 orbits, are initiated by the selection of a schematic Hamiltonian and effective electromagnetic operators. An application to the Z-even N=52 isotones gives a good description for energies of both low-spin and yrast high-spin levels, and for E2 and M1 transition strengths and moments where these have been measured, over the entire range of Z. The calculated E2 matrix elements for the lower-spin yrast states in 98Pd and 100Cd suggest, within the context of a simple collective-model description, a finite and stable prolate deformation with B approximately 0.1, in marked contrast to previous collective interpretations.

Octupole collectivity in the ground band of 148Nd.

E3 matrix elements have been determined for transitions between states up to ${\mathit{I}}^{\mathrm{\ensuremath{\pi}}}$=${13}^{\mathrm{\ensuremath{-}}}$ in $^{148}\mathrm{Nd}$, providing direct evidence for strong octupole collectivity in a rotational band. These values, as well as the corresponding E1 and E2 matrix elements, were obtained from Coulomb excitation data using $^{58}\mathrm{Ni}$ and $^{92}\mathrm{Mo}$ ions, projectile excitation using a $^{208}\mathrm{Pb}$ target, and from recoil distance lifetime measurements. The results are consistent with collective model predictions for $^{148}\mathrm{Nd}$ assuming intrinsic quadrupole and octupole moments of ${\mathit{Q}}_{20}$\ensuremath{\approxeq}400 e ${\mathrm{fm}}^{2}$ and ${\mathit{Q}}_{30}$\ensuremath{\approxeq}1500 e ${\mathrm{fm}}^{3}$. The E3 strength coupling the negative-parity states to the \ensuremath{\beta} band is found to be appreciable.

Structure of intermediate states in the photoexcitation of the 89Y isomer

Resonant photon scattering off 89Y was investigated in a measurement of the 89Y m excitation function for bremsstrahlung endpoint energies E 0 = 2−5 MeV and in a nuclear-resonance-fluorescence experiment with E 0 = 5 MeV. The results are compared to a quasiparticle-phonon model calculation. Besides a well-known single-particle M1 transition at low energies, the photoexcitation spectrum is governed by transitions to states built by coupling of the dominant p 1 2 hole ground-state configuration to collective quadrupole phonons in the neighbouring 90Zr. The detailed decay cascade to the isomer reproduces the experimental finding of only two intermediate states with about equal strength and explains the suppression of other possible transitions due to the nature of the particular E1 matrix element. The theoretical isomer branching ratios are small compared to the experiment, but depend critically on details of the model.