Quadrupole Boson Research Articles

E2 decay characteristics of the M1 scissors mode of Sm152

The $E2/M1$ multipole mixing ratio of the ${1}_{\text{sc}}^{+}\ensuremath{\rightarrow}{2}_{1}^{+}\phantom{\rule{4pt}{0ex}}\ensuremath{\gamma}$-ray transition and therefore the $F$-vector $E2$ decay of the scissors mode of the shape-phase transitional nucleus $^{152}\mathrm{Sm}$ have been measured with the nuclear resonance fluorescence method at the High-Intensity $\ensuremath{\gamma}$-ray Source ($\mathrm{HI}\ensuremath{\gamma}\mathrm{S}$). Furthermore, parity quantum numbers of several dipole-excited states have been remeasured, making use of the polarized $\ensuremath{\gamma}$-ray beam at $\mathrm{HI}\ensuremath{\gamma}\mathrm{S}$, and partially reassigned. The new data enable an unambiguous determination of the proton and neutron effective boson quadrupole charges within the proton-neutron interacting boson model. With the well-constrained parameter set, a new mixed-symmetry state is predicted.

Boson Description of Band Crossing in Even Barium Isotopes

Properties of the states of yrast bands in even Ba isotopes are studied on the basis of the boson representation of fermion operators. The parameters of the boson Hamiltonian and the interaction between collective quadrupole bosons, positive parity bosons, and spins $${{J}^{\pi }}$$ from $${{0}^{ + }}$$ to $${{10}^{ + }}$$ are calculated microscopically. This approach describes band crossing in the yrast bands and reproduces their energies and $$B(E2)$$ within these bands up to $${{I}^{\pi }} = {{18}^{ + }}.$$

E2 decay strength of the M1 scissors mode of ^{156}Gd and its first excited rotational state.

The E2/M1 multipole mixing ratio δ_{1→2} of the 1_{sc}^{+}→2_{1}^{+} γ-ray decay in ^{156}Gd and hence the isovector E2 transition rate of the scissors mode of a well-deformed rotational nucleus has been measured for the first time. It has been obtained from the angular distribution of an artificial quasimonochromatic linearly polarized γ-ray beam of energy 3.07(6)MeV scattered inelastically off an isotopically highly enriched ^{156}Gd target. The data yield first direct support for the deformation dependence of effective proton and neutron quadrupole boson charges in the framework of algebraic nuclear models. First evidence for a low-lying J^{π}=2^{+} member of the rotational band of states on top of the 1^{+} band head is obtained, too, indicating a significant signature splitting in the K=1 scissors mode rotational band.

Semi-phenomenological description of the chiral bands in 188,190Os

A set of interacting particles are coupled to a phenomenological core described using the generalized coherent state model. Among the particle–core states, a finite set which have the property that the angular momenta carried by the proton and neutron quadrupole bosons and the particles, separately, are mutually orthogonal are identified. The magnetic properties of such states are studied. All terms of the model Hamiltonian exhibit chiral symmetry except the spin–spin interaction one. There are four bands of the type with two-quasiparticle–core dipole states, exhibiting properties which are specific to magnetic twin bands. An application is presented, for the isotopes Os.

A new picture for the chiral symmetry properties within a particle–core framework

The generalized coherent state model (GCSM), proposed previously for a unified description of magnetic and electric collective properties of nuclear systems, is extended to account for the chiral-like properties of nuclear systems. To a phenomenological core described by the GCSM, a set of interacting particles is coupled. Among the particle–core states, one identifies a finite set which has the property that the angular momenta carried by the proton and neutron quadrupole bosons and the particles, respectively, are mutually orthogonal. All terms of the model Hamiltonian satisfy the chiral symmetry except for the spin–spin interaction. The magnetic properties of the particle–core states, where the three mentioned angular momenta are orthogonal, are studied. A quantitative comparison of these features with the similar properties of states, where the three angular momenta belong to the same plane, is performed.

A semi-microscopic approach to the backbending phenomena in even-even nuclei

The phenomenon of backbending in rare earth nuclei is described by means of the band hybridization mechanism. The hybridization of two rotational bands is performed by treating a model Hamiltonian associated with a set of interacting particles moving in a deformed mean field and coupled to a phenomenological core described in terms of quadrupole boson operators, in a product space of angular momentum projected states. The model states for the ground and S bands are obtained by angular momentum projection of a deformed product state with an axially symmetric coherent boson state standing for the core factor function and a BCS type single-particle factor function defining the nature of the band. The yrast band energies are defined by the lowest eigenvalues of the model Hamiltonian in an orthogonal basis constructed by diagonalizing the overlap matrix of the ground and S band states, which are not mutually orthogonal. The formalism is positively tested on six deformed even-even nuclei known to exhibit a backbending behavior in their moment of inertia.

Closed formulas for ground band energies of nuclei with various symmetries

A time-dependent variational principle is used to dequantize a second-order quadrupole boson Hamiltonian. The classical equations for the generalized coordinate and the constraint for the angular momentum are quantized and then analytically solved. A generalized Holmberg–Lipas formula for energies is obtained. A similar J(J + 1) dependence is provided by the coherent state model in the large deformation regime, by using an expansion in powers of 1/x for energies, with x denoting a deformation parameter squared. A simple compact expression is also possible for the near-vibrational regime. These three expressions have been used for 44 nuclei covering regions characterized by different dynamic symmetries or in other words belonging to all the known nuclear phases. Nuclei satisfying the specific symmetries of the critical point in the phase transitions O(6) → SU(3), SU(5) → SU(3) have also been considered. The agreement between the results and the corresponding experimental data is very good. This is reflected in very small root mean square values of deviations.

Coherent state model extension for the description of positive and negative parity bands in even-odd nuclei

A particle-core Hamiltonian is used to describe the lowest parity partner bands $K^{\pi}=1/2^{\pm}$ in $^{237}$U and $^{239}$Pu. The quadrupole and octupole boson Hamiltonian associated to the core is identical to the one previously used for the description of four positive and four negative parity bands in the neighboring even-even isotopes. The single particle space for the odd nucleon consists of three spherical shell model states, two of positive and one of negative parity. The particle-core Hamiltonian consists of four terms: a quadrupole-quadrupole, an octupole-octupole, a spin-spin and a rotational $\hat{I}^2$ interaction, with $\hat {I}$ denoting the total angular momentum. The parameters involved in the particle-core coupling Hamiltonian were fixed by fitting four particular energies in the two bands. The calculated excitation energies are compared with the corresponding experimental data as well as with those obtained with other approaches. Also, we searched for some signatures for a static octupole deformation in the considered odd isotopes.

Phenomenological description of the states 0+ and 2+ in some even–even nuclei

A sixth-order quadrupole boson Hamiltonian is used to describe the states 0+ and 2+ identified in several nuclei by various types of experiments. Two alternative descriptions of energy levels are proposed. One corresponds to a semi-classical approach of the model Hamiltonian while the other one provides the exact eigenvalues. Both procedures yield close formulae for energies. The first procedure involves four parameters, while the second involves a compact formula with five parameters. In each case the parameters are fixed by a least-square fit procedure. Applications are performed for eight even–even nuclei. Both methods yield results which are in surprisingly good agreement with the experimental data. We give also our predicted reduced transition probabilities within the two approaches, although the corresponding experimental data are not yet available.

A simple description of the states 0+ and 2+ in 168Er

A sixth-order quadrupole boson Hamiltonian is used to describe 26 states 0+ and 67 states 2+ which have been recently identified in 168Er. Two closed expressions are alternatively used for energy levels. One corresponds to a semi-classical approach while the other one represents the exact eigenvalue of the model Hamiltonian. The semi-classical expression involves four parameters, while the exact eigenvalue is determined by five parameters. In each of the two descriptions a least-squares fit procedure is adopted. Both expressions provide a surprisingly good agreement with the experimental data. Very simple expressions for the reduced E2 transition probabilities are derived within the two approaches.

Semiclassical description of a sixth-order quadrupole boson Hamiltonian

A sixth-order quadrupole boson Hamiltonian is treated through a time dependent variational principle approach choosing as trial function a coherent state with respect to zeroth b 0 † and second b 2 † + b −2 † components of the quadrupole bosons. The coefficients involved in the model Hamiltonian are chosen so that the classical effective potential energy term has two distinct minima. The classical system is described by two degrees of freedom and has two constants of motion. The equation of motion for the radial coordinate is analytically solved and the resulting trajectories are extensively studied. One distinguishes three energy regions exhibiting different types of trajectories. When one passes from the region characterized by two wells to the region of energies higher than the maximum value of the effective potential the trajectories period exhibits a singularity which reflects in fact a phase transition. The classical trajectories are quantized by a constraint of the integral action similar to the well known Bohr–Sommerfeld quantization condition. The semiclassical spectra corresponding to the two potential wells have specific properties. The tunneling process through the potential barrier is also studied. It is a remarkable fact that the transmission coefficients exhibit jumps in magnitude when the intrinsic momentum acquires certain values. It is an open question whether such discontinuities in the transmission coefficients show up also when the final states are not bound. Semiclassical energy levels are compared also with the exact eigenvalues of the initial Hamiltonian, obtained through a diagonalization procedure. The present approach suggests that the energy levels in the laboratory frame can be written as a sum of energy associated to the intrinsic degrees of freedom and a rotational term obtainable through a projection method. This is numerically confirmed for the ground band.

New results for the missing quantum numbers labelling the quadrupole and octupole boson basis

The many $2^k$-pole boson states, $|N_kv_k\alpha_k I_kM_k>$ with $k=2,3$, realize the irreducible representation (IR) for the group reduction chains $SU(2k+1)\supset R_{2k+1}\supset R_3\supset R_2$. They have been analytically studied and widely used for the description of nuclear systems. However, no analytical expression for the degeneracy $d_v(I)$ of the $R_{2k+1}$'s IR, determined by the reduction $R_{2k+1}\supset R_3$, is available. Thus, the number of distinct values taken by $\alpha_k$ has been so far obtained by solving some complex equations. Here we derive analytical expressions for the degeneracy $d_v(I)$ characterizing the octupole and quadrupole boson states, respectively. The merit of this work consists of the fact that it completes the analytical expressions for the $2^k$-pole boson basis.

Unified description of three positive and three negative parity interacting bands

The coherent state model (CSM) is extended so that three negative parity bands are treated on equal footing with three positive parity bands. The six rotational bands are generated by projecting out angular momenta and parities from three intrisic orthogonal states which exhibit both quadrupole and octupole deformations. The projected states are, by construction, mutually orthogonal. In the space of projected states, a sub-space of a quadrupole and octupole multi-boson states, an effective quadrupole and octupole boson Hamiltonian is solved. The eigenstates of the model Hamiltonian are linked by multipole transition operators for which lowest order boson expressions are considered. The calculations involve 6 structure coefficients and two deformation parameters. All of them are fixed by a least square fit of the known experimental energies. Applications are made to Gd158, Yb172, Ra218, Ra226, Th232, U238, Pu238. Very good agreement is obtained for both excitation energies and transition probabilities. New signatures for octupole deformation, manifested in excited bands, are pointed out.

Isovector dipole and quadrupole excitations in66Zn

The nucleus ${}^{66}\mathrm{Zn}$ is investigated using $\ensuremath{\gamma}\ensuremath{\gamma}$-coincidence studies following the $\ensuremath{\beta}$ decay of the ${0}^{+}$ ground state of ${}^{66}\mathrm{Ga}.$ The data survey short-lived ${J}^{\ensuremath{\pi}}{=1}^{+}{,2}^{+}$ states, $E2/M1$ mixing ratios, and, together with the previously known lifetimes, also absolute transition strengths $B(\ensuremath{\sigma}L).$ The proton-neutron mixed-symmetry states ${2}_{\mathrm{ms}}^{+}$ (isovector quadrupole excitation) and ${1}_{\mathrm{ms}}^{+}$ (``scissors mode'') were identified from their decay patterns and absolute transition strengths. A $\ensuremath{\gamma}$ transition between these mixed-symmetry states was observed and its rate is discussed. We additionally present a candidate for the two-phonon ${2}_{2,ms}^{+}$ state, whose structure is closely related to the ${1}_{\mathrm{ms}}^{+}$ excitation. Fifteen $M1$ and $E2$ strengths involving mixed-symmetry states show a reasonable agreement with the prediction of the pure O(6) limit of the proton-neutron extension of the interacting boson model (IBM-2) using the effective quadrupole boson charges ${e}_{\ensuremath{\pi}}{+e}_{\ensuremath{\nu}}$ and the g-factor difference ${g}_{\ensuremath{\pi}}\ensuremath{-}{g}_{\ensuremath{\nu}}$ as the only free parameters.

Interrelation between the isoscalar octupole phonon and the proton–neutron mixed-symmetry quadrupole phonon in near-spherical nuclei

The interrelation between the octupole phonon and the low-lying proton–neutron mixed-symmetry quadrupole phonon in near-spherical nuclei is investigated. The one-phonon states decay by collective E3 and E2 transitions to the ground state and by relatively strong E1 and M1 transitions to the isoscalar 2 + 1 state. We apply the proton–neutron version of the interacting boson model including quadrupole and octupole bosons ( sdf -IBM-2). Two F -spin symmetric dynamical symmetry limits of the model, namely the vibrational and the γ -unstable ones, are considered. We derive analytical formulae for excitation energies as well as B( E1) , B( M1) , B( E2) and B( E3) values for a number of transitions between low-lying states.

Classical and Quantal Chaos Described by a Fourth Order Quadrupole Boson Hamiltonian

A quadrupole boson Hamiltonian is considered within a time dependent variational formalism. The trial functions are coherent states with respect to the quadrupole bosons [Formula: see text] and [Formula: see text]. The character, chaotic or regular, of the classical trajectories are studied, in terms of both the Poincaré surface of sections and maximal Lyapunov exponents, as a function of energy and an order parameter B, the coefficient of the third order boson term. A peculiar feature concerning the behavior of chaos and order in the range of large energies is pointed out. Quantizing the classical energy function one obtains an operator whose spectrum is analyzed in terms of level spacing distribution, Some classical and quantal features of the chaotic motion are possibly related.

Detailed spectroscopy and IBFFM interpretation of the odd-odd nuclei 132Pr and 130Pr

High-spin rotational structures in the odd-odd nuclei 130,132Pr have been studied via the 104Pd + 32S reaction at 160 MeV. Several states below the previously reported negative- and positive-parity states have been established through the observation of low-energy transitions, leading to new spin assignments for all observed levels. The bands based on the πh 11 2 ⊗ vg 7 2 and πh 11 2 ⊗ vh 11 2 configurations have been extended at much higher spins. Detailed interacting boson-fermion-fermion model calculations have been performed for the two nuclei, which reveal the complex structure of the new observed low-lying states. The change from inverted to normal signature at high spin and the staggering pattern of all observed bands are well reproduced, being explained by the interplay between a new term introduced in the boson quadrupole operator and the residual quadrupole-quadrupole interaction.

Low-lying negative-parity bands in heavy nuclei: A phenomenological approach

A phenomenological analysis of theKπ=0−, 1−, 2− band spectra in actinides and few rare-earth nuclei is carried out within the coherent state model using an interacting boson Hamiltonian which includes quadrupole and octupole bosons. The bands are described as octupole boson excitations of intrinsic coherent wave functions associated to the model ground, beta and gamma bands. The calculation reproduces all available experimental energy spectra up to high angular momenta. Also theE1 branching ratios are fairly well reproduced. Different signatures of static octupole deformation are discussed.

Regular and irregular features of classical motion described by a quadrupole boson Hamiltonian.

A fourth-order quadrupole boson Hamiltonian H is treated semiclassically through a time-dependent variational principle. The variational functions are coherent states for boson operators. In the parameters space of H there are regions, conventionally called ``nuclear phases,'' determining specific static properties. Several ground states corresponding to different equilibrium shapes are found as static solutions of classical equations of motion. The mechanism of destroying the tori of regular orbits and the onset of chaos depend on the nuclear phase. The regular and chaotic motions are analyzed in terms of Poincar\'e sections and the largest Lyapunov exponent. \textcopyright{} 1996 The American Physical Society.

First Observation of the Scissors Mode in aγ-Soft Nucleus: The Case of196Pt

A nuclear resonance fluorescence experiment with one of the newly developed highly efficient EUROBALL cluster detectors has been performed on the $\ensuremath{\gamma}$-soft nucleus ${}^{196}\mathrm{Pt}$. Magnetic dipole excitations were observed between 2 and 3.5 MeV excitation energy. They are interpreted as the main fragments of the scissors mode based on the measured excitation strengths and branching ratios. A strong $\ensuremath{\gamma}$ decay, which we believe to be an $E2$ decay of the scissors mode to the ${2}_{1}^{+}$ state, is observed. It allows an extraction of the effective boson quadrupole charges in the interacting boson model.