Quadrupole Vibrations Research Articles

Induced effects of a rigid boundary on a pulsating spherical source

This work develops a comprehensive analytical formalism to study the boundary-induced effects on a pulsating spherical radiating source near a rigid planar boundary in an inviscid fluid. Using the modal multipole expansion method, the image method and the translational addition theorem for spherical wave functions, exact closed-form expressions are derived for the dimensionless axial acoustic radiation force function as well as the radiation, amplification and extinction energy efficiency factors. The source-boundary distance and the dimensionless size parameter are especially addressed in the numerical simulations for spherical sources with different vibration modes. It is shown that the axial radiation force is positive or negative, depending on the choice of parameters. Moreover, cases when the extinction acoustic power completely vanishes are predicted theoretically for dipole and quadrupole vibrations, which can be used to achieve acoustic invisibility in acoustic cloaking devices and acoustic stealth technologies. Combination of more than one vibration modes is also investigated to better simulate the acoustic manipulation in practical applications. This work can help us understand the underlying mechanism in manipulating the active spherical radiating sources in a bounded field.

Core of F25 studied by the F25(−p ) proton-removal reaction

The $^{9}\mathrm{Be}(^{25}\phantom{{}^{B}e}\mathrm{F}(5/{2}^{+}),^{24}\phantom{{}^{B}e}\mathrm{O}$)X proton-removal reaction was studied at the NSCL using the S800 spectrometer. The experimental spectroscopic factor for the ground-state to ground-state transition indicates a substantial depletion of the proton ${d}_{5/2}$ strength compared to shell-model expectations, similar to the findings of an inverse-kinematics $(p,2p)$ measurement performed at RIBF. The $^{25}\phantom{{}^{B}e}\mathrm{F}$ to $^{24}\phantom{{}^{B}e}\mathrm{O}$ ground-states overlap is considerably less than anticipated if the core nucleons behaved as rigid, doubly-magic $^{24}\phantom{{}^{B}e}\mathrm{O}$ within $^{25}\phantom{{}^{B}e}\mathrm{F}$. We interpret the new results within the framework of the Particle-Vibration Coupling (PVC) model, of a ${d}_{5/2}$ proton coupled to a quadrupole phonon of an effective core. This approach provides a good description of the experimental data, requiring an effective $^{24}\phantom{{}^{B}e}\mathrm{O}^{*}$ core with a phonon energy of $\ensuremath{\hbar}{\ensuremath{\omega}}_{2}$= 3.2 MeV and a $B(E2)\ensuremath{\approx}2.7$ W.u. -- softer and more collective than a bare $^{24}\phantom{{}^{B}e}\mathrm{O}$. Both the Nilsson deformed mean field and the PVC models appear to capture the properties of the effective core of $^{25}\phantom{{}^{B}e}\mathrm{F}$, suggesting that the additional proton polarizes $^{24}\phantom{{}^{B}e}\mathrm{O}$ in such a way that it becomes either slightly deformed or a quadrupole vibrator.

Triaxiality-induced monopole-quadrupole-hexadecupole coupling in the isoscalar giant resonances ofGe86

The isoscalar giant resonances for $^{86}\mathrm{Ge}$ are studied by the quasiparticle finite amplitude method based on the covariant density functional theory. In addition to the well-known monopole-quadrupole coupling that splits the isoscalar giant monopole resonance in axially deformed nuclei, a monopole-quadrupole-hexadecupole coupling is identified in the neutron-rich triaxially deformed nucleus $^{86}\mathrm{Ge}$, leading to the emergence of a distinct resonance peak at the low energy side of the isoscalar monopole strength function. The transition density of the triaxiality-induced resonance peak shows a strong interplay among monopole, quadrupole, and hexadecupole vibrations. The resonance peak responds to monopole, quadrupole, and hexadecupole perturbations simultaneously, which could be regarded as a fingerprint of the triaxiality in $^{86}\mathrm{Ge}$.

Applications of the dynamical generator coordinate method to quadrupole excitations

We apply the dynamical generator coordinate method (DGCM) with a conjugate momentum to a nuclear collective excitation. To this end, we first discuss how to construct a numerically workable scheme of the DGCM for a general one-body operator. We then apply the DGCM to the quadrupole vibration of $^{16}$O using the Gogny D1S interaction. We show that both the ground state energy and the excitation energies are lowered as compared to the conventional GCM with the same number of basis functions. We also compute the sum rule values for the quadrupole and monopole operators, and show that the DGCM yields more consistent results than the conventional GCM to the values from the double commutator. These results imply that the conjugate momentum is an important and relevant degree of freedom in collective motions.

Investigation of isotopic dependence on the O + Ni fusion cross section near barrier energies

Fusion excitation functions have been measured for $^{16}\mathrm{O}+^{61}\mathrm{Ni}$ and $^{18}\mathrm{O}+^{61,62}\mathrm{Ni}$ systems around the Coulomb barrier ($\ensuremath{\approx}0.7{V}_{B}$--$1.3{V}_{B}$) using the recoil mass spectrometer, heavy ion reaction analyzer. The ground state $Q$ value for two neutron stripping is positive for both systems with $^{18}\mathrm{O}$ as the projectile. Strong enhancement of the experimental fusion cross sections were observed below the barrier for all the systems compared to that of the predictions of the one-dimensional barrier penetration model. To understand such enhancement, a coupled-channels formalism has been used. A comparative study of these systems indicated that the coupling of two neutron transfer channels with the collective excitations could play the role behind the sub-barrier fusion enhancement for $^{18}\mathrm{O}$ induced reactions. However, the sub-barrier enhancement for $^{16}\mathrm{O} + ^{61}\mathrm{Ni}$ is found to be due to the coupling of quadrupole vibrations of both the interacting nuclei. Also after comparing these systems with other systems of different Ni isotopes, we have found that the signature of the role of coupling to neutron transfer channels due to ground state positive $Q$ value for neutron transfer is ambiguous.

Finite amplitude method on the deformed relativistic Hartree-Bogoliubov theory in continuum: The isoscalar giant monopole resonance in exotic nuclei

Finite amplitude method based on the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc-FAM) is developed and applied to study isoscalar giant monopole resonance in exotic nuclei. Validation of the numerical implementation is examined for $^{208}\textrm{Pb}$. The isoscalar giant monopole resonances for even-even calcium isotopes from $^{40}\textrm{Ca}$ to the last bound neutron-rich nucleus $^{80}\textrm{Ca}$ are calculated, and a good agreement with the available experimental centroid energies is obtained for $^{40-48}\textrm{Ca}$. For the exotic calcium isotopes, e.g., $^{68}\textrm{Ca}$ and $^{80}\textrm{Ca}$, the DRHBc-FAM calculated results are closer to the energy weighted sum rule than the calculations on the harmonic oscillator basis, which highlights the advantages of DRHBc-FAM in describing giant resonances for exotic nuclei. In order to explore the soft monopole mode in the exotic nuclei, the giant monopole resonance for the deformed exotic nucleus $^{200}\textrm{Nd}$ is investigated, where the prolate shape and the oblate shape coexist. A soft monopole mode near 6.0 MeV is found in the prolate case, and another one near 4.5 MeV is found in the oblate case. The transition density of the soft monopole mode shows in phase or out-of-phase vibrations near the surface region, which is generated by quadrupole vibrations.

New insights into sub-barrier fusion of 28Si + 100Mo

Fusion cross sections of the 28Si + 100Mo system have been measured near and below the Coulomb barrier by detecting the evaporation residues at forward angles. The excitation function has an overall smoother trend than what obtained in a previous experiment, and a large discrepancy is found for the lowest-energy region, where we observe a tendency of the S factor to develop a maximum, which would be a clear indication of hindrance. The results have been compared with the theoretical prediction of coupled-channels calculations using a Woods–Saxon nuclear potential, and including the low-energy excitation modes of both nuclei. Good agreement with data is found by including, in the coupling scheme, the three lowest members of the ground state rotational band of the oblate deformed 28Si, and two-phonons of the strong quadrupole vibration of 100Mo. The additional coupling, in a schematic way, of the two-neutron pick-up between ground states (Q-value = +4.86 MeV) has a minor effect on calculated cross sections, and does not essentially improve the data fit. The excitation function of 28Si + 100Mo has been compared with that of (1) the heavier system 60Ni + 100Mo having analogous features, and (2) several near-by 28Si, 32S + Zr, Mo systems with various nuclear structures and transfer Q-values. The role of quadrupole and octupole excitation modes, as well as of transfer channels, in affecting the fusion dynamics, are clarified to some extent. Systematic measurements of fusion barrier distributions and CC calculations properly including transfer couplings, are necessary, in order to shed full light on the influence of the various coupled channels on the fusion cross sections.

Li9(d,p) reaction as a specific probe of Li10 , the paradigm of parity-inverted nuclei around the N=6 closed shell

We show, within the framework of renormalized nuclear field theory and of the induced reaction surrogate formalism, that the highly debated $^{10}\mathrm{Li}$ structure, studied in a recent high statistics $^{9}\mathrm{Li}(d,p)\phantom{\rule{0.16em}{0ex}}^{10}\mathrm{Li}$ one-neutron transfer experiment, is consistent with, or better, requires, the presence of a virtual $1/{2}^{+}$ state of similar single-particle strength than that of the $1/{2}^{\ensuremath{-}}$ resonance at $0.45\ifmmode\pm\else\textpm\fi{}0.03\phantom{\rule{0.28em}{0ex}}\mathrm{MeV}$. Based on continuum spectroscopy self-energy techniques, we find that the physical mechanism responsible for parity inversion in $_{3}^{10}\mathrm{Li}$ is the same as that at the basis of the similar phenomenon observed in $_{4}^{11}\mathrm{Be}$ and as that needed in $^{11}\mathrm{Li}$ to have an important $s$-wave ground-state component. In particular the strong dynamical coupling between the ${s}_{1/2}$ and the ${d}_{5/2}$ states, mediated by the quadrupole vibration of the core $^{9}\mathrm{Li}$. A phenomenon which also affects the strength distribution of the ${d}_{5/2}$ state, in particular, in the energy range of 3--4.5 MeV. Furthermore, this mechanism is also consistent with the (normal) sequence of the $1{p}_{1/2}$ and $2{s}_{1/2}$ levels in the $N=7$ isotones $_{5}^{12}\mathrm{B}$ and $_{6}^{13}\mathrm{C}$. The main aim of the present Rapid Communication is that of treating structure and reactions on equal footing and in a common language. In other words, the calculation of the $^{9}\mathrm{Li}(d,p)^{10}\mathrm{Li}$ reaction as a single conceptual step from individual single-particle motion and collective vibrations to absolute double differential cross sections of renormalized virtual and resonant final states, which can be directly compared with experiment.

Абляция и фрагментация золотых наночастиц под действием интенсивного лазерного облучения в спектральных областях дипольного и квадрупольного плазмонных резонансов

Methods of vacuum deposition and subsequent heat treatment of thin gold films are proposed, which allow one to obtain both extremely small gold nanoparticles with spectrally narrow dipole plasmon resonances and larger nanoparticles that support quadrupole vibrations. The effect of high-power laser radiation on both small and large metal nanoparticles in the regions of dipole and quadrupole resonances, respectively, was experimentally studied. Powerful laser irradiation leads to the ablation of small particles, while large particles decay into smaller particles during irradiation.

Doubly critical character of the N=90 isotones 150Nd, 152Sm, 154Gd, and 156Dy

The N=90 isotones 150Nd, 152Sm, 154Gd, and 156Dy, which are known to provide the best examples of the X(5) critical point symmetry between quadrupole vibrations [U(5)] and axial quadrupole deformation [SU(3)], are proved to lie on the border between the regions of stable axial octupole deformation and octupole vibrations, described in terms of an Analytic Quadrupole Octupole Axially symmetric (AQOA) model including tunneling effects.

Measurement of the 21+ level lifetime in Sn120 by the Doppler shift attenuation method: Evidence of enhanced collectivity

Background: There has been a considerable interest focused on the study of enhancement or suppression in collectivity of the excited ${2}_{1}^{+}$ states in stable Sn isotopes. Independent measurements of Coulomb excitation cross sections and ${2}_{1}^{+}$ level lifetimes report discrepant transition probabilities. Existing estimates for ${2}_{1}^{+}$ lifetime indicate reduced collectivity.Purpose: A reexamination of lifetime of the ${2}_{1}^{+}$ state in the most abundant $^{120}\mathrm{Sn}$ isotope is thus warranted. The same has been carried out in the present work and the result has been used to determine the transition probability as an indicative of the underlying collectivity.Methods: Low-lying levels in the vibrational $^{120}\mathrm{Sn}$ nucleus have been excited by inelastic scattering with $^{32}\mathrm{S}$ beam at ${E}_{\mathrm{lab}}=120$ MeV. Level lifetime measurements have been carried out using the Doppler shift attenuation method, wherein the Doppler affected $\ensuremath{\gamma}$-ray peaks have been analyzed using updated methodologies.Results: From the measured lifetime of the ${2}_{1}^{+}$ state (${E}_{x}=1171$ keV) in $^{120}\mathrm{Sn}, {\ensuremath{\tau}}_{{2}_{1}^{+}}=0.{863}_{\ensuremath{-}0.036}^{+0.029}$ ps, a value of $B(E2;{0}_{\mathrm{g}.\mathrm{s}.}^{+}\ensuremath{\rightarrow}{2}_{1}^{+})=0.{215}_{+0.009}^{\ensuremath{-}0.007}$ is deduced. An estimate of the ${4}_{1}^{+}$ (feeder) level lifetime, ${\ensuremath{\tau}}_{{4}_{1}^{+}}=1.{77}_{\ensuremath{-}0.089}^{+0.084}$ ps, is also reported from lineshape analysis of $\ensuremath{\gamma}$ rays in cascade.Conclusions: An enhancement in collectivity for the ${2}_{1}^{+}$ state is confirmed, following an improved determination of the level lifetime, with reduced uncertainties. The excited ${2}_{1}^{+}$ state is also found to have a nonvanishing moment of inertia, suggesting anharmonic nature of quadrupole vibrations.

Proton-neutron multiplet states and isomers in the odd-odd nucleus I122

The level structure of the odd-odd iodine nucleus I122 has been studied by the Sn120 (Li7, 5n)I122 reaction at Elab=58 MeV. Two chopped beams from the 14UD (Unit Doubled) tandem heavy-ion accelerator at the Australian National University were used; one was a nanosecond (ns) pulsed-beam with separation of 1.7 microseconds (μs) and the second one was a 60-μs beam separated by 880μs. Through in-beam and out-of-beam γ-ray spectroscopy, the complex proton-neutron multiplet states, including isomers with half-lives of a few tens of ns to a few tens of μs, have been identified. Collective states based on the proton h11/2 orbital exhibit two kinds of structure depending upon the coupled neutron orbital: a quadrupole vibration when involving the h11/2 orbital and a γ-soft deformed rotation when involving the g7/2 orbital. With spherical shell model and deformed shell-model calculations, the observed level structures are explained.

Low-lying quadrupole and octupole collective excitations in the Sn112,116,118,120,122,124 isotopes

Background: In recent years, considerable interest has been focused on the study of the effect of neutron skin on the collective properties of vibrational nuclei. This can be best evidenced by exclusively determining neutron and proton transition matrix elements involved in a particular excitation by investigating the Coulomb-nuclear interference feature of inelastic scattering.Purpose: Measurement of angular distributions of the inelastic scattering cross sections for excitations to low-lying ${2}_{1}^{+}$ and ${3}_{1}^{\ensuremath{-}}$ states in $^{112,116,118,120,122,124}\mathrm{Sn}$ using $^{7}\mathrm{Li}$ beam as probe at ${E}_{\mathrm{lab}}=28\phantom{\rule{0.28em}{0ex}}\mathrm{MeV}$ and determination of neutron and proton transition matrix elements involved in each excitation.Methods: Projectilelike fragments were detected using six sets of Si-surface barrier detector telescopes to measure the cross sections for elastic and inelastic scattering channels. Optical model analysis of elastic scattering data, coupled reaction channels, and continuum discretized coupled channels calculations were performed to understand the measured differential cross sections. An attempt has been made to extract the microscopic mass and charge deformation lengths.Results: For the ${2}_{1}^{+}$ state, experimental $B(E2)$ values are in good agreement with existing results obtained by electromagnetic methods. Charge and mass quadrupole vibrations are homogeneous. Significant differences are observed for excitation to the ${3}_{1}^{\ensuremath{-}}$ state across the Sn isotopic chain. Available structural information for collective octupole vibrations could not reproduce the present data for this excitation. Results show much lower values of octupole mass deformation parameters.Conclusions: Isoscalar nature of surface vibrations for the ${2}_{1}^{+}$ state in Sn isotopes is verified, with ${M}_{n}/{M}_{p}\ensuremath{\sim}N/Z$. For the ${3}_{1}^{\ensuremath{-}}$ state, damped mass vibration is the primary observation. On comparison with existing estimates, a significant deviation from isoscalar nature is conjectured for this excitation, when probed with $^{7}\mathrm{Li}$.

“Stiff” deformed nuclei, configuration dependent pairing and the $\beta$β and $\gamma$γ degrees of freedom

We review the current experimental data on collective structures within the pairing gap of even-even deformed nuclei, with emphasis on nuclei near mass number $A \sim 150$. The essential physics that determines the characteristics of the first excited 0+ (02+) state in these nuclei has been in dispute for several decades. Interpretation of these states in terms of surface $\beta$ quadrupole vibrations has often been challenged. We examine the role that configuration dependent pairing can play in these levels particularly at the onset of deformation as major shells fill. In all deformed nuclei rotational bands are found experimentally, starting at a state with spin 2+ with excitation energies near the middle of the pairing gap. These rotational bands, with quantum number $K^{\pi} = 2^{+}$, are usually referred to as $\gamma$ bands and have been identified with quadrupole surface vibrations in the plane perpendicular to the major axis of deformation. However $K^{\pi} = 2^{+}$ bands can also arise due to the breaking of axial symmetry of the quadrupole shape. We discuss data that can help with these different interpretations.

Investigation and elimination of orbit distortion sources to achieve a high-quality photon beam at the Taiwan Photon Source

High beam stability is important to obtain a brilliant photon beam. There are many sources which can negatively affect beam stability, such as mechanical vibrations, electrical noise and malfunction of power supplies, etc. These sources are hard to avoid completely by careful design and may appear any time after installation work. To identify possible sources and study their effect to beam motion, a systematic study was performed starting with specific hardware installations. A method was developed to quickly identify locations of multiple perturbations and to observe vibrations and low-level undesired alternating magnetic fields. As for mechanical vibrations, a severe beam motion during beam commissioning was caused by quadrupole and vacuum chamber vibrations. The transfer of such vibrations to the beam were simulated by forced-shaking. Inversion of the response matrix together with singular value decomposition could identify perturbations by electrical noise originating from a nearby radio frequency system and cooling fans. After eliminating these detrimental factors, a high-quality beam, stable to less than 10% of the beam size, was achieved at the Taiwan Photon Source.

Finite Element Method for Solving the Collective Nuclear Model with Tetrahedral Symmetry

We apply a new calculation scheme of a finite element method (FEM) for solving an elliptic boundary-value problem describing a quadrupole vibration collective nuclear model with tetrahedral symmetry. We use of shape functions constructed with interpolation Lagrange polynomials on a triangle finite element grid and compare the FEM results with obtained early by a finite difference method.

Solution of Quantum Mechanical Problems Using Finite Element Method and Parametric Basis Functions

New computational schemes, symbolic-numerical algorithms and programs implementing the high-accuracy finite element method (FEM) for the solution of quantum mechanical boundary-value problems (BVPs) are reviewed. The elliptic BVPs in 2D and 3D domains are solved using the multivariable FEM and Kantorovich method using parametric basis functions. We demonstrate and compare the efficiency of the proposed calculation schemes, algorithms, and software by solving the benchmark BVPs that describe the scattering on a barrier and a well, the bound states of a helium atom, and the quadrupole vibration in a collective nuclear model.

Алгоритмы для решения параметрической самосопряжённой эллиптической краевой задачи в двумерной области методом конечных элементов высокого порядка точности

We consider the calculation schemes for solving elliptic boundary-value problems (BVPs) within the framework of the Kantorovich method that provides the reduction of an elliptic BVP to a system of coupled second-order ordinary differential equations (ODEs). The surface basis functions of the expansion depend on the independent variable of the ODEs parametrically. Here we use the basis functions calculated by means of the finite element method(FEM), as well as the probe parametric surface basis functions calculated in the analytical form. We propose new calculation schemes and algorithms for solving the parametric self-adjoint elliptic boundary-value problem (BVP) in a 2D finite domain, using high-accuracy finite element method (FEM) with rectangular and triangular elements. The algorithm and the programs calculate with the given accuracy the eigenvalues, the surface eigenfunctions and their first derivatives with respect to the parameter of the BVP for parametric self-adjoint elliptic differential equation with the Dirichlet and/or Neumann type boundary conditions on the 2D finite domain, and the potential matrix elements, expressed as integrals of the products of surface eigenfunctions and/or their first derivatives with respect to the parameter. The parametric eigenvalues (potential curves) and the potential matrix elements computed by the program can be used for solving bound-state and multi-channel scattering problems for systems of coupled second-order ODEs by means of the Kantorovich method. We demonstrate the efficiency of the proposed calculation schemes and algorithms in benchmark calculations of 2D elliptic BVPs describing quadrupole vibrations of a collective nuclear model.

Yang–Mills equation for the nuclear geometrical collective model connexion

The Bohr–Mottelson collective model of rotations and quadrupole vibrations is a foundational model in nuclear structure physics. A modern formulation using differential geometry of bundles builds on this legacy collective model to allow a deformation-dependent interaction between rotational and vortical degrees of freedom. The interaction is described by the bundle connexion. This article reports the Yang–Mills equation for the connexion. For a class of solutions to the Yang–Mills equation, the differential geometric collective model attains agreement between experiment and theory for the moments of inertia of deformed isotopes. More generally, the differential geometric framework applies to models of emergent phenomena in which two interacting sets of degrees of freedom must be unified.

Elastic response of the atomic nucleus in gauge space: Giant Pairing Vibrations

Due to quantal fluctuations, the ground state of a closed shell system $ A_{0}$ can become virtually excited in a state made out of the ground state of the neighbour nucleus $ \vert gs(A_0+2) \rangle $ ( $ \vert gs(A_0-2) \rangle $ ) and of two uncorrelated holes (particles) below (above) the Fermi surface. These $ J^{\pi} = 0^{+}$ pairing vibrational states have been extensively studied with two-nucleon transfer reactions. Away from closed shells, these modes eventually condense, leading to nuclear superfluidity and thus to pairing rotational bands with excitation energies much smaller than $ \hbar\omega_{0}$ , the energy separation between major shells. Pairing vibrations are the plastic response of the nucleus in gauge space, in a similar way in which low-lying quadrupole vibrations, i.e. surface vibrations with energies much smaller than $ \hbar\omega_{0}$ whose eventual condensation leads to quadrupole deformed nuclei, provide an example of the plastic nuclear response in 3D space. While much is known, in particular concerning its damping, regarding the counterpart of quadrupole plastic modes, i.e. regarding the giant quadrupole resonances (GQR), $ J^{\pi} = 2^{+}$ elastic response of the nucleus with energies of the order of $ \hbar\omega_{0}$ , little is known regarding this subject concerning pairing modes (giant pairing vibrations, GPV). Consequently, the recently reported observation of L = 0 resonances, populated in the reactions 12C(18O,16O)14C and 13C(18O,16O)15C and lying at an excitation energy of the order of $ \hbar\omega_{0}$ , likely constitutes the starting point of a new field of research, that of the study of the elastic response of nuclei in gauge space. Not only that, but also the fact that the GPV have likely been serendipitously observed in these light nuclei when it has failed to show up in more propitious nuclei like Pb, provides unexpected and fundamental insight into the relation existing between basic mechanisms --Landau, doorway, compound damping-- through which giant resonances acquire a finite lifetime, let alone the radical difference regarding these phenomena displayed by correlated (ph) and (pp) modes.