Collective Motion In Nuclei Research Articles

Dynamics in Nuclei

The nucleus is a complex many-body system with some remarkable emergent collective properties of multiple nucleons acting together. Bohr and Mottelson [1] provided a description of collective motion in nuclei based on geometrical shapes with superimposed oscillations around those shapes. Later, Lie algebras and symmetries were used to describe nuclear dynamics [2], followed by advances in the shell model approach [3] with new effective nucleon-nucleon two- and three-body interactions, and more recently with Hartree-Fock-Bogoliubov approximations within the extended generator coordinate method [4]. Yet, the underlying science question has remained the same. In nuclei, where there is explicit deformation in the ground state, “are the low-lying 0+ states collective vibrations built on the ground state or are they minima of a coexisting shape?” Ref. [4] has shown that for a significant percentage of K = 0+ excitations built on the deformed ground state (g.s.) should, in fact, be a collective vibration. The question has remained open due to sufficiently convincing experimental data with lifetimes, transfer reaction cross sections, and E0 transitions [5]. This paper summarizes the experimental situation regarding the lifetimes of 0+ states.

Mapping the demise of collective motion in nuclei at high excitation energy

High energy gamma-rays from the 116Sn + 24Mg reaction at 23A MeV were measured using the MEDEA detector at LNS – INFN Catania. Combining this new data with previous measurements yields a detailed view of the quenching of the Giant Dipole Resonance as a function of excitation energy in nuclei of mass A in the range 120÷132. The transition towards the disappearance of the dipole strength, which occurs around 230 MeV excitation energy, appears to be remarkably sharp. Current phenomenological models give qualitative explanations for the quenching but cannot reproduce its detailed features.

FELIX-2.0: New version of the finite element solver for the time dependent generator coordinate method with the Gaussian overlap approximation

The time-dependent generator coordinate method (TDGCM) is a powerful method to study the large amplitude collective motion of quantum many-body systems such as atomic nuclei. Under the Gaussian Overlap Approximation (GOA), the TDGCM leads to a local, time-dependent Schrödinger equation in a multi-dimensional collective space. In this paper, we present the version 2.0 of the code FELIX that solves the collective Schrödinger equation in a finite element basis. This new version features: (i) the ability to solve a generalized TDGCM+GOA equation with a metric term in the collective Hamiltonian, (ii) support for new kinds of finite elements and different types of quadrature to compute the discretized Hamiltonian and overlap matrices, (iii) the possibility to leverage the spectral element scheme, (iv) an explicit Krylov approximation of the time propagator for time integration instead of the implicit Crank–Nicolson method implemented in the first version, (v) an entirely redesigned workflow. We benchmark this release on an analytic problem as well as on realistic two-dimensional calculations of the low-energy fission of 240Pu and 256Fm. Low to moderate numerical precision calculations are most efficiently performed with simplex elements with a degree 2 polynomial basis. Higher precision calculations should instead use the spectral element method with a degree 4 polynomial basis. We emphasize that in a realistic calculation of fission mass distributions of 240Pu, FELIX-2.0 is about 20 times faster than its previous release (within a numerical precision of a few percents). Program summaryProgram Title: FELIX-2.0Program Files doi:http://dx.doi.org/10.17632/t8b4h9g88r.1Licensing provisions: GPLv3Programming language: C++Journal reference of previous version: Computer Physics Communications 200, 350–363 (2016)Does the new version supersede the previous version?: YesReasons for the new version: FELIX-2.0 extends the physics capabilities of the previous version, since it includes the option to have a metric term in the collective Schrödinger equation. The computational efficiency of the code is considerably increased thanks to the implementation of several new numerical methods. This version also provides a more flexible and robust workflow to perform calculation of fission fragment distributions more efficiently.Summary of revisions: The new version includes the ability to use a metric term in the TDGCM+GOA equation. Numerical integration methods (both spatial and time integration) have been updated to more efficient schemes: matrix elements are now computed with Gauss–Legendre or Gauss–Legendre–Lobatto quadratures, the time propagation of the wave packet is computed with the Krylov approximation of the propagator instead of the previous Crank–Nicolson scheme. A new type of finite element based on n-dimensional orthotope is implemented, which enables a spectral element scheme for spatial discretization. Finally, the workflow and inputs/outputs of the code were entirely redesigned to provide more flexibility and be more user-friendly.Nature of problem: The Gaussian overlap approximation to the time-dependent generator coordinate method [1,2] yields a local, time-dependent Schrödinger equation in a small, multi-dimensional collective space. Its solution provides the time-evolution of the collective probability amplitude. For applications to nuclear fission, scission configurations are defined by a hyper-surface in this collective space referred to as the frontier. Distributions of nuclear observables such as charge or mass distributions of the fission fragments are extracted from the probability for the system to pass through any given section of the frontier. This probability is computed by first solving the evolution equation up to times greater than 10−20s and then integrating the flux of probability through the frontier over the entire timerange.Solution method: FELIX-2.0 solves the time-dependent Schrödinger equation by first discretizing the N-dimensional collective space with the continuous Galerkin Finite Element Method. This produces a large set of coupled, time-dependent Schrödinger equations characterized by the sparse overlap and Hamiltonian matrices. The solution is evolved in small time steps by applying an explicit and unitary propagator built as a Krylov approximation [3] of the exponential of the Hamiltonian.Additional comments including Restrictions and Unusual features: Although the implementation of the program gives it the ability to solve the TDGCM+GOA equation in a generic N-dimensional collective space, it has only been tested on 1-, 2- and 3-dimensional meshes.[1] J.J. Griffin, J. A. Wheeler, Collective Motions in Nuclei by the Method of Generator Coordinates, Phys. Rev. C 108, 311-327 (1975)[2] P. G. Reinhard, K. Goeke, The Generator-Coordinate Method and Quantized Collective Motion in Nuclear Systems, Rep. Prog. Phys. 50, 1-64 (1987)[3] Y. Saad, Iterative Methods for Sparse Linear Systems: Second Edition, SIAM, 2003

Investigation of the disappearance of collective motion in nuclei of mass A~120-130

A study of hot Giant Dipole Resonance (GDR) in nuclei of mass A=120130 in an excitation energy range from 150 to 330 MeV, where the GDR quenching is expected to arise, has been undertaken using the MEDEA multi-detector system. Hot nuclei were populated using complete and incomplete fusion reactions. The characterization of hot system was performed through the study of residue time of flight combined with the analysis of light charged energy spectra detected in coincidence. Gamma-ray energy spectra show an evolution of the GDR main features both in terms of width and multiplicity. Evidences of a saturation of gamma multiplicity appear at high excitation energy at variance with predictions of statistical model calculations. Gamma- ray energy spectra can been reproduced in a phenomenological way introducing in the statistical model a sharp suppression of the gamma-ray emission above E* = 240 MeV. A comparison of experimental data to models describing the GDR disappearance will be presented.

Self-consistent pairing interaction and collective motion in nuclei: a semiclassical approach

The solutions of the semiclassical time–dependent Hartree–Fock-Bogoliubov equations of motion are studied in a linear approximation in which the pairing field is allowed to oscillate and to become complex. The pairing field fluctuations are derived from the self–consistent relation (the gap equation of the BCS type), while the static pairing field is approximated with a phenomenological constant Δ. The self–consistent pairing-field fluctuations introduce possibility of new collective modes of the system. We have found out the dispersion relation which determines possible collective modes of the system generated by the pairing interaction. The obvious solution of our dispersion relation at ω=0 corresponds to the Anderson–Goldstone–Nambu mode and is related to gauge symmetry. The solutions of dispersion relation have been studied in a simple model, in which nuclei are represented as homogeneous spheres of symmetric nuclear matter characterized by parameters (size, density, pairing gap) typical of heavy nuclei. We have found that the dispersion relation has approximate solution at ħω≈2Δ for the monopole channel. Our semiclassical approach is valid for both weak and strong pairing. Here we focus on weak pairing which is suitable for systems with size and pairing parameter appropriated to nuclei.

Restoration of rotational symmetry in deformed relativistic mean-field theory

We report on a very recently developed three-dimensional angular momentum projected relativistic mean-field theory with point-coupling interaction (3DAMP+RMF-PC). Using this approach the same effective nucleon-nucleon interaction is adopted to describe both the single-particle and collective motions in nuclei. Collective states with good quantum angular momentum are built projecting out the intrinsic deformed mean-field states. Results for 24Mg are shown as an illustrative application.

Gauge-Invariant Formulation of the Adiabatic Self-Consistent Collective Coordinate Method

The adiabatic self-consistent collective coordinate (ASCC) method is a practical microscopic theory of large-amplitude collective motion in nuclei with superfluidity. We show that its basic equations are invariant under transformations involving the gauge angle in particle-number space. By virtue of this invariance, a clean separation of the large-amplitude collective motion and the pairing rotational motion can be made, and this allows us to restore the particle-number symmetry broken by the Hartree-Fock-Bogoliubov (HFB) approximation. We formulate the ASCC method explicitly in a gauge-invariant form. In solving the ASCC equations, it is necessary to fix the gauge. Applying this new formulation to the multi-O(4) model, we compare different gauge-fixing procedures and demonstrate that calculations using different gauges indeed yield the same results for gauge-invariant quantities, such as the collective path and quantum spectra. We propose a gauge-fixing prescription that seems most convenient in realistic calculations.

Evolution of the giant dipole resonance properties with excitation energy

The studies of the evolution of the hot Giant Dipole Resonance (GDR) properties as a function of excitation energy are reviewed. The discussion will mainly focus on the A ∼ 100-120 mass region where a large amount of data concerning the width and the strength evolution with excitation energy are available. Models proposed to interpret the main features and trends of the experimental results will be presented and compared to the available data in order to extract a coherent scenario on the limits of the development of the collective motion in nuclei at high excitation energy. Experimental results on the GDR built in hot nuclei in the mass region A ∼ 60-70 will be also shown, allowing to investigate the mass dependence of the main GDR features. The comparison between limiting excitation energies for the collective motion and critical excitation energies extracted from caloric curve studies will suggest a possible link between the disappearance of collective motion and the liquid-gas phase transition.

High-spin collectivity in ? 40 nuclei

Recent experiments with large gamma-ray detector arrays coupled to 4π charge-particle detectors have enabled the detailed spectroscopic study of highly- and super-deformed rotational bands in a number of N ≈ Z nucleic in the A ∼ 40 mass region. In the context of the Nilsson scheme, these rotational structures are associated with the shell gaps at large prolate deformation for particle numbers N, Z = 16, 18, 20, while in the spherical shell model context they are described in terms of multi-particle multi-hole excitations across the N, Z = 20 spherical shell gap. In analogy with rotational bands in heavy nuclei, these highly collective structures involve active valence particles in two major shells for both protons and neutrons, yet their valence space dimension remains small enough to be approached from a shell-model perspective. These rotational bands thus provide an ideal opportunity for detailed comparison between experimental data and both microscopic and macroscopic models of collective motion in nuclei.

High-spin collectivity in A ∼ 40 nuclei

Recent experiments with large gamma-ray detector arrays coupled to 4π charge-particle detectors have enabled the detailed spectroscopic study of highly- and super-deformed rotational bands in a number of N ≈ Z nucleic in the A ∼ 40 mass region. In the context of the Nilsson scheme, these rotational structures are associated with the shell gaps at large prolate deformation for particle numbers N, Z = 16, 18, 20, while in the spherical shell model context they are described in terms of multi-particle multi-hole excitations across the N, Z = 20 spherical shell gap. In analogy with rotational bands in heavy nuclei, these highly collective structures involve active valence particles in two major shells for both protons and neutrons, yet their valence space dimension remains small enough to be approached from a shell-model perspective. These rotational bands thus provide an ideal opportunity for detailed comparison between experimental data and both microscopic and macroscopic models of collective motion in nuclei.

Goldstone-Brueckner perturbation theory extended in terms of mixed nonorthogonal Slater determinants

The Goldstone-Brueckner perturbation theory is extended to incorporate in a simple way correlations associated with large amplitude collective motions in nuclei. The new energy expansion making use of non-orthogonal vacua still allows to remove the divergences originating from the hard-core of the bare interaction. This is done through the definition of a new Brueckner matrix summing generalized Brueckner ladders. At the lowest-order, this formalism motivates variational calculations beyond the mean-field such as the Generator Coordinate Method (GCM) and the Projected Mean-Field Method from a perturbative point of view for the first time. Going to higher orders amounts to incorporate diabatic effects in the GCM and to extend the projection technique from product states to well-defined correlated states.

Nature of low-energy dipole strength in nuclei: the case of a resonance at particle threshold in 208Pb.

A high-resolution (gamma,gamma') study of the electric dipole response in 208Pb at the S-DALINAC reveals a resonance structure centered around the neutron emission threshold. Microscopic quasiparticle phonon model calculations in realistic model spaces including the coupling to complex configurations are able to describe the data in great detail. The resonance is shown to result from surface density oscillations of the neutron skin relative to an approximately isospin-saturated core. It also forms an integral part of a toroidal E1 mode representing an example of vortex collective motion in nuclei.

Quadrupole-Octupole Collectivity and Fine Structure of Nuclear Rotational Spectra

We apply a point-symmetry based quadrupole-octupole rotation model to study the collective motion of nuclei with simultaneous presence of octupole and quadrupole deformations. We show that it is capable to reproduce the fine odd-even staggering effects in nuclear octupole bands.

Fast rotation of the N = Z nucleus 36Ar

A highly-deformed rotational band has been identified in the N=Z nucleus 36Ar. At high spin the band is observed to its presumed termination at I=16+, while at low spin it has been firmly linked to previously known states in 36Ar. Spins, parities, and absolute excitation energies have thus been determined throughout the band. Lifetime measurements establish a large low-spin quadrupole deformation (beta_2=0.46+-0.03) and indicate a decreasing collectivity as the band termination is approached. With effectively complete spectroscopic information and a valence space large enough for significant collectivity to develop, yet small enough to be meaningfully approached from the shell model perspective, this rotational band in 36Ar provides many exciting opportunities to test and compare complementary models of collective motion in nuclei.

Hadron Structure Studied with the Electromagnetic Probe - from Giant Resonances to Meson Production

The development of theoretical photonuclear physics over the last 40 years is illustrated by a few selected examples highlighting a number of important issues like collective motion in nuclei, the role of subnuclear degrees of freedom, relativity and meson production.

Adiabatic Selfconsistent Collective Coordinate Method for Large Amplitude Collective Motion in Nuclei with Pairing Correlations

An adiabatic approximation to the selfconsistent collective coordinate method is formulated in order to describe large amplitude collective motions in superconducting nuclei on the basis of the time-dependent Hartree-Fock-Bogoliubov equations of motion. The basic equations are presented in a local harmonic form which can be solved in a similar way as the quasiparticle RPA equations. The formalism guarantees the conservation of nucleon number expectation values. An extension to the multi-dimensional case is also discussed.

The Structure of S Pair in Collective States (I)**The project supported by National Natural Science Foundation, of China

A new S pair and seniority particle operator used to overcome the difficulties due to seniority mixing are defined in collective motion of nuclei. The structure amplitude of S pair is determined in a self-consistent manner.

Coherent nuclear resonant scattering of X-rays: Time and space picture

The problem of coherent resonant scattering of X-rays by an ensemble of nuclei is solved directly in time and space. In a first step the problem with a single coherently scattered beam is considered – nuclear forward scattering. The wave equation describing the propagation of the radiation through the nuclear ensemble is derived. It is a first order integro-differential equation. Its kernel is a double time function $$K(t,\tilde t)$$ which represents the coherent single scattering response of the nuclear system at time t to excitation at $$\tilde t$$ . The kernel is defined by the character of the interactions the nuclei experience with the environment and by the character of their motion. A general procedure of solution of the wave equation is introduced which is independent of the type of kernel. In a second step the wave equation is generalized to the case of many coherently scattered beams, which is, e.g., the case of nuclear Bragg diffraction. Kernels of the wave equations are derived for some particular cases: collective motion of nuclei in space, thermal lattice vibrations, time-independent hyperfine interactions, and time-dependent hyperfine interactions due to external magnetic-field switching.

Fractional Parentage Coefficients for Composite Pairs**The project supported by National Natural Science Foundation of China

A new set of fractional parentage coefficients is introduced to build the orthogonal basis of states with many and pairs in the microscopic theory of low-lying collective motion in nuclei.

Interplay between one-body and collisional damping of collective motion in nuclei.

Damping of giant collective vibrations in nuclei is studied within the framework of the Landau-Vlasov kinetic equation. A phenomenological method of independent sources of dissipation is proposed for taking into account the contributions of one-body dissipation, the relaxation due to the two-body collisions and the particle emission. An expression for the intrinsic width of slow damped collective vibrations is obtained. In the general case, this expression cannot be represented as a sum of the widths associated with the different independent sources of the damping. This is a peculiarity of the collisional Landau-Vlasov equation where the Fermi-surface distortion effect influences both the self-consistent mean field and the memory effect at the relaxation processes. The interplay between the one-body, the two-body, and the particle emission channels which contribute to the formation of the total intrinsic width of the isoscalar ${2}^{+}$ and ${3}^{\mathrm{\ensuremath{-}}}$ and isovector ${1}^{\mathrm{\ensuremath{-}}}$ giant multipole resonances in cold and hot nuclei is discussed. We have shown that the criterion for the transition temperature ${\mathit{T}}_{\mathrm{tr}}$ between the zero-sound and first-sound regimes in hot nuclei is different from the case of infinite nuclear matter due to the contribution from the one-body relaxation and the particle emission. In the case of the isovector GDR the corresponding transition can be reached at temperature ${\mathit{T}}_{\mathrm{tr}}$=4--5 MeV. \textcopyright{} 1996 The American Physical Society.