Model for collective motion

Collective motion is a manifestation of emergent phenomena in mediumheavy and heavy nuclei. A relatively large number of constituent nucleons contribute coherently to nuclear excitations (vibrations, rotations) that are characterized by large electromagnetic moments and transition rates. Basic features of collective excitations are reviewed, and a simple model introduced that describes large-amplitude quadrupole and octupole shape dynamics, as well as the dynamics of induced fission. Modern implementations of the collective Hamiltonian model are based on the microscopic framework of energy density functionals, that provide an accurate global description of nuclear ground states and collective excitations. Results of illustrative calculations are discussed in comparison with available data. Outline • Introduction • Nuclear Shape Parameters • Nuclear Surface Oscillations • The Rotation-Vibration Model • Microscopic Derivation of the Collective Hamiltonian • Microscopic Collective Hamiltonian Based on Density Functional Theory Z. P. Li School of Physical Science and Technology, Southwest University, Chongqing 400715, China, email: zpliphy@swu.edu.cn D. Vretenar Department of Physics, Faculty of Science, University of Zagreb, HR-10000 Zagreb, Croatia; State Key Laboratory of Nuclear Physics and Technology, School of Physics, Peking University, Beijing 100871, China, e-mail: vretenar@phy.hr ∗ zpliphy@swu.edu.cn 1 ar X iv :2 20 3. 07 60 8v 1 [ nu cl -t h] 1 5 M ar 2 02 2 2 Z. P. Li and D. Vretenar Introduction A medium-heavy or heavy atomic nucleus presents a typical example of a complex quantum system, in which different interactions between a relatively large number of constituent nucleons give rise to physical phenomena that are qualitatively different from those exhibited by few-nucleon systems. There are a number of features that characterize complex systems, but for the topic of the present chapter of particular interest is the emergence of collective structures and dynamics that do not occur in light nuclei composed of only a small number of nucleons. Collective motion is the simplest manifestation of emergent phenomena in atomic nuclei. It can be interpreted as a kind of motion in which a large number of nucleons contribute coherently to produce a large amplitude oscillation of one or more electromagnetic multipole moments. Collective motion gives rise to excited states that are characterized by large electromagnetic transition rates to the ground state, that is, rates that correspond to many single-particle transitions [1]. This chapter will mainly focus on low-energy large-amplitude collective motion (LACM), such as surface vibrations, rotations, and fission. Theoretical studies of LACM started as early as in the 1930s. Based on the liquid drop model of the atomic nucleus [2], Flügge applied Rayleigh’s normal modes [3] to a classical description of low-energy excitations of spherical nuclei [4]. The model of Flügge was quantized by Bohr [5], who formulated a quantum model of surface oscillations of spherical nuclei, and also introduced the concept of intrinsic frame of reference for a quadrupole deformed nuclear surface characterized by the Euler angles and the shape parameters β and γ (nowadays often called the Bohr deformation parameters). Subsequently, Bohr and Mottelson [6] generalized the model to vibrations and rotations of deformed nuclei. A generalization of the Bohr Hamiltonian to describe large-amplitude collective quadrupole excitations of even-even nuclei of arbitrary shape was introduced by Belyaev [7] and Kumar and Baranger [8]. Several specific forms of the collective Hamiltonian, designed to describe collective excitations in nuclei of particular shape were also considered [9–14]. In the past several decades, enormous progress has been made in developing microscopic many-body theories of nuclear systems. However, a description of collective phenomena starting from single-nucleon degrees of freedom still presents a considerable challenge. One of the methods that has been used to obtain such a description is the adiabatic time-dependent HFB theory (ATDHFB) [7, 15, 16] which, in the case of quadrupole collective coordinates, leads to the Bohr Hamiltonian. Another approach to collective phenomena that is based on microscopic degrees of freedom, is the generator coordinate method (GCM) [17]. In the Gaussian overlap approximation (GOA) [18, 19], the GCM also leads to the Bohr collective Hamiltonian [20, 21]. Model for collective motion 3 Nuclear Shape Parameters Excitation spectra of even-even nuclei in the energy range of up to ≈ 3 MeV, exhibit characteristic band structures that are interpreted as vibrations and rotations of the nuclear surface in the geometric collective model, first introduced by Bohr and Mottelson [5, 6], and further elaborated by Faessler and Greiner [12, 13]. For low-energy excitations the compression mode is not relevant because of high incompressibility of nuclear matter, and the diffuseness of the nuclear surface layer can also be neglected to a good approximation. One therefore starts with the model of a nuclear liquid drop of constant density and sharp surface. With these assumptions, the time-dependent nuclear surface can, quite generally, be described by an expansion in spherical harmonics with shape parameters as coefficients: R(θ ,φ ; t) = R0 ( 1+ ∞ ∑ λ=0 λ ∑ μ=−λ αλ μ(t)Yλ μ(θ ,φ) ) (1) where R(θ ,φ ; t) denotes the nuclear radius in spherical coordinates (θ ,φ), and R0 is the radius of a sphere with the same volume. The shape parameters αλ μ(t) play the role of collective dynamical variables, and their physical meaning will be discussed for increasing values of λ . Fig. 1 Nuclear shapes with dipole (λ = 1), quadrupole (λ = 2), octupole (λ = 3), and hexadecupole (λ = 4) deformations. To lowest order, the dipole mode λ = 1 corresponds to a translation of the nucleus as a whole and, therefore, is not considered for low-energy excitations. Dynamical quadrupole deformations, that is, the mode with λ = 2, turn out to be the most relevant low-lying collective excitations. Most of the following discussion of collective models will focus on this case, so a more detailed description is included below. Octupole dynamical deformations, λ = 3, are the principal asymmetric modes of a nucleus associated with negative-parity bands. While there is no evidence for pure hexadecupole excitations in low-energy spectra, this mode plays an important role as admixture to quadrupole excitations, and for fission dynamics. Shape oscillations of higher multipoles are not relevant for low-energy excitations. For the case of pure quadrupole deformation the nuclear surface is parameterized R(θ ,φ) = R0 ( 1+ 2 ∑ μ=−2 α∗ μY2μ(θ ,φ) ) (2) 4 Z. P. Li and D. Vretenar where the time dependence is implicit for dynamical variables. If the shape of the nucleus is an ellipsoid, its three principal axes (x, y, z) are linked with the (X , Y , Z) axes of a Cartesian coordinate system in the laboratory frame. From the symmetry of the ellipsoid, it follows that a1 = a−1 = 0,a2 = a−2, where aμ are the shape parameters in the principal-axis system. Obviously, the five coefficients αμ in the laboratory frame reduce to two real independent variables a0 and a2 in the principal-axis system, which, together with the three Euler angles, provide a complete parameterization of the nuclear surface. The details of the transformation between αμ and aμ are included below. In the principal-axis system, the nuclear radius is given by R(θ ′,φ ′) = R0 [ 1+a0Y20(θ ′,φ )+a2Y22(θ ′,φ )+a2Y2−2(θ ′,φ ′) ] . (3) Two parameters (a0, a2) are generally used to describe quadrupole deformations but, instead of a0 and a2, the polar coordinates β and γ are usually employed. They are defined as follows: a0 = β cosγ, a2 = 1 √ 2 β sinγ. (4) Using Eq. (3) and Eq. (4), we can express the increments of the three semi-axes in the principal-axis system: δRκ = R0 √ 5 4π β cos ( γ− 2π 3 κ ) , κ = 1,2,3 (5) where κ = 1,2,3 correspond to x,y,z, respectively. The parameters β and γ only describe exactly ellipsoidal shapes in the limit of small β -values.