Large-amplitude collective dynamics of shape phase transition in the low-lying states of 30−36 Mg is investigated by solving the five-dimensional (5D) quadrupole collective Schrödinger equation. The collective masses and potentials of the 5D collective Hamiltonian are microscopically derived with use of the constrained Hartree-Fock-Bogoliubov plus local quasiparticle random phase approximation method. Good agreement with the recent experimental data is obtained for the excited 0 + states as well as the ground bands. For 30 Mg, the shape coexistence picture that the deformed excited 0 + state coexists with the spherical ground state approximately holds. On the other hand, large-amplitude quadrupole-shaped fluctuations dominate in both the ground and the excited 0 + states in 32 Mg, providing a picture that is different from the interpretation of the "coexisting spherical excited 0 + state" based on the naive inversion picture of the spherical and deformed configurations.Nuclei exhibit a variety of shapes in their ground and excited states. A feature of the quantum phase transition of a finite system is that the order parameters (shape deformation parameters) always fluctuate and vary with the particle number. Especially, the large-amplitude shape fluctuations play a crucial role in transitional (critical) regions. Spectroscopic studies of low-lying excited states in transitional nuclei are of great interest to observe such unique features of the finite quantum systems.Low-lying states of neutron-rich nuclei at approximately N = 20 attract great interest, as the spherical configurations associated with the magic number disappear in the ground states. In neutron-rich Mg isotopes, the increase of the excitation energy ratio E(4 1 + )/E(2 1 + ) [1-3] and the enhancement of B(E2; 2 1 + → 0 1 + ) from 30 Mg to 34 Mg [4-6] indicate a kind of quantum phase transition from spherical to deformed shapes taking place around 32 Mg. These experiments stimulate microscopic investigations on quadrupole collective dynamics unique to this region of the nuclear chart with various theoretical approaches: the shell model [7-10], the Hartree-Fock-Bogoliubov (HFB) method [11,12], the parityprojected Hartree-Fock (HF) [13], the quasiparticle random phase approximation (QRPA) [14,15], the angular-momentum projected generator coordinate method (GCM) with [16] and without [17,18] restriction to the axial symmetry, and the antisymmetrized molecular dynamics [19]. Quite recently, excited 0 + states were found in 30 Mg and 32 Mg at 1.789 MeV and 1.058 MeV, respectively, and their characters are presently under hot discussion [20-23]. For 30 Mg, the excited 0 2 + state is interpreted as a prolately deformed state which coexists with the spherical ground state. For 32 Mg, from the observed population of the excited 0 2 + state in the (t,p) reaction on 30 Mg, it is suggested [22] that the 0 2 + state is a spherical state coexisting with the deformed ground state and that their relative energies are inverted at N = 20. However, available shell-mod...

The oblate-prolate shape coexisting/mixing phenomena in proton-rich 68,70,72 Se are investigated by means of the adiabatic self-consistent collective coordinate (ASCC) method. The one-dimensional collective path and the collective Hamiltonian describing the large-amplitude shape vibration are derived in a fully microscopic way. The excitation spectra, B(E2) and spectroscopic quadrupole moments are calculated by requantizing the collective Hamiltonian and solving the collective Schrödinger equation. The basic properties of the coexisting two rotational bands in low-lying states of these nuclei are well reproduced. It is found that the oblate-prolate shape mixing becomes weak as the rotational angular momentum increases. We point out that the rotational energy plays a crucial role in causing the localization of the collective wave function in the (β, γ ) deformation space.

We treat proton-neutron pairing amplitudes, in addition to the nuclear deformation, as generator coordinates in a calculation of the neutrinoless double-beta decay of 76 Ge. We work in two oscillator shells, with a Hamiltonian that includes separable terms in the quadrupole, spin-isospin, and pairing (isovector and isoscalar) channels. Our approach allows larger single-particle spaces than the shell model and includes the important physics of the proton-neutron quasiparticle random-phase approximation (QRPA) without instabilities near phase transitions. After comparing the results of a simplified calculation that neglects deformation with those of the QRPA, we present a more realistic calculation with both deformation and proton-neutron pairing amplitudes as generator coordinates. The future should see proton-neutron coordinates used together with energy-density functionals.

We investigate the extent to which theories of collective motion can capture the physics that determines the nuclear matrix elements governing neutrinoless double-beta decay. To that end we calculate the matrix elements for a series of isotopes in the full pf shell, omitting no spin-orbit partners. With the inclusion of isoscalar pairing, a separable collective Hamiltonian that is derived from the shell model effective interaction reproduces the full shell-model matrix elements with good accuracy. A version of the generator coordinate method that includes the isoscalar pairing amplitude as a coordinate also reproduces the shell model results well, an encouraging result for theories of collective motion, which can include more single-particle orbitals than the shell model. We briefly examine heavier nuclei relevant for experimental double-beta decay searches, in which shell-model calculations with all spin-orbit partners are not feasible; our estimates suggest that isoscalar pairing also plays a significant role in these nuclei, though one we are less able to quantify precisely.