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Triaxiality and shape dynamics in $^{70}$Ge

T. M. Kowalewski, A. D. Ayangeakaa, N. Sensharma, R. V. F. Janssens, Y. M. Wang, Q. B. Chen, J. M. Allmond, C. M. Campbell, S. Carmichael, M. P. Carpenter, P. Copp, C. Cousins, M. Devlin, U. Garg, C. Müller-Gatermann, T. J. Gray, D. J. Hartley, J. Heery, J. Henderson, H. Jayatissa, S. R. Johnson, S. P. Kisyov, F. G. Kondev, T. Lauritsen, S. Nandi, R. Rathod, W. Reviol, M. Rocchini, E. Rubino, R. Russell, A. Saracino, D. Seweryniak, M. Siciliano, C. Y. Wu

TL;DR

Using multi-step Coulomb excitation of $^{70}$Ge on $^{208}$Pb, the study determines a comprehensive set of electromagnetic matrix elements for eleven low-lying states. The data are interpreted with the generalized triaxial rotor model, large-scale shell-model calculations in the jj44 space (JUN45 and jj44b), and relativistic density functional theory coupled to the five-dimensional collective Hamiltonian, revealing significant triaxiality and strong mixing between the $0^+_1$ and $0^+_2$ states, with similar $\beta_{\mathrm{rms}}$ for these two states and notable $\gamma$-softness. Invariants derived from the data point to a predominantly oblate-triaxial ground state and a prolate-triaxial, but highly $\gamma$-soft, $0^+_2$ state, indicating nuanced shape coexistence and evolution along the Ge isotopic chain. The RDFT+5DCH framework reproduces key features of the low-spin spectrum and quadrupole properties, including a bimodal collective distribution for $0^+_2$ and an $E0$ strength of $\rho^2(E0,0^+_2\to0^+_1) \approx 29\times10^{-3}$, reinforcing the interpretation of shape coexistence in $^{70}$Ge.

Abstract

The electromagnetic properties of low-lying states in $^{70}$Ge were investigated via multi-step Coulomb excitation of a $^{70}$Ge beam impinging on a $^{208}$Pb target at the ATLAS facility of the Argonne National Laboratory. A total of 27 transitional elements and six diagonal matrix elements coupling 11 low-lying states, were extracted from the measured cross sections. These were used to calculate reduced transition probabilities, spectroscopic quadrupole moments, and rotational invariant shape parameters, providing enhanced precision and expanding on previous studies. The experimental data were compared within several theoretical frameworks, including the generalized triaxial rotor model, configuration interaction shell-model calculations, and computations within the combined frameworks of relativistic density functional theory and the five-dimensional collective Hamiltonian. The results demonstrate a good agreement with the experimental data and, in conjunction with calculations using a two-state mixing model, support significant triaxiality and strong mixing between the $0^+_1$ and $0^+_2$ states. This results in the magnitudes of their respective quadrupole deformations $[β_\text{rms}(0^+_1) = 0.228\,(3),\,β_\text{rms}(0^+_2) = 0.273\,(1)]$ being more similar than previously observed. The implications of these results for understanding the complex shape coexistence phenomena, the role of triaxiality, and shape evolution along the Ge isotopic chain are discussed.

Triaxiality and shape dynamics in $^{70}$Ge

TL;DR

Using multi-step Coulomb excitation of Ge on Pb, the study determines a comprehensive set of electromagnetic matrix elements for eleven low-lying states. The data are interpreted with the generalized triaxial rotor model, large-scale shell-model calculations in the jj44 space (JUN45 and jj44b), and relativistic density functional theory coupled to the five-dimensional collective Hamiltonian, revealing significant triaxiality and strong mixing between the and states, with similar for these two states and notable -softness. Invariants derived from the data point to a predominantly oblate-triaxial ground state and a prolate-triaxial, but highly -soft, state, indicating nuanced shape coexistence and evolution along the Ge isotopic chain. The RDFT+5DCH framework reproduces key features of the low-spin spectrum and quadrupole properties, including a bimodal collective distribution for and an strength of , reinforcing the interpretation of shape coexistence in Ge.

Abstract

The electromagnetic properties of low-lying states in Ge were investigated via multi-step Coulomb excitation of a Ge beam impinging on a Pb target at the ATLAS facility of the Argonne National Laboratory. A total of 27 transitional elements and six diagonal matrix elements coupling 11 low-lying states, were extracted from the measured cross sections. These were used to calculate reduced transition probabilities, spectroscopic quadrupole moments, and rotational invariant shape parameters, providing enhanced precision and expanding on previous studies. The experimental data were compared within several theoretical frameworks, including the generalized triaxial rotor model, configuration interaction shell-model calculations, and computations within the combined frameworks of relativistic density functional theory and the five-dimensional collective Hamiltonian. The results demonstrate a good agreement with the experimental data and, in conjunction with calculations using a two-state mixing model, support significant triaxiality and strong mixing between the and states. This results in the magnitudes of their respective quadrupole deformations being more similar than previously observed. The implications of these results for understanding the complex shape coexistence phenomena, the role of triaxiality, and shape evolution along the Ge isotopic chain are discussed.

Paper Structure

This paper contains 10 sections, 2 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Experimental Details
  3. Data Analysis and Results
  4. Discussion
  5. Matrix Elements and Transition Probabilities
  6. Rotor Model Calculations
  7. Large-Scale Shell-Model Calculations and Rotational Invariants
  8. Shape Coexistence and Two-State Mixing
  9. Relativistic Density Functional Theory and the Five Dimensional Collective Hamiltonian
  10. Conclusions

Figures (13)

  • Figure 1: (Color online) Difference in the time-of-flight between the scattered reaction participants as a function of the laboratory scattering angle measured with the CHICO2 detector, illustrating the clear separation between the detected $^{208}$Pb and $^{70}$Ge ions. The color scale (right side) provides the range of z-axis values (counts/$1.55^\circ$) in this plot. The CHICO2 detector system is composed of left and right hemispheres. The plot above shows the time-of-flight difference $\Delta T = t_\mathrm{left} - t_\mathrm{right}$ between the hemispheres for sequential hits at forward scattering angles. A similar pattern and separation is achieved in the plot for the other hemisphere, while, due to the scattering kinematics, $^{208}$Pb is absent from the backward angles in both hemispheres.
  • Figure 2: (Color online) Doppler- and efficiency-corrected $\gamma$-ray spectrum measured with GRETINA following the Coulomb excitation of a $^{70}$Ge beam incident on a $^{208}$Pb target. This spectrum shows $\gamma$ rays that were coincident with $^{70}$Ge recoils detected by CHICO2 at laboratory scattering angles between $130^\circ$ and $160^\circ$; it spectrum contains all transitions observed in the experiment. The typical energy resolution obtained ranges from a full width at half maximum (FWHM) of $\approx 3$ keV at 668 keV to $\approx 7$ keV at 2157 keV. The $994$-keV peak (shown in blue and marked with a caret) was excluded from the analysis, as it was not observed in previous works and its placement in the level scheme is uncertain. The $1215$-keV peak (shown in red and marked with an asterisk) was also excluded, as it is believed to correspond to the sum peak resulting from simultaneous detection of $1039.5$-keV ($2^+_1\to0^+_1$) and $176.1$-keV ($0^+_2\to2^+_1$) $\gamma$ rays.
  • Figure 3: (Color online) Partial level scheme of $^{70}$Ge including all the transitions and levels observed in the present study. Transitions in black are those observed in Ref. Sugawara2003 while those in red were known in the literature, but observed in Coulomb excitation for the first time in the present work. Spin, parity, and energy assignments for the observed levels were adopted from the most recent ENSDF evaluation Mukhopadhyay2017, with tentative values enclosed in parentheses and energies rounded to the nearest keV. Note that the 3295-keV state is listed as $3^+,4^+$ in the latest ENSDF evaluation, but taken to be $4^+$ in the present analysis as Coulomb excitation preferentially excites through $E2$ transitions and GOSIA requires a definite spin assignment.
  • Figure 4: Doppler-corrected $\gamma$-ray decay spectra produced from the Coulomb excitation of $^{70}$Ge, shown in five subsets corresponding to different projectile laboratory scattering angles. At backward angles more $\gamma$ rays are seen, particularly from transitions including higher energy and spin states. This qualitatively shows the dependence of the excitation cross section on scattering angle, energy, and spin.
  • Figure 6: (Color online) Comparison between the measured differential $\gamma$-ray yields (black points) and the differential yields calculated using the set of matrix elements obtained from the GOSIA fit (red dashed curve) for several representative transitions. Each point corresponds to the center of the five aforementioned angle partitions, converted into the center of mass (CoM) frame. In this frame, the characteristic shape of the differential yields -- which depends on energy, spin, and parity of the initial and final levels -- can be more distinctly seen.
  • ...and 8 more figures