Electromagnetic observables are able to give insight into collective and emergent features in nuclei, including nuclear clustering. These observables also provide strong constraints for ab initio theory, but comparison of these observables between th
eory and experiment can be difficult due to the lack of convergence for relevant calculated values, such as $E2$ transition strengths. By comparing the ratios of $E2$ transition strengths for mirror transitions, we find that a wide range of ab initio calculations give robust and consistent predictions for this ratio. To experimentally test the validity of these ab initio predictions, we performed a Coulomb excitation experiment to measure the $B(E2; 3/2^- rightarrow 1/2^-)$ transition strength in $^7$Be for the first time. A $B(E2; 3/2^- rightarrow 1/2^-)$ value of $26(6)(3) , e^2 mathrm{fm}^4$ was deduced from the measured Coulomb excitation cross section. This result is used with the experimentally known $^7$Li $B(E2; 3/2^- rightarrow 1/2^-)$ value to provide an experimental ratio to compare with the ab initio predictions. Our experimental value is consistent with the theoretical ratios within $1 sigma$ uncertainty, giving experimental support for the value of these ratios. Further work in both theory and experiment can give insight into the robustness of these ratios and their physical meaning.
The large reported $E2$ strength between the $2^+$ ground state and $1^+$ first excited state of $^8$Li, $B(E2; 2^+ rightarrow 1^+)= 55(15)$ e$^2$fm$^4$, presents a puzzle. Unlike in neighboring $A=7-9$ isotopes, where enhanced $E2$ strengths may be
understood to arise from deformation as rotational in-band transitions, the $2^+rightarrow1^+$ transition in $^8$Li cannot be understood in any simple way as a rotational in-band transition. Moreover, the reported strength exceeds textit{ab initio} predictions by an order of magnitude. In light of this discrepancy, we revisited the Coulomb excitation measurement of this strength, now using particle-$gamma$ coincidences, yielding a revised $B(E2; 2^+ rightarrow 1^+)$ of $25(8)(3)$ e$^2$fm$^4$. We explore how this value compares to what might be expected in rotational, Elliott SU(3), and textit{ab initio} descriptions, including no-core shell model (NCSM) calculations with various internucleon interactions. While the present value is a factor of $2$ smaller than previously reported, it remains anomalously enhanced.
Electric quadrupole (E2) matrix elements provide a measure of nuclear deformation and related collective structure. Ground-state quadrupole moments in particular are known to high precision in many p-shell nuclei. While the experimental electric quad
rupole moment only measures the proton distribution, both proton and neutron quadrupole moments are needed to probe proton-neutron asymmetry in the nuclear deformation. We seek insight into the relation between these moments through the ab initio no-core configuration interaction (NCCI), or no-core shell model (NCSM), approach. Converged ab initio calculations for quadrupole moments are particularly challenging, due to sensitivity to long-range behavior of the wave functions. We therefore study more robustly-converged ratios of quadrupole moments: across mirror nuclides, or of proton and neutron quadrupole moments within the same nuclide. In calculations for mirror pairs in the p-shell, we explore how well the predictions for mirror quadrupole moments agree with experiment and how well isospin (mirror) symmetry holds for quadrupole moments across a mirror pair.
Ab initio nuclear theory provides not only a microscopic framework for quantitative description of the nuclear many-body system, but also a foundation for deeper understanding of emergent collective correlations. A symplectic Sp(3,R)$supset$U(3) dyna
mical symmetry is identified in ab initio predictions, from a no-core configuration interaction approach, and found to provide a qualitative understanding of the spectrum of 7Be. Low-lying states form an Elliott SU(3) spectrum, while an Sp(3,R) excitation gives rise to an excited rotational band with strong quadrupole connections to the ground state band.
The need to enforce fermionic antisymmetry in the nuclear many-body problem commonly requires use of single-particle coordinates, defined relative to some fixed origin. To obtain physical operators which nonetheless act on the nuclear many-body syste
m in a Galilean-invariant fashion, thereby avoiding spurious center-of-mass contributions to observables, it is necessary to express these operators with respect to the translational intrinsic frame. Several commonly-encountered operators in nuclear many-body calculations, including the magnetic dipole and electric quadrupole operators (in the impulse approximation) and generators of SU(3) and Sp(3,R) symmetry groups, are bilinear in the coordinates and momenta of the nucleons and, when expressed in intrinsic form, become two-body operators. To work with such operators in a second-quantized many-body calculation, it is necessary to relate three distinct forms: the defining intrinsic-frame expression, an explicitly two-body expression in terms of two-particle relative coordinates, and a decomposition into one-body and separable two-body parts. We establish the relations between these forms, for general (non-scalar and non-isoscalar) operators bilinear in coordinates and momenta.
Ab initio theory describes nuclei from a fully microscopic formulation, with no presupposition of collective degrees of freedom, yet signatures of clustering and rotation nonetheless arise. We can therefore look to ab initio theory for an understandi
ng of the nature of these emergent phenomena. To probe the nature of rotation in 10Be, we examine the predicted rotational spectroscopy from no-core configuration interaction (NCCI) calculations with the Daejeon16 internucleon interaction, and find spectra suggestive of coexisting rotational structures having qualitatively different intrinsic deformations: one triaxial and the other with large axial deformation arising primarily from the neutrons.
Structural phenomena in nuclei, from shell structure and clustering to superfluidity and collective rotations and vibrations, reflect emergent degrees of freedom. Ab initio theory describes nuclei directly from a fully microscopic formulation. We can
therefore look to ab initio theory as a means of exploring the emergence of effective degrees of freedom in nuclei. For the illustrative case of emergent rotational bands in the Be isotopes, we establish an understanding of the underlying oscillator space and angular momentum (orbital and spin) structure. We consider no-core configuration interaction (NCCI) calculations for 7,9,11Be with the Daejeon16 internucleon interaction. Although shell model or rotational degrees of freedom are not assumed in the ab initio theory, the NCCI results are suggestive of the emergence of effective shell model degrees of freedom (0 hbar-omega and 2 hbar-omega excitations) and LS-scheme rotational degrees of freedom, consistent with an Elliott-Wilsdon SU(3) description. These results provide some basic insight into the connection between emergent effective collective rotational and shell model degrees of freedom in these light nuclei and the underlying ab initio microscopic description.
No-core configuration interaction (NCCI) calculations for p-shell nuclei give rise to rotational bands, identified by strong intraband E2 transitions and by rotational patterns for excitation energies, electromagnetic moments, and electromagnetic tra
nsitions. However, convergence rates differ significantly for different rotational observables and for different rotational bands. The choice of internucleon interaction may also substantially impact the convergence rates. Consequently, there is a substantial gap between simply observing the qualitative emergence of rotation in ab initio calculations and actually carrying out detailed quantitative comparisons. In this contribution, we illustrate the convergence properties of rotational band energy parameters extracted from NCCI calculations, and compare these predictions with experiment, for the isotopes 7-11Be, and for the JISP16 and Daejeon16 interactions.
The nuclear collective models introduced by Bohr, Mottelson and Rainwater, together with the Mayer-Jensen shell model, have provided the central framework for the development of nuclear physics. This paper reviews the microscopic evolution of the col
lective models and their underlying foundations. In particular, it is shown that the Bohr-Mottelson models have expressions as macroscopic limits of microscopic models that have precisely-defined expressions in many-nucleon quantum mechanics. Understanding collective models in this way is especially useful because it enables the analysis of nuclear properties in terms of them to be revisited and reassessed in the light of their microscopic foundations.
The Laguerre functions constitute one of the fundamental basis sets for calculations in atomic and molecular electron-structure theory, with applications in hadronic and nuclear theory as well. While similar in form to the Coulomb bound-state eigenfu
nctions (from the Schroedinger eigenproblem) or the Coulomb-Sturmian functions (from a related Sturm-Liouville problem), the Laguerre functions, unlike these former functions, constitute a complete, discrete, orthonormal set for square-integrable functions in three dimensions. We construct the SU(1,1)xSO(3) dynamical algebra for the Laguerre functions and apply the ideas of factorization (or supersymmetric quantum mechanics) to derive shift operators for these functions. We use the resulting algebraic framework to derive analytic expressions for matrix elements of several basic radial operators (involving powers of the radial coordinate and radial derivative) in the Laguerre function basis. We illustrate how matrix elements for more general spherical tensor operators in three dimensional space, such as the gradient, may then be constructed from these radial matrix elements.
