Statistical correlations of nuclear quadrupole deformations and charge radii

Background: Shape deformations and charge radii, basic properties of atomic nuclei, are influenced by both the global features of the nuclear force and the nucleonic shell structure. As functions of proton and neutron number, both quantities show regular patterns and, for nuclei away from magic numbers, they change very smoothly from nucleus to nucleus.

Purpose: In this paper, we explain how the local shell effects are impacting the statistical correlations between quadrupole deformations and charge radii in well-deformed even-even Er, Yb, and Hf isotopes. This implies, in turn, that sudden changes in correlations can be useful indicators of underlying shell effects.

Methods: Our theoretical analysis is performed in the framework of self-consistent mean-field theory using quantified energy density functionals and density-dependent pairing forces. The statistical analysis is carried out by means of the linear least-square regression.

Results: The local variations of nuclear quadrupole deformations and charge radii, explained in terms of occupations individual deformed Hartree-Fock orbits, make and imprint on statistical correlations of computed observables. While the calculated deformations or charge radii are, in some cases, correlated with those of their even-even neighbors, the correlations seem to deteriorate rapidly with particle number.

Conclusions: The statistical correlations between nuclear deformations and charge radii of different nuclei are affected by the underlying shell structure. Even for well deformed and superfluid nuclei for which these observables change smoothly, the correlation range is rather short. This result suggests that the frequently made assumption of reduced statistical errors for the differences between smoothly-varying observables cannot be generally justified.

Figure 1

Proton quadrupole ground-state deformations ${\beta }_2$ for even-even Er, Yb, and Hf isotopes with $98\le N\le 106$ calculated with (a) SV-min and (b) Fy($\mathrm{\Delta }r$,BCS) EDFs compared to empirical values [24] (dashed lines). Statistical model uncertainties and experimental errors are marked.

Figure 2

Charge radii for even-even Er, Yb, and Hf isotopes with $98\le N\le 106$ calculated with (a) SV-min and (b) Fy($\mathrm{\Delta }r$,BCS) EDFs compared to empirical values [25] (dashed lines). Statistical model uncertainties and experimental errors are marked.

Figure 3

The CoD $R_{{\beta }_2(Z,N),{\beta }_2(Z^{\prime },N^{\prime })}^2$ between the quadrupole deformation of the nucleus indicated and ${\beta }_2$ of other Er, Yb, and Hf isotopes with $98\le N\le 106$ calculated with (left) SV-min and (right) Fy($\mathrm{\Delta }r$,BCS) EDFs.

Figure 4

Similar as in Fig. 3 but for the charge radii.

Figure 5

Proton (top) and neutron (bottom) single-particle energies of ${}^{172}\mathrm{Yb}$ calculated with SV-min (left) and Fy($\mathrm{\Delta }r$,BCS) (right) EDFs as functions of the proton quadrupole deformation. The asymptotic Nilsson labels $\left[N_{\mathrm{osc}}{n}_z\mathrm{\Lambda }\right]{\mathrm{\Omega }}^{\pi }$ are marked. The positions of proton Fermi levels for the $N=102$ isotones is indicated by stars in panel (b) and the neutron Fermi levels along the Yb isotopic chain—in (d).

Figure 6

Proton (solid lines) and neutron (dashed lines) pairing energies (expectation values of the pairing EDF) for even-even Er, Yb, and Hf isotopes with $98\le N\le 106$ calculated with (a) SV-min and (b) Fy($\mathrm{\Delta }r$,BCS) EDFs.