Signatures of shape phase transitions in odd-mass nuclei

Quantum phase transitions between competing ground-state shapes of atomic nuclei with an odd number of protons or neutrons are investigated in a microscopic framework based on nuclear energy density functional theory and the particle-plus-boson-core coupling scheme. The boson-core Hamiltonian, as well as the single-particle energies and occupation probabilities of the unpaired nucleon, are completely determined by constrained self-consistent mean-field calculations for a specific choice of the energy density functional and paring interaction, and only the strength parameters of the particle-core coupling are adjusted to reproduce selected spectroscopic properties of the odd-mass system. We apply this method to odd-$A$ Eu and Sm isotopes with neutron number $N\approx 90$, and explore the influence of the single unpaired fermion on the occurrence of a shape phase transition. Collective wave functions of low-energy states are used to compute quantities that can be related to quantum order parameters: deformations, excitation energies, $E2$ transition rates, and separation energies, and their evolution with the control parameter (neutron number) is analyzed.

Figure 1

Self-consistent RHB triaxial quadrupole binding energy maps of the even-even ${}^{148–154}\mathrm{Sm}$ isotopes in the $\beta -\gamma$ plane ($0\le \gamma \le {60}^{\circ }$). For each nucleus the energy surface is normalized with respect to the binding energy of the absolute minimum.

Figure 2

The lowest three negative- and positive-parity bands of ${}^{149}\mathrm{Sm}$. Available data are compared to model calculation in which states are classified in bands according to dominant $E2$ transitions. The data are from Ref. [47], and parentheses denote states with only a tentative assignment of spin and parity.

Figure 3

Same as in the caption to Fig. 2, but for ${}^{151}\mathrm{Sm}$.

Figure 4

Same as in the caption to Fig. 2, but for ${}^{153}\mathrm{Sm}$.

Figure 5

Same as in the caption to Fig. 2, but for ${}^{155}\mathrm{Sm}$.

Figure 6

Evolution of excitation energies of low-lying [(a) and (b)] positive- ($\pi =+1$) and [(c) and (d)] negative-parity ($\pi =-1$) yrast states as functions of neutron number in the isotopes ${}^{147–155}\mathrm{Eu}$, in comparison with available data taken from Ref. [47].

Figure 7

Same as in the caption to Fig. 6, but for the isotopes ${}^{147–155}\mathrm{Sm}$.

Figure 8

Evolution of characteristic mean-field quantities calculated for the lowest three positive- ($\pi =+1$) and negative-parity ($\pi =-1$) bands of the odd-$A$ Eu and Sm nuclei, and $K^{\pi }=0^{+}$ bands in the even-$A$ Sm isotopes, as functions of mass number: the mean value $\overline{\beta }$ (a) and variance $\mathrm{\Delta }\beta$ (b) of the equilibrium deformation parameter for the bandhead state. “band 1,” “band 2,” and “band 3” denote the lowest, second-lowest, and third-lowest bands, respectively. See the main text for the definition of each quantity.

Figure 9

Left: calculated values of $\overline{B\left(E2\right)}$ for the two lowest $0^{+}$ states (a), and the difference $\overline{B\left(E2\right)}\left(0_2^{+}\right)-\overline{B\left(E2\right)}\left(0_1^{+}\right)$ (b) for the even-even Sm isotopes. Right: the excitation energies $E(2^{+},0_k^{+})$ ($k=1,2$) of the states $2^{+}$ (c) and the energy ratio of the $4^{+}$ to the $2^{+}$ excited states $R(4^{+},2^{+},0_k^{+})$ (d) for the $K^{\pi }=0_1^{+}$ and $0_2^{+}$ bands of the even-even core nuclei ${}^{146–154}\mathrm{Sm}$. In panel (d) the corresponding values of the energy ratio $E\left(4_1^{+}\right)/E\left(2_1^{+}\right)$ in the U(5), X(5), and SU(3) limits: 2.00, 2.91 ($k=1$), 2.79 ($k=2$), and 3.33, respectively, are also indicated by dotted lines.

Figure 10

Dependence on mass number of the calculated spectroscopic quantities for the odd-$A$ Eu and Sm isotopes: average $B(E2;J_0+\mathrm{\Delta }J\to J_0)$ (in units of $e^2{\mathrm{b}}^2$) for transitions between the bandhead $J_0$ of a given band and the lowest five states with $J_0+\mathrm{\Delta }J$, with $\mathrm{\Delta }J=1$ [(a) and (b)] and 2 [(c) and (d)]; the excitation energy $E(J_1,J_0)$ ($J_1=J_0+\mathrm{\Delta }J$, in MeV) of the second-lowest state in a band relative to the bandhead [(e) and (f)], and the energy ratio $R(J_2,J_1,J_0)$ with $J_2=J_0+2\mathrm{\Delta }J$ [(g) and (h)]. See the main text for the definition of each quantity.

Figure 11

Differentials of the mean value $\delta \overline{\beta }$ [(a-1) and (a-2)] and variance $\delta \mathrm{\Delta }\beta$ [(b-1) and (b-2)] of the quadrupole deformation parameter $\beta , q$ invariants $\overline{B\left(E2\right)}$ with $\mathrm{\Delta }J=1$ [(c-1) and (c-2)], and $\mathrm{\Delta }J=2$ [(d-1) and (d-2)], excitation energy $\delta E(J_1,J_0)$ [(e-1) and (e-2)], and the energy ratio $\delta R(J_2,J_1,J_0)$, for the odd-$A$ Eu and Sm isotopes, as functions of the mass number.

Figure 12

Same as in the caption to Fig. 11, but for the even-even Sm core nuclei.

Figure 13

Proton (a) and neutron (b) separation energies ($s_p$ and $s_n$), and their differentials $\delta s_p$ (c) and $\delta s_n$ (d), for the odd-$A$ Eu and Sm isotopes, respectively.