Large-scale shell-model calculations for unnatural-parity high-spin states in neutron-rich Cr and Fe isotopes

We investigate unnatural-parity high-spin states in neutron-rich Cr and Fe isotopes using large-scale shell-model calculations. These shell-model calculations are carried out within the model space of $fp\text{-shell}+0g_{9/2}+1d_{5/2}$ orbits with the truncation allowing $1\hslash \omega$ excitation of a neutron. The effective Hamiltonian consists of GXPF1Br for $fp$-shell orbits and $V_{\mathrm{MU}}$ with a modification for the other parts. The present shell-model calculations can describe and predict the energy levels of both natural- and unnatural-parity states up to the high-spin states in Cr and Fe isotopes with $N\le 35$. The total energy surfaces present prolate deformations on the whole and indicate that the excitation of one neutron into the $0g_{9/2}$ orbit plays the role of enhancing prolate deformation. For positive (unnatural)-parity states in odd-mass Cr and Fe isotopes, their energy levels and prolate deformations indicate the decoupling limit of the particle-plus-rotor model. The sharp drop of the $9/2_1^{+}$ levels in going from $N=29$ to $N=35$ in odd-mass Cr, Fe, and Ni isotopes is explained by the Fermi surface approaching the $\nu 0g_{9/2}$ orbit.

Figure 1

Shell-model results and experimental data on the energy levels of ${}^{57}\mathrm{Cr}$. Experimental data, labeled “${}^{57}\mathrm{Cr}(\exp$),” are taken from [28].

Figure 2

Calculated and experimental energy levels of the lowest states for each spin and parity in odd-mass Cr and Fe isotopes. In each plot, the left and right panels present the calculated and experimental energy levels, respectively. Experimental data are taken from [28, 29, 30, 31, 32, 47, 48, 49, 51].

Figure 3

Calculated and experimental energy levels of the lowest states for each spin and parity in even-mass Cr and Fe isotopes. The notation is the same as in Fig. 2. Experimental data are taken from [30, 51, 52, 53, 54, 55]. In ${}^{58}\mathrm{Fe},$ experimental negative-parity states, except for the $3_1^{-}$ and $4_1^{-}$ states, are taken from bands 2 and 3 in Ref. [52], where these states are discussed based on the projected shell-model calculation [56] interpreting them as negative-parity states.

Figure 4

Comparison of the level spacing of the yrast bands in ${}^{56,58}\mathrm{Cr}$ versus the bands built on $9/2_1^{+}$ in ${}^{57,59}\mathrm{Cr}$.

Figure 5

Total energy surfaces for the negative (natural)- and positive (unnatural)-parity states in odd-mass nuclei. Energy minima are represented by white circles. Energy contours are based on the energy minima starting from 0 MeV.

Figure 6

Total energy surfaces for positive(natural)- and negative(unnatural)-parity states in even-mass nuclei. Notation is the same as in Fig. 5.

Figure 7

Electric spectroscopic and intrinsic quadrupole moments for members of the band built on $9/2_1^{+}$ in odd-mass Cr and Fe nuclei. Horizontal axes denote the spin $I$ values of the states as $2I$. Upper panels show the electric spectroscopic quadrupole moments $Q_S.$ Middle and lower panels show the intrinsic quadrupole moments $Q_0$ solved by Eq. (3) assuming $K=1/2$ and the absolute values of $Q_0$ solved by Eq. (4), respectively. The experimental spectroscopic quadrupole moment $Q_S$ of ${}^{61}\mathrm{Fe}$ [36] labeled “${}^{61}\mathrm{Fe}(\exp$)” is assumed to be negative. The absolute value of $Q_0$ labeled “${}^{59}\mathrm{Fe}(\exp$)” is evaluated with the experimental $B\left(E2\right)$ values [49] assuming $K=1/2$.

Figure 8

Excitation energy of $9/2_1^{+}$ in odd-mass Cr, Fe, and Ni isotopes from $N=29$ to $N=35$. Plots labeled “cal” and “exp” represent the present results and the experimental data, respectively. Plots labeled “cal not modified” show the results without modification of the two-body matrix elements described in Sec. 2b. Experimental data are taken from [27, 28, 29, 30, 31, 47, 48, 49, 62, 63, 64]. In ${}^{55}\mathrm{Fe},$ the state of $3.86$-MeV excitation energy in Ref. [47] is adopted as the experimental $9/2^{+}$ state because it has the largest spectroscopic factor of $9/2^{+}$ in the $(d,p)$ reaction.

Figure 9

Effective single-particle energies (ESPEs) of a neutron for Cr and Fe isotopes. Open circles represent ESPEs calculated as the hole states to specify the occupied states.

Figure 10

Plots of $-S_n+E_x(9/2_1^{+})$. Experimental one-neutron separation energies $S_n$ are taken from [47, 48, 49, 62, 63, 64]. The Coulomb energy is corrected in the calculated $S_n$ according to [9].