Rotation and alignment of high-$j$ orbitals in transfermium nuclei

The structure of nuclei with $Z\sim100$ is investigated systematically by the Cranked Shell Model (CSM) with pairing correlations treated by a Particle-Number Conserving (PNC) method. In the PNC method, the particle number is conserved and the Pauli blocking effects are taken into account exactly. By fitting the experimental single-particle spectra in these nuclei, a new set of Nilsson parameters ($\kappa$ and $\mu$) is proposed. The experimental kinematic moments of inertia and the band-head energies are reproduced quite well by the PNC-CSM calculations. The band crossing, the effects of high-$j$ intruder orbitals and deformation are discussed in detail.


Introduction
The exploration of the island of stability with high mass and charge, i.e., the region of superheavy elements (SHE), has been one of the fundamental questions in natural science. Great experimental progress has been made in synthesizing the superheavy elements. Up to now, superheavy elements with Z ≤ 118 have been synthesized via cold and hot fusion reactions [1][2][3]. However, due to the extremely low production cross-sections, these experiments can rarely reveal the detailed spectroscopic information. One indirect way is to study lighter nuclei in the deformed region with Z ≈ 100 and N ≈ 152, which are the heaviest systems accessible in present in-beam experiments (see Refs. [4][5][6] and references therein). The strongly downsloping orbitals originating from the spherical subshells active in the vicinity of the predicted shell closures come close to the Fermi surface of transfermium nuclei due to deformation effect. The rotational properties of transfermium nuclei will be strongly affected by these spherical orbitals. The proton 1/2 − [521] orbital is of particular interest since it stems from the spherical 2 f 5/2 orbital. The spin−orbit interaction strength of 2 f 5/2 − 2 f 7/2 partner governs the size of the possible Z = 114 shell gap. The Cranked Shell Model (CSM) with the pairing correlations treated by a Particle-Number Conserving (PNC) method [7,8] is used to study the rotational and single-particle properties of Z ∼ 100 nuclei. a e-mail: hext@nuaa.edu.cn EPJ Web of Conferences

Theoretical framework
The Cranked Shell Model Hamiltonian of an axially symmetric nucleus in the rotating frame is expressed as: where h Nil is the Nilsson Hamiltonian [11], −ω j x is the Coriolis force with the cranking frequency ω about the x axis (perpendicular to the nuclear symmetry z axis). H P is the pairing including monopole and quadrupole pairing correlations, with ξ and η being the time-reversal states of a Nilsson state ξ and η, respectively. The quantity q 2 (ξ) = √ 16π/5 ξ| r 2 Y 20 |ξ is the diagonal element of the stretched quadrupole operator, and G 0 and G 2 are the effective strengths of monopole and quadrupole pairing interactions, respectively.
In our calculation, h 0 (ω) = h Nil − ω j x is diagonalized firstly to obtain the cranked Nilsson orbitals. Then, H CSM is diagonalized in a sufficiently large Cranked Many-Particle Configuration (CMPC) space to obtain the yrast and low-lying eigenstates. Instead of the usual single-particle level truncation in common shell-model calculations, a cranked many-particle configuration truncation (Fock space truncation) is adopted which is crucial to make the PNC calculations for low-lying excited states both workable and sufficiently accurate [9,10] . The eigenstate of H CSM is expressed as: where |i is a cranked many-particle configuration (an occupation of particles in the cranked Nilsson orbitals) and C i is the corresponding probability amplitude. The angular momentum alignment J x of the state |ψ is given by: The kinematic moment of inertia is J (1) = ψ| J x |ψ /ω.

Results and discussions
The Nilsson parameters (κ, µ) proposed in Refs. [11,12] cannot well describe the experimental level schemes of transfermium nuclei while it is optimized to reproduce the experimental level schemes for the rare-earth and actinide nuclei. By fitting the experimental single-particle levels in the odd-A nuclei with Z ≈ 100, we obtained a new set of Nilsson parameters κ and µ (see Table 1) which are dependent on the main oscillator quantum number N as well as on the orbital angular momentum l [13,14]. The CMPC space in the work of Ref. [13] is constructed in the proton N = 4, 5, 6 shells and the neutron N = 6, 7 shells. The dimensions of the CMPC space for the nuclei with Z ≈ 100 are about 1000 both for protons and neutrons. Figure 1 gives the experimental and calculated moments of inertia INPC 2013 Table 1. Nilsson parameters κ and µ proposed for the nuclei with Z ≈ 100. Taken from Ref. [13,14]. of excited 1-qp bands in the odd-Z Bk, Es, and Md isotopes (taken from Ref. [13]). The data are well reproduced by the PNC calculations. Only one signature band was observed in 251 Md. We calculated J (1) 's for two signature partner bands which vary smoothly with frequency in 251 Md.
In order to investigate the effect of the proton N = 7 shell on the rotational properties of the transfermium nuclei, the proton N = 7 shell is included to construct the CMPC space [15]. We find that the 1/2 − [770] orbital plays an important role in the rotational properties of 251 Md.  [16], we calculate 251 Md for ε 2 = 0.28 and 0.255 with and without the proton N = 7 shell, respectively. There is no signature splitting when the proton N = 7 shell is not included whether we take ε 2 = 0.28 or ε 2 = 0.255. The signature splitting occur at ω ≈ 0.225 ( ω ≈ 0.275) for ε 2 = 0.28 (ε 2 = 0.255) when the effect of the proton N = 7 shell is considered [17]. 521](α = +1/2) band is denoted by solid circles. Solid and dashed lines are used for the calculated α = +1/2 and α = −1/2 bands, respectively. Taken from Ref. [15].