Nilsson diagrams

The Nilsson diagrams, shown in Figs. 1-3, have been obtained by diagonalizing the self-consistent mean-field Hamiltonians corresponding to the states that were constraint to a sequence of values of the axial mass quadrupole moment [60], $Q=\langle{2z^2-x^2-y^2}\rangle$. Then, by using a simple phenomenological formula [61],

$$\beta_2 \equiv \frac{4\pi}{3R^2 A}\sqrt{\frac{5}{16\pi}}Q \simeq Q\times0.009\,\mbox{b}^{-1},$$ (5)

values of the average quadrupole moments were translated into values of the Bohr parameters $\beta_2$. The numerical factor of $0.009\,\mbox{b}^{-1}$ corresponds to $R=1.2$fm$A^{1/3}$ and $A=254$. Note that the Nilsson diagrams obtained in this way constitute only an illustration of single-particle properties of the nuclei in the nobelium region. Indeed, going in either direction in the chart of nuclei, the relative distances between spherical and deformed levels change, closing some of the gaps visible in Figs. 1-3 while opening others. Examples for such evolution of the shell structure of spherical states and the deformed minima can be found in Refs. [43,62,63,64]. Small, but clearly visible changes of the Nilsson diagrams can already be spotted when just adding a few neutrons or protons, compare for example the Nilsson diagram obtained with SLy4 for $^{254}$ No, Fig. 1, with the one for $^{250}$ Fm presented in Refs [43], or the Nilsson diagrams obtained with NL1 and NL3* for $^{254}$ No, Fig. 3, with those for $^{244}$ Cm presented in Ref. [20].

Figure 1: Proton (left panels) and neutron (right panels) Nilsson diagrams of $^{254}$ No obtained for the Skyrme EDF SLy4 (upper panels) and UNEDF2 (lower panels). At spherical shapes, the orbitals are labelled with spherical quantum numbers. For SLy4, at large deformations, the deformed single-particle orbitals are labelled by the expectation values $\langle \hat{j}_\parallel \rangle$ of the projection of the angular momentum on the axial-symmetry axis. For UNEDF2, these orbitals are labelled by the Nilsson labels $\Omega [Nn_z\Lambda ]$ determined using code HFODD. Solid and dashed lines are used for the positive and negative parity states, respectively.

FigSHE01.eps

Figure 2: Same as in Fig. 1 but for the Gogny EDF D1S and D1M. All results and dominant Nilsson labels were determined using the axial-symmetry code HFBAXIAL.

FigSHE02.eps

Figure 3: Same as in Fig. 1 but for the covariant EDFs NL1 and NL3*. All results and dominant Nilsson labels were determined using axially symmetric code RHB.

FigSHE03a.eps FigSHE03b.eps

FigSHE03c.eps FigSHE03d.eps

As we can see, for all considered EDFs, the overall positions and deformation-dependence of single-particle levels is fairly similar. In particular, deformed shell gaps, which appear near ground-state deformations of $\beta_2=0.3$, occur at particle numbers of $Z=98$, $Z=104$ and $N=150$ and/or $N=152$ in the majority of the functionals. The only exception is the functional NL3*, which is characterized by additional gaps at $Z=102$ and $N=148$. Another deformed gap at $\beta_2=0.2$ is observed for $Z=110$. Moreover, significant differences in important details are also visible. The proton deformed shell gaps appear consistently above the one at $Z=100$ that is tentatively inferred from the experimental data, see discussion in Refs. [8,21].

Interestingly, although different EDFs show similar deformed neutron shell closures, we observe dramatic differences in the shell structure at sphericity between the Skyrme EDF UNEDF2 and the other ones, as shown in Fig. 1. Compared to SLy4, spherical orbital $1j_{15/2^-}$ ($1i_{11/2^+}$) is lowered (raised) by about 2MeV, and thus their relative positions are inverted, resulting in the spherical shell gaps at $N=152$ and 170, whereas other EDFs predict shell gaps at $N=164$.

The strong rearrangement of spherical neutron shells observed for UNEDF2 as compared to all other EDFs is a consequence of its rather large $C^{JJ}_0$ and $C^{JJ}_1$ coupling constants of the spin-current tensor terms of this parameterization [27]. In the $N \approx 152$ region, the inversion of the spherical level sequence substantially increases the number of filled spherical shells for which the spin-orbit partner is empty, thereby increasing the size of the spin-current terms. In fact, such behaviour is often found at mid-shell for parameterizations with large attractive tensor terms [65,66].

The relativistic NL1 and NL3* functionals have the unique feature that they predict a large spherical $N=138$ gap of about 3MeV that is absent in all non-relativistic calculations. As the sequence of spherical subshells is different, for NL1 this gap is located between the $1i_{11/2^+}$ and $2g_{9/2^+}$ levels, whereas for NL3* it is found between the $1i_{11/2^+}$ and $1j_{15/2^-}$ levels.