Carbon-oxygen-neon mass nuclei in superstrong magnetic fields

The properties of ${}^{12}\mathrm{C},{}^{16}\mathrm{O}$, and ${}^{20}\mathrm{Ne}$ nuclei in strong magnetic fields $B\simeq {10}^{17}$ G are studied in the context of strongly magnetized neutron stars and white dwarfs. The sky3d code is extended to incorporate the interaction of nucleons with the magnetic field and is utilized to solve the time-independent Hartree-Fock equations with a Skyrme interaction on a Cartesian three-dimensional grid. The numerical solutions demonstrate a number of phenomena, which include a splitting of the energy levels of spin-up and -down nucleons, spontaneous rearrangement of energy levels in ${}^{16}\mathrm{O}$ at a critical field, which leads to jump-like increases of magnetization and proton current in this nucleus, and evolution of the intrinsically deformed ${}^{20}\mathrm{Ne}$ nucleus toward a more spherical shape under increasing field strength. Many of the numerical features can be understood within a simple analytical model based on the occupation by the nucleons of the lowest states of the harmonic oscillator in a magnetic field.

Figure 1

Dependence of proton energy levels of ${}^{16}\mathrm{O}$ on magnetic field (left panels) and the projections of the angular momentum and spin on the $z$ axis. The lower panels show the $s$ states, the upper ones the $p$ states and a $d$ state for large $B$ fields. A rearrangement of the energy levels is seen in the upper left panel at the magnetic field $B\ge 4\times {10}^{17}$ G. This is related to the rearrangement of the proton levels as discussed in the text.

Figure 2

Same as Fig. 1, but for neutrons.

Figure 3

Same as Fig. 1, but for protons in ${}^{12}\mathrm{C}$. No rearrangement of levels is observed for this nucleus.

Figure 4

Same as Fig. 3, but for neutrons.

Figure 5

Velocity distribution for neutrons (upper panel) and protons (middle and bottom panels) in ${}^{16}\mathrm{O}$. The upper and middle panels have the same color coding show on the left and correspond to magnetic field $B=3.9\times {10}^{17}$ G. The bottom panel with its color coding corresponds to magnetic field $B=4.1\times {10}^{17}\mathrm{G}>B_c\left[{}^{16}\mathrm{O}\right]$. The background shows the density distribution with maximum $0.155 {\mathrm{fm}}^{-3}$ at the center (yellow color) to the boundary where the density drops to zero (dark blue). The velocity is shown in units of speed of light.

Figure 6

Same as Fig. 5, but for protons in ${}^{20}\mathrm{Ne}$ and for fields $B=4.0\times {10}^{13}$ G (top panel) and $B=4.9\times {10}^{17}$ G (bottom panel).

Figure 7

The ratio of the spin density to the particle density for neutrons (upper panel) and protons (middle and bottom panel) for ${}^{16}\mathrm{O}$ nucleus. The magnetic field $z$ axis is directed from bottom to top. The upper and middle panel have the same color coding shown on the left and correspond to magnetic field $B=3.9\times {10}^{17}$ G. The bottom panel with its color coding corresponds to magnetic field $B=4.1\times {10}^{17}$ G.

Figure 8

Same as Fig. 7, but for protons in ${}^{20}\mathrm{Ne}$ and for fields $B=4.0\times {10}^{13}$ G (top panel) and $B=4.9\times {10}^{17}$ G (bottom panel).