The table on the next slide illustrates how single-particle energies fill energy shells in a two-dimensional neutron box. Here \( n_{x} \) and \( n_{y} \) are the momentum quantum numbers, \( n_{x}^{2} + n_{y}^{2} \) determines the single-particle energy level, \( N_{\uparrow \downarrow } \) represents the cumulated number of spin-orbitals in an unpolarized spin phase, and \( N_{\uparrow \uparrow } \) stands for the cumulated number of spin-orbitals in a spin-polarized system.