\qquad (10)$

the spatial distribution of the wave function $\phi _{l0}$ depends on $l$. The smaller the $l$, the smaller the size.  Thus, for the $\psi _{\alpha }$ belonging to the LLL, $l_{in}$ is  just equal to $l_{1}+l_{2}+\cdot \cdot
\cdot +l_{{N}_{in}}$ .  If $l_{1}=0$,  there must be an electron staying at the center, because the $\phi _{00}$ wave function is distributed closely surrounding the center.  Thus, a (N$_{in})_{c}$-N$_{out} $ structure must be contributed by the $\psi _{\alpha }$ with $l_{1}=0$ ,  while a (N$%
_{in})_{h}$-N$_{out}$ structure is contributed by those with $l_{1}>0$ . When all the $l_{i}$ of the inner electrons satisfies  $l_{i}+1=l_{i+1}$ , the inner electrons are said to be compactly aligned. Meanwhile, $l_{in}$ would arrive at its lower bound $(l_{in})_{b}=$N$_{in}($N$_{in}-1)$ $/2$ , if $l_{1}=0$. In this case, we say that the core is inert (not excited). Otherwise, we have $l_{in}>$ $(l_{in})_{b}$, and we say that the core is excited. Evidently, all the hollow states must have $l_{1}>0$, thus they have an excited core.

When $L$ $\leq 90$, core excitation is not possible (unless the electrons jump to higher LLL), therefore the first-states would have a (N$%
_{in})_{c}-$N$_{out}$ structure with the core inert. However, when $L>90$, core excitation might 