them is a two-bunched BF belonging to the LLL.  Thus, a simple way to find out the CFS is to look for the two-bunched BFs of a state.

\(v) [**Particle separation**]{}

It is noted that the $U$ in eq.(1.3) can be exactly rewritten as

$U=\frac{1}{2}M\omega _{o}^{2}R_{c}^{2}+\sum\limits_{j<k}
u_{eff}(r_{jk})\qquad \qquad \qquad (11.1)$

Where $R_{c}$ is the radial distance of the c.m., and $u_{eff}(r_{jk})$ is the effective pairwise interaction

$u_{eff}(r_{jk})=\frac{(m\ast \omega _{o})^{2}}{2M}r_{jk}^{2}+\frac{%
e^{2}}{4\pi \varepsilon _{r}\varepsilon _{0}}\cdot \frac{1}{r_{jk}}\qquad
\qquad \qquad (11.2)$

There is a minimum in $u_{eff}$ located at $r_{jk}=r_{u}=(\frac{e^{2}M}{4\pi
\varepsilon _{r}\varepsilon _{0}(m\ast \omega _{o})^{2}})^{1/3}$ . Evidently, if each electron separates from all its adjacent electrons by this distance, the potential energy can be minimized.  Therefore, in low-lying states,  adjacent electrons would roughly keep the separation $%
r_{u}$ .  With the above parameters, $r_{u}=576.3\stackrel{\circ }{A}$ .The $r_{u}$ is a basic measure and is useful for the understanding of electronic correlation and the size of the system.  Obviously, for a (N$%
_{in})-$N$_{out}$ structure, the distance between the ring and the core depends closely on $r_{u}$ .

\(vi) [**Core-ring separation**]{}

It is recalled that, in order to minimize the potential energy, the ring should separate from the core by an appropriate distance. In this subsection, we 