content_image/371442.jpg content_image/371443.jpg content_image/371444.jpg content_image/371445.jpg
the consecutive occupied levels of the N particles (shell gap) 

<img src='content_image/371442.jpg'>

This G value is a more accurate measure of the shell structure of the single particle energies than the ℏ Ω$_{0}$ in Eq.(18). In order to estimate a reasonable γ$_{min}$ value, we have to determine the effective width of the smoothing function with a given p . The effective width corresponds to the central peak of h$_{p}$ ( x ) in the interval x ∈ [ − x ( p ) 1 , x ( p ) 1 ]. Since the effective range of the smoothing function decreases for increasing p , therefore, for larger p value one should use larger γ values for having the same smoothing effect. In order to compensate this effect, it is worthwhile to introduce a renormalized smoothing range as follows: 

in which the p dependence of the smoothing is considerably reduced. 

<img src='content_image/371443.jpg'>

In order to smooth the fluctuations due to the major shells this Γ$_{p}$ range should be larger than the shell gap Γ$_{p}$ > G . To achieve this we introduce a factor F > 1, and calculate a minimal value for the renormalized range Γ$_{p,min}$ = F G . (We observed that the optimal value for the factor F is F = 1 . 5 − 2 for light and F = 2 . 5 − 3 . 5 for heavier nuclei.) Having fixed this minimum we search for the first minimum of δE ( γ, p ) for 

<img src='content_image/371444.jpg'>

This criteria serves as a guide to select the proper minimum of the plateau curve δE ( γ$_{p}$, p ). For most nuclei the plateau curves have multiple minima at γ$_{p,}$$_{1}$ < γ$_{p,}$$_{2}$ <, .., < γ$_{p,l}$ . The number of minima l generally increases when p increases. We observed that for p = 2 we have at most two minima, i.e. l = 1 or l = 2 and one of them satisfies the following condition: 

<img src='content_image/371445.jpg'>

For higher p values the proper minimum should be close to this value since we reduced the p dependence considerably by using the renormalized smoothing range. Therefore, we have to select the k -th minimum, for which Γ$_{p,k}$ = x ( p ) 1 γ$_{p,k}$ ≈ Γ$_{2}$$_{,l}$ . If we select the smoothing range according to this criteria then the variation of the corresponding δE ( γ$_{p,k}$, p ) values will be small. 

