Uniform neighbour-shell spacing $(l=0)$

Consider the ensemble average of the $s-$ASDOS (10) if only one $s-$scattering event in the environment is included. With $l=l_1=l_2=0$ the average reflection amplitude (12) reduces to

$$\overline{T_0} = e^{i2\delta_0} \left( \frac {2\pi N_0 f_... ...nt {\rm d}r\; rG(r) \; \left(h_0^{(1)}(k_Fr))\right)^2\right).$$ (13)

The first term in the brackets corresponds to a point source inside of an infinitesimal vacuum sphere which is embedded in a homogeneous medium ( $n_F = 1 + 2\pi N_0 f_0/k_F^2$, Fermi index of refraction). The source emits a unit $s-$wave and receives the inward-reflected amplitude $n_F-1$. This term is not important in the present context, because, it does not depend on $G(r)$. The really interesting term is the second one. Its real part,

$$\sin(\delta_0) \int {\rm d}r\; \frac{G(r)}{rk_F^2} \; \sin(2k_Fr+3\delta_0),$$ (14)

indicates that only asymptotic information on the neighbour-shell sequence is fed back to the $s-$ASDOS. However, in contrast to the Fraunhofer diffraction (7) the correlation shell is now represented by $G(r)/r = 4\pi N_0 (g(r)-1)$ instead of the reduced pair correlation function $G(r)$. Note also that the present ASDOS-based treatment provides immediately the sine function as the correct standard to be contrasted with the shell structure. The $s-$ASDOS will be efficiently reduced at $E_F$ if the neighbour-shell radii tend to coincide with maxima (sgn$\delta_0=-1$) or with minima (sgn$\delta_0=+1$) of $\sin(2k_Fr+3\delta_0)$. Low $s-$ASDOS due to $s-$scattering in the environment requires the shell sequence

$$R_n = \frac{2\pi}{2k_F} \left(\frac{6+sgn\delta_0}{4} -\frac{3\delta_0}{2\pi}+(n-1)\right).$$ (15)

The same result is obtained on moving (8) as a rigid entity towards larger distances. Hence, stabilization which is confined to the angular momentum $l=0$ will never generate deviations from the uniform shell spacing.