Non-uniform neighbour-shell spacing $(l>0)$

In this section both the $s-$ and the $p-$scattering in the environment are considered to act on both the $s-$ and the $p-$ASDOS. Again, we start from the single-scattering approximation (12) to the ${\rm Re}\overline{T_l}$ and obtain

$${\rm Re}\overline{T_l}=\int{\rm d}r\frac{4\pi N_0}{k_F^2}\,g(r)\, S_l(k_Fr).$$ (16)

The standards,

$$S_0(k_Fr) = \sin(\delta_0)S_{00}(k_Fr)+\sin(\delta_1)S_{01}(k_Fr),$$ (17)

$$S_1(k_Fr) = \sin(\delta_1)S_{11}(k_Fr)+\sin(\delta_0)S_{10}(k_Fr),$$

arise from scattering-path loops with definite sequences of the angular-momentum quantum number $l$: $0-0-0$ in $S_{00}$, $1-1-1$ in $S_{11}$, $0-1-0$ in $S_{01}$ and $1-0-1$ in $S_{10}$. Note that $S_{01}$ and $S_{10}$ are composed of just the elements which produce hybridization effects in the higher-order terms of the multiple-scattering series. In detail we obtain

$$S_{00}(k_Fr) = \sin(2k_Fr+3\delta_0),$$ (18)

$$S_{01}(k_Fr) = -\left(1-\frac{1}{(k_Fr)^2}\right)\sin(2k_Fr+2\delta_0+\delta_1) -\frac{2}{k_Fr}\cos(2k_Fr+2\delta_0+\delta_1),$$

$$S_{11}(k_Fr) = \left(1-\frac{10}{(k_Fr)^2}+\frac{6}{(k_Fr)^4}\right) \sin(2k_Fr+3\delta_1)+\left(\frac{4}{k_Fr} -\frac{12}{(k_Fr)^3}\right)\cos(2k_Fr+3\delta_1),$$

$$S_{10}(k_Fr) = -3\left(1-\frac{1}{(k_Fr)^2}\right)\sin(2k_Fr+2\delta_1+ \delta_0)-\frac{6}{k_Fr}\cos(2k_Fr+2\delta_1+\delta_0).$$

There is no doubt that (16,17,18) must provide a poor approximation to the ASDOS (10) if numerical accuracy is aimed at. However, this is not our task. We argue as follows: Pseudogaps of the total ASDOS at $E_F$ indicate that stability requires highly constructive interference of the electrons at the Fermi level. Hence, with the stable neighbour-shell sequence, the contributions to the back-scattering (11) which emerge from all shells must interfere constructively with each other and destructively with the electron states of the bare atomic sphere (cf. (10)). Suppose the $\delta_l$ are the scattering phase shifts of the pseudoatoms in the stable configuration. Then the formulas (16,17,18) are really useful. They show how the electronic interference and the radial structure of the correlation shell interrelate in order to achieve low ASDOS.

The following general conclusions can be drawn: (i) According to (10,16) low $\overline{D_l}(E_F)$ is obtained if the shells (the maxima of $(g(r)-1)$) coincide with the minima of $S_l(k_Fr)$. However, stability requires low total ASDOS, $\overline{D_0}(E_F)+\overline{D_1}(E_F)$. The really stable shell sequence will thus result from the competition between $\overline{D_0}(E_F)$ and $\overline{D_1}(E_F)$. (ii) Asymptotically, the shells must be uniformly spaced with the Friedel wave length, $\lambda_{FR} = 2\pi/(2k_F)$. This is just the demand of the Nagel-Tauc condition. The distant shells act "in phase" on the central atomic sphere. (iii) At smaller distances, the polynomials of powers of $1/(k_Fr)$ (18) give rise to deviations from the extrapolated asymptotic bahaviour. This is due to the $p-$scattering in the environment. Note that the cosine terms decay only as $1/(k_Fr)$ and act thus up to considerable distances. The short-range sensitivity is increased due to the higher powers of $1/(k_Fr)$. On the other hand, contraction and dilation of the close neighbour shells may also arise from the involved higher electronic angular momenta. (iv) The condition $\delta_0=\delta_1$ defines the remarkable situation where the standards (18) have no long-range parts. The asymptotic contributions due to $s-$ and $p-$scattering in the environment cancel out each other. On such conditions, the system gets short-range confined. It must be stressed again that the above conclusions are drawn from single-scattering formulas. They are derived from the state of the electronic interference which in turn is a guide line for the multiple scattering.