Introduction

Since the mid-twentieth continuous interest has been devoted to the electron-ion interrelation in metallic alloys. A crude model treats the alloy as a system of correlated ions immersed in a gas of nearly free electrons (the electrons, for shortness). Model parameters are the number of electrons per ion, $\overline{Z}$, the diameters of the pseudoatoms and the atom number density, $N_0$. Hume-Rothery [1] has established the investigation of the electron-ion interrelation in metallic alloys as a research topic in metal physics. He found that the lattice types and other properties of certain alloys are related to $\overline{Z}$. Mott and Jones [2] have suggested a heuristically important generalization of the Hume-Rothery concept: Crystalline alloys are stable if the Fermi sphere touches $\bf k$-space faces which correspond to reflections with high structure factors (Jones zones). After Nagel and Tauc [3], metallic glasses are stabilized against crystallization by a minimum of the electronic density of states at the Fermi level. This is accompanied by a peak of the structure factor at twice the Fermi momentum ($K_p = 2 k_F$). They estimated $\overline{Z} \approx 1.7$ as the critical electron concentration which ensures this electron-ion interrelation to act most efficiently.

Haeussler [4] reported experimental results which qualify amorphous alloys with $\overline{Z} \approx 1.8$ as interesting in two respects: (i) The neighbour-shell radii follow the sequence $R_n$ = (2$\pi$/2$k_F$) (5/4 + (n-1)) which makes the diffraction peak at $2k_F$ most pronounced. Beck and Oberle [5]) as well as Hafner and v. Heimendahl [6] have suggested a physically appealing interpretation of the Hume-Rothery concept in real space: In stable configurations, the average neighbour shells about fixed atoms match just the Friedel minima of the effective interatomic potential. (ii) Such alloys show marked anomalies of the electronic conductivity, the Hall coefficient, the thermopower and the crystallization temperature which cannot be explained by free-electron arguments. Alloys with $\overline{Z} \approx 1.8$ exist obviously on the conditions of strong interrelation between the electrons and the ions due to highly constructive electronic interference. The present paper examines how the character of the electronic interference at the Fermi level depends on the ionic configuration. It turns out that aspects of the stability of the system can also be treated upon employing the single-scattering approximation to the electron wave.

First we describe a model of an isotropic topologically disordered one-component system which provides most-constructive interference for electronic transitions between surface points of the Fermi sphere at a critical electron concentration of $\overline{Z} \approx 1.81$. Our input assumptions: (i) The measure of length carried by the electrons is the Fermi wave length, 2$\pi$/$k_F$. The electron states are planar only within the bounds of the mean elastic free path. (ii) Transitions between electron states are due to electronic diffraction from stacks of atomic planes which are regular in the same bounds. The measure of length carried by the atoms is $a_0$ $\equiv$ $N_0^{-1/3}$. (iii) Both measures of length are related to one another according to the Mott-Jones version of the Hume-Rothery concept. We obtain a solving $k_F$-$a_0$ relation together with the corresponding critical electron concentration. Note that the above conditions ensure only that scattered electron waves add up to make a strong field, no matter how this field acts.

A second topic to be dealt with in this paper is the interrelation between the radial sequence of neighbour shells and the stability of the configuration. Here, additional demands are required which ensure the scattered field to act stabilizing. We follow Nagel and Tauc in that stability is ascribed to low electronic density of states at the Fermi level. In contrast to common Fraunhofer diffraction experiments, the electronic density of states involves Fresnel-type diffraction where the atomic spheres themselves are both the emitters and the detectors of waves. Thus, the valence electrons extract short-range properties from the sequence of neighbour shells employing higher angular momenta. We will analyze, mainly by analytical means, how short- and long-range properties of the radial sequence of neighbour shells interrelate with low electronic density of states at the Fermi energy. We confine to single-scattering in the environment of the considered atomic sphere which includes primary interference effects without preventing ensemble averaging. Single-scattering formulas are not capable providing precise numerical results. They show where there is constructive interference of waves and indicate thus where something happens in multiple scattering. This is just the goal of the present paper.