## 2 The Hume-Rothery Electron Concentration Rule
The Hume-Rothery rule can be explained as resulting from a perturbation of the kinetic energy of the valence electrons by their diffraction by the crystal lattice when an electron has such a wave length \(\lambda\) = h(2 mT\({}_{\rm e}\))\({}^{-1/2}\) and a direction so as to permit Bragg reflection from an important crystallographic plane. Due to this perturbation, a rearrangement of electronic states to both higher and, more importantly, lower energies take place. This leads to the occurrence of gaps or pseudogaps (that is, local deep minima) in the electronic density of states (DOS). Thus, special stability would be expected for metals with just the right number of electrons. In that case, E\({}_{\rm F}\) lies close to a DOS minimum, so that only the states shifted to lower energies are occupied and the energy of the electronic system is decreased. This electron number is proportional to the volume of a polyhedron in the reciprocal space (the so-called Brillouin-Jones zone), corresponding to the crystallographic planes giving rise to the perturbation. In alloys with high enough symmetry, this zone is quite close to a spherical shape, and the diffraction condition can be conveniently expressed in the form K\({}_{\rm hkl}\) = 2k\({}_{\rm F}\), where K\({}_{\rm hkl}\) is the reciprocal vector of the considered diffraction plane, k\({}_{\rm F}\) = (3\(\pi^{2}\)n)\({}^{1/3}\)/a\({}_{0}\) is the radius of the Fermi sphere in reciprocal space, and n = (e/a)N is the electron density, where N is the number of atoms in the unit cell. A long-term study aimed at calculating the electron per atom ratio, e/a, in structurally-complex metallic alloys was undertaken by [PERSON] and co-workers, who have shown that this magnitude plays a key role in the cohesion of diverse classes of solids and it is closely linked with the more usual valence concept. To this end, they exploited the so-called full-potential linearized augmented plane wave (FLAPW)-Fourier theory in order to study the electronic structure of many pseudogap-bearing binary and ternary intermetallic compounds, including Al-, Zn-, and Cd-based compounds, Zintl polar compounds, phosphorus-based compounds, and inter-transition metal compounds. In this way, reliable e/a values for 54 elements in the periodic table were obtained, including representatives from group 1 up to group 15 (Figure 4) [28]. Thus, the origin of a pseudogap at the Fermi level for a large number of materials has been successfully interpreted in terms of electronic interference conditions, regardless of the bond-types involved. The Hume-Rothery criterion can be applied to QCs by introducing a pseudo-Brillouin zone related to the most intense diffraction reflections. In doing so, we note that the high-order symmetry of i-QCs (as compared to periodic crystalline phases) generally provides an enhanced overlapping of the Fermi  
Figure 4: Electron per atom concentration ratio e/a for 54 elements in the periodic table. The value in the top (bottom) level for Ca, Sc, and Y is used when these atoms are alloyed with non-transition metal (transition metal) elements, respectively. (Reprinted from [PERSON] and [PERSON], The physics of the Hume-Rothery electron concentration rule, Crystals 7, 9 (2017); doi:10.3390/cryst010009, Creative Commons Attribution License CC BY 4.0). surface with the pseudo-Brillouin zone in the reciprocal space. In fact, it has been confirmed that for QCs containing elements with a full d-band, such as i-Zn\({}_{43}\)Mg\({}_{37}\)Ga\({}_{20}\) (e/a = 2.221), i-Al\({}_{56}\)Li\({}_{33}\)Cu\({}_{11}\) (e/a = 2.129), i-Zn\({}_{60}\)Mg\({}_{30}\)(RE)\({}_{10}\) (e/a = 2.127, where RE stands for rare-earth atoms) or i-Zn\({}_{80}\)Sc\({}_{15}\)Mg\({}_{5}\) (e/a = 2.039), Fermi sphere-pseudo-Brillouin zone interaction gives rise to a significant reduction of the DOS (pseudogap) close to E\({}_{\rm F}\), ultimately leading to the stabilization of these compounds as a result of distributing electrons with the highest kinetic energies into deeper states in the energy spectrum.