Electrons per Atom Ratio Determination and Hume-Rothery Electron Concentration Rule for P-Based Polar Compounds Studied by FLAPW–Fourier Calculations

Abstract

The extent to which reliable electrons per atom ratio, e/a, are determined and the validity of the Hume-Rothery stabilization mechanism are ensured upon increasing ionicity are studied by applying first-principles full potential linearized augmented plane wave (FLAPW)-Fourier band calculations to as many as 59 binary compounds formed by adding elements from periods 2-6 to phosphorus in group 15 of the Periodic Table. Van Arkel-Ketelaar triangle maps were constructed both by using the Allen electronegativity data and by using an energy difference between the center-of-gravity energies of FLAPW-derived s and p partial densities of states (DOSs) for the equiatomic compounds studied. The determination of e/a and the test of the interference condition, both of which play a key role in the Hume-Rothery stabilization mechanism, were reliably made for all intermetallic compounds, as long as the ionicity is less than 50%. In the A-P (A = Li, Na, K, Rb, and Cs) compounds with ionicity exceeding 50%, however, e/a determination becomes unstable, as reflected in its P concentration dependence. New Hume-Rothery electron concentration rules were theoretically found in two families of polar compounds: skutterudite compounds TMP(3), TMAs(3), and TMSb(3) (TM = Co, Ni, Rh, and Ir; cI32) with e/a = 4.34 and TM(3)P (TM = Cr, Mn, Fe, and Ni; tI32) with e/a = 2.20.

The Physics of the Hume-Rothery Electron Concentration Rule

“…The FLAPW separation energy E 3s − E 3p or the energy difference between the center of gravity energies of s-and p-partial DOSs is included in Figure 28. It amounts to only 0.48 eV for Na (cI2), whereas it is increased to 11.35 eV for Cl (oS8) [11,12].…”

“…It allows us to efficiently perform the FLAPW electronic structure calculations even for a compound with a giant unit cell like 2/1-2/1-2/1 approximants containing 680 atoms per unit cell. The FLAPW-Fourier theory discussed in the present Section has been elaborated since 2011 by using the WIEN2k program package [11][12][13][14][15][16][17][18][19][20]. In the WIEN2k-FLAPW electronic structure calculations, we have exclusively employed the GGA-PBE (Generalized Gradient Approximation-Perdew, Burke and Ernzerhof) for exchange-correlation energy and dealt with only non-magnetic states even for 3d-TM elements like Fe, Co and Ni.…”

“…In the present article, both electron wave vector k and reciprocal lattice vector G in the reciprocal space are expressed in units of 2π a for a cubic lattice with the lattice constant a [11][12][13][14][15][16][17][18][19][20]. Hence, the reciprocal lattice vector G = 2π a (h k l) is simply given by G = (h k l), where integers h, k and l represent the set of Miller indices and…”

“…We consider it to originate from a repulsive interaction due to the orthogonality between sp-and d-like wave functions and have called it the "d-states-mediated splitting" [11][12][13][14][15][16][17][18]. repulsive interaction due to the orthogonality between sp-and d-like wave functions and have called it the "d-states-mediated splitting" [11][12][13][14][15][16][17][18]. The center of gravity energy for each electronic state 2 k + G ( ) ZSPs 2 was calculated and plotted in Figure 16c, using three colored circles at symmetry points Γ, N and H. Now, the complicated behavior of the energy dependent Fourier coefficients is straightened and the square of the wave-…”

Wetting and adhesion properties of quasicrystals and complex metallic alloys

“…Figure 5a shows a revised, so far unpublished, version of n(E F ) plotted as a function of the average number of electrons per atom e/a computed for many compounds (not necessarily identical to those used for contact angle measurements in the present study). In this figure, the x-axis represents the average number of electrons per atoms computed for each compound by referring to a recently published list of elemental values of e/a [20]. The same data is shown in Fig.…”