Hume-Rothery Electron Concentration Rule Research Articles

The Hume-Rothery Electron Concentration Rule Extended to Alloys and Compounds whose Ionic and/or Covalent Bondings are Increased

The Hume-Rothery Electron Concentration Rule Extended to Alloys and Compounds whose Ionic and/or Covalent Bondings are Increased

The Hume-Rothery electron concentration rule studied through interference phenomena in a series of cP12 compounds with either a true gap or pseudogap across the Fermi level

ABSTRACTAmong binary MX2-type cP12 compounds with space group , there are many compounds with either a true gap or pseudogap at the Fermi level. They are grouped into two with respect to the total valence electrons per atom (VEC), the first being at 14/3, while the second at 20/3. The ratio of the number of itinerant valence electrons per atom e/a over VEC is used as a measure to evaluate a drastic change in the electronic structure from the nearly free electron-like to the tight binding-like. The Full-potential Linearized Augmented Plane Wave (FLAPW) band calculations were performed to study to what extent the value of e/a can be well defined. We claim that the Hume-Rothery electron concentration rule holds for As2Si, P2Si and MgTe2, where the ratio above is higher than 90%. For compounds with M = Fe, Ru and Os and X = S, Se and Te, where the ratio is between 60 and 70%, the discussion based on a large Brillouin zone is still meaningful but the validity of Hume-Rothery electron concentration rule is judged to be marginal. The value of e/a is lowered below 20% and the Hume-Rothery electron concentration rule fails in MgO2 and O2Os.

The Physics of the Hume-Rothery Electron Concentration Rule

For a long time we have shared the belief that the physics of the Hume-Rothery electron concentration rule can be deepened only through thorough investigation of the interference phenomenon of itinerant electrons with a particular set of lattice planes, regardless of whether d-states are involved near the Fermi level or not. For this purpose, we have developed the FLAPW-Fourier theory (Full potential Linearized Augmented Plane Wave), which is capable of determining the square of the Fermi diameter, ( 2 k F ) 2 , and the number of itinerant electrons per atom, e/a, as well as the set of lattice planes participating in the interference phenomenon. By determining these key parameters, we could test the interference condition and clarify how it contributes to the formation of a pseudogap at the Fermi level. Further significant progress has been made to allow us to equally handle transition metal (TM) elements and their compounds. A method of taking the center of gravity energy for energy distribution of electrons with a given electronic state has enabled us to eliminate the d-band anomaly and to determine effective ( 2 k F ) 2 , and e/a, even for systems involving the d-band or an energy gap across the Fermi level. The e/a values for 54 elements covering from Group 1 up to Group 16 in the Periodic Table, including 3d-, 4d- and 5d-elements, were determined in a self-consistent manner. The FLAPW-Fourier theory faces its limit only for elements in Group 17 like insulating solids Cl and their compounds, although the value of e/a can be determined without difficulty when Br becomes metallic under high pressures. The origin of a pseudogap at the Fermi level for a large number of compounds has been successfully interpreted in terms of the interference condition, regardless of the bond-types involved in the van Arkel-Ketelaar triangle map.

Determination of electrons per atom ratio for transition metal compounds studied by FLAPW-Fourier calculations

Among 54 elements in the periodic table, whose electrons per atom ratio e/a has been consistently determined within the framework of the Full-potential Linearised Augmented Plane Wave-Fourier theory, the present authors have claimed the need of properly differentiating two distinct e/a values for Ca in the Group 2 and for Sc and Y in Group 3 in the periodic table, depending on whether they are dissolved in transition metal (TM) (TM from Group 4–10) matrix or in non-TM (Al, Zn, etc.) matrix. It is also emphasised that this unique alloying environment effects are essentially absent in other elements Sr and Ba in Group 2 and La in Group 3. We could also show that the linear interpolation rule stating that e/a of any binary compounds is estimated by taking a composition average of those of constituent elements, holds within the accuracy of ±20%, even when elements in Groups 2 and 3 are involved. The Hume-Rothery electron concentration rule for Laves compounds with Pearson symbol cF24 has been successfully elucidated by classifying them into subgroups with respect to square of critical reciprocal lattice vector serving as a key parameter in the interference condition.

Electrons per atom ratio determination and Hume-Rothery electron concentration rule for P-based polar compounds studied by FLAPW-fourier calculations.

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.

Theoretical Foundation for the Hume-Rothery Electron Concentration Rule for Structurally Complex Alloys

Theoretical Foundation for the Hume-Rothery Electron Concentration Rule for Structurally Complex Alloys

E/a classification of Hume–Rothery Rhombic Triacontahedron-type approximants based on all-electron density functional theory calculations

There are three key electronic parameters in elucidating the physics behind the Hume–Rothery electron concentration rule: the square of the Fermi diameter (2kF)2, the square of the critical reciprocal lattice vector and the electron concentration parameter or the number of itinerant electrons per atom e/a. We have reliably determined these three parameters for 10 Rhombic Triacontahedron-type 2/1–2/1–2/1 (N = 680) and 1/1–1/1–1/1 (N = 160–162) approximants by making full use of the full-potential linearized augmented plane wave-Fourier band calculations based on all-electron density-functional theory. We revealed that the 2/1–2/1–2/1 approximants Al13Mg27Zn45 and Na27Au27Ga31 belong to two different sub-groups classified in terms of equal to 126 and 109 and could explain why they take different e/a values of 2.13 and 1.76, respectively. Among eight 1/1–1/1–1/1 approximants Al3Mg4Zn3, Al9Mg8Ag3, Al21Li13Cu6, Ga21Li13Cu6, Na26Au24Ga30, Na26Au37Ge18, Na26Au37Sn18 and Na26Cd40Pb6, the first two, the second two and the last four compounds were classified into three sub-groups with = 50, 46 and 42; and were claimed to obey the e/a = 2.30, 2.10–2.15 and 1.70–1.80 rules, respectively.

E/a determination for 4d- and 5d-transition metal elements and their intermetallic compounds with Mg, Al, Zn, Cd and In

The present work is devoted to the determination of the effective electrons per atom ratio e/a by means of first-principles full-potential linearized augmented plane wave-Fourier method for elements from Rb to Ag in Period 5 and from Cs to Au in Period 6 of the periodic table and is regarded as a continuation of the preceding work done for elements from K to Cu in Period 4. The value of e/a was determined by reading off the square of the Fermi diameter, from the dispersion relation for electrons outside the Muffin-Tin spheres. A straightforward reading of the ordinate at the Fermi level, i.e. local reading method was validated for Rb and Cs in Group 1, Sr in Group 2, Y in Group 3, Pd and Pt in Group 10 and Ag and Au in Group 11. Instead, the nearly free electron (NFE) method was found to be indispensable for TM elements from Zr to Rh in Period 5 and those from Ba to Ir in Period 6. The composition dependence of e/a values for intermetallic compounds in X–TM (X = Mg, Al, Zn, Cd and In) alloy systems was also studied. The new Hume–Rothery electron concentration rule was established by constructing e/uc, the number of electrons per unit cell, vs. square of critical reciprocal lattice vector, , diagram for structurally complex metallic alloys having a pseudogap at the Fermi level. A proper use of either the local reading- or the NFE-e/a for the elements as indicated above is found to be essential.

Hume-Rothery stabilization mechanism and e/a determination in MI-type Al–Mn, Al–Re, Al–Re–Si, Al–Cu–Fe–Si and Al–Cu–Ru–Si 1/1-1/1-1/1 approximants – a proposal for a new Hume-Rothery electron concentration rule

Full-potential linearized augmented plane wave (FLAPW) band calculations with subsequent FLAPW-Fourier analyses have been performed for five MI-type Al–TM (transition metal)-based 1/1-1/1-1/1 approximants in order to elucidate the origin of a pseudogap from the point of view of the Fermi surface–Brillouin zone (FsBz) interactions. The square of the Fermi diameter is determined from the Hume-Rothery plot to be close to 50 without exception in units of , where is the lattice constant. The FsBz interactions involving several reciprocal lattice vectors are claimed to be responsible for constituting the pseudogap structure across E F. This is referred to as the multi-zone effect. Among them, the wave, in which the Fourier coefficients are most evenly and densely distributed across E F, is selected as the critical one satisfying the matching condition . The d-states-mediated-splitting appears to be absent in spite of substantial occupations of TM-d states in the valence band. All MI-type approximants studied are found to obey the Hume-Rothery stabilization mechanism. A new Hume-Rothery electron concentration rule linking the number of atoms per unit cell, e/uc, with a critical reciprocal lattice vector is found to hold well for structurally complex intermetallic compounds obeying the Hume-Rothery stabilization mechanism.

Hume-Rothery stabilization mechanism and e/a determination for RT- and MI-type 1/1-1/1-1/1 approximants studied by FLAPW-Fourier analyses

Full-potential linearized augmented plane wave (FLAPW) electronic band calculations were performed for two RT- (rhombic triacontahedron) and five MI- (Mackay icosahedron) type 1/1-1/1-1/1 approximants plus several complex metallic compounds in Al-TM (TM = transition metal element) binary alloy systems in order to elucidate the origin of a pseudogap from the viewpoint of Fermi surface-Brillouin zone (FsBz) interactions. The square of the Fermi diameter (2k(F))(2) and square of the critical reciprocal lattice vector |G|(2) or the critical set of lattice planes, with which electrons at the Fermi level E(F) are interfering, can be extracted from the FLAPW-Fourier method. We revealed that a pseudogap in both RT- and MI-type 1/1-1/1-1/1 approximants universally originates from interference phenomenon satisfying the matching condition (2k(F))(2) = |G|(2) equal to 50 in units of (2π/a)(2), where a is the lattice constant. The multi-zone effect involving not only |G|(2) = 50 but also its neighboring ones is also claimed to be responsible for constituting a pseudogap across E(F). The value of e/a for Mn, Fe, Re and Ru elements in the periodic table is deduced to be positive in the neighborhood of unity. All 1/1-1/1-1/1 approximants, regardless of RT- or MI-type atomic cluster involved, are stabilized at around e/a= 2.7, while their counterpart quasicrystals are at around e/a= 2.2. A new Hume-Rothery electron concentration rule linking the number of atoms per unit cell, e/uc, with a critical|G|(2) holds well for all complex intermetallic compounds characterized by a pseudogap at E(F).

Hume-Rothery electron concentration rule across a whole solid solution range in a series of gamma-brasses in Cu–Zn, Cu–Cd, Cu–Al, Cu–Ga, Ni–Zn and Co–Zn alloy systems

The aim of the present work is to examine if the Hume-Rothery stabilisation mechanism holds across whole solid solution ranges in a series of gamma-brasses with especial attention to the role of vacancies introduced into the large unit cell. The concentration dependence of the number of atoms in the unit cell, N, for gamma-brasses in the Cu–Zn, Cu–Cd, Cu–Al, Cu–Ga, Ni–Zn and Co–Zn alloy systems was determined by measuring the density and lattice constants at room temperature. The number of itinerant electrons in the unit cell, e/uc, is evaluated by taking a product of N and the number of itinerant electrons per atom, e/a, for the transition metal element deduced earlier from the full-potential linearised augmented plane wave (FLAPW)-Fourier analysis. The results are discussed within the rigid-band model using as a host the density of states (DOS) derived earlier from the FLAPW band calculations for the stoichiometric gamma-brasses Cu5Zn8, Cu9Al4 and TM2Zn11 (TM = Co and Ni). A solid solution range of gamma-brasses in Cu–Zn, Cu–Cd, Cu–Al, Cu–Ga and Ni–Zn alloy systems is found to fall inside the existing pseudogap at the Fermi level. This is taken as confirmation of the validity of the Hume-Rothery stability mechanism for a whole solute concentration range of these gamma-brasses. An exception to this behaviour was found in the Co–Zn gamma-brasses, where orbital hybridisation effects are claimed to play a crucial role in stabilisation.

Failure of the Hume-Rothery stabilization mechanism in theAg5Li8 gamma-brass studied by first-principles FLAPW electronic structure calculations

The first-principles FLAPW (full potential linearized augmentedplane wave) electronic structure calculations were performed for theAg5Li8 gamma-brass, which contains 52 atoms in a unit cell and has been known for many years asone of the most structurally complex alloy phases. The calculations were also made for itsneighboring phase AgLi B2 compound. The main objective in the present work is to examine if theAg5Li8 gamma-brass is stabilized at the particular electrons per atom ratioe/a = 21/13 in the same way as some other gamma-brasses likeCu5Zn8 and Cu9Al4, obeying the Hume-Rothery electron concentration rule. For this purpose, thee/a valuefor the Ag5Li8 gamma-brass as well as the AgLi B2 compound was first determined by means of the FLAPW-Fouriermethod we have developed. It proved that both the gamma-brass and the B2 compound possess ane/a value equal tounity instead of 21/13. Moreover, we could demonstrate why the Hume-Rothery stabilization mechanism fails for theAg5Li8 gamma-brass and proposed a new stability mechanism, in which the unique gamma-brassstructure can effectively lower the band-structure energy by forming heavily populatedbonding states near the bottom of the Ag-4d band.

Mediated resonance effect of the vanadium 3dstates on phase stability in theAl8V5γ-brass studied by first-principles FLAPW and LMTO-ASA electronic structure calculations

The mechanism for the stability of the ${\mathrm{Al}}_{8}{\mathrm{V}}_{5}\phantom{\rule{0.3em}{0ex}}\ensuremath{\gamma}$-brass containing 52 atoms in its cubic unit cell has been investigated by means of first-principles full-potential linearized augmented plane wave (FLAPW) and linearized muffin-tin orbital-atomic sphere approximation (LMTO-ASA) electronic structure calculations. The LMTO-ASA identified a deep valley at $0.5\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ above the Fermi level in its density of states (DOS) as arising from orbital hybridizations between V $3d$ and Al $3p$ states. On the other hand, the FLAPW revealed the V $3d$ states mediated resonance of electrons with different sets of lattice planes. The resonance involved is found to be substantial not only at ${\ensuremath{\mid}\mathbf{G}\ensuremath{\mid}}^{2}=18$ or {330} and {411} zones but also at those in the range $14\ensuremath{\leqslant}{\ensuremath{\mid}\mathbf{G}\ensuremath{\mid}}^{2}\ensuremath{\leqslant}30$. A comparison with the electronic structure of the CsCl-type AlV compound proved that the V $3d$ states mediated resonance occurs only in ${\mathrm{Al}}_{8}{\mathrm{V}}_{5}$ but not in AlV compound. The V $3d$ states mediated resonance is proved to result in a significant suppression of the $sp$-partial DOS over the energy range from the Fermi level up to $+2.2\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. A gain in the electronic energy has been attributed to the formation of highly condensed bonding states below the Fermi level, again caused by the V $3d$ states mediated resonance. It is also proposed that the ${\mathrm{Al}}_{8}{\mathrm{V}}_{5}$ is stabilized at $e∕a=1.94$ rather than $21∕13$ as is expected from the Hume-Rothery electron concentration rule.

The Hume–Rothery electron concentration rule for a series of gamma-brasses studied by full-potential linearized augmented plane wave (FLAPW)

A theoretical interpretation of the Hume–Rothery electron concentration rule has been attempted by performing full-potential linearized augmented plane wave (FLAPW) band calculations for a series of gamma-brasses: Cu5Zn8, Cu9Al4 and TM2Zn11 (TM = Ni, Pd, Co and Fe). The Fourier spectrum of the FLAPW wavefunction was evaluated for states at point N on the set of {110} zone planes of the reduced bcc Brillouin zone at energy eigenvalues sustaining the pseudogap. It turned out that electrons near the Fermi level resonate with periodic potentials due to {330} and {411} lattice planes to open the pseudogap for all gamma-brasses studied. The effective electron per atom ratio (e/a) was evaluated by averaging the wave vector 2(k i + G) having the largest Fourier component of the FLAPW wavefunction at energy E over selected electronic states k i in the Brillouin zone, with subsequent construction of the dispersion relation E versus in the extended zone scheme. The e/a value proved to be close to e/a = 21/13 = 1.62, the value empirically assigned for the gamma-brass in the Hume–Rothery electron concentration rule, for Cu5Zn8, Cu9Al4, Ni2Zn11 and Pd2Zn11, but to be as small as 1.43 and 1.31 for Co2Zn11 and Fe2Zn11, respectively.

Interpretation of the Hume-Rothery electron concentration rule in theT2Zn11(T=Ni, Pd, Co, and Fe)γbrasses based on first-principles FLAPW calculations

The first-principles full-potential augmented plane wave (FLAPW) band calculations were performed for a series of ${T}_{2}{\mathrm{Zn}}_{11}$ ($T=\mathrm{Ni}$, Pd, Co, and Fe) $\ensuremath{\gamma}$ brasses to elucidate the Hume-Rothery electron concentration rule. The pseudogap is found immediately below the Fermi level ${E}_{F}$ in the ${\mathrm{Ni}}_{2}{\mathrm{Zn}}_{11}$ and ${\mathrm{Pd}}_{2}{\mathrm{Zn}}_{11}\phantom{\rule{0.3em}{0ex}}\ensuremath{\gamma}$ brasses. A resulting gain in the electronic energy is attributed to their stabilization in the same way as in ${\mathrm{Cu}}_{5}{\mathrm{Zn}}_{8}$ and ${\mathrm{Cu}}_{9}{\mathrm{Al}}_{4}$ previously studied. However, the pseudogap is essentially shifted above ${E}_{F}$ in both ${\mathrm{Co}}_{2}{\mathrm{Zn}}_{11}$ and ${\mathrm{Fe}}_{2}{\mathrm{Zn}}_{11}$. The Fourier analysis of the FLAPW wave function was made at the symmetry point $N$ of the reduced Brillouin zone in the energy range involving the pseudogap. It is found that the plane wave giving rise to the largest Fourier component always resonates with the {330} and {411} zone planes to produce the pseudogap near ${E}_{F}$. Moreover, a single-branch energy dispersion relation was constructed in the extended zone scheme by averaging the wave vector $2(\mathbf{k}+\mathbf{G})$ having the largest Fourier component of the FLAPW wave function over selected electronic states in the Brillouin zone. The $e∕a$ value thus deduced is found to be close to $21∕13=1.615$ for ${\mathrm{Cu}}_{5}{\mathrm{Zn}}_{8}$, ${\mathrm{Cu}}_{9}{\mathrm{Al}}_{4}$, ${\mathrm{Ni}}_{2}{\mathrm{Zn}}_{11}$, and ${\mathrm{Pd}}_{2}{\mathrm{Zn}}_{11}\phantom{\rule{0.3em}{0ex}}\ensuremath{\gamma}$ brasses but to be only 1.4 and 1.3 for ${\mathrm{Co}}_{2}{\mathrm{Zn}}_{11}$ and ${\mathrm{Fe}}_{2}{\mathrm{Zn}}_{11}$, respectively.

Verification of Hume-Rothery electron concentration rule inCu5Zn8andCu9Al4γbrasses byab initioFLAPW band calculations

Both ${\mathrm{Cu}}_{5}{\mathrm{Zn}}_{8}$ and ${\mathrm{Cu}}_{9}{\mathrm{Al}}_{4}$ $\ensuremath{\gamma}$ brasses containing 52 atoms in its cubic cell have been recognized as a typical Hume-Rothery alloy phase, since these structurally complex compounds are stabilized at the same electron concentration of 1.61 electrons per atom in spite of different amounts of solute elements Zn and Al involved. Both compounds are found to possess the pseudogap at the Fermi level. Its origin has been studied by performing the full-potential linearized augmented plane-wave (FLAPW) band calculations to lay a firm foundation on the empirical Hume-Rothery electron concentration rule in a straightforward way. This was accomplished through the study of the Fourier coefficient in the plane-wave expansion of the FLAPW wave function in the intermediate region outside the muffin-tin potential for states near the Fermi level. It is concluded that electrons contributing to forming the pseudogap resonate with crystal planes associated with {330} and {411} zones and give rise to standing waves for both ${\mathrm{Cu}}_{5}{\mathrm{Zn}}_{8}$ and ${\mathrm{Cu}}_{9}{\mathrm{Al}}_{4}$ $\ensuremath{\gamma}$ brasses. This clearly explains why these two compounds having different solute concentrations are stabilized at the same $e∕a$ value of 1.61.

軽水炉用制御材Ag-In-Cd合金の押出しに関する研究

To obtain fundamental informations on the fabrication technology of a control rod absorber of Ag-15 wt% In-5 wt% Cd alloy, which was developed considering the HumeRothery electron concentration rule, some experiments were performed. By the hardness tests on alloys obtained in several steps of cold rolling and subsequent annealing, the alloy was confirmed to have good workability. The cast alloy was able to be extruded at 700 deg C in round rods. The value of extrusion coefficient K was determined to be about 15 kg/mm/sup 2/ at 700 deg C. The tensile properties and microstructures of extruded rods were also studied.