Hume-Rothery Rule Research Articles

Hume-Rothery controlled formation of structurally complex alloy phases in the ternary Ga–Mg–Pd system

The Hume-Rothery rule has been used as a guide to search for new approximants of the Mackay icosahedron and Bergman cluster type in the Ga–Mg–Pd system. Ternary alloys with target compositions close to the e/ a ratios 1.67 and 2.09 were synthesized by melting the elements and subsequent annealing in sealed tantalum ampoules. Six new intermetallic phases have been characterized: a cubic face-centered Mackay icosahedron approximant Ga 15− x− y Mg 38+ y Pd 13+ x , a rhombohedral Mackay icosahedron approximant Ga 30− x Mg 32+ x Pd 21, a cubic Bergman cluster approximant Ga 12+ x+ y Mg 11− y Pd 6− x , a cubic phase with interpenetrating I3 and Frank–Kasper networks Ga 32.6Mg 43.4Pd 29, an ordered ternary variant of the NiTi 2 structure type GaMg 3Pd 2 and a ternary Laves phase. The e/ a ratios for the approximants with Mackay icosahedra or Bergman clusters show a strong experimental correlation indicating that the valence electron concentration is an important factor concerning the stability of the phases. The approximant phases exist over a wide range of composition. The structure of the rhombohedral phase is based on the geometry of the canonical cell tiling CB 2C. The results support the proposal by Henley and Mihalkovic on the geometrical structures of Mackay icosahedron-type quasicrystals in ternary systems with one transition and two main group metals.

New group of stable icosahedral quasicrystals: structural properties and formation conditions

Structural studies on the icosahedral quasicrystals in Zn–Mg–Sc, Cu–Ga–Mg–Sc, and Zn–Mg–Ti alloys as well as their corresponding 1/1 cubic approximants, have revealed that these quasicrystals belong to a new structural group similar to Cd-based quasicrystals. This group is characterized by a triple-shell icosahedral cluster different from both Mackay- and Bergman-types. The presence of the atomic cluster has been deduced from the structure model of the approximant crystal, Zn 17Sc 3, in which the clusters are embedded in a periodic network of so-called ‘glue atoms’. Density measurement suggested the presence of at least 2.7 Zn atoms in the first shell of the cluster in this approximant. The substitutional relationship in these three quasicrystals indicates the important role of Hume-Rothery rule for the formation of this type of quasicrystal. The occurrence of a P-type icosahedral quasicrystal in Zn–Mg–Yb alloy is also reported.

Composition rule of bulk metallic glasses and quasicrystals using electron concentration criterion

This paper aims at establishing a number of electrons/atom (e/a)-based criterion for searching bulk metallic glasses (BMGs) and quasicrystals with large forming abilities in the Zr-based multicomponent alloy systems. After discussions on the diffraction characteristics corresponding to the Fermi surfaces-Brillouin zone interaction in the Zr-based Hume-Rothery phases, the Hume-Rothery matching rule is well explained when the effective e/a value of the matrix element Zr is taken as 1.5. The BMG- and quasicrystal-related phases are pointed out to be a family of nearly e/a-constant phases in a given alloy system. An e/a-constant criterion is then used to predict the ideal composition of the quasicrystals and BMGs in the Zr-Ti-Ni, Zr-Al-Ni, and Zr-Al-Ni-Cu systems, respectively. Nearly pure bulk Zr-Ti-Ni quasicrystals and a series of BMGs with glass-forming abilities greater than that of the known Zr65Al7.5Ni10Cu17.5 alloy are found.

Search and Synthesis of New Family of Quasicrystals

ABSTRACTStarting from the Zn17Sc3 cubic approximant, new icosahedral quasicrystal was searched by substituting Zn by other metals, M, at the alloy composition of Zn75M10Sc15. In the cases of M = Mn, Fe, Co, Ni, Pd, Pt, Ag and Au, new P-type quasicrystals were discovered in as-cast alloys. In the cases of M = Fe, Co, Ni, Pd and Ag, the quasicrystals are thermodynamically stable at approximately 700 °C. This result indicates that use of an approximant crystal as a starting material is very efficient way to search new quasicrystal alloy, and many kinds of metals stabilize the quasicrystal structures; i.e. noble metals and transition elements including Mn, Fe, Co and Ni in addition to Mg. Taking the variety in base metals of Tsai-type approximants into account, this variety in additional components suggests many possibilities of undiscovered quasicrystals. The equality ofthe electron concentration, ela ≈ 2.1, in Hume-Rothery rule may be a guide to these quasicrystals.

Phase Transformations in Binary Alloys under High Pressure

High-pressure studies of binary alloy systems display some specific features connected with composition referring as a variable parameter. In a two-component system under pressure there are observed transformations, whichdo not occur in a one-component system. In particular, there is a formation of a single phase from a mixture of two phases or a decomposition of a uniform phase into two phases of different compositions. Alloys of In and Sn with neighbouring elements give us examples of these kinds of transformations. Studies on binary alloys with application of pressure provide experimental evidences that the alloy composition or electron concentration is one of the main factor for structural phase stability following from Hume-Rothery rule.

The Hume-Rothery size rule and double-well microstructures ingold-nickel

The miscibility gap and high critical temperature in Au-Ni together withcoherency strain effects in decomposition arise primarily due to the Pauliprinciple, whereby the fcc lattice parameter of Au is 1.15 times that of Ni,thus acting maximally according to Hume-Rothery against extensive solidsolution formation. Since the critical point is only about 100 °Cbelow the solidus, this simplifies the theoretical evaluation of thesolubility limitation. It also offers a bridge to spinodal kineticsencompassing a dominant solution effect of local Vegard's law strain,which acts repulsively in bulk clustering and attractively in gradient energyrelaxation. Low and high temperature TEM hot stage observations of continuousmodulated decomposition spanning the miscibility gap of Au-Ni(Tc≃1080 K) can be interpreted according to the time-dependentGinzburg-Landau (TDGL) reaction-diffusion theory modified to include soluteconservation and a diffusion-relaxable effect of strain while maintaining fulllow amplitude coherency. Modulated decomposition structures of order 3 nm,proven through lattice imaging to be spinodal in character by Sinclair et al, have often been reported near the so-called upper limit, coherentcritical temperature (~420 K), where the wavelength is supposed to beinfinite, while we report here well advanced 10 nm modulations at 350 °C abovethis. As a crystallographic template, the fine three-dimensional interleaveddendrite-like tweed pattern first decoheres, then locally transforms at 773 Kinto ~100 nm lamellar spacing intragranular nodules. These coarsenedbulk products are modulation-templated, faceted, near-spherical structures,first observed by Underwood between 673 and 973 K and by Cahn at 993 K in1954, and prove to be consistent with steady state lamellar solutions of theTDGL equation. This three-stage decomposition mode in ~100 nm thickfilms competes with the well known grain-boundary-nucleated short-circuitedlamellar product, which was originally deemed to be unique in bulk samples athigher temperatures.

Identification of the Brillouin zone planes in the Hume-Rothery matching rule and their role in the formation of the pseudogap fromab initioband calculations for theAl−Mg−Zn1/1-1/1-1/1 approximant

The Hume-Rothery matching rule has been widely used for many years as a practically useful guide to search for new quasicrystals and their approximants. In this work, we have performed the linear muffin-tin orbital atomic-sphere approximation band calculations for the nearly-free-electron-like ${\mathrm{Al}}_{30}{\mathrm{Mg}}_{40}{\mathrm{Zn}}_{30}$ 1/1-1/1-1/1 approximant. It is shown that highly degenerate free-electron states in the vicinity of the center of (543), (710), and (550) planes, whose reciprocal lattice vectors well coincide with the Fermi diameter ${2k}_{\mathrm{F}}$ in the extended zone scheme, are all reduced to the regions centered at the point N corresponding to the center of the (110) zone planes in the reduced-zone scheme and that the lifting of these degenerate states leads to the sizable pseudogap at the Fermi level, thereby lowering the electronic energy in this system. This is, to our knowledge the first attempt to identify the Brillouin zone planes in the empirical Hume-Rothery rule and to extract their role in the formation of the pseudogap from ab initio band calculations.

Interplay of atomic and electronic structures in RT- and MI-type 1/1-approximants: why does the Hume-Rothery rule work?

Interplay of atomic and electronic structures in RT- and MI-type 1/1-approximants: why does the Hume-Rothery rule work?

Direct Evidence of the Hume-Rothery Stabilization Mechanism in Al-Mn-Fe-Si Mackay-Type 1/1-Cubic Approximants

The detailed structure analysis for a series of the Al–Mn–Fe–Si Mackay-type 1/1-cubic approximants has been performed by applying the Rietveld method to the x-ray powder diffraction spectra taken with a synchrotron radiation x-ray source. We found that identical Mackay clusters occupy both the body-center and vertex positions of the cubic lattice but that atom distributions in glue sites connecting the Mackay clusters are unique to each alloy. The number of s, p-electrons per unit cell is calculated by assigning the appropriate number of valence electrons to constituent atoms involved in the refined atomic structure. It is shown that, when the Fe atoms are substituted for Mn atoms, the total number of s, p-electrons per unit cell remains unchanged and is always centered at a value satisfying the so called Hume-Rothery matching rule, where the Fermi sphere well coincides with the relevant Brillouin zone planes. We conclude that the Hume-Rothery matching condition stabilizes the ternary Al–Mn–Fe–Si approximants by compensating an increase in the valence electrons due to the replacement of Mn atoms by Fe atoms by an increase in the number of vacancies in the glue sites.

Electronic characterization of quasicrystalline surface transformations

When ion bombarded at room temperature, the surface of the icosahedral quasicrystal Al 70Pd 20Mn 10 converts to a cubic structure. Bombarded at elevated temperatures, the surface remains quasicrystalline but converts to a tenfold-symmetric structure. In this paper, we search for a change in the electronic properties as the structure is modified and argue a case against the Hume-Rothery rule for these alloys.

Determination of Spatially Hybridized Charge Distribution and its Effect on Electron Transport in the Al-Cu-Ru-Si 1/1-Approximant ---Theoretical Basis for the Hume-Rothery rule---

AbstractThe origin of the pseudogap across the Fermi level was investigated by analyzing the electronic structure calculated in the framework of the LMTO-ASA method for the RT-type Al-Mg-Zn and MI-type Al-Cu-Ru-Si 1/1-approximants. The pseudogap in the former is proved to originate from the interaction of electronic states with the Brillouin zone planes associated with reciprocal lattice vectors matching with the Fermi sphere in the extended zone scheme. In the latter, the Fermi surface-Brillouin zone interaction coupled with the hybridization effect between the Al-3p and transition metal d-states produces a deep pseudogap at the Fermi level. The real-space charge distribution for electrons at the Fermi level is calculated for the Al-Cu-Ru-Si 1/1-approximant. The charge distribution thus obtained could explain not only the possession of a large resistivity of this approximant but also evidenced that the icosahedral clusters play a key role in reducing the electronic energy in favor of quasicrystalline and approximant phases.

Role of point defects on the optical properties of Pd-based Hume-Rothery alloys

Cubic Pd 1 − x In x and Pd 1 − x Al x (0.4 <x < 0.56) intermetallic compound films have been deposited on oxidized silicon wafers by r.f. sputtering using either a multizone target or two independent targets. The actual composition was determined by electron-probe microanalysis. The characterizations by X-ray diffraction show that the films crystallize in a β-CsCl structure. This is the structure prescribed by the Hume-Rothery rule for an electron concentration of 1.5 valence electrons per atom. In non-stoichiometric compounds, constitutional point defects keep the rule enforced. The optical properties have been studied by normal incidence reflectivity and ellipsometry and have been interpreted in terms of a Lorenz-Drude model. The analyzed results show that the variations of the optical properties can be assigned to changes in the electronic density of states. These changes are caused by the point defects imposed by the deviation from stoichiometry.

自然銅と炭化水素を包有する斜長石巨晶 マグマ混合の最盛期を凍結か （２） 炭化水素を包有する斜長石巨晶とその地球化学的意義

Part 2. Hydrocarbons, verified by high-temperature Micro-FTIR, are found to be included by the same anorthite and labradorite megacrysts containing native coppers, as described in Part 1, in basaltic rocks, from Japanese island arc and in the western continental margin of North America, respectively. A finding of Muntz metal (Cu60Zn40) within the anorthite megacryst offerred the establishment of Hume-Rothery's rule in nature. Concerning the genesis of native coppers coexistent with the hydrocarbon, the exsolution model fortified by their purity and oriented distribution has been suggested and can account for crystal growth of native coppers based on the theory of Ostwald ripening. While continental margins and Island arcs have geological and tectonical backgrounds common to each other, remarkable difference in morphology between fractured labradorites and euhedral anorthites seems to flare up in the power of the volcanic eruption. Moreover it proved that the corroded olivines within anorthite megacrysts hereupon have also a trace amount of hydrocarbons, detection of which discloses that the magmas crystallizing these anorthite and olivines contained the same hydrocarbons. Fe/Mg ratio of the corroded olivines, Sr-content and Sr-isotope ratio of anorthite megacrysts typical of arc-volcanism in Japan afford the lateral variations from the Japan trench to the Back arc side, and their geochemical behaviour witnesses the rapid crystallization of anorthite megacrysts and their corroded olivines in close association with the crust components and the slab-sediments subducted beneath the island arc. Review (2) must be highly speculative because of the paucity of experimental and natural evidences on the megacrysts and their inclusions. Nevertheless it is upbeat, and the key point is that island arc magmatism should be reconstructed as the petrochemical system containing transition metals (e.g. Cu and Zn), hydrocarbons and chlorines. This reconstruction must elicit the far-reaching effect of mixing the basaltic magma with the sea water and slab-sediments rich in metals. Advances in understanding the geochemical system that formed anorthite megacrysts have come from the mineralogical study of their active inclusions providing more questions than answers.

Singularities of the electronic structure and pre-martensitic anomalies of lattice properties in β-phases of metals and alloys

Abstract Basic ideas concerning the effect of singularities in the electron energy spectrum on martensitic transitions and pre-martensitic anomalies in elastic moduli and phonon spectra of metals and alloys are reviewed. The microscopic mechanisms of the pre-martensitic anomalies in the lattice properties are thoroughly analyzed. It is shown that in some cases investigation of the singularities in the behaviour of the electron state density near the Fermi level is an efficient tool for analyzing qualitatively the stability of crystalline structures. With alkali and alkaline-earth metals taken as an example, the Hume-Rothery rule is generalized to transitions under pressure and it is shown that the loss of the structural stability can be a consequence of the Fermi level approaching a peak in the electron state density. The contribution of the singularities in the shape of the Fermi level (“nesting”) to martensitic transitions and pre-martensitic anomalies in βl-phases of alloys and intermetallic compounds ...

Formation of three types of quasicrystal in the Al-Pd-Mg system

Abstract In the Al-Pd-Mg system, a Mackay icosahedron (MI) type of icosahedral phase (I-phase) with the F-type superlattice order and a decagonal phase (D-phase) have been found to form in a melt-quenched state, in addition to the Frank-Kasper (FK) type of stable I-phase reported previously. This is the first example in which the three types of quasicrystalline phase are formed in the same alloy system. The formation range of the FK-type I-phase in the melt-quenched state is rather wide; 5–20 at.%Pd and 20–45 at.%Mg. In contrast, the MI-type I-phase and D-phase are formed in small composition ranges around Al52Pd31Mg17 and Al74Pd21Mg5 respectively. Electron diffraction studies have shown that the period along the tenfold axis of the D-phase is about 1.6 nm. The formation and stability of the MI-type and FK-type I-phases have been discussed in terms of a Hume-Rothery rule.

Hume-Rothery rule in V–VI compounds

The crystal structures of compounds of the right hand side of the Mendeleiev table can be easily understood in terms of their electronic structure. The bonding mechanism is of ppσ type and the expected simple cubic structure is subsequently deformed by a Peirls transition: some bonds are shortened and some others are elongated among the directions of the axes of the cube. The alternation of short and long bonds is driven by the electron/atom ratio.Using simple theoretical arguments, we show that the periodicity of deformation is simply related to the average number Np of p electrons. If the filling of the p band isNp6=nmwhere n and m are prime numbers, the structure shows a m-fold periodicity. We show that, up to very large m values, this rule is confirmed without any exception by an important series of compounds including AsxTe1-x, SbxTe1-x, BixSe1-x, BixTe1-x and As2GexTe3+x.

Correlation of 3d transition metal carbide phase formation with 3d electrons by carbon ion implantation

A systematic study of carbide phase formation by carbon ion implantation showed NaCl type f.c.c. carbide formation in Ti, V and Cr, and hexagonal carbide formation in Fe, Co and Ni. No carbide formation was observed in the system Cu, Ag, Au and Zn. This indicates that 3d transition metal carbide phase formation by carbon ion implantation correlates with incomplete 3d shell electrons. The s shell electrons do not contribute. The electronic factor was further demonstrated in the structural alternation from f.c.c. in systems of Ti, V and Cr to hexagon in Fe, Co and Ni with increasing 3d electron concentration, which follows the Hume-Rothery rule.

Hard-sphere properties of solid argon-xenon mixtures probed by recoilless resonant absorption.

The Mo\ifmmode\ddot\else\textasciidieresis\fi{}ssbauer effect in $^{119}\mathrm{Sn}$ was applied to obtain information on the local dynamical properties in frozen Ar-Xe mixtures. The results indicate a softening of the lattice in a composition range between 5% and 15% Xe in the mixture, and possible dealloying at higher concentrations of Xe. These results are in excellent agreement with hard-sphere model calculations and with the Hume-Rothery size rule for alloys.

Hume-Rothery stable Fibonacci alloys

Several experimental results 1,2,3,4,5 suggest that stable quasicrystals are Hume- Rothery phases 6. The singularly continuous character of the total density of states (DOS) of one dimensional quasiperiodic systems makes difficult to calculate the total energies E and occupation numbers n needed to study their stability. Reported calculations of the DOS of Fibonacci chains by decimation 7,8,9,10 are not exact and give wrong results regarding the stability. In this paper we obtain by decimation exact results for n(ε F) and E(ε F) for periodic approximants that converge rapidly to the true Fibonacci chain. It is shown that the maxima of stability fulfils the Hume-Rothery rule. The effect of disorder is also studied.

Freezing of binary hard-disc alloys. II. The fluid-disordered crystal transition

For pt.I see ibid., vol.21, p.3165 (1988). The density-functional theory of freezing is extended to the case of binary mixtures of hard discs viewed as a simple model for monolayers adsorbed on a smooth substrate. The theory is applied to the fluid-disordered crystal transition. The results are similar to those already obtained for the three-dimensional case: when the ratio of diameters is lowered, the phase diagram changes rapidly from a spindle, over a lower azeotrope to an eutectic diagram. Evidence is also found for the existence of a two-dimensional analogue of the empirical Hume-Rothery rule.