Hume-Rothery electron phases

The most famous example of the crystal structure correlating with the average number of valence electrons per atom or band filling, N, is the Hume-Rothery alloy system of noble metals with sp-valent elements, such as Zn, Al, Si, Ge, and Sn. Assuming that Cu and Ag have a valence of 1, then the fee -phase is found to extend to a value of N around 1.38, the bcc / -phase to be stabilized around 1.48, the -phase around 1.62, and the hep e-phase around 1.75, as illustrated for the specific case of Cu-Zn alloys in Fig. 6.15. In 1936 Mott and Jones pointed out that the fee and bcc electron per atom ratios correlate with the number of electrons required for a free-electron Fermi sphere first to make contact with the fee and bcc Brillouin zone faces. The corresponding values of the Fermi vector, fcF, are given by 

the shape of the band energy difference curves in Fig. 6.16(a) can be understood in terms of the relative behaviour of the densities of states in the middle panel. In particular, from eqn (6.111), the stationary points in the upper curve correspond to band occupancies for which A F vanishes in panel (c). Moreover, whether the stationary point is a local maximum or minimum depends on the relative values of the density of states at the Fermi level through eqn (6.113). Thus, the bcc-fcc energy difference curve has a minimum around N = 1.6 where the bcc density of states is lowest, whereas the hcp-fcc curve has a minimum around N = 1.9, where the hep density of states is lowest. The fee structure is most stable around N = 1, where A F fts 0, and the fee density of states is lowest. 

We see that the structural trend from fee - bcc hep is driven by the van Hove singularities in the densities of states. These arise whenever the band structure has zero slope as occurs at the bottom or top of the energy gaps at the Brillouin zone boundaries. The van Hove singularities at the bottom of the band gap at X and at the top of the band gap at L in fee copper are marked X4. and Ly, respectively, in the middle panel of Fig. 6.16. It is, thus, not totally surprising that the reciprocal-space representation 

Ducastelle, F. (1991). Order and phase stability in alloys. North-Holland, Amsterdam. 

Hafner, J. (1987). From hamiltonians to phase diagrams the electronic and statistical mechanical theory of sp-bonded metals and alloys. Springer, Berlin. 

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Figure 4.40. Valence electron (s,p, d) concentration ranges for different types of phases as reported by Lee and Hoistad (1995). Experimental average values are compared with those computed. (For the conventional names of the phases, compare with Table 4.6.) On the left transition metal binary alloys, on the right the Hume-Rothery electron phases are shown.

Discuss the origin of the Hume-Rothery electron phases within the framework of Jones original rigid-band analysis. How does second-order perturbation theory help quantify Mott and Jones earlier supposition on the importance of the free electron sphere touching a Brillouin zone boundary ... 

The simple model fits observation remarkably well for pure metals. However, this model begins to fail for alloys (solutions of one metal in another), such as brass and bronze, where for certain stoichiometries the material has anomalous physical properties and behaves almost as a componnd (e.g., CuZn and Cu5Sn, termed Hume-Rothery electron phases). 

The phase stability of crystalline electron phases or Hume-Rothery (HR) phases has been explained by the afore-mentioned band-structure effects. For this purpose, the k- as well as r-space representation have been used successfully [5.45,46]. Crystalline HR-phases are well documented and excellent text books or reviews exist in this field [5.13,14, 35]. In the present section, only a few facts are mentioned in order to show how glassy metals belong to this class of phases. 

By crystallographic transformations one generally means that, when a new phase (product) forms from the old (parent), it bears certain deHnite geometrical relationships. The most widely studied crystallographic transformation is the martensitic transformation, the prototype of which occurs in quenched steels. Actually, martensite (in honor of Professor A. Martens) was the name given by Osmond in 1895 to the microstructure observed in quenched steels, but in more modern times the word martensite designates a transformation mechanism, now known to be associated with many metals, alloys, ceramics, and even some polymers. These transformations occur in a variety of intermetallics, most notably the Hume-Rothery electron compounds as found, for example, in /3-brass of near-equiatomic composition, and many similar alloys of Cu, Ag, and Au. 

Hume-Rothery s rule The statement that the phase of many alloys is determined by the ratio.s of total valency electrons to the number of atoms in the empirical formula. See electron compounds. 

Hume-Rothery phases (brass phases, electron compounds ) are certain alloys with the structures of the different types of brass (brass = Cu-Zn alloys). They are classical examples of the structure-determining influence of the valence electron concentration (VEC) in metals. VEC = (number of valence electrons)/(number of atoms). A survey is given in Table 15.1. 

The components of polar intermetallics generally include an active metal from the group 1 or 2 or the rare-earth series plus, sometimes, a late-transition metal, and a metal from the p-block. Because of the presence of an electron-poorer late transition metal, polar intermetallics generally have lower e/a values (about 2.0-4.0) than classic Zintl phases (>4.0) [45], Note these values are traditionally calculated over only electronegative atoms [45], in contrast to those of Hume-Rothery phases (<2.0) [45] and QC/ACs (2.0 0.3) [25], for which electron counts are considered to be distributed over all atoms. The former two higher values are decreased to about 1.5-2.5 and >2.5, respectively, when counted over all atoms (but with omission of any dw shells). For comparison purposes, Fig. 3 sketches the distribution of all these intermetallic phases according to e/a counted over all atoms, as we will use hereafter. 

Since all known QC systems, with e/a of about 1.75-2.20 [25], lie close to the approximate border between the Hume-Rothery and polar intermetallic phase regions, a reasonable starting place for development of new QC/AC systems is to study selected polar intermetallic systems with nearby e/a values. Synthetic explorations of such polar intermetallics have been significant only in the past few decades [42,45], Knowledge and insights developed about the diverse interplays between composition-structure-electronic structure-physical properties for these phases were expected to be a considerable aid to the discovery of novel QC/ACs. 

An important class of intermetallic phases (generally showing rather wide homogeneity ranges) are the Hume-Rothery phases, which are included within the so-called electron compounds . These are phases whose stable crystal structures may be supposed to be mainly controlled by the number of valence electrons per atom, that is, by the previously defined VEC. 

The Hume-Rothery phases constitute an interesting and ubiquitous group of binary and complex intermetallic substances it was indeed Hume-Rothery who, already in the twenties, observed that one of the relevant parameters in rationalizing compositions and structures of a number of phases is the average number of valence electrons per atom (nJnM). An illustration of this fact may be found in Table 4.6, where a number of the Hume-Rothery structure types have been collected, together with a few more major structure types relevant to transition metal alloys. For each phase the corresponding VEC has been reported as njnai ratio, both calculated on the basis of the s and p electrons and of s, p and d electrons. 

Remarks on the alloy crystal chemistry of the 11th group metals. A selection of the phases formed in the binary alloys of Cu, Ag and Au and of their crystal structures is shown in Tables 5.54a and 5.54b. For a number of these phases, more details (and a classification in terms of Hume-Rothery Phases ) are given in 4.4.5 and in Table 4.5 (structure types, valence electron concentration, etc.). Table 5.54a and 5.54b show the formation of several phases having a high content... 

The Cu5Zn8 ( Cu5Zn6 9 — Cu5Zn9 7) phase is a classical example of a Hume-Rothery phase ( electron compounds , brass-type phases) that is of a phase in which there is a structure-determining influence of the VEC (valence electron concentration, see 4.4.5). 

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