Electron, atom ratio

The y-brass structure is adopted by a series of alloys with an electron atom ratio of 21 13, e.g. CUjZng, CugAl4., Cu3Sna. 

Reference has already been made to the high melting point, boiling point and strength of transition metals, and this has been attributed to high valency electron-atom ratios. Transition metals quite readily form alloys with each other, and with non-transition metals in some of these alloys, definite intermetallic compounds appear (for example CuZn, CoZn3, Cu3,Sng, Ag5Al3) and in these the formulae correspond to certain definite electron-atom ratios. 

It should not be thought that the structure of every intermetallic compound can be treated so simply the discussion of such struetural features as the transfer of electrons between atoms, the occurrence of strained bonds, the significance of relative atomic sizes, and the electron-atom ratio (Hume-Rothery ratio) must, however, be postponed to later papers. 

The occurrence of binary and ternary ReB2 phases is regulated by a constant electron-atom ratio e a = 4.4. 

Figure 8.6. Variation of SE between two competing magnetic states with electron/atom ratio for some 3d elements (from de Fontaine el al. 1995). Weiss (1963), A Miodownik (1978b), Bendick et al. (1977), Bendick and Pepperhoff (1978), + Bendick ef al. (1978), Moruzzi and Marcus (1990a), A Moroni and Jarlsberg (1990), O Asada and Terakuia (1995).

Nonstoichiometric compounds are mixed-valence compounds with nonintegral electron/atom ratios. Electronic properties of these compounds depend crucially on the nature and magnitude of nonstoichiometry. Electronic conduction in many such compounds occurs by hopping between the cations of different valencies (e.g. Pr " " and Pr" " in Pri2022)- Nonstoichiometry with a wide range of compositions is more common in oxides, sulphides, and related materials where the bonding is not completely ionic. In ionic nonstoichiometric compounds, structural rearrangements... 

Hume-Rothery (12,13) has pointed out that in some alloys the structure of the intermetallic phases are determined by the electron concentration (E.C.). The work of Hume-Rothery and others has shown that the series of changes (i.e. a phase —> (3 phase —> 7 phase — phase), which occurs as the composition of an alloy is varied continuously, takes place at electron-atom ratios of 3/2, 21/13, and 7/4, respectively. The interpretation of these changes in terms of the Brillouin zone theory has been made by H. Jones (14) and can be understood from the A (E)-curves for typical face centered cubic (a) and body centered cubic (6) structures as... 

HL ME-KOTHERY RULES. When alloy systems form distinct phases, it is found that the ratio or the number of valence electrons to the numher of atoms is characteristic of Lhe phase (e.g., /), y-. s- whatever the actual elements making up the alluy. Thus, both Na, Phs and NisZiii are y-structures, with the electron-atom ratio 21 13. The rules are explained by the tendency to form a structure in which all the Brillouin rones are nearly Tull, or else entirely empty. 

Figure 9.2. Distribution of energy states for an electron/atom ratio of 3/2 favoring the bcc structure.

Ordered TiFe apparently contains no localized moment, or p < 0.061 hb (480a), which is anticipated for an electron/atom ratio of 6 if the relative positions of the d and s-p bands are similar to that found in metallic chromium. 

Assuming a fixed band structure (the rigid band model), a decrease in the density of states is predicted for an increase in the electron/atom ratio for a Fermi surface that contacts the zone boundary. It will be recalled that electrons are diffracted at a zone boundary into the next zone. This means that A vectors cannot terminate on a zone boundary because the associated energy value is forbidden, that is, the first BZ is a polyhedron whose faces satisfy the Laue condition for diffraction in reciprocal space. Actually, when a k vector terminates very near a BZ boundary the Fermi surface topology is perturbed by NFE effects. For k values just below a face on a zone boundary, the electron energy is lowered so that the Fermi sphere necks outwards towards the face. This happens in monovalent FCC copper, where the Fermi surface necks towards the L-point on the first BZ boundary (Fig. 4.3f ). For k values just above the zone boundary, the electron energy is increased and the Fermi surface necks down towards the face. 

The experimental values of JV/jV for the p phase vary from 1-48 to 1-50 with the exception of the Gu-Al alloy which gives 1 37. An analogous treatment for the y and e phases may be made, but is more complex. In these cases also the agreement between the theoretical values and the experimental data is good. The electron atom ratio is not always correct, however, because other factors, such as the dimensions of the atoms and other energy relationships, may introduce complicating factors. 

A striking feature of this table is the variety of formulae of alloys with a particular structure. Hume-Rothery first pointed out that these formulae could be accounted for if we assume that the appearance of a particular structure is determined by the ratio of valence electrons to atoms. Thus for all the formulae in the first two columns we have an electron atom ratio of 3 2, for the third column 21 13, and for the fourth 7 4, if we assume the normal numbers of valence electrons for all the atoms except the triads in Group VIII of the Periodic Table. These fit into the scheme only if we assume that they contribute no valence electrons, as may be seen from the following examples ... 

Relation between electron atom ratio and crystal structure... 

Electron atom ratio 3 2 Electron atom ratio 21 13 Electron atom ratio 7 4... 

This is invariably the case with phases, the range of composition over which the phase is stable decreasing at lower temperatures. In Table 29.11 are given the experimental values of the electron atom ratios for four systems in which both (5 and 7 phases occur. The second column gives the electron atom ratio for maximum solubility in the a phase, the third for the (5 phase boundary with smallest electron concentration, and the fourth for the boundaries of the y phase. It will be seen that there is general agreement with the theoretical values, particularly in the second and third columns, though all the values for the y phase boundaries exceed the theoretical electron atom ratio. 

Absorbed hydrogen dissolves as protons with a closely associated, somewhat delocalized electron, but among the metals of greatest interest as catalysts the process is exothermal only for palladium, some palladium-rich alloys, transitional elements, and alloys with [Pg.145]

---