Alloys, Brillouin zone

The resonating-valence-bond theory of metals discussed in this paper differs from the older theory in making use of all nine stable outer orbitals of the transition metals, for occupancy by unshared electrons and for use in bond formation the number of valency electrons is consequently considered to be much larger for these metals than has been hitherto accepted. The metallic orbital, an extra orbital necessary for unsynchronized resonance of valence bonds, is considered to be the characteristic structural feature of a metal. It has been found possible to develop a system of metallic radii that permits a detailed discussion to be given of the observed interatomic distances of a metal in terms of its electronic structure. Some peculiar metallic structures can be understood by use of the postulate that the most simple fractional bond orders correspond to the most stable modes of resonance of bonds. The existence of Brillouin zones is compatible with the resonating-valence-bond theory, and the new metallic valencies for metals and alloys with filled-zone properties can be correlated with the electron numbers for important Brillouin polyhedra. 

In this chapter we will have a closer look at the methods of the reconstruction of the momentum densities and the occupation number densities for the case of CuAl alloys. An analogous reconstruction was successfully performed for LiMg alloys by Stutz etal. in 1995 [3], It was found that the shape of the Fermi surface changed and its included volume grew with Mg concentration. Finally the Fermi surface came into contact with the boundary of the first Brillouin zone in the [110] direction. Similar changes of the shape and the included volume of the Fermi surface can be expected for CuAl [4], although the higher atomic number of Cu compared to that of Li leads to problems with the reconstruction, which will be examined. 

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... 

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... 

Fig. 5. Interpretation of Hume-Rothery alloys in terms of Brillouin zone theory.

If a Brillouin zone is nearly full or nearly empty, the electron concentration may become an important factor in determining alloy stability. This factor forms the basis of the Hume-Rothery rules (8) and is of great importance in intermetallie semiconductors. 

The Brillouin zones also play an important part in inter-metallic compounds (p. 318). In the alloy y-brass, for example, the coincident reflection 330 and 411 is seen in the X-ray diagram to be extraordinarily strong. The inscribed sphere in... 

As a simple (indeed over-simplified) example, we take (as did Clapp) the case of the alloy CuAu we employ an essentially geometrical treatment of strain and ignore electronic effects (the effect of apbs on the Brillouin zones of the alloy, which controls the scale of the final periodicity). CuAu I has a simple superstructure of the cubic-close-packed (c.c.p.) arrangement of metal atoms in pure Cu or Au metals. In the parent, f.c.c. unit cell alternate (001) A layers of atoms contain exclusively Au and exclusively Cu (Fig. 23). Au is larger than Cu and hence, pmely in terms of size effects, the Cu layers must be under tension and the Au layers under compression if the layers are to be perfectly commensurate (as they are). The size effect is in fact seen, for the structure is metrically as well as symmetrically tetragonal the (now distorted) f.c.c. unit cell is face-centred tetragonal, with da = 0.93s instead of 1.000. In the CuAu II structure this strain is relieved (in one direction only ) by the introduction of apbs at every fifth cube plane normal to the layers (Fig. 24). 

The next advance came from the application of Fermi-Dirac statistics to the electrons in metals, which led to the band theory of a quasi-continu-ous series of energy levels, and to the concept of Brillouin zones, which is of special value for alloys. This sets the stage for a detailed study of the electronic factor in catalysis on metals. 

Apart from the decomposition products, formic acid itself can alter the resistance. As Schwab 4) found, the activation energy of the decomposition of formic acid with Hume-Rothery alloys increases if the Brillouin zone is filled up with electrons by changing the composition of the alloys. He concluded that on adsorption of formic acid, that is to say, in its activation, electrons pass over to the catalyst. This electron transfer would make the resistance decrease if formic acid does not decompose. 

In Fig. 1 the susceptibility % q) of the Ag3Mg alloy ordered as LI2 is shown. It was calculated along the F-X direction of the Brillouin zone (F-X corresponding to the... 

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