Brillouin polyhedra

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. 

Electron numbers for cubic Brillouin polyhedra are given by D. P. Shoemaker and T. C. Huang, Acta Cryst. 7, 249 (1954). 

In the papers referred to above it is pointed out that the mechanical properties of the transition elements and the distances between atoms in metals and intermetallic compounds are well accounted for by these considerations. In the following sections of the present paper a discussion is given of the number of valence electrons by the Brillouin polyhedron method, and it is shown that the calculations for the filled-zone alloys such as the 7-alloys provide further support for the new system of metallic valences. 

Fig. 2. The Brillouin polyhedron for forms [4421 and (600, an important one for y-brass and other y-alloys.

Inasmuch as the inscribed sphere corresponds to only 226 electrons per unit cube, it seems likely that the density of energy levels in momentum space has become small at 250.88, possibly small enough to provide a satisfactory explanation of the filled-zone properties. However, there exists the possibility that the Brillouin polyhedron is in fact completely filled by valence electrons. If there are 255.6 valence electrons per 52 atoms at the composition Cu6Zn8, and if the valence of copper is one greater than the valence of zinc, then it is possible to determine values of the metallic valences of these elements from the assumption that the Brillouin polyhedron is filled. These values are found to be 5.53 for copper and 4.53 for zinc. The accuracy of the determination of the metallic valences... 

It may be noted that the Brillouin polyhedron used in this calculation corresponds to the only strongly reflecting planes in the entire range 262/F 46. Thus the Brillouin polyhedron is, in effect, isolated from interference by any neighboring crystallographic planes. 

The uncertainties given are calculated standard deviations. Analysis of the interatomic distances yields a selfconsistent interpretation in which Zni is assumed to be quinquevalent and Znn quadrivalent, while Na may have a valence of unity or one as high as lj, the excess over unity being suggested by the interatomic distances and being, if real, presumably a consequence of electron transfer. A valence electron number of approximately 432 per unit cell is obtained, which is in good agreement with the value 428-48 predicted on the basis of a filled Brillouin polyhedron defined by the forms 444, 640, and 800. ... 

The brittleness of these intermetallic compounds suggests an electronic structure involving a filled Brillouin zone. It was pointed out by Ketelaar (1937) that the strongest reflection, that of form 531, corresponds to a Brillouin polyhedron for which the inscribed sphere has a volume of 217 electrons per unit cube, which agrees well with the value 216 calculated on the assumption that the sodium atom is univalent and the zinc atoms are bivalent that is, calculated in the usual Hume-Rothery way. It has also been... 

It was pointed out by Jones87 that the first important Brillouin polyhedron for the 7-alloys (bounded by the forms 330 and 411 ) contains 22.5 electrons per 13 atoms, and he proposed that some effect determined by the shape of the Brillouin polyhedron could reduce this number to 21 electrons per 13 atoms and thus explain the stability of the 7-alloys. ... 

There is, in fact, one other important Brillouin polyhedron for these crystals.88 It is bounded by the only other strongly reflecting crystallographic forms, 600 and 442, and its volume is 63.90 electrons per 13 atoms, very nearly equal to that given by the metallic valences of Table 11-1. The number 63.90 corresponds exactly to valences... 

Because (k) = (k + G), a knowledge of (k) within a given volume called the Brillouin zone is sufficient to detennine (k) for all k. In one dimension, G = Imld where d is the lattice spacing between atoms. In this case, E k) is known once k is detennined for -%ld < k < %ld. (For example, m the Kronig-Peimey model (fignre Al.3.6). d = a + b and/rwas defined only to within a vector 2nl a + b).) In tlnee dimensions, this subspace can result in complex polyhedrons for the Brillouin zone. 

The region within which k is considered (—n/a first Brillouin zone. In the coordinate system of k space it is a polyhedron. The faces of the first Brillouin zone are oriented perpendicular to the directions from one atom to the equivalent atoms in the adjacent unit cells. The distance of a face from the origin of the k coordinate system is n/s, s being the distance between the atoms. The first Brillouin zone for a cubic-primitive crystal lattice is shown in Fig. 10.11 the symbols commonly given to certain points of the Brillouin zone are labeled. The Brillouin zone consists of a very large number of small cells, one for each electronic state. 

The first Brillouin zones for the SC, BCC, and FCC lattices are shown in Figure 4.1. The inner symmetry elements for each BZ are the center, F the three-fold axis, A the four-fold axis, A and the two-fold axis, S. The symmetry points on the BZ boundary (faces) (X, M, R, etc.) depend on the type of polyhedron. The reciprocal lattice of a real-space SC lattice is itself a SC lattice. The Wigner-Seitz cell is the cube shown in Figure 4.1a. Thus, the first BZ for the SC real-space lattice is a cube with the high symmetry points shown in Table 4.3. 

The COR is constructed to calculate those extra structure constants which may be used to correct the ASA for approximate treatment of the region between the sphere and the atomic polyhedron, and for the neglect of higher components as described in Sect.6.9. The programme produces and stores on disk or tape a set of correction-term matrices distributed on a suitable grid in the irreducible wedge of the Brillouin zone. Whenever requested the correction matrices may be retrieved by LMTO and used together with the canonical structure constants to set up the corrected LMTO matrices. 

The smallest polyhedron centered at the origin, (000) in k-space, that is enclosed by the perpendicular bisectors of G to each nearest reciprocal lattice point is called the first Brillouin zone. Similarly, the smallest polyhedron enclosed by the perpendicular bisectors of the next set of reciprocal lattice vectors is the second Brillouin zone, etc. 

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