Tetrahedrally close-packed structures

Tetrahedrally close-packed structures. Frank-Kasper structures. A... 

Table 3.6. Examples of tetrahedral close-packed structures.

The holes in the close-packed structure of a metal can be filled with smaller atoms to form alloys (alloys are described in more detail in Section 5.15). If a dip between three atoms is directly covered by another atom, we obtain a tetrahedral hole, because it is formed by four atoms at the corners of a regular tetrahedron (Fig. 5.30a). There are two tetrahedral holes per atom in a close-packed lattice. When a dip in a layer coincides with a dip in the next layer, we obtain an octahedral hole, because it is formed by six atoms at the corners of a regular octahedron (Fig. 5.30b). There is one octahedral hole for each atom in the lattice. Note that, because holes are formed by two adjacent layers and because neighboring close-packed layers have identical arrangements in hep and ccp, the numbers of holes are the same for both close-packed structures. 

Many metals have close-packed structures, with the atoms stacked in either a hexagonal or a cubic arrangement close-packed atoms have a coordination number of 12. Close-packed structures have one octahedral and tivo tetrahedral holes per atom. 

Structure types have been established. Similar to Al, the M2X3 crystals (M = Ga, In, Tl X = S, Se, Te) are mostly based on M-defect tetrahedral structures, namely W (Ga2S3, In2Se3) and ZB (Ga2Se3, Ga2Te3, In2Te3). At atmospheric pressure, 283 can be present in three modifications. The low-temperature a-form is a cubic close-packed structure of S atoms, where 70% of the In atoms are randomly distributed on octahedral sites and the rest remain on tetrahedral sites. The P-form is related to the spinel structure, and the y-modification is hexagonal. 

When the atomic size ratio is near 1.2 some dense (i.e., close-packed) structures become possible in which tetrahedral sub-groups of one kind of atom share their vertices, sides or faces to from a network. This network contains holes into which the other kind of atoms are put. These are known as Laves phases. They have three kinds of symmetry cubic (related to diamond), hexagonal (related to wurtzite), and orthorhombic (a mixture of the other two). The prototype compounds are MgCu2, MgZn2, and MgNi2, respectively. Only the simplest cubic one will be discussed further here. See Laves (1956) or Raynor (1949) for more details. 

Several cubic structures, therefore, in which (besides 0, 0, 0 0, K, M M, 0, M M, M, 0) one or more of the reported coordinate groups are occupied could be considered as filled-up derivatives of the cubic close-packed structures. The NaCl, CaF2, ZnS (sphalerite), AgMgAs and Li3Bi-type structures could, therefore, be included in this family of derivative structures. For this purpose, however, it may be useful to note that the radii of small spheres which fit exactly into tetrahedral and octahedral holes are, respectively, 0.225. and 0.414... if the radius of the close-packed spheres is 1.0. For a given phase pertaining to one of the aforementioned types (NaCl, ZnS, etc.) if the stated dimensional conditions are not fulfilled, alternative descriptions of the structure may be more convenient than the reported derivation schemes. 

Similar considerations may be made with reference to the other simple close-packed structure, that is to the hexagonal Mg-type structure. In this case two basic derived structures can be considered the NiAs type with occupied octahedral holes and the wurtzite (ZnS) type with one set of occupied tetrahedral holes (compare with the data given with an origin shift in 7.4.2.3.2). For a few more comments about interstices and interstitial structures see 3.8.4. See Fig. 3.35. 

An important contribution to the structure analysis of intermetallic phases in terms of the coordination polyhedra has been carried out by Frank and Kasper (1958). They described several structure types (Frank-Kasper structures) as the result of the interpenetration of a group of polyhedra, which give rise to a distorted tetrahedral close-packing of the atoms. Samson (1967, 1969) developed the analysis of the structural principles of intermetallic phases having giant unit cells (Samson phases). These structures have been described as arrangements of fused polyhedra rather than the full interpenetrating polyhedra. 

The scheme of cluster condensation or cluster fragment condensation leads eventually to structures observed in bulk metals. Particularly through extensive condensation of tetrahedral and octahedral clusters, arrangements closely related to the hexagonal and cubic close-packed structures can be obtained. Condensation also of icosahedral five-fold symmetrical clusters may be related to crystalline and quasicrystalline metallic structures. 

Notes on the alloy crystal chemistry of the 6th group metals. A selection of the intermetallic phases, and of their structures, formed by Cr, Mo and W is shown in Table 5.35. Attention has been given in this list to the presence of several tetrahedrally close-packed alloys, often corresponding to ranges of solid solutions. 

Sphalerite and wurtzite structures general remarks. Compounds isostructural with the cubic cF8-ZnS sphalerite include AgSe, A1P, AlAs, AlSb, BAs, GaAs, InAs, BeS, BeSe, BeTe, BePo, CdS, CdSe, CdTe, CdPo, HgS, HgSe, HgTe, etc. The sphalerite structure can be described as a derivative structure of the diamond-type structure. Alternatively, we may describe the same structure as a derivative of the cubic close-packed structure (cF4-Cu type) in which a set of tetrahedral holes has been filled-in. This alternative description would be especially convenient when the atomic diameter ratio of the two species is close to 0.225 see the comments reported in 3.7.3.1. In a similar way the closely related hP4-ZnO... 

As pointed out in the description of the cubic close-packed structure (cF4-Cu type), this structure may be described (especially for certain values of the atomic diameter ratio) as a derivative of the Cu-type structure in which two sets of tetrahedral holes have been filled-in. 

Zr fAl3, hP7, structural type (a tetrahedrally close-packed phase) Hexagonal, space group P6, N. 174. 

Cristobahte is the third crystaUine sihca form stable at high temperature. It exists between 1,470 to 1,723°C. A metastable form may exist below 1,470°C. Cristobahte has three-layer sequences of Si04. The oxygen atoms of the tetrahedral Si04 have cubic close-packed structure. Cristobahte is found in some volcanic rocks. 

The octahedral holes in a close-packed structure are much bigger than the tetrahedral holes—they are surrounded by six atoms instead of four. It is a matter of simple geometry to calculate that the radius of a sphere that will just fit in an... 

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