Close packing tetrahedral hole

Zinc Blende (Cubic Close Packed, Tetrahedral Holes Half Occupied)... 

Compounds that have the empirical formulas MCr02 and DCr204 where M is a monovalent and D a divalent cation, are known as chromites. These are actually mixed oxides and probably are better written as M20-Cr203 and D0-Cr203, respectively. The oxides of D are largely spinels, ie, the oxygen atoms define a close-packed cubic array having the octahedral holes occupied by the Cr(III) cation and the tetrahedral holes occupied by D (54). Chromite ore is an important member of this class of oxides. 

Fig. 16.2. Both the f.c.c. and the c.p.h. structures are close-packed. Both contain one octahedral hole per atom, and two tetrahedral holes per atom. The holes in the f.c.c. structures ore shown here.

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. 

FIGURE 5.30 The lot ations of (a) tetrahedral and (b) octahedral holes Note that both types of holes are defined by two neighboring close-packed layers, so they are present with equal abundance in both hep and ccp 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. 

When the radius ratio of an ionic compound is less than about 0.4, corresponding to cations that are significantly smaller than the anion, the small tetrahedral holes may be occupied. An example is the zinc-blende structure (which is also called the sphalerite structure), named after a form of the mineral ZnS (Fig. 5.43). This structure is based on an expanded cubic close-packed lattice of the big S2 anions, with the small Zn2+ cations occupying half the tetrahedral holes. Each Zn2+ ion is surrounded by four S2 ions, and each S2" ion is surrounded by four Zn2+ ions so the zinc-blende structure has (4,4)-coordination. 

Buckminsterfullerene is an allotrope of carbon in which the carbon atoms form spheres of 60 atoms each (see Section 14.16). In the pure compound the spheres pack in a cubic close-packed array, (a) The length of a side of the face-centered cubic cell formed by buckminsterfullerene is 142 pm. Use this information to calculate the radius of the buckminsterfullerene molecule treated as a hard sphere, (b) The compound K3C60 is a superconductor at low temperatures. In this compound the K+ ions lie in holes in the C60 face-centered cubic lattice. Considering the radius of the K+ ion and assuming that the radius of Q,0 is the same as for the Cft0 molecule, predict in what type of holes the K ions lie (tetrahedral, octahedral, or both) and indicate what percentage of those holes are filled. 

Tetrahedral and octahedral interstitial holes are formed by the vacancies left when anions pack in a ccp array, (a) Which hole can accommodate the larger ions (b) What is the size ratio of the largest metal cation that can occupy an octahedral hole to the largest that can occupy a tetrahedral hole while maintaining the close-packed nature of the anion lattice (c) If half the tetrahedral holes are occupied, what will be the empirical formula of the compound MVAV, where M represents the cations and A the anions ... 

Ziegler-Natta catalyst A stereospecific catalyst for polymerization reactions, consisting of titanium tetrachloride and triethylaluminum. zinc-blende structure A crystal structure in which the cations occupy half the tetrahedral holes in a nearly close packed cubic lattice of anions also known as sphalerite structure. 

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. 

This structure could also be described as derived from a cubic close-packed array of atoms (Bi atoms) by filling all the tetrahedral and octahedral holes with Li atoms. 

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. 

Nitrogen Compounds.—The formation of ZrNj 9 by a new method of evaporation synthesis has been described. The compound LiZrN2 has been prepared from LijN and ZrNi 22 and X-ray crystallographic studies have shown that the crystals contain hexagonally close-packed nitride ions, with Zr" and Li ions in octahedral and tetrahedral holes, respectively. ... 

The smallest A-ions, Li+ and Mg +, favour crystal structures in which they are octahedrally coordinated. This is the case in compounds of e.g. the LiSbFa-, LiaZrFe- and Na2SiF6-types. In the only fluoride spinel, Li2NiF4, half of the Li-ions is even tetrahedrally coordinated (page 34). In all cases these smallest A-ions, similar to the transition ions themselves, occupy the holes between the closed-packed fluoride layers. 

Similarly, Figure 1.6(a) shows two close-packed layers, now with the second type of space, tetrahedral holes, shaded. Four spheres surround each of these holes with centres at the corners of a tetrahedron (Figure 1.6(b)). If n spheres are in the array, then there are 2n tetrahedral holes. 

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... 

FIGURE 1.6 (a) Two layers of close-packed spheres with the tetrahedral holes shaded (h) a computer representation of a tetrahedral hole. 

Below 146°C, two phases of Agl exist y-Agl, which has the zinc blende structure, and (3-Agl with the wurtzite structure. Both are based on a close-packed array of iodide ions with half of the tetrahedral holes filled. However, above 146°C a new phase, a-AgI, is observed where the iodide ions now have a body-centred cubic lattice. If you look back to Figure 5.7, you can see that a dramatic increase in conductivity is observed for this phase the conductivity of a-Agl is very high, 131 S m , a factor of 10 higher than that of (3- or y-AgI, comparable with the conductivity of the best conducting liquid electrolytes. How can we explain this startling phenomenon ... 

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