Face-centered cubic lattice holes

The geometry of ionic crystals, in which there are two different kinds of ions, is more difficult to describe than that of metals. However, in many cases the packing can be visualized in terms of the unit cells described above. Lithium chloride, LiCl, is a case in point Here, the larger Cl- ions form a face-centered cubic lattice (Figure 9.18). The smaller Li+ ions fit into holes between the Cl- ions. This puts a Li+ ion at the center of each edge of the cube. 

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. 

Figure 5.18.1 The NaCl crystal structure consisting of two interpenetrating face-centered cubic lattices. The face-centered cubic arrangement of sodium cations (the smaller spheres) is readily apparent with the larger spheres (representing chloride anions) filling what are known as the octahedral holes of the lattice. Calcium oxide also crystallizes in the sodium chloride structure.

BaTiOs crystallizes in the perovskite structure. This structure may be described as a barium-oxygen face-centered cubic lattice, with barium ions occupying the corners of the unit cell, oxide ions occupying the face-centers, and titanium ions occupying the centers of the unit cells, (a) If titanium is described as occupying holes in the Ba-O lattice, what type of hole does it occupy (b) What fraction of the holes of this type does it occupy (c) Suggest a reason why it occupies those holes of this type but not the other holes of the same type ... 

The compound Ceo is not itself a superconductor, but when alkali metals are added it becomes superconducting. The doped compound forms a face-centered cubic lattice with a lattice constant of 10.04 A, and this structure has two tetrahedral holes and one octahedral hole per Ceo molecule. If all of these holes are occupied by alkali metals A, the resulting compound is A Ceo- An example of such a compound is K2RbC6o with potassium in the smaller tetrahedral holes and rubidium in the larger octahedral holes. The transition temperatures of these doped fullerenes range from 19 to 47 K. The compound RbsCeo was found to have an isotope effect exponent a = 0.37, somewhat less than the BCS value 0.5. 

The octahedral holes at the centers of unit cells constitute just one-fourth of all the octahedral holes in a face-centered cubic lattice. (See Problem 10.2.)... 

Only those atoms that form four covalent bonds produce a repetitive three-dimensional structure using only covalent bonds. The diamond structure. Fig. 27.11, is one of several related structures in which only covalent bonds are used to build the solid. The diamond structure is based on a face-centered cubic lattice wherein four out of the eight tetrahedral holes are occupied by carbon atoms. Every atom in this structure is surrounded tetrahedrally by four others. No discrete molecule can be discerned in diamond. The entire crystal is a giant molecule. 

The zinc blende or sphalerite structure in Table 12.1 consists of a face-centered cubic lattice of the anions, but with the cations occupying only every other tetrahedral hole. Compounds that assume this structure therefore have a I I ratio and include the following species BeO, BeS, ZnO, ZnS, ZnSe, MnS, CdS, HgS, SiC, GaP, AlP, InAs, CuF, and CuCI. 

A second common type of defect in crystals is known as a Frenkel defect Frenkel defects occur when one of the ions (usually the smaller ion) becomes displaced from its normal position and occupies an interstitial site in the crystalline lattice. This occurs more frequently when there is a large difference in size between the cations and the anions. For example, the Ag+ ions in AgBr usually sit in the octahedral holes formed by a face-centered cubic lattice of Br ions. However, every so often, one of these Ag+ ions might find itself displaced to one of the smaller tetrahedral holes in the lattice. In the zinc blende ionic lattice, where every other tetrahedral hole is... 

KEY TERMS face-centered cubic crystal lattice octahedral hole... 

Spinels have a crystal structure in which there is a face-centered cubic arrangement of O2 ions. There are two types of structures in which cations have octahedral or tetrahedral arrangements of anions surrounding them. In the spinel structure, it is found that the +3 ions are located in octahedral holes and the tetrahedral holes are occupied by the +2 ions. A different structure is possible for these ions. That structure has half of the +3 metal ions located in the tetrahedral holes while the other half of these ions and the +2 ions are located in the octahedral holes. In order to indicate the population of the two types of lattice sites, the formula for the compound is grouped with the tetrahedral hole population indicated first (the position normally occupied by the +2 ion, A) followed by the groups populating the octahedral holes. Thus, the formula AB204 becomes B(AB)04 in order to correctly... 

Hagg found that metals can accommodate interstitial nonmetal atoms of radius up to 59% of that of the metal atoms. Show that, in this limiting case, accommodation of the nonmetal atoms in the octahedral holes of a face-centered cubic metal lattice should result in an expansion of the unit cell dimension by 12.4%. [Hint Review the radius ratio rules in Section 4.5.]... 

Where the lithium ions fit best will be determined by their size relative to the iodide ions. Note from above that there are two types of interstices in a closest packed structure. These represent tetrahedral (f) and octahedral (o) holes because the coordination of a small ion fitted into them is either tetrahedral or octahedral (see Fig. 4.12). The octahedral holes are considerably larger than the tetrahedral holes and can accommodate larger cations without severe distortion of the structure. In lithium iodide the lithium ions fit into the octahedral holes in a cubic closest packed lattice of iodide ions. The resulting structure is the same as found in sodium chloride and is face-centered (note that face-centered cubic and cubic closest packed describe the same lattice). 

D20.6 In a face-centered cubic close-packed lattice, there is an octahedral hole in the center. The rock-salt structure can be thought of as being derived from an fee structure of Cl ions in which Na+ ions have filled the octahedral holes. 

Face-centered cubic crystal lattice. Burns when heated with a hot enough flame (over 800, oxygen torch), df 3.513. rt 2 4173. Hardness — 10 (Mohs scale), Sp heat at 100°K 0.606 cal/g-atom/ K. Entropy at 298.I6 K 0,5684 cal/g-atom/ K. Band gap energy 6.7 ev. Dielectric constant 5.7. Electron mobility —1800 cm1/v-sec. Hole mobility 1200 cmz/v-sec. Can be pulverized in a steel mortar. Attacked by laboratory -type cleaning soln (potassi um dichromate + coned HiSO ), In the jewelry trade the unit of weight for diamonds is one carat — 200 mg. Ref Wall Street J. 164, no. 36, p 10 (Aug 19, 1964),... 

The thermochromism of Ag2[HgI4] is due to an order-disorder transition which involves no less than three phases. According to Ketalaar (33), both the yellow low-temperature 0 modification and the red high-temperature oc form contain iodide ions which are cubic close-packed, while the silver and mercury ions occupy some of the tetrahedral holes. The 0 form has tetragonal symmetry, with the mercury ion situated at the corners of a cubic unit cell and the silver ions at the midpoints of the vertical faces. As the temperature is increased it becomes possible for the silver and mercury ions to occupy each others lattice sites and also the two extra lattices sites Hop and bottom face centers of the unit cubel which were unoccupied at lower temperatures. Above 52°C. the mercury and silver ions are completely disordered. The a modification has. therefore, averaged face-centered cubic symmetry. More recently, magnetic (39) and dielectric polarization (37, 39) measurements confirm the presence of a third phase, the 0 modification. With an increase... 

The Ca " ions in fluorite are in a face-centered cubic arrangement. This lattice has, in addition to the octahedral holes mentioned earlier, holes that are tetrahedrally coordinated. The tetrahedral holes of the fee structure are occupied by F ions in fluorite. Each F" ion is tetrahedrally coordinated to Ca " ions. Figure 27.10(a) also shows that the Ca " ion on the top face is connected to four F ions below it it is similarly connected to four F ions (not shown) lying above it. The coordination of the Ca " ion is eight, and the fluorite structure is described as having 8-4 coordination. Fluorite may be considered as a face-centered cubic array of Ca " ions interpenetrated by a simple cubic array of F ions. 

In close-packed structures, for example, face centered cubic (fee) austenite, the formation of a vacancy can be envisaged as a definite process in which an anpty hole is left in an otherwise not-greatly-disturbed lattice. This concept, as has been pointed out by Lomer [78], is much less tenable for the bcc structure. In such structures, the formation of relaxed vacancies has been postulated, allowing relatively easy atomic movanent inside the disturbed region. 

The three categories differ in how they represent the compressibility and expansion of the polymer systems under scrutiny. Volumetric changes are restricted to a change in cell volume in cell models. Lattice vacancies are allowed in lattice-fluid theory, and the cell volume is assumed constant. Cell expansion and lattice vacancies are allowed by hole models. The models also differ on the lattice type, such as a face-centered cubic, orthorhombic, hexagonal, and also in their selection of interpolymer/interoligomer potential such as Lennard-Jones potential, hard-sphere, or square-well. 

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