Face-centered-cubic lattice

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. 

LiCl. the Cl- ions aie in contact with each other, forming a face-centered cubic lattice. In NaCl, the Cl- ions are forced slightly apart by the larger Na+ ions. In CsCl. the large Cs+ ion at the center touches the Cl ions at each comer of the cube. 

To conclude this section it may be worthwhile to remark that, although the L-J-D method is generally applied to a face-centered cubic lattice (z = 12), it is equally valid for the cavities in a clathrate (z = 20—28), as long as one restricts oneself to first-neighbor interactions. 

In hydrates with their open structure the relative contribution of second and third neighbor solvent molecules to w(r) is only of the order of i of that in the much denser face-centered cubic lattice. It is therefore a better approximation to neglect second and third neighbors altogether than to use the functions derived by Wen tor f et al.u for the face-centered cubic lattice including contributions due to second and third shell neighbors. 

Magnesia forms solid solutions with NiO. Both MgO and NiO have face-centered cubic lattices with NaCl-type structures. The similarity between the ionic radii of the metals (Ni2+ = 0.69 A, Mg2+ = 0.65 A) allows interchangeability in a crystal lattice, and thus the formation of solid solutions with any proportion of the two oxides is possible. Such solid solutions are more difficult to reduce than NiO alone. Thus Takemura et al. (I) demonstrated that NiO reduced completely at 230°-400°C (446°-752°F) whereas a 10% NiO-90% MgO solid solu-... 

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. 

Hauke (1937) reported the same structure for NaZn13 and two other compounds (KZn13 and KCd13), and later (1938) for two more (RbCd13 and CsCd13). The structure is based on a face-centered cubic lattice. 

Reciprocal spacings (1 jdm — qm = (2 sin 6)1 X) were calculated from the positions of the powder lines, and the lines were indexed on the basis of a face-centered cubic lattice. Analysis of a few lines in the back-reflection region gave a preliminary value of 0-081409 A-1 for 1 ja0. A refinement of l/a0 was carried out by... 

It has been found that adamantane crystallizes in a face-centered cubic lattice, which is extremely unusual for an organic compound. The molecule therefore should be completely free from both angle strain (since all carbon atoms are perfectly tetrahedral) and torsional strain (since all C—C bonds are perfectly staggered), making it a very stable compound and an excellent candidate for various applications, as wUl be discussed later. 

The face-centered cubic lattice can be viewed as a simple cube with each face (a) expanded just enough to fit an additional atom in the center of each face giving eight atoms at the comers and six atoms... 

Figure 4.3.2 The diamond crystalline lattice structure composed of two interpenetrating face-centered cubic lattices.

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.

Figure 9.2 is schematic diagram of the crystal structure of most of the alkali halides, letting the black circles represent the positive metal ions (Li, Na, K, Rb, and Cs), and the gray circles represent the negative halide ions (F, Cl, Br, and I).The ions lie on two interpenetrating face-centered-cubic lattices. Of the 20 alkali halides, 17 have the NaCl crystal structure of Figure 9.1. The other three (CsCl, CsBr, and Csl) have the cesium chloride structure where the ions lie on two interpenetrating body-centered-cubic lattices (Figure 9.3). The plastic deformation on the primary glide planes for the two structures is quite different.

The fourth and final crystal structure type common in binary semiconductors is the rock salt structure, named after NaCl but occurring in many divalent metal oxides, sulfides, selenides, and tellurides. It consists of two atom types forming separate face-centered cubic lattices. The trend from WZ or ZB structures to the rock salt structure takes place as covalent bonds become increasingly ionic [24]. 

The ion lattice model of Ref.20, applied to metals, leads to the values of 360, 210, and 730 erg/cm2 for the 111 face of lithium, sodium, and aluminum, respectively, all of which crystallize in the face-centered cubic lattices. See also Ref.21. ... 

The shape of the single crystal obtained by the method described above is a sphere with several flat facets as drawn in Fig. 2-6. Usually seven large facets, which are assigned to seven of possible eight (111) surfaces, are seen on the apex positions of a cube. Five small facets, which are assigned to five of possible six (100) surfaces, are also seen on the center of the faces of a cube. The missing (111) and (100) facets are supposed to be located where the shaft is attached. Figure 2—7 shows the relative positions of the three low index surfaces of platinum, which is a face-centered cubic lattice. 

Fig. 2-7 Three low index surfaces of Pt (face centered cubic lattice) si e crystal and their surface atom arrangement and atomic density.

The clean siuface of solids sustains not only surface relaxation but also surface reconstruction in which the displacement of surface atoms produces a two-dimensional superlattice overlapped with, but different from, the interior lattice structure. While the lattice planes in crystals are conventionally expressed in terms of Miller indices (e.g. (100) and (110) for low index planes in the face centered cubic lattice), but for the surface of solid crystals, we use an index of the form (1 X 1) to describe a two-dimensional surface lattice which is exactly the same as the interior lattice. An index (5 x 20) is used to express a surface plane in which a surface atom exactly overlaps an interior lattice atom at every five atomic distances in the x direction and at twenty atomic distances in the y direction. 

The side chain separation varies in a range of 1 nm or slightly above. The network of aqueous domains exhibits a percolation threshold at a volume fraction of 10%, which is in line with the value determined from conductivity studies. This value is similar to the theoretical percolation threshold for bond percolation on a face-centered cubic lattice. It indicates a highly interconnected network of water nanochannels. Notably, this percolation threshold is markedly smaller, and thus more realistic, than those found in atomistic simulations, which were not able to reproduce experimental values. 

As stated in Problem 3.1, copper crystallizes into a face-centered cubic lattice with a unit cell edge length a = 3.62 A. Calculate the number of Cu atoms per cm exposed on each surface of the (100), (110), and (111) planes. 

Studies on crystalline CggO [39] using calorimetry and high-resolution X-ray powder diffraction show a face centered cubic lattice (a = 14.185 A) with an orientational disorder at room temperature. An orientational ordering transition occurs at 278 K, upon which a simple cubic phase develops. At 19 K this phase, which is similar to the orientational ordered phase of Cgg itself, shows additional randomness due to a distribution of orientation of the oxygens in CggO. 

Reaction with methane at 2,100°C produces hafnium carbide, a dark-gray, brittle solid, which is not a true stoichiometric compound. It probably is a homogeneous mixture in which carbon impregnates interstitial sites in the face-centered cubic lattice of hafnium. 

As a example, for the face centered cubic lattice the (100), (010) and (001) planes look identical and the (110), (101) and (Oil) planes look identical. But the sphere also has a backside with a (-100) plane, a (0-10) plane etc what about them The answer comes in the form of the stereograpic map. The stereographic map is an area on the unit sphere, which contains one representative for each equivalent surface orientation. 

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