Cubic lattice unit cell

Potassium crystallizes in a body-centered cubic lattice (unit cell length a = 520 pm). 

Using your largest spheres, construct a single simple cubic lattice unit cell. Using dots to represent centers of atoms, draw a diagram to represent your model. 

Construct a single body-centered cubic lattice unit cell. 

A primitive cubic lattice unit cell has atoms at the corners but each one of them is shared by eight neighboring unit cells and therefore the total contribution of corner atoms is equivalent to only one. A body-centered cubic unit cell has only two (/ = 0) and a face-centered cubic unit cell has four, one due to eight corner points and three due to centre points on each of six faces. These face centered points are shared by two neighboring unit cells. 

The initial configuration is set up by building the field 0(r) for a unit cell first on a small cubic lattice, A = 3 or 5, analogously to a two-component, AB, molecular crystal. The value of the field 0(r) = at the point r = (f, 7, k)h on the lattice is set to 1 if, in the molecular crystal, an atom A is in this place if there is an atom B, 0, is set to —1 if there is an empty place, j is set to 0. Fig. 2 shows the initial configuration used to build the field 0(r) for the simple cubic-phase unit cell. Filled black circles represent atoms of type A and hollow circles represent atoms of type B. In this case all sites are occupied by atoms A or B. 

In this diagram, a series of hexagon-shaped planes are shown which are orthogonal, or 90 degrees, to each of the corners of the cubic cell. Each plane connects to cuiother plane (here shown as a rectangle) on each fiace of the unit-cell. Thus, the faces of the lattice unit-cell and those of the reciprocal unit-cell can be seen to lie on the same pltme while those at the corners lie at right angles to the corners. 

Two modifications are known for polonium. At room temperature a-polonium is stable it has a cubic-primitive structure, every atom having an exact octahedral coordination (Fig. 2.4, p. 7). This is a rather unusual structure, but it also occurs for phosphorus and antimony at high pressures. At 54 °C a-Po is converted to /3-Po. The phase transition involves a compression in the direction of one of the body diagonals of the cubic-primitive unit cell, and the result is a rhombohedral lattice. The bond angles are 98.2°. 

There are very many unit cell structures if we consider all the atoms or ions in the crystal. However, if we focus on just one atom or ion, we can reduce the number to just 14 primitive cells. Three of these, the simple cubic, face-centered cubic (fee), and body-centered cubic (bcc) unit cells, are shown in Figure 9-2. The lattice points, represented by small spheres in the drawings, correspond to the centers of the atoms, ions, or molecules occupying the lattice. 

Properties of cubic system unit cells One metal atom occupies each lattice point Primitive... 

The face-centered cubic (fee) unit cell common to the alkali halides except for CsCl, CsBr, and Csl is shown in Fig. 13 these crystals all cleave fairly easily along the (100) planes. One should note that the unit surface cell for the (001) plane has lattice vectors rotated by 45° from those of the bulk. The generic direct and reciprocal lattices for the ((X)l) surfaces of these materials are shown in Fig. 14. 

The bcc structures of iron, molybdenum, and tungsten are not close-packed as are those of fee and hep, and more interstitial sites are available for the free valence electrons. In the bcc lattice the octahedral holes are in the middle of the faces and in the center of the edges of the unit cell. Four tetrahedral sites form a circle in the eight faces of the unit cell. A second group of four tetrahedral sites forms a circle around all twelve edges of a cubic bcc unit cell. A maximum of two free electrons per face and per edge can be accommodated in the octahedral sites and two electrons... 

In a cubic giant unit cell 24 tetrameric (NaMe)4 units form a zeolitic host lattice. This structure is related to that of methyllithium, but the arrangement of the tet-ramers is more complicated. It results in the presence of large cavities, in which (LiMe)4 units can be intercalated up to the structurally predetermined molar ratio of 3 1. In this sense the host-guest compound (NaMe)4 [(LiMe)4]4 x < 0.333), 39, was termed the first organometallic supramolecular compound . 

The FCC structure is illustrated in figure Al.3.2. Metallic elements such as calcium, nickel, and copper fonu in the FCC structure, as well as some of the inert gases. The conventional unit cell of the FCC structure is cubic with the lengdi of the edge given by the lattice parameter, a. There are four atoms in the conventional cell. In the primitive unit cell, there is only one atom. This atom coincides with the lattice pomts. The lattice vectors for the primitive cell are given by... 

The rocksalt stmcture is illustrated in figure Al.3.5. This stmcture represents one of the simplest compound stmctures. Numerous ionic crystals fonn in the rocksalt stmcture, such as sodium chloride (NaCl). The conventional unit cell of the rocksalt stmcture is cubic. There are eight atoms in the conventional cell. For the primitive unit cell, the lattice vectors are the same as FCC. The basis consists of two atoms one at the origin and one displaced by one-half the body diagonal of the conventional cell. 

Figure Bl.21.1. Atomic hard-ball models of low-Miller-index bulk-temiinated surfaces of simple metals with face-centred close-packed (fee), hexagonal close-packed (licp) and body-centred cubic (bcc) lattices (a) fee (lll)-(l X 1) (b)fcc(lO -(l X l) (c)fcc(110)-(l X 1) (d)hcp(0001)-(l x 1) (e) hcp(l0-10)-(l X 1), usually written as hcp(l010)-(l x 1) (f) bcc(l 10)-(1 x ]) (g) bcc(100)-(l x 1) and (li) bcc(l 11)-(1 x 1). The atomic spheres are drawn with radii that are smaller than touching-sphere radii, in order to give better depth views. The arrows are unit cell vectors. These figures were produced by the software program BALSAC [35]-...

The summation is over the different types of ion in the unit cell. The summation ca written as an analytical expression, depending upon the lattice structure (the orij Mott-Littleton paper considered the alkali halides, which form simple cubic lattices) evaluated in a manner similar to the Ewald summation this typically involves a summc over the complete lattice from which the explicit sum for the inner region is subtractec... 

Below a temperature of Toi 260 K, the Ceo molecules completely lose two of their three degrees of rotational freedom, and the residual degree of freedom is a ratcheting rotational motion for each of the four molecules within the unit cell about a different (111) axis [43, 45, 46, 47]. The structure of solid Ceo below Tqi becomes simple cubic (space group Tji or PaS) with a lattice constant ao = 14.17A and four Ceo molecules per unit cell, as the four oriented molecules within the fee structure become inequivalent [see Fig. 2(a)] [43, 45]. Supporting evidence for the phase transition at Tqi 260 K is... 

Fig. 2. Structures for the solid (a) fee Cco, (b) fee MCco, (c) fee M2C60 (d) fee MsCeo, (e) hypothetical bee Ceo, (0 bet M4C60, and two structures for MeCeo (g) bee MeCeo for (M= K, Rb, Cs), and (h) fee MeCeo which is appropriate for M = Na, using the notation of Ref [42]. The notation fee, bee, and bet refer, respectively, to face centered cubic, body centered cubic, and body centered tetragonal structures. The large spheres denote Ceo molecules and the small spheres denote alkali metal ions. For fee M3C60, which has four Ceo molecules per cubic unit cell, the M atoms can either be on octahedral or tetrahedral symmetry sites. Undoped solid Ceo also exhibits the fee crystal structure, but in this case all tetrahedral and octahedral sites are unoccupied. For (g) bcc MeCeo all the M atoms are on distorted tetrahedral sites. For (f) bet M4Ceo, the dopant is also found on distorted tetrahedral sites. For (c) pertaining to small alkali metal ions such as Na, only the tetrahedral sites are occupied. For (h) we see that four Na ions can occupy an octahedral site of this fee lattice.

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