Cell unit

Rather than having to visualize an infinitely extended lattice, we can focus our attention on a small portion of it, namely a unit cell, which is a parallelopiped (or box) with lattice points located at its eight corners. The entire lattice can then be re-generated by stacking identical unit cells in three dimensions. A unit 

The choice of a unit cell depends on the best description of the symmetry elements present in the crystal structure. In practice, as far as it is possible, the unit cell is chosen such that some or all of the angles are 90° or 120°. Repetition of the unit cell by translations in three non-coplanar directions regenerates the whole space lattice. 

The unit cell is the smallest volume that contains the most essential structural information required to describe the crystal structure. For a particular crystalline compound, each lattice point is associated with a basic structural unit, which may be an atom, a number of atoms, a molecule, or a number of molecules. In other words, if the coordinates of all atoms within the unit cell are known, complete information is obtained on the atomic arrangement, molecular packing, and derived parameters such as interatomic distances and bond angles. 

In general, the size and shape of a unit cell is specified by six parameters  

The magnitudes of these lattice vectors are called the lattice parameters. They and the angles between them are unique to each element and are used for material identification. The smallest volume of repeating volume of atoms is called a primitive unit cell however, for reasons that will become apparent later, it is often convenient to define a larger unit cell that has a higher symmetry than the primitive unit cell. 

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There has been much activity in the study of monolayer phases via the new optical, microscopic, and diffraction techniques described in the previous section. These experimental methods have elucidated the unit cell structure, bond orientational order and tilt in monolayer phases. Many of the condensed phases have been classified as mesophases having long-range correlational order and short-range translational order. A useful analogy between monolayer mesophases and die smectic mesophases in bulk liquid crystals aids in their characterization (see [182]). 

This region has been divided into two subphases, L and S. The L phase differs from the L2 phase in the direction of tilt. Molecules tilt toward their nearest neighbors in L2 and toward next nearest neighbors in L (a smectic F phase). The S phase comprises the higher-ir and lower-T part of L2. This phase is characterized by smectic H or a tilted herringbone structure and there are two molecules (of different orientation) in the unit cell. Another phase having a different tilt direction, L, can appear between the L2 and L 2 phases. A new phase has been identified in the L 2 domain. It is probably a smectic L structure of different azimuthal tilt than L2 [185]. 

LS. In the LS phase the molecules are oriented normal to the surface in a hexagonal unit cell. It is identified with the hexatic smectic BH phase. Chains can rotate and have axial symmetry due to their lack of tilt. Cai and Rice developed a density functional model for the tilting transition between the L2 and LS phases [202]. Calculations with this model show that amphiphile-surface interactions play an important role in determining the tilt their conclusions support the lack of tilt found in fluorinated amphiphiles [203]. 

S. Chains in the S phase are also oriented normal to the surface, yet the unit cell is rectangular possibly because of restricted rotation. This structure is characterized as the smectic E or herringbone phase. Schofield and Rice [204] applied a lattice density functional theory to describe the second-order rotator (LS)-heiTingbone (S) phase transition. 

Metals A and B form an alloy or solid solution. To take a hypothetical case, suppose that the structure is simple cubic, so that each interior atom has six nearest neighbors and each surface atom has five. A particular alloy has a bulk mole fraction XA = 0.50, the side of the unit cell is 4.0 A, and the energies of vaporization Ea and Eb are 30 and 35 kcal/mol for the respective pure metals. The A—A bond energy is aa and the B—B bond energy is bb assume that ab = j( aa + bb)- Calculate the surface energy as a function of surface composition. What should the surface composition be at 0 K In what direction should it change on heaf)pg, and why ... 

Fig. VIII-8. Surface structures (a) (1 x 1) structure on the (100) surface of a FCC crystal (from Ref. 76) (b) C(2 x 1) surface structure on the (100) surface of a FCC ciystal (from Ref. 76). In both cases the unit cell is indicated with heavy lines, and the atoms in the second layer with pluses. In (b) the shaded circles mark shifted atoms. (See also Ref. 77.)...

Fig. XVII-18. Contours of constant adsorption energy for a krypton atom over the basal plane of graphite. The carbon atoms are at the centers of the dotted triangular regions. The rhombuses show the unit cells for the graphite lattice and for the commensurate adatom lattice. (From Ref. 8. Reprinted with permission from American Chemical Society, copyright 1993.)...

The unit cell can be defined in tenns of tluee lattice vectors (a, b, c). In a periodic system, the point x is equivalent to any point x, provided the two points are related as follows ... 

Knowing the lattice is usually not sufficient to reconstruct the crystal structure. A knowledge of the vectors (a, b, c) does not specify the positions of the atoms within the unit cell. The positions of the atoms withm the unit cell is given by a set of vectors i., = 1, 2, 3... u where n is the number of atoms in the unit cell. The set of vectors, x., is called the basis. For simple elemental structures, the unit cell may contain only one atom. The lattice sites in this case can be chosen to correspond to the atomic sites, and no basis exists. 

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. 

Copper has a FCC structure with one atom m the primitive unit cell. From simple orbital counting, one might... 

In describing a particular surface, the first important parameter is the Miller index that corresponds to the orientation of the sample. Miller indices are used to describe directions with respect to the tluee-dimensional bulk unit cell [2]. The Miller index indicating a particular surface orientation is the one that points m the direction of the surface nonual. For example, a Ni crystal cut perpendicular to the [100] direction would be labelled Ni(lOO). 

The second important parameter to consider is the size of the surface unit cell. A surface unit cell caimot be smaller than the projection of the bulk cell onto the surface. However, the surface unit cell is often bigger than... 

In addition, the surface unit cell may be rotated with respect to the bulk cell. Such a rotated unit cell is notated as... 

Most metal surfaces have the same atomic structure as in the bulk, except that the interlayer spaciugs of the outenuost few atomic layers differ from the bulk values. In other words, entire atomic layers are shifted as a whole in a direction perpendicular to the surface. This is called relaxation, and it can be either inward or outward. Relaxation is usually reported as a percentage of the value of the bulk interlayer spacing. Relaxation does not affect the two-dimensional surface unit cell synuuetry, so surfaces that are purely relaxed have (1 x 1) synuuetry. 

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