The unit cell hexagonal and cubic close-packing

The unit cell hexagonal and cubic close-packing 

A unit cell is a fundamental concept in solid state chemistry, and is the smallest repeating unit of the structure which carries all the information necessary to construct unambiguously an infinite lattice. 

Close-packing of spheres results in the most efficient use of the space available 74% of the space is occupied by the spheres. 

The smallest repeating unit in a solid state lattice is a unit 

Place one sphere over every other hollow in layer A 

LayerB cemtains two diffMent types of hollow (see text) 

By placmg spheres in one or other of these different hollows, two new layers of qtheres can be produced. 

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The best way to determine the type of unit cell adopted by a metal is x-ray diffraction, which gives a characteristic diffraction pattern for each type of unit cell (see Major Technique 3 following his chapter). However, a simpler procedure that can be used to distinguish between close-packed and other structures is to measure the density of the metal we then calculate the densities of the candidate unit cells and decide which structure accounts for the observed density. Density is an intensive property, which means that it does not depend on the size of the sample (Section A). Therefore, it is the same for a unit cell and a bulk sample. Hexagonal and cubic close packing cannot be distinguished in this way, because they have the same coordination numbers and therefore the same densities (for a given element). 

In this experiment you will build models of the simple cubic, body-centered cubic, and face-centered cubic unit cells, of hexagonal close-packed structures and cubic close-packed structures, and of simple ionic solids. 

The differing malleabilities of metals can be traced to their crystal structures. The crystal structure of a metal typically has slip planes, which are planes of atoms that under stress may slip or slide relative to one another. The slip planes of a ccp structure are the close-packed planes, and careful inspection of a unit cell shows that there are eight sets of slip planes in different directions. As a result, metals with cubic close-packed structures, such as copper, are malleable they can be easily bent, flattened, or pounded into shape. In contrast, a hexagonal close-packed structure has only one set of slip planes, and metals with hexagonal close packing, such as zinc or cadmium, tend to be relatively brittle. 

The corresponding unit cells are shown in Figure 1.1 and an examination of simple ball-and-stick models (which the reader is strongly urged to carry out) shows that the face-centred cubic (fee) and hexagonal close-packed (hep) structures correspond to the only two possible ways of close-packing spheres, in which each sphere has twelve nearest neighbours. 

Answer (c) is correct atoms at the comers, along the edges, and on the faces of a unit cell are shared with adjacent unit cells, (a) is incorrect often there are several formula units within a unit cell, as in the case for NaCl. (b) is incorrect the unit cell need not be cubic that of hexagonal close packing is not. (d) is incorrect a unit cell will not contain the same number of cations as anions if their numbers are not equal in the formula of the compound. 

In the sphalerite structure the anions form a cubic close packed array. The structure has a single adjustable parameter, the cubic cell edge. The 0 ions are too small for them to be in contact in this structure (see Fig. 6.4) so ZnO adopts the lower symmetry hexagonal wurtzite structure which has three adjustable parameters, the a and c unit cell lengths and the z coordinate of the 0 ion, allowing the environment around the Zn " ion to deviate from perfect tetrahedral symmetry. In the sphalerite structure the ZnX4 tetrahedron shares each of its faces with a vacant octahedral cavity (one is shown in Fig. 2.6(a)), while in the wurtzite structure one of these faces is shared with an empty tetrahedral cavity which places an anion directly over the shared face as seen in Fig. 2.6(b). The primary coordination number of Zn " in sphalerite is 4 and there are no tertiary bonds, but in wurtzite, which has the same primary coordination number, there is an additional tertiary bond with a flux of 0.02 vu through the face shared with the vacant tetrahedron. 

The concept of close packing is particularly useful in describing the crystal structures of metals, most of which fall into one of three classes hexagonal close packed, cubic close packed (i.e., fee), and body-centered cubic (bcc). The bcc unit cell is shown in Fig. 4.8 its structure is not close packed. The stablest structures of metals under ambient conditions are summarized in Table 4.1. Notable omissions from Table 4.1, such as aluminum, tin, and manganese, reflect structures that are not so conveniently classified. The artificially produced radioactive element americium is interesting in that the close-packed sequence is ABAC..., while one form of polonium has... 

Wurtzite structure. Zinc sulfide can also crystallize in a hexagonal form called wurtzite that is formed slightly less exothermically than the cubic zinc blende (sphalerite) modification (Afff = —192.6 and —206.0 kJ mol-1, respectively) and hence is a high temperature polymorph of ZnS. The relationship between the two structures is best described in terms of close packing (Section 4.3) in zinc blende, the anions (or cations) form a cubic close-packed array, whereas in wurtzite they form hexagonal close-packed arrays. This relationship is illustrated in Fig. 4.13 note, however, that this does not represent the actual unit cell of either form. 

Figure 6. TEM images of STAC-1 viewed down the a axis of a hexagonal unit cell (indicated by [M/]h) or the [110] direction of a cubic unit cell (indicated by [M/]c). The crystal is dominated by ABCABC close packing (indicated on (a)) with one stacking fault (marked by a horizontal line). A Fourier transform optical diffraction pattern with both Miller-Bravais indices to the hexagonal unit cell and Miller indices (in parentheses) to the cubic unit cell is inserted in (b). Simulated images based on a proposed model (right) are also inserted with specimen thickness of 30 nm, and lens focuses of—30 nm (a) and —10 nm (b).

Various schemes have been proposed for the classification of the different alumina structures (Lippens and Steggerda, 1970). One approach was to focus attention on the temperatures at which they are formed, but it is perhaps more logical to look for differences in the oxide lattice. On this basis, one can distinguish broadly between the a-series with hexagonal close-packed lattices (i.e. ABAB...) and the y-series with cubic close-packed lattices (i.e. ABCABC...). Furthermore, there is little doubt that both y- and j/-A1203 have a spinel (MgAl204) type of lattice. The unit cell of spinel is made up of 32 cubic close-packed O2" ions and therefore 21.33 Al3+ ions have to be distributed between a total of 24 possible cationic sites. Differences between the individual members of the y-series are likely to be due to disorder of the lattice and in the distribution of the cations between octahedral and tetrahedral interstices. 

Clearly, there are two choices for placing each plane, and an infinite number of crystal structures can be generated that have the same atomic packing density. The two simplest such structures correspond to the periodic layer sequences abcabcab. . . and abababa.. . . The first of these is the fee structure already discussed, and the second is a close-packed structure in the hexagonal crystal system termed hexagonal close-packed (hep). In each of these simple structures, atoms occupy 74.0% of the unit cell volume, as the following example shows. (Atoms that crystallize in the bcc structure occupy only 68.0% of the crystal volume, and the packing fraction for a simple cubic array is only 52.4%.)... 

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