Filling the Unit Cell

The lattice by itself does not define a crystal structure. To generate a crystal structure, we need to associate an atom or group of atoms with each lattice point. In the simplest case, the crystal stmcture consists of identical atoms, and each atom lies directly on a lattice point. When this happens, the crystal structure and the lattice points have identical patterns. Many metallic elements adopt such structures, as we will see in Section 12.3. Keep in mind that only elements can form structures of this type. For compounds, even if we were to put an atom on every lattice point, the points would not be identical because the atoms are not all of the same. 

The crystal structure of graphene illustrates two important characteristics of crystals. First, we see that no atoms lie on the lattice points. While most of the structures we discuss in this chapter do have atoms on the lattice points, there are many examples, like graphene, where this is not the case. Thus, to build up a structure you must know the location and orientation of the atoms in the motif with respect to the lattice points. Second, we see that bonds can be formed between atoms in neighboring unit cells. This happens in many crystals, particularly metallic, ionic, and network-covalent solids. 

Lattice points at comers plus one lattice point in center of unit cell 

Lattice points at Motif inside comers of unit cell unit cell 

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Four molecules, i.e., 80 atoms fill the unit cell. The space group is Pbcn. The bond characteristics are similar to those of Sxg, S12, and Sfibrousi and lie between the values for Sg and Sg ... 

Because the diffraction pattern of a crystal is the periodic superposition (or product, or convolution) of the continuous transform of the unit cell contents with the lattice transform, other interesting consequences follow. For example, the locations of reflections in the diffraction pattern of a crystal, the net or lattice on which they fall, is entirely determined by the lattice properties of the crystal, namely the unit cell vectors. They in no way depend on the structure or properties of the molecules that fill the unit cells. On the other hand, the intensity we measure at each point in the diffraction pattern, and its associated phase, is entirely determined by the distribution of electrons, the positions of atoms xj, yj, Zj, within the unit cells. 

FIGURE 5.13 The structure factor Fhki is a wave, and it therefore has an amplitude and phase. It can also be described as a vector in the complex plane, as was seen in Chapter 4. The individual wave contributions to F%ki by each atom in the molecule can also be described by vectors. The sum of these vectors for all atoms yields Fhki The vectors added here correspond to contributions from the five atoms of the molecule filling the unit cell in Figure 5.12. 

Although we have repeatedly discussed the interplay of real space and diffraction space, described many of their properties, and seen many examples in Chapter 6, it may be useful to review their relationship once again before proceeding. Table 7.1 outlines the principle relationships, some in terms of precise quantitative or mathematical terms and others in terms of what one observes in diffraction space as a consequence of properties or events in real space. In simplest terms we may think of real space as the physical crystal, the unit cells which make it up, and the distribution of atoms that fill the unit cells. We should in practice think of reciprocal space as diffraction space, or the pattern of diffraction intensities produced by a crystal exposed to X-rays, and thepoints in space where they may be observed. 

In Figure 11.35 the Na and Q ions have been moved apart so the symmetry of the stmcture can be seen more clearly. In tiiis representation no attention is paid to the relative sizes of the ions. The representation in Figure 11.36 , on flie other hand, shows the relative sizes of the ions and how they fill the unit cell. Notice that the particles at comers, edges, and faces are shared by other unit cells. 

When the nodes for one net are connected check for other, interpenetrating, nets in the structure. This is done by filling the unit cell with nodes and expanding. At this point, start using lots of different colours for the individual... 

Let us now look at the c.p.h. unit cell as shown in Fig. 5.4. A view looking down the vertical axis reveals the ABA stacking of close-packed planes. We build up our c.p.h. crystal by adding hexagonal building blocks to one another hexagonal blocks also stack so that they fill space. Here, again, we can use the unit cell concept to open up views of the various types of planes. 

Fig. 3.—Parallel packing arrangement of the 2-fold helices of cellulose I (1). (a) Stereo view of two unit cells approximately normal to the ac-plane. The two comer chains (open bonds) in the back, separated by a, form a hydrogen-bonded sheet. The center chain is drawn in filled bonds. All hydrogen bonds are drawn in dashed lines in this and the remaining diagrams, (b) Projection of the unit cell along the c-axis, with a down and b across the page. No hydrogen bonds are present between the comer and center chains.

Fig. 8.—Packing arrangement of four symmetry-related 2-fold helices of mannan II (6). (a) Stereo view of two unit cells approximately normal to flic frc-plane. The two chains in the back (open bonds) and the two in the front (filled bonds) are linked successively by 6-0H-- 0-6 bonds. The front and back chains, both at left and right, are further connected by 0-2 -1V -0-2 bridges, (h) Projection of the unit cell along the c-axis the a-axis is down the page. This highlights the two sets of interchain hydrogen bonds between antiparallel chains, distinguished by filled and open bonds. The crossed circles are water molecules at special positions.

Fig. 9. — Antiparallel packing arrangement of the 3-fold helices of (1— 4)-(3-D-xylan (7). (a) Stereo view of two unit cells roughly normal to the helix axis and along the short diagonal of the ab-plane. The two helices, distinguished by filled and open bonds, are connected via water (crossed circles) bridges. Cellulose type 3-0H-0-5 hydrogen bonds stabilize each helix, (b) A view of the unit cell projected along the r-axis highlights that the closeness of the water molecules to the helix axis enables them to link adjacent helices.

Fig. 14.—Antiparallel packing arrangement of extended, 4-fold, 2,3,6-tri-O-ethylamylose (12) helices, (a) Stereo view of two unit cells approximately normal to the lie-plane. The helix at the center (filled bonds) is antiparallel to the two helices (open bonds) at the comers in the back. There is no intra- or inter-chain hydrogen bond, and only van der Waals forces stabilize the helices, (b) A e-axis projection of the unit cell shows that the ethyl groups extend into the medium in radial directions. 

Fig. 20. (continued)—(b) A oaxis projection of the unit cell shows large gaps between the helices, which are allegedly filled by 250 water molecules. 

Fig. 21.—Structure of the 6-fold anhydrous curdlan III (19) helix, (a) Stereo view of a full turn of the parallel triple helix. The three strands are distinguished by thin bonds, open bonds, and filled bonds, respectively. In addition to intrachain hydrogen bonds, the triplex shows a triad of 2-OH - 0-2 interchain hydrogen bonds around the helix axis (vertical line) at intervals of 2.94 A. (b) A c-axis projection of the unit cell contents illustrates how the 6-0H - 0-4 hydrogen bonds between triple helices stabilize the crystalline lattice.

As schematically represented in Fig. 3 the structure can be considered two interpenetrating fee lattices of 8,2(8,2)12 units the 8,2(8,2)12 units of each fee lattice differ only by the 90° rotation of these units. Thus there are eight of these 8,2(8,2)12 units or 1248 8 atoms in the unit cell. The metal atom positions and the location of the remaining 8 atoms in the structure can be pictured in the octant of the cell shown in Fig. 3. Six metal atom sites exist in each octant of the ceil, and these are statistically half-filled. The sites are located 1.27 10 pm (for YB g) inside the cell from the center of each face of an octant one such site is depicted in Fig. 3. The center of each octant is occupied by either a 36- or a 48-8 atom group, which are labeled, respectively, configurations I and II (Fig. 4). Half of the octants contain configuration I, and half contain 11 in a random fashion. ... 

Symmetry axes can only have the multiplicities 1,2,3,4 or 6 when translational symmetry is present in three dimensions. If, for example, fivefold axes were present in one direction, the unit cell would have to be a pentagonal prism space cannot be filled, free of voids, with prisms of this kind. Due to the restriction to certain multiplicities, symmetry operations can only be combined in a finite number of ways in the presence of three-dimensional translational symmetry. The 230 possibilities are called space-group types (often, not quite correctly, called the 230 space groups). 

Facility is provided to bring the equipment up to the desired absolute pressure without subjecting the test unit to excessive pressure difference between its interior and exterior. This is accomplished by first filling the test cell with water by means of a hand operated hydraulic pump(17 in Figure 1) to a suitable value as indicated on the Bourdon dial gauges(P in Figure 1). The pressure thus developed is used also to control the appropriate back-pressure relief valve... 

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