Crystal Systems and Unit Cells

S1 CRYSTAL STRUCTURES S1.1 Crystal Systems and Unit Cells 

In crystals of any material, the atoms present are always arranged in exactly the same way, over the whole extent of the solid, and exhibit long-range translational order. A crystal is conventionally described by its crystal structure, which comprises the unit cell, the symmetry of the unit cell, and a list of the positions of the atoms that lie in the unit cell. 

Crystal structures and crystal lattices are different, although these terms are frequently (and incorrectly) used as synonyms. A crystal structure is built of atoms. A crystal lattice is an infinite pattern of points, each of which must have the same surroundings in the same orientation. A lattice is a mathematical concept. 

The unique axis in the monoclinic unit cell is mosdy taken as the b axis. Rhombohedral unit cells are often specified in terms of a bigger hexagonal unit cell. 

Different compounds that crystallize with the same crystal structure, for example, the two alums, NaAl(SC 4)2 12H20 and NaFe(SC 4)2 12H20, are said to 

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The unit cell content. To complete the description of the crystal structure, the list of the atoms contained in the unit cell and their coordinates (fractional coordinates related to the adopted system and unit cell edges) are then reported. These are usually presented in a format such as M El in n x, y, z. In the MoSi2 structure, also reported in Table 3.2, and in Fig. 3.7, for instance, four silicon atoms... 

What are the spacings of the 100, 110, and 111 planes in a cubic crystal system of unit cell dimension a In what sequence would you expect to find these reflections in a powder diffraction photograph ... 

The x-ray diffraction powder pattern for Mg2NiH4 was indexed on the basis of a tetragonal crystal system with unit cell dimensions of a = 6.464 A and c = 7.033 A. The measured density was 2.57 g/cm3, compatible with four formula units of Mg2NiH4 per unit cell. 

Within a given crystal system, there are in some cases several different types of crystal lattice, depending upon the type of minimum-size unit cell that corresponds to a choice of axes appropriate to the given crystal system. This unit cell may be primitive P or in certain cases body-centered I, face-centered F, or end-centered A, B, or Q depending on which pair of end faces of the unit cell is centered. The lattices are designated as primitive, body-centered, face-centered, or end-centered depending on whether the smallest possible unit cell that corresponds to the appropriate type of axes is primitive, body-centered, face-centered, or end-centered. There are in all 14 types of lattice, known as Bravais lattices. In the cubic system there are three primitive, body-centered, and face-centered these are shown in Fig. 2. 

The X-ray diffraction of a bundle of ten pieces of P(3HB) fibers (beam size 300 pm) as shown in Fig. 2c includes reflections simultaneously from both the a-structure (2/1 helix conformation Okamura and Marchessault 1967 Yokouchi et al. 1973) and the p-structure (planar zigzag conformation Orts et al. 1990) of P(3HB) that are drawn in Fig. 3. It is well known that P(3HB) crystallizes as an orthorhombic crystal system with unit cell parameters of a = 0.576 nm, ft = 1.320 nm, and c(fiber axis) = 0.598 nm and space group (a-structure ... 

The a-phase PLLA is most common structure that can be obtained fi om crystallization in melt, solution and fiber spinning with low strain rate and temperature. Santis and Kovacs [213] first determined the conformation of a-phase PLLA and PDLA. PLLA is left handed IO7 helix, and a right handed IO3 helix for the i/-isomer PDLA. Both PLA are orthorhombic crystal systems with unit-cell parameters of a = 1.06 nm 6=0.610 nm and c=2.88 nm. The ratio of a and b is 1.737, which is nearly equal to /3 indicating an almost hexagonal packing of helices. Figure 4.46 shows the crystal structure of a phase PLLA. 

In certain of the seven crystal systems, the unit cells can contain face- or body-centered building blocks. For the cubic lattice, the body- and face-centered unit cell is shown in Figure 2.7. 

Macroscopic morphology, crystal system information Unit cell and space group Site symmetry in crystalline and amorphous materials Position of atoms, thermal vibration amplitudes Imperfections... 

The problems already mentioned at the solvent/vacuum boundary, which always exists regardless of the size of the box of water molecules, led to the definition of so-called periodic boundaries. They can be compared with the unit cell definition of a crystalline system. The unit cell also forms an "endless system without boundaries" when repeated in the three directions of space. Unfortunately, when simulating hquids the situation is not as simple as for a regular crystal, because molecules can diffuse and are in principle able to leave the unit cell. 

A unit cell for the calcite structure can be found, on the Web site for this book. From this structure, determine (a) the crystal system and (b) the number of formula units present in the unit cell. 

Figure 3.4. The crystal systems and the Bravais lattices illustrated by a unit cell of each. All the points which, within a unit cell, are equivalent to each other and to the cell origin are shown. Notice that, in the primitive lattices the unit cell edges are coincident with the smallest equivalence distances. For the rhombohedral lattice, described in terms of hexagonal axis, the symbol hR is used instead of a symbol such as rP. In the construction of the so-called Pearson symbol ( 3.6.3), oS and mS will be used instead of oC and mC.

In the nineteenth century, when crystal morphology was systematized to fourteen types of unit cells, seven crystal systems and thirty-two crystal groups, the following two macroscopic treatments on the morphology of crystals emerged. 

The next step is for a protein crystallographer to mount a small perfect crystal in a closed silica capillary tube and to use an X-ray camera to record diffraction patterns such as that in Fig. 3-20. These patterns indicate how perfectly the crystal is formed and how well it diffracts X-rays. The patterns are also used to calculate the dimensions of the unit cell and to assign the crystal to one of the seven crystal systems and one of the 65 enantiomorphic space groups. This provides important information about the relationship of one molecule to another within the unit cell of the crystal. The unit cell (Fig. 3-21) is a parallelopiped... 

Crystal system and space group Unit cell dimensions... 

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