Systems, crystal

Crystal System Unit Cell Symmetry Bravais Latt 

FIGURE 6.4 (a) CsCl structure (primitive cubic) and (b) NaCl structure. The larger spheres 

Another property of each crystal system that distinguishes one system from another is called symmetry. There are four types of symmetry operations reflection, rotation, inversion, and rotation-inversion. If a lattice has one of these types of symmetry, it means that after the required operation, the lattice is superimposed upon itself. This is easy to see in the cubic system. If we define an axis normal to any face of a cube and rotate the cube about that axis, the cube will superimpose upon itself after each 90 of rotation. If we divide the degrees of rotation into 360°, this tells us that a cube has three fourfold rotational symmetry axes (on axes normal to three pairs of parallel faces). Cubes also have threefold rotational symmetry using an axis along each body diagonal (each rotation is 

System Axial Lengths and Angles Bravais Lattice 

Cubic Three equal axes at right angles a—b — c, q = — 7 — 90° Simple Body-centered Face-centered 

Tetragonal Three axes at right angles, two equal a = b c, a — /3 = y = 90° Simple Body-centered 

The symmetries of crystals can be divided into six basic types or crystal systems (Table 6)  

The symmetry increases from the triclinic system to the cubic system. 

Identify the crystal system to which the following octahedral crystal belongs  

Isometric. The crystal has three four-fold rotational axes that run from an apex to its opposing counterpart and four three-fold axes that run from the center of each face to the center of the opposite face. One of the four-fold axes and one of the three-fold axes are shown on the next page  

Seven different point lattices can be obtained simply by putting points at the corners of the unit cells of the seven crystal systems. However, there are other arrangements of points which fulfill the requirements of a point lattice, namely, that each point have identical surroundings. The French crystallographer Bravais worked on this problem and in 1848 demonstrated that there are fourteen possible point lattices and no more this important result is commemorated by our use of the terms Bravais lattice and point lattice as synonymous. For example, if a point is placed at the center of each cell of a cubic point lattice, the new array of points also forms a point lattice. Similarly, another point lattice can be based on a cubic unit cell having lattice points at each corner and in the center of each face. 

The fourteen Bravais lattices are described in Table 2-1 and illustrated in Fig. 2-3, where the symbols P, F, /, etc., have the following meanings. We must first distinguish between simple, or primitive, cells (symbol P or R) and nonprimitive cells (any other symbol) primitive cells have only one lattice point per cell while nonprimitive have more than one. A lattice point in the interior of a cell belongs to that cell, while one in a cell face is shared by two cells and one at a corner is shared by eight. The number of lattice points per cell is therefore given by 

System Axial lengths and angles Bravais lattice Lattice symbol 

On this basis there are seven different possible combinations of a, ft, and c and a, )3, and y, each of which represents a distinct crystal system. These seven crystal systems are cubic, tetragonal, hexagonal, orthorhombic, rhombohedral, monoclinic, and triclinic. The lattice parameter relationships and unit cell sketches for each are represented in Table 3.2. The cubic system, for which a = b = c and a = /3 = y = 90°, has the greatest degree of symmetry. The least symmetry is displayed by the triclinic system, because a = b = c and a y. 

From the discussion of metallic crystal structures, it should be apparent that both FCC and BCC structures belong to the cubic crystal system, whereas HCP falls within the hexagonal system. The conventional hexagonal unit cell really consists of three parallelepipeds situated as shown in Table 3.2. 

Concept Check 3.2 What is the difference between crystal structure and crystal system [The answer may be found at www.wiley.com/college/callister (Student Companion Site).] 

It is important to note that many of the principles and concepts addressed in previous discussions in this chapter also apply to crystalline ceramic and polymeric systems (Chapters 12 and 14). For example, crystal structures are most often described in terms of unit cells, which are normally more complex than those for FCC, BCC, and HCP. In addition, for these other systems, we are often interested in determining atomic packing factors and densities, using modified forms of Equations 3.3 and 3.8. Furthermore, according to unit cell geometry, crystal structures of these other material types are grouped within the seven crystal systems. 

System Other names Angles between Length of Examples 

Trigonal Rhombohedral a = l3 — j 90° JC — y = z Sodium nitrate Ruby Sapphire 

Hexagonal None z axis is perpen- X — y — uf z Silver iodide 

The crystal system favoured by a substance is to some extent dependent on the atomic or molecular complexity of the substance. More than 80 per cent of the crystalline elements and very simple inorganic compounds belong to the regular and hexagonal systems. As the constituent molecules become more complex, the orthorhombic and monoclinic systems are favoured about 80 per cent of the known crystalline organic substances and 60 per cent of the natural minerals belong to these systems. 

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The nematic to smectic A phase transition has attracted a great deal of theoretical and experimental interest because it is tire simplest example of a phase transition characterized by tire development of translational order [88]. Experiments indicate tliat tire transition can be first order or, more usually, continuous, depending on tire range of stability of tire nematic phase. In addition, tire critical behaviour tliat results from a continuous transition is fascinating and allows a test of predictions of tire renonnalization group tlieory in an accessible experimental system. In fact, this transition is analogous to tire transition from a nonnal conductor to a superconductor [89], but is more readily studied in tire liquid crystal system. 

A5 The Gay-Beme model for liquid crystal systems and some typical arrangements. 

Three different crystallization systems show in values of 2, 3, and 4. Calculate the value required for K in each of these systems so that all will show 6 = 0.5 after 10 sec. Use these m and K values to compare the development of crystallinity with time for these three systems. 

CAS Registry Number mol wt crystal system space group lattice constants, nm... 

Compound CAS Registry Formula Appearance Crystal system Density, Mp,°C Bp, °C Solubihty... 

Compound CAS Registry Number Formul a Appearance Crystal system and space group Density, g/cm Mp, °C Solubihty... 

Optical signs of micas are negative, crystal system is monoclinic, and the streak is colorless. 

Mineral Chemicalcomp 0 sition, w t% CaO 3 Meltingpoint, K Molecularweight Density,g/cm Crystal system... 

Material Crystal system Space group a b c Angle Density g/cm ... 

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