Basic Crystal Systems

Now we may ask, what types of lattices are possible Think of the unit cells as a bimch of identical building blocks that must fill all space when assembled. There are only seven basic shapes that meet this requirement and these shapes constitute the seven basic crystal 

---

The combination of primitive or centered cell settings with the seven basic crystal systems leads to the 14 Bravais lattices, which are depicted in Figure 10.26. All other combinations can be reduced to smaller cells without loss of symmetry. 

Crystal Systems in Minerals Shown are the seven basic crystal systems and representative crystals of each. 

The symmetries for the seven basic crystal systems described above assume the full symmetry or holohedry of each of the lattices. When the basis is added, some of these symmetries may be restricted. For example, a face-centered cubic (fee) crystal such as A1 has the full cubic symmetry. However, diamond also has the fee structure with atoms occupying the lattice points as well as every other tetrahedral interstitial point. Its point group is 43m, which implies a rotation-inversion on the fourfold axes. The threefold symmetry is preserved without the threefold rotation-inversion. The twofold symmetry is no longer preserved and the only mirror symmetry is along the 110 planes. 

There are seven different polyhedra with different symmetries that can fill all 3-D space. These polyhedra form the primitive cells of seven basic crystal systems. It is also possible to form an additional seven nonprimitive systems with higher symmetry by adding lattice points in the center or on the faces of the basic systems, thus forming the 14 Bravais lattices that describe all crystals. 

General discussions of the basic ideas have been presented in (33) and (34). Their applications to crystallization systems were reported in (3 ) and (36), and to polymerization reactors in (5) and (37) to (4J) (see also Table I). 

A single substance may crystallise in more than one of seven crystal systems, all of which differ in their lattice arrangement, and exhibit not only different basic shapes but also different physical properties. A substance capable of forming more than one different crystal is said to exhibit polymorphism, and the different forms are called polymorphs. Calcium carbonate, for example, has three polymorphs — calcite (hexagonal),... 

For this study (IS) a batch crystallization system was employed. The experimental set-up is shown in figure 1. The basic parts of the apparatus were the following ... 

Diffraction patterns can be described in terms of three-dimensional arrays called lattice points.33 The simplest array of points from which a crystal can be created is called a unit cell. In two dimensions, unit cells may be compared to tiles on a floor. A unit cell will have one of seven basic shapes (the seven crystal systems), all constructed from parallelepipeds with six sides in parallel pairs. They are defined ac-... 

A crystal lattice is an array of points arranged according to the symmetry of the crystal system. Connecting the points produces the lattice that can be divided into identical parallelepipeds. This parallelepiped is the unit cell. The space lattice can be reproduced by repeating the unit cells in three dimensions. The seven basic primitive space lattices (P) correspond to the seven systems. There are variations of the primitive cells produced by lattice points in the center of cells (body-centered cells, I) or in the center of faces (face-centered cells, F). Base-centered orthorhombic and monoclinic lattices are designated by C. Primitive cells contain one lattice point (8 x 1/8). Body-centered cells... 

The basic requirements of a crystallization system are (1) a vessel to provide sufficient residence time for crystals to grow to a desired size, (2) mixing to provide a uniform environment for crystal growth, and (3) a means of generating supersaturation. Crystallization equipment is manufactured and sold by several vendors, but some chemical companies design their own crystallizers based on expertise developed within their organizations. Rather than attempt to describe the variety of special crystallizers that can be found in the marketplace, this section will provide a brief general survey of types of crystallizers that utilize the modes outlined above. 

The most basic photochromic systems are those that undergo a light-induced structural rearrangement. Isomerizations often involve large nuclear rearrangements which, for example, can change the symmetry or convert from a linear to bent structure. This property is especially useful for doping of liquid crystals and thin films, in which the microscopic structure of individual dopant molecules can be used to modulate the macroscopic properties of a host system. 

Although there are thousands of different minerals, the shapes of their crystals can be described using just six basic geometric forms. These are called crystal systems. To determine what crystal system a mineral belongs to, it is nesessary to obtain a well-formed specimen, then observe the number and shape of the faces and the angles at which they meet. This task may be complicated by the fact that each crystal system includes several different forms, and a single crystal may combine several forms in its shape. 

The above practice is rather straightforward for a single-crystal experiment, but often provides doubtful results when only powder diffraction data are available. The basic reason is that the powder diffraction pattern is onedimensional, owing to the collapse of the reciprocal lattice of the individual crystallites onto the 26 axis. Consequently, reflections with the same r/ / modulus (i.e. with the same interplanar spacing d ki- indeed dhki= l/ rM /l) will overlap on the 29 axis. For convenience, we quote in Table 7.2 the algebraic expressions of d ki for the various crystal systems. 

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

We describe each crystal system and include crystallographic notation to help distinguish the characteristics of each type of crystal system. Figure 8-4 shows a diagram with a box that represents the basic unit of a crystal... 

Although there are only 14 basic lattices, interpenetration of lattices can occur in actual crystals, and it has been deduced that 230 combinations are posible which still result in the identity of environment of any given point. These combinations are the 230 space groups, which are divided into the 32 point groups, or classes, mentioned above in connection with the seven crystal systems. The law of Bravais has been extended by Donnay and Marker in 1937 into a more generalized form (the Bravais-Donnay-Harker Principle) by consideration of the space groups rather than the lattice types. 

---