Crystal system and lattices

1 Introduction

A large majority of substances around us are solids. The distinctive features of solids are:

• They have a definite shape.
• They are rigid and hard.
• They have fixed volume.

These characteristics can be explained on the baisi of following facts:

• The costituent units of solids are held very close to each other so that the packing of the costituents is very efficient. Consequently solids have high densities.
• Since the constiuents of solids are closely packed, it imparts rigidity and hardness to solids.
• The costituents of solids are held toghetes by strong forces of actraction. This results in their having define shape and fixed volume.

Information regarding the nature of chemical forces in solids can be obtained by the study of the structure of solids, i.e. arrangements of atoms in space.

2 Classsification of solids

Solids are classified into categories:

• Amorphous solids
• Crystalline solids

The two types of solids have different characteristics.

• Amorphous Solids. An amporphous solidd is a substance whose constituents do not possess an orderly arangement. Important examples of amorphous solids are glass and plastics. Although amorphous solids consist of microcrystalline substance but the orderly arrangement is restricted to very short distances. These distances are of the same order of magnitude as the interatomic distances.
• Crystalline Solids. A crystalline solid is a substance whose constituents possess an orderly arrangement in a definite geometric pattern. Some very common examples of crystalline substances are sodium chloride, sugar and diamond. The main characteristica of crystalline substances are:
1. Orderly arrangement. The costituent units of crystalline solids are arranged in an orderly fashion which repeats itself over very long distances as compared to interatomic distances. The arrangement of bricks in a wall can be considered as an example.  The arrangement is so well defined that the entire pattern can be repeated provided the arrangement of a few atoms is known.
2. Crystals are always bounded by plane faces.
3. The faces of crystals always meet at some fixed angles. 
4. Crystalline solids exibit anisotropy in many of their properties.
   It means all those properties wich depend upon direction or angular orientation of crystals. These show different behaviour in non-parallel directions. One such consequence of anisotropy is the phenomenon of cleavage. In crystals the splitting is easier in some directions than others
5. The transition from the solid to liquid (i.e. melting point) for crystalline solids is sharp and distinct. An amorphous substance, on the other hand, has no sharp melting point. The transition fron solid to liquid in an amorphous solid does not take place at a define point but extends over a long range. The absence of sharp melting point suggests that most of amorphous solids may be best thought of as liquids.
6. Crystalline solids exhibit definite heats of fusion.

Crystalline – periodic arrangement of atoms: definite repetitive pattern

 Non-crystalline or Amorphous – random arrangement of atoms
.
The periodicity of atoms in crystalline solids can be described by a network of points in space called lattice

Atomic arrangement
 Space lattice
  A space lattice can be defined as a three dimensional array of points, each of which has identical surroundings.
  If the periodicity along a line is a, then position of any point along the line can be obtained by a simple translation, ru = ua.
 Similarly ruv = ua + vb will repeat the point along a 2D plane, where u and v are integers.

The Seven Crystal Systems

First notice: The intention of the following listing is to give you an overview rather than making you feel required to learn them by heart!

In a first step one divides the Bravais lattices into 7 crystal systems which are defined by the lengths $a$, $b$, $c$ and angles $\alpha$, $\beta$, $\gamma$ between the primitive translation vectors. The resulting crystal systems are listed and visualised below.

Crystal System	Lengths	Angles
cubic	$a=b=c$	$\alpha =\beta =\gamma ={90}^{\circ }$
trigonal	$a=b=c$	$\alpha =\beta =\gamma <{120}^{\circ }$, $\ne {90}^{\circ }$
hexagonal	$a=b\ne c$	$\alpha =\beta ={90}^{\circ }$, $\gamma ={120}^{\circ }$
tetragonal	$a=b\ne c$	$\alpha =\beta =\gamma ={90}^{\circ }$
orthorhombic	$a\ne b\ne c$	$\alpha =\beta =\gamma ={90}^{\circ }$
monoclinic	$a\ne b\ne c$	$\alpha =\beta ={90}^{\circ }\ne \gamma$
triclinic	$a\ne b\ne c$	$\alpha \ne \beta \ne \gamma$

The 14 Bravais Lattices

So one classifies different lattices according to the shape of the parallelepiped spanned by its primitive translation vectors.

However, this is not yet the best solution for a classification with respect to symmetry. Consider for example the unit cells (a) and (b) presented before: While cell (a) is the actual unit cellspanned by the primitive translation vectors, it does not show the symmetry of the latticeproperly whereas cell (b) clearly shows the two axes of rotation.

So sometimes it makes sense not to use a primitive unit cell but one which fits better to the symmetry of the problem. This idea leads to the 14 Bravais Lattices which are depicted below ordered by the crystal systems:

Cubic

There are three Bravais lattices with a cubic symmetry. One distinguishes the simple/primitive cubic (sc), the body centered cubic (bcc) and the face centered cubic (fcc) lattice.

Tetragonal

There are two tetragonal Bravais lattices with $a=b\ne c$ and $\alpha =\beta =\gamma ={90}^{\circ }$. One is primitive and the other body centered.

Orthorhombic

There are four orthorhombic Bravais lattices with $a\ne b\ne c$ and $\alpha =\beta =\gamma ={90}^{\circ }$: Primitive, body centered, face centered and base centered.

Hexagonal

When two sides are of equal length with an enclosed angle of ${120}^{\circ }$ the crystal has a hexagonal structure and thus a 6-fold rotary axis.

Monoclinic

As in the orthohombric structure, all edges are of unequal length. However, one of the three angles is $\ne {90}^{\circ }$.

Trigonal and Triclinic

The trigonal (or rhombohedral) lattice has three edges of equal length and three equal angles ($\ne {90}^{\circ }$). In the triclinic lattice, all edges and angles are unequal.