< Engineering Physics -I > < Crystal Physics – Lattice, Unit Cell and Bravais lattices >
Introduction (Attention Grabber)
Learning Objectives
On completion of this chapter you will be able to know about:
Crystalline and non crystalline solids
Crystal system
A solid is that form of matter that possesses rigidity and hence possesses a
definite shape and a definite volume. There are two types of solids:
Crystalline solids
Solids with a definite geometric pattern.
Examples: Iron, copper, silver, sulphur etc. are some elements which form crystalline
solids. Potassium chloride, sodium nitrate etc are some of the compounds, which are
crystalline.
Amorphous solids
Solids with particles not arranged in a regular fashion. They have only short
range order or even the particles are disordered in some cases.
Examples: Glass. Rubber and Plastics
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Classification of Crystalline Solids
Ionic
Molecular
Covalent
and
Metallic
Classification of Crystalline solids
Some substances adopt different structural arrangements under different
conditions. Such arrangements are called Polymorphs. Example: Diamond and graphite
are two different polymorphic forms of carbon.
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The study of the geometric form and other physical properties of crystalline solids by
using X-rays, neutron beams and electron beams constitute the science of
crystallography.
Crystallography is mainly used to determine the internal atomic arrangements in
crystal, bonding and their strength.
Lattice Points:
Lattice points denote the position of atoms or molecules in the crystals.
Space Lattice:
The three dimensional space lattice may be defined as an infinite array of points in
three dimensions in which every point has an identical environment as any other point in
the array (Fig.1).
Fig.1.
Thus a crystal lattice refers to the geometry of set of points in space. As infinite three
dimensional arrays of points showing how atoms or molecules are arranged in a crystal is
known as space lattice or lattice array. In the lattice array, every point has surroundings
or environment identical to that every point in the array.
Basis and Crystal Structure:
In order to convert the geometrical array of points (i.e. the lattice) in a crystal
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structure, we must locate atoms or molecules on the lattice points. The repeating unit
assembly- atom, molecule, ion or radical – that is located at each lattice point is called
basis. Thus the basis is an assembly of atoms identical in composition, arrangement and
orientation. Thus a crystal structure is formed by the following logical relation.
Space lattice + basis = crystal structure
Unit Cell:
The crystal structure of a material or the arrangement of atoms in a crystal structure
can be described in terms of its unit cell. The unit cell is a tiny box containing one or
more motifs, a spatial arrangement of atoms. The units cells stacked in three-dimensional
space describes the bulk arrangement of atoms of the crystal (Fig.2.).
Fig.2.
Unit cell definition using parallelepiped with lengths a, b, c and angles between the
sides given by α, β,γ. The size and shape of a unit cell is determined by the lengths of the
edges of the unit cell (a, b and c) and by the angles α, β and γ between the edges b and c,
c and a, and a and b respectively.
The intercepts a, b and c define the dimensions of a unit cell and are known as its
primitives or characteristic intercepts on the axes. These three quantities a,b and c are
also called the fundamental translational vectors.
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< Engineering Physics -I > < Crystal Physics – Lattice, Unit Cell and Bravais lattices >
The primitives a, b and c and the interfacial angles α, β and γ are the basis lattice
parameters because they determine the form and actual size of the unit cell and hence the
space lattice. The unit cell formed by the primitives a,b and c is called primitive cell. In a
primitive cell, there is only one lattice point. If there are two or more lattice points then it
is not a primitive cell. In case of simple cubic crystal lattice, the primitive cell and unit
cell are equal since it has only one lattice point in its unit cell. But most of the unit cells
of various crystal lattice contain two or more lattice points and it is not necessary that
the unit cell should be equal to the primitive cell.
Crystal Systems
The symmetry of the axial distances (a, b, c) and also the axial angles between
the edges (α, β and γ) the various crystals (Table-1) can be divided into seven systems
(Fig.3) These are also called crystal habits.
Table -1
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Fig.3.
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Bravais lattices
When the crystal systems are combined with the various possible lattice
centerings, we arrive at the Bravais lattices. They describe the geometric arrangement of
the lattice points, and thereby the translational symmetry of the crystal. In three
dimensions, there are 14 unique Bravais lattices which are distinct from one another in
the translational symmetry they contain. All crystalline materials recognized until now
(not including quasicrystals) fit in one of these arrangements. The Bravais lattices are
sometimes referred to as space lattices.
The crystal structure consists of the same group of atoms, the basis, positioned
around each and every lattice point. This group of atoms therefore repeats indefinitely in
three dimensions according to the arrangement of one of the 14 Bravais lattices. The
characteristic rotation and mirror symmetries of the group of atoms, or unit cell, is
described by its crystallographic point group. The following fourteen different types of
lattices are known as Bravais lattices (Fig.4).
Fig.4
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Fig.4
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Check your understanding
1. The three axes of a crystal lattice are mutually perpendicular and two of the lattice
parameters are equal. The crystal systems is
(a) Cubic (b) Tetragonal (c) Orthorhombic (d) hexagonal
2. Which of the following is an example of tetragonal lattice?
(a) Calcite (b) Potassium (c) Mercury Chloride (d) None
Check the correct answers on page: 9
Summary
On completion of this chapter you have learned that:
1. Crystals have directional properties and are anisotropic substances
2. The regular arrangement of the space positions of the atoms in a crystal is called
space lattice.
3. A crystal structure is developed by the combination of space lattice and basis.
4. A unit cell is defined as the volume of a solid from which the entire crystal can be
constructed by translational repetition in three dimensions.
5. The intercepts on the crystallographic axes a, b and c which define the dimension
of a unit cell and the interfacial angles α, β and γ are the basic lattice parameters.
6. There are seven crystal systems which can form fourteen Bravais lattices in the
three dimensions.
Activity
Prepare the models of seven crystal systems
Suggested Reading
1. Engineering Physics by Dr.P.K.Palanisamy, Scitech Publishers, Chennai-17
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2. Engineering Physics by Dr.G.Senthilkumar, VRB Publishers, Chennai-92
Answers to CYU.
1. (b)
2. (d)
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