UNIT I
CRYSTAL PHYSICS
INTRODUCTION
Materials are differs from one another in their properties. Some solids are brittle,
some are ductile, some are weak, some are malleable, some are good conductors of electricity
and heat, some are magnetic and so on. But all these materials are composed of atoms and
molecules. These atoms are held together by the forces of attraction. The attractive forces
which hold the particles of substance together are “bonds”. The differences in the properties
of the solids are due to their structure.
CLASSIFICATION OF SOLIDS
Most of the materials do not have any characteristic difference in their outward
appearance. But if we examine them under a microscope we shall find these materials to have
different internal atomic structures. Based on internal structures, the solids can be classified
into two categories namely
i) Crystalline solids
ii) Non – crystalline solids or amorphous materials
CRYSTALLINE SOLIDS
Crystals are those in which the atoms are arranged in an orderly fashion throughout in
a three dimensional pattern. Each atom is fixed at a definite point in space, at a definite
distance from each other and in a definite angular orientation to all other atoms surrounding
it. Therefore solids have well defined geometrical form. Further when crystal breaks, all the
broken pieces will have a regular shape. It is called as anisotropic substances.
These
crystalline solids are classified into two such as,
i) Single crystal
ii) Poly crystal
Single crystal:
The crystalline solid which contains only one crystal, it is called as single crystal. Fig
(1.1) represents the schematic structure of single crystal.
Fig 1.1 Schematic structure of single crystal
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Poly crystal:
The polycrystalline materials are aggregate of many grains separated by well defined
grain boundary. Fig (1.2) represents the schematic structure of poly crystal.
Fig 1.2 Schematic structure of poly crystal
NON – CRYSTALLINE SOLIDS (AMORPHOUS MATERIALS)
Amorphous means without form. The materials in which atoms are arranged in an
irregular fashion are known as amorphous materials. Example: rubber, glass and plastics.
The schematic representation of amorphous materials as shown in fig 1.3
Fig 1.3 Structure of amorphous materials
DIFFERENCES BETWEEN CRYSTALLINE AND NON – CRYSTALLINE
MATERIAL
S. No
1.
Non – Crystalline Material
Crystalline Material
They have a definite and regular They do not have definite and regular
geometrical shapes which extend geometrical shape
throughout the crystal
2.
They are anisotropic
They are isotropic
3.
They are most stable
They are less stable
4.
They have sharp melting point
They do not have sharp melting point
5.
Examples: NaCl, KCl
Examples: Glasses, Rubber
FUNDAMENTALS OF CRYSTALS AND ITS STRUCTURES
Crystal:
A crystal is a three dimensional solid which consists of periodic arrangement of
atoms. Crystal is regular polyhedral form bounded by smooth surfaces, which is formed by
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chemical compound under the action of its inter atomic forces, when passing from the state of
liquid to that of a solid, under suitable condition. X –rays are most widely used to study the
crystal structures, because the wavelength of X –rays are almost equal to that of the inter
atomic distances.
Crystallographic terms:
A crystal is a collection of atoms in three dimensions. As a matter of convenience,
these atoms are considered as points to study the crystal structure. The representation of
atoms in the crystals as points in three dimensions is known as space lattice.
Lattice:
Lattice is a geometrical concept. It is defined as an array of points which are
imaginarily kept to represent the position of atoms in the crystal such that every lattice point
has got the same environment as that of the other and hence one lattice point cannot be
distinguished from the other point.
Lattice plane:
A set of parallel and equally spaced plane in space lattice is defined as lattice plane
and is as shown in figure (1.4).
Lattice
point
Lattice
line
Fig 1.4
Fig 1.5
Lattice point:
The atom in the crystal is replaced by the point is called as lattice point and is as
shown in figure (1.5).
Lattice line:
The lattice points are joined with the lines as shown in figure (1.5). These lines are
known as lattice line.
Basis (motif):
The crystal structure is obtained by adding a unit assembly of atoms to each lattice
point. This unit assembly is called as motif (or) basis. The number of atoms in the basis may
be 1 or 2 or 3.etc and it may be go even above 1000 which are identical in
composition,arrangement and orientation. Example, Aluminium and Barium has the basis of
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the one atom, NaCl and KCl has the basis of the two atoms and CaF2 has the basis of the
three atoms.
Crystal structure:
When the basis is repeated in a space lattice with correct periodicity in all directions,
then it gives the actual crystal structure.
Therefore, a space lattice combines with a basis gives a crystal structure.
(i.e.,) Space lattice + Basis = Crystal Structure.
Unit cell:
It is defined as the smallest volume of a solid from which the entire crystal structure is
constructed by translation repetition in the three dimensions. The unit cell fully represents the
characteristics of entire crystal. A unit cell in three dimensions is shown in fig. 1.6.
Fig. 1.6
Lattice parameters of the unit cell:
A unit cell is constructucted if the distance between two neighbouring lattice points
along three dimensions and angle between them are known.
The distance between two neighbouring lattice point is nothing but the edges of the
unit cell. The lengths OA, OB, OC in three axes OX, OY and OZ are the axial lengths or
intercepts.(fig 1.7). In fig 1.7 the axial lengths OA = a, OB = b and OC=c are known as
intercepts a, b and c along three axes. The angles between three intercepts (α, β and γ) are
called interfacial angles. Therfore, the both intercepts and interfacial angles are the lattice
parameters of the unit cell. They determine the actual shape and size of the unit cell.
Fig 1.7
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Primitive cell & Non - Primitive cell:
A primitive cell is the simplest type of unit cell which contains only one lattice points
or atom per unit cell. Example: simple cubic.
Non - primitive cell:
If there are more than one lattice points in a unit cell, it is called a non - primitive cell.
Example: BCC and FCC.
THE CRYSTAL SYSTEMS
On the basis of lattice parameters such as intercepts or axial lengths (a, b & c) and
interfacial angles (α, β and γ) the crystals are classified into 7 crystal system.
The 7 basic crystal systems are
1. Triclinic
2. Monoclinic
3. Orthorhombic
4. Tetragonal
5. Hexagonal
6. Trigonal (or) Rhombohetral
7. Cubic
1. Triclinic crystal system:
In this crystal system, all three axial lengths
of unit cell are not equal and all the axes are
inclined obliquely to each other (fig 1.8 a).
Example: Copper Sulphate (CuSo4),
Potassium dichromate (K2Cr2O7)
2. Monoclinic crystal system:
In this crystal system, all three axial lengths
of unit cell are not equal. Two axes are
perpendicular to each other and third is
obliquely inclined (fig 1.8 b).
Example: Sodium Sulphate (Na2So3),
Ferrous Sulphate (FeSo4)
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3. Orthorhombic crystal system:
In this crystal system, all three axial lengths
of unit cell are not equal but they are
perpendicular to each other (fig 1.8 c).
Example: Sulphur, Topaz
4. Tetragonal crystal system:
In this crystal system, two axial lengths of
the unit cell are equal and third axial length is
either longer or shorter (fig 1.8 d). but they are
perpendicular to each other
Example: Ordinary white tin, Indium
5. Hexagonal crystal system:
In this crystal system, two axial lengths of
the unit cell are equal and lying in one plane at
120˚ with each other. The third axial length is
either longer or shorter than the other two and it
is perpendicular to this plane (fig 1.8 e).
Example: Quartz, Tourmaline.
6. Trigonal (or rhombohedral) crystal system:
In this crystal system, all three axial lengths
of unit cell are equal. And also they are equally
inclined to each other at an angle other than 90˚
(fig 1.8 f).
Example: Calcite
7. Cubic crystal system:
In this crystal system, all three axial lengths of
unit cell are equal and they are perrpendicular to
each other (fig 1.8 g).
Example: Sodium Chloride (NaCl),
Calcium Fluoride (CaF2)
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Table 1.1
Seven crystal systems
S.No
Crystal systems
Axial
lengths
(a, b, c)
Interfacial angles
(α, β, γ)
Example
1
Triclinic
a ≠b ≠c
α≠ β ≠ γ ≠ 90
CuSo4, K2Cr2O7
2
Monoclinic
a ≠b ≠c
α = β = 90˚ , γ ≠ 90˚
Na2So3, FeSo4
3
Orthorhombic
a ≠b ≠c
α =β = γ = 90˚
Sulphur, Topaz
4
Tetrahedral
a =b ≠c
α =β = γ = 90˚
Ordinary white tin,
Indium
5
Hexagonal
a =b ≠c
α = β = 90˚ , γ = 120˚
Quartz, Tourmaline
6
Trigonal (or)
Rhombohedral
a=b=c
α = β = γ ≠ 90˚
Calcite
7
Cubic
a=b=c
α = β = γ = 90˚
NaCl, CaF2
BRAVAIS LATTICE
In 1880 Bravais introduced the concept of space lattice. He showed that there are only
14 ways of arranging points in space such that the environment looks same from each point.
Hence, there are only 14 types of space lattices which can be possibly developed from
7 crystal systems as shown in table (1.2). These 14 types of space lattices are known as
Bravais lattice.
Table 1.2
Possible Bravais Lattice
Crystal
No.of possible
S.No
systems
Bravais Lattices
1
2
3
Triclinic
Monoclinic
Orthorhombic
1
2
4
4
5
Tetrahedral
Hexagonal
Trigonal (or)
Rhombohedral
Cubic
2
1
Total
14
6
7
1
3
The 14 possible Bravais lattices drawn from the crystal systems are shown in table 1.3.
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Table 1.3
Bravais lattices of seven crystal systems
S.
No
Crystal systems
1
Triclinic
( a ≠ b ≠ c,
α ≠ β ≠ γ ≠ 90 )
2
Monoclinic
(a ≠ b ≠ c,
α = β = 90 ,
γ ≠ 90 )
3
Orthorhombic
( a ≠ b ≠ c,
α = β = γ = 90 )
4
Tetrahedral
(a = b ≠ c,
α = β = γ = 90 )
5
6
7
Bravais Lattices
Hexagonal
(a = b ≠ c,
α = β = 90 ,
γ ≠ 120 )
Trigonal (or)
Rhombohedral
(a = b = c,
α = β = γ ≠ 90 )
Cubic
(a = b = c,
α = β = γ = 90 )
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