crystal structure

1. Dept of
Education
K.SATHIS
H
M.Ed

2.  A Crystal is a three-dimensional solid which consists of a periodic arrangement of
atoms.
 A crystal is a solid form of substance (ice)
 Some crystals are very regularly shaped and can be classified into one of several shape
categories such rhombic, cubic, hexagonal, tetragonal, orthorhombic, etc.
 With pharmaceuticals, crystals normally have very irregular shapes due to dendritic
growth which is a spiky type appearance like a snowflake. It can be difficult to
characterise the size of such a crystal.
 Crystals are grown to a particular size that is of optimum use to the manufacturer.
Typical sizes in pharmaceutical industry are of the order of 50m.
What is Crystal

3. Classification of Solids
Crystalline materials
Non-crystalline
materials
 Glass
 Rubber and
 Plastics
Poly-crystal
(It has an aggregate of many
small crystals that are
separated by well-defined
boundaries)
Single crystal
(The entire solid consists of only
one crystal)
Example
Non-metallic
crystals
Metallic crystals
 Germanium
 Silicon
 crystalline carbon etc.,
 Iron
 Copper
 Silver
 Aluminium
 Tungsten etc.,

4. Crystal Symmetry
Crystals have inherent symmetry
The definite ordered arrangement of the faces and edges of a crystal is known as
`crystal symmetry’.
It is a powerful tool for the study of the internal structure of crystals.
Crystals possess different symmetries or symmetry elements.
The seven crystal systems are characterized by three symmetry elements. They are
Centre of symmetry
Planes of symmetry
Axes of symmetry.

5. Centre of Symmetry
It is a point such that any line drawn through it will meet the
surface of the crystal at equal distances on either side.
Since centre lies at equal distances from various symmetrical
positions it is also known as `centre of inversions’.
It is equivalent to reflection through a point.
A Crystal may possess a number of planes or axes of symmetry but
it can have only one centre of symmetry.
For an unit cell of cubic lattice, the point at the body centre
represents’ the `centre of symmetry’ and it is shown in the figure.

6. Plane of Symmetry
A crystal is said to have a plane of symmetry, when it is divided by an imaginary plane into two
halves, such that one is the mirror image of the other.
In the case of a cube, there are three planes of symmetry parallel to the faces of the cube and six
diagonal planes of symmetry .

7. Axis of Symmetry
This is an axis passing through the crystal such that if the crystal is rotated around it through some
angle, the crystal remains invariant. The axis is called `n-fold, axis’.
If equivalent configuration occurs after rotation of 180º, 120º and 90º, the axes of rotation are
known as two-fold, three-fold and four-fold axes of symmetry respectively.
If equivalent configuration occurs after rotation of 180º, 120º and 90º, the axes of rotation are
known as two-fold, three-fold and four-fold axes of symmetry.
If a cube is rotated through 90º, about an axis normal to one of its faces at its mid point, it brings
the cube into self coincident position.
Hence during one complete rotation about this axis, i.e., through 360º, at four positions the cube is
coincident with its original position. Such an axis is called four-fold axes of symmetry or tetrad
axis.

8. Axis of Symmetry
If n=1, the crystal has to be rotated through an angle = 360º, about an axis to achieve self
coincidence. Such an axis is called an `identity axis’. Each crystal possesses an infinite number
of such axes.
If n=2, the crystal has to be rotated through an angle = 180º about an axis to achieve self
coincidence. Such an axis is called a `diad axis’. Since there are 12 such edges in a cube, the
number of diad axes is six.
If n=3, the crystal has to be rotated through an angle = 120º about an axis to achieve self
coincidence. Such an axis is called is `triad axis’. In a cube, the axis passing through a solid
diagonal acts as a triad axis. Since there are 4 solid diagonals in a cube, the number of triad axis is
four.

9. Axis of Symmetry
If n=4, for every 90º rotation, coincidence is achieved and the axis is termed `tetrad axis’. It is
discussed already that a cube has `three’ tetrad axes.
If n=6, the corresponding angle of rotation is 60º and the axis of rotation is called a hexad axis.
A cubic crystal does not possess any hexad axis.
Crystalline solids do not show 5-fold axis of symmetry or any other symmetry axis higher than
`six’, Identical repetition of an unit can take place only when we consider 1,2,3,4 and 6 fold
axes.

10. Symmetrical Elements of cube
(a) Centre of symmetry 1
(b) Planes of symmetry 9 (Straight planes - 3, Diagonal planes - 6)
(c) Diad axes 6
(d) Triad axes 4
(e) Tetrad axes 3
----
Total number of symmetry elements = 23
----
Thus the total number of symmetry elements of a cubic structure is 23.

11.  Lattice is as an array of points in space in which the environment about each point is the same i.e., every point
has identical surroundings to that of every other point in the array.
Lattice or Space lattice
 The crystal structure is obtained by adding a unit assembly of atoms to each lattice point. This unit assembly
is called basis.
 A basis may be a single atom or assembly of atoms which is identical in composition, arrangement and orientation.
Space lattice + Basis Crystal structure
Example
• Aluminium and Barium
•Sodium chloride NaCl
• Potassium chloride KCl
Basis

12.  The unit cell is defined as the smallest geometric figure which is repeated to drive the actual crystal structure
Unit cell
Primitive cell
 A primitive cell is the simplest type of unit cell which contains only one lattice point per unit cell (contains lattice
points at its corner only)
Example:
 SC – Simple cubic
Non-primitive cell
 If there are more than one lattice point in an unit cell, it is called a non-primitive cell (contains more than one
lattice point per unit cell)
Example:
 BCC and FCC

14. Cubic (3) Simple, Body-centred, Face-centred
Tetragonal (2) Simple, Body-centred,
Orthorhombic (4) Simple, Body-centred, Face-centred, Base-centred
Monoclinic (2) Simple, Base-centred
Triclinic (1) Simple
Rhombohedral (1) Simple
Hexagonal (1) Simple
Types of Crystal systems

15. Cubic crystal system
In this crystal system, all the three axial lengths of
the unit cell are equal and they are perpendicular to
each other
a = b = c and α = β = γ = 90o
Example:
Iron, copper, Sodium Chloride (NaCl),
Calcium Fluoride (CaF2)
a
a
a
90o
90o
90o
Cubic crystal system
Y
X
Z

16. Simple Cubic Body-centred Cubic Face-centred Cubic

17. 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. All the three axes are perpendicular to
each other.
a = b ≠ c and α = β = γ = 90o
Example:
Ordinary white tin, Indium.
Tetragonal crystal system
90o
90o
90o
b
a
c
Y
X
Z

18. Body-centred Tetragonal
Simple Tetragonal

19. Orthorhombic crystal system
In this system, three axial lengths of the unit cell are
not equal but they are perpendicular to each other
a ≠ b ≠ c and α = β = γ = 90o
Example:
Sulphur, Topaz
a
b
c
90o
90o 90o
Orthorhombic crystal system
Y
X
Z

20. Simple Body-centred Face-centred Base-centred

21. Monoclinic crystal system
In this system, three axial lengths of the unit cell are
not equal. Two axes are perpendicular to each other
and third axis is obliquely inclined.
a ≠ b ≠ c and α = β = 90o, γ ≠ 90o
Example:
Sodium sulphite (Na2SO3),
Ferrous sulphate (FeSO4).
a
b
c
β ꞊ 90o
γ ≠ 90o
Monoclinic crystal system
Y
X
Z

22. Simple
Base-centred

23. Triclinic crystal system
In this system, three axial lengths of the unit cell are
not equal and all the three axes are inclined
obliquely to each other.
a ≠ b ≠ c and α ≠ β ≠ γ ≠ 90o
Example:
Copper sulphate (CuSO4),
Potassium dichromate (K2Cr2O7).
a
b
c β
γ
Triclinic crystal system
Y
X
Z

24. Rhombohedral crystal system
In this system, three axial lengths of the unit cell are
equal. They are equally inclined to each other at an
angle other than 90o.
a = b = c and α = β = γ ≠ 90o
Example:
Calcite.
a
a
a
β γ
Rhombohedral crystal system
Y
X
Z

25. Hexagonal crystal system
In this crystal system, two axial lengths of the unit cell
(horizontal) are equal and lying in one plane at angle 120o
with each other
 The third axial length (vertical) is either longer or shorter
than other two and it is perpendicular to this plane.
a = b ≠ c and α = β = 90o
γ = 120o
Example:
Quartz, Tourmaline
b
c
a
90o
120o
90o
Hexagonal crystal system
120o

26. Crystal planes
Miller introduced a set of three numbers to designate a
plane in a crystal. This set of three numbers is called
Miller indices of the concerned plane.
A
B
C
Y
X
Z
2
2
1