The document provides an overview of crystallography, detailing the geometrical properties, structures, and classifications of crystals and their systems. It explains key concepts such as unit cells, crystal symmetry, coordination number, and packing factors, alongside methodologies for determining Miller indices and interplanar spacing. Additionally, it discusses various crystal structures like simple cubic, body-centered cubic, and face-centered cubic, outlining their atomic arrangements and characteristics.

2. Crystal
System
Crystallography
It is a branch of science which deals with geometrical properties and structure of
crystal and crystalline substance.
Interfacial angle
The angle between any two faces of a crystal is called interfacial angle.

3. Crystal symmetry:
Definite ordered arrangement of atoms in a crystal is known as crystal symmetry.
Every crystal must possess three types of symmetry:
(a) Plane of symmetry
(b) Axis of symmetry
(c) Centre of symmetry
Plane of symmetry:
It is an imaginary plane which can
divide the crystal into half such that one
is mirror image of the other
Axis or line of symmetry:
A symmetry axis is a line passing through the crystal so
that the definite angular rotation of the crystal produces
exactly same original appearance.

4. Centre of symmetry:
A centre of symmetry is a point in the crystal such that any
straight line through it passes through a pair of similar
points situated at a same distance but on the opposite
side of centre of symmetry
Lattice Crystal
Structure
space lattice : the regular arrangement of an
infinite set of points (atom, ions or molecules) in
space is called space lattice.
Basis :
The crystal structure is formed by associating
every lattice point with an assembly of atoms or
molecules or ions, which are identical in
composition, arrangement and orientation, is
called as the basis

5. In copper and sodium crystals, the basis is single atoms;
in NaCl, the basis is diatomic and in CaF2 the basis is
triatomic.

6. Unit Cell
The smallest repeating pattern (unit) from which the lattice is known is called unit cell.
The unit cells are repeated over and over again in three dimensions, and as a result into the whole of
space lattice of the crystal.
For describing unit cell, we must know
(a) The distance a, b and c, i.e. length of edge of unit cell (called lattice parameter)
(b) The angles α, β and γ between three imaginary lines
Types of unit cell
The unit cells are of four types:
1. Primitive or simple
2. Face-centred
3. Body-centred
4. End-centred

7. Primitive Cell
A unit cell which has only one atom or one lattice point is known as a primitive cell
Thus, a primitive cell can be a unit cell, but a
unit cell need not always be a primitive cell.
A primitive unit cell contains only one lattice point. If a unit
cell contains more than one lattice point, then it is called
non-primitive or multiple cells. For example, BCC and
FCC are non-primitive unit cells.

8. Classification
of Crystal
Systems

9. Classification
of Crystal
Systems

13. Crystal Parameters
Nearest neighbour distance (2r)
The distance between the centres of two nearest neighbour atoms is known as the nearest neighbour
distance
a = 2r, where r is the radius of the atom.
Co-ordination number (CN)
In a crystal, the number of equidistant
nearest neighbouring atoms that a
reference atom has in the lattice is known
as the co-ordination number, N.

14. Atomic packing factor (packing fraction)
Atomic packing factor is defined as the ratio of the volume (v) occupied by the effective
number of atoms in a unit cell to the total volume (V) of the unit cell, i.e.,
Calculation of Atomic Packing Fraction for Various Systems
•• Find the relationship between the radius (r) and the lattice constant (a) of the structure.
•• Determine the effective number of atoms within the structure.
•• Find the total volume of the effective number of atoms.
•• Divide the effective atom volume with the total volume of the structure.
•• Convert the ratio in terms of percentage by multiplying with 100.

15. Simple Cubic
A simple cubic structure is formed by arranging eight atoms touching each other at the eight corners of a
cube.
Since the atoms touch each other in simple cubic, the nearest neighbour distance is 2r = a
Relation between lattice constant and density
Let
a = Lattice constant of material (i.e., cubic crystal)
δ = ∗Density of the material of the crystal
n = Number of atoms per unit cell (i.e., atoms in volume a3 of the unit cell)
M = Atomic weight of the material
Na = Avogadro’s number = 6.023 × 1026/k mole

17. Simple Cubic (SC) Structure
• a = b = c and α = β = γ = 90°.
• Atoms are present only at the corners of this unit cell.
• A corner atom is shared by eight-unit cells, so that the contribution of a corner atom to a unit cell is 1/8.
• The cube has eight corners; hence, the contribution of eight corner atoms to a unit cell or the number of
atoms per unit cell =1/8×8 = 1.
The volume occupied by atoms in the unit cell (v) = 1 × (4/3) × πr3
and
The volume of unit cell (V) = a3.
Hence, the packing factor or density of packing in the unit cell

18. Body-Centered Cubic (BCC) Structure
• Atoms are present at the corners of the cube and one atom is completely present at the centre of the
unit cell.
• The centre of the unit cell is defined as the intersecting point of two body diagonals (AD and BE).
• A corner atom is shared by eight-unit cells so that the contribution of a corner atom to a unit cell is 1/8.
The number of atoms per unit cell = (1/8) × 8 +1 = 2.
The centre atom is surrounded by eight corner atoms, so the co-ordination number is 8.
The length of the body diagonal AD = 4r
∴ AD2 = AC2 + CD2 = AB2 + BC2 + CD2
= a2 + a2 + a2 = 3a2

20. Face-Centered Cubic (FCC) Structure
• Atoms are present at the corners and at the face centres of this cubic structure.
• The intersection of face diagonals represents face centre of the cube.
• A corner atom is shared by eight-unit cells and a face-centred atom is shared by two unit cells.
• The cube has eight corners and bounded by six faces;
The number of atoms per unit cell = (1/8)×8 + (1/2)×6 =4.

22. Miller Indices
Miller Indices for Direction in Crystal
Crystal planes are defined as some imaginary planes inside a crystal in which large concentration of atoms is
present. Inside the crystal, there exists certain directions along which large concentration of atoms exists. These
directions are called crystal directions.
• Crystal planes and directions can be represented by a set of
three small integers called Miller indices.
• These integers are represented in general as h, k and l.
• If these integers are enclosed in round brackets as (hkl),
then it represents a plane.
• If they are enclosed in square brackets as [hkl], then it
represents crystal direction perpendicular to the above-said
plane.

23. Way of obtaining Miller indices for a plane
1. Find the intercepts on the axes in terms of the lattice
constants a, b and c. For example, consider 2a, 3b
and 4c as the intercepts of a plane.
2. Express the intercepts as multiplies of lattice
parameters along the respective axes. For the
plane, these are 2a/a, 3b/b and 4c/c that is 2, 3 and
4.
3. Take the reciprocal of these numbers, that is, 1/2,
1/3 and 1/4.
4. Reduce these fractions to the smallest triad of
integers h, k, l having the same ratio. The quantity (h
k l ) is then the Miller index of that system of planes.

24. Important Features of Miller
Indices
• Miller indices represent a set of equidistant parallel planes.
• If the Miller indices of a plane represent some multiples of Miller indices of another plane, then these
planes are parallel. For example (844) and (422) or (211) are parallel planes.
• If (hkl) are the Miller indices of a plane, then the plane divides the lattice constant ‘a’ along the X-axis into
h equal parts, ‘b’ along the Y-axis into k equal parts and ‘c’ along the Z-axis into l equal parts.
• If a plane is parallel to one of the crystallographic axes, then the plane intersects that axis, at infinity and
the Miller indices along that direction is zero.
• If a plane cuts an axis on the negative side of the origin, then the corresponding index is negative and is
indicated by placing a minus sign above the index. For example, if the plane cuts on the negative Y-axis,
then the Miller indices of the plane is (hkl).

25. Interplaner Spacing in
Terms of Miller Indices

27. Diffraction of X-Rays by
Crystal

28. Powder or Debye-Scherrer Method