SREENIDHI INSTITUTE OF SCIENCE & TECHNOLOGY

(An Autonomous Institution approved by UGC and ‘A’ Grade Awarded by NAAC)

APPLIED PHYSICS –I (Code: 6H223)

I - B.Tech II Semester
( for ECE)

L T P/D C
3 1 0 3
 UNIT – I
Crystallography and Crystal Structures
 Space Lattice, Unit Cell, Lattice
Parameters, Crystal Systems, Bravais Lattices,
Miller Indices, Crystal: Planes and Directions,
Inter Planar Spacing of Orthogonal Crystal
Systems, Atomic Radius, Co-ordination
Number and Packing Factor of SC, BCC,
FCC and HCP Structures. Atomic densities
along various important directions and planes
and comparison of deformation of BCC, FCC
and HCP metals.
Introduction:
Solids can be broadly classified into two
categories based on the arrangement of
atoms or molecules as crystalline and non
crystalline (amorphous).

Crystalline Solids:
In crystalline solids the atoms or
molecules are arranged in a periodic manner
in all three directions and further those are
classified as mono (single) crystals and
polycrystalline solids.
 Single Pyrite
Single Crystal
Crystal
Polycrystal Polycrystalline
Pyrite form
(Grain)
 C r y s t a l s w h i c h h a v e d i ff e r e n t p e r i o d i c
arrangements in all the three directions exhibit
varying physical properties with directions and
they are called anisotropic substances.
Ex: Al, Cu, Ag, Ge, Si, Diamond etc…
Non crystalline Solids:
Non crystalline substances are also called
amorphous. In amorphous solids the atoms or
molecules arranged randomly and which have no
regular structure.

They have no directional properties and therefore

they are called as isotropic substances.
Ex: Rubber, Glass, Wood, Plastic etc…
Crystallography:
Study of the geometrical form and other physical properties of
crystalline solids by using x-rays, neutron beams and electron
beams.
Crystallography is mainly used to determine the internal atomic
arrangement in crystals, bonding and their strength.
Lattice: The periodic arrangement of atoms in a crystal is
called lattice.
Lattice Point: which denote the position of atoms or
molecules (Imaginary points).
Each lattice point in space may represent a single atom or group
of atoms.
Ex: In Ge & Si, each lattice point is 1 atom.
In NaCl, each lattice point is 1 NaCl molecule.
Space lattice:
A space lattice is defined as an infinite array of points
in three dimensions in which every point has an
identical environment to that of every other point in the
array.

P
b
a
Where a and b are called the repeated
translation vectors.
The position vector of any lattice point for two dimensional
space lattice is

The position vector of any lattice point for three dimensional

space lattice is

Where n1,n2& n3 are the integers ,

a, b & c are the translational vectors on along the
crystallography axes.
Three dimensional lattice
Lattice planes

Lattice lines

Lattice points
BASIS :
A group of atoms or molecules identical in composition,
arrangement or orientation is called the Basis.
The basis provides the number of atoms per lattice point,
their types and orientation.
Lattice + basis = crystal structure

Lattice + basis = crystal structure

UNIT CELL:
The unit cell is a smallest block or geometric figure from
which the entire crystal is built by repetition in three
dimensions.
Z

α γ b
Y
a β

X
CRYSTALLOGRAPHIC AXES:
The lines drawn parallel to the lines of intersection
of any three faces of the unit cell which do not lie in
the same plane are called crystallographic axes.
PRIMITIVES:
a, b and c are the dimensions of an unit cell and are
known as Primitives.
INTERFACIAL ANGLES:-
The angles between three crystallographic axes are
known as Interfacial angles α ,β and γ.
PRIMITIVE CELL:-
The unit cell is formed by primitives is called primitive cell.
A primitive cell will have only one lattice point.
LATTICE PARAMETERS:-
The primitives and interfacial angles are called lattice
parameters.

NOTE:
i). Primitives decides the size of the unit cell.
ii). Interfacial angles and primitives together decides the shape
and size of the unit cell.
Volume of the unit cell = a3
Therefore, mass of the unit cell = volume x density
m = a3ρ -------------(1)

Let say ‘n’ molecules / unit cell , M is the

molecular weight and N is the Avogadro number.

Then mass of the each molecule = M/N

Mass of the each unit cell = n x M/N -----------(2)

 Therefore, Eq.(1) = Eq.(2)

a3ρ = n x M/N

Lattice constant(a) = (nM/Nρ)1/3 ---------(3)

Eq.(3) gives the relation between lattice constant & crystal

density.
Crystal Systems:
Crystals are classified into seven crystal systems on the
basis of shape of the unit cell or lattice parameters in
terms of lengths and the angle of inclination between
them.

The seven crystal systems are i). Cubic ii). Tetragonal

iii). Orthorhombic iv). Monoclinic v). Triclinic
vi). Trigonal (or) Rhombohedral vii). Hexagonal
Crystal System Unit Vector Angles Examples
— Cubic a=b=c α = β = γ =90˚ Cu,Ag,Na,Po

— Tetragonal a=b≠c α = β = γ =90˚ SnO2, TiO2

— Orthorhombic a≠b≠c α = β = γ =90˚ BaSO4,

MgSO4

— Monoclinic a≠b≠c α = β = 90 ≠γ CaSO4

2H2O

— Triclinic a ≠ b ≠c α ≠ β ≠ γ ≠90˚ CuSO4 5H2O

— Trigonal/ a=b=c α = β = γ ≠90˚

As,Bi,Sb,Calcite
Rhombohedral
BRAVAIS LATTICES:-

According to Bravais, there are only 14 distinguishable ways

of arranging lattice points in a three dimensional space so
that the environment looks the same from each point. The
lattices are called Bravais lattices.

The classification of Bravais lattices is based on the following

crystal lattices.
i). Primitive Lattice(P)
ii). Body Centered Lattice(I)
iii). Face Centered Lattice(F) and
iv). Base Centered Lattice(C) .
 i). Primitive Lattice(P):

Primitive Lattice structure has 8 lattice points at the eight

corners of the unit cell and each point share with another 8
unit cells

No. of lattice points in unit cell=(1/8)*8=1 lattice point/cell

ii). Body Centered Lattice(I):

Body Centered Lattice structure has 8 lattice points at the

eight corners of the unit cell with addition of one center
lattice point. Corner lattice point share with another 8
unit cells and center point is independent to other unit
cells
No. of lattice points in unit cell=(1/8)*8+1=2 lattice
points/cell
iii). Face Centered Lattice(F):
6

1
2 4
3

Face Centered Lattice structure has 8 lattice points at

the eight corners of the unit cell with addition of these
points six faces having 6 lattice points. Corner lattice
point share with another 8 unit cells and face centered
lattice point share with 2 unit cells

No. of lattice points in unit cell=(1/8)*8+(1/2)*6=4 lattice

points/cell
iv). Base Centered Lattice(C):

Base Centered Lattice structure has 8 lattice points at the

eight corners of the unit cell with addition of these points
two bases having 2 lattice points. Corner lattice point
share with another 8 unit cells and base centered lattice
point share with 2 unit cells
No. of lattice points in unit cell=(1/8)*8+(1/2)*2=2 lattice
points/cell
CRYSTAL TYPE BRAVAIS LATTICE SYMBOLS

•Cubic Simple P
Body Centered I
Face Centered F

•Tetragonal Simple P
Body Centered I

•Orthorhombic Simple P
Base Centered C
Body Centered I
Face Centered F

•Monoclinic Simple P
Base Centered C

•Triclinic Simple P

•Trigonal Simple P

•Hexgonal Simple P
7P+3I+2F+2C = 14
Thus the total number of Bravais Lattices are 14
Cubic crystal system:

P I F

1
2 4
3

a=b=c α = β = γ =90˚
Tetragonal crystal system

P I

a=b≠c α = β = γ =90˚
Ortho Rhombic crystal system
P I F

1
2 4
3

a≠b≠c α = β = γ =90˚
Monoclinic crystal system
C
P

a≠b≠c α = β = 90 ≠ γ
Triclinic clinic crystal system :
P

a≠b≠c α ≠ β ≠ γ ≠90˚
Trigonal crystal system:
P

a = b = c α = β = γ ≠ 90˚
Hexagonal crystal system:
P

a=b≠c α =β=90˚,γ=120˚
NEAREST NEIGHBOR DISTANCE(2r):
The distance between the centers of two nearest neighboring
atoms is called nearest neighbor distance.

r r

2r

ATOMIC RADIUS(r ):
Atomic radius is defined as half of the distance between the
nearest neighboring atoms in an unit cell of a crystal
atomic radius= r
CO-ORDINATION NUMBER:
Co-ordination number is defined as the number of
equidistance nearest neighbors that an atom has in its unit
cell.

Its signifies the tightness of packing of atoms in the

crystal. More closely packed structure has greater co-
ordination number

Ex: Simple cubic=6, Body Centered=8, Face centered=12

ATOMIC PACKING FACTOR:
Atomic packing factor is the ratio of volume occupied by
the atoms in a unit cell to the total volume of the unit cell.
It is also called packing fraction.
Atomic Packing factor = volume occupied by the
atoms in a unit cell / Total volume of a unit cell.

!The packing factor varies from 0.34 to 0.74 for different

crystal structures

! High value indicates that very closely packed and low

value Indicates loosely packed structure.
Void space:
Void space in the unit cell is the vacant space
left or unutilized space in unit cell and more
commonly known as interstitial space.

Void space = (1-APF )X100

 SIMPLE CUBIC STRUCTURE:

1. Effective number of atoms per

unit cell (8 x 1/8) =1

2. Atomic radius r = a / 2

3. Nearest neighbor distance

2r = a
4. Lattice constant a=2r r r
ā
5. co-ordination number = 6
 6. Atomic packing factor

7. Void space = (1-APF) X

100

= (1-0.52)X 100

=48%
Ex: Polonium.
 BCC STRUCURE:

1. Effective number of atoms per

unit cell (8 x 1/8) + I =2
D
2. Atomic radius r = √3a /4
4r
3. Nearest neighbor distance ā

2r =√3a/2 √2ā
C

4. Lattice constant a=4r/√3 ā

A B
ā
5.Co-ordination number = 8
6. Atomic packing factor

7. Void space = (1-APF) x 100

= (1-0.68) x 100
= 32%

• Ex: Na, lithium and

Chromium.
 FCC Crystal Structure :

1. Effective number of atoms per unit

cell = (8 x 1/8) + 1/2 X 6 = 4

C
2.Atomic radius r = a / 2√2
4r
ā
3.Nearest neighbor distance

2r = a /√2 A ā B

4.Co-ordination number = 12
5.Atomic packing factor

6. Void space = (1-APF) X 100

= (1-0.74) X 100
= 26%

Ex: Cupper , Aluminum, Silver and

Led
 Hexagonal close packed structure:

1. Effective number of atoms per unit

cell

2 x (6x 1/6) + 2 x 1/2 + 3 = 6.

2. Nearest neighbor distance

2r = a

3. Atomic radius r = a / 2.

4. Co-ordination number = 12.

5.Volume of the HCP unit cell:
The volume of the unit cell determined by computing
the area of the base of the unit cell and then by
multiplying it by the unit cell height.
Volume = (Area of the hexagon) x (height of the cell)

Area of the hexagon:

o

60°
A O B

If c is the height of the unit cell

c/a ratio:

The three body atoms lie in a horizontal

plane at a height c/2 from the base or at
top of the Hexagonal cell.
C

c/2

o
A
30°
X N
a
a

B
 6. Atomic packing factor

7. Void space = (1-APF) x 100

= (1-0.74) x 100
= 26%

Ex: Mg, Cd and Zn.

Diamond structure:

Diamond is a combination of two

interpenetrating
FCC - sub lattices along the body
diagonal by 1/4th
Cube edge.
 6

4
1 2
3

5
a y

a/4

x a/4 p a/2

a
 Diamond - APF
y
1. Effective number of atoms per unit
cell (8 x 1/8) + 1/2 X 6 + 4 = 8.
a/4
2r
2.Atomic radius r = √3a / 8.

z
3.Nearest neighbor distance
a/4
2r = √3a / 4.
x a/4 p
4.Co-ordination number = 4.
* The fraction such as 1/4,1/2,3/4 denote
the height above the base in units of cube edge.

0 1/2 0

3/4 1/4

0 1/2
1/2

3/4
1/4

0 1/2 0
 5. Atomic packing factor

6. Void space = (1-APF) x 100

= (1-0.34) x 100
= 66%
Ge, Si and Carbon atoms are
possess this structure
Directions and Planes in Crystals:
In crystals atoms are regularly arranged in space
such that environment about any atom is same.
There exists some directions and planes which
c o n t a i n s l a rg e c o n c e n t r a t i o n o f a t o m s . F o r
understanding crystallography, the concept of
directions and planes play an important role.

Directions :
In crystal analysis, it is essential to indicate certain
directions inside the crystal and which is a line
joining any two points of the lattice.
Two dimensional crystal directions representation:
y
A [1 2 ] B[1 1 ]

C[3 2 ]

D[3 1 ]

O x
Let ‘o’ be the origin of the pattern. Consider the
directions OA,OB,OC & OD. The direction is
described by giving the first integer point (x,y)
through which the line passes.
 • The line OA is passing through the first lattice
point at x=1 & y=2. Therefore, the direction OA is
given by [1 2].
• A convention square brackets are used to represent
directions.
Three dimensional crystal directions
representation: z
P[u v w]
Suppose to indicate the
c
direction ‘OP’ as shown in
wc
figure. Consider x. y & z are o
b
the crystallography axes. a
ua vb
y
x
• If a, b and c represent unit translational vectors along
x,y and z directions, then moving ‘u’ times ‘a’ along
x-axis, ‘v’ times ‘b’ along y-axis and ‘w’ times ‘c’
along z-axis, we can reach ‘P’.

• If u, v and w are the smallest integers, the direction

‘OP’ is indicated by [u v w]. z

[001]
Ex: Some important [011]

directions in a cubic crystal [111]

[000] [010]
y
[100] [110]
x
Some important directions in cubic crystal:-
• Square brackets [ ] are used to indicate the directions

• The digits in a square brackets indicate the indices of

that direction.

• A negative index is indicated by a ‘bar’ over the

digit .

• Ex: for positive x-axis→[ 100 ]

• for negative x-axis→[ 100 ]

 Crystal planes:
A crystal is made up of a large number of parallel
equidistant planes passing through lattice points are
called Lattice planes or crystal planes.

"The perpendicular distance

between adjacent planes is
called inter planar spacing.
"The distance between parallel
planes of any these sets is called
the lattice spacing.
 Miller indices:
The Miller Indices are the three smallest possible
integers (h k l), which have the same ratio as the
reciprocals of the intercepts of the crystal plane
having on the three crystallographic axes.

• These indices are used to indicate the different sets

of parallel planes in a crystal.
Procedure for finding Miller indices:

z
C

r
o q B
y
p
A

x
Choose system of three coordinate axes x,y & z i.e.,
crystallographic axes

Determine the intercepts p, q & r of the required plane ‘ABC’

on these axes i.e., OA = p, OB = q & OC = r.

Take ratio of reciprocals of the Intercepts i.e., 1/p:1/q:1/r.

Convert these reciprocals into integers by multiplying each

one of them with their L.C.M .

Enclose these integers in smaller parenthesis i.e., Miller

indices (h k l) of the crystal.
Important features of miller indices:
• When a plane is parallel to any axis, the intercept of the
plane on that axis is infinity. Hence its miller index for
that axis is zero.
• When the intercept of a plane on any axis is negative a
bar is put on the corresponding miller index.
• All equally spaced parallel planes have the same index
number (h k l).
• If a plane passes through origin, it is defined in terms
of a parallel plane having non-zero intercept.
• If a normal is drawn to a plane (h k l), the direction of
the normal is [h k l].
 Construction of (100) plane

x
y
(1 0 0) plane
 z

y
Set of (100) parallel planes
Construction of (010) plane

(010) plane.
y
 z

y Set of ( 0 1 0 ) parallel planes

Construction of (001) plane

1 x

1
( 001 ) plane
y
Set of ( 0 0 1 ) parallel planes:
z

y
Construction of (110) plane

(110)

y
 z

(110)

y
Set of (110) parallel planes
 Construction of ( ī 0 0) Planes

y
Construction of (111) plane

Intercepts of the
z
planes are 1,1,1

Reciprocals of intercepts
are 1/1,1/1,1/1

Miller indices:(111)

(111) plane
y
Inter planner spacing of orthogonal crystal system:
(in 3 dimensions)
‘The distance ‘d’ between a series of planes in a
crystal is known as the ‘d’ spacing or inter planar
spacing’.
• Let ( h ,k, l ) be the miller indices of the plane ABC.
• Let ON=d be a normal to the plane passing through
the origin ‘0’.
• Let this ON make angles α, β and γ with x, y and z
axes respectively.
• Imagine the reference plane passing through the
origin “o” and the next plane cutting the intercepts a/h,
b/k and c/l on x, y and z axes.
 Z

C M
c/l γ N
α A X
O β
a/h
b/k
B

Y
• Let the direction cosines of ON be cosα, cosβ & cos
γ
But law of direction cosines
• Let OM be the perpendicular distance of the next
plane from the origin and its intercepts 2a/h, 2b/k &
2c/l.

• Therefore, the spacing between the adjacent planes

dhkl = OM-ON
Note: The inter planar spacing of Simple Cubic Structure
a=b=c