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CRYSTAL
STRUCTURE
Introduction

A crystal is a solid composed of atoms or other microscopic
particles arranged in an orderly repetitive array.

Further Solids can be broadly classified into Crystalline and
Non-crystalline or Amorphous.

In crystalline solids the atoms are arranged in a periodic
manner in all three directions, where as in non crystalline
the arrangement is random.

Non crystalline substances are isotropic and they have no
directional properties.

Crystalline solids are anisotropic and they exhibit varying
physical properties with directions.

Crystalline solids have sharp melting points where as
amorphous solids melts over a range of temperature.
Space lattice
A Space lattice is defined as an infinite array of points in three
dimensions in which every point has surroundings identical to
that of every other point in the array.
X
X
X
X
X
X
X
X
X
X
a
X
X
X
X
X
X
b
X
X
Where a and b are
X
called the repeated
translation vectors.
X
Three dimensional lattice
Lattice planes
Lattice lines
Lattice points
UNIT CELL
The unit cell is a smallest unit which is repeated
in space indefinitely, that generates the spacelattice.
BASIS
A group of atoms or molecules identical in
composition is called the Basis.
Lattice + basis = Crystal structure
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:
The 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.
Z
c
α
a
NOTE
γ
b
β
Y
1. Primitives decides the size of the unit cell.
2. Interfacial angles decides the shape of the unit cell.
X
PRIMITIVE CELL
The unit cell is formed by primitives is called primitive cell.
A primitive cell will have only one lattice point.
Z
LATTICE PARAMETERS
c
The primitives and interfacial
angles together called as lattice
parameters.
α
a
Y
γ
β
b
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
BRAVIAS LATTICES
There are only fourteen distinguishable ways of arranging
the points independently in three dimensional space and
these space lattices are known as Bravais lattices and they
belong to seven crystal systems
CRYSTAL TYPE
BRAVAIS LATTICE
1. Cubic
Simple
Body centered
Face centered
2. Tetragonal
Simple
Body centered
3. Orthorhombic
Simple
Base centered
Body centered
Face centered
4. Monoclinic
Simple
Base centered
SYMBOLS
Simple Cubic
P
Base Centered
C
5. Triclinic
Simple
6. Trigonal
Simple
Body Centered
I
7. Hexgonal
Simple
Face Centered
F
Crystal System
Unit Vector
Angles

Cubic
a=b=c
α = β = γ = 90˚

Tetragonal
a=b≠c
α = β = γ = 90˚


Ortho rhombic
a≠b≠c
α = β = γ = 90˚

Mono clinic
a≠b≠c
α = β = 90 ≠ γ

Triclinic
a≠b≠c
α ≠ β ≠ γ ≠ 90˚

Trigonal
a=b=c
α = β = γ ≠ 90˚

Hexagonal
a=b≠c
α = β = 90˚,γ =120˚
Cubic Crystal System
6
2
1
3
5
a = b = c & α = β = γ =90˚
4
Tetragonal Crystal System
a = b ≠ c & α = β = γ =90˚
Ortho Rhombic Crystal System
6
2
1
3
5
a ≠ b ≠ c & α = β = γ =90˚
4
Monoclinic Crystal System
a ≠ b ≠ c & α = β = 90 ≠ γ
Triclinic clinic Crystal System
a ≠ b ≠ c & α ≠ β ≠ γ ≠90˚
Trigonal Crystal System
a = b = c & α = β = γ ≠90˚
Hexagonal Crystal System
a = b ≠ c & α = β =90˚,γ =120˚
NEAREST NEIGHBOUR DISTANCE
The distance between the centers of two nearest neighboring
atoms is called nearest neighbor distance.
CO – ORDINATION NUMBER
Co-ordination number is defined as the number of
equidistance nearest neighbors that an atom has in a given
structure.
ATOMIC PACKING FACTOR
Atomic packing factor is the ratio of volume occupied by the
atoms in an unit cell to the total volume of the unit cell. It is
also called packing fraction.
Vol occupied by the atoms in an unit cell
Atomic Packing factor 
Total volume of a unit cell
Void Space


Vacant space left or unutilized space in unit
cell , and more commonly known as interstitial
space.
Void space = ( 1-APF ) X 100
SIMPLE CUBIC STRUCTURE - PACKING FACTOR
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. Co-ordination number = 6
r
r
a
5. Atomic packing factor
4 3
1  r
3

3
a
wherea  2r
4 3
1  r
3

6.Void space = (1-APF) X 100
(2r )3
= (1-0.52)X 100
 0.52
= 48%
 (52%)
Example: Polonium.
BCC STRUCURE – PACKING FACTOR
1. Effective number of atoms per
unit cell (8 x 1/8) + 1 =2
2. Atomic radius r = √3a /4
3. Nearest neighbor distance
2r =√3a/2
4. Co-ordination number = 8
D
3a
a
4r
C
2a
A
a
a
B
5.Atomic packing factor
4 3
2  r
3

a3
3
wherer 
a
4
6.Void space = (1-APF) x 100
= (1-0.68) x 100
= 32%
4
3 3
2   ( a)
3
4

( a )3
 0.68

Ex: Na, lithium and
Chromium.
 (68%)
FCC Crystal Structure – APF
1. Effective number of atoms per unit
cell (8 x 1/8) + 1/2 X 6 = 4
2. Atomic radius r = a / 2√2
3. Nearest neighbor distance
2r = a /√2
4. Co-ordination Number = 12
C
2a
a
4r
A
a
B
1
2
4
3
5.Atomic packing factor
4
4   r3
3

a3
r
a
2 2
4
a 3
4  (
)
3 2 2

( a )3
6.Void space = (1-APF) X 100
= (1-0.74) X 100
= 26%
 0.74
 (74%)
Ex: Cupper , Aluminum, Silver and Lead
Diamond Structure:
Diamond is a combination of interpenetrating Fcc - Sub lattices
along the body diagonal by 1/4th Cube edge.
6
4
1
2
3
5
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
4. Co-ordination number = 4.
a/4
p
y
a
a/4
z
x
a/4
p
a/2
a
5. Atomic packing factor
4 3
8  r
3

a3
3
r
a
8
6. Void space = (1-APF) x 100
= (1-0.34) x 100
= 66%
Ge, Si and Carbon atoms are
possess this structure
4
3 3
8  (
a)
3
8

( a )3
 0.34
 (34%)
Hexagonal Close Packed Structure
1. Effective number of atoms per
unit cell
2 x (6x 1/6) + 2 x 1/2 + 3 = 6.
2. Atomic radius r = a / 2.
3. Nearest neighbor distance 2r = a
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
 6  (ABC )
1
 6  ( a)(a sin 600 )
2
C
 3a 2 sin 600
3 3a

2
2
If c is the height of the unit cell
60°
A
a
B
3 3 2
V
ac
2
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 2
( 2r )  x  ( )
2
2
2
APQ  cos 30 
0
a 2 c 2
( 2r )  ( )  ( )
2
3
a 2 c 2
2
a ( ) ( )
2
3
2r
c/2
o
30°
x
q
p
B
N
a
2
c2
a
 a2 
4
3
c2 8
c
8
  
2
a
3 a
3
2
x
a
x
3
2
A
a
5. Atomic packing factor
4
6   r3
3

3 2a 3
a
r
2
6. Void space = (1-APF) x 100
= (1-0.74) x 100
= 26%
Ex: Mg, Cd and Zn.
4
a 3
6  ( )
3
2

3 2 ( a )3
 0.74
 (74%)
Sodium Chloride Structure

Nacl Crystal is an ionic crystal. It consists of two
FCC sub lattices.

One of the chlorine ion having its origin at the
(0, 0, 0) point and other of the sodium ions having its
origin at (a/2,0,0).

Each ion in a NaCl lattice has six nearest neighbor
ions at a distance a/2. i,e its Co-ordination number is
6.
Sodium Chloride structure
Na
Cl

Each unit cell of a sodium chloride as four sodium ions
and four chlorine ions. Thus there are four molecules in
each unit cell.

Cl : (0,0,0) (1/2,1/2,0) (1/2,0,1/2) (0,1/2,1/2)
Na : (1/2,1/2,1/2),(0,0,1/2) (0,1/2,0)(1/2,0,0)
Structure of Cesium chloride:
• Cscl is an ionic Compound.
Cs
• The lattice points of CsCl are two
interpenetrating simple cubic lattices.
Cl
• One sub lattice occupied by cesium
ions and another occupied by Cl ions.
• The co-ordinates of the ions are
Cs : (000),(100),(010),(001),(110),
(110),(011),(111).
Cl : (1/2,1/2,1/2).
Some important directions in Cubic Crystal

Square brackets [ ] are used to indicate the directions

The digits in a square bracket indicate the indices of that
direction.

A negative index is indicated by a ‘bar’ over the digit .

Ex: for positive x-axes→[ 100 ]

for negative x-axes→[ 100 ]
Fundamental directions in crystals
z
[001]
[000]
[010]
y
[100]
x
Crystal planes & Miller indices

Reciprocals of intercepts made by the plane which are
simplified into the smallest possible numbers or integers
and represented by (h k l ) are known as Miller Indices.
(or)
The miller indices are the three smallest integers which
have the same ratio as the reciprocals of the intercepts
having on the three axes.

These indices are used to indicate the different sets of
parallel planes in a crystal.
Procedure for finding Miller indices
Find the intercepts of desired plane on the three Co-ordinate axes.
Let they be (pa, qb, rc).
Express the intercepts as multiples of the unit cell dimensions i.e. p,
q, r. (which are coefficients of primitives a, b and c)
Take the ratio of reciprocals of these numbers i.e. a/pa : b/qb : c/rc.
which is equal to 1/p:1/q:1/r.
Convert these reciprocals into whole numbers by multiplying each
with their L.C.M , to get the smallest whole number.
These smallest whole numbers are 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.

The numerical parameters of
the plane ABC are (2,2,1).

The reciprocal of these
values are given by
(1/2,1/2,1).

LCM is equal to 2.
z
1
2

Multiplying the reciprocals
with LCM we get Miller
indices [1,1,2].
2
y
x
Construction of [100] plane
z
Intercepts of the Plane are  (1, , )
1 1 1
R`eciproca ls of intercepts are  ( , , )
1  
Miller indices : (1,0,0)
x
y
[1 0 0]
plane
z
x
y
Set of [100] parallel planes
Intercepts of the Plane are  (,1, )
1 1 1
R`eciproca ls of intercepts are  ( , , )
 1 
Miller indices : (0,1,0)
z
x
[010]
y
plane.
z
x
y
Set of ( 0 1 0 ) parallel planes
z
Intercepts of the Plane are  (, ,1)
1 1 1
R`eciproca ls of intercepts are  ( , , )
  1
Miller indices : (0,0,1)
x
y
[ 001 ]
plane
Set of ( 0 0 1 ) parallel planes
z
[ 001 ]
x
y
Construction of [110] plane
Intercepts of the Plane are  (1,1, )
1 1 1
R`eciproca ls of intercepts are  ( , , )
1 1 
Miller indices : (1,1,0)
z
[110]
x
y
z
x
[110]
y
Set of [110] parallel planes
Construction of ( ī 0 0) Planes
Intercepts of the Plane are  (1, , )
z
1 1 1
R`eciproca ls of intercepts are  ( , , )
1  
Miller indices : (1,0,0)
x
y
Intercepts of the
planes are 1,1,1
z
Reciprocals of intercepts
are 1/1,1/1,1/1
Miller indices:(111)
x
(111)
y
plane
Inter planner spacing of orthogonal crystal
system:




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 C
l

o

a
h
x
A
d
o
N
B
b
k
y

OA = a/h, OB = b/k, OC = c/l
A normal ON is drawn to the plane ABC from the
origin “o”. the length “d” of this normal from the
origin to the plane will be the inter planar separation.

from∆ ONA

from∆ ONB

from∆ ONC
ON
d

OA ( a )
h
ON
d
cos  

OB ( b )
k
ON
d
cos  

OC ( c )
l
cos  
Where cosα, cosβ ,cosγ are directional cosines of α,β,γ
angles.
According to law of directional cosines
cos 2   cos 2   cos 2   1
d 2
d 2
d 2
[
] [
] [ ] 1
a
b
c
( )
( )
( )
h
k
l
2
2
2
h
k
l
d 2{ 2  2  2 }  1
a b c
d
1
h2 k 2 l 2
 2 2
2
a b c
This is the general expression for inter planar separation for
any set of planes.
In cubic system as we know that a = b = c, so the
expression becomes
d
a
h k l
2
2
2