Crystal structure

CRYSTAL
STRUCTURE
PRESENTED BY
SANDHYA R,
M.TECH(ECE)

AGENDA
1. INTRODUCTION
2. CLASSIFICATION OF SOLID
2.1. CRYSTALLINE SOLID
2.2. NON-CRYSTALLINE (OR) AMORPHOUS SOLID
3. BASIC TERMS OF CRYSTALLOGRAPHY
3.1. LATTICE POINT AND SPACE LATTICE
3.2. BASIS
3.3. LATTICE PARAMETERS
3.4. UNIT CELLAND ITS PROPERTIES
3.5 PRIMITIVE CELL
3.6 NON-PRIMITIVE CELL

4. TYPES OF UNIT CELL
4.1. SIMPLE CUBIC CRYSTAL STRUCTURE (SC)
4.2. BASE CENTERED CUBIC CRYSTAL STRUCTURE (BCC)
4.3. FACE CENTERED CUBIC CRYSTAL STRUCTURE (FCC)
5. SEVEN CRYSTAL SYSTEM
6. BRAVAIS LATTICE
7. MILLER INDICES
7.1. DEFINITION
7.2. PROCEDURE FOR FINDING MILLER INDICES
7.3. FEATURES
8. SUMMARY
9. REFERENCE

1. INTRODUCTION
WHAT IS CRYSTAL?
• Crystals is a solid form of substance and are made of small
basic building block
• It is also defined as a three-dimensional solid structure which
consists of a periodic arrangement of atoms
• Some crystals are very regularly shaped and can be classified
into one of several shape categories such as rhombic, cubic,
hexagonal, triclinic, etc

2.1 CRYSTALLINE SOLID
• Crystalline solid is one in which the atoms, ions or molecules
are arranged in regular order with uniform intervals in all
directions as shown in the below fig.1.1
• There are metallic and non metallic crystals and can be
mono- crystal or poly-crystal
Fig.1.1. Crystalline Solid

CHARACTERISTICS OF CRYSTALINE SOLIDS
• Crystalline solids have periodical array of atoms arranged in
all directions
• They have Sharp melting point and boiling point
• Crystalline solids are also called as anisotropic solids since
they have directional properties
Example: Copper, Aluminium, Diamond, Graphite, etc.

2.2 AMORPHOUS SOLID
• In Amorphous solids the arrangements of atoms, ions or
molecules is not in regular order as shown in the below
fig.1.2
• Hence their structure is non-crystalline
Fig.1.2. Amorphous Solid

CHARACTERISTICS OF AMORPHOUS SOLIDS
• Amorphous solids have irregular arrangement of atoms.
• They do not have sharp melting and boiling point
• Amorphous solids are also called as isotropic solids since
they have no directional properties
Example: Glass, Rubber, Polymers, etc.

3. BASIC TERMS OF CRYSTALLOGRAPHY
3.1. LATTICE POINTS AND SPACE LATTICE
• These are imaginary points in space which denote the
position of atoms, ions or molecules in a crystal
• Each lattice point has identical surroundings as a model of
crystal structure. The 2-D space lattice is shown in the fig.1.3
Fig.1.3 2-D Space Lattice

3.2. BASIS
• A group of molecules or atom attached to each lattice point
which are identical in composition and orientation is called
basis
• It is also called as motif and it is represented as shown in the
fig.1.4
Fig.1.4. Basis

• To obtain a crystal structure, an atom or a group of atoms
(i.e.) a basis, must be placed on each lattice point as
illustrated below
Fig.1.5. Formation of Crystal structure

3.3. LATTICE PARAMETERS
• Consider a unit cell consisting of three mutually
perpendicular edges OA, OB and OC and draw parallel lines
along the three edges as of the fig.1.8
• These lines are taken as crystallographic axes and they are
denoted as x, y and z axes
Fig.1.6. Lattice parameters

• The angles between x, y and z axes given as 𝜶, 𝜷 and 𝜸 are
called as interaxial or interfacial angles
• Therefore, the three coordinate axes x, y and z and the three
interfacial angles 𝛼, 𝛽 and 𝛾 are together known as lattice
parameters
• If the value of these intercepts and interaxial angles are
known one can easily determine the form and actual size of
the unit cell

3.4. UNIT CELL
• All crystalline solids indicate that there is smallest group of
atoms, ions or molecules which when repeated in all the
direction will give the entire crystal structure
• This smallest unit is known as Unit Cell
• Thus a unit cell is defined as
fundamental block of a
crystal structure
Fig.1.7. Unit Cell

PROPERTIES OF UNIT CELL
ATOMIC RADIUS
• Atomic radius is half the distance between nearest neighbours
in a crystal of a pure element
• It is usually expressed in terms of cube edge ‘a’(lattice
parameter)
Fig.1.8. Atomic radius

CO-ORDINATION NUMBER
• It is defined as the number of nearest atoms, which are
directly surrounding a given atom
• If the coordination number is high, then the structure will be
more closely packed
NUMBER OF ATOMS PER UNIT CELL
• The number of atoms possessed by a unit cell can be
determined if the arrangement of atoms inside the unit cell is
known

ATOMIC PACKING FACTOR OR PACKING
DENSITY(APF)
• Atomic packing factor (also known as density of packing) is
defined as the ratio of the volume of atoms per unit cell to the
total volume occupied by the unit cell
APF =
𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒂𝒕𝒐𝒎𝒔 𝒑𝒆𝒓 𝒖𝒏𝒊𝒕 𝒄𝒆𝒍𝒍 (𝒗)
𝑽𝒐𝒍𝒖𝒎𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒖𝒏𝒊𝒕 𝒄𝒆𝒍𝒍 (𝑽)
=
𝑣
𝑉

3.5. PRIMITIVE CELL
• Primitive cell is the simplest type of unit cell which contains
only one lattice points per unit cell
• Example: Simple Cubic (SC), Which contains lattice points
at its corner only
3.6. NON-PRIMITIVE CELL
• The unit cells, which contain more than one lattice point, are
called non-primitive cell
• Example: Body centred and Face centred structures

4. TYPES OF UNIT CELL
4.1. SIMPLE CUBIC CRYSTAL STRUCTURE(SCC)
• The simplest structure of cubic crystal system is simple cubic
crystal structure
• The unit cell consists of eight corner atoms which are in
contact with each other along the sides of the cube
Fig.1.9. Simple cubic crystal structure

• ATOMIC RADIUS
Since the atoms along the sides of the cube are in contact
with each other, the distance between any two nearest atoms is
calculated as,
2r = a
Where ‘a’ be the side of the cube Fig.1.10. Calculation of r
• CO-ORDINATION NUMBER
Each corner in a simple cubic unit cell is surrounded by
four nearest neighbours in its own plane. One atom exactly
above and one atom below
∴ The co-ordination number for SCC = 4+1+1 = 6
r =
𝒂
𝟐

• NUMBER OF ATOMS INA UNIT CELL
In a crystal structure, each and every atom at the unit cell
is shared by eight adjacent unit cells and hence each corner
atom contributes 1/8 of its part to one unit cell
∴The total number of atoms present in a unit cell = 1/8×8 = 1
• ATOMIC PACKING FACTOR
No. of atoms per unit cell = 1
Volume of atom per unit cell (v) = 1
4
3
𝜋𝑟3
Volume of the unit cell (V) = 𝑎3
Side of the unit cell (a) = 2r
∴ atomic radius r =
𝑎
2

∴Atomic packing factor =
𝑣
𝑉
=
4
3
𝜋𝑟3
𝑎3
=
4
3
𝜋𝑟3
(2𝑟)3
=
𝜋
6
= 0.5236
It shows that 52% of the volume of unit cell is occupied by
atoms and remaining 48% is vacant. Therefore simple cubic
system is a loosely packed system. Example: Polonium.
APF = 52%

4.2. BODY CENTERED CUBIC STRUCTURE(BCC)
• The unit cell of body centered has eight corner atoms and one
more at the centre of the body of the body of the unit cell as
shown in the fig.1.11
• In BCC unit cell, the atoms touches along the body diagonal
of the cube and there is no contact between atoms along side
of unit cell
Fig.1.11. Body Centered Cubic Structure

• ATOMIC RADIUS
In BCC structure since there is contact between diagonal
atoms the atomic radius can be calculated as follows
In fig. 1.11. let ‘a’ be the side of the unit cell
∴DH = a
AH = 4r
In triangle ADH, (𝐴𝐷)2
+(𝐷𝐻)2
= (𝐴𝐻)2
Fig.1.12 Calculation of atomic radius

Also, (𝐴𝐷)2
= (𝐴𝐵)2
+(𝐵𝐷)2
∴ (𝐴𝐵)2
+(𝐵𝐷)2
+(𝐷𝐻)2
=(𝐴𝐻)2
𝑎2
+𝑎2
+𝑎2
= 16𝑟2
16𝑟2
= 3𝑎2
∴ 𝑟2
=
3𝑎2
16
r =
3𝑎2
16
r =
𝒂 𝟑
𝟒

In bcc, The nearest neighbour for a body centered atom is
corner atom. Thus a body centered atom is surrounded by eight Corner
atoms. Similarly, each corner atom is shared by eight unit cells
∴ The co-ordination number of a BCC unit cell = 8
• NUMBER OF ATOMS / UNIT CELL
In BCC, the body centered atom is not shared by any other unit cell
but the eight corner atoms are shared by eight other unit cell. Hence the
contribution of each corner atom is 1/8 part of it to that unit cell
∴ The total number of a atoms in unit cell = 1+
𝟏
𝟖
× 8= 1+1= 2

Number of atoms per unit cell = 2
Volume of atoms per unit cell = 2×
4
3
𝜋𝑟3
Side of the unit cell(a) =
4𝑟
3
(Since r =
𝒂 𝟑
𝟒
)
∴APF =
𝑣
𝑉
=
2×
4
3
𝜋𝑟3
4𝑟
3
3
=
𝜋 3
8
= 0.68
It shows that 68% of the volume is occupied by atoms and
remaining 32% is vacant. Example : Iron, barium, Chromium.
APF = 68%

4.3. FACE CENTERED CUBIC SYSTEM (FCC)
• The unit cell of face centered has one atom at the center of
each face of the cubic unit cell
• Therefore there are six face centered atom along with eight
corner atoms as shown in the fig.1.13
• The atom along the face diagonal are in contact with each
other and there is no contact between the corner atoms
Fig.1.13. Face centered Cubic Structure

• ATOMIC RADIUS
In fig.1. . Consider the ∆ ABC,
(𝐴𝐶)2
= (𝐴𝐵)2
+(𝐵𝐶)2
(4𝑟)2
= 𝑎2
+𝑎2
16𝑟2 = 2𝑎2
𝑟2
=
𝟐𝑎2
𝟏𝟔
Fig.1.14 Calculation of atomic radius
In order to find the coordination number let us consider the
corner atom
It is surrounded by twelve face centered atoms
r =
𝒂 𝟐
𝟒

In face centered atoms it is surrounded by four corner atoms
in its own plane, four face centered atoms in the unit cell
above the plane and four centered atoms below the plane
∴ The coordination number in this case = 4+4+4 = 12
• NUMBER OF ATOMS PER UNIT CELL
Each face centered atoms is shared by two unit cells.
Therefore a face centered atom contributes half of its portion to
one unit cell.
Thus the total contribution of all face centered atoms =
1
2
×6 = 3
The contribution of corner atoms =
1
8
×8 = 1
∴ The total number of atoms in unit cell = 3+1= 4 atoms

Number of atoms per unit cell = 4
Volume of 4 atom (v) = 4×
4
3
𝜋𝑟3
Side of the unit cell (a) =
4𝑟
2
(Since r =
𝒂 𝟐
𝟒
)
∴APF =
𝑣
𝑉
=
4×
4
3
𝜋𝑟3
4𝑟
2
3
=
𝜋 2
6
= 0.74
Therefore 74% of the volume is occupied and remaining 26% is
vacant. Example: Al, Cu, Pb, Gold.
APF = 74%

5. SEVEN CRYSTAL SYSTEM
SL.
NO.
CRYSTAL
SYSTEM
AXIAL LENGTH
(a, b & c)
INTERAXIAL ANGLES
(𝜶, 𝜷 & 𝜸)
EXAMPLE
1 Cubic a = b = c 𝜶 = 𝜷 = 𝜸 NaCl
2 Tetragonal a = b ≠ c 𝜶 = 𝜷 = 𝜸 Ti𝑶𝟐
3 Orthorhombic a ≠ b ≠ c 𝜶 = 𝜷 = 𝜸 Ba𝑺𝑶𝟒
4 Monoclinic a ≠ b ≠ c 𝜶 = 𝜷 = 𝜸 Fe𝑺𝑶𝟒
5 Triclinic a ≠ b ≠ c 𝜶 = 𝜷 = 𝜸 Cu𝑺𝑶𝟒
6 Rhombohedral
(or)
Trigonal
a = b = c 𝜶 = 𝜷 = 𝜸 Calcite
7 Hexagonal a = b ≠ c 𝜶 = 𝜷 = 𝜸 Quartz

• Bravai’s who introduced the space lattice idea in 1848,
showed that there are only 14 ways of arranging points in
space
• So that the environment looks the same from each point
• That is, there are 14 possible types of space lattice in these
seven crystal system which are tabulated as follows
6. BRAVAIS LATTICE

14 POSSIBLE TYPES OF SPACE LATTICE IN 7 CRYSTAL SYSTEM
CRYSTAL SYSTEMS NUMBER OF POSSIBLE TYPES
Cubic 3 (Simple, Body-centered, Face-centered)
Tetragonal 2(Simple, Body-centered)
Orthorhombic 4 (Simple, Body-centered, Base-centered, Face-
centered)
Monoclinic 2(Simple, Base-centered)
Triclinic 1(Simple)
Rhombohedral
(or)
Trigonal
1(Simple)
Hexagonal 1(Simple)
TOTAL 14

7. MILLER INDICES
7.1. DEFINITION
Miller indices are the three smallest possible integers
enclosed in parenthesis which have the same ratios as the
reciprocals of the intercept of the plane concerned in the three
axes.
Fig.1.16. Miller Indices of a Plane

7.2. PROCEDURE FOR FINDING MILLER INDICES OF
CRYSTAL PLANES
Step 1: Find the intercepts of the plane along the coordinate
axes x, y and z
Step 2: Express the intercepts in terms of axis units (i.e. x=2a,
y=3b, z=2c)
Step 3: Find the reciprocals of the integer’s 2, 3 and 2 (i.e. 1/2:
1/3 : 1/2)
Step 4: Convert these reciprocals into whole numbers by
multiplying each one of them with their L.C.M. i.e. 323
Step 5: Enclose these integers in parenthesis (323). This
represents the indices of the given plane

7.3. FEATURES OF MILLER INDICES OF CRYSTAL
PLANES
• All the parallel planes have the same Miller indices. Thus the
Miller indices define a set of parallel planes
• A plane parallel to one of the co-ordinates axes has an
intercept of infinity
• If the Miller indices of two planes have the same ratio (i.e.
484 and 242 or 121) then the planes are parallel to each other

8. SUMMARY
• Crystalline structure is present in solid rather than in fluids
• There are crystalline and non-crystalline solids
• The crystal structure is composed from space lattice and basis
• The smallest component in the crystal is called “Unit Cell”
• The types of Unit cell include Simple cubic (SC), Face
centred cubic (FCC) and body centered cubic structure(BCC)
• These unit cell is governed by six lattice parameters : a, b, c,
𝜶, 𝜷 & 𝜸

• These parameters paved way for 7 Crystal systems.
• According to the possible arrangement of lattice points in
space lattice there are 14 possible types of crystalline lattice
introduced by Bravai’s introduced called “Bravai’s Lattice”
• The notation system in crystallography where a family of
lattice planes is determined by any three integers is defined as
Miller Indices.

9. REFERENCE
• ‘Material Science’ by K. Oudayakumar and Dr. D.
Sendhilnathan, 2nd edition August 2013.
• Askeland R. Donald, ‘The science and engineering of
materials’, Publisher : Bill Stenquist, fourth edition.
• Nesse D. William, ‘Introduction to Mineralogy’,
Publication: Oxford University Press, 2nd edition.
• http://www.brainkart.com/article/Primitive-and-non-
primitive-unit-cell_38660/

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