�ݺ�ߣ

Engineering Materials
ME 1113
Lecture - 03
Metal structure and
Crystallization
Prepared by
Md. Helal Hossain

Crystal Geometry
To study the crystal geometry, we have to have knowledge about
following topics.
• Crystal.
• Lattice.
• Motif.
• Unit cell.
• 7 crystal Systems.
• 14 Bravais lattice.
• Miller Indices.
ME 1113 MD. HELAL HOSSAIN
ENGINEERING MATERIALS

Crystal
Crystal is one dimensional (1D), two
dimensional (2D) three dimensional (3D)
periodic arrangement of atoms or ions in
a space. Let’s consider a 3D NaCl crystal
structure as shown in figure. The red
atom is chlorine and white atom is
sodium. Now three dimensional periodic
arrangement of atoms means that these
chlorine and sodium atoms are repeating
in each direction at equal distances.
ME 1113 ENGINEERING MATERIALS
3D NACL Crystal
Structure
MD. HELAL HOSSAIN

Lattice
Lattice is one dimensional (1D), two dimensional (2D) three
dimensional (3D) periodic arrangement of points in a space. The
difference between crystal and lattice is whether the points are being
considered or atoms are being considered otherwise both are one and
three dimensional periodic arrangement.
Infinite lattice
MD. HELAL HOSSAIN

Crystal and Lattice
Difference:
Relation between crystal and lattice:
The relationship between them is expressed by following equation.
Crystal = Lattice + Motif or Basis
So the linking bridge between crystal and lattice is a motif.
ME 1113 ENGINEERING MATERIALS MD. HELAL HOSSAIN

Lattice Translation: Lattice translation is defined as any vector from one
lattice point to another lattice point. In the figure, blue color arrow
indicates the lattice translation but red color arrow does not represent
lattice translation since there is one lattice point in-between first and
last lattice point.
Lattice
MD. HELAL HOSSAIN

Unit Cell: Unit cell is defined as a region of space which can generate
the entire lattice or crystal by repetition through lattice translations.
The common definition of unit cell is that it is a parallelogram (in 2D) or
a parallelepiped (in 3D).
Unit Cell
2D Unit Cell 3D Unit Cell
MD. HELAL HOSSAIN

Based on the unit cell geometry, there are seven basic crystal system:
7 Crystal System
Tetragonal
Orthorhombic
Monoclinic
Triclinic
Trigonal Hexagonal Cubic
MD. HELAL HOSSAIN

14 Bravais Lattice Systems
Bravais Lattice refers to the 14 different 3-
dimensional configurations into which atoms
can be arranged in crystals. They are:
Cubic System: In Bravais lattices with cubic
systems, the following relationships can be
observed.
a = b = c
𝛂 = 𝞫 = 𝝲 = 90°
The 3 possible types of cubic cells.
Examples: Polonium has a simple cubic
structure, iron has a body-centered cubic
structure, and copper has a face-centered
cubic structure.
MD. HELAL HOSSAIN

Orthorhombic Systems: The Bravais lattices with orthorhombic systems obey the
following equations:
a ≠ b ≠ c
𝛂 = 𝞫 = 𝝲 = 90°
The four types of orthorhombic systems are illustrated here.
Examples: Rhombic Sulphur has a simple orthorhombic structure, Magnesium sulfate
heptahydrate (MgSO4.7H2O) is made up of a base centered orthorhombic structure,
Potassium Nitrate has a structure which is body-centered orthorhombic.
MD. HELAL HOSSAIN

Tetragonal Systems: In tetragonal Bravais lattices, the following relations are observed:
a = b ≠ c
𝛂 = 𝞫 = 𝝲 = 90°
The two types of tetragonal systems are simple tetragonal cells and body-centered
tetragonal cells, as illustrated here.
Examples: Examples of tetragonal Bravais lattices are – stannic oxide (simple
tetragonal) and titanium dioxide (body-centered tetragonal)
MD. HELAL HOSSAIN

Monoclinic Systems: Bravais lattices having monoclinic systems obey the following
relations:
a ≠ b ≠ c
𝞫 = 𝝲 = 90° and 𝛂 ≠ 90°
The two possible types of monoclinic systems are primitive and base centered
monoclinic cells, as illustrated below.
Examples: Cubic cells are – Monoclinic Sulphur (simple monoclinic) and sodium sulfate
decahydrate (base centered monoclinic)
MD. HELAL HOSSAIN

Triclinic System: There exists only one type of triclinic
Bravais lattice, which is a primitive cell. It obeys the
following relationship.
a ≠ b ≠ c
𝛂 ≠ 𝞫 ≠ 𝝲 ≠ 90°
Examples: potassium dichromate (Chemical formula
K2Cr2O7).
Rhombohedral System: Only the primitive unit cell for a
rhombohedral system exists. Its cell relation is given by:
a = b = c
𝛂 = 𝞫 = 𝝲 ≠ 90°
Examples: Calcite and sodium nitrate are made up of
simple rhombohedral unit cells.
MD. HELAL HOSSAIN

Hexagonal System: The only type of hexagonal
Bravais lattice is the simple hexagonal cell. It has
the following relations between cell sides and
angles.
a = b ≠ c
𝛂 = 𝞫 = 90o and 𝝲 = 120°
An illustration of a simple hexagonal cell is
provided below.
Examples: Zinc oxide and beryllium oxide are
made up of simple hexagonal unit cells.
Thus, it can be noted that all 14 possible Bravais
lattices differ in their cell length and angle
relationships. It is important to keep in mind
that the Bravais lattice is not always the same as
the crystal lattice.
MD. HELAL HOSSAIN

To fix the coordinate system of a crystal, first a unit cell has to be
selected. Then a corner of unit cell has to be selected as the origin (O).
The vectors from the origin corner along the edges of the unit cell
define the basis vectors of the lattice. The basis vectors of lattice are
denoted ¯a and ¯b. Y OX forms the 2D crystal coordinate system. Let
consider the 3D crystal coordinate system as shown in figure below.
The basis vectors of lattice are denoted ¯a, ¯b and ¯c. OXYZ forms the
3D crystal coordinate. system.
Crystal Coordinate System
MD. HELAL HOSSAIN

Lattice Parameters: The lengths of two edges of unit cell or basis
vectors (in 2D unit cell) and the lengths of three edges of unit cell or
basis vectors (in 3D unit cell); and the inter-axial angles between basis
vectors are defined as the lattice parameters.
• Lattice Parameters of 2D Unit Cells: In case of 2D unit cells, the
number of lattice parameters are three: two length of edges and one
angle between them. They are denoted as
a = length of edge
b = length of edge
γ = angle between two a and b
These parameters in different 2D unit cells are shown in figure
Lattice Parameters
MD. HELAL HOSSAIN

• Lattice Parameters of 3D Unit Cells: In case of 3D unit cells, the
number of lattice parameters are six: three length of edges and three
angles between them. They are denoted as
a¯ = length of edge
¯b = length of edge
c¯ = length of edge
γ = angle between two ¯ a and ¯b
β = angle between two ¯ a and ¯ c
α = angle between two ¯b and ¯ c
These parameters in different 3D unit cells are shown in figure.
Lattice Parameters
MD. HELAL HOSSAIN

In a crystal lattice, planes are defined by
the arrangement of atoms, and
directions are defined by the path
between atoms. Miller indices provide a
concise and systematic way to describe
these planes and directions.
To determine the Miller indices of a
plane, you take the reciprocals of the
intercepts made by the plane on the
crystallographic axes and multiply them
by a common denominator to get integer
values. The resulting integers are then
enclosed in parentheses, written as (hkl)
and for the direction it is written as [hkl]
Miller Indices
MD. HELAL HOSSAIN

Why is Miller indices important:
• Miller indices play a crucial role in crystallography as they help in
identifying and characterizing different crystallographic planes and
directions.
• They provide a universal language for scientists to communicate and
understand the structural properties of crystals, enabling the study of
various physical and chemical phenomena in materials science, solid-
state physics, and mineralogy.
Miller Indices
MD. HELAL HOSSAIN

How to determine Miller Indices:
Step 01: Identify the intercepts on the x-, y-
and z- axes.
In this case the intercept on the x-axis is at x
= a ( at the point (a,0,0) ), but the surface is
parallel to the y- and z-axes - strictly
therefore there is no intercept on these two
axes but we shall consider the intercept to be
at infinity ( ∞ ) for the special case where the
plane is parallel to an axis. The intercepts on
the x-, y- and z-axes are thus
Intercepts: a, ∞, ∞
Miller Indices
MD. HELAL HOSSAIN

Step 02:Specify the intercepts in fractional
co-ordinates
Co-ordinates are converted to fractional co-
ordinates by dividing by the respective cell-
dimension - for example, a point (x,y,z) in a
unit cell of dimensions a x b x c has
fractional co-ordinates of ( x/a, y/b, z/c ). In
the case of a cubic unit cell each co-ordinate
will simply be divided by the cubic cell
constant, a . This gives
Fractional Intercepts: a/a, ∞/a, ∞/a i.e. 1, ∞,
∞
Miller Indices
MD. HELAL HOSSAIN

Step 03: Take the reciprocals of the fractional intercepts
This final manipulation generates the Miller Indices which (by
convention) should then be specified without being separated by any
commas or other symbols. The Miller Indices are also enclosed within
standard brackets (hkl) when one is specifying a unique surface such as
that being considered here.
The reciprocals of 1 and ∞ are 1 and 0 respectively, thus yielding
Miller Indices: (100)
So the surface/plane illustrated is the (100) plane of the cubic crystal.
Miller Indices
MD. HELAL HOSSAIN

Miller Indices - Examples
MD. HELAL HOSSAIN

Miller Indices - Examples
DO IT
YOURSEL
F
MD. HELAL HOSSAIN

Polymorphism and Allotropy
Polymorphism: Polymorphism in general means having many forms. The
term is used in chemistry to describe the ability of a chemical
compound to crystallize in multiple unit cell configurations.
Types of polymorphism:
1. Monotropic System: In a monotropic system, only one
polymorph is stable at all temperature ranges. A very good example
of a monotropic system is metolazone.
2. Enantiotropic System: In an enantiotropic system, different
polymorphs are stable across different temperature ranges. Examples
of an enantiotropic system include carbamazepine and
acetazolamide.
MD. HELAL HOSSAIN

Polymorphism and Allotropy
Allotropy: While polymorphism is the ability of a solid material to exist
in more than one crystal structure, allotropy is the ability of chemical
elements to exist in two or more different forms in the same physical
state.
Difference between polymorphism and allotropy:
Polymorphism is the ability of a solid material to exist in more than one
crystal structure. Allotropy is the ability of chemical elements to exist in
two or more different forms in the same physical state.
In the case of crystal solids, allotropy is a particular case of
polymorphism.
MD. HELAL HOSSAIN

End of Lecture – 03
Thank you
ME 1113 ENGINEERING MATERIALS MD. HELAL HOSSAIN

�ݺ�ߣ

Lecture 03_Metal structure and Crystallization.pptx

Recommended

More Related Content

Similar to Lecture 03_Metal structure and Crystallization.pptx (20)

More from MdHelalHossain6 (20)

Recently uploaded (20)

Lecture 03_Metal structure and Crystallization.pptx