Elementary Material Science Concept
CHAPTER 01 LECTURE 01
Farhan Sadik Sium Lecturer, EEE
1.8.1 Type of Crystals
1. Classification of materials in terms of crystallization.
There are two types of materials in terms of crystallization: Crystalline and Non- crystalline materials. Non-crystalline material also called Amorphous material.
2. Crystalline and Amorphous solids
Crystalline solids: A crystalline solid is a solid in which the atoms (or ions or molecules) bond with each other in a regular pattern to form a periodic collection (or array) of atoms, as shown for the copper crystal in Figure 1. The most important property of a crystal is periodicity, which leads to what is termed long-range order. Most of the materials are crystalline materials.
Crystalline materials are of two types: Single crystal and Polycrystal.
Amorphous solids: Amorphous solid is a solid that exhibits
no
crystalline structure or long-range order. It only possesses a short-range order in the sense that the nearest neighbors of an atom are well defined by virtue of chemical bonding requirements. Glass is an amorphous solid which shows the continuous random network structure of amorphous silicon dioxide.2. Crystalline and Amorphous solids
Fig. 1 Single crystal, Polycrystal and Amorphous materials
3. Difference between
polycrystal and single crystal.
Polycrystalline materials are composed of many small crystals that are periodic across grain. On the other hand, single crystals have only a single big crystal that is periodic across the whole volume of the crystal.
4. Definition of crystal
Crystal is a three-dimensional periodic arrangement of atoms, molecules, or ions. A characteristic property of the crystal structure is its periodicity and a degree of symmetry. For each atom, the number of neighbors and their exact orientations are well defined; otherwise the periodicity will be lost.
Fig. 2. A standard crystal structure.
5. Lattice, Basis and Unit cell.
Lattice: Lattice is a regular array of points in space with a discernible periodicity. There are 14 distinct lattices possible in three-dimensional space. When an atom or molecule is
placed at each lattice point, the resulting regular structure is a crystal structure.
Basis: Basis represents an atom, a molecule, or a collection of atoms, that is placed at each lattice point to generate the true crystal structure of a substance. All crystals are thought of as a lattice with each point occupied by a basis.
Unit cell: Unit cell is the most convenient small cell in a crystal structure that carries the characteristics of the crystal. The repetition of the unit cell in three dimensions generates the whole crystal structure.
All crystal structures can be described in terms of lattice and basis as depicted in the following figure.
Figure: Crystal structure is composed of lattice and basis.
Figure: Unit cell and Lattice points.
6. Lattice parameters
As a convention, we generally represent the geometry of the unit cell as a parallelepiped with sides a, b and c and angles alpha, beta and gama as shown in the figure. The sides a, b and c and angles alpha, beta and gama are referred to as lattice parameters. To establish a reference frame and to apply three-dimensional geometry, we
insert a xyz coordinate system. Figure 4. Demonstration of lattice parameters.
7. Crystal System and Bravais lattice
There are seven (7) crystal systems in terms of unit cell geometry. The seven crystal system includes fourteen (14) lattices (called Bravais lattice) in total considering simple, face centered and body centered for some of the unit cell geometries.
The seven crystal systems are depicted in the figure below in terms of their parametric conditions. The fourteen Bravais lattices are also shown in the same figure.
Figure: The seven crystal systems (unit cell geometries) and fourteen Bravais lattices.
8. Type of structures of a unit cell in terms of number of atoms and their position
There may be three types of structures of a specific geometry of unit cell in terms of.
number of atoms and their position in the crystal. They are called Simple, Face- centered and Body-centered. For example, cubic unit cell has all the three structures namely:
Simple cubic (SC): When all the atoms take their position during solidification from gas or liquid phases at the eight corners of the unit cell. Practically, there is no crystal with SC unit cell.
Face-centered Cubic (FCC): In FCC unit cell, atoms take their positions at eight corners as well as one atom at the center of each of six faces. Cu has a FCC unit cell structure.
Body-centered Cubic (BCC): In BCC unit cell, atoms take their positions at eight corners as well as one atom at the center of the body of unit cell. Fe has a BCC unit cell structure.
Figure 6. SC, FCC and BCC unit cells.
9. Determination of relation between atomic radius (R) and lattice
parameter (a).
Relation between atomic radius (r) and lattice parameter (a) is calculated as follows for SC, FCC and BDD unit cells.10. Determination of number of atoms (N) per unit cell.
Number of atoms in a unit cell is determined considering the sharing of atoms with the neighboring unit cells as shown in the figure.
11. Determination of Atomic
Packing Factor or Efficiency or Density (APF).
Atomic Packing Factor or Density (APF) is defined by the following expression.APF =
Let us, for example, determine the APF for SC unit cell. Note that atoms are always considered to be a solid sphere. For SC unit cell, N = 1 and r = a/2.
Therefore, APF for SC unit cell:
Assignment: Determine APF for BCC and FCC unit cells and show that APF of FCC is greater than BCC.
12. Table showing number of atoms, relation of
‘a’ with ‘R’ and Atomic Packing Factor or
Density.
13. Close-packed crystal structure (CP) and
Hexagonal close-packed (HCP) crystal structure
The FCC crystal structure of Cu is known as a Close-packed (CP) crystal structure because the Cu atoms are packed as closely as possible. The volume of the FCC unit cell is 74% full of atoms, which is the maximum packing possible with identical spheres.
By comparison, iron has a body-centered cubic (BCC) crystal structure and its unit cell has Fe atoms at its comers and one Fe atom at the center of the cell. The volume of the BCC unit cell is 68% full of atoms, which is lower than the maximum possible packing.
However, the FCC crystal structure is only one way to pack the atoms as closely as possible. On the other hand, in Zinc, the atoms are arranged as closely as possible in a hexagonal symmetry, to form the Hexagonal close-packed (HCP) crystal structure. This structure corresponds to packing spheres as closely as possible also makes 74% full of atoms.
1. Significance of crystal directions and planes
In explaining crystal properties, we must frequently specify a direction in a crystal, or a particular plane of atoms. Many
properties, for example, the elastic modulus, electrical resistivity, magnetic susceptibility, etc., are directional within the crystal.
We use the convention described here for labeling crystal
directions based on three-dimensional geometry. Note that all parallel vectors have the same indices.
2. Determination of crystal direction
Any direction of a particular point from the origin of a crystal is determined three-dimensionally based on the intercepts or
projections of the point on xyz co-ordinates as follows. The
indices for negative x, y and z directions are written with a bar on the top as shown in the figure.
Table: General convention to determine crystal direction
Steps Action
1 Three coordinates x y z
2 Side parameters a b c
3 Intercepts x1 y1 z1
4 Clear fraction u v w
5 Indices with square bracket (without comma) [ u v w ]
Example: Find direction of P in the following figure (b).
Table: Determination of crystal direction
Steps Action
1 Three coordinates x y z
2 Side parameters a b c
3 Intercepts 1/2 1 1/2
4 Clear fraction (multiplying by 2) 1 2 1
5 Indices with square bracket [ 1 2 1 ]
3. Determination of crystal
planes
We also frequently need to describe a particular atomic plane in a crystal. We use the following convention depicted in the tablebelow, called the Miller indices of a plane, for this purpose. Note that if the plane passes through the origin, we can use another convenient parallel plane, or simply shift the origin to another
point. In fact, all parallel planes have identical Miller indices. Note that plane parallel to any axis will have infinite intercept for that axis.
Table: General convention to determine crystal planeSteps Action
1 Three coordinates x y z
2 Side parameters a b c
3 Intercepts x1 y1 z1
Reciprocal of intercepts 1/x1 1/y1 1/z1
4 Clear fraction h k l
Reduced to smallest integer h k l
5 Indices with square parenthesis (without
comma) ( h k l )
Example: Find orientation of following plane in the figure (a).
Step
s Action
1 Three coordinates x y z
2 Side parameters a b c
3 Intercepts 1/2 1
Reciprocal of intercepts 2 1 0
4 Clear fraction (no fraction) 2 1 0
Reduced to smallest integer (nothing to reduce) 2 1 0
5 Indices with parentheses (First bracket) ( 2 1 0 ) Step
s Action
1 Three coordinates x y z
2 Side parameters a b c
3 Intercepts 1/2 1
Reciprocal of intercepts 2 1 0
4 Clear fraction (no fraction) 2 1 0
Reduced to smallest integer (nothing to reduce) 2 1 0
5 Indices with parentheses (First bracket) ( 2 1 0 )
4. Planner concentration
Planar concentration of atoms is the number of atoms per unit area, that is, the surface concentration of atoms, on a given plane in the crystal.
1.8.3. Allotropy or
Polymorphism
1. Allotropy and Allotrope
The behavior of certain materials showing more than one crystal structure is termed as Polymorphism or Allotropy.
Each of such multi crystal structures of a material is called Allotrope and the mother material is called Allotropic
material. Many substances have allotropes that exhibit widely different properties.
Some of the Allotropic materials and their Allotropes are shown in the table below. Materials
Heating temperature Allotropes Structures Properties
Iron < 912 deg. C α-Fe BCC Metal
912.– 1400 deg. C γ-Fe FCC Metal
> 1400 deg. C δ-Fe BCC Metal
Carbon Diamond Insulator
Graphite Conductor
BuckminsterFullerene
(C60) Semiconduct
or
Three Allotropes of Carbon
1.9. Crystalline Defects and Their
Significance
1. Crystalline Defect
When all the atoms are brought together to form a perfect crystal, the total potential energy of the atoms are lowered as much as possible for that particular structure. However, when crystal is grown from a liquid or vapor, quite often some abnormalities in the crystal structure occurs that is called crystalline defect. Usually all the defects act as scattering centers for electron in the metal crystal.
2. Significance of crystalline defect.
Quite often crystalline defects are of great importance as the key physical properties like mechanical, electrical and magnetic
properties are controlled by these defects.
3. Classification of Defects
The defects exist in the crystal and their causes and impacts are summarized in the following table.
Defects
Type Defect causes
Definitions
Point defect Vacancies Vacancy is a point defect in a crystal, where a normally occupied lattice site is missing an atom.
Impurities
If the impurity atom is doped in the host atom,
point defect may arise due to lattice change or distortions. Figure
Line defect Edge dislocations A line defect is formed in a crystal when an atomic plane terminates within the crystal instead of passing all the way to the end of the crystal, we therefore call this type of defect an edge dislocation
Screw dislocations The screw dislocation, which is essentially a shearing of one portion of the crystal with respect to another.
Planner
defect Grain boundaries Grain boundary is a surface region between differently oriented adjacent grain crystals that contains a lattice mismatch between adjacent grains.
Figure. Point defects
Figure. Planner defect
4. Grain and Grain Boundary
Grain is an individual crystal within a polycrystalline material.
Within a grain, the crystal structure and orientation are the same everywhere and the crystal is oriented in one direction only.
Grain boundary is a surface region between differently oriented adjacent grain crystals. The grain boundary contains a lattice mismatch between adjacent grains.
1.12.2. Phase Diagrams and Other Alloys
1. What is phase diagram for materials? 2. What is the significance of phase diagram?
3. Draw and explain phase diagram for Pb-Sn alloy
