1. CRYSTAL STRUCTURES AND BONDS, ME 3002 3rd semester notes

1. CRYSTAL STRUCTURES AND BONDS

1.1 Unit Cell and Space Lattice

Unit Cell: The smallest repeating unit of a crystal structure that reflects the full symmetry of the crystal. It's a box-like structure that, when repeated in space, forms the entire crystal lattice. Each unit cell represents a portion of the entire crystal structure and contains the arrangement of atoms, ions, or molecules in the crystal.
Space Lattice: The three-dimensional arrangement of points (atoms, ions, or molecules) in space. These points form the periodic pattern of the crystal structure. A space lattice is the geometric arrangement of unit cells in space.

Diagram of Unit Cell in a Crystal Structure:

In the image above, you see a simple cubic unit cell, where each corner of the cube contains an atom. The unit cell is repeated in three-dimensional space to form the crystal lattice.

1.2 Crystal System

The crystal system refers to the classification of crystals based on the symmetry of their unit cells. There are seven basic crystal systems, which are based on the lengths and angles of the unit cell's axes.

Cubic (Isometric): All three axes are of equal length, and the angles between them are all 90°. E.g., Sodium chloride (NaCl).
Tetragonal: Two axes are equal, but the third is different. The angles are 90° between the axes. E.g., Tin (Sn).
Orthorhombic: All three axes are different in length, but the angles are 90° between the axes. E.g., Sulfur (S).
Rhombohedral (Trigonal): All three axes are equal in length, but the angles between them are not 90° (they are oblique). E.g., Quartz (SiO2).
Hexagonal: Two axes are of equal length, with the third axis perpendicular to them. The angle between the two equal axes is 120°, and between the other axes is 90°. E.g., Graphite.
Monoclinic: All three axes are of different lengths, and two axes are at an angle of 90°, while the third is oblique. E.g., Gypsum.
Triclinic: All three axes are of different lengths, and none of the angles are 90°. E.g., Turquoise.

1.2.1 The Seven Basic Crystal Systems

The seven crystal systems mentioned above can be differentiated by the lengths and angles of their unit cell's axes. These systems form the basis for how atoms arrange in space and influence the properties of materials.

1.2.2 Atomic Radius and Atomic Radius for Simple Cubic, BCC, and FCC

Atomic Radius: The atomic radius refers to the distance from the nucleus of an atom to the outermost electron shell. It plays a critical role in determining the packing and overall density of materials.

For different types of unit cells, the atomic radius varies:

Simple Cubic (SC): In a simple cubic structure, each corner of the cube has an atom, and atoms touch along the edge of the cube. If the edge length of the unit cell is 'a', then the atomic radius $r$ is:
$r = \frac{a}{2}$
Body-Centered Cubic (BCC): In a BCC structure, atoms are positioned at each corner of the cube and one atom in the center. The atoms touch along the body diagonal. If the edge length is 'a', the atomic radius $r$ is:
$r = \frac{a}{\sqrt{3}} \times \frac{1}{2}$
Face-Centered Cubic (FCC): In an FCC structure, atoms are at the corners and the centers of the faces of the cube. The atoms touch along the face diagonal. The atomic radius $r$ for FCC is:
$r = \frac{a}{\sqrt{2}} \times \frac{1}{2}$

Diagram:

Simple Cubic, BCC, and FCC structures differ in the arrangement and number of atoms per unit cell.
For Simple Cubic, the atoms are at each corner.
For BCC, one atom is at the center and corners.
For FCC, atoms are at corners and face centers.

1.2.3 Atomic Packing Factor (APF) for Simple Cubic, BCC, FCC, and HCP

The Atomic Packing Factor (APF) is a measure of how efficiently atoms are packed into a unit cell. It is defined as the fraction of volume in a unit cell that is occupied by atoms.

Simple Cubic (SC):
- In a simple cubic structure, atoms are only located at the corners. Each corner atom is shared by 8 unit cells, so the number of atoms per unit cell is 1.
- APF for SC is:
  $APF = \frac{\text{Volume of atoms in unit cell}}{\text{Volume of unit cell}} = \frac{1 \times \frac{4}{3} \pi r^3}{a^3} \approx 0.52$
Body-Centered Cubic (BCC):
- In a BCC structure, atoms are at the corners and one at the center. There are 2 atoms per unit cell.
- APF for BCC is:
  $APF = \frac{2 \times \frac{4}{3} \pi r^3}{a^3} \approx 0.68$
Face-Centered Cubic (FCC):
- In FCC, atoms are at the corners and face centers, contributing 4 atoms per unit cell.
- APF for FCC is:
  $APF = \frac{4 \times \frac{4}{3} \pi r^3}{a^3} \approx 0.74$
Hexagonal Close-Packed (HCP):
- In HCP, the atoms are arranged in a hexagonal pattern with two atoms in the center of each unit cell.
- APF for HCP is approximately:
  $APF = 0.74$

1.3 Simple Problems on Finding the Number of Atoms for a Unit Cell

To calculate the number of atoms in a unit cell, we consider how atoms are shared between adjacent unit cells. The number of atoms is a result of adding the contribution of atoms at the corners, faces, edges, or centers.

Example 1: Simple Cubic (SC)

Atoms are located only at the corners of the cube.
Each corner atom is shared by 8 unit cells, so the contribution of each corner atom to the unit cell is $\frac{1}{8}$ .
There are 8 corners, so the total number of atoms in a simple cubic unit cell is:
$8 \times \frac{1}{8} = 1 \text{ atom per unit cell}$

1.3.1 Bonds in Solids: Primary and Secondary Bonds

Primary Bonds: These are the strong, fundamental bonds that hold atoms or ions together in solids. They are formed due to the electrostatic interaction between atoms or molecules.
- Ionic Bond: Formed when one atom transfers electrons to another atom, resulting in positive and negative ions that attract each other (e.g., NaCl).
- Covalent Bond: Formed when two atoms share electrons to fill their outer electron shells (e.g., diamond, H2O).
- Metallic Bond: Involves the sharing of free electrons among a lattice of metal atoms, which allows metals to conduct electricity and have malleability (e.g., copper, gold).
Secondary Bonds: These are weaker bonds that arise due to interactions between molecules or atoms, not involving the transfer or sharing of electrons.
- Dispersion Bond (London Dispersion Force): These are weak attractions due to temporary dipoles created when electrons move around in atoms or molecules (e.g., noble gases).
- Dipole Bond: Occurs between molecules that have a permanent dipole moment (e.g., HCl).
- Hydrogen Bond: A specific type of dipole-dipole interaction where hydrogen is bonded to electronegative atoms like oxygen or nitrogen (e.g., H2O, DNA strands).

1. CRYSTAL STRUCTURES AND BONDS, ME 3002 3rd semester notes