Crystal Structures_Module_I

Module-I: Crystal Structures

1. Introduction

  • Definition of crystal structures: Arrangement of atoms in crystalline materials with ordered patterns.

  • Importance in materials science and solid-state physics.

  • Impacts on properties like strength, conductivity, and appearance.

2. States of Matter

  • Definition: Matter that has mass and occupies space (volume).

  • States: Solid, Liquid, Gas, Plasma.

3. Types of Solids

  • Single Crystalline: Atoms arranged in a single uniform lattice.

  • Polycrystalline: Composed of many small crystals or grains.

  • Amorphous: Non-crystalline solids without long-range order.

4. Lattice Structure

  • Lattice: A 3D translationally periodic arrangement of points in space.

    • Lattice points: Identical environments for atoms in the crystal.

    • Crystal Structure: Lattice + Basis

  • Basis: Group of atoms associated with each lattice point.

5. Unit Cell

  • Definition: The smallest repetitive unit that builds the crystal.

  • Types of Unit Cells:

    • Primitive Cell: Atoms only at corners.

    • Body-Centered Cube (BCC): Atoms at corners and one at the center.

    • Face-Centered Cube (FCC): Atoms at corners and centers of each face.

6. Crystal Systems and Bravais Lattices

  • Lattice Parameters: a, b, c (lattice constants) and angles α, β, γ.

  • Bravais Lattices: 14 distinguishable arrangements.

    • 7 Crystal Systems: Cubic, Tetragonal, Orthorhombic, Hexagonal, Trigonal, Monoclinic, Triclinic.

7. Packing Fraction and Atomic Arrangement

  • Atomic Packing Factor (APF): Ratio of volume occupied by atoms in a unit cell to total volume.

    • Example for SC, BCC, FCC: 0.52, 0.68, 0.74 respectively.

  • Void Space: Unoccupied volume in a unit cell, calculated as (1 - APF).

  • Coordination Number: Number of nearest neighbors of an atom (6 for SC, 8 for BCC, 12 for FCC).

8. Miller Indices

  • Definition: A notation to identify crystallographic planes and directions.

    • Notation: (h, k, l) for planes, [hkl] for directions, <hkl> for families.

  • Determination: Based on intercepts on axes.

9. Interplanar Spacing

  • Formula: dhkl = a / √(h² + k² + l²)

    • Used to calculate distance between crystal planes in cubic structures.

10. Problem Solving Examples

  • Calculate number of atoms/unit cell, densities, Miller indices, and interplanar spacings for various structures using given formulas and example problems.