Study Notes on Solid State Physics and Crystallography

MODULE 2: SOLID STATE PHYSICS

2.1 CRYSTALLOGRAPHY

INTRODUCTION

• Matter usually exists in solid or fluid (liquid, gas) states.

• Modern classification of matter specifies:

  • Condensed State (includes solids and liquids)

  • Gaseous State

• Definition of Solids:

  • Any material whose position of constituent particles is fixed is regarded as a solid.

• Characteristics of Solids:

  • Incompressibility

  • Rigidity

  • Mechanical Strength

  • Indicates closely packed molecules, atoms, or ions leading to well-ordered molecular, atomic, or ionic arrangement.

Classification of Solids

• Solids can be classified into two main categories:

  • Crystalline Solids:

  • Particles are orderly arranged (long-range order).

  • Amorphous Solids:

  • Particles are randomly oriented.

• Definition of Crystal Structure:

  • If atoms or molecules are uniquely arranged, the solid or liquid is referred to as having a crystal structure.

  • Key properties include:

  • Long-range order

  • Symmetry

• Main property of a crystal structure is its periodicity, resulting from the arrangement of atoms/molecules at the lattice points.

  • The entire crystal structure can be seen as the repetition of the unit cell.

• For given crystal structures, the shape of the unit cell is consistent but can vary between different types of crystals.

• X-ray diffraction studies demonstrate that constituent particles are arranged in a specific pattern within crystals, requiring the wavelength of light used to be comparable with the atomic spacing.

LATTICE AND BASIS

Crystal Structure Formation

• A crystal structure is formed when a group of atoms is identically arranged at lattice points.

• Space Lattice Definition:

  • An array of imaginary points arranged in space where each point has identical surroundings.

• Basis:

  • The group of atoms or molecules acting as the building unit or structural unit for the complete crystal structure.

  • Lattice + Basis = Crystal Structure

• Translation in Lattice:

  • The line joining any two lattice points is a translation.

  • Non-collinear translations lead to a plane lattice, while three non-coplanar translations lead to a space lattice.

UNIT CELL

Definition and Characteristics

• A Unit Cell is the building block of a crystal.

  • It retains the same symmetry as the entire crystal.

  • When arranged in 3D, it forms the bulk crystal.

  • It can be described as the smallest volume which, when repeated in all directions, generates the crystal structure.

  • The three edges of the unit cell are denoted as a, b, c, and the angles between them as α, β, γ (lattice parameters).

• A parallelepiped is generally considered as a unit cell in 3D.

Types of Unit Cells

• Unit cells are categorized into:

  • Primitive: Minimum volume unit cell with one lattice point.

  • Non-Primitive: Contains more than one lattice point.

  • Note: Not all primitive cells are unit cells. Some coincide.

  • Example:

  • Simple Cube is a primitive cell with 1 atom per unit cell.

  • Other types: Non-Primitive cells with 2 and 4 atoms per unit cell.

BRAVAIS LATTICE IN 2-D

Space Lattice Characteristics

• The number of lattices obtained in 2-D is unlimited, with no restrictions on lengths (a, b) or angles between them.

• An arbitrary lattice characterized by arbitrary lengths is known as oblique lattice.

  • Invariant under rotation of $2 heta/n$ (with $n=1, 2$).

  • Also invariant under rotation $2 heta/n$ for $n=3, 4, 6$.

COMMON UNIT CELL TYPES

Types and Examples

• Simple (Primitive) Cubic:

  • Atoms at corners, 1 atom per unit cell.

• Body-Centered Cubic (BCC):

  • Atoms at each corner + 1 atom at the center.

  • Coordination Number: 8.

  • Total atoms: 2.

• Face-Centered Cubic (FCC):

  • Atoms at each corner + atoms in the center of each face.

  • Coordination Number: 12.

  • Total atoms: 4.

• Hexagonal Close-Packed (HCP):

  • Structure with three layers of atoms where atoms nest in the gaps, forming hexagonal layers.

COORDINATION NUMBER

• Defined as the number of nearest neighbors to a central atom in a molecule or crystal.

• Coordination Numbers for Common Unit Cells:

  • Body-Centered Cubic: 8 (total 2 atoms)

  • Simple Cubic: 6 (total 1 atom)

ATOMIC PACKING FRACTION (APF)

Definition

• APF measures the arrangement efficiency of atoms/ions in solids.

• Defined as the ratio of the volume of atoms occupying the unit cell to the volume of the unit cell:

  • $APF = \frac{Volume \ of \ atoms \ in \ unit \ cell}{Volume \ of \ unit \ cell}$

Examples of APF Calculation

1. Simple Cubic:

   • Consider cube of side 'a'. Radius 'r' placed at corners, such that $a = 2r$.

   • Volume of atoms in unit cell: $1 \times \frac{4}{3} \pi r^3$

   • Total atoms per unit cell: $ rac{1}{8} \times 8 = 1$.

   • Volume of unit cell: $a^3 = (2r)^3 = 8r^3$.

   • $APF = \frac{\frac{4}{3} \pi r^3}{8r^3} = 0.524 \ (52\%)$

2. Body-Centered Cube:

   • Cube of side 'a', radius 'r' at corners and center. Body diagonal: $\sqrt{3}a = 4r$

   • Volume of atoms in unit cell: $2 \times \frac{4}{3} \pi r^3$

   • Total atoms per unit cell: $\frac{1}{8} \times 8 + 1 = 2$.

   • Volume of unit cell: $a^3$, with $a = \frac{4r}{\sqrt{3}}$.

   • $APF = \frac{volume \ of \ atoms}{volume \ of \ unit \ cell}$, calculated as $68\%$.

3. Face-Centered Cube:

   • Cube of side 'a', radius 'r' at corners and face center. Face diagonal: $\sqrt{2}a = 4r$

   • Volume of atoms in unit cell: $4 \times \frac{4}{3} \pi r^3$.

   • Total atoms per unit cell: $\frac{1}{8} \times 8 + \frac{1}{2} \times 6 = 4$.

   • Volume of unit cell: $\sqrt{2}a$, calculated as $0.74 \ (74\%)$.

MILLER INDICES

Definition and Use

• Miller Indices specify directions and planes within a crystal lattice.

• Defined by finding intercepts of the plane with axes along primitive translation vectors a, b, and c (with intercepts x, y, z).

• Calculate indices:

  1. Form fractional triplet (x/a, y/b, z/c).

  2. Take reciprocals of the values.

  3. Reduce to smallest integers, creating parenthesis notation (hkl).

• Example: If a plane intercepts at x=1, y=3, z=1, the Miller indices would be (313).

• Orientation specifies the crystal plane determined by three non-collinear points or coordinates in terms of lattice constants a, b, and c, ensuring not collinear.

Steps to Find Miller Indices

1. Determine intercepts on the axes along the basis vectors a, b, c concerning lattice constants.

2. Form the fractional triplet $(x/a, y/b, z/c)$.

3. Take reciprocals of fractions.

4. Reduce results to three smallest integers by multiplying with a common factor.

5. Final notation is enclosed in parentheses (hkl).

Example of Miller Indices Calculation

• Intercepts are considered as x=2a, y=3/2b, z=c.

  1. Form set: $(2, 3/2, 1)$.

  2. Invert to obtain $(1/2, 2/3, 1)$.

  3. Use common factors, e.g., multiplying by 6 to get (346).

• Certain figures illustrate Miller indices of different planes relative to the simple cubic structure: (100), (010), (111), etc.

FIGURES AND DIAGRAMS

• Various figures illustrate different parameters and types of unit cells, as described throughout the notes, including diagrams for the lattice parameters, unit cell types, packing structures, and Miller indices exemplification.