Unit 4: Solid State

Table of Contents

Crystal Lattices and Systems

Solids are characterized by strong intermolecular forces and fixed positions of particles. They can be crystalline (ordered) or amorphous (disordered).

Basic Terminology

Seven Crystal Systems and Fourteen Bravais Lattices

Unit cells are defined by their edge lengths (a, b, c) and the angles between them (α, β, γ). Based on these parameters, all crystals can be grouped into 7 Crystal Systems.

These systems can have variations (e.g., body-centered, face-centered), leading to a total of 14 Bravais Lattices.

The 7 Crystal Systems and 14 Bravais Lattices
Crystal System Axial Parameters Angular Parameters Bravais Lattices (Unit Cell Types)
Cubic a = b = c α = β = γ = 90° Primitive, Body-Centered (BCC), Face-Centered (FCC)
Tetragonal a = b ≠ c α = β = γ = 90° Primitive, Body-Centered
Orthorhombic a ≠ b ≠ c α = β = γ = 90° Primitive, Body-Centered, Face-Centered, End-Centered
Monoclinic a ≠ b ≠ c α = γ = 90°, β ≠ 90° Primitive, End-Centered
Triclinic a ≠ b ≠ c α ≠ β ≠ γ ≠ 90° Primitive
Hexagonal a = b ≠ c α = β = 90°, γ = 120° Primitive
Rhombohedral
(or Trigonal)		 a = b = c α = β = γ ≠ 90° Primitive

Laws of Crystallography and Miller Indices

Laws of Crystallography

  1. Law of Constancy of Interfacial Angles (Steno's Law): For a given crystalline substance, the angles between adjacent corresponding faces (interfacial angles) are always constant, regardless of the size or shape of the crystal.
  2. Law of Rational Indices (Haüy's Law): The intercepts made by any crystal face on the crystallographic axes can be expressed as small, rational (whole-number) multiples of the unit intercepts (a, b, c). This leads to the concept of Miller Indices.

Miller Indices (hkl)

A notation system used to specify crystal planes and directions. They are derived as follows:

  1. Step 1: Find the intercepts. Determine the intercepts of the crystal plane on the a, b, c axes. (e.g., 2a, 3b, 1c).
  2. Step 2: Write the coefficients. Write down the numerical coefficients of the intercepts. (e.g., 2, 3, 1).
  3. Step 3: Take the reciprocals. Invert the coefficients. (e.g., (1) / (2), (1) / (3), (1) / (1)).
  4. Step 4: Clear fractions. Multiply by the least common multiple to get the smallest set of whole numbers. (e.g., Multiply by 6 → 3, 2, 6).
  5. Step 5: Write the indices. Enclose in parentheses (hkl). (e.g., (326)).

Special Cases:

  • An intercept at infinity (∞) corresponds to a reciprocal of 0. (e.g., a plane parallel to the b-axis (101)).
  • A negative intercept is shown with a bar over the number. (e.g., (11̄0)).

X-ray Diffraction and Bragg's Law

X-ray Diffraction

Since the wavelength of X-rays (≈ 1 \AA) is comparable to the spacing between atoms in a crystal, a crystal can act as a 3D diffraction grating for X-rays. This phenomenon is used to determine the atomic structure of crystalline solids.

Bragg's Law

W.L. Bragg and W.H. Bragg treated X-ray diffraction as a "reflection" from successive planes of atoms in the crystal.

Derivation:

  1. Consider two parallel X-rays (1 and 2) striking two parallel crystal planes (A and B) separated by a distance d.
  2. Both rays strike the planes at the same glancing angle θ.
  3. Ray 1 reflects from plane A. Ray 2 travels an extra distance to reflect from plane B.
  4. The extra path length traveled by Ray 2 is XY + YZ.
  5. From trigonometry, XY = YZ = d sinθ.
  6. Total path difference = 2d sinθ.
  7. For constructive interference (which produces a diffraction spot), this path difference must be an integer multiple (n) of the X-ray wavelength (λ).
Bragg's Law: nλ = 2d sinθ
Where:
  • n = Order of reflection (1, 2, 3, ...)
  • λ = Wavelength of the X-rays
  • d = Interplanar spacing
  • θ = Glancing angle of incidence

Crystal Defects and Band Theory

Defects in Crystals

An ideal crystal has a perfect, repeating structure. A crystal defect is any deviation from this perfect order. These imperfections are crucial for many properties of solids.

  • Schottky Defect: (A vacancy defect). To maintain neutrality, a pair of oppositely charged ions (one cation, one anion) are missing from their lattice sites.
    • Found in: Highly ionic compounds with similar ion sizes (e.g., NaCl, KCl).
    • Effect: Decreases the density of the crystal.
  • Frenkel Defect: (A dislocation defect). A smaller ion (usually the cation) leaves its normal lattice site and occupies a small interstitial site.
    • Found in: Compounds with a large difference in ion sizes (e.g., AgCl, ZnS).
    • Effect: Does not change the density.

Colour Center (F-Center)

This is a type of non-stoichiometric defect (metal excess defect).

  • How it forms: An anion is missing from its lattice site (anion vacancy). To maintain charge neutrality, this "hole" is occupied by an unpaired electron.
  • Name: F-Center comes from the German Farbe (color).
  • Effect: This trapped electron can absorb energy from visible light and jump to an excited state, causing the crystal to appear colored (e.g., NaCl + Na vapor → Yellow).

Energy Band Theory of Conductor, Semiconductors and Insulators

This model (based on MOT) explains the electrical conductivity of solids.

  • Valence Band (VB): The energy band formed from the filled valence atomic orbitals.
  • Conduction Band (CB): The next higher, empty energy band.
  • Band Gap (Eg): The energy difference between the top of the VB and the bottom of the CB.
  1. Conductors (Metals):
    • Band Gap: Eg = 0.
    • The valence band is either partially filled (like Na, 3s1) or it overlaps with the empty conduction band (like Mg, 3s2 overlaps 3p0).
    • Electrons can move freely into higher energy states with a tiny electrical potential, allowing high conductivity.
  2. Insulators:
    • Band Gap: Eg is very large (e.g., > 3 eV).
    • The valence band is completely full, and the conduction band is empty.
    • It is too difficult for an electron to jump the gap, so no current flows.
  3. Semiconductors (e.g., Si, Ge):
    • Band Gap: Eg is small (e.g., ~1 eV).
    • At 0 K, they are insulators. At room temperature, thermal energy is sufficient to promote a few electrons from the VB to the CB.
    • This creates mobile electrons in the CB and "holes" in the VB, both of which contribute to a small conductivity.
    • Key: Conductivity of semiconductors increases with temperature.

Other States: Glasses and Liquid Crystals

Glasses

  • Definition: Glasses are amorphous solids (non-crystalline).
  • Structure: They have short-range order (like liquids) but no long-range order (like crystals).
  • Formation: They are formed by cooling a liquid very rapidly, "freezing" the disordered liquid structure in place. They are often called supercooled liquids.
  • Properties: They do not have a sharp, defined melting point; instead, they soften over a range of temperatures (a "glass transition temperature").

Liquid Crystals

  • Definition: A state of matter (a mesophase) with properties intermediate between a crystalline solid and an isotropic liquid.
  • Properties: They can flow like a liquid but exhibit optical properties (like birefringence) of a crystal due to long-range molecular order. They are typically composed of long, rod-shaped molecules.

Phases of Liquid Crystals

  1. Nematic Phase:
    • Order: Molecules have orientational order (they all point in the same general direction, defined by a "director" axis).
    • Disorder: They have no positional order (their centers of mass are randomly distributed, like a liquid).
  2. Smectic A Phase:
    • Order: Molecules have orientational order AND 1D positional order.
    • They are arranged in well-defined layers. The molecules are, on average, perpendicular (normal) to the layer planes.
    • They are random *within* the layer.
  3. Smectic C Phase:
    • Order: Same as Smectic A (orientational order + layers).
    • Difference: The molecules are tilted at an angle with respect to the layer planes.