Comprehensive Notes: Crystal Structure and Types of Crystals
Crystal Structure
A crystalline solid possesses rigid, long-range order; its atoms, molecules, or ions occupy specific positions.
A unit cell is the basic repeating structural unit of a crystalline solid.
There are 7 types of unit cells (based on edge lengths a, b, c and interaxial angles α, β, γ):
Simple Cubic (SC): a = b = c; α = β = γ = 90°
Tetragonal: a = b ≠ c; α = β = γ = 90°
Orthorhombic: a ≠ b ≠ c; α = β = γ = 90°
Rhombohedral (Trigonal): a = b = c; α = β = γ ≠ 90°
Monoclinic: a ≠ b ≠ c; α ≠ 90°, β ≈ 90°, γ ≈ 90° (often α = γ = 90°, β ≠ 90°)
Triclinic: a ≠ b ≠ c; α ≠ β ≠ γ; none are 90°
Hexagonal: a = b ≠ c; α = β = 90°, γ = 120°
The coordination number CN is the number of atoms surrounding an atom in the lattice; CN indicates how tightly packed the atoms are.
The basic repeating unit in the array of atoms is the simple cubic cell (SC) for foundational counting.
Unit Cells and Cubic Cells
Three main cubic cell types:
Primitive cubic (SC)
Body-centered cubic (BCC)
Face-centered cubic (FCC)
In a BCC cell, spheres in each layer rest in depressions between spheres in the previous layer; CN = 8.
In an FCC cell, CN = 12.
Most cells’ atoms are shared with neighboring cells:
Corner atoms are shared by 8 unit cells → contribution per corner = 1/8
Face-centered atoms are shared by 2 unit cells → contribution per face = 1/2
Edge atoms are shared by 4 unit cells → contribution per edge = 1/4 (noting typical cubic cells have no edge-only sites full counting here for standard SC/BCC/FCC discussions)
A simple cubic cell contains the equivalent of 1 complete atom.
A body-centered cubic (BCC) cell contains 2 equivalent atoms.
A face-centered cubic (FCC) cell contains 4 complete atoms.
Edge length a and atomic radius r are related (assuming atoms touch along certain directions):
Simple cubic: a = 2r
Body-centered cubic (BCC): a = rac{4r}{\,\sqrt{3}\,}
Face-centered cubic (FCC): a = rac{4r}{\sqrt{2}}
Crystal problems:
If atoms occupy a face-centered cubic lattice, there are 4 atoms per unit cell.
Examples and Worked Problems
Potassium crystallizes in a body-centered cubic lattice with density \rho = 0.856\ \text{g/cm}^3 at 25°C.
(a) How many atoms are in a unit cell? → 2\,\text{atoms}
(b) What is the edge length of the cell? → a = 0.533\ \text{nm}
Ionic crystals:
Ionic crystals are composed of charged ions held together by Coulombic attraction.
The unit cell of an ionic compound can be defined by the positions of the anions or the positions of the cations.
12.4 Types of Crystals: Ionic Crystals
Crystal structures of three ionic compounds:
CsCl: Simple cubic lattice (CsCl-type)
ZnS: Zinc blende structure (based on FCC)
CaF2: Fluorite structure (based on FCC)
ZnS (Zinc blende) in a unit cell:
The unit cell has four Zn²⁺ ions completely contained inside, and S²⁻ ions at the corners and faces.
Corner contributions: 8 corners × 1/8 = 1 S²⁻
Face contributions: 6 faces × 1/2 = 3 S²⁻
Total S²⁻ in unit cell: 4
Therefore: 4 Zn²⁺ (interior) and 4 S²⁻ (corner/face) → ZnS has 4 Zn²⁺ and 4 S²⁻ per unit cell.
NaCl unit cell density problem (Worked Example 12.5):
Each unit cell contains 4 Na⁺ and 4 Cl⁻ ions.
Mass of Na⁺ ion: m_{Na^+} = 22.99\ ext{amu} \times \left(\frac{1\ \text{g}}{6.022\times 10^{23}\ \text{amu}}\right) = 3.818\times 10^{-23}\ \text{g}
Mass of Cl⁻ ion: m_{Cl^-} = 35.45\ \text{amu} \times \left(\frac{1\ \text{g}}{6.022\times 10^{23}\ \text{amu}}\right) = 5.887\times 10^{-23}\ \text{g}
Edge length from problem: a = 564\ \text{pm} = 5.64\times 10^{-8}\ \text{cm}
Number of NaCl formula units per unit cell: 4 Na⁺ and 4 Cl⁻ → 4 formula units.
Mass of unit cell: m{cell} = 4\times m{Na^+} + 4\times m_{Cl^-} = 3.882\times 10^{-22}\ \text{g}
Volume of unit cell: V_{cell} = a^3 = (5.64\times 10^{-8}\ \text{cm})^3 = 1.794\times 10^{-22}\ \text{cm}^3
Density: \rho = \frac{m{cell}}{V{cell}} = \frac{3.882\times 10^{-22}\ \text{g}}{1.794\times 10^{-22}\ \text{cm}^3} \approx 2.16\ \text{g/cm}^3
Think About It: Unit conversions are common sources of error; verify dimensions; a wrong cm/m conversion could yield an incorrect density by orders of magnitude (e.g., 10^12 g/cm³).
Iridium (Ir) density problem (Worked Example 12.6):
A metal with FCC lattice; 4 atoms per unit cell; edge length a = 383\ \text{pm}
Mass of Ir atom: M{Ir} = 192.2\ \text{amu} \Rightarrow m{Ir} = 3.192\times 10^{-22}\ \text{g}
Edge length in cm: a = 3.83\times 10^{-8}\ \text{cm}
Volume: V_{cell} = a^3 = 5.618\times 10^{-23}\ \text{cm}^3
Mass per unit cell: m{cell} = 4\times m{Ir} = 1.277\times 10^{-21}\ \text{g}
Density: \rho = \frac{m{cell}}{V{cell}} = 22.7\ \text{g/cm}^3
Metallic crystals:
In metallic crystals, every lattice point is occupied by an atom of the same metal.
Valence electrons are delocalized over the entire crystal, creating a “sea” of electrons.
Delocalized electrons make metals good conductors of heat and electricity.
Large cohesive forces from delocalization make metals strong.
Summary of Crystals
Table 12.4 (Types of Crystals and Their General Properties):
Ionic Crystals
Cohesive forces: Coulombic attraction
General properties: Hard, brittle, high melting point, poor conductor of heat and electricity
Examples: NaCl, LiF, MgO, CaCO₃
Covalent Crystals
Cohesive forces: Covalent bonds
General properties: Hard, brittle, high melting point, poor conductor of heat and electricity
Examples: Diamond, SiO₂ (quartz)
Molecular Crystals
Cohesive forces: Dispersion and dipole-dipole forces, hydrogen bonds
General properties: Soft, low melting point, poor conductor of heat and electricity
Examples: Ar, CO₂, I₂, H₂O, C₁₂H₂₂O₁₁
Metallic Crystals
Cohesive forces: Metallic bonds
General properties: Variable hardness and melting point, good conductor of heat and electricity
Examples: All metallic elements (Na, Mg, Fe, Cu, etc.)
Note: Diamond is a good conductor of heat (table footnote).
*Included in this category are crystals made up of individual atoms.
Amorphous Solids
Amorphous solids lack a regular three-dimensional arrangement of atoms.
Glass is an amorphous solid and is a fusion product; SiO₂ is the chief component.
Na₂O and B₂O₃ are typically fused with molten SiO₂ and allowed to cool without crystallizing.
12.5
Amorphous Solids — Types of Glass (Table 12.5)
Pure quartz glass
Composition: 100% SiO₂
Properties: Low thermal expansion; transparent across a wide wavelength range; used in optical research.
Pyrex glass
Composition: 60–80% SiO₂, 10–25% B₂O₃, some Al₂O₃
Properties: Low thermal expansion; transparent to visible and infrared, but not UV; used in cookware and laboratory glassware.
Soda-lime glass
Composition: 75% SiO₂, 15% Na₂O, 10% CaO
Properties: Easily attacked by chemicals; transmits visible light but absorbs ultraviolet; used in windows and bottles.
Crystalline Quartz vs. Amorphous Glass
Crystalline quartz: well-ordered crystal structure of SiO₂.
Noncrystalline (amorphous) quartz glass: irregular structure with no long-range order.
Close-Packed Structures: Hexagonal vs Cubic Close Packing
Hexagonal close-packed (hcp)
Stacking sequence: ABAB…
Layer B fits into depressions of layer A.
Site directly over an atom in layer A is characteristic of the hcp arrangement.
Cubic close-packed (ccp)
Stacking sequence: ABCABC…
Site directly over an atom in layer A for ccp is not directly above; structure corresponds to a face-centered cubic (FCC) cell.
Relationship: Hexagonal close-packing (hcp) and cubic close-packing (ccp) are two ways to achieve close packing; ccp corresponds to the FCC cell in 3D.
12 Key Concepts (Overview)
Intermolecular forces: Dipole-dipole interactions; Hydrogen bonding; Dispersion forces; Ion-dipole interactions.
Properties of liquids: Surface tension; Viscosity; Vapor pressure.
Crystal structure: Unit cells; Packing of spheres; Closest packing.
Types of crystals: Ionic crystals; Covalent crystals; Molecular crystals; Metallic crystals.
Amorphous solids; Phase changes; Phase diagrams.
Ionic crystals: Coulombic attractions; High melting points; Generally brittle.
Covalent crystals: Extensive covalent bonds; Very hard; Often poor conductors.
Molecular crystals: Intermolecular forces; Soft; Low melting points; Poor conductors.
Metallic crystals: Delocalized electrons; Good conductors; Metallic bonding.
Practical problems: Density calculations; Unit cell edge lengths; Real-world materials.
Close packing: Rules for CN and atoms per unit cell; SC, BCC, FCC.
Crystal problems and estimations are common in determining properties from lattice geometry.
Phase changes and phase diagrams connect temperature, pressure, and phases of matter.
Hexagonal Close-Packing (hcp) vs Cubic Close-Packing (ccp)
hcp structure:
ABAB stacking; site directly over A in layer A is part of the description.
ccp structure (FCC):
ABCABC stacking; site directly over A for hcp is not directly over A for ccp.
Summary: hcp and ccp are two distinct close-packed arrangements; ccp is equivalent to a face-centered cubic cell.