Chapter 3: The Structures of Simple Solids

3.1 Unit Cells and Crystal Structure Description

A crystal of an element or compound can be regarded as constructed from regularly repeating structural elements, which may be atoms, molecules, or ions. The crystal lattice is the geometric pattern formed by the points that represent the positions of these repeating structural elements.

Key Point: The lattice defines a network of identical points that has the translational symmetry of a structure. A unit cell is a subdivision of a crystal that when stacked together following translations reproduces the crystal.

Unit Cell:

An imaginary parallel-sided region (a 'parallelepiped') from which the entire crystal can be built up by purely translational displacements. Unit cells fit perfectly together with no space excluded.

The Seven Crystal Systems

All ordered structures adopted by compounds belong to one of seven crystal systems distinguished by their rotational symmetry:

System	Lattice Parameters	Essential Symmetry
Cubic	a = b = c, α = β = γ = 90°	Four 3-fold axes (tetrahedral)
Tetragonal	a = b ≠ c, α = β = γ = 90°	One 4-fold axis
Orthorhombic	a ≠ b ≠ c, α = β = γ = 90°	Three perpendicular 2-fold axes
Hexagonal	a = b ≠ c, α = β = 90°, γ = 120°	One 6-fold axis
Trigonal	a = b = c, α = β = γ ≠ 90°	One 3-fold axis
Monoclinic	a ≠ b ≠ c, α = γ = 90°, β ≠ 90°	One 2-fold axis
Triclinic	a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90°	None

Lattice Types

Primitive (P)

1 lattice point per cell

Translational symmetry of unit cell only

Body-centred (I)

2 lattice points per cell

Additional (+½,+½,+½) translation

Face-centred (F)

4 lattice points per cell

Additional face-centre translations

Example 3.1: Identifying Lattice Types

Problem: Determine the translational symmetry in cubic ZnS and identify its lattice type.

Answer: The displacements (0, +½, +½), (+½, +½, 0), and (+½, 0, +½) move each atom to an equivalent position. These translations correspond to the face-centred lattice, so the lattice type is F.

Determine the lattice type of CsCl (Cs⁺ at corners, Cl⁻ at body centre).

Fractional Coordinates

The position of an atom in a unit cell is described by fractional coordinates (x, y, z), expressed as fractions of the cell edge lengths. For example, an atom at the body centre has coordinates (½, ½, ½).

Counting Atoms in Unit Cells:

Atom fully inside cell: counts as 1
Atom on face (shared by 2 cells): counts as ½
Atom on edge (shared by 4 cells): counts as ¼
Atom at corner (shared by 8 cells): counts as ⅛

3.2–3.3 Close Packing of Spheres

Key Point: The close packing of identical spheres can result in a variety of polytypes, of which hexagonal and cubic close-packed structures are the most common.

When there is no directional covalent bonding, spheres pack together as closely as geometry allows, adopting a close-packed structure—the structure with least unfilled space.

Building Close-Packed Layers

A single close-packed layer has each sphere with six nearest neighbours in a hexagonal pattern
The second layer occupies half the dips in the first layer (insufficient space for all)
The third layer can be placed in two different ways, generating different polytypes

⬡

Hexagonal Close-Packed (hcp)

ABAB... stacking

Be, Mg, Zn, Ti, Co

◇

Cubic Close-Packed (ccp)

ABCABC... stacking

Cu, Ag, Au, Al, Ni, Pt

The coordination number in both close-packed arrangements is 12: 6 in the same layer + 3 above + 3 below. The packing efficiency is 74%.

Example 3.3: Calculating Packing Efficiency

For ccp: Consider the fcc unit cell. Face diagonal = 4r, so cell edge a = √8r.

Cell volume = (√8r)³ = 8^3/2r³

4 spheres per cell, total sphere volume = 4 × (4/3)πr³ = (16/3)πr³

Packing fraction = (16π/3) / 8^3/2 = 0.740 = 74%

Calculate the packing fraction for (a) primitive cubic and (b) body-centred cubic structures.

Holes in Close-Packed Structures

⬢

Octahedral Holes

N holes for N spheres

Max radius: 0.414r

△

Tetrahedral Holes

2N holes for N spheres

Max radius: 0.225r

Example 3.4: Size of Octahedral Hole

For spheres of radius r in contact, an octahedral hole can accommodate a sphere of radius r_h:

From geometry: (r + r_h)² + (r + r_h)² = (2r)²

Therefore: r_h = (√2 − 1)r = 0.414r

Show that the maximum radius of a sphere in a tetrahedral hole is 0.225r.

3.4–3.8 Metals and Alloys

X-ray diffraction studies reveal that many metallic elements have close-packed structures, indicating that the bonds between atoms have little directional covalent character.

Common Metal Structures

Structure	Coordination	Packing	Examples
hcp	12	74%	Be, Ca, Co, Mg, Ti, Zn
ccp (fcc)	12	74%	Ag, Al, Au, Cu, Ni, Pb, Pt
bcc	8 (+6)	68%	Ba, Cr, Fe, W, alkali metals
Primitive cubic	6	52%	Po

3.6 Polymorphism

Key Point: The lack of directionality in metallic bonding accounts for the wide occurrence of polymorphism—the ability to adopt different crystal forms under different conditions.

Example: Iron Polymorphism

α-Fe (bcc): stable up to 906°C

γ-Fe (ccp): stable from 906°C to 1401°C

α-Fe (bcc): stable from 1401°C to melting point (1530°C)

β-Fe (hcp): forms at high pressures

3.8 Alloys

Alloy:

A blend of metallic elements prepared by mixing molten components and cooling. Alloys may be homogeneous solid solutions or compounds with definite composition.

Substitutional Solid Solutions

Atoms of solute metal replace atoms of parent metal randomly. Requirements (Hume-Rothery rules):

Atomic radii differ by less than ~15%
Same crystal structure preferred
Similar electronegativities
Same valence

Example: Brass (Cu-Zn): Cu_1-xZn_x with 0 ≤ x ≤ 0.38

Interstitial Solid Solutions

Small atoms (B, C, N) occupy interstitial holes in the metal structure. The host structure is preserved.

Example: Carbon steel—C atoms in octahedral holes of bcc Fe. Carbon content 0.2–1.6 atom %.

Intermetallic Compounds

Definite structures often unrelated to parent metals. Examples include:

β-brass (CuZn): bcc at high T, hcp at low T
Zintl phases: Electropositive + less electropositive metals (e.g., KGe)

3.9–3.10 Ionic Solids

Key Point: The ionic model treats a solid as an assembly of oppositely charged spheres that interact by nondirectional electrostatic forces.

Ionic solids are recognized by their brittleness, high melting points, and solubility in polar solvents. Binary ionic materials are typical of elements with large electronegativity differences (Δχ > 3).

AX Structures

▣

Rock Salt (NaCl)

6:6 coordination

ccp anions, all octahedral holes filled

▢

Caesium Chloride (CsCl)

8:8 coordination

Primitive cubic, cubic holes filled

◈

Sphalerite (ZnS)

4:4 coordination

ccp anions, half tetrahedral holes filled

⬡

Wurtzite (ZnS)

4:4 coordination

hcp anions, half tetrahedral holes filled

AX₂ Structures

⬢

Fluorite (CaF₂)

8:4 coordination

ccp cations, all tetrahedral holes filled

⬡

Rutile (TiO₂)

6:3 coordination

Distorted hcp, half octahedral holes

ABX₃ Structures

◇

Perovskite (CaTiO₃)

A: 12-coord, B: 6-coord

A+X close-packed, B in octahedral holes

△

Spinel (MgAl₂O₄)

A: 4-coord, B: 6-coord

ccp O²⁻, A in ⅛ tet, B in ½ oct holes

Radius Ratio Rules

Key Point: The radius ratio γ = r₊/r₋ helps predict which structure a compound will adopt based on geometric packing considerations.

Radius Ratio (γ)	Coordination	Structure Type
0.225–0.414	4:4	Sphalerite, Wurtzite
0.414–0.732	6:6	Rock salt
0.732–1.0	8:8	Caesium chloride

Example 3.12: Predicting Structures

TlCl: r(Tl⁺) = 159 pm, r(Cl⁻) = 181 pm

γ = 159/181 = 0.88

Prediction: CsCl structure (8:8) ✓ Correct!

Predict structures for RbI, BeO, and PbF₂ using radius-ratio rules.

Structure Maps

The reliability of radius-ratio rules is only ~50%. Structure maps based on electronegativity difference and average principal quantum number provide better predictions.

IONIC METALLIC COVALENT

3.11–3.15 Energetics of Ionic Bonding

Lattice Enthalpy (Δ_LH°):

The standard molar enthalpy change for the formation of a gas of ions from the solid: MX(s) → M⁺(g) + X⁻(g). Always positive (endothermic).

The Born–Haber Cycle

Key Point: Lattice enthalpies are determined from enthalpy data using a Born–Haber cycle; the most stable crystal structure commonly has the greatest lattice enthalpy.

Born–Haber Cycle for KCl

1 K(s) → K(g) +89 kJ/mol

2 K(g) → K⁺(g) + e⁻ +425 kJ/mol

3 ½Cl₂(g) → Cl(g) +122 kJ/mol

4 Cl(g) + e⁻ → Cl⁻(g) −355 kJ/mol

5 K⁺(g) + Cl⁻(g) → KCl(s) −Δ_LH°

Σ K(s) + ½Cl₂(g) → KCl(s) −438 kJ/mol

From the cycle: Δ_LH° = 719 kJ mol⁻¹

The Born–Mayer Equation

Key Point: The Born–Mayer equation estimates lattice enthalpy from electrostatic interactions; the Madelung constant reflects the geometry of the lattice.

Δ_LH° = (N_A|z_Az_B|e²/4πε₀d) × (1 − d*/d) × A

where A = Madelung constant, d = r₊ + r₋, d* ≈ 34.5 pm

Madelung Constants

Structure	Madelung Constant (A)
Caesium chloride	1.763
Rock salt	1.748
Sphalerite	1.638
Wurtzite	1.641
Fluorite	2.519
Rutile	2.408

3.14 The Kapustinskii Equation

Δ_LH° = (N_ion|z_Az_B|/d) × κ × (1 − d*/d)

where κ = 1.21 × 10⁵ kJ pm mol⁻¹, N_ion = number of ions per formula unit

3.15 Consequences of Lattice Enthalpies

Thermal Stability of Carbonates

Decomposition: MCO₃(s) → MO(s) + CO₂(g)

Large cations stabilize large anions. Decomposition temperature increases with cation radius:

MgCO₃: ~300°C
CaCO₃: ~840°C
SrCO₃: ~1100°C
BaCO₃: ~1300°C

Oxidation State Stability

Small anions (especially F⁻) stabilize high oxidation states because of their large lattice enthalpies.

Only fluorides exist for: Ag(II), Co(III), Mn(IV)

Solubility

Rule: Compounds with ions of widely different radii are more soluble in water.

Examples:

Group 2 sulfates: solubility decreases from MgSO₄ to BaSO₄
Group 2 hydroxides: solubility increases from Mg(OH)₂ to Ba(OH)₂

3.16–3.17 Defects and Nonstoichiometry

Key Point: Defects are present in all solids because their formation increases entropy. The Gibbs energy G = H − TS has a minimum at a nonzero defect concentration.

Intrinsic Point Defects

Schottky Defect

Paired cation and anion vacancies maintaining charge neutrality. Common in ionic materials with high coordination numbers (e.g., NaCl, MgO).

Frenkel Defect

Ion displaced from normal site to interstitial position. Common in open structures with low coordination (e.g., AgCl, ZnS).

Extrinsic Defects

Extrinsic defects arise from intentionally added impurities (dopants). Very low concentrations (~1 per 10⁹ atoms) can significantly alter properties.

Example: Doped Zirconia

Replacing Zr⁴⁺ with Ca²⁺ in ZrO₂ creates O²⁻ vacancies to maintain charge neutrality. These vacancies allow oxide ion migration, increasing ionic conductivity.

Colour Centres (F-centres)

An F-centre is an electron trapped in an anion vacancy. Heating alkali halides in alkali metal vapour produces characteristic colours:

NaCl → orange
KCl → violet
KBr → blue-green

Nonstoichiometric Compounds

Key Point: Nonstoichiometric compounds have compositions that are not simple integer ratios, typically occurring for d- and f-block elements with multiple oxidation states.

Example: Iron(II) oxide exists as Fe_1-xO with x = 0.04–0.11 (i.e., Fe_0.89O to Fe_0.96O). Fe²⁺ vacancies are compensated by Fe³⁺ ions.

3.18–3.20 Electronic Structure of Solids

Key Point: A metallic conductor has electric conductivity that decreases with increasing temperature; a semiconductor has conductivity that increases with increasing temperature.

Band Formation

When a large number of atomic orbitals overlap in a solid, they form an almost continuous band of energy levels. Bands are separated by band gaps—energy ranges with no available states.

Fermi Level:

The highest occupied energy level at T = 0. In metals, it lies within a band; electrons near this level can easily be promoted to nearby empty levels.

Metals, Semiconductors, and Insulators

Metal

Overlapping bands
or partially filled

~1 eV

Semiconductor

Small band gap
Si: 1.1 eV

~7 eV

Insulator

Large band gap
NaCl: 7 eV

Intrinsic Semiconductors

Key Point: In an intrinsic semiconductor, conductivity arises from thermal excitation of electrons from the valence band to the conduction band.

σ = σ₀ exp(−E_g/2kT)

where E_g = band gap energy

Extrinsic Semiconductors

−

n-Type

Donor doping

Si doped with As (extra electron)
Electrons are charge carriers

+

p-Type

Acceptor doping

Si doped with Ga (electron deficit)
Holes are charge carriers

Example: n-Type Silicon

When As ([Ar]4s²4p³) substitutes for Si ([Ne]3s²3p²), one extra electron is available. At T > 0, this electron can be thermally promoted into the conduction band, greatly increasing conductivity.

Would VO or NiO be expected to show metallic properties? Why?

Oxide Semiconductors

For metal oxides:

p-type: Metals in low oxidation states (MnO, Cr₂O₃). Holes form by oxidation of metal.
n-type: Metals in high oxidation states (Fe₂O₃, MnO₂). Electrons occupy conduction band by reduction.

Chapter Summary

Crystal Structure

Seven crystal systems, three lattice types (P, I, F). Unit cells describe repeating structural units.

Close Packing

hcp (ABAB) and ccp (ABCABC) with 74% efficiency. Octahedral and tetrahedral holes.

Ionic Structures

Described by hole-filling in close-packed arrays. Radius-ratio rules predict coordination.

Lattice Enthalpy

Born–Haber cycle for experimental values. Born–Mayer equation for calculations.

Defects

Schottky (vacancies) and Frenkel (interstitials) defects. Doping creates extrinsic defects.

Band Theory

Band gaps determine conductivity. Metals, semiconductors, and insulators distinguished by gap size.