Crystal Systems, Unit Cells and Packing Efficiency

• Distinguish crystalline from amorphous solids and define a unit cell
• Identify the three cubic unit cells: SC, BCC, FCC/CCP
• Count atoms per unit cell using corner/face/body contribution rules
• Calculate packing efficiency for SC, BCC, and FCC
• Use the density formula ρ = ZM/(NAa³) to find density or edge length
• Relate radius to edge length for each cubic system

Level 1 — Foundations

Crystalline vs Amorphous

• Crystalline: long-range order, sharp melting point, anisotropic (NaCl, quartz, diamond)
• Amorphous: short-range order only, no sharp melting point, isotropic (glass, rubber, plastic)

Unit Cell
Smallest repeating unit that, when translated in 3D, generates the entire crystal lattice.

7 Crystal Systems: Cubic, Tetragonal, Orthorhombic, Hexagonal, Rhombohedral, Monoclinic, Triclinic
(JEE focuses almost entirely on cubic)

Atom Contribution per Unit Cell

• Corner atom: shared by 8 unit cells → contributes 1/8
• Face-centre atom: shared by 2 unit cells → contributes 1/2
• Body-centre atom: entirely inside → contributes 1
• Edge-centre atom: shared by 4 unit cells → contributes 1/4

Three Cubic Systems Summary

Level 2 — JEE Depth

Packing Efficiency Derivations

Simple Cubic:
Atoms touch along edge: 2r = a → r = a/2
Volume of 1 atom = (4/3)πr³ = (4/3)π(a/2)³
PE = (Z × volume of atom)/(volume of unit cell) = [1 × (4/3)π(a/2)³]/a³
= (4/3)π(1/8) = π/6 = 0.5236 → 52.36%

BCC:
Atoms touch along body diagonal: 4r = a√3 → r = a√3/4
PE = [2 × (4/3)π(a√3/4)³]/a³ = 2 × (4/3)π × (3√3/64)
= 8π√3/64 × (1/3) × 3 = π√3/8 = 0.6802 → 68.02%

FCC:
Atoms touch along face diagonal: 4r = a√2 → r = a√2/4 = a/(2√2)
PE = [4 × (4/3)π(a/(2√2))³]/a³ = 4 × (4/3)π × (1/(16√2))
= π/(3√2) = 0.7405 → 74.05%
FCC has the most efficient packing — same as HCP (both are closest packed, 74%).

Density Formula
$\rho = \frac{Z \times M}{N_A \times a^3}$

• Z = atoms per unit cell
• M = molar mass (g/mol)
• NA = 6.022×10²³ mol⁻¹
• a = edge length in cm (if ρ in g/cm³), or convert Å to cm: 1 Å = 10⁻⁸ cm

Voids in Close Packing (CCP/FCC)
For N spheres in CCP:

• Number of octahedral voids = N (one per sphere)
• Number of tetrahedral voids = 2N (two per sphere)
• Radius ratio for octahedral void: r_void/r_atom ≈ 0.414
• Radius ratio for tetrahedral void: r_void/r_atom ≈ 0.225

JEE Traps

• a must be in cm for ρ formula when M is g/mol; 1 Å = 10⁻⁸ cm
• BCC: atoms DON'T touch along edge; they touch along body diagonal
• FCC: atoms DON'T touch along edge; they touch along face diagonal
• HCP also has 74% packing efficiency — same as FCC, different arrangement

Example 1: Density of Aluminium (FCC)

Given: FCC structure, edge length a = 4.05 Å, M = 27 g/mol
Find: density ρ

Step 1: Identify Z for FCC
  Z = 4 atoms/unit cell

Step 2: Convert a to cm
  a = 4.05 Å = 4.05×10⁻⁸ cm
  a³ = (4.05×10⁻⁸)³ = 66.43×10⁻²⁴ cm³ = 6.643×10⁻²³ cm³

Step 3: Apply density formula
  ρ = ZM/(NA × a³)
  ρ = (4 × 27) / (6.022×10²³ × 6.643×10⁻²³)
  ρ = 108 / (6.022 × 6.643)
  ρ = 108 / (39.99)
  ρ = 2.70 g/cm³

Answer: ρ = 2.70 g/cm³
(This matches the known density of aluminium — good verification!)

Example 2: Confirming BCC Packing Efficiency

In BCC: atoms touch along body diagonal
Body diagonal = a√3, occupied by 4 radii: 4r = a√3 → r = a√3/4

Volume of 2 atoms (Z=2 for BCC):
  V_atoms = 2 × (4/3)πr³ = 2 × (4/3)π(a√3/4)³

Calculate (a√3/4)³:
  = a³ × 3√3/64

So V_atoms = 2 × (4/3)π × a³ × 3√3/64
           = 2 × 4π × 3√3 × a³ / (3 × 64)
           = 8π√3a³ / 64
           = π√3a³ / 8

Volume of unit cell = a³

Packing efficiency = (π√3/8) = 3.14159 × 1.7321 / 8
                   = 5.4414 / 8
                   = 0.6802 → 68.02% ✓

• Q1: How many atoms are in one unit cell of FCC gold?
• Q2: If the edge length of a BCC iron crystal is 2.87 Å, what is the radius of an Fe atom?
• Q3: Which has higher packing efficiency — BCC sodium or SC polonium?
• Q4: An element has density 8.9 g/cm³, FCC structure, and M = 63.5 g/mol. Find edge length a.
• Stretch: Q5: NaCl has a face-centred arrangement of Cl⁻ ions with Na⁺ in octahedral voids. If the edge length is 5.64 Å and M(NaCl) = 58.5 g/mol, calculate the density. (Hint: Z = 4 for NaCl in terms of formula units)

NCERT Chapter 1 link: Chapter 1 "The Solid State" Class 12 — Sections 1.5 to 1.8 cover unit cells, types of cubic structures, packing in crystals, and density calculations. NCERT examples 1.1–1.3 are direct density and radius calculation problems.

Exam connections: JEE Mains regularly tests: (a) Z for SC/BCC/FCC, (b) density calculation given a or vice versa, (c) packing efficiency values, (d) coordination numbers. JEE Advanced may test NaCl/CsCl structure analysis, void counting, or radius ratio applications to predict structure type.

Study strategy: Draw SC, BCC, FCC unit cells repeatedly until atom positions are automatic. Memorise packing efficiency as 52:68:74 (SC:BCC:FCC). Practise density calculations end-to-end with unit conversion — losing marks on unit errors is avoidable with habit.

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