Crystal Lattice & Unit Cell for JEE Main 2026

Solid State is a chapter where JEE Main is remarkably formulaic — the same 8–10 numerical types recycle year after year, always using the same unit cell parameters. Students who learn the BCC, FCC, and simple cubic results once, along with the packing efficiency and density formula, can solve any Solid State JEE question. This guide gives you the complete framework with the exact numbers you need.

Simple Cubic, BCC, and FCC: The Critical Numbers

Simple cubic (SC): 1 atom per unit cell (8 corners × 1/8), radius r = a/2, packing efficiency = 52.4%. Body-centred cubic (BCC): 2 atoms per unit cell (8 × 1/8 + 1 body), radius r = a√3/4, packing efficiency = 68%. Face-centred cubic (FCC): 4 atoms per unit cell (8 × 1/8 + 6 × 1/2), radius r = a√2/4 = a/(2√2), packing efficiency = 74%. Hexagonal close-packed (HCP): also 74% packing efficiency, same as FCC. Packing efficiency = (volume of atoms in unit cell / volume of unit cell) × 100. These six numbers — atoms per cell, radius, and PE for SC, BCC, FCC — must be known cold.

Density of a crystal: ρ = (Z × M)/(Nₐ × a³), where Z is atoms per unit cell, M is molar mass, Nₐ is Avogadro's number (6.022 × 10²³), and a is the edge length. JEE gives three of these quantities and asks for the fourth. The most common formats: given ρ, M, and a → find Z (and hence identify the crystal type); given Z, M, and a → find ρ. Dimensional consistency: ρ in g/cm³ requires M in g/mol and a in cm. Take a free Solid State mock to practise density calculations under time pressure. For deeper Solid State coverage, see our solid state complete guide.

Ionic Crystal Structures and Coordination Numbers

NaCl structure (Rock Salt): FCC arrangement of Cl⁻ ions with Na⁺ in octahedral voids. Coordination number of each ion = 6. Formula unit per cell: 4. Examples: NaCl, KCl, MgO, FeO. ZnS (Zinc Blende): FCC arrangement of S²⁻ with Zn²⁺ in alternate tetrahedral voids. Coordination number = 4. Formula unit per cell: 4. Examples: ZnS, CuCl, AgI. CsCl structure: simple cubic arrangement of Cl⁻ with Cs⁺ at body centre. Coordination number = 8. Formula unit per cell: 1. JEE asks which structure a given ionic compound has based on the radius ratio rule: r+/r− < 0.414 → tetrahedral holes (ZnS type); 0.414–0.732 → octahedral holes (NaCl type); > 0.732 → cubic holes (CsCl type).

Void fractions: in a close-packed structure, octahedral voids = N (one per sphere), tetrahedral voids = 2N (two per sphere). For an ionic compound with formula MX where cations occupy all octahedral voids: 1 M per 1 X → NaCl structure. For MX₂ where cations occupy all tetrahedral voids: 1 M per 2 X → CaF₂ (Fluorite) structure. This stoichiometry–structure link is a favourite JEE conceptual question. For the complete solution and solid state framework, see our solid state and solutions guide.

Point Defects in Crystals

Stoichiometric defects: (1) Schottky defect — equal number of cation and anion vacancies, maintains electrical neutrality, lowers density (examples: NaCl, KBr, CsCl). (2) Frenkel defect — ion displaced from lattice to interstitial site (usually smaller cation), maintains density (examples: ZnS, AgBr, AgCl). Non-stoichiometric defects: (3) Metal excess defect — extra cations with electrons in interstitial sites (F-centres, give colour to crystals, e.g., NaCl yellow in Na vapour, KCl violet in K vapour); (4) Metal deficiency defect — cation vacancies compensated by higher oxidation state ions (e.g., FeO). These four defects and their examples are tested every second session.

Impurity defects: substitution of host lattice ions by aliovalent impurity ions — e.g., adding SrCl₂ to NaCl introduces Sr²⁺ in Na⁺ sites, creating cation vacancies for charge balance. This type of defect increases conductivity. AgBr shows both Schottky and Frenkel defects — a fact JEE uses in multiple correct options. For related topics, see our solid state guide.

Packing in Layers and ABAB/ABCABC Stacking

Square packing (2D) gives 52.4% efficiency; hexagonal close packing (2D) gives 90.7%. In 3D, layers can stack as: ABAB (HCP, hexagonal axes, trigonal bipyramidal unit cell shape) or ABCABC (CCP = FCC, cubic unit cell). Both give 74% packing efficiency. The void structure differs: HCP has octahedral and tetrahedral voids in a hexagonal pattern; CCP has them in a cubic pattern. Exam question: "A mineral with ABCABC stacking" → CCP/FCC structure → coordination number = 12 for each atom. "ABAB stacking" → HCP → coordination number = 12 as well. Both are equally closely packed — a key conceptual distinction.