3.2 Defects in Solids, Non-stoichiometry and Semiconductors
★★★ 2021–2025: 24 parts, 177 marks (Civil Services 5 parts / 35 m · IFoS 4 / 41 m · Geo-Scientist P-I 12 / 86 m, P-II 3 / 15 m). Geo-Scientist Paper I asks it every year.
Syllabus lines. Civil Services: “stoichiometric and nonstoichiometric defects, impurity defects, semi-conductors”. IFoS: “Imperfections in crystals, stoichiometric and nonstoichiometric defects, impurity defects, semi-conductors”. Geo-Scientist P-I: “Defects, non-stoichiometric compounds and solid solutions, atom and ion diffusion, solid electrolytes … semiconductors”; P-II: “Band theory of solids: qualitative idea of band theory, conducting, semiconducting and insulating properties”. Crystal systems and X-ray diffraction have their own notes.
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CSE 2024 Q7(a) = asked there; tap to open the solved answerSN = line kept in the short notes
3.2.1 Why every real crystal has defects ★★
- Making a defect breaks bonds: ΔH > 0. But n defects can sit on N sites in W = N!/[n!(N − n)!] ways, so the crystal gains configurational entropy S = k ln W CSE 2025 Q3(e) 5m.SN
- G = nΔH − TS has a minimum at a finite n: above 0 K a perfect crystal is not the equilibrium state, and the defect fraction rises exponentially with T.SN
Solved PYQ IFoS 2024 Q2(b) 15m
Derive the number of vacancies per mole of metal atoms at temperature T (neglect vibrational entropy; Ev per mole of vacancies).
- N atoms and n vacancies on N + n sites: . Stirling: .
- with εv = Ev/NA; set : .
- So ; for n ≪ N, n ≈ NA e−Ev/RT per mole.SN
3.2.2 Stoichiometric defects: Schottky and Frenkel ★★★
| Schottky | Frenkel | |
|---|---|---|
| What happens | equal (charge-balanced) cation and anion vacancies | ion (usually the cation) leaves its site for an interstitial site |
| Density | falls | unchanged |
| Favoured by | high coordination number, ions of similar size | low coordination number, small polarisable cation, large anion |
| Examples | NaCl, KCl, CsCl, KBr; CaF₂ (1 Ca²⁺ + 2 F⁻ vacancies) | AgCl, AgBr, AgI, ZnS; AgBr shows both |
- Both keep the formula and electrical neutrality; both give a small ionic conductivity (ions hop into vacancies) CSE 2024 Q2(c)(ii) 10m Geo 2024 Q2(a)(ii) 5m Geo 2023 Q3(c) 5m.SN
- AgCl shows Frenkel: Ag⁺ (115 pm) is much smaller than Cl⁻ (181 pm); its filled d¹⁰ shell is soft and polarisable, so covalent bonding stabilises it in the odd site; the high dielectric constant screens the defect charges Geo 2025 Q1(a) 5m.SN
- Matching: Frenkel – AgCl; perovskite – CaTiO₃; spinel – MgAl₂O₄; ilmenite – FeTiO₃; corundum – Al₂O₃ (also Ti₂O₃); Schottky – CaF₂; non-stoichiometric – Ti₁₋ₓO; semiconductor – GaAs; metallic oxide – ReO₃ Geo 2023 Q1(a) 5m Geo 2022 Q2(d) 6m.
Solved PYQ IFoS 2021 Q1(a) 8m
AgBr: which vacancies does Frenkel disorder make? Temperature at which the Frenkel fraction exceeds 1 ppb (ΔHF = 1.16 eV per defect, A = 3.091).
- Ag⁺ (1.15 Å) fits an interstice; Br⁻ (1.96 Å) does not: only Ag⁺ vacancies + Ag⁺ interstitials.
- f = A e−ΔH/2kT = 10⁻⁹ ⇒ T = ΔH/[2k ln(A/10⁻⁹)] = (1.16 × 1.602 × 10⁻¹⁹)/(2 × 1.381 × 10⁻²³ × 21.85) = 308 K.SN
3.2.3 Impurity defects and charge balance ★★
- An aliovalent ion must be compensated. Ca²⁺ in NaCl: each Ca²⁺ on a Na⁺ site needs one Na⁺ vacancy. 1 in 10 000 Na sites as Ca²⁺: Na₀.₉₉₉₈Ca₀.₀₀₀₁Cl (9998 + 2 = 10 000 positive charges) CSE 2025 Q2(d) 5m.SN
- The extra vacancies raise ionic conductivity: doped NaCl conducts better than pure NaCl.
- Kröger–Vink notation: Mg′Al (Mg²⁺ on an Al³⁺ site, one negative effective charge), VO•• (oxide vacancy, +2), Ali•••. MgO in Al₂O₃: 2 Mg per O vacancy, or 3 Mg per Al interstitial IFoS 2022 Q1(c) 8m.SN
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