Solid State

1) Solid State – Key Points

• Particles are closely packed and intermolecular forces are very strong.
• Particles can only vibrate about their mean positions.
• Characteristics: definite shape & volume, high rigidity & density, low compressibility.

2) Types of Solids

Crystalline Solids

• Sharp edges and flat faces
• Sharp melting point
• Long-range order (basic unit repeats infinitely)
• Anisotropic in nature
• Made of many tiny units called crystals

Amorphous Solids

• Irregular (haphazard) arrangement
• No sharp melting point
• Isotropic in nature
• Also called supercooled liquids

3) Space Lattice & Unit Cell

• Space lattice: regular repeating arrangement of points in 3D space.
• Unit cell: smallest complete unit which when repeated generates the crystal.

Types of Unit Cells

• Primitive: atoms only at corners.
• Non-primitive (centred): atoms also at face/edge/body centre in addition to corners.

4) Cubic Unit Cells – Atoms & Radius Relation

• Simple Cubic (sc): atoms/unit cell = 1, r = a/2
• Body Centred Cubic (bcc): atoms/unit cell = 2, 4r = √3 a ⇒ r = (√3/4)a
• Face Centred Cubic (fcc): atoms/unit cell = 4, 4r = √2 a ⇒ r = (√2/4)a

5) X-ray Diffraction & Bragg’s Law

• Crystal structure can be determined using X-ray diffraction.
• Bragg’s Law: 2d sinθ = nλ
• d = interplanar distance, θ = glancing angle, λ = wavelength, n = 1,2,3…

6) Ionic Crystal Structures

Coordination number: number of nearest oppositely charged ions around a central ion.

(i) NaCl Type (Rock Salt, AB)

• Coordination number: 6 : 6
• Formula units per unit cell: 4
• Distance relation: r₊ + r_– = a/2
• Examples: KCl (also similar salts)

(ii) CsCl Type (AB)

• Coordination number: 8 : 8
• Formula units per unit cell: 1
• Distance relation: r₊ + r_– = (√3/2)a
• Examples: CsCl, CsBr

(iii) ZnS Type (Sphalerite)

• Coordination number: 4 : 4
• Formula units per unit cell: 4
• Examples: ZnS, CuCl, CuBr

(iv) Fluorite Type (AB₂)

• Coordination number: 8 : 4
• Formula units per unit cell: 4
• Examples: CaF₂, SrF₂, BaF₂, PbF₂

(v) Anti-fluorite Type (A₂B)

• Coordination number: 4 : 8
• Formula units per unit cell: 4
• Examples: Na₂O, Li₂O, K₂O

7) Density of a Unit Cell

d = (zM) / (a³N_A)

• z = number of atoms / formula units per unit cell
• M = molar mass (or formula mass)
• a = edge length of unit cell
• N_A = Avogadro’s number

8) Packing & Packing Fraction

Types of Packing

• Simple cubic (square packing): coordination number within layer = 4
• bcc packing: coordination number = 8
• hcp / ccp: coordination number (3D) = 12
• hcp: ABABAB…
• ccp: ABCABC…
• Packing fraction for hcp/ccp = 0.74

Packing Fractions

• Simple cubic: φ = π/6 ≈ 0.52
• bcc: φ ≈ 0.68
• fcc/ccp: φ ≈ 0.74

9) Voids in Close Packing

• Voids (interstices): unoccupied spaces in lattice.
• Octahedral void: coordination number = 6
• Tetrahedral void: coordination number = 4 (smaller than octahedral)
• In ccp: octahedral voids are at edge centres and body centre.
• In ccp: tetrahedral voids lie along body diagonal (2 per body diagonal).

10) Radius Ratio Rule (Ionic Solids)

Radius ratio = r₊ / r_–

• 0.155 to 0.225 → CN = 3 (planar triangle)
• 0.225 to 0.414 → CN = 4 (ZnS type)
• 0.414 to 0.732 → CN = 6 (NaCl type)
• 0.732 to 1.00 → CN = 8 (CsCl type)

11) Crystal Defects

A) Point Defects

• Stoichiometric defects: cation:anion ratio remains 1:1
• Non-stoichiometric defects: ratio not 1:1

Stoichiometric Defects

• Schottky defect: one cation + one anion missing; density decreases; solids remain neutral.
• Examples: NaCl, KCl
• Frenkel defect: cation moves to interstitial site; density unchanged.
• Examples: AgBr, ZnS
• AgBr shows both Schottky and Frenkel defects.

Non-stoichiometric Defects

• Metal excess (anion vacancy): missing anion, electron occupies site → F-centre; coloured, paramagnetic.
• Metal excess (interstitial cation): extra cation + electron in interstitial site; coloured, semiconducting.
• Example: ZnO white at low temperature, yellow at high temperature.
• Metal deficiency: missing cation; nearby cation changes oxidation state; semiconducting.
• Examples: NiO, FeO

Impurity Defects

• Defects arising due to presence of impurities in the crystal.