Solid State

Solids: Substances with definite shape, volume, rigid. Particles closely packed; only vibrational motion.

Classification of Solids:

Properties of crystalline solids:

• Definite geometric shape
• Sharp melting point
• Anisotropic (properties depend on direction)
• Show cleavage along definite planes

Amorphous solids: Isotropic (same in all directions); also called pseudo-solids or supercooled liquids.

Classification of Crystalline Solids by Bonding:

Molecular Solids subtype:

• Non-polar molecular: $H_2, CO_2, I_2$ — only London forces, low MP
• Polar molecular: $HCl, SO_2$ — dipole-dipole
• Hydrogen-bonded: $H_2O$ (ice), $NH_3$ — H-bonding, harder

Crystal Lattice: Regular 3D arrangement of points in space. Unit Cell: Smallest repeating unit; when repeated in 3D gives the entire lattice.

Parameters of Unit Cell:

• Edge lengths: $a, b, c$
• Angles: $\alpha$ (between b, c), $\beta$ (between a, c), $\gamma$ (between a, b)

Seven Crystal Systems:

Bravais Lattices: $14$ total (combinations of $7$ systems with positions of points: primitive, body-centred, face-centred, end-centred).

Types of Cubic Unit Cells:

Contribution to Unit Cell:

• Corner atom: $1/8$
• Face-centred: $1/2$
• Edge-centred: $1/4$
• Body-centred: $1$ (full)

For SC: $8 \times (1/8) = 1$ atom For BCC: $8 \times (1/8) + 1 = 2$ atoms For FCC: $8 \times (1/8) + 6 \times (1/2) = 4$ atoms

Relation between Edge Length ($a$) and Atomic Radius ($r$):

Packing Efficiency:

$$\text{PE} = \frac{n \times \frac{4}{3}\pi r^3}{a^3} \times 100\%$$

FCC and HCP are closest packings (maximum, $74\%$).

Closest Packing of Spheres:

• 2D layer: each sphere in contact with 6 others (hexagonal layer)
• 3D layered: layers stacked in pattern
• HCP: ABAB... (hexagonal)
• CCP/FCC: ABCABC... (cubic)

Tetrahedral and Octahedral Voids:

In closest packing:

• Tetrahedral void: Surrounded by $4$ spheres
• Octahedral void: Surrounded by $6$ spheres

Number of voids: If $N$ = number of atoms in closest packing:

• Tetrahedral voids = $2N$
• Octahedral voids = $N$

Filling of Voids by Cations:

• Radius ratio rules:
• $r_+/r_- < 0.225$: linear (CN 2)
• $0.225 - 0.414$: tetrahedral (CN 4)
• $0.414 - 0.732$: octahedral (CN 6)
• $0.732 - 1.0$: BCC (CN 8)

Density of Unit Cell: $$\rho = \frac{Z \times M}{N_A \times a^3}$$

• $Z$: atoms per unit cell
• $M$: molar mass
• $N_A$: Avogadro
• $a$: edge length

Important Ionic Crystal Structures:

Defects are deviations from perfect crystal arrangement. Real crystals always have defects.

Types of Defects:

• Point defects: Around a single point
• Line defects: Along a line/edge
• Plane defects: Across a plane

Point Defects:

A. Stoichiometric Defects (do not change the ratio of cations to anions):

1. Schottky Defect:

• Equal pairs of cations and anions missing from lattice
• Maintains electrical neutrality
• Decreases density
• Found in: NaCl, KCl, CsCl, AgBr — ionic compounds with similar-sized ions and high CN

2. Frenkel Defect:

• Cation displaced from regular site to interstitial site
• Density remains unchanged
• Found in: AgCl, AgBr, AgI, ZnS — ionic compounds with low CN and high size difference between cation and anion (small cation, large anion)

B. Non-Stoichiometric Defects (ratio of ions differs from formula):

1. Metal Excess Defect:

• (a) Due to anion vacancy: missing anion; electron occupies the vacant site → F-centre (Farben = color)
• NaCl heated with Na vapor → yellow color due to F-centres (e⁻ traps absorb light)
• Similarly, KCl → violet, LiCl → pink
• (b) Due to extra cation in interstitial position: e.g., $ZnO$ heated → loses O, gives $Zn^{2+}$ in interstitial, with electron(s); becomes yellow when hot, white when cool. ZnO is a non-stoichiometric solid.

2. Metal Deficiency Defect:

• Less cations than required by stoichiometry
• Some cations exhibit higher oxidation state to compensate
• Example: $Fe_{0.95}O$, $Cu_{1.97}S$. The $Fe^{2+}$ deficiency is compensated by $Fe^{3+}$ ions.

Effect of Defects on Properties:

• Schottky: lowers density, affects ionic conductivity
• Frenkel: increases ionic conductivity
• F-centres: cause color (excited electrons absorb visible light)
• Metal deficiency: causes semiconducting behavior

Impurity Defects: Adding aliovalent ions creates vacancies (e.g., $SrCl_2$ in NaCl creates one cation vacancy per $Sr^{2+}$ added).

Electrical Conductivity: Variation based on solid type.

Band Theory of Solids:

• Atomic orbitals combine to form molecular orbitals, then bands of energy levels
• Valence band: Lower energy filled band
• Conduction band: Higher energy band, may be empty or partially filled
• Energy gap: Between valence and conduction bands

Classification:

• Conductors: Bands overlap; electrons free to move; very high conductivity
• Insulators: Large band gap (> $5$ eV); electrons cannot reach conduction band
• Semiconductors: Small band gap (< $3$ eV); electrons thermally excited to conduction band

Intrinsic Semiconductors: Pure Si, Ge — low conductivity at low T, increases with T.

Extrinsic Semiconductors (Doped):

Magnetic Properties of Solids:

Curie temperature: Above this $T$, ferromagnetic materials become paramagnetic (thermal energy disrupts domain alignment).

• Fe: $1043$ K
• Co: $1394$ K
• Ni: $631$ K

Dielectric Properties: Insulators that align dipoles in electric field. Used in capacitors.

Piezoelectricity: Some crystals (e.g., quartz, Rochelle salt) generate electric current when pressed; conversely vibrate when current applied.

Pyroelectricity: Generate small electric current on heating.

Ferroelectricity: Have permanent dipoles even without external field; align in same direction. Example: $BaTiO_3$, Rochelle salt. (Above critical T, become paraelectric.)

Antiferroelectricity: Dipoles align in alternating opposite directions; net dipole zero. Example: $PbZrO_3$.