Solid State

1. Crystalline vs. Amorphous

Crystalline solids have a regular arrangement (long-range order), while amorphous solids like glass are 'pseudo-solids' or supercooled liquids.

Unit Cells and Packing

• Simple Cubic (SCC): $Z=1$, Packing Efficiency = 52.4%.
• Body-Centered (BCC): $Z=2$, Packing Efficiency = 68%.
• Face-Centered (FCC/CCP): $Z=4$, Packing Efficiency = 74%.

Interactive diagram: open this page in a browser to use it.

2. Density of a Unit Cell

Relations between edge length $a$ and atomic radius $r$: simple cubic $a = 2r$; BCC $\sqrt{3}a = 4r$; FCC $\sqrt{2}a = 4r$.

3. Voids

In close packing of $N$ spheres there are $N$ octahedral and $2N$ tetrahedral voids. In an FCC unit cell ($Z = 4$) there are 4 octahedral voids (edge centres and body centre) and 8 tetrahedral voids (inside the cell). The radius ratio $r_+/r_-$ decides which void a cation fits: 0.225–0.414 tetrahedral, 0.414–0.732 octahedral, 0.732–1 cubic.

4. Common Ionic Structures

• NaCl (rock salt): FCC $Cl^-$ with $Na^+$ in all octahedral voids; coordination 6:6.
• ZnS (zinc blende): $Zn^{2+}$ in half the tetrahedral voids; 4:4.
• CsCl: simple cubic arrangement; 8:8.
• $CaF_2$ (fluorite): $F^-$ in all tetrahedral voids; 8:4.

5. Point Defects

• Schottky: equal numbers of cations and anions missing; density decreases. Seen in NaCl, KCl, CsCl, AgBr.
• Frenkel: a smaller ion (usually the cation) moves to an interstitial site; density unchanged. Seen in AgCl, ZnS, AgBr.
• Metal-excess (F-centres): anion vacancies occupied by electrons give colour, e.g. NaCl heated in Na vapour turns yellow.
• AgBr shows both Schottky and Frenkel defects.

6. Electrical and Magnetic Properties

Doping silicon with a group 15 element gives an n-type semiconductor (extra electrons); doping with a group 13 element gives p-type (holes). Ferromagnetic substances (Fe, Co, Ni, $CrO_2$) have aligned domains; antiferromagnetic ones (MnO) have equal opposing domains; ferrimagnetic ones ($Fe_3O_4$) have unequal opposing domains.

Practice questions

1. The number of atoms per unit cell in an FCC lattice is:

   1. 1
   2. 2
   3. 4
   4. 6
   Answer

   C. 8 × 1/8 (corners) + 6 × ½ (faces) = 4.

2. In a BCC lattice, the edge length a and atomic radius r are related by:

   1. a = 2r
   2. √(2)a = 4r
   3. √(3)a = 4r
   4. a = 4r
   Answer

   C. Atoms touch along the body diagonal: √(3)a = 4r.

3. Schottky defects in a crystal cause:

   1. An increase in density
   2. A decrease in density
   3. No change in density
   4. Colour in the crystal
   Answer

   B. Ions are missing from the lattice, so mass per unit volume drops.

4. Silicon doped with boron is:

   1. An n-type semiconductor
   2. A p-type semiconductor
   3. An insulator
   4. A superconductor
   Answer

   B. Boron has 3 valence electrons; each dopant atom creates a hole, giving p-type conduction.

5. The coordination number of Na⁺ in the NaCl structure is:

   1. 4
   2. 6
   3. 8
   4. 12
   Answer

   B. Na⁺ occupies octahedral voids, surrounded by six Cl⁻ ions.