From salt crystals to silicon chips — the chemistry of ordered matter. With interactive packing fraction calculator, unit cell diagrams, defects, magnetic and electrical properties, and 20+ real-life examples.
Before diving in, here's the entire chapter on one page. How solids are classified, their structure, properties, and defects.
Solids are classified as crystalline (ordered, sharp MP) or amorphous (disordered, glassy). Crystals have repeating lattice + basis. Unit cells (primitive, BCC, FCC) determine packing efficiency. Density is calculated from unit cell parameters. Defects (point, line, impurity) affect properties. Magnetic and electrical properties arise from electron arrangement.
Solid: State of matter with definite shape and volume. Particles are closely packed and held by strong intermolecular forces.
Two types: Crystalline and amorphous.
Crystalline: Long-range order (atoms regularly arranged over long distances). Sharp melting point. Anisotropic (properties depend on direction). Examples: NaCl, diamond, quartz, metals, ice.
Properties:
Amorphous: Short-range order (no long-range arrangement). No sharp melting point — soften over a range. Isotropic. Examples: glass, rubber, plastics, amorphous silicon.
Properties:
| Type | Particles | Forces | Properties | Examples |
|---|---|---|---|---|
| Molecular | Molecules | Van der Waals / H-bonds | Soft, low MP, non-conducting | Ice, dry ice, iodine |
| Ionic | Ions (+, −) | Strong electrostatic (Coulomb) | Hard, brittle, high MP, conduct in solution/melt | NaCl, CaO, MgO |
| Covalent (network) | Atoms (covalent) | Covalent bonds | Very hard, very high MP, non-conducting (except graphite) | Diamond, quartz, SiC |
| Metallic | Metal cations + sea of electrons | Metallic bonds | Malleable, ductile, conductive, high MP | Fe, Cu, Al, Na |
Crystalline: long-range order, sharp MP, anisotropic. Amorphous: short-range order, gradual softening, isotropic. By bonding: molecular (weak forces), ionic (electrostatic), covalent (network), metallic (sea of electrons).
Space lattice: Regular 3D arrangement of points representing the positions of atoms/ions/molecules in a crystal.
Lattice point: Each point in the space lattice. Represents identical environment.
Basis: The atom/molecule/ion placed at each lattice point.
Unit cell: Smallest repeating portion of the lattice that, when repeated in 3D, gives the entire crystal.
Lattice parameters: Edge lengths (a, b, c) and angles (α, β, γ).
| System | a, b, c | α, β, γ | Example |
|---|---|---|---|
| Cubic | a = b = c | α = β = γ = 90° | NaCl, Cu, Fe |
| Tetragonal | a = b ≠ c | α = β = γ = 90° | White tin (Sn) |
| Orthorhombic | a ≠ b ≠ c | α = β = γ = 90° | Rhombic sulfur |
| Monoclinic | a ≠ b ≠ c | α = γ = 90°, β ≠ 90° | Monoclinic sulfur |
| Triclinic | a ≠ b ≠ c | α ≠ β ≠ γ | CuSO₄·5H₂O |
| Hexagonal | a = b ≠ c | α = β = 90°, γ = 120° | Graphite, ZnO |
| Rhombohedral | a = b = c | α = β = γ ≠ 90° | Calcite, quartz |
14 Bravais lattices: The 7 crystal systems combined with different centering (P, I, F, C) give 14 unique lattices.
1. Primitive (Simple Cubic, SC):
2. Body-Centered Cubic (BCC):
3. Face-Centered Cubic (FCC) — also called Cubic Close Packed (CCP):
Choose unit cell type — see Z, coordination number, and packing efficiency.
Crystal = lattice + basis. 7 crystal systems + 14 Bravais lattices. Cubic unit cells: SC (Z=1, PF=52.4%), BCC (Z=2, PF=68%), FCC (Z=4, PF=74%). Packing efficiency: SC < BCC < FCC (max).
Close packing: Arrangement of spheres in a way that maximizes packing efficiency (fills space most densely).
1D: Atoms in a row — coordination number 2
2D: Atoms in a layer — square close packing (CN=4, PF=52.4%) or hexagonal close packing (CN=6, PF=60.4%)
3D: Layers stacked to give higher packing
1. Simple cubic (SC) packing: Layers directly on top → packing efficiency 52.4%
2. Hexagonal close packing (HCP): Layer B sits in depressions of layer A; layer C sits in depressions of B directly above A → ABAB... stacking. Packing efficiency 74%. Example: Mg, Zn, Ti.
3. Cubic close packing (CCP) = FCC: Layer B in depressions of A; layer C in depressions of B but not above A → ABCABC... stacking. Packing efficiency 74%. Example: Cu, Ag, Au, Al.
Void types:
Number of voids:
Close packing in 3D: SC (52.4%), BCC (68%), FCC/HCP (74% — most efficient). FCC = ABC stacking, HCP = AB stacking. Voids: tetrahedral (2 per atom) and octahedral (1 per atom).
Density (ρ) = mass of unit cell / volume of unit cell
Formula: ρ = Z × M / (N_A × a³)
Where:
For edge length in picometers (pm): 1 pm = 10⁻¹⁰ cm, so a³ (in pm³) × 10⁻³⁰ = a³ (cm³)
For edge length in Angstroms (Å): 1 Å = 10⁻⁸ cm, so a³ (in ų) × 10⁻²⁴ = a³ (cm³)
| Unit Cell | Z | r and a relation | a from r |
|---|---|---|---|
| Simple Cubic (SC) | 1 | a = 2r | a = 2r |
| BCC | 2 | 4r = √3 a → r = √3 a/4 | a = 4r/√3 ≈ 2.31 r |
| FCC | 4 | 4r = √2 a → r = a/(2√2) | a = 2√2 r ≈ 2.83 r |
Enter unit cell parameters — calculate crystal density.
Density = Z × M / (N_A × a³). SC: Z=1, a=2r. BCC: Z=2, a=4r/√3. FCC: Z=4, a=2√2 r. Higher Z or smaller a → higher density. Used to identify metals and determine Avogadro's number.
1. Point defects (0D): At a single atomic position. Types:
2. Line defects (1D): Dislocations — edge or screw. Cause plastic deformation.
3. Interfacial defects (2D): Grain boundaries, twin boundaries, stacking faults.
4. Volume defects (3D): Voids, cracks, foreign inclusions.
| Feature | Frenkel Defect | Schottky Defect |
|---|---|---|
| What happens | Ion moves to interstitial site | Cation + anion vacancies |
| Stoichiometry | Preserved | Preserved |
| Density | Unchanged | Decreases |
| Common in | Large cation, small anion, low CN | High CN, similar-sized ions |
| Examples | AgCl, AgBr, ZnS, AgI | NaCl, KCl, CsCl, KBr |
| Cause | Large size difference | High coordination, ionic |
Substitutional impurity: Foreign atom replaces a host atom. Example: SrCl₂ in NaCl — Sr²⁺ replaces 2 Na⁺ (creates 1 cation vacancy per Sr²⁺).
Interstitial impurity: Small foreign atom in interstitial space. Example: C in Fe (steel).
Doping: Deliberate impurity addition in semiconductors.
Non-stoichiometric defects: Compounds with variable composition (e.g., FeO₁₋ₓ). Due to variable oxidation states of transition metals.
F-centers: Anionic vacancies filled with electrons → color (e.g., NaCl is yellow when F-centers present).
Point defects: Frenkel (ion to interstitial, density unchanged) vs Schottky (missing pairs, density decreases). Impurity defects: substitutional or interstitial. Non-stoichiometric compounds (variable composition). F-centers cause color. Used in electronics (doping) and materials science.
1. Diamagnetic: All electrons paired. Weakly repelled by magnetic field. Examples: NaCl, benzene, H₂O.
2. Paramagnetic: Unpaired electrons. Weakly attracted by magnetic field. Examples: O₂, Cu²⁺ salts, TiO₂.
3. Ferromagnetic: Unpaired electrons aligned in domains → strong permanent magnetism. Examples: Fe, Co, Ni, Gd.
4. Antiferromagnetic: Adjacent spins antiparallel → net zero. Example: MnO.
5. Ferrimagnetic: Unequal antiparallel spins → net non-zero. Example: Fe₃O₄ (magnetite).
| Property | Diamagnetic | Paramagnetic | Ferromagnetic |
|---|---|---|---|
| Unpaired e⁻ | None | Present | Present + aligned |
| Response | Weakly repelled | Weakly attracted | Strongly attracted |
| Permeability | μ < 1 | μ > 1 (small) | μ >> 1 |
| Examples | Diamond, H₂O | TiO₂, CuSO₄ | Fe, Co, Ni |
Diamagnetic (all paired, weakly repelled), paramagnetic (unpaired, weakly attracted), ferromagnetic (strong permanent, aligned domains), antiferromagnetic (antiparallel, net zero), ferrimagnetic (unequal, net non-zero). Ferromagnetism → permanent magnets and data storage.
Valence band: Highest occupied band (full of electrons)
Conduction band: Lowest empty band (empty or partly filled)
Band gap (Eg): Energy difference between valence band and conduction band.
Classification:
Intrinsic semiconductor: Pure Si or Ge. Small conductivity due to thermal excitation (electron + hole pairs).
Extrinsic semiconductor: Doped with impurities.
n-type: Doped with pentavalent element (P, As, Sb) → extra electron → electron carriers. Donor doping.
p-type: Doped with trivalent element (B, Al, Ga) → missing electron (hole) → hole carriers. Acceptor doping.
Band theory: conductors (no gap), semiconductors (~1 eV gap), insulators (>3 eV gap). Intrinsic (pure) vs extrinsic (doped). n-type (electron carriers, pentavalent dopant). p-type (hole carriers, trivalent dopant). Foundation of all electronics.
Q1. Why does NaCl have a sharp melting point while glass softens gradually?
NaCl is a crystalline solid with long-range order. All ionic bonds have the same energy → uniform melting requires same amount of energy throughout → sharp melting point (801°C).
Glass is amorphous — short-range order only. Different regions have different strengths → melt at different temperatures → gradual softening over a range.
Q2. A compound forms an FCC crystal with edge length 4.07 Å. If density is 10.5 g/cm³, calculate the molar mass.
ρ = Z·M / (N_A · a³)
a = 4.07 Å = 4.07 × 10⁻⁸ cm
a³ = (4.07 × 10⁻⁸)³ = 6.74 × 10⁻²³ cm³
FCC → Z = 4, N_A = 6.022 × 10²³
10.5 = (4 × M) / (6.022 × 10²³ × 6.74 × 10⁻²³)
M = (10.5 × 6.022 × 10²³ × 6.74 × 10⁻²³) / 4
= (10.5 × 6.022 × 6.74) / 4 ≈ 106.8 g/mol
This matches silver (M_Ag = 108 g/mol) ✓
Q3. Calculate the packing efficiency of a simple cubic unit cell.
In SC, atoms are at 8 corners. Z = 1 (8 × 1/8). Edge length a = 2r (atoms touch along edge).
Volume of 1 atom = (4/3)πr³
Total volume of atoms = Z × (4/3)πr³ = 1 × (4/3)πr³
V_cell = a³ = (2r)³ = 8r³
PF = (Volume of atoms) / (Volume of cell) × 100
= [(4/3)πr³] / (8r³) × 100
= (π/6) × 100
≈ 52.4%
SC has the lowest packing efficiency of the three cubic structures. HCP/FCC has 74% (highest).
Q4. (a) What is a Schottky defect? (b) How does it affect density? (c) Which crystals show this defect? (d) Compare with Frenkel defect.
Equal numbers of cations and anions are missing from the crystal lattice, creating paired vacancies. This maintains the electrical neutrality of the crystal.
Since atoms/ions are removed from the lattice, mass decreases while volume stays nearly the same → density decreases.
Common in ionic compounds with high coordination number and similar-sized ions:
| Feature | Schottky | Frenkel |
|---|---|---|
| What | Missing ion pairs | Ion to interstitial site |
| Density | Decreases | Unchanged |
| Stoichiometry | Preserved | Preserved |
| Occurs in | NaCl, KCl (high CN, similar size) | AgCl, ZnS (large size difference) |
| Cause | High coordination number | Large cation/small anion mismatch |
Q5. Calculate the density of copper. Copper crystallizes in FCC with a = 3.61 Å, M = 63.5 g/mol.
FCC → Z = 4
a = 3.61 Å = 3.61 × 10⁻⁸ cm
M = 63.5 g/mol
N_A = 6.022 × 10²³
a³ = (3.61 × 10⁻⁸)³ = 47.04 × 10⁻²⁴ cm³
= 4.704 × 10⁻²³ cm³
ρ = Z × M / (N_A × a³)
= 4 × 63.5 / (6.022 × 10²³ × 4.704 × 10⁻²³)
Numerator: 4 × 63.5 = 254
Denominator: 6.022 × 4.704 = 28.32
ρ = 254 / 28.32 ≈ 8.97 g/cm³
Accepted density of Cu ≈ 8.96 g/cm³. Our calculation gives 8.97 — excellent agreement!
This method is used to determine unknown crystal parameters or identify materials from their crystal data.
Q6. What are the differences between n-type and p-type semiconductors?
n-type: Semiconductor doped with pentavalent impurity (P, As, Sb) → extra electrons → electron carriers.
p-type: Semiconductor doped with trivalent impurity (B, Al, Ga) → electron holes → hole carriers.
n-type: majority carriers = electrons, minority = holes
p-type: majority carriers = holes, minority = electrons
n-type: donor atom (gives extra electron). E.g., P has 5 valence e⁻ — 4 form bonds with Si, 1 free.
p-type: acceptor atom (creates hole). E.g., B has 3 valence e⁻ — needs 1 more electron → creates hole.
n-type: Fermi level closer to conduction band (electrons easier to excite)
p-type: Fermi level closer to valence band
n-type + p-type → p-n junction (diode)
Used in: LEDs, solar cells, transistors, integrated circuits, rectifiers
Real devices often use p-n-p or n-p-n configurations (transistors).
Selected exercise questions from the LEGACY NCERT Chapter 1 (The Solid State). Removed from current syllabus but useful for competitive exams.
Amorphous solid: A solid in which constituent particles (atoms, ions, molecules) are arranged in a short-range order but lack long-range order. They do not have definite geometry, have no sharp melting point, and are isotropic.
Examples: Glass, rubber, plastics, amorphous silicon, pitch (tar).
(a) HCP vs CCP:
HCP: ABABAB... stacking of layers. Coordination number 12, packing efficiency 74%. Examples: Mg, Zn, Ti.
CCP (= FCC): ABCABC... stacking. Coordination number 12, packing efficiency 74%. Examples: Cu, Ag, Au, Al.
Both have the same packing efficiency, only the layer arrangement differs.
(b) Crystal lattice vs unit cell:
Crystal lattice: The 3D arrangement of lattice points (all equivalent) that represents the overall structure.
Unit cell: The smallest portion of the lattice that, when repeated in all 3D, generates the entire crystal. Contains all the structural information.
(a) FCC: 8 corners (each 1/8) + 6 face centers (each 1/2) = 8(1/8) + 6(1/2) = 1 + 3 = 4 lattice points
(b) BCC: 8 corners (each 1/8) + 1 body center (whole) = 1 + 1 = 2 lattice points
(c) Simple cubic: 8 corners (each 1/8) = 1 lattice point
Coordination number: Number of nearest neighbour atoms surrounding a given atom in the crystal.
(a) FCC = 12 (each atom has 12 nearest neighbors)
(b) BCC = 8 (each atom has 8 nearest neighbors)
(c) Simple cubic = 6 (each atom has 6 nearest neighbors)
Given: BCC → Z = 2, M = 93 g/mol, ρ = 8.55 g/cm³
Formula: a³ = Z × M / (N_A × ρ)
= 2 × 93 / (6.022 × 10²³ × 8.55)
= 186 / (5.148 × 10²⁴)
= 3.61 × 10⁻²³ cm³
a = (3.61 × 10⁻²³)^(1/3) = 3.31 × 10⁻⁸ cm = 331 pm
For BCC: r = √3 a / 4
r = (1.732 × 331) / 4 = 143 pm
Atomic radius of niobium ≈ 143 pm
Given: FCC → Z = 4, a = 4.07 × 10⁻⁸ cm, ρ = 10.5 g/cm³
Formula: M = ρ × N_A × a³ / Z
a³ = (4.07 × 10⁻⁸)³ = 6.74 × 10⁻²³ cm³
M = (10.5 × 6.022 × 10²³ × 6.74 × 10⁻²³) / 4
= (10.5 × 6.022 × 6.74) / 4
= 426.2 / 4 = 106.6 g/mol
Atomic mass of Ag ≈ 107 g/mol ✓ (literature: 107.87)
(a) Frenkel defect: When an ion (usually a cation) leaves its normal lattice site and occupies an interstitial site, creating a vacancy + interstitial pair. Stoichiometry preserved. Density unchanged. Common in AgCl, AgBr, AgI, ZnS. Caused by large size difference between cation and anion.
(b) Schottky defect: Equal number of cations and anions are missing from lattice, creating vacancy pairs. Stoichiometry preserved. Density decreases. Common in NaCl, KCl, KBr, CsCl. Caused by high coordination number and similar-sized ions.
(c) F-centre: An anionic vacancy filled with one or more electrons (F = color center, from German "Farbe"). These trapped electrons absorb visible light → give color to the crystal. Example: NaCl heated in Na vapor becomes yellow because Cl⁻ vacancies trap electrons. Alkali metal halides with F-centers are colored.
(a) Ferromagnetism: Strong permanent magnetism due to aligned magnetic moments in domains. Examples: Fe, Co, Ni, Gd. These metals are strongly attracted to magnets and can be permanently magnetized.
(b) Paramagnetism: Weak attraction due to unpaired electrons. No long-range alignment (thermal energy disrupts). Examples: O₂, CuSO₄, TiO₂, some transition metal complexes.
(c) Diamagnetism: Weak repulsion due to all paired electrons. No net magnetic moment per atom. Examples: NaCl, benzene, H₂O, most organic compounds.
Answer: When a solid (especially an ionic solid) is heated:
Schottky defect can arise — some cations and anions leave their lattice sites, creating vacancy pairs. This is more common at high temperatures because thermal energy allows ions to escape from their sites.
Property affected: Density decreases (mass lost, volume nearly constant).
Also, if the defect involves trapped electrons (F-centers), the crystal may become colored.
Conductivity may also increase due to increased ion mobility in the defective crystal.
Semiconductor: A material with electrical conductivity between that of conductors and insulators. Band gap ~1 eV. Conductivity increases with temperature (unlike metals).
Two main types:
(a) Intrinsic (pure) semiconductors: Pure Si or Ge. Small conductivity due to thermal excitation of electrons across the band gap. Both electrons and holes contribute. Conductivity is low but increases with T.
(b) Extrinsic (doped) semiconductors:
• n-type: Doped with pentavalent element (P, As, Sb). Extra electron per dopant atom → electron-rich → electrons are majority carriers. Examples: Si doped with P.
• p-type: Doped with trivalent element (B, Al, Ga). Creates a hole per dopant atom → hole-rich → holes are majority carriers. Examples: Si doped with B.
n-type + p-type → p-n junction → basis of diodes, transistors, LEDs, solar cells.
Given: CCP = FCC. r = 125 pm. M = 27 g/mol.
(a) Side length:
For FCC: a = 2√2 r = 2 × 1.414 × 125 = 353.5 pm ≈ 354 pm
(b) Number of unit cells in 1 cm³:
a = 354 pm = 3.54 × 10⁻⁸ cm
Volume of unit cell = a³ = (3.54 × 10⁻⁸)³ = 4.44 × 10⁻²³ cm³
Number of cells in 1 cm³ = 1 / (4.44 × 10⁻²³) = 2.25 × 10²² unit cells
| N_A | 6.022 × 10²³ mol⁻¹ | Avogadro's number |
| 1 pm | 10⁻¹⁰ cm = 10⁻¹² m | Picometer |
| 1 Å | 10⁻⁸ cm = 10⁻¹⁰ m | Angstrom |
| SC efficiency | 52.4% | Simple cubic |
| BCC efficiency | 68% | Body-centered |
| FCC efficiency | 74% | Face-centered (max) |
| Diamond band gap | ~5.5 eV | Insulator |
| Si band gap | ~1.1 eV | Semiconductor |
| Ge band gap | ~0.7 eV | Semiconductor |
| Z | Number of atoms per unit cell |
| a, b, c | Unit cell edge lengths |
| α, β, γ | Unit cell angles |
| r | Atomic radius |
| M | Molar mass (g/mol) |
| N_A | Avogadro's number |
| ρ | Density (g/cm³) |
| PF | Packing fraction / efficiency |
| SC | Simple cubic |
| BCC | Body-centered cubic |
| FCC | Face-centered cubic |
| HCP | Hexagonal close packed |
| CN | Coordination number |
| E_g | Band gap energy |
(A) Both true, R explains A. (B) Both true, R does NOT explain A. (C) A true, R false. (D) A false, R true.
All key formulas at one place — perfect for quick revision before competitive exams.