UNIT 2: Crystal Structures and Defects
A. Crystal Lattices (SC, FCC, BCC, HCP) and Co-ordination Number
A crystal lattice is a 3D periodic arrangement of atoms, ions, or molecules. The co-ordination number is the number of nearest neighbours to an atom.
| Crystal Structure | Lattice Points per Unit Cell | Co-ordination Number | Atomic Arrangement |
|---|---|---|---|
| Simple Cubic (SC) | 1 | 6 | Atoms at corners only |
| Body-Centered Cubic (BCC) | 2 | 8 | Atoms at corners + 1 at body centre |
| Face-Centered Cubic (FCC) | 4 | 12 | Atoms at corners + 1 at each face centre |
| Hexagonal Close-Packed (HCP) | 2 (effective) | 12 | Two-layer ABAB... stacking; 6 in plane, 3 above, 3 below |
[!TIP] Exam Focus: You may be asked to calculate/state co-ordination numbers for these structures. Remember FCC and HCP both have CN=12 (highest packing), BCC has CN=8, SC has CN=6 (lowest).
B. Atomic Packing Factor (APF) for FCC and BCC
Atomic Packing Factor (APF) = (Volume of atoms in unit cell) / (Volume of unit cell). It measures packing efficiency.
For BCC:
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Atoms touching along body diagonal: \(4R = \sqrt{3}a\) → \(a = \frac{4R}{\sqrt{3}}\)
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Volume of atoms = \(2 \times \frac{4}{3}\pi R^3\)
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Volume of cell = \(a^3\)
$$\text{APF}_{\text{BCC}} = \frac{2 \times \frac{4}{3}\pi R^3}{\left(\frac{4R}{\sqrt{3}}\right)^3} = \frac{\sqrt{3}\pi}{8} \approx \boxed{0.68}$$
For FCC:
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Atoms touching along face diagonal: \(4R = \sqrt{2}a\) → \(a = \frac{4R}{\sqrt{2}} = 2\sqrt{2}R\)
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Volume of atoms = \(4 \times \frac{4}{3}\pi R^3\)
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Volume of cell = \(a^3\)
$$\text{APF}_{\text{FCC}} = \frac{4 \times \frac{4}{3}\pi R^3}{(2\sqrt{2}R)^3} = \frac{\pi}{3\sqrt{2}} \approx \boxed{0.74}$$
[!TIP] Common Pitfall: For FCC, remember there are 4 atoms/unit cell (8 corners × 1/8 + 6 faces × 1/2). HCP also has APF ≈ 0.74. APF(SC) = 0.52.
C. Miller Indices for Crystal Planes
Miller Indices (hkl) are a set of three integers used to designate crystal planes and directions.
Procedure for Planes:
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Find intercepts of plane with crystallographic axes (in terms of lattice parameters a, b, c).
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Take reciprocals of intercepts.
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Clear fractions to smallest integers.
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Enclose in parentheses (hkl). Negative intercepts indicated with bar (e.g., \(\bar{1}\)).
Example: Plane intercepts at a, 2b, ∞c → reciprocals: 1, 1/2, 0 → clear: (210).
Directions: [uvw] are indices parallel to axes, found by projecting vector onto axes and reducing to smallest integers.
[!TIP] Exam Tip: You may be asked to determine Miller indices from a given sketch. Practice with planes parallel to one or two axes (intercept = ∞, reciprocal = 0).
D. Crystal Imperfections and Defects
Real crystals contain deviations from perfect periodicity. They critically influence mechanical properties.
Classification:
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Point Defects (0D):
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Vacancy: Missing atom.
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Interstitial: Extra atom in space.
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Substitutional: Foreign atom replaces host.
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Frenkel defect: Vacancy + interstitial pair (common in ionic solids).
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Schottky defect: Cation-anion vacancy pair.
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Line Defects (1D):
- Dislocations: Edge (extra half-plane) and Screw (helical ramp). Primary carriers of plastic deformation.
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Surface/Planar Defects (2D):
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Grain Boundaries: Interface between misoriented grains.
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Twin Boundaries: Mirror symmetry.
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Stacking Faults: Error in atomic layer sequence (e.g., ABCAB → ABCACB in FCC).
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Volume Defects (3D): Pores, cracks, inclusions.
[!TIP] Key Insight: Defects increase electrical resistivity (electron scattering) but are essential for diffusion and plastic deformation. Hume-Rothery rules (Unit 3) govern substitutional solid solutions, a type of point defect.
E. Grain Boundaries and Effect of Grain Size
Grain Boundaries are 2D interfacial regions where crystals of different orientations meet. They are high-energy regions with atomic mismatch.
Effect of Grain Size:
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Smaller grain size → More grain boundary area per volume.
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Hall-Petch Relationship: Yield strength increases with decreasing grain size.
$$\sigma_y = \sigma_0 + \frac{k}{\sqrt{d}}$$
where \(\sigma_y\) = yield strength, \(\sigma_0\) = friction stress, \(k\) = strengthening coefficient, \(d\) = average grain diameter.
Why? Grain boundaries hinder dislocation motion (dislocations pile up at boundaries). Finer grains mean shorter slip distances and more barriers.
Other Effects:
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↑ Strength & Hardness
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↑ Toughness (at room temp, due to crack deflection)
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↓ Creep rate (boundaries block diffusion)
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May ↓ corrosion resistance (boundaries are chemically active)
[!TIP] Exam Focus: Be ready to explain the Hall-Petch equation and the mechanism (dislocation pile-up). Also, note that at very high temperatures, coarse grains are better for creep resistance (grain boundary sliding becomes dominant).