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CRYSTAL STRUCTURES & PROPERTIES - Coggle Diagram
CRYSTAL STRUCTURES & PROPERTIES
SOLID STRUCTURE & LATTICE
Noncrystalline / amorphous: no periodic repeating packing or long-range order; may form from complex structures or rapid cooling.
Dense, regular packing tends to have lower energy than nondense, random packing.
Crystalline: periodic 3-D atomic arrangement with long-range order; common in metals, many ceramics and some polymers.
Crystal structure = lattice + basis; the unit cell is the repeating building block. Lattice parameters: a, b, c and angles α, β, γ.
Atomic hard-sphere model treats atoms / ions as spheres that touch nearest neighbours.
Structure influences density and thermal, mechanical, electrical, magnetic and optical properties; compound examples include NaCl, Al2O3 and AlN.
Three-dimensional crystals are described by 7 crystal systems and 14 Bravais lattices.
UNIT CELL, DENSITY & PACKING
Atoms per unit cell: SC = 1; BCC = 2; FCC = 4; HCP conventional cell = 6.
Atomic packing factor (APF) = (total volume of atoms in unit cell) / (unit-cell volume).
Atom sharing corner = 1/8; face-centred = 1/2; edge-centred = 1/4; body-centred atom = 1.
Theoretical Values: SC 0.52; BCC 0.68; FCC / HCP 0.74.
For a cubic cell, Vc = a³. First use the structure-specific radius–edge relation to find a, then calculate cell volume and density.
Linear density (LD) = atoms centred on a specified direction ÷ direction length.
Planar density (PD) = atoms centred on a specified plane ÷ planar area.
METALLIC CRYSTAL STRUCTURES
SC (simple cubic): 8 corners → 1 atom / cell; coordination number (CN) 6; APF 0.52; atoms touch along cube edges, a = 2R; rare (Po).
BCC (body-centred cubic): corners + 1 body-centre → 2 atoms / cell; CN 8; APF 0.68; body diagonal √3a = 4R, so a = 4R / √3.
FCC (face-centred cubic): corners + 6 face centres → 4 atoms / cell; CN 12; APF 0.74; face diagonal √2a = 4R, so a = 4R / √2.
HCP (hexagonal close-packed): 6 atoms / conventional cell; CN 12; APF 0.74; stacking sequence ABAB…
FCC close-packed planes stack ABCABC…; the third layer differs from the first. FCC and HCP are close-packed; BCC is intermediate; SC is least efficiently packed.
CN = nearest-neighbour count. APF = atomic volume in a cell ÷ total unit-cell volume.
SINGLE / POLYCRYSTALS & PROPERTIES
Randomly oriented grains often give isotropic bulk behaviour
Isotropic = properties are the same in all directions.
Polycrystalline material
Many grains; each grain is a single crystal and neighbouring grains are separated by grain boundaries. Most engineering materials are polycrystals.
Anisotropic = properties depend on sample / crystal orientation.
Single crystal
one continuous crystal orientation; properties may change with direction → anisotropic (e.g. directional elastic modulus in BCC iron).
Crystal size can vary widely (lecture gives approximately 1 nm to 2 cm). Grain orientation and packing affect deformation and measured properties.
The crystal structure / arrangement helps explain differences in density and deformation behaviour among metals.
CRYSTALLOGRAPHIC DIRECTIONS & PLANES
Direction is a vector / line between two points; indices use square brackets [uvw]. Find coordinate differences, reduce to smallest integers; negative values use an overbar.
Parallel vectors have the same direction indices. A family of equivalent directions uses angle brackets, e.g. (\langle100\rangle) in cubic crystals; hexagonal directions use Miller–Bravais four-axis notation.
Plane indices use Miller notation (hkl). Find intercepts along a, b, c → take reciprocals → clear fractions to smallest integers → write (hkl).
If a plane crosses the chosen origin, shift to a parallel plane or choose another origin. An axis-parallel plane has an infinite intercept, giving an index of 0.
Close-packed plane examples: FCC 111 and the HCP basal plane.
X-RAY DIFFRACTION (XRD)
XRD uses X-rays scattered by periodic crystal planes to help determine crystal structure and interplanar spacing.
Bragg’s law: nλ = 2dhkl sin θ. Constructive interference occurs when path difference equals an integer number of wavelengths; otherwise the diffracted intensity is low.
For cubic crystals: dhkl = a / √(h^2 + k^2 + l^2). With known X-ray wavelength lambda and measured angle theta, calculate d and analyse the structure.
Reflection rule — BCC: diffraction allowed when h + k + l is even; odd sums are forbidden.
Reflection rule — FCC: diffraction allowed only when h, k, l are all even or all odd; mixed parity is forbidden.
Diffraction peak positions identify plane spacings and help distinguish crystal structures; lecture also introduces diffraction selection rules for common metals.