Atomic Bonding, Crystal Structures & Crystallography

How atoms arrange themselves. Learn atomic bonding (metallic, ionic, covalent, secondary) and bond energy curves, crystalline vs amorphous solids, unit cells and lattices, the common metallic structures (BCC, FCC, HCP) with coordination numbers and packing factors, theoretical density, allotropy (α-Fe, γ-Fe), Miller indices for directions and planes, linear and planar density, slip systems, single crystals vs polycrystals and anisotropy, and X-ray diffraction with Bragg's law — with Python computing densities, packing factors and XRD peak positions.

Most metals and ceramics are crystalline: atoms repeat in a regular 3D pattern. The pattern (crystal structure) controls density, how easily layers slide (ductility), and how the material diffracts X-rays.

Iron changes from BCC to FCC when heated above 912 °C — this allotropic change is the basis of heat-treating steel.

An analogy

Crystal structures are like ways of stacking oranges at a fruit stall: some stacks pack tightly (FCC, HCP — 74% of space filled), others leave more gaps (BCC — 68%). Tightly packed layers slide over each other more easily, making metals like copper and aluminium very ductile.

Bonding

Bond types and consequences:

BCC, FCC and HCP

FCC (Cu, Al, Ni, Au, γ-Fe, austenitic stainless): CN 12, APF 0.74, 12 slip systems on {111} planes — ductile even at low temperature. BCC (α-Fe, Cr, W, Mo): CN 8, APF 0.68 — strong, with a ductile-to-brittle transition at low temperature. HCP (Ti, Mg, Zn): CN 12, APF 0.74 but few easy slip systems — limited ductility at room temperature.

Miller indices

Directions [uvw]: vector components reduced to smallest integers. Planes (hkl): take reciprocals of intercepts on the axes, clear fractions. Families {hkl} and ⟨uvw⟩ group equivalent planes/directions. Slip happens on the most densely packed planes along the most densely packed directions (FCC {111}⟨110⟩, BCC {110}⟨111⟩).

Polycrystals, anisotropy and diffraction

Most engineering metals are polycrystalline — many grains with different orientations separated by grain boundaries; random texture makes them nearly isotropic, while rolling creates texture (used in electrical steels and deep-drawing sheet). Single crystals (turbine blades, silicon wafers) exploit anisotropy. Amorphous solids (glass, many polymers, metallic glasses) lack long-range order. XRD identifies phases, lattice parameters, texture and residual stress; peak broadening gives crystallite size.

Formulas

Worked example: Atomic radius from lattice parameter

Given: Cu FCC, a = 0.3615 nm

Find: R.

Answer: 0.128 nm

Worked example: Miller indices

Given: Plane intercepts x = 1, y = ½, z = ∞

Find: (hkl).

Answer: (120)

Worked example: Bragg angle

Given: d = 0.2087 nm, λ = 0.15406 nm

Find: 2θ.

Answer: 43.3°

Density, packing factor and XRD peaks

import math
# Crystal structures: density, atomic packing factor and XRD peak positions
NA = 6.022e23
for name, n, a_nm, M in (("alpha-Fe (BCC)", 2, 0.2866, 55.85), ("Cu (FCC)", 4, 0.3615, 63.55), ("Al (FCC)", 4, 0.4050, 26.98)):
    a = a_nm * 1e-7                          # cm
    rho = n * M / (NA * a ** 3)
    print(f"{name:15s}: density {rho:.2f} g/cm3")
print(f"APF: BCC = {math.pi*math.sqrt(3)/8:.3f}, FCC = {math.pi/(3*math.sqrt(2)):.3f}")
# First XRD peaks of Cu (FCC: h,k,l all odd or all even) with Cu K-alpha
lam, a = 0.15406, 0.3615
for hkl in ((1, 1, 1), (2, 0, 0), (2, 2, 0)):
    d = a / math.sqrt(sum(i * i for i in hkl))
    print(f"Cu {hkl}: d = {d:.4f} nm, 2-theta = {2*math.degrees(math.asin(lam/(2*d))):.2f} deg")
# alpha-Fe (BCC) : density 7.88 g/cm3
# Cu (FCC)       : density 8.94 g/cm3
# Al (FCC)       : density 2.70 g/cm3
# APF: BCC = 0.680, FCC = 0.740
# Cu (1, 1, 1): d = 0.2087 nm, 2-theta = 43.32 deg
# Cu (2, 0, 0): d = 0.1807 nm, 2-theta = 50.45 deg
# Cu (2, 2, 0): d = 0.1278 nm, 2-theta = 74.13 deg

Knowing only the crystal structure, lattice parameter and atomic mass, we predict densities matching handbook values (iron 7.87, copper 8.96, aluminium 2.70 g/cm³). The XRD peak positions are the fingerprint used to identify phases in real samples — copper's strongest peak at 2θ ≈ 43.3° is the (111) plane.

Common mistakes

Applications

Key takeaways

Going further