Crystallography & Phase Diagrams — Materials Science Reference

Crystallography & Phase Diagrams

A Tier-1 reference covering the geometric description of crystalline matter (crystallography, defects, diffraction) and the thermodynamic + kinetic description of multi-phase systems (phase diagrams, CALPHAD, TTT/CCT). These two pillars together constitute the structural backbone of physical metallurgy and materials science: structure tells you what is there, phase thermodynamics tells you what should be there at equilibrium, and kinetics tells you what you actually get in a finite process time.

1. At a glance

Materials properties derive from structure (atomic arrangement), composition, and processing. The PSPP tetrahedron (Processing ↔ Structure ↔ Properties ↔ Performance) is the canonical reasoning frame in materials science. Crystallography defines the geometry of ordered atomic arrangements; phase diagrams define the thermodynamic stability and multi-component equilibria. Together they explain:

• Why steel hardens (γ → martensite, a diffusionless lattice shear).
• Why aluminum alloys age-harden (GP zones + coherent precipitates obstruct dislocations).
• Why ceramics fracture brittlely (ionic/covalent bonding + limited slip systems).
• Why semiconductors conduct selectively (band-gap engineering grounded in periodic potentials).
• Why nickel superalloys creep so little (γ’ L1₂ precipitates in γ FCC matrix; coherency).
• Why glass is amorphous (kinetic suppression of crystallization below T_g).

This note collapses the introductory through intermediate content of a standard graduate physical-metallurgy + crystallography sequence (Callister, Reed-Hill, Porter & Easterling, Hammond) into a working reference.

2. Crystal systems + Bravais lattices

A crystal is a solid with long-range translational order. The translational symmetry is captured by a Bravais lattice — an infinite array of points generated by integer linear combinations of three primitive vectors a, b, c.

2.1 The 7 crystal systems

Defined by the symmetry constraints on the unit-cell parameters (a, b, c, α, β, γ):

System	Constraints	Example
Triclinic	a ≠ b ≠ c; α ≠ β ≠ γ ≠ 90°	K₂Cr₂O₇
Monoclinic	a ≠ b ≠ c; α = γ = 90°, β ≠ 90°	β-Sn (alternative setting), monoclinic ZrO₂
Orthorhombic	a ≠ b ≠ c; α = β = γ = 90°	α-S (orthorhombic sulfur), Ga, Fe₃C
Tetragonal	a = b ≠ c; α = β = γ = 90°	β-Sn, TiO₂ rutile, martensite (BCT)
Trigonal/rhombohedral	a = b = c; α = β = γ ≠ 90°	α-Al₂O₃ (corundum), calcite, quartz
Hexagonal	a = b ≠ c; α = β = 90°, γ = 120°	Mg, Zn, Ti-α, graphite, ZnO wurtzite
Cubic	a = b = c; α = β = γ = 90°	Fe-α (BCC), Cu/Al/Ni (FCC), NaCl, diamond

2.2 The 14 Bravais lattices (Bravais, 1850)

Within the 7 crystal systems, only 14 distinct translation lattices satisfy the symmetry constraints:

• P — primitive (one lattice point per cell).
• I — body-centered (additional point at ½, ½, ½).
• F — face-centered (additional points at ½, ½, 0; ½, 0, ½; 0, ½, ½).
• C (or A/B) — base-centered (one face only).

Enumerated: triclinic P; monoclinic P, C; orthorhombic P, C, I, F; tetragonal P, I; trigonal R + hexagonal P (sharing the hexagonal system); cubic P, I, F. That gives 14.

2.3 Point groups, space groups, notation

• 32 crystallographic point groups — the distinct combinations of rotation, reflection, inversion, and rotoinversion symmetries compatible with periodic translation (Hessel 1830, Bravais 1849).
• 230 space groups — point groups combined with the translation lattice plus glide planes and screw axes (Fedorov 1891, Schoenflies 1891, Barlow 1894).
• Schoenflies notation (D₄ₕ, O_h, etc.) — favored by spectroscopy + molecular chemistry.
• Hermann-Mauguin (international) notation (Pm-3m, Fm-3m, P6₃/mmc, I4₁/amd, etc.) — favored by crystallography. The IUCr International Tables for Crystallography Vol. A is the canonical reference.

2.4 Common metal lattices

Lattice	Coordination #	APF	Examples
BCC (Im-3m)	8	0.68	α-Fe (≤912°C), Cr, W, Mo, V, Nb, Ta, K, Na
FCC (Fm-3m)	12	0.74	γ-Fe (912-1394°C), Cu, Al, Ni, Pt, Au, Ag, Pb, γ-Mn
HCP (P6₃/mmc)	12	0.74	Mg, Zn, Cd, Ti-α, Co, Zr, Be

APF = atomic packing fraction. FCC and HCP are both close-packed (74%) but with different stacking sequences (FCC: ABCABC; HCP: ABABAB). The stacking-fault energy (SFE) controls whether a metal slips in FCC or partial-dislocation HCP-like ways — low SFE (austenitic stainless, brasses) → easy twinning + extended dissociated dislocations; high SFE (Al, Ni) → cross-slip dominates.

3. Miller indices

Miller indices (William Hallowes Miller, 1839) are a compact integer labeling of crystallographic planes and directions.

3.1 Planes (hkl)

Take reciprocals of the fractional intercepts the plane makes on the three crystallographic axes, then clear common factors. Parentheses (hkl) denote a specific plane; braces {hkl} denote the family of symmetry-equivalent planes.

3.2 Directions [uvw]

Vector components in the crystallographic basis, reduced to smallest integers. Square brackets [uvw] specific; angle brackets family.

3.3 Hexagonal (hkil) Miller-Bravais

For hexagonal systems, the four-index (hkil) convention enforces i = −(h+k), making the symmetry-equivalent prism planes obviously related: (10-10), (01-10), (-1100), etc.

3.4 Plane spacing d_hkl

For cubic: d_hkl = a / √(h² + k² + l²). For tetragonal: 1/d² = (h² + k²)/a² + l²/c². For hexagonal: 1/d² = (4/3)·(h² + hk + k²)/a² + l²/c². For orthorhombic: 1/d² = h²/a² + k²/b² + l²/c². General triclinic: closed-form using the reciprocal metric tensor (see Cullity & Stock 2014).

3.5 Bragg’s law

nλ = 2 d_hkl · sin θ (W. L. Bragg, 1913; W. H. Bragg + W. L. Bragg Nobel 1915).

n is the diffraction order (almost always taken as 1, with higher orders absorbed into higher-index reflections), λ the wavelength, θ the Bragg angle (half the scattering angle 2θ). Constructive interference from parallel lattice planes of spacing d_hkl produces a diffraction peak.

4. Diffraction

4.1 X-ray diffraction (XRD)

The workhorse of phase identification + structural refinement.

• Bragg-Brentano θ-2θ powder geometry — focusing geometry; flat sample; standard for polycrystalline phase ID.
• Single-crystal diffractometers — 4-circle (κ, ω, φ, 2θ) for structure solution.
• Source wavelengths — Cu Kα (1.5406 Å, most common), Cu Kα₁/Kα₂ doublet, Mo Kα (0.7107 Å, lower absorption, better for heavy elements), Co Kα (1.7902 Å, avoids Fe fluorescence), Cr Kα (2.2909 Å, residual stress).
• Hardware: Bruker D8 (Advance, Discover, Venture), Rigaku SmartLab + MiniFlex, PANalytical / Malvern Empyrean + Aeris, Stoe Stadi P/MP.

4.2 Electron diffraction

• Selected-area electron diffraction (SAED) in TEM — spot patterns from single grains; convergent-beam electron diffraction (CBED) gives 3D point-group/space-group information.
• Electron backscatter diffraction (EBSD) in SEM — Kikuchi-band indexing maps orientation across polycrystals (texture, grain boundaries, misorientation distributions). Vendors: Oxford Instruments AZtec, EDAX OIM/APEX, Bruker Esprit.

4.3 Neutron diffraction

Sensitive to light elements (H, Li, O) that X-rays barely see, and to magnetic structure (magnetic neutron scattering from unpaired electron spins). Facilities: ORNL HFIR + SNS (USA), ILL (Grenoble), J-PARC (Japan), ISIS (UK), FRM-II (Germany), ANSTO (Australia). Two flavors: reactor-source (continuous, monochromated) and spallation (pulsed, time-of-flight).

4.4 Synchrotron XRD

High-brilliance, tunable-wavelength X-rays from storage rings. Used for high-resolution powder diffraction, time-resolved studies (sub-millisecond phase transformations), in-situ mechanical + thermal experiments, pair distribution function (PDF) for amorphous + nanocrystalline materials. Facilities: APS (Argonne), ALS + LCLS (Berkeley + SLAC), NSLS-II (Brookhaven), ESRF-EBS (Grenoble), Diamond Light Source (UK), SOLEIL (France), SPring-8 + SACLA (Japan), MAX IV (Sweden), PETRA III (DESY), SSRL (Stanford).

4.5 Rietveld refinement

Whole-pattern fitting (Hugo Rietveld 1969) — minimizes a weighted sum of squared residuals between observed and calculated powder pattern, refining lattice parameters, atomic positions, site occupancies, thermal parameters, phase fractions, preferred orientation, and instrument broadening simultaneously. Quantitative phase analysis (QPA) of multi-phase mixtures with ~1-2% accuracy in best cases.

4.6 Modern XRD software stack

• GSAS-II (Toby + Von Dreele, APS — open source, Python).
• TOPAS (Bruker; the de-facto industrial Rietveld engine).
• FullProf Suite (Rodríguez-Carvajal, ILL — open source).
• HighScore Plus (Malvern PANalytical).
• Match! + Crystal Impact Diamond (visualization).
• ICDD PDF-4+ (Powder Diffraction File — the canonical reference patterns database, ~500k entries).
• Crystallographic Open Database (COD) + ICSD (Karlsruhe inorganic) + CSD (Cambridge organic, CCDC) — structure databases.

5. Structure factor + intensity

The structure factor F_hkl describes how the atoms within the unit cell coherently scatter:

F_hkl = Σ_j f_j · exp(2πi(h·x_j + k·y_j + l·z_j))

where f_j is the atomic scattering factor of atom j (a function of sinθ/λ for X-rays; a constant for neutrons set by the nuclear scattering length b_j), and (x_j, y_j, z_j) the fractional coordinates.

Observed integrated intensity:

I_hkl ∝ |F_hkl|² · m_hkl · LP(θ) · A(θ) · exp(−2M)

with m_hkl multiplicity, LP the Lorentz-polarization factor (geometry of the diffractometer), A absorption, and exp(−2M) the Debye-Waller factor (thermal vibration damping, M = 8π²<u²>sin²θ/λ²).

Systematic absences arise when F_hkl = 0 identically due to lattice centering or glide/screw symmetry — they are the fingerprint that lets you read the space group off a powder pattern (e.g. BCC reflections require h+k+l even; FCC requires h, k, l all even or all odd; diamond-structure additionally requires h+k+l ≠ 4n+2).

6. Defects

Real crystals are not perfect — they contain a hierarchy of defects that control most engineering properties (yield strength, ductility, diffusion, electronic carrier mobility, etc.).

6.1 Point defects (0-D)

• Vacancy — missing atom; equilibrium concentration n/N ≈ exp(−E_v/kT). For metals near melting, ~10⁻⁴.
• Interstitial — extra atom in an interstitial site (self-interstitial rare in metals due to large E_f; foreign interstitials like C, N, H, O common).
• Substitutional — solute atom replacing solvent on a lattice site.
• Schottky defect — paired cation + anion vacancy in an ionic crystal (preserves charge neutrality).
• Frenkel defect — paired vacancy + interstitial of the same ion.

6.2 Line defects: dislocations (1-D)

Predicted independently by Orowan, Polanyi, and Taylor (all 1934) to reconcile observed yield stresses with theoretical (~G/10) values; directly imaged by TEM in the 1950s. A dislocation is a line discontinuity in the displacement field around which the lattice is reconstructed.

• Edge dislocation — extra half-plane of atoms terminating at the dislocation line; Burgers vector b perpendicular to the line.
• Screw dislocation — helical lattice distortion; b parallel to the line.
• Mixed — most real dislocations are partly edge and partly screw along their length.

Burgers circuit + vector: trace a closed circuit in a perfect crystal, repeat in the defective crystal; the closure failure is b. For metals, b is the shortest lattice translation in the densest direction.

Slip systems = slip plane × slip direction:

Lattice	Slip plane(s)	Slip direction(s)	# systems
FCC	{111}	<110>	12
BCC	{110}, {112}, {123}	<111>	12 (primary) + many secondary
HCP basal	(0001)	<11-20>	3
HCP prismatic	{10-10}	<11-20>	3
HCP pyramidal	{10-11}	<11-20>	6
HCP pyramidal <c+a>	{11-22}	<11-23>	6

Critical resolved shear stress (CRSS) — Schmid’s law: τ_CRSS = σ · cos φ · cos λ (Schmid 1924). Slip begins on the system with the highest Schmid factor when τ reaches CRSS.

Strengthening mechanisms all work by impeding dislocation motion: solid-solution (lattice distortion), grain-boundary (Hall-Petch), precipitation/dispersion (Orowan looping or shearing), strain hardening (dislocation forests + Taylor relation σ_y = α·G·b·√ρ).

6.3 Planar defects (2-D)

• Grain boundaries — interface between misoriented grains; described by 5 macroscopic DOFs (3 misorientation + 2 plane normal). Low-angle (<~15°) = dislocation-array (Read-Shockley); high-angle = coincidence-site lattice (CSL) special boundaries like Σ3 (twin, prevalent in low-SFE FCC).
• Twin boundaries — mirror-symmetric across a plane (annealing twins in brass; deformation twins in low-SFE FCC + HCP + martensitic steels).
• Stacking faults — local deviation from the perfect stacking sequence (ABCABC vs ABCBCA). SFE controls dislocation dissociation (Shockley partials) and twin propensity.
• Phase boundaries + antiphase boundaries (in ordered alloys).

6.4 Volume defects (3-D)

• Precipitates — second-phase particles (intentional, for strengthening — Al-Cu θ’, Ni-base γ’).
• Inclusions — generally unintentional (oxides, sulfides from melting).
• Voids + porosity — from solidification shrinkage, gas evolution, sintering kinetics, creep cavitation.

7. Diffusion

Mass transport in solids governs every microstructural evolution process at elevated temperature.

7.1 Fick’s laws

Fick’s 1st law (steady state): J = −D · ∇c. Flux is proportional to the negative concentration gradient; D is the diffusivity (m²/s).

Fick’s 2nd law (transient): ∂c/∂t = ∇·(D·∇c), reducing to ∂c/∂t = D·∂²c/∂x² for 1-D and constant D.

7.2 Arrhenius temperature dependence

D(T) = D₀ · exp(−Q / (R·T))

with D₀ a pre-exponential (m²/s), Q the activation energy (J/mol), R the gas constant. A semilog plot of D vs 1/T is linear; the slope = −Q/R.

7.3 Diffusion mechanisms

• Vacancy-mediated substitutional diffusion — solvent + substitutional solute hop into adjacent vacancies; rate-limited by vacancy concentration × jump frequency.
• Interstitial diffusion — C, N, H, O in metals; much faster (smaller atoms, more vacant interstitial sites).
• Grain-boundary diffusion — D_gb / D_lattice typically 10³-10⁶ at low T; dominant in fine-grained polycrystals.
• Pipe diffusion — along dislocation cores; intermediate enhancement.
• Surface diffusion — fastest of all; controls early-stage sintering, evaporation-condensation.

7.4 Kirkendall effect

Smigelskas + Kirkendall (1947, Cu-Zn): unequal partial diffusivities of two species in a binary alloy cause a net flux of vacancies and physical migration of inert markers (the Kirkendall plane). Established that substitutional diffusion is vacancy-mediated. Produces Kirkendall porosity if vacancy supersaturation precipitates as voids (a real failure mode in solder joints, plated coatings).

8. Gibbs phase rule

For a system at equilibrium:

F = C − P + 2

with C the number of components, P the number of phases coexisting, F the degrees of freedom. The “+2” counts T and P; at constant P (common in condensed-system phase diagrams) the rule becomes F = C − P + 1.

A single-phase region in a binary T-x diagram (C=2, P=1) has F = 2 (T and composition); a two-phase region (P=2) has F = 1 (pick T, composition is fixed by tie line); a three-phase invariant (eutectic, peritectic) has F = 0 — fixed T and composition.

9. Unary (single-component) systems

9.1 Water

The textbook unary diagram: solid (ice Ih + many high-pressure polymorphs), liquid, vapor. Triple point 273.16 K, 611.657 Pa (the definition of the kelvin until 2019 SI redefinition). Critical point 647.096 K, 22.064 MPa.

9.2 Iron

Iron’s P-T phase diagram is the centerpiece of physical metallurgy because every steel sees it:

• α-Fe (ferrite) — BCC, stable < 912°C (1184 K). Ferromagnetic below Curie 770°C.
• γ-Fe (austenite) — FCC, stable 912-1394°C. Higher solubility for C (up to 2.1 wt% at 1147°C eutectic).
• δ-Fe — BCC again, stable 1394-1538°C melting.
• ε-Fe — HCP, stable only at very high pressure (>~13 GPa); inferred to be the structure of Earth’s inner core.

The α ↔ γ transformation drives the entire heat-treatment toolkit for steels (austenitize, quench, temper).

9.3 Polymorphism + allotropy

Different crystal structures of the same composition. Polymorphism = general term; allotropy = same for an element. Examples: α-Fe / γ-Fe / δ-Fe / ε-Fe; α-quartz / β-quartz / tridymite / cristobalite / stishovite / coesite (SiO₂); diamond / graphite / lonsdaleite / fullerenes / nanotubes / graphene (C); α-Sn (gray, diamond, semiconductor) / β-Sn (white, tetragonal, metal) — the “tin pest” transformation at 13.2°C.

10. Binary phase diagrams

The richest part of equilibrium thermodynamics in materials science. Read by combining the lever rule (mass balance between two phases on a tie line) with the Gibbs phase rule (degrees of freedom). Classical types:

10.1 Isomorphous

Complete liquid + solid solubility across the whole composition range. The two endpoint metals share crystal structure, similar atomic radii (Hume-Rothery <15%), valences, and electronegativities. Classic systems: Cu-Ni, Au-Ag, Ge-Si, NiO-MgO. Two-phase region between liquidus (above) and solidus (below); lever-rule tie lines give phase fractions.

10.2 Eutectic

A liquid decomposes into two distinct solid phases simultaneously at a fixed eutectic temperature + composition (F = 0):

L → α + β at T_E

Classic systems: Pb-Sn (61.9 wt% Sn, 183°C — the historical solder eutectic), Al-Si (12.6 wt% Si, 577°C — casting aluminum), Ag-Cu (28.1 wt% Cu, 779°C — brazing fillers), Bi-Sn (low-melt eutectic), Cu-Ag.

10.3 Peritectic

A liquid + a solid react to form a new solid at fixed T (F = 0):

L + α → β at T_P

Classic example: Fe-C at 1493°C, where δ-ferrite + L → γ-austenite. Peritectics are notoriously hard to traverse to equilibrium because the product phase β surrounds α and chokes off further reaction (peritectic envelope, microsegregation).

10.4 Eutectoid

Solid-state analog of eutectic — one solid decomposes into two solids:

γ → α + Fe₃C at 727°C (Fe-Fe₃C — the pearlite reaction)

Eutectoid 0.77 wt% C in steel. The microstructure is pearlite, alternating lamellae of α-ferrite and Fe₃C (cementite).

10.5 Monotectic

A liquid decomposes into a different liquid + a solid (miscibility gap in the liquid):

L₁ → L₂ + α

Classic system: Cu-Pb (immiscibility above the monotectic). Relevant for bearing alloys and for some cast irons.

10.6 Order-disorder transitions

A random solid solution orders into a superlattice on slow cooling. Classic: Cu-Au L1₂ ordering at Cu₃Au composition, and Fe-Al B2 → DO₃ ordering. Critical T_c is a second-order or weakly first-order transition (Bragg-Williams 1934).

10.7 Intermetallic compounds

Stoichiometric or near-stoichiometric phases with distinct crystal structures, often line compounds in the binary diagram. Examples: Ni₃Al (γ’, L1₂ — the strengthening phase of Ni-superalloys), TiAl (γ-TiAl, L1₀ — aerospace), NiTi (B2, the shape-memory Nitinol), Fe₃C (cementite), Mg₂Si, Laves phases (MgZn₂ C14, MgCu₂ C15, MgNi₂ C36 — common in superalloys, often deleterious topologically close-packed phases).

10.8 Solidus / liquidus / solvus

• Liquidus — boundary above which everything is liquid.
• Solidus — boundary below which everything is solid.
• Solvus — solubility limit of a solute in a solid solvent (sets max solid-solution composition; below it precipitates form).

The solvus is the line you traverse when designing a precipitation-hardening (age-hardening) heat treatment.

11. The iron-carbon (Fe-Fe₃C) diagram

The foundational phase diagram of physical metallurgy. Technically a metastable diagram — the truly stable iron-carbon system is Fe-graphite — but Fe₃C is so kinetically favored at the relevant temperatures that the metastable diagram is the engineering reference for steels. Gray cast iron uses the stable (graphitic) eutectic; white cast iron freezes to the metastable (cementite) eutectic.

11.1 Key phases

• α-Ferrite — BCC, max C solubility 0.022 wt% at 727°C. Soft, ductile, ferromagnetic.
• γ-Austenite — FCC, max C solubility 2.11 wt% at 1147°C. Non-magnetic, soft at T, dissolves more C than ferrite by ~100×.
• δ-Ferrite — BCC, stable 1394-1538°C, max C 0.09 wt%.
• Cementite (Fe₃C) — orthorhombic, 6.67 wt% C; very hard (~800 HV) and brittle.
• Pearlite — eutectoid two-phase aggregate of α + Fe₃C lamellae, formed by γ → α + Fe₃C at 727°C, 0.77 wt% C.
• Ledeburite — eutectic aggregate γ + Fe₃C, formed by L → γ + Fe₃C at 1147°C, 4.3 wt% C (basis of white cast iron).

11.2 Steel classifications by composition

• Hypoeutectoid (<0.77 wt% C) — pro-eutectoid α + pearlite.
• Eutectoid (≈0.77 wt% C) — 100% pearlite.
• Hypereutectoid (0.77-2.11 wt% C) — pro-eutectoid Fe₃C (grain-boundary cementite) + pearlite.
• Cast iron (>2.11 wt% C, usually 2-4 wt%) — gray (graphite flakes) / ductile (graphite nodules) / malleable (annealed) / white (Fe₃C eutectic).

11.3 White vs gray cast iron

White irons solidify on the metastable Fe-Fe₃C eutectic — very hard, brittle, wear-resistant (used for high-wear surfaces). Gray irons solidify on the stable Fe-graphite eutectic — graphite flakes act as crack initiators but provide damping and machinability. Ductile (nodular) iron uses Mg or Ce inoculation to nucleate graphite as spheroids instead of flakes, dramatically improving toughness (invented Morrogh + Williams 1948, INCO Millis 1948).

12. Ternary phase diagrams

Three components → 4-dimensional thermodynamic space (T + 2 independent composition variables, since x_A + x_B + x_C = 1). Visualized as:

• Gibbs composition triangle — equilateral, vertices = pure components. Isothermal sections show phase regions at one T.
• Liquidus projection — top-down view of the liquidus surface onto the composition triangle; lines = univariant valleys (binary eutectics extending into ternary); points = ternary invariants.
• Tie triangles — three-phase regions in an isothermal section; phase fractions by the ternary lever rule.

Examples:

• Al₂O₃-SiO₂-ZrO₂ — refractory ceramics, mullite + zirconia formation.
• Fe-Cr-Ni — stainless steels (austenitic 304/316, duplex, ferritic).
• Cu-Ni-Zn — nickel silvers.
• Ti-Al-V — Ti-6Al-4V and other aerospace Ti alloys.

Real engineering alloys are multi-component (7+ in a typical Ni-superalloy), and the only practical way to compute their phase equilibria is CALPHAD.

13. CALPHAD (CALculation of PHAse Diagrams)

Founded by Larry Kaufman + Harold Bernstein (1970 book Computer Calculation of Phase Diagrams). CALPHAD parameterizes the Gibbs energy of each phase as a function of composition and temperature (using sub-lattice models, Redlich-Kister polynomials, magnetic Inden-Hillert-Jarl models, etc.), fits the parameters to experimental + ab-initio data, and minimizes total G to compute equilibrium phase fractions, compositions, driving forces.

13.1 Software

• Thermo-Calc (Thermo-Calc Software AB, Sweden) — industry standard; modules: TC-Python, DICTRA (diffusion), TC-PRISMA (precipitation), Property models.
• Pandat (CompuTherm, USA) — competitive feature parity; integrated with PanDiffusion, PanPrecipitation.
• FactSage (Thermfact + GTT-Technologies) — strong in slag + oxide systems.
• MTDATA (NPL UK).
• JMatPro (Sente Software) — fast property prediction for steels + Al + Ni alloys; uses CALPHAD database internally.
• OpenCalphad (Hillert + Sundman, open source).
• PyCalphad (Otis + Liu, open-source Python toolkit; integrates with ESPEI parameter assessment).

13.2 Major databases

TCFE / TCNI / TCAL / TCMG / TCSLD / TCTI / TCCU / TCSI / TCAUS (Thermo-Calc); PanIron / PanNickel / PanAluminum / PanTitanium (Pandat); FSstel / FTlite / FToxid (FactSage). The CALPHAD assessment community produces thousands of binary + ternary assessments per year, published primarily in Calphad journal (Pergamon → Elsevier).

13.3 Applications

• Alloy design — predict phase fractions, solidification path, freezing range, solvus temperature, scheil simulation for casting microsegregation.
• High-entropy alloys (HEAs) — multi-principal-element alloys (Cantor 2004, Yeh 2004); CALPHAD predicts single-phase vs multi-phase, σ + Laves phase formation.
• Additive manufacturing — predict solidification cracking susceptibility, residual stress, partitioning during rapid solidification.
• Welding metallurgy — HAZ phase prediction, Schaeffler/WRC-1992 diagram replacement.
• Hot-cracking criteria — Scheil + freezing-range based.

14. Kinetics + time-temperature

Equilibrium phase diagrams tell you what should be there given infinite time. Real heat treatments take seconds to hours, and the kinetic outcome is generally far from equilibrium. Two complementary diagrams capture this for steels (and increasingly for Al, Ti, Ni alloys with the same formalism).

14.1 TTT (Time-Temperature-Transformation) diagram

For a given austenitized steel, plot the time required to reach a given fraction of transformation (typically 1% start, 50% mid, 99% finish) versus isothermal hold temperature. The result is a set of C-curves — fast transformation at intermediate undercooling (compromise between driving force and atomic mobility), slow at small undercooling (low ΔG) and large undercooling (low D). Pioneered by Davenport + Bain at U.S. Steel (1930s).

Typical steel TTT shows two distinct C-curves:

• Upper bay — diffusional pearlite (lamellar α + Fe₃C, rate-controlled by C diffusion).
• Lower bay — bainite (acicular α plates + carbides, intermediate kinetics, displacive + diffusional).
• Below the M_s (martensite start) temperature, the diffusionless martensite transformation begins on cooling — not on isothermal hold — and proceeds athermally as T drops to M_f (finish).

14.2 CCT (Continuous Cooling Transformation) diagram

Practical heat treatments cool continuously, not isothermally. CCT diagrams replot the same kinetics in terms of cooling-rate trajectories. Critical cooling rate to fully bypass pearlite + bainite and produce 100% martensite is a key design parameter.

14.3 Hardenability

The depth to which a steel can be hardened by quenching. Quantified by the Jominy end-quench test (ASTM A255): a 25.4 × 102 mm round bar is austenitized and water-quenched from one end; hardness is then measured along the length. Hardenability rises with alloying elements that suppress diffusional transformations (Mn, Cr, Mo, Ni, V, B). Stein + Grossmann ideal-diameter calculations + DI (ideal diameter) values feed into selection. See [[Engineering/Tier3/steel-grades]] for grade-by-grade hardenability data.

14.4 Martensite

Bain (1924) — a diffusionless, displacive, shear transformation of FCC austenite to a body-centered-tetragonal (BCT) lattice that traps the dissolved C in octahedral sites. The C-trapping is what makes it hard (and brittle as-quenched). M_s drops with C and most alloying additions; for high-C steels M_s can drop below room temperature, requiring cryogenic treatment to complete the transformation.

Microstructurally, martensite is:

• Lath martensite — packets of parallel laths in low-C (<~0.6 wt%) steels.
• Plate martensite — lenticular plates in high-C steels; high residual austenite content possible.

14.5 Tempering

The brittle as-quenched martensite is reheated below A_c1 (~600-650°C max) to allow controlled decomposition: ε-carbide → cementite + ferrite; trapped C precipitates progressively; dislocation density decreases. Result: a tempered-martensite or tempered-bainite microstructure with tunable strength/toughness balance. Tempered-martensite embrittlement (TME, 250-350°C) and temper embrittlement (375-575°C, slow cooling, P + Sb + Sn segregation to prior-γ grain boundaries) are notorious failure-mode windows to avoid.

15. Solidification + microstructure

The casting + solidification step sets the as-cast grain structure and segregation profile, which subsequent thermomechanical processing can refine but never fully erase.

15.1 Constitutional supercooling

Tiller, Jackson, Rutter, Chalmers (1953): solute rejected at the solid-liquid interface raises the liquidus locally, so the liquid just ahead of the interface is below its (local) liquidus even though it is above the bulk T_L. The instability criterion is G/v < m·C₀(1−k)/(k·D), where G is the thermal gradient, v interface velocity, m the liquidus slope, k partition coefficient, D solute diffusivity. Below this criterion the interface destabilizes from planar → cellular → dendritic.

15.2 Solidification morphologies

• Planar — high G/v, low alloying. Pure metals, low-supercooling castings.
• Cellular — moderate G/v.
• Dendritic — low G/v, high alloying; columnar-to-equiaxed transition (CET) in castings; primary + secondary + tertiary arms; arm spacing scales as λ ~ (G·v)^(−n) with n ~ 0.3-0.5.

15.3 Eutectic morphology

Regular (lamellar or rod) vs irregular (flake graphite in gray iron, divorced eutectic in Al-Si modified with Sr). Jackson + Hunt (1966) coupled-growth theory.

15.4 Microsegregation + macrosegregation

• Microsegregation — solute partitioning during dendritic solidification leaves dendrite cores solute-lean and interdendritic regions solute-rich. Quantified by the Scheil-Gulliver equation: C_S = k·C₀·(1−f_S)^(k−1), assuming complete liquid mixing, no solid back-diffusion.
• Coring — the resulting compositional variation across a dendrite, visible in optical micrographs after etching.
• Macrosegregation — compositional gradient on the scale of the casting itself, driven by interdendritic flow, solidification shrinkage, sedimentation of free crystals.

See [[Engineering/Tier3/casting-processes]] for processing-side detail.

16. Precipitation hardening + age hardening

Discovered by Alfred Wilm (1906, Duralumin) when he accidentally let a quenched Al-Cu-Mg alloy age at room temperature and found it hardened over days. Now the dominant strengthening mechanism for Al, Mg, Ti, Ni, Cu-Be, and precipitation-hardening stainless steels.

16.1 Process

1. Solution anneal — heat above the solvus, dissolve all alloying elements into single-phase solid solution.
2. Quench — rapid cool to lock in supersaturated solid solution + excess vacancies.
3. Age — hold at intermediate T (room T to ~250°C for Al; ~700-900°C for Ni-superalloys) to nucleate and grow a fine dispersion of coherent or semi-coherent precipitates.

16.2 Precipitation sequence (Al-Cu, the canonical case)

SSSS → GP zones (Guinier + Preston 1938, copper-rich monolayer discs on {100}_Al) → θ” (coherent BCT) → θ’ (semi-coherent) → θ (Al₂Cu, incoherent, BCT/CT C16). Peak hardness at θ’/θ” mixture (T6 temper). Overaging coarsens to θ and softens.

16.3 Dislocation-precipitate interaction

• Shearing — small coherent precipitates are cut by passing dislocations (work-hardens, increases ductility).
• Orowan looping (Orowan 1948) — large incoherent precipitates are bypassed, leaving dislocation loops around each particle; τ_Orowan = G·b / L (L = inter-particle spacing).

Peak strength is at the crossover.

16.4 Engineering examples

• Al 2024 (Al-Cu-Mg) — T3/T351/T4 natural age; T6/T81 artificial age. Aircraft skin.
• Al 7075 (Al-Zn-Mg-Cu) — peak T6; T73/T74 overaged for SCC resistance.
• 17-4 PH stainless (Fe-Cr-Ni-Cu) — Cu precipitation, H900-H1150 tempers.
• Maraging steels (18 Ni 250/300/350) — intermetallic Ni₃(Ti, Mo) precipitation in low-C martensite matrix.
• Inconel 718 — γ” (Ni₃Nb, D0₂₂) primary + γ’ (Ni₃(Al,Ti), L1₂) secondary; the workhorse aerospace Ni-superalloy.
• Single-crystal turbine blades (CMSX-4, René N5, CMSX-10) — 60-70 vol% γ’ (Ni₃Al-based) cuboids in γ matrix.

See [[Engineering/Tier3/aluminum-alloys]] and [[Engineering/materials-steel]] for grade-level detail.

17. Recovery, recrystallization, grain growth

Cold-worked metal stores ~5-10% of the input plastic work as elastic strain energy in the dislocation forest (ρ ~ 10¹⁵-10¹⁶ m⁻² after heavy cold work). Annealing reduces this energy in three sequential regimes:

1. Recovery — dislocation rearrangement into low-angle subgrain boundaries; some annihilation; minor hardness drop, residual stress relief.
2. Recrystallization — nucleation + growth of new strain-free grains, consuming the deformed matrix. Driven by stored energy. Avrami kinetics (JMAK 1939-41). The recrystallization temperature scales as ~0.4 T_m for pure metals, ~0.5-0.6 T_m for alloys.
3. Grain growth — once recrystallized, grains coarsen to reduce total boundary area; d^n − d₀^n = K·t (n typically 2-3).

17.1 Hall-Petch grain-size strengthening

σ_y = σ₀ + k · d^(−1/2)

Hall (1951) + Petch (1953). σ₀ is the friction stress (single-crystal lattice resistance + solid-solution), k the Hall-Petch slope (material-dependent, higher for low-symmetry crystals + low-SFE alloys), d the grain size. Hall-Petch holds robustly from millimeter down to ~50 nm; at finer grain sizes inverse Hall-Petch (softening) is observed due to grain-boundary sliding + diffusion-mediated mechanisms.

18. Electronic-structure introduction

A full treatment lives in [[MaterialsScience/electronic-structure-band-theory]] (future Tier 1 note); this is the orientation.

Electrons in a periodic potential have Bloch wavefunctions ψ_k(r) = exp(i k·r) · u_k(r) with u_k periodic. Energy eigenvalues form bands E_n(k) over the Brillouin zone. The number of filled bands and the band gap (energy between highest filled and lowest empty bands at 0 K) classify materials:

• Metal — partially filled band, no gap. Cu, Al, Fe, Au.
• Semimetal — bands touch at isolated points/lines (Bi, Sb, graphene Dirac points).
• Semiconductor — narrow gap (≤~3 eV), thermally + dopant-accessible. Si (1.12 eV), Ge (0.66), GaAs (1.42), SiC (3.0-3.3), GaN (3.4).
• Insulator — wide gap (≥~4 eV). Al₂O₃ (~8.8 eV), SiO₂ (~9 eV), diamond (5.5).
• Superconductor — gap of Cooper-paired condensate below T_c (BCS, 1957).

DFT computes electronic structure ab initio (Hohenberg + Kohn 1964; Kohn + Sham 1965; Kohn Nobel 1998). Standard software: VASP (Vienna Ab-initio Simulation Package, plane-wave PAW), Quantum Espresso (open source, plane-wave), ABINIT (open source), CASTEP (Cambridge, plane-wave PAW), FHI-aims (NAO all-electron, Berlin), ORCA (molecular, post-HF + DFT, Neese), NWChem (PNNL), Gaussian (molecular). For very large systems, CP2K (mixed Gaussian-plane wave) and SIESTA (NAO). Modern functionals: PBE (GGA, default), SCAN (meta-GGA), HSE06 + PBE0 (hybrids, better gaps), GW + BSE (many-body, accurate gaps + optical).

19. Modern + ML for materials (2024-2026)

The data-driven revolution of the past decade has shifted materials discovery from one-by-one synthesis to high-throughput simulation + ML + autonomous experimentation.

19.1 Databases

• Materials Project (Ceder/Persson Berkeley/LBNL, 2011+) — ~150k DFT-computed inorganic compounds; pymatgen + custodian + atomate + FireWorks tooling.
• OQMD (Open Quantum Materials Database, Wolverton Northwestern) — ~1M DFT entries.
• AFLOWlib (Curtarolo Duke) — ~3.5M entries, strong on alloys + Heusler compounds.
• NOMAD Repository + Archive (Scheffler FHI Berlin) — federated raw + curated DFT data.
• Materials Cloud (Marzari EPFL) — Quantum Espresso + AiiDA-driven workflows.
• Citrination (Citrine Informatics) — commercial materials informatics platform.
• Open Catalyst Project (OC20/OC22/OCx24) (Meta + CMU) — ~265M DFT relaxations for catalysis; ML benchmark.

19.2 ML potentials + foundation models

• SchNet (Schütt 2017) — early continuous-filter convolutional network.
• M3GNet (Chen + Ong UCSD 2022) — universal graph network potential.
• CHGNet (Deng + Ceder Berkeley 2023) — charge-aware graph network.
• MACE (Batatia + Csányi Cambridge 2022) — high-order equivariant message passing.
• Allegro (Musaelian + Kozinsky Harvard 2023) — local equivariant, strict locality.
• NequIP (Batzner + Kozinsky 2022) — E(3)-equivariant.
• Equiformer / EquiformerV2 (Liao + Smidt MIT/Google 2022/2023) — equivariant transformers; top of OC20 leaderboard.
• MatterSim (Microsoft Research AI4Science 2024) — universal MLP across the periodic table, T- + P-aware.
• Orb (Orbital Materials 2024) — production-targeted universal MLP.

19.3 Generative + autonomous discovery

• GNoME (Google DeepMind, Merchant et al. Nature 2023) — graph-network active-learning loop predicted 2.2 million new candidate stable inorganic crystals; ~380k below convex hull; verified subset.
• MatterGen (Microsoft, Zeni et al. preprint 2024 → Nature 2025) — diffusion-based generative model for stable + property-conditioned inorganic crystals.
• A-Lab (Berkeley, Szymanski + Ceder Nature 2023) — autonomous synthesis robot that made and characterized 41 novel inorganic compounds in 17 days.
• Acceleration Consortium (Aspuru-Guzik, Toronto / Mila) — broader self-driving-lab framework.
• Aionics, Citrine, Kebotix, Uncountable, Polymathic — commercial autonomous + informatics players.

19.4 CALPHAD + ML hybrids

ML used to fit Gibbs-energy models from DFT + experiment (ESPEI; Otis 2017), to assess uncertainty in CALPHAD parameters, and to extrapolate to high-entropy multicomponent space where assessments are sparse.

20. Common pitfalls

• Confusing equilibrium phase diagram with kinetic outcome. The Fe-Fe₃C diagram says pearlite forms below 727°C; reality is that a fast quench gives bainite or martensite instead. The phase diagram is a constraint, not a prediction of microstructure.
• Lever-rule misuse off a tie line. Lever rule applies only within a two-phase region along an isothermal tie line. Applying it across three-phase regions or to non-equilibrium structures is a common student error.
• Ignoring cooling rate. A nominal “anneal” or “normalize” without specifying cooling rate makes the resulting microstructure ambiguous; CCT diagram is needed.
• Off-stoichiometry shifts intermetallic stability. Slight deviation from line-compound stoichiometry can destabilize the phase entirely (e.g. Ni₃Al composition window).
• GGA-DFT underestimates band gaps (typically by 30-50%). Use hybrid functionals (HSE06, PBE0) or GW for quantitative gaps; PBE is fine for total energies + relaxed structures but quantitatively wrong for excited-state properties.
• Treating ML-potential predictions as ground truth outside the training distribution. Universal MLPs are sharply more accurate inside their training set than in extrapolation regimes (e.g. exotic stoichiometries, magnetic transitions, charged defects).
• Conflating polymorphism with allotropy. Allotropy is reserved for elements; polymorphism is the general term for compounds + elements alike.
• Forgetting metastability of Fe₃C. Cementite is not the thermodynamic ground state of Fe-C; graphite is. Long-time exposure at elevated T (graphitization in pearlitic steel piping after decades in service) is a real, slow degradation mode.

21. Cross-references

• [[MaterialsScience/_index]]
• [[Engineering/_index]]
• [[Engineering/materials-steel]], [[Engineering/materials-aluminum]], [[Engineering/materials-polymers]], [[Engineering/materials-composites]], [[Engineering/materials-ceramics]]
• [[Engineering/Tier3/steel-grades]], [[Engineering/Tier3/aluminum-alloys]]
• [[Engineering/Tier3/casting-processes]] (solidification + segregation)
• [[Engineering/Tier3/surface-treatments]] (heat-treatment, tempering, induction hardening)
• [[Engineering/Tier3/semiconductor-materials]] (band structure in practice)
• [[Math/eigenvalue-problems]] (band-structure diagonalization, stiffness + elasticity tensors, reciprocal-space transforms)
• Future Tier-1 partners: [[MaterialsScience/electronic-structure-band-theory]], [[MaterialsScience/mechanical-behavior-fundamentals]], [[MaterialsScience/characterization-techniques]], [[MaterialsScience/computational-materials]].

22. Citations

• Callister, W. D., Rethwisch, D. G. Materials Science and Engineering: An Introduction, 10th ed., Wiley, 2018.
• Reed-Hill, R. E., Abbaschian, R., Abbaschian, L. Physical Metallurgy Principles, 4th ed., Cengage, 2009.
• Porter, D. A., Easterling, K. E., Sherif, M. Y. Phase Transformations in Metals and Alloys, 4th ed., CRC Press, 2021.
• Hammond, C. The Basics of Crystallography and Diffraction, 4th ed., Oxford / IUCr Texts on Crystallography, 2015.
• Cullity, B. D., Stock, S. R. Elements of X-Ray Diffraction, 3rd ed., Pearson, 2014.
• Kelly, A., Knowles, K. M. Crystallography and Crystal Defects, 3rd ed., Wiley, 2020.
• ASM Handbook Vol. 3: Alloy Phase Diagrams, ASM International, 1992 (and ASM Alloy Phase Diagram Center online updates).
• Hillert, M. Phase Equilibria, Phase Diagrams and Phase Transformations: Their Thermodynamic Basis, 2nd ed., Cambridge, 2007.
• Lukas, H. L., Fries, S. G., Sundman, B. Computational Thermodynamics: The Calphad Method, Cambridge, 2007.
• Kaufman, L., Bernstein, H. Computer Calculation of Phase Diagrams, Academic Press, 1970.
• Bragg, W. L. “The diffraction of short electromagnetic waves by a crystal.” Proc. Cambridge Phil. Soc. 17 (1913) 43.
• Rietveld, H. M. “A profile refinement method for nuclear and magnetic structures.” J. Appl. Cryst. 2 (1969) 65-71.
• Kohn, W., Sham, L. J. “Self-consistent equations including exchange and correlation effects.” Phys. Rev. 140 (1965) A1133.
• Hohenberg, P., Kohn, W. “Inhomogeneous electron gas.” Phys. Rev. 136 (1964) B864.
• Merchant, A. et al. (GNoME, DeepMind). “Scaling deep learning for materials discovery.” Nature 624 (2023) 80-85.
• Zeni, C. et al. “MatterGen: a generative model for inorganic materials design.” Nature 639 (2025).
• Szymanski, N. J. et al. (A-Lab, Berkeley). “An autonomous laboratory for the accelerated synthesis of novel materials.” Nature 624 (2023) 86-91.
• Schmid, E., Boas, W. Plasticity of Crystals, F. A. Hughes, 1950 (original 1935).
• Hall, E. O. Proc. Phys. Soc. B 64 (1951) 747; Petch, N. J. J. Iron Steel Inst. 174 (1953) 25.
• Bain, E. C. “The nature of martensite.” Trans. AIME 70 (1924) 25.
• IUCr International Tables for Crystallography, Vol. A (Space-Group Symmetry), 6th ed., Wiley/IUCr, 2016.
• ICDD Powder Diffraction File (PDF-4+), International Centre for Diffraction Data, current release.