Foundations

Silicate structures organize silicon–oxygen tetrahedra as isolated units, chains, sheets or frameworks.

Rock can begin melting without every component entering the liquid.

Crystallizing minerals remove selected chemical components from a melt.

Metamorphism changes mineral assemblage and texture under changed pressure, temperature, fluid and deformation conditions without requiring bulk melting.

Mechanical weathering changes fragment size; chemical weathering transforms or dissolves minerals through reactions such as hydrolysis and oxidation.

A distribution coefficient relates one species' concentrations in two specified coexisting phases.

In homogeneous constant-D batch melting, mass balance and equilibrium give melt enrichment Cℓ/C₀=1/[D+F(1−D)].

A petrographic microscope combines observations in plane-polarized and crossed-polarized light: relief, colour or pleochroism, cleavage, extinction behaviour and interference response help distinguish minerals.

In a suitable transmitted-light grain–medium boundary test, increasing the objective-to-specimen separation moves the Becke bright line into the material of higher refractive index.

For a homogeneous plane-parallel transparent plate at normal incidence with two effective indices for the common wave-normal direction, optical path retardation is Γ=d(n_slow−n_fast).

The optical indicatrix represents directional refractive behaviour in an ideal transparent anisotropic crystal.

For two equilibrium phases α and β with compositions x_α<x_β and bulk composition x₀ between them, conservation gives f_α=(x_β−x₀)/(x_β−x_α) and f_β=1−f_α.

In a simple binary eutectic system at fixed pressure, liquid can coexist with two distinct solid phases at the eutectic temperature and liquid composition.

In a binary peritectic equilibrium at fixed pressure, an earlier solid and liquid react to form a different solid at an invariant temperature.

A solid solution accommodates component substitution in one crystal phase.

Three-component compositions occupy a normalized triangle: x_A+x_B+x_C=1 with nonnegative fractions.

For closed mixing by mass of two endmembers, concentration is C_mix=fC₁+(1−f)C₂.

For parent P decaying into daughter D at constant λ, normalization to a stable isotope S gives D/S=(D/S)₀+(P/S)(e^(λt)−1).

For a closed system with corrected common lead and retained decay-chain products, radiogenic ratios satisfy ²⁰⁶Pb*/²³⁸U=e^(λ₂₃₈t)−1 and ²⁰⁷Pb*/²³⁵U=e^(λ₂₃₅t)−1.

Magma ascent can lower confining pressure enough for dissolved volatiles to exsolve and expand.

A mineral assemblage at specified pressure, temperature and bulk composition minimizes Gibbs energy subject to component balance.

Combining mineral phase fractions with elastic properties can predict seismic velocities, but equilibrium databases and shear-modulus data have separate calibration domains.

An apparent thermochronological age depends on diffusion kinetics, grain geometry and cooling history.

U–Pb discordance can reflect common lead, radiogenic lead loss, initial daughter disequilibrium or mixtures of differently aged growth zones.

Reactive transport couples species fluxes to kinetic/equilibrium reactions while enforcing elemental and charge balance.

Richards flow conserves water in a variably saturated porous medium while Darcy flux uses saturation-dependent permeability.

Coupled groundwater and deformation models account for pore pressure, body forces, elastic response and boundary conditions.

Shared stream-power models combine detachment and transport limits instead of identifying sediment flux with erosion.

Ice thermal models combine advection, diffusion and deformation heating; enthalpy tracks sensible heat and liquid fraction at the pressure-melting boundary.

Mode-I stress intensity depends on load and crack geometry.