Foundations

A periodic crystal potential organizes energies by wavevector and band.

Band occupancy and doping control mobile carriers.

Ionic, metallic, molecular and network solids organize common bonding patterns.

Polymers are substances composed of macromolecules with recurring structural units.

Ligand donor atoms surround a central metal in a coordination sphere.

Macroscopic electric polarization P gives bound volume charge ρ_b=−∇·P and surface charge σ_b=P·n at a material boundary.

A damped bound-electron oscillator gives ε_r(ω)=1+Ne²/[ε₀m(ω₀²−ω²−iγω)] for the exp(−iωt) convention.

A Bravais lattice consists of translations R=Σ_i n_i a_i with integer n_i and independent primitive vectors.

Reciprocal vectors obey a_i·b_j=2πδ_ij and G=Σ_i h_i b_i with integer h_i.

For a cell basis, the kinematic elastic scattering amplitude contains F(G)=Σ_α f_α(G)exp(iG·d_α), and peak intensity is proportional to |F(G)|² with experiment-specific factors.

A monatomic chain with mass m, spacing a and nearest-neighbour spring constant κ has ω(k)=2√(κ/m)|sin(ka/2)|.

For a stable three-dimensional harmonic crystal with s atoms per primitive cell, the displacement dynamical matrix has 3s branches.

The three-dimensional Debye approximation assigns g(ω)=9Nω²/ω_D³ for 0≤ω≤ω_D so that the mode count is 3N.

A single electron-carrier Drude model has dp/dt=−e(E+v×B)−p/τ.

In nondegenerate three-dimensional parabolic bands, n=N_c exp[−(E_c−μ)/(k_BT)] and p=N_v exp[−(μ−E_v)/(k_BT)], where N_c,v=2(2πm_c,v*k_BT/h²)³ᐟ² for one isotropic valley including two spins.

A semicrystalline polymer contains ordered crystallites and amorphous material.

For an ideal linear step-growth model, the number-average length Xn counts monomer-derived units per product molecule, including unreacted monomers in the counted population.

Green chemistry evaluates product and process design to reduce hazardous inputs, releases and waste over a stated lifecycle.

Theoretical atom incorporation and actual process material use answer different questions.

Molar-mass dispersity is the ratio of mass-average to number-average molar mass.

Size-exclusion results depend on column, solvent and calibration, so retention-based molar masses are not intrinsically absolute.

Branching models locate gelation when continuing connections become self-sustaining.

A Maxwell spring and dashpot in series predicts exponential stress relaxation with a characteristic time.

For linear harmonic loading, the complex modulus separates in-phase elastic storage from quadrature viscous loss.

Corrected diffraction intensities constrain squared structure-factor amplitudes, while phases remain essential for reconstructing electron density.

Phase probabilities and map weighting affect reconstructed density, and model-derived phases can bias the apparent support for that same model.

The Br2+ attosecond study reports coherent vibronic signals and interprets their evolution using electronic/nuclear modeling with anharmonic and spin–orbit contributions.