Foundations

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.

A weak coupling from an initial state to a sufficiently dense continuum gives a leading transition rate Γ≈(2π/ħ)|V_fi|²ρ(E_f) at the energy-conserving final energy, for slowly varying matrix elements and density of states.

Near a stationary equilibrium, a conservative system with quadratic kinetic matrix M and potential Hessian K has Mq̈+Kq=0.

A reference material is sufficiently homogeneous and stable for specified properties and an intended measurement or examination use.

A detection limit depends on the measurement procedure's false-presence probability α and false-absence probability β.

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 nonrelativistic point-Coulomb two-body bound state separates as ψ_nlm=R_nl(r)Y_lm(θ,φ).

An isotope delta compares a specified sample isotope ratio with the same ratio in a stated reference.

An ion's equilibrium potential balances its chemical and electrical driving terms.

The canonical glycolytic pathway converts one six-carbon glucose into two three-carbon pyruvates.

Pyruvate dehydrogenase converts pyruvate to acetyl-CoA with CO₂ release and NADH formation.

A respiratory electron-transport chain couples favourable electron transfers to an electrochemical proton gradient across a membrane.

Fermentation regenerates oxidized electron carriers by transferring electrons to metabolic organic intermediates, permitting substrate-level ATP production without the aerobic respiratory chain.

In oxygenic photosynthesis, photosystems II and I use absorbed light to drive electron transfer from water toward reduction of NADP⁺.

The Calvin–Benson cycle carboxylates ribulose-1,5-bisphosphate through RuBisCO, reduces the resulting carbon compounds using ATP and NADPH, and regenerates the acceptor.

NAD(H) and NADP(H) commonly carry hydride equivalents, while flavins can participate in one- or two-electron chemistry.

For a typical saturated even-chain fatty acyl-CoA, β-oxidation repeats oxidation, hydration, oxidation and thiolytic cleavage.

In the canonical fatty-acid-synthase pathway, ATP-dependent acetyl-CoA carboxylation supplies malonyl units.

Gluconeogenesis makes glucose from suitable noncarbohydrate precursors.

Glycogen synthase uses activated UDP-glucose to extend glucan chains, while branching creates additional nonreducing ends.

The oxidative pentose-phosphate route converts glucose-6-phosphate toward pentose phosphate with CO₂ release and NADPH formation.

PLP-dependent transamination transfers an amino group between an amino acid and a keto acid without requiring free ammonia as the immediate product.

The mammalian urea cycle combines nitrogen from ammonia and aspartate with bicarbonate-derived carbon to form urea.

De novo purine synthesis builds a ring on activated ribose toward IMP, then branches toward AMP and GMP.

In the stated reversible single-substrate Michaelis–Menten model, competitive inhibition increases apparent Km while leaving Vmax unchanged.

Metabolic flux responds to substrate/product availability, allosteric effectors, covalent enzyme modification and cellular demand.