Foundations

At normal incidence on a lossless nonmagnetic dielectric interface, the electric-amplitude reflection coefficient is r=(n₁−n₂)/(n₁+n₂).

A nonrelativistic accelerated point charge radiates total power P=q²a²/(6πε₀c³).

For inertial frames in standard configuration with S′ moving at v along S's x axis, x′=γ_v(x−vt), t′=γ_v(t−vx/c²), y′=y and z′=z, where γ_v=(1−v²/c²)⁻¹ᐟ².

Along a timelike worldline in an inertial coordinate frame, dτ=dt√(1−u²/c²), and elapsed proper time is its integral.

With S′ moving at v along x, a particle's velocities obey u′_x=(u_x−v)/(1−vu_x/c²) and u′_y,z=u_y,z/[γ_v(1−vu_x/c²)].

A particle of invariant mass m has p=γ_u m u, E=γ_u mc² and K=(γ_u−1)mc²; E²−|p|²c²=m²c⁴.

For a light source and observer receding directly from each other at relative speed v, f_obs=f_source√[(1−v/c)/(1+v/c)].

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.

For noninteracting spin-½ electrons in a three-dimensional volume, k_F=(3π²n)¹ᐟ³ and E_F=ħ²k_F²/(2m).

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

For independent spin-½ moments of magnitude μ in magnetic flux density B, a moment has energies ±μB, partition sum 2cosh(μB/(k_BT)) and mean moment μtanh(μB/(k_BT)).

Ferromagnetic order aligns moments, antiferromagnetic order has opposing ordered sublattices and ferrimagnetic order has unequal opposing sublattice moments.

For a uniform three-dimensional ideal gas of spinless bosons in the thermodynamic limit, the excited-state capacity reaches n=ζ(3/2)(mk_BT/(2πħ²))³ᐟ² at T_c.

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.

For two Newtonian particles with equal opposite internal force and no external force, the centre of mass moves uniformly.

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

A measurand is the quantity intended to be measured, including the relevant system state, components and conditions.

Accuracy describes closeness of a measured value to a true measurand value; trueness concerns agreement of the limiting replicate mean with a reference value; precision concerns agreement among repeated values under specified conditions.

Measurement error is measured value minus reference value.

An influence quantity changes the relation between instrument indication and measurement result without changing the quantity actually measured in a direct measurement.

Verification provides objective evidence that specified requirements are fulfilled.

An uncertainty budget states the measurement model, input estimates, standard uncertainties, distributions, covariances, sensitivities, evaluation methods and degrees of freedom, together with how they combine and any coverage factor.

For n independent identically distributed observations of a stable quantity with finite variance, the replicate-mean standard uncertainty is estimated by s/√n, where s²=Σ_j(x_j−x̄)²/(n−1).

An assigned rectangular distribution on estimate±a gives u=a/√3, while a symmetric triangular distribution on the same interval gives u=a/√6.

For independent input standard-uncertainty estimates in a linearized model, Welch–Satterthwaite gives ν_eff=u_c⁴/Σ_i[(c_i u_i)⁴/ν_i].