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.

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 primary caesium frequency standard relates its oscillator to the defining unperturbed Cs-133 hyperfine transition by estimating and correcting design-dependent shifts, including atomic-motion Doppler and thermal blackbody effects.

With calibrated light frequency f, vacuum wavelength is λ₀=c/f.

A Kibble balance combines weighing mg=I Bℓ with velocity-mode U=vBℓ to eliminate the common geometry-field factor: mgv=IU.

For a conservative central interaction U(r), relative motion is planar and angular momentum L=μr²θ̇ is conserved.

For a frame rotating at constant Ω about a common inertial origin, a_inertial=a_rot+2Ω×v_rot+Ω×(Ω×r).

For ideal independent holonomic constraints f_α(q,t)=0, constrained Lagrange equations can retain redundant coordinates through d(∂L/∂q̇_i)/dt−∂L/∂q_i=Σ_α λ_α∂_i f_α.

In canonical coordinates, {A,B}=Σ_i[(∂_q_iA)(∂_p_iB)−(∂_p_iA)(∂_q_iB)] and dA/dt={A,H}+∂_tA.

A canonical transformation changes phase-space coordinates and the Hamiltonian while preserving Hamilton's description of the same physical trajectories.

A thin spherical lens in a common surrounding medium has 1/f=(n_lens/n_medium−1)(1/R₁−1/R₂).

For scalar co-polarized fields, averaged intensity includes a cross-correlation term.

For equal-phase coherent illumination of narrow slits separated by d, far-field bright directions satisfy d sinθ=mλ and dark directions d sinθ=(m+½)λ.

A monochromatic scalar field in a uniform medium propagates paraxially as E_z(x,y)≈e^(ikz)/(iλz)∫∫E₀(ξ,η)exp[ik((x−ξ)²+(y−η)²)/(2z)]dξdη, with k=2π/λ.

When the input-aperture quadratic phase in the Fresnel integral is negligible, the far-field amplitude is proportional to the two-dimensional Fourier transform of the aperture field evaluated at spatial frequencies x/(λz), y/(λz).

At normal incidence on a periodic grating of pitch d, principal propagating orders satisfy d sinθ_m=mλ.

A uniform ideal circular pupil of diameter D gives an Airy intensity pattern whose first dark ring has small angle θ≈1.22λ/D.

For a linear shift-invariant scalar imaging system, coherent image amplitude is E_image=E_object*h.

For a narrow-band wave packet on a smooth real dispersion branch, its leading envelope velocity is v_g=dω/dk at the carrier, while phase velocity is ω/k.

For V(x)=−V₀ inside |x|<a/2 and zero outside, bound states have −V₀<E<0.