Accelerating from rest to rapidity $\eta$ and braking to rest requires two separate photon-rocket burns. Their mass ratios multiply, giving $M_f/M_i=e^{-2\eta }$. For ideal equal-mass matter–antimatter annihilation, the initial antimatter mass fraction is $(1-e^{-2\eta })/2$. Braking fuel must be carried and accelerated during the first burn.

Two head-on photons can create an electron-positron pair at the limiting product $E_1{E}_2=(m_e{c}^2{)}^2$. A target at the peak of the frequency-density blackbody spectrum has $E_1=2.82144k_{B}T$, whereas the wavelength-density peak corresponds to $E_1=4.96511k_{B}T$. At $T=3\mathrm{K}$ these conventions give incident thresholds approximately $3.58\times 10^{14}\mathrm{eV}$ and $2.03\times 10^{14}\mathrm{eV}$ respectively. A real blackbody has a continuum of target energies and angles, so an absorption calculation integrates the spectrum and cross section rather than imposing a sharp peak-photon opacity boundary.

An acoustic phonon is a quantum of vibration on an acoustic phonon branch. At long wavelengths the atoms in each primitive cell move together and translational invariance makes the frequency approach zero. A three-dimensional crystal has three acoustic branches, corresponding to longitudinal and transverse polarizations. A linear small-wavevector dispersion has slope equal to the appropriate sound speed.

An optical phonon is a quantum of vibration on an optical phonon branch, involving relative motion of atoms within a primitive cell. Its zone-center frequency can remain nonzero. Some such modes couple strongly to light when the relative displacement changes electric polarization; the name does not imply that every optical phonon is optically active. Their finite energies produce thresholds for electron emission and thermally suppressed absorption at low temperature.

The displacement eigenvector of a lattice normal mode, including relative amplitudes and phases for the atoms in a primitive cell. Longitudinal and transverse describe its orientation relative to the phonon wavevector in simple isotropic cases.

Separation of a biomolecular mixture into compositionally distinct phases. Concentration dependence, partitioning and reversibility should be tested; a fluorescent punctum or recovery after bleaching alone does not establish the mechanism.

Formation of finite-scale composition domains when complete macroscopic separation is frustrated, for example by the covalent connection of incompatible blocks in a diblock copolymer. Interfacial cost and chain entropy jointly select the domain scale.

The distance over which a constant phase mismatch accumulates phase difference $\pi$. In unseeded, undepleted sum-frequency generation, the generated intensity first reaches a maximum at this length and returns to zero at $2{\ell }_c$. This length concerns relative phases of interacting waves, rather than the temporal optical coherence length of one source.

For $L>0$, extend $f_L(x)=\sin x-x\sin L/L$ from $[-L,L]$ periodically. The linear term makes both endpoint values zero and the endpoint first derivatives match. The function is odd, so its Fourier series has only sine terms, with the displayed Fourier coefficients. Indeed, for $k_n=n\pi /L\ne 1$,subtracting and multiplying by $2/L$ proves the formula. The coefficients decay as $O(n^{-3})$. At $L=m\pi$, the single coefficient $b_m$ has removable limit $1$, with all others zero. Multiplying $b_n$ by $e^{-\kappa k_n^{2}t}$ solves the corresponding heat equation on the circle.

The proportions of ferrite and cementite within the two-phase pearlite microstructure. A lever rule on a weight-percent-carbon diagram gives mass fractions; converting those to volume fractions requires the phase densities. Proeutectoid phases outside pearlite must be treated separately.

The charge relation combines left and right weak isospin with the generator distinguishing quarks from leptons inside the extended color charge group.