An interface whose lattice planes match continuously across the boundary. Elastic misfit strain can accumulate when the unstrained lattice parameters differ.

An interface with largely matched regions separated by a network of misfit dislocations.

An interface without extended one-to-one lattice registry; its local structure and energy differ from those of a coherent crystal interface.

The excess free energy per interface area, specified for the phases, orientation, temperature and constraints.

The change in d-electron energy relative to the unsplit barycentre in a crystal-field model, before separately accounting for pairing and other bonding contributions.

In the ideal tetrahedral point-charge model, the five metal d orbitals split into a lower twofold $e$ set and an upper threefold $t_2$ set. Relative to their weighted barycentre their energies are $-3{\Delta }_t/5$ and $+2{\Delta }_t/5$, where ${\Delta }_t>0$ is the separation. For occupancies $e^{n_e}{t}_2^{n_t}$, the crystal field stabilization energy is $(-0.6n_e+0.4n_t){\Delta }_t$. Under weak-field high-spin filling it is $-0.6{\Delta }_t$, $-1.2{\Delta }_t$, $-0.8{\Delta }_t$ and zero for $d^6,d^7,d^8,d^{10}$ respectively. Shorter metal–ligand distances generally increase the splitting, so orbital filling can superimpose nonmonotonic bond-length changes on an ionic-size trend. The model alone does not determine an absolute bond length.

Separation of metal d-orbital energies by the electrostatic field of surrounding coordination ligands. An octahedral field puts the three $t_{2g}$ orbitals below the two $e_g$ orbitals.

A crown ether with fifteen ring atoms, including five oxygen donor atoms.

A fifteen-membered crown with one nitrogen donor and four oxygen donors; nitrogen substitution can attach a signaling group.

A transition-metal-catalyzed carbon–carbon coupling of an Grignard reagent with an organic halide.

The gradient of this polynomial vanishes exactly when $x=0,\pm \sqrt{a/2}$ and $y=0,\pm \sqrt{b/2}$, giving nine stationary points. Its diagonal Hessian matrix has entries $12x^2-2a$ and $2b-12y^2$. The second-derivative test therefore gives two strict local maxima at $(0,\pm \sqrt{b/2})$, with value $b^2/4$, and two strict local minima at $(\pm \sqrt{a/2},0)$, with value $-a^2/4$. The origin and the four points $(\pm \sqrt{a/2},\pm \sqrt{b/2})$ are saddle points of a scalar function, with values zero and $(b^2-a^2)/4$, respectively. Every stationary point is nondegenerate.

Completing the squares organizes the contour plot:The level set through the four outer saddle points of a scalar function factors asIt is a circle together with a hyperbola when $a\ne b$; for $a=b$, the latter factor's complementary zero set is the pair of lines $y=\pm x$. Neither the local maxima nor the local minima are global, since the restrictions to the two coordinate axes are unbounded in opposite directions. The case $a=1$, $b=4$ gives four outer saddle values $15/4$ and an exact circle-hyperbola contour crossing at all four outer saddle points of a scalar function.

A schematic statistical equilibrium is $n_l{n}_e{q}_{lu}=n_u(A_{ul}+n_e{q}_{ul})$. The radiative yield per excitation is $A_{ul}/(A_{ul}+n_e{q}_{ul})$. It is near unity below critical density $A_{ul}/q_{ul}$ and decreases above it. Quenching suppresses emission relative to naive low-density excitation scaling; it need not make the absolute emissivity decrease when density rises.

At fixed density below critical density, optically thin two-level cooling scales as $\mathcal{C}\propto T^{-1/2}{e}^{-\theta /T}$, with $\theta =\Delta E/k_B$. Under temperature-independent heating, $d\ln \mathcal{C}/d\ln T=-1/2+\theta /T$ gives restoring thermal balance for $T<2\theta$ and instability for $T>2\theta$. This boundary assumes fixed ionic abundance and does not describe a constant-pressure perturbation.

Let $A_b^{\prime }=J^d{}_b{A}_d$, where $J^d{}_b=\partial x^d/\partial x^{\prime b}$. ThenThe second term prevents the ordinary derivative from being a generally covariant tensor. It vanishes for affine coordinate changes, including Cartesian Lorentz transformations, so the restricted claim of Lorentz covariance is valid. Antisymmetrization cancels the same term for arbitrary smooth coordinate changes and gives the exterior derivative of a one-form.

An accelerating classical charge radiates according to the Larmor formula. For a slowly evolving nonrelativistic circular electron orbit, energy balance gives $d(r^3)/dt=-4c{r}_0^2$, with $r_0$ the classical electron radius. The finite formal collapse time exposes classical atomic instability; it is not a model of a stable quantum atom.

In a stable-particle approximation with negligible later annihilation, comoving particle number is conserved and physical number density falls as $a^{-3}$. Replacing scale-factor ratios by photon-temperature ratios additionally assumes no change in the entropy degrees of freedom.

The effective count appearing in thermal entropy density $s\propto g_{\ast s}{T}^3$. Conservation of comoving entropy implies $g_{\ast s}{T}^3{a}^3$ is constant, so stable relic dilution includes $g_{\ast s,0}/g_{\ast s,f}$ when expressed using temperatures.

In a supplied equilibrium-cutoff model with $P=m{c}^2/(k_B{T}_f)$, present number density can contain $P^{3/2}{e}^{-P}$. The logarithmic derivative is $3/(2P)-1$, making small changes in a large cosmological particle freeze-out parameter important for the inferred abundance.

Number-changing reactions involving a relic candidate and another nearly degenerate species. If reactions interchange the species rapidly enough, the effective annihilation rate depends on their joint abundances and channels. A single-species relic estimate can then fail even for fixed candidate mass.

For $u=A{T}^4$, $P=u/3$ and physical cell volume $R^3$, the first law integrates to $S=4u{R}^3/(3T)$ with zero chemical potentials and a fixed relativistic species set. Entropy conservation through a species threshold instead requires tracking $g_{\ast s}{T}^3{R}^3$.