For a Fermi liquid, $C_{\mathrm{el}}=\gamma T+o(T)$ at low temperature. In a simple quasiparticle description, $\gamma =({\pi }^2/3)k_B^{2}g(E_F)$, where the density of states includes both spin states and uses the same per-volume or per-mole normalization as $\gamma$. A fit $C/T=\gamma +\beta T^2$ separates this contribution from a Debye lattice term over an appropriate low-temperature range. Magnetic or other low-energy excitations can require extra fit terms.

Transmission-electron diffraction from a region selected by an aperture in an image plane. Downstream lenses image the objective back focal plane to the detector, producing reciprocal-space rather than real-space information.

diffraction in which repeated scattering and coupling between transmitted and diffracted beams significantly affect their amplitudes. Electron-diffraction intensity can then vary strongly with thickness and excitation error; it is not generally just the squared single-scattering crystal basis structure factor.

With $(+---)$ signature and $c=1$, $T^0{}_0=(E^2+B^2)/(2{\mu }_0)$ while $T^0{}_i=-(\mathbf{E}\times \mathbf{B}{)}_i/{\mu }_0$. Raising the second index gives $T^{0i}=S_i$, where $\mathbf{S}$ is the Poynting vector, and symmetry gives the physical energy-flux components $T^{i0}$. The negative mixed component is a metric-sign effect, not reversed energy transport. In SI coordinates $x^0=ct$, $T^{i0}=S_i/c$.

The resistance per square of a uniform thin conducting sheet. For bulk resistivity $\rho$ and film thickness $t$, $R_{\square }=\rho /t$ and a rectangular stripe has $R=R_{\square }L/W$. The number of squares is dimensionless and independent of an overall change of lateral scale. Networks and grids use an effective sheet resistance that may depend on finite-size connectivity and contacts.

Electrical resistance introduced by a connection between conducting regions. Interface chemistry, contact geometry and microscopic barriers can dominate over the bulk resistance of the connected materials. Four-terminal measurements help separate contact drops from the resistance of a test section.

The scale $R_K=h/e^2\simeq 25.8\mathrm{k\Omega }$. Spin degeneracy perfectly transmitting channels instead have conductance $2e^2/h$. Suppressing charge fluctuations in a simple tunnel-island picture requires nanowire junction resistances large relative to $h/e^2$.

Reduction of an effective electric dipole moment by the opposing local field of neighboring dipoles.

The point-dipole energy is $[{\mathbf{p}}_1\cdot {\mathbf{p}}_2-3({\mathbf{p}}_1\cdot \hat{r})({\mathbf{p}}_2\cdot \hat{r})]/(4\pi {ϵ}_0{r}^3)$. If the separation is perpendicular to the first dipole, the second term vanishes. Parallel dipoles then have positive energy $p_1{p}_2/(4\pi {ϵ}_0{r}^3)$, while mutually perpendicular dipoles give zero in this approximation. Real molecular interactions can include multipoles, induction and short-range contributions beyond this point-dipole term.

Ehrhart's theorem says that the lattice-point count $L_P(n)$ of a $d$-dimensional lattice polytope is a polynomial in the nonnegative integer dilation factor, of degree $d$. For full-dimensional $P\subset {\mathbb{R}}^d$, its leading coefficient is the ordinary volume and its constant coefficient is one. Rational vertices instead generally produce a quasipolynomial.^[ref]

For a $d$-dimensional lattice polytope $P$ and positive integer $n$, Ehrhart reciprocity stateswith relative interior when needed. Negative input means polynomial evaluation, not a negative dilation count. For the unit square, $(1-n{)}^2=(n-1{)}^2$ counts the interior grid points of its $n$th dilation.^[ref]

For radiatively efficient release of Newtonian surface binding energy, $L\sim GM\dot{M}/R_{\ast }$ and $L_E=4\pi GMc/\kappa$ imply the displayed rate. Its efficiency is $\eta \sim GM/(R_{\ast }{c}^2)$, so at fixed opacity the maximum rate scales with stellar radius. This differs from defining an Eddington supply rate with an arbitrarily fixed reference efficiency. Geometry, advection and mass loss can modify the simple limit.

An effective relation between deviatoric stress and deformation rate in mantle rocks. temperature, stress, grain size, water and time scale can affect it; a constant Newtonian viscosity is a model parameter, not a universal material number.

A series spring of modulus $G$ and dashpot of viscosity $\eta$ have $\tau =\eta /G$ and $\dot{\gamma }=\dot{\sigma }/G+\sigma /\eta$. With $e^{i\omega t}$ amplitudes, $G^{\prime }=G(\omega \tau {)}^2/[1+(\omega \tau {)}^2]$ and $G^{\prime \prime }=G\omega \tau /[1+(\omega \tau {)}^2]$. Low frequencies are fluid-like; high frequencies retain an elastic plateau. The loss peaks at $\omega \tau =1$.

A spherical black grain of radius $a$ absorbs luminosity $L$ through projected area $\pi a^2$ and emits over $4\pi a^2$. Radiative balance at sublimation temperature gives $r_{\mathrm{sub}}=[L/(16\pi {\sigma }_{\mathrm{SB}}{T}_{\mathrm{sub}}^4){]}^{1/2}$. Grain size cancels only under the stated unit-efficiency approximation; shielding and wavelength-dependent emissivity change the result.

In oriented Minkowski coordinates with $(+---)$ signature and ${ϵ}^{0123}=+1$, the conventional dual is ${\tilde{F}}^{ab}=\frac{1}{2}{ϵ}^{abcd}{F}_{cd}$. For $F_{0i}=E_i$ and $F_{ij}=-{ϵ}_{ijk}{B}_k$ in $c=1$ units, ${\tilde{F}}^{ab}{F}_{ab}=-4\mathbf{E}\cdot \mathbf{B}$. A definition without the half produces twice this dual and contraction $-8\mathbf{E}\cdot \mathbf{B}$. The factor accounts for summing both orders of each antisymmetric index pair.

For a composition-fixed equation of state, let $\dot{\theta }=D\theta /Dt$ and let $\mathbf{F}$ be nonconservative force per unit mass. The vorticity and scalar-gradient evolution equations give the source terms in the title. Spatially varying heating projected along absolute vorticity and the curl of friction can generate or destroy Ertel potential vorticity. Heating that changes potential temperature does not automatically change PV everywhere: its spatial gradient and orientation matter. Unresolved irreversible mixing can also alter the parcelwise conservation picture.

Two equally illuminated slits of different widths have field amplitudes $A_j(\theta )\propto w_j\mathrm{sinc}(\pi w_j\sin \theta /\lambda )$. Their interference intensity is $A_1^2+A_2^2+2A_1{A}_2\cos (2\pi d\sin \theta /\lambda +\delta )$. Near the centre, $A_2/A_1=w_2/w_1$, so the local maximum-to-minimum ratio is $(w_1+w_2{)}^2/(w_2-w_1{)}^2$. At one slit envelope's zero the interference visibility vanishes even if the other slit still transmits.

Doppler spectroscopy measures the line-of-sight component of a host star's stellar reflex motion, rather than the full companion mass. Let the measured orbital period, orbital eccentricity and stellar radial-velocity semi-amplitude be $P,e,K$, with host mass $M_{\star }$ and sky-plane orbital inclination $i$. The exact two-body relation isCubing and rearranging defines the observed binary mass functionFor fixed $M_{\star }$, the function $M_p^3/(M_{\star }+M_p{)}^2$ is strictly increasing: its logarithmic derivative is $3/M_p-2/(M_{\star }+M_p)>0$. Since $\sin i\le 1$, the smallest compatible companion mass occurs for an edge-on orbit, $i=\pi /2$.

Only when $M_p\ll M_{\star }$ does this simplify toA small inferred projected mass alone therefore does not exclude a more massive companion at small orbital inclination. An exoplanet transit constrains orbital inclination and helps remove this degeneracy. When the companion mass is appreciable, solve the exact binary mass function instead of inserting a total-mass correction on the wrong side of the approximate formula.

A stellar binary identified through periodic Doppler shifts of spectral lines. A single-lined low-inclination binary can share a small projected velocity amplitude with an edge-on planetary system; the companion luminosity, line shapes and independent inclination constraints help distinguish them.