For molar equation $p=RT/(v-b)-a/v^2$, the critical values are $v_c=3b$ and $T_c=8a/(27Rb)$. At $v=v_c$, $(\partial p/\partial v{)}_T=-R(T-T_c)/(4b^2)$. Thus ${\kappa }_T=-v_c^{-1}(\partial v/\partial p{)}_T=4b/[3R(T-T_c)]$ on the stable branch above $T_c$. The mean-field compressibility exponent is one; the continuation below the critical temperature is unstable rather than an equilibrium negative compressibility.

Counting two transverse polarizations and zero-point energy $\hslash ck/2$ per mode gives $u=\hslash c{k}_{\max }^4/(8{\pi }^2)=2{\pi }^2\hslash c/{\lambda }_{\min }^4$. This is a bare cutoff-dependent mode sum, not a classical incident radiation bath or a renormalization-independent absolute observable.

For identical equal-frequency electric-dipole photons, exchange symmetry removes the antisymmetric rank-one transition tensor. Ranks zero and two survive. A zero-to-zero angular-momentum transition can occur, while zero-to-one is forbidden in this approximation; the rank-two triangle condition still permits some nonzero-to-nonzero transitions with a change of one.

Absorbing one photon from each of two opposing equal-frequency beams cancels the first-order Doppler shift through their nearly zero summed wavevector. Absorbing both photons from the same beam does not share this cancellation and can create a broad background.

An intermediate contribution to a second-order transition amplitude need not be an occupied, on-shell stationary atomic state. Both time orderings must be summed with their energy denominators. Large detuning can suppress real intermediate occupation while preserving a coherent two-photon coupling. Near resonance, finite linewidth, saturation and sequential population transfer must be treated explicitly.

A lattice represented by a rectangular conventional cell with corner points and one center point. For conventional sides $2a,2b$, primitive vectors can be $(a,b)$ and $(-a,b)$. The special case $a=b$ is a square lattice with a rotated primitive basis.

A two-dimensional Bravais lattice with perpendicular primitive vectors of the same length $\ell$. An orthogonal matrix $R$ gives its orientation, and its primitive area is ${\ell }^2$. Its nearest-neighbor graph is a square lattice; the set of crystal translations and the graph with chosen adjacency are related but distinct objects.

Taking the polar axis along the separation vector reduces the solid-angle integral to $2\pi {\int }_{-1}^1{e}^{ikdu}du$. Thus the total scattering cross-section contains the interference term $2\mathrm{Re}(c_1{c}_2^{\ast })\sin (kd)/(kd)$ as well as the two self terms. This is an outgoing-angle integral for fixed incident direction, rather than an average over the incoming beam.

An antiperiodic boundary condition is a phase twist of $\pi$ across the boundary. A Fourier mode then has $k=2\pi (n+1/2)/L$. It occurs in the even-fermion sector of a periodic Jordan–Wigner transformation and in imaginary-time fermionic thermal fields, for different underlying reasons.

A planetary boundary layer is the part of an atmosphere strongly affected by exchanges of momentum, heat and material with the underlying surface on relatively short timescales. Surface friction generates shear, while heating or cooling changes buoyancy and hence vertical mixing. Its thickness and internal structure vary with the forcing rather than being fixed by one universal altitude.

A sun-heated surface can drive vigorous convection, whereas nocturnal cooling can create stable density stratification and suppress mixing. This distinction matters for interpreting a weather station: a near-surface wind or temperature observation need not represent the air above the boundary layer. Moisture and suspended material are likewise sensitive to the strength of vertical transport.

The lowest part of the terrestrial atmosphere that responds directly to surface fluxes on relatively short timescales through turbulence. Its depth changes with heating, stability and weather; a sharp composition or temperature change can mark its top. A uniform-layer chemical budget converts a surface flux into a volume tendency by dividing by layer depth.

The troposphere above the directly surface-coupled atmospheric boundary layer. Entrainment between these regions can change pollutant abundances even when local chemical production is unchanged.

In a parabolic band, transport mobility and mean free path obey $\ell =\mu \hslash k_F/e$. The rough condition $\ell \gtrsim L$ gives $\mu \gtrsim eL/(\hslash k_F)$, whereas surviving a first semicircle uses path length $\pi L/2$. Clear peaks usually need a larger margin and a specular boundary, so this is an observability scale rather than a universal sharp threshold.

For a matrix over the real numbers or complex numbers, applying the transpose twice to $A=\lambda A^T$ gives $A={\lambda }^{2}A$. A nonzero matrix is therefore possible only for $\lambda =1$ or $\lambda =-1$. These cases are precisely the symmetric matrices and skew-symmetric matrices, respectively. For every other scalar, the only solution is the zero matrix.

Image-intensity variation produced when scattered and unscattered electron-wave amplitudes interfere after phase shifts from the specimen and imaging optics. It depends on defocus and aberrations.

Mass–thickness contrast is image contrast caused by different amounts or strengths of electron scattering along the beam path. In bright-field transmission electron microscopy, a region that scatters more electrons outside the accepted transmitted beam generally appears darker.

A thicker part of one material and an equally thick region containing more strongly scattering atoms can therefore produce similar darkening. The image alone does not separate thickness from composition. Changing the collection aperture changes which scattered electrons are rejected, while electron diffraction in a crystal can add orientation-dependent image contrast that is not explained by mass and thickness alone.

In a dark-matter-dominated growing mode with a constant relative offset and initially negligible baryon density contrast, ${\Delta }_B/{\Delta }_D=1-R_r/R=1-(1+z)/(1+z_r)$. Replacing this by $1-z/z_r$ requires high redshift. The initial relative velocity and finite baryon self-gravity alter transients, although the relative difference becomes small as the common growing mode increases.

Here $x=m_W^2/m_t^2$, the bottom quark mass is neglected, and the W boson is on shell. The weak charged current gives the spin-averaged squared scattering amplitude $g_W^2|V_{tb}{|}^2(m_t^2-m_W^2)(m_t^2+2m_W^2)/(4m_W^2)$. The two-body Lorentz-invariant phase space has $p^{\ast }=(m_t^2-m_W^2)/(2m_t)$; multiplying by $p^{\ast }/(8\pi m_t^2)$ gives the displayed decay width. Its longitudinal-vector contribution explains the large-$m_t$ cubic growth, and its threshold factor makes the width vanish as $m_t\downarrow m_W$. The formula requires $m_t>m_W$ and receives finite-mass and radiative corrections.

A semidiurnal tide has two principal oscillations per day. The dominant lunar constituent has a period of approximately 12 hours 25 minutes; the solar semidiurnal constituent has a period of 12 hours.^[ref]

The Bay of Fundy is a coastal basin in eastern Canada notable for large tidal ranges. Its geometry and the dynamical response of the connected bay system strongly amplify the incoming semidiurnal tide.

The important physical idea is forced motion of a basin: water depth, shape and friction determine its natural response, while the external astronomical tide supplies the drive. Large range is not a direct local measurement of an unusually strong gravitational field. Range also varies with location and the spring-neap cycle, so one quoted maximum does not characterize every observation.