For an on-shell massive vector with $k^{\mu }=(E,|\mathbf{k}|\hat{\mathbf{k}})$ in Minkowski metric signature $+---$, the displayed vector satisfies $k\cdot {ϵ}_L=0$ and ${ϵ}_L^2=-1$. It specifies the longitudinal one of the three physical polarization states. It is not a replacement for a final polarization sum when other states are unobserved.

2016 paper: Public original PDF, physical page 829. Student-answer availability for this paper remains unverified.

For polymer mass concentration $c$, the leading ideal chain pressure is $RTc/M_n$. Plotting $\Pi /c$ against $c$ and extrapolating removes the leading virial correction. The intercept has units of energy per mass; without the measurement temperature it determines $M_n/T$, not an absolute molar mass.

For a regular chart of orthogonal coordinates $\mathbf{r}(q)$, define $h_i=|{\partial }_{q_i}\mathbf{r}|>0$ and ${\mathbf{e}}_i=h_i^{-1}{\partial }_{q_i}\mathbf{r}$. The tangent unit vectors form an orthonormal basis. Matching $d\Phi =\nabla \Phi \cdot d\mathbf{r}$ with $d\mathbf{r}={\sum }_i{h}_i{\mathbf{e}}_{i}d{q}_i$ proves the formula. Dividing by a scale factor converts a derivative per coordinate increment into a derivative per physical distance in that direction.

For a right-handed regular system of orthogonal coordinates, the gradient in orthogonal coordinates gives $\nabla u_i={\mathbf{e}}_i/h_i$. Thus $\varphi {\mathbf{e}}_1=(\varphi h_1)\nabla u_1$, and the curl product rule givesAlso $\nabla u_2\times \nabla u_3={\mathbf{e}}_1/(h_2{h}_3)$ and the divergence of this cross product is zero, by the divergence-of-a-cross-product identity and the vanishing curl of each gradient. Applying the divergence product rule to $\varphi {\mathbf{e}}_1=(\varphi h_2{h}_3)(\nabla u_2\times \nabla u_3)$ givesCyclic permutation and addition give the full component formulas for any vector field. This derivation accounts for the varying coordinate basis without differentiating its unit vectors explicitly. The factors $h_i>0$ are scale factors of orthogonal coordinates; the chart must avoid coordinate singularities.

These orthogonal coordinates extend two confocal families of parabolas parallel to an axis. In the displayed scaling their scale factors of orthogonal coordinates are $h_u=h_v=2\sqrt{u^2+v^2}$ and $h_w=1$, so the Laplacian is $\Delta \varphi =({\varphi }_{uu}+{\varphi }_{vv})/[4(u^2+v^2)]+{\varphi }_{ww}$ away from $u=v=0$. With $u\ge 0$, signed $v$ distinguishes the upper and lower half-planes; the two signs at $u=0$ describe the same negative $x$ half-axis. Fixed positive $u^2$ or $v^2$ describes a parabolic cylinder. This differs from the rotational three-dimensional parabolic coordinates, whose coordinate surfaces are elliptic paraboloids.

An interpenetrating donor-acceptor morphology in a photovoltaic absorber. Small domains facilitate exciton access to interfaces, while continuous pathways in each phase allow separated charges to reach selective contacts.

For a spherical body of radius $R$, mass $M$ and mean density $\bar{\rho }=3M/(4\pi R^3)$, a low circular orbit has period $P\simeq 2\pi \sqrt{R^3/(GM)}$, by equating the gravitational acceleration from Newton's law of universal gravitation to the centripetal acceleration. Substitution proves the displayed relation. Thus similar low-orbit periods indicate similar mean densities, even for bodies with very different masses and radii. At altitude $h$, the right side is multiplied by $(1+h/R{)}^3$; the simple density inference requires $h\ll R$ and neglects rotation, nonsphericity and other gravitating bodies.

The single-mode squeeze operator changes occupation in pairs. Consequently an optical squeezed vacuum has only even photon numbers, with mean $s={\sinh }^{2}r$ and variance $2s(s+1)$. Its large number fluctuations are compatible with quadrature squeezing: they concern different observables.

A rattling mode is a relatively soft, often localized vibration of a guest atom within a more rigid surrounding cage. An approximately harmonic narrow-frequency contribution can be modeled as an Einstein solid term with frequency $\Omega$ and degeneracy $g$, giving the heat capacity in the title. It freezes out exponentially at low temperature and approaches $gR$ per mole at high temperature, so it can bend a plot of $C/T$ versus $T^2$ away from its pure electronic-plus-acoustic straight line.

Calorimetry of Ba4Na2Ge25 shows an additional low-energy contribution fitted by an Einstein energy $\hslash \Omega /k_B\simeq 32\mathrm{K}$, assigned to a barium guest site.^[ref] This is a measured-material example; the narrow-mode description remains an approximation to a crystal's full vibrational spectrum.

A relatively short potential coding region under an explicitly chosen length convention. Evidence of biological translation and evidence for a stable functional peptide are different.

Formation of a critical nucleus within the bulk parent phase without assistance from a pre-existing interface. Its interfacial cost competes with the bulk free-energy gain.

Formation of a critical nucleus at a pre-existing surface or interface. Favorable wetting can replace part of the cost of a free nucleus and lower its barrier; later deposition of a free particle on a support is a different process.

A schematic time history of growth-species concentration showing buildup, a short nucleation interval above an effective nucleation threshold, then growth above the equilibrium solubility but below substantial new nucleation. The concentration history suggests how to separate particle birth from growth; it does not establish that every particle synthesis follows that mechanism.

Formation of most nuclei within an interval short compared with subsequent particle growth. A common birth period can help narrow a particle size distribution if mixing and later growth are sufficiently uniform and secondary nucleation is suppressed.

2016 paper: Public original PDF, physical page 826. Student-answer availability for this paper remains unverified.