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An equilibrium stage models a contact whose outgoing liquid and vapor satisfy vapor-liquid equilibrium at the stage's temperature and pressure. It is a composition model, not a claim that each piece of packing reaches thermodynamic equilibrium.^[ref]

A fractionating column provides contact between rising vapor and returning liquid during fractional distillation. Packing or trays provide contact area; the resulting separation depends on both vapor-liquid equilibrium and the actual flow and exchange between the thermodynamic phases.^[ref]

For a loop of area $A$ and self-inductance $L$, negligible resistance conserves $LI+{\Phi }_{\mathrm{ext}}$. With zero trapped total flux and ${\Phi }_{\mathrm{ext}}=A{B}_0\sin (\omega t)\cos \theta$, the electric current is $I=-A{B}_0\sin (\omega t)\cos \theta /L$. The magnetic torque generalized to the angle between loop normal and field is ${\tau }_{\theta }=-IAB\sin \theta$. For an orientation varying slowly over a field cycle, averaging gives the displayed torque and effective potential $U_{\mathrm{av}}=A^2{B}_0^2{\cos }^2\theta /(4L)$. Its stable orientations have loop normal perpendicular to the applied field. Without damping, a freely released loop librates around such an orientation instead of settling there.

A nozzle shapes a passage to control the speed and direction of a fluid flow. In compressible flow, converging-diverging passages can accelerate a gas through a sonic flow throat into supersonic flow.

An orifice is a localized opening through which a fluid passes, often from a vessel into a region of lower pressure. A simple inviscid incompressible estimate for an opening of area $A$ iswhen the reservoir velocity and elevation change are negligible and $\Delta p$ is the driving pressure difference. The relation follows from a Bernoulli equation estimate of exit speed.

Real jets contract and dissipate energy, so a discharge coefficient $C_d$ is often introduced through $Q=C_d{Q}_{\mathrm{ideal}}$. The vena contracta can have a smaller area than the opening itself. Measuring collected fluid per unit time and the driving head tests the coefficient for a specified geometry and regime. For substantial gas compression or choked flow, the incompressible formula must be replaced by an appropriate compressible-flow model.

The Magnus effect is a transverse force on a spinning body in relative motion through a fluid. Rotation changes the surrounding circulation, pressure distribution and wake. In a two-dimensional cylinder model, the Kutta–Joukowski theorem gives force per unit span ${\mathbf{F}}^{\prime }=\rho \mathbf{U}\times (\Gamma \hat{\mathbf{z}})$, with relative fluid velocity $\mathbf{U}$ and circulation $\Gamma$ positive anticlockwise viewed from $+z$. Real viscosity and boundary-layer behavior help determine the circulation; impermeability alone does not determine it from spin rate.

The spin-orbit interaction couples orbital angular momentum to spin. In chemical atoms it helps determine fine structure and the coupling scale that separates weak-field Zeeman splitting from the Paschen-Back effect.

By William Lawrence Bragg, credited as W. L. Bragg, pp. 531–539 in Fifty Years of X-ray Diffraction, edited by Paul Peter Ewald. The chapter author is distinct from the volume editor. First published in 1962; digitized in 1999.

Institutional PDF: www.iucr.org/__data/assets/pdf_file/0014/761/bragg_wl.pdf

Institutional transcription: www.iucr.org/what-we-do/publications/books/Fifty-Years-of-X-ray-Diffraction/w-l-bragg

Cited passage: printed p. 533, PDF p. 3, retrospective account of sodium chloride and potassium chloride analysis using Laue photographs in late 1912 and early 1913, followed by work with the spectrometer. This is a first-person retrospective, not the original structure-determination paper.

For refractive index $n(y)=c\sqrt{y}$ with $c>0$, the optical-length integrand is $c\sqrt{y}\sqrt{1+y^{\prime 2}}$. The Beltrami identity gives $\sqrt{y}/\sqrt{1+y^{\prime 2}}=\sqrt{h}$, while the full Euler-Lagrange equation gives $2y{y}^{\prime \prime }=1+y^{\prime 2}$. Together these yield the displayed parabolas and exclude a spurious constant solution of the squared first integral. Equal-height endpoints $(\pm L,Y)$ force $x_0=0$ and $h^2-Yh+L^2/4=0$. There are two positive stationary rays for $Y>L$, one coalesced ray for $Y=L$, and none of this smooth family for $Y<L$. For $L=1,Y=5/4$, the two paths are $y=1+x^2/4$ and $y=1/4+x^2$. Fermat principle requires stationary optical length, not a global minimum over every competing path.

For a positive refractive index depending only on $x$, Fermat principle applied to $\int \mu (x)\sqrt{1+y^{\prime 2}}dx$ has $y$ as a cyclic coordinate. The Euler-Lagrange equation therefore makes the displayed quantity constant along a stationary ray. When ${\mu }^2>p^2$, its slope is $y^{\prime }=p/\sqrt{{\mu }^2-p^2}$. If $\theta$ is the angle to the $x$ axis, this is conservation of $\mu \sin \theta$, the stratified-medium form of Snell's law. For fixed endpoint values, the integrand has second derivative $\mu (1+y^{\prime 2}{)}^{-3/2}>0$ with respect to $y^{\prime }$. Its strict convexity and the constant momentum show, by the tangent-line inequality and integration, that any such ray is the unique minimum among competing graph paths with the same endpoints.

For speed of light $c(y)=c_0/\sqrt{1-ky}$, where $c_0>0$ and $1-ky>0$, the travel-time integrand in Fermat's principle is $c_0^{-1}\sqrt{(1-ky)(1+y^{\prime 2})}$. For nonzero $k$, the Beltrami identity giveswhere $y_0$ is the turning height. The corresponding stationary rays are the parabolasFor equal-height endpoints $(\pm x_0,0)$ with $x_0>0$, symmetry gives $x_c=0$ and $(k{x}_0{)}^2=4k{y}_0(1-k{y}_0)$. Real turning heights therefore require $|k|x_0\le 1$, not merely a one-sided condition on signed $k$. When the inequality is strict, the two algebraic values are $y_0=[1\pm \sqrt{1-k^2{x}_0^2}]/(2k)$; only rays remaining inside the specified optical layer are admissible. Equality merges the two branches. At $k=0$, the medium has constant speed of light and the ray joining these endpoints is the straight horizontal segment; the formulas containing $1/k$ must be replaced by that limit.

Online text: mathcs.clarku.edu/~djoyce/elements/bookI/defI7.html

Definition from Euclid's Elements by Euclid. The linked English adaptation and guide are by David E. Joyce; Clark University is the web host. No publication year is established from this page.

Cited passages: Book I, Definition 7; Joyce's guide distinguishing bounded figures from the infinite plane surface.

Molecular beam epitaxy grows crystalline films by directing controlled beams of material toward a crystalline substrate in a vacuum system. A typical apparatus has source cells, shutters, a heated substrate holder and diagnostic tools such as electron diffraction. Opening or closing shutters changes which species reach the surface, enabling layered heterostructures. Growth quality depends on surface diffusion, incorporation and desorption as well as the arriving flux. The method's value is precise control of composition and interfaces; depositing material in vacuum alone does not ensure epitaxial crystal ordering.