Take equal masses $m$, atomic spacing $a$, and alternating springs $K\gg \kappa >0$. The primitive unit cell contains a strong-bonded pair, so its length is $2a$ and the Brillouin zone ends at $\pm \pi /(2a)$. Expanding the exact alternating-spring chain dispersion relation givesThe low-frequency acoustic phonon branch moves almost rigid dimers of mass $2m$ coupled by weak springs; its speed of sound tends to $a\sqrt{2\kappa /m}$. The high-frequency optical phonon branch describes internal dimer stretch. At the zone edge the exact frequencies are $\sqrt{2\kappa /m}$ and $\sqrt{2K/m}$.

For unstressed central Hookean springs, the quadratic elastic energy is $\frac{1}{2}{\sum }_{ij}{k}_{ij}[{\mathbf{n}}_{ij}\cdot ({\mathbf{u}}_i-{\mathbf{u}}_j){]}^2$, with unit edge direction ${\mathbf{n}}_{ij}$. A common translation gives zero displacement difference. An infinitesimal rotation ${\mathbf{u}}_i=\mathbf{\Omega }\times {\mathbf{r}}_i$ also gives zero extension because a vector is perpendicular to its cross product with $\mathbf{\Omega }$. Thus a noncollinear three-dimensional free framework has three translational and three rotational zero modes. Extra mechanisms can give additional zero modes; six is a lower bound until rigidity is established.

For a fixed homogeneous absorber in a narrow-beam geometry, define the X-ray spectral attenuation operator on L1 space over a bounded positive energy interval. A physical input $f(E)\ge 0$ is an integrable function giving incident photon rate density. Let its linear attenuation coefficient $\mu (E)\ge 0$ be a measurable function, with $x\ge 0$. The transmitted uncollided density isHere $w(E)\ge 0$ is a fixed bounded detector response described by a measurable function; background has been subtracted. This is a positive linear operator on L1 space. Since $0\le e^{-\mu (E)x}\le 1$, it cannot increase the integrated absolute incident rate. Multiplication of exponentials proves $A_{x+y}=A_x{A}_y$ for successive layers of the same material. Ignore secondary radiation entering the detector, saturation and changes to the source. Discrete spectral lines use the finite-sum version below instead of an ordinary rate density.

For a chosen filter, stipulate ${\mu }_{\alpha }=0.20\mathrm{m}{\mathrm{m}}^{-1}$, ${\mu }_{\beta }=0.80\mathrm{m}{\mathrm{m}}^{-1}$ and input line rates $(N_{\alpha },N_{\beta })=(4,1)$ in a common arbitrary rate unit. Assume equal calibrated detector weight for both lines. At $x=2.0\mathrm{mm}$,The selected $8.5\mathrm{keV}$ line's fraction rises from $4/5=0.800$ to $0.929974$, but its absolute rate falls by about $33.0\%$. Filtering improves purity at the cost of signal. It multiplies existing components and cannot create a new emitted spectral line.

With just one integrated reading, $S=N_{\alpha }+N_{\beta }$, the two unknown line rates are underdetermined. If the source is unchanged between an empty-path reading and a filtered reading, the ideal inverse problem becomesThe determinant $e^{-1.6}-e^{-0.4}\simeq -0.468424$ is nonzero, so these two ideal readings determine the two stipulated line rates uniquely. Identical coefficients would give zero determinant and destroy that separation. This small linear algebra example explains why absorber contrast supplies information, while also showing its limit: unknown scattered components or unknown detector weights add unknowns that two readings cannot determine.

Precision X-ray emission spectroscopy turns closely spaced spectral lines into quantitative constraints on atomic energy levels. Bragg's law makes a crystal an angular ruler for wavelength: resolving nearby directions can reveal components hidden inside an apparently single line. Accurate line positions and the ability to separate two lines are related, but distinct, experimental achievements.

Karl Manne Georg Siegbahn received the 1924 Nobel Prize in Physics; his lecture was delivered on 11 December 1925.^[ref]

His son Kai Siegbahn received the 1981 Nobel Prize in Physics for electron spectroscopy.^[ref]

The Barkla characteristic X-ray experiment distinguished element-dependent K and L groups through absorption and excitation. Moseley's law related corresponding line frequencies across chemical elements to atomic number. Measuring the components within those groups asks a further question: which distinct atomic energy differences produce the neighbouring spectral lines?

Siegbahn's lecture organizes measured spectral lines into energy levels and resolved double levels.^[ref] The payoff is a map of atomic energy differences as well as an element's position in the periodic table.

Charles Glover Barkla's 1911 paper describes irradiating a target with an X-ray tube and observing secondary X-rays using an electroscope. For soft Characteristic X-rays from tin, he used a soft primary beam and placed the electroscope close to the target to limit loss in air. Successive aluminium absorbers changed the gas-ionization signal. Thick absorbers suppressed the soft component; a carbon radiator supplied a comparison for the surviving scattered component. Subtracting this estimated contribution left approximately equal fractional losses through successive sheets, evidence for a nearly homogeneous component. Changing primary-beam hardness could suppress excitation of the more penetrating characteristic component.^[ref]

An electroscope reads loss of stored electric charge; ionization in its detecting gas enables discharge. An ionization chamber with an electrometer instead measures the collected electric charge or electric current. These are integrating readouts: neither directly assigns an energy to each arriving photon. The diagram separates these functions without claiming a surviving apparatus drawing.

The K and L components were distinguished by their penetrating power and dependence on the emitting chemical element. Barkla's Nobel lecture, delivered on 3 June 1920 for the 1917 prize, explains that absorber-defined K radiation had seemed homogeneous, whereas later diffraction experiments resolved neighbouring spectral lines in both K and L groups. Absorber tests therefore distinguished groups without resolving every line.^[ref]

Henry Gwyn Jeffreys Moseley subsequently related the frequencies of corresponding spectral lines to atomic number. The approximate relation $\sqrt{\nu }=a(Z-b)$, with series-dependent constants $a,b$, is Moseley's law; its discovery should be distinguished from Barkla's identification of characteristic secondary radiation.^[ref]

A continuous incident frequency spectrum can contribute a continuous scattered background alongside discrete Characteristic X-rays. Here “scattered” identifies a production process, not a universal spectral shape: scattering of a narrow incident line need not make a continuum. In particular, the characteristic component is photon emission after vacancy creation, rather than simply the primary beam redirected. Source hardness, target substitution and absorber thickness probe different parts of that distinction. A surviving signal alone establishes neither an additional electron shell nor a fully resolved spectral line.

For a repulsive contact-interacting Bose-Einstein condensate, the classical Hamiltonian density generating evolution in a frame rotating at the chemical potential is $\mathcal{K}={\hslash }^2|\nabla \psi {|}^2/(2M)-\mu |\psi {|}^2+g|\psi {|}^4/2$. Its integral is the grand canonical ensemble generator $H-\mu N$, not the physical energy $H$ alone. For $g,\mu >0$ the uniform mean-field theory minima satisfy $|{\psi }_0{|}^2=\mu /g$, with a freely chosen constant phase. In the physical frame, the corresponding field has time factor $e^{-i\mu t/\hslash }$.

Expand a repulsive Bose-Einstein condensate around the real uniform saddle ${\psi }_0=\sqrt{\mu /g}$ as $\psi ={\psi }_0+\chi$. After removing a constant and a temporal boundary term, the quadratic Lagrangian density is the displayed expression. Its equation $i\hslash \dot{\chi }=-{\hslash }^2{\nabla }^2\chi /(2M)+\mu (\chi +{\chi }^{\ast })$ mixes a fluctuation and its complex conjugate. Writing $\chi ={\chi }_1+i{\chi }_2$ gives canonically paired amplitude and phase fluctuations and one positive Bogoliubov quasiparticle dispersion branch, $\hslash \omega =\sqrt{{ϵ}_k({ϵ}_k+2\mu )}$ with ${ϵ}_k={\hslash }^2{k}^2/(2M)$. The phase direction is a Goldstone mode; its zero uniform restoring force does not produce a second independent positive-frequency amplitude branch in this first-order-in-time theory.

For the faces of a closed polyhedron, take outward unit normal vectors. Applying the divergence theorem to an arbitrary constant vector field $c$ givesSince $c$ is arbitrary, the vector areas sum to zero. This is an oriented statement; the positive scalar surface areas do not cancel. The same proof works for every piecewise smooth closed boundary to which the divergence theorem applies.

Choose a hypothetical fluid with molar constants $a_m=0.360\mathrm{Pa}{\mathrm{m}}^6\mathrm{mo}{\mathrm{l}}^{-2}$ and $b_m=4.00\times 10^{-5}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$, using $R=8.314462618\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$. These stipulated constants are not a fit to carbon dioxide. The molar critical formulas giveAt $T=360\mathrm{K}$ and $v_m=240\mathrm{c}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$, first convert $v_m$ to $2.40\times 10^{-4}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$. The repulsive-volume term and attractive correction areso $p=8.716033\mathrm{MPa}$. The ideal gas law would instead give $RT/v_m=12.471694\mathrm{MPa}$. The excluded volume raises pressure relative to that ideal value, but attraction more than offsets the increase here. Neither correction alone describes this example.

For $n_{\mathrm{mol}}=0.500\mathrm{mol}$, the same state occupies $V=n_{\mathrm{mol}}{v}_m=120\mathrm{c}{\mathrm{m}}^3$. The total-volume formula gives $n_{\mathrm{mol}}{b}_m=20\mathrm{c}{\mathrm{m}}^3$ and the same $8.716033\mathrm{MPa}$. Halving the sample has halved both its physical volume and its excluded volume, while retaining its pressure. Conversely, if the two critical measurements $T_c,p_c$ are used to determine model constants, inversion givesAn independent critical molar volume then tests the prediction $v_{m,c}=3R{T}_c/(8p_c)$; agreement with two fitted numbers does not ensure agreement with the third.

At $t=0.9$, the Van der Waals liquid-gas coexistence example below becomes $T\simeq 288.6537\mathrm{K}$, $p_{\ast }\simeq 5.391653\mathrm{MPa}$, $v_{m,l}\simeq 72.40823\mathrm{c}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$ and $v_{m,g}\simeq 281.8611\mathrm{c}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$. For a mean molar volume of $120\mathrm{c}{\mathrm{m}}^3\mathrm{mo}{\mathrm{l}}^{-1}$, the lever rule gives gas mole fraction $f_g\simeq 0.227219$. Its volume fraction is instead $f_g{v}_{m,g}/v_m\simeq 0.533703$: less than one quarter of the molecules occupy more than half the volume. The flat equilibrium pressure corresponds to changing these proportions. It does not describe a homogeneous sample sliding along the unstable loop.

For one mole, liquid–vapour phase coexistence requires equal pressure and chemical potential. At $\tau =RTb/a\ll 1$, the saturation pressure is exponentially small. Neglecting it in the dense-phase equation of state gives $RT{V}_l^2-a{V}_l+ab=0$. The root near the excluded volume isThis is an asymptotic low-temperature result, not an exact coexistence volume at every subcritical temperature. Away from that limit, solve equal pressure together with the Maxwell equal-area construction.

A Rabi oscillation is a coherent oscillation of populations in a driven two-level quantum system.

A resonant drive can transfer a quantum state from one energy eigenstate to another and then bring its population back. This is unitary time evolution: the drive exchanges energy with the selected two-level system, while total probability stays one. One complete Rabi cycle has population period $2\pi /\Omega$; halfway through, a Rabi pi pulse transfers the whole population in the ideal model. This makes Rabi oscillations useful for calibrating resonant quantum control, preparing quantum superpositions and supplying the pulses of Ramsey interferometry.

For two equal-intensity mutually incoherent narrow spectral lines with wavenumbers $k_1,k_2$, a Michelson interferometer with optical path difference $x$ has normalized intensity $I/I_0=1+\cos (\bar{k}x)\cos (\Delta kx/2)$, where $\bar{k}=(k_1+k_2)/2$ and $\Delta k=|k_1-k_2|$. With mirror displacement $d=x/2$, the fast period is $\pi /\bar{k}\simeq \bar{\lambda }/2$; successive interference visibility revivals are separated by $\pi /\Delta k\simeq {\bar{\lambda }}^2/(2\Delta \lambda )$. The first visibility zero is halfway to the first revival. A signed envelope has twice the period of its absolute value. Unequal line strengths prevent perfect zeros.

Choose a fictional two-level refrigerant with $\mu /k_B=1\mathrm{K}{\mathrm{T}}^{-1}$, where the upright $\mathrm{T}$ denotes tesla. This chosen moment is a convenient model parameter, not a fitted property of the historical salt. At bath temperature $T_b=1\mathrm{K}$, raise $B$ from $0.20\mathrm{T}$ to $1.00\mathrm{T}$. Then $x$ increases from $0.20$ to $1.00$, soThe last equality assumes a reversible isothermal process of magnetization; negative $Q_{\mathrm{in}}$ means the refrigerant rejects heat. The bath gains $0.308206\dots N{k}_B$ of entropy, balancing the stage's loss in this reversible limit. An appreciable temperature difference during real heat transfer adds entropy production.

Now open the heat switch and reduce $B$ from $1.00\mathrm{T}$ to $0.10\mathrm{T}$. Constant $x=1$ gives $T_f=0.10\mathrm{K}$. Throughout this ideal cooling leg, $p_{-}=0.880797\dots$, $p_{+}=0.119203\dots$, and $S_{\mathrm{mag}}=0.365334\dots N{k}_B$. Neither state becomes more probable. Field reduction shrinks the energy gap by ten while preserving the Boltzmann distribution's dimensionless ratio of gap to temperature.

There is a subtle energy check. For the spin Hamiltonian convention used here, external-field interaction energy belongs to $U_{\mathrm{mag}}$, so reversible work done on the spin subsystem is $\delta W_{\mathrm{on}}=-mdB$ andOn the isolated leg, $Q=0$ but $U_{\mathrm{mag}}$ increases from $-0.761594\dots N{k}_B\mathrm{K}$ to $-0.0761594\dots N{k}_B\mathrm{K}$, becoming less negative as the gap shrinks. Integrating $-mdB$ at constant $x=1$ supplies exactly $+0.685435\dots N{k}_B\mathrm{K}$. Cooling therefore need not decrease this chosen spin internal energy. An account of work exchanged with a magnet circuit must specify where field interaction energy is assigned before comparing signs. The occupation ratio and temperature prediction do not depend on that bookkeeping choice.

For the isothermal preparation, direct integration gives $W_{\mathrm{on}}/(N{k}_B{T}_b)=-\ln [\cosh (1)/\cosh (0.20)]=-0.413913\dots$. The endpoint internal energy difference is $-0.722119\dots N{k}_B{T}_b$, equal to $Q_{\mathrm{in}}+W_{\mathrm{on}}$. This independently checks both the rejected heat and the magnetic-work sign in the chosen convention.

Diffraction contrast is image-intensity variation caused by spatially varying electron diffraction conditions. JEOL documents a silicon semiconductor-interconnect specimen at $200\mathrm{kV}$: its differently oriented crystals are imaged in bright-field transmission electron microscopy and same-area dark-field transmission electron microscopy, alongside an electron diffraction pattern. Selecting the marked diffracted beam makes the corresponding crystal bright after it was dark. The preparation route, thickness and exposure are not supplied. The changed objective aperture selection provides a comparison within one specimen; the readout supports orientation-dependent contrast rather than a map of surface height or chemical composition.^[ref]

This comparison is useful precisely because the material has not been replaced between the two images. It does not prove that thickness and chemical composition never influence transmission electron microscopy contrast. Instead, it supplies a concrete counterexample to assigning either quantity uniquely from brightness. A proposed extension is to repeat the first aperture setting after the second: recovery of the original contrast tests whether specimen motion or irradiation changed the comparison.

An objective aperture selects outgoing electron directions in the objective lens's back focal plane. Passing the central transmitted beam gives bright-field transmission electron microscopy; selecting a separated diffracted beam gives dark-field transmission electron microscopy.^[ref]

In a paraxial approximation, an outgoing angle $\theta$ maps approximately to position $r=f\theta$ in this focal plane, where $f$ is the objective focal length. A centred opening of radius $a$ therefore admits $|\theta |\lesssim a/f$. Selecting a direction selects a component of the outgoing wavefunction; the subsequent image still assigns intensity to specimen positions. The opening is not a hole cut into a real-space specimen image. Its angular selection must be distinguished from selecting a specimen region for electron diffraction.

A circular shaft twisted uniformly through small end angle $\varphi$ over length $L$ has local engineering shear strain $r\varphi /L$ at radius $r$. Its shear stress is $Gr\varphi /L$, so the integrated torque is $C={\int }_{A}r(Gr\varphi /L)dA=G{J}_p\varphi /L$, using the polar second moment of area. The restoring torque has the opposite sign to the imposed twist. The stored elastic energy is $C\varphi /2=G{J}_p{\varphi }^2/(2L)$. The circular cross-section is important: general noncircular sections can warp and have a different torsion constant.

A neap tide has a reduced high-to-low tidal range when the directions to the Moon and Sun make approximately a right angle at the Earth. Their principal tidal contributions then partly oppose one another. The solar and lunar contributions are not equal, so this does not mean that tides disappear.

A spring tide has an enhanced high-to-low tidal range because the Moon and Sun are approximately aligned as seen from the Earth, so their principal tidal contributions reinforce. Such alignments occur twice in a lunar phase cycle.

A superleak is a narrow or porous connection that strongly suppresses the viscous normal component of a superfluid while allowing appreciable superfluid flow. Treating ${\mathbf{v}}_n\simeq 0$ is a regime assumption, not a perfect material filter for particular atoms. The thermomechanical effect uses that distinction to maintain a temperature difference during chemical equilibration. The idealization fails when normal leakage or dissipative quantum vortex motion becomes appreciable.

The Wiedemann-Franz law relates the electronic thermal conductivity of a normal degenerate metal to its electrical conductivity:An isotropic kinetic theory estimate gives $K_e=c_e{v}_F^2{\tau }_Q/3$. With electronic heat capacity density $c_e={\pi }^{2}n{k}_B^{2}T/(2E_F)$, $v_F^2=2E_F/m^{\ast }$ and $\sigma =n{e}^2{\tau }_{\sigma }/m^{\ast }$, the ratio is $L_0{\tau }_Q/{\tau }_{\sigma }$. Approximately elastic scattering with a smooth energy dependence gives equal heat and charge relaxation times. Inelastic electron-phonon scattering can make them different; the law also concerns the electronic contribution, not additional lattice heat conduction.^[ref]