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Hot electrons in galaxy clusters scatter photons of the cosmic microwave background, producing a characteristic spectral distortion. The thermal contribution measures a line-of-sight integral of electron pressure, whereas X-ray emission depends approximately on the integral of electron density squared times a temperature-dependent emissivity. Combining the two with a geometrical model can constrain a cluster's physical size and hence its angular diameter distance. Temperature structure and departures from the assumed geometry introduce systematic uncertainty.

An alloy developed to sustain mechanical load at high temperatures while resisting creep and environmental attack.

In a local London equation description, the measured London penetration depth satisfies ${\lambda }_L^2=m/({\mu }_0{n}_s{e}^2)$ when $n_s$ counts superconducting electrons and $m$ is their effective mass. Thus screening measures $n_s/m$. If instead one counts Cooper pairs, use pair mass $2m$ and pair electric charge $-2e$, giving pair density $n_s/2$ for the same screening length. Identifying this superconducting carrier density with the total conduction-electron density requires the additional assumption that the relevant carriers all participate in screening at the measurement temperature.

A superconducting critical current $I_c$ is the limiting electric current, for a specified sample, temperature, external magnetic field and measurement criterion, beyond which the chosen state of superconductivity no longer sustains dissipation-free transport. A weak connection can limit a loop before the rest of its material does. A magnetic field generated by the electric current itself also contributes to the sample's conditions. A limit on the initial superconducting persistent current and a bound on its subsequent electrical resistance answer different experimental questions.

A limiting transport current per conductor cross-sectional area for specified temperature, magnetic field and dissipation criterion. Local current distribution, flux motion and weak links can make it distinct from an intrinsic depairing limit. Its variation with the operating conditions is summarized by a superconducting critical surface.

The boundary of a stated superconducting transport criterion in temperature, magnetic field and current-density space. A value measured under one combination does not specify a winding's usable current at another field or temperature. Strain and sample geometry can require further parameters.

In this fully gapped isotropic BCS quasiparticle spectrum, $\Delta$ is the minimum single-quasiparticle excitation energy. The interval between $-\Delta$ and $+\Delta$ has width $2\Delta$. A process that creates two quasiparticles from the BCS ground state has an energetic threshold $2\Delta$, subject also to its transition-selection rules.^[ref]

For example, two independently created excitations with ${\xi }_1={\xi }_2=0$ cost $E_1+E_2=2\Delta$. With the chosen cutoff example this is $1.46574\mathrm{meV}$, while adding one excitation costs at least $0.732871\mathrm{meV}$. This factor of two matters when comparing photon absorption with the Giaever superconducting tunnelling experiment. Quantum tunnelling from a normal metal can add one excitation to a sample exhibiting superconductivity; with two gapped electrodes, their excitation costs add. A minimum excitation cost is an energetic condition; the measured onset can additionally depend on temperature and on the available transitions.

A change between normal and superconducting phases. Its existence gives a physical reason why an experimentally successful normal-state resistance model need not extrapolate through every lower temperature.

The superconducting transition temperature $T_c$ is the temperature at which the material enters superconductivity on cooling in zero applied magnetic field and with negligible measuring electric current. A measured transition at nonzero magnetic field or appreciable electric current need not occur at this zero-field $T_c$. If a resistance-versus-temperature experiment for superconductivity gives a transition spread over several temperatures, report whether its quoted value is the onset, midpoint, or disappearance of detectable electrical resistance, together with the experimental conditions. This operational distinction matters when comparing samples. ^[ref]

A liquid maintained above its equilibrium boiling temperature at the imposed pressure. A barrier to nucleation can delay its conversion into vapor, making this a metastable state.

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.

A vapor whose partial pressure exceeds its vapor pressure at the local temperature; nucleation can permit condensation.

The 1935 primary paper The Spin-Isomerization of Hydrogen at Paramagnetic and Diamagnetic Surfaces, pp. 1251–1252, describes evacuated catalyst samples cooled before admitting molecular hydrogen. Gas samples were analyzed in a glass cell in liquid air: a heated platinum wire and Wheatstone bridge converted changes in thermal conductivity into electrical resistance readings, calibrated against normal hydrogen.^[ref]

Its comparison of oxide surfaces supplied a control: specimens exhibiting strong paramagnetism converted the gas much faster than the comparison exhibiting diamagnetism despite all adsorbing molecular hydrogen. Contact time, pressure, and temperature were recorded. This supports a role for surface magnetic properties beyond adsorption alone; it does not identify every microscopic surface site or directly measure nuclear-spin orientations.^[ref]

The inference chain is preparation, contact with a specified surface, then a calibrated bulk transport readout of composition. It is distinct from a direct calorimetry measurement of heat capacity: transport and heat capacity can both depend on the spin populations, but they are different measured quantities. Changing the catalyst's own surface structure, introducing magnetic impurities, or changing the detector's gas pressure can confound an uncalibrated comparison. The free-rotor partition functions predict equilibrium populations and stored energy, not the conversion rate on a particular surface.

For finite-range interactions and bounded wall couplings, only sites within an interaction range of the wall have altered local environments. In a regular container of linear size $L$ and dimension $d$, the surface area scales as $A\sim L^{d-1}$ while volume scales as $V\sim L^d$. A boundary contribution of order $A$ therefore gives an order-$1/L$ correction to free-energy density. Away from a critical phenomenon, a regular surface contribution often has the form $F=Vf+A\gamma +\cdots$, with $\gamma$ a wall Helmholtz free energy per area. Critical finite-size rounding, edges and long-range interactions require separate care; vanishing relative boundary size does not guarantee a simple surface expansion in every regime.

An exact check uses the zero-field nearest-neighbour Ising model on a chain of $N\ge 3$ sites with fixed coupling $J>0$ and $K=J/(k_{B}T)$. The open chain has $N-1$ bonds, while the periodic ring has $N$. Their finite partition functions and dimensionless Helmholtz free energies per spin $g_N=F_N/(N{k}_{B}T)$ areFor the open chain, sum the last Ising spin successively: each removed bond contributes $2\cosh K$, leaving a factor of two. For the ring, the transfer matrix for the one-dimensional Ising model has eigenvalues $2\cosh K$ and $2\sinh K$, and its $N$th power's trace gives the displayed sum. The two ends are a constant-size boundary in one dimension, so the open-chain total correction is $k_{B}T\log \cosh K$, and its per-spin correction is exactly proportional to $1/N$.

Both sets of finite results approach the same $g_{\infty }$ at the fixed plotted $K=0.8$. Multiplying the correction by $N$ exposes the open-chain boundary term, while the ring correction vanishes exponentially at fixed finite $K$. This model's thermodynamic limit remains a real analytic function at every positive temperature: having a thermodynamic limit does not require a phase transition.

For a liquid interface with constant surface tension $\gamma$, the mechanical surface energy is the energy associated with its surface area $A$:The additive constant $C$ does not affect the work needed to change surface area. For several interfaces with the same surface tension, $A$ means their total surface area; the two faces of a thin soap film must both be counted. ^[ref]

With assumed $\gamma =0.072\mathrm{N}{\mathrm{m}}^{-1}$, creating $1\mathrm{c}{\mathrm{m}}^2=10^{-4}{\mathrm{m}}^2$ of one interface requires $7.2\mu \mathrm{J}$ of reversible work. In the sliding-rim example illustrated under surface tension, moving a $20\mathrm{mm}$ rim outwards by $1\mathrm{mm}$ creates $4.0\times 10^{-5}{\mathrm{m}}^2$ across the two faces and requires $2.88\mu \mathrm{J}$. Independently, multiplying its $2.88\mathrm{mN}$ holding force by the displacement gives the same work.

This makes the preference for a round isolated droplet precise. At fixed volume $V$, the three-dimensional isoperimetric inequality giveswith equality for a sphere. Thus a sphere minimizes surface energy among sufficiently regular closed shapes enclosing the same volume, when surface tension is uniform and other energy contributions are negligible. This is an application of the isoperimetric inequality, rather than a proof of it. Walls, wetting, and gravity can change the shape by adding constraints or other energy contributions. ^[ref]

The differential rule $d{E}_{\mathrm{s}}=\gamma dA$ integrates directly to $E_{\mathrm{s}}=\gamma A+C$ because surface tension is constant. At fixed liquid volume, decreasing surface area can drive viscous fluid flow; the released surface energy can supply viscous dissipation. This mechanical bookkeeping is useful for predicting which deformations can release energy. If surface tension depends on temperature or composition, distinguish this mechanical area-work relation from the complete interfacial thermodynamic free energy and internal energy.