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A solid can locally exceed its solidus because a thin solute-diffusion layer changes composition faster than thermal conduction changes temperature. Near a melting phase boundary, a depleted solid can join the equilibrium solidus at the interface while the adjacent, more concentrated solid remains above its own lower solidus. The assumed single sharp interface is then thermodynamically unstable to additional melting and may be replaced by a mushy layer.

At fixed temperature and pressure, a closed reacting system reaches thermochemical equilibrium by minimizing Gibbs free energy over nonnegative species amounts subject to conserved elemental abundances. For an allowed stoichiometric vector $\nu$, an interior minimum has ${\sum }_i{\nu }_i{\mu }_i=0$. More generally the chemical potential vector lies in the row span of the elemental conservation matrix at positive concentrations, with the appropriate one-sided inequalities for absent species. The first law of thermodynamics gives the thermodynamic differential, while the Second law of thermodynamics supplies the minimization direction.

The experimental distinction is concrete. A 1962 gallium arsenide p-n junction laser used top and bottom contacts, parallel polished faces, and current pulses with the device in liquid nitrogen. An infrared image converter recorded the radiation pattern; a spectrometer measured narrowing above laser threshold. Comparing the same device below and above laser threshold supplied a control. Pulsed, cooled operation did not demonstrate continuous operation at room temperature.^[ref]

Zhores I. Alferov's 1970 team photolithographically patterned stripe double heterostructure lasers on silver-covered copper plates. He reports sustained laser emission at room temperature; Fig. 4 shows the device structure. Independently, Izuo Hayashi, Morton B. Panish and collaborators demonstrated continuous double heterostructure laser operation with diamond heat sinks.^[ref]

The below-threshold comparison tests the onset of stimulated emission; continuous operation additionally requires sustained removal of waste heat. Neither observation alone measures the separate band offsets, optical overlap, or every loss mechanism. The schematic above therefore explains a design principle rather than reconstructing the historical apparatus.

A convection roll is a circulating cell pattern whose ideal velocity and temperature fields are periodic across a chosen horizontal direction and independent of the coordinate along the convection roll axis. In a two-dimensional stream function representation, its velocity field is $(-{\psi }_z,0,{\psi }_x)$. Rayleigh-Bénard convection can select stationary or oscillatory convection rolls; a two-dimensional convection roll may be unstable to three-dimensional disturbances, a different convection roll orientation, or slow amplitude and phase modulations. Existence of a saturated Landau amplitude equation branch does not alone establish its stability.

Energy transport by bulk fluid motion together with local thermal conduction. A prescribed adiabatic process profile constrains temperature but does not require zero convergence of this energy transport. A cloud held at a specified temperature can therefore have nonzero net radiative heating balanced by convection.

Two added electrons move above a frozen filled Fermi sea and attract within a thin energy shell. The Pauli-blocked continuum has a logarithmic pair susceptibility, so an arbitrarily weak net attractive constant interaction produces a state below the two-particle threshold at zero temperature. This two-added-particle result is not by itself a full many-electron superconducting state.

Core body temperature refers to internal temperature, distinguished from skin temperature. Different measurement sites are proxies with their own limitations, not interchangeable readings of a perfectly uniform interior. ^[ref]

The interface between Earth's metallic core and silicate mantle, approximately 2890 km beneath the surface. It is a major material interface and a thermal boundary across which core heat enters the mantle. Its temperature is distinct from mantle potential temperature because pressure and the lower thermal boundary layer matter.

The Cosmic Background Explorer connected two questions: does the cosmic microwave background follow a Planck spectrum, and how does its temperature vary across the sky? John C. Mather and George F. Smoot shared the 2006 Nobel Prize in Physics for these discoveries.^[ref] The FIRAS and COBE DMR comparisons below explain why measuring spectral shape and measuring angular structure require different information.

The cosmic microwave background temperature is about $2.725\mathrm{K}$, corresponding to the displayed thermal energy. The conversion uses $k_B\simeq 8.61733\times 10^{-5}eV/K$. Thus $2.35\mathrm{meV}$ would correspond to about $27.3\mathrm{K}$, rather than the present CMB. Primary references are NASA's CMB temperature measurements and NIST's Boltzmann constant in electronvolts per kelvin.

Two separately mixed layers with an interfacial thermal bottleneck require separate energy budgets. For lower input $F$, interlayer transfer $J$, surface loss $E$, and per-area heat capacities $H_l,H_u$, these are $H_l{\dot{T}}_l=F-J$ and $H_u{\dot{T}}_u=J-E$. At equilibrium all three fluxes agree, but a finite interface resistance retains a temperature difference. Thus layer count alone does not specify the cooling rate.

Confining a near-critical fluctuating order parameter changes its free energy. Its range is governed by the growing correlation length, and its sign depends on the boundary preferences. This is a thermal collective effect, distinct from the zero-temperature ideal-conductor electromagnetic formula.

The critical micelle concentration is the approximate surfactant concentration above which additional dissolved amphiphile predominantly enters micelles rather than remaining as isolated molecules. It is a concentration crossover governed by equality of the molecular chemical potential in the two environments, not a claim that absolutely no aggregates exist below a sharp threshold. Temperature, ionic conditions and molecular structure affect its value.

For fixed temperatures and stress-free impermeable boundaries, linear marginal Rayleigh-Bénard convection with dimensionless horizontal wavenumber $q$ obeys $\mathrm{Ra}(q)=(q^2+{\pi }^2{)}^3/q^2$. Differentiation gives $q_c=\pi /\sqrt{2}$ and the displayed minimum. The threshold depends on the mechanical boundary conditions: two rigid no-slip boundaries instead give approximately $1708$.

For a system coupled to a thermal reservoir at fixed temperature $T$, prepare the forward process in the initial canonical ensemble and the reverse process in the final control parameter's canonical ensemble. Reverse the protocol, assume microscopic reversibility and matching path probability support, and count $W$ as work done on the system. With Helmholtz free energy difference $\Delta F=F_{\mathrm{final}}-F_{\mathrm{initial}}$ and $\beta =1/(k_{B}T)$,The final forward distribution need not already be in thermal equilibrium. Its reverse preparation is prescribed equilibrium preparation, rather than the actual forward final distribution used in finite trajectory entropy production. Thus the paired path logarithm $\beta (W-\Delta F)$ is dissipated work in thermal units, not automatically the instantaneous total stochastic entropy production at the end of an arbitrary driven protocol. ^{[ref] [ref]}

cryo-electron microscopy images specimens at cryogenic temperature. In biological single-particle analysis, many weak projections of vitrified particles are combined to estimate a three-dimensional scattering electrostatic potential map; specimen heterogeneity and radiation damage limit the recoverable detail.

In NIST's 2001 comparison, the Laser-Optimized Cryogenic Radiometer held its cavity near $5.4\mathrm{K}$ and controlled the heat sink separately. Opening the beam reduced heater power at fixed cavity temperature; an AC bridge read the temperature. Tests used stabilized, spatially filtered lasers at $488$, $514$ and $633\mathrm{nm}$, with horizontal input and a movable photodiode transfer detector in the cavity's optical plane. The paper contrasts this with the High Accuracy Cryogenic Radiometer's optical heating followed by two electrical heating cycles and interpolation.^[ref]

The two laboratories compared calibrations of the same transfer detectors, and checked their spatial response. Alignment differences and transported dust limited that comparison; its agreement did not prove universal equivalence. Its correction tables retain window transmission, cavity radiometric absorptance, alignment, electrical readout and optical-electrical heating non-equivalence. The heat sink was controlled, but §3 does not specify its numerical setpoint: the $5.0\mathrm{K}$ reference in the preceding example is stipulated.^[ref]

For a reproducible substitution measurement, record the beam's wavelength and entrance boundary, receiver and reference temperatures, heater terminal voltage and electric current, dark readings before and after illumination, and the settling rule. Check beam collection at the aperture, window transmission, radiometric absorptance, deposited lead heating, and sensitivity to beam position. Repeatability alone misses a stable correction error. Photometric calibration can subsequently use this radiant flux scale, but photometry still needs the appropriate spectral visual weighting; the cold receiver does not itself measure luminous flux.

A detection probe with cooled electronic components to reduce noise and improve sensitivity. The sample need not itself be cooled to the electronics temperature.

A cryogenic radiometer uses low-temperature operation to improve electrical substitution radiometry through particular material and thermal properties. Lower heat capacity of a copper receiver permits a substantial cavity without proportionately slow relaxation; cold shields reduce unwanted thermal radiation, and heater leads with superconductivity can suppress resistive lead heating. These features do not make radiometric absorptance exactly one or abolish optical-electrical heating non-equivalence.^[ref]

Cold surroundings matter as much as a cold receiver. In a simplified gray-surface model, net outward thermal radiation is $ϵ\sigma A(T^4-T_s^4)$, using emissivity $ϵ$, area $A$, shield temperature $T_s$ and the Stefan-Boltzmann constant $\sigma$. Cold shields reduce the incoming term as well as the receiver's emission; cooling the receiver alone while leaving it exposed to warm surroundings does not remove incoming radiation. An entrance window still provides an optical and thermal path. Vacuum reduces gas-mediated heat transfer, while an intentional radiometer thermal link remains necessary. This model explains specific losses, rather than promising a universal improvement factor or perfect accuracy.

Cryogenics studies the production and use of very low temperatures. Its experimental challenge is a balance of rates: a refrigerator must remove energy from a cold region faster than unwanted heat transfer supplies it. A low temperature reading alone does not establish a useful cooling capacity. A maintained cold liquid bath is especially valuable because latent heat allows it to absorb heat while its phase fraction changes at nearly fixed temperature and pressure.

Helium liquefaction illustrates why staged cooling and recovery of a returning stream's cooling capacity matter. A counterflow heat exchanger transfers heat from the incoming stream to the colder returning stream without mixing them. This reduces the incoming specific enthalpy before expansion. The heat exchanger recovers cooling already produced; the surrounding refrigerator still requires input work and rejection of heat to a warmer environment.