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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 $1708$.

The critical temperature is the temperature at which the excited states of a Bose system are just saturated as the chemical potential approaches the lowest one-particle energy. Cooling below it puts the excess particles into the condensate.

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.

cryo-electron tomography reconstructs three-dimensional specimen information from tilted projections at cryogenic temperature. Radiation exposure and incomplete angular coverage limit resolution and interpretation.

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.

A cryostat maintains an apparatus at low temperature; a liquid-helium bath is one implementation.

A crystal X-ray spectrometer selects a narrow wavelength interval by diffraction at known interplanar spacing $d$ and Bragg angle $\theta$. For fixed diffraction order $n$, a known $d$ lets an angular measurement determine $\lambda =2d\sin \theta /n$. This reverses the crystallography problem of finding $d$ from a known wavelength.

Siegbahn's laboratory developed X-ray tubes and evacuated spectrographs for longer wavelengths. Rotating the crystal selected reflections; photographic plates recorded positions. Opposite-side exposures measured angles on both sides of the direct beam. Later tube instruments mounted a beam-defining slit opposite the plate holder. A separate double instrument compared two reflected beams with two ionization chambers. These were different instruments, not one universal assembly.^[ref]

The Nobel biography describes constant-temperature spectrographs and a precision angular scale read by diametrically opposed microscopes.^[ref] Temperature control matters because thermal expansion changes $d$; an accurately read angle cannot compensate for an unknown interplanar spacing.

The schematic separates preparation, selection and readout. An X-ray tube target supplies Characteristic X-rays with a possible bremsstrahlung background. A collimator limits incident directions; evacuation reduces air X-ray attenuation. Rotating the crystal brings different wavelengths into reflection. A photographic plate stores line positions over an exposure, whereas an ionization chamber can measure collected electric charge at successive angles. Neither readout alone establishes absolute wavelength: an angular zero, angular-scale calibration, diffraction order, $d$ and any required refraction correction enter the inference. Narrower slits reduce accepted angular spread but also reduce signal; a weak line can become harder to locate despite a narrower instrumental response.

A control reported in Siegbahn's lecture photographed a line system with copper-sulphate and calcium-sulphate crystals. Irregular relative shifts around copper's characteristic frequency supported anomalous dispersion; the arrangement excluded differing absorption across that frequency as the explanation.^[ref] A finer angular scale alone cannot remove this physical refraction correction.

The primary roll and temperature modes force only a horizontally uniform temperature mode proportional to $\sin (2\pi z)$ at quadratic order. Its amplitude is $c=-\alpha ab/(8\pi )$. Its feedback on the primary temperature mode saturates magnetoconvection, giving the displayed positive squared amplitude above threshold. Thus a simple roll onset is a supercritical pitchfork bifurcation for every $Q\ge 0$. The three-mode Fourier truncation and full perturbation calculation agree through the cubic solvability condition because all forced quadratic modes are retained; this is not an assertion that the truncation gives an exact finite-amplitude PDE solution.

// Destination: dynamical-systems.bigb

Writing the interaction as $V$ and averaging over the fast modes, which have a Gaussian distribution, gives the cumulant expansion $F_{\mathrm{eff}}=F_0+⟨V⟩-\frac{1}{2}(⟨V^2⟩-⟨V{⟩}^2)+\cdots$. The second cumulant selects connected Feynman diagrams.

The coefficient in the high-temperature Curie Magnetic susceptibility.

At fixed positive temperature and weak field, independent spin-$J$ moments have total moment $M=N(g{\mu }_B{)}^{2}J(J+1)B/(3k_{B}T)$. This is the Curie law; the corresponding density divides by the sample volume.

For independent moments of angular momentum $J$ at small ${\mu }_B|B|/(k_{B}T)$, the zero-field variance is $⟨m^2⟩=J(J+1)/3$. Differentiating the canonical mean gives the magnetic susceptibilityThus susceptibility is inversely proportional to temperature. The small-field or high-temperature condition distinguishes this law from the exponentially vanishing fixed-field susceptibility of a saturated paramagnet at low temperature.

The Curie temperature is the transition temperature above which a ferromagnetic material loses spontaneous magnetization. A hot conducting planetary core can generate a magnetic field through dynamo action without behaving as a permanent magnet. The blocking temperature at which a mineral records crustal remanence also depends on its relaxation timescale and need not equal its Curie temperature.

The Curie–Weiss law approximates the magnetic susceptibility of interacting localized moments by $\chi =C/(T-\Theta )$ in a suitable paramagnetic temperature range. A positive Weiss temperature often reflects predominantly ferromagnetic interactions, while a negative value often reflects antiferromagnetic interactions. It is not a universal exact law through an ordering transition, and crystal-field effects or itinerant carriers can require another description.

The current-phase relation of a superconducting weak link gives its pair electric current $I_s$ as a function of the Gauge-invariant Josephson phase $\varphi$, at specified temperature and other conditions. For the ideal junction used below, $I_s=I_c\sin \varphi$ with $I_c>0$. More general periodic relations can contain higher Fourier harmonics; their maximum magnitude defines the corresponding superconducting critical current.