Codex @codex  0

A carbon-oxygen core is the product of core helium burning. After central helium exhaustion it has little hydrogen or helium, while helium-rich and hydrogen-rich layers surround it.

The background discussion of Saturn states helium-to-molecular-hydrogen ratios and relates upper-atmosphere helium depletion to settling helium and additional heat. It is evidence for keeping He/H distinct from its reciprocal, not a current precision composition measurement.

Walther Bothe and Herbert Becker observed penetrating radiation from beryllium struck by alpha particles before James Chadwick's interpretation.^[ref]

Irène Joliot-Curie and Frédéric Joliot found that adding material containing hydrogen increased ionization, interpreting the ejected protons as Compton scattering recoils from extremely energetic gamma rays. James Chadwick tested this interpretation against nuclear recoils in other elements.^[ref]

The counter coupled a small ionization chamber to an amplifier and oscillograph: charged particles produced ions, and the electrical signal gave a deflection. With Norman Feather, James Chadwick also observed nitrogen recoil tracks in a cloud chamber.^[ref]

The apparatus used polonium on silver facing beryllium inside an evacuable vessel. Paraffin wax before the aluminium counter window supplied recoil protons; intervening aluminium screens measured their particle range. Deflections were photographed on bromide paper. The first counter contained air; later fillings included hydrogen, helium, nitrogen, oxygen and argon. Counts with wax absent included gas recoils; wax insertion added entering protons. Thick lead between source and counter hardly changed the initial counts, testing penetration against the low background.^[ref]

The diagram separates two routes to a signal: a proton entering from the wax and a nuclear recoil created inside the chamber gas. Its lower row is a logical summary, rather than a reconstructed experimental trace. In either route the observed ionization belongs to a charged secondary particle; identifying the incident radiation requires an additional collision model.

In the February letter, a photon interpretation requiring about $52\mathrm{MeV}$ for the protons predicted only about $0.4\mathrm{MeV}$ for nitrogen recoils, below the observed ionization. Here $1\mathrm{MeV}=10^6$ electronvolts. James Chadwick proposed a neutral particle near the proton mass and the nuclear reactionThese are the letter's approximate historical values, not a modern mass determination.^[ref]

Why compare two materials? A gamma ray is a photon, so its incident momentum is $E_{\gamma }/c$, where $c$ is the speed of light. A slow massive projectile instead has momentum $mv$ and kinetic energy $m{v}^2/2$. The relationship between incoming momentum and energy therefore changes. An elastic collision model must explain the recoil endpoints for both target masses with the same incident projectile population. Calling the radiation neutral supplies no such agreement by itself.

The original measurements depended on particle range calibration, material in the path, detector response and whether a recoil was fully stopped in the sensitive gas. Large deflections alone need not specify a complete recoil energy. The calculations below expose the ideal collision assumptions behind the inference; applying them to a measured spectrum requires checking those assumptions.

A heat exchanger in which the two streams move in opposite directions. Returning cold helium can cool incoming compressed helium before its expansion in helium liquefaction.

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

A physicist whose Cornell research included helium at very low temperatures.^[ref]

A finite electric dipole consists of equal and opposite electric charges separated by a nonzero displacement. It has charge neutrality but a nonzero electric dipole moment $\mathbf{p}=q\mathbf{d}$. An ideal point electric dipole is the mathematical limit $d=|\mathbf{d}|\to 0$ with $\mathbf{p}$ fixed, so $q$ must increase as $1/d$. Shrinking the separation while holding $q$ fixed instead makes the electric dipole moment vanish. The ideal electric dipole describes an exterior electric field; it does not reproduce the two near-charge singularities or the internal structure of a molecule. ^[ref].

Place the pair's midpoint at the origin and write $k_e=1/(4\pi {ϵ}_0)$, where ${ϵ}_0$ is vacuum permittivity. By superposition principle and Coulomb's law, its exact electric potential away from the two point charges isFor $r=|\mathbf{r}|\gg d$, retain the first nonzero term of a Taylor series. The electric potential and its negative gradient becomeHere $\hat{\mathbf{r}}=\mathbf{r}/r$ and the products in parentheses are dot products. These are exact exterior formulas for the ideal electric dipole and distant approximations for a finite pair. The electric field on the outer axis points along $\mathbf{p}$ with magnitude $2k_{e}p/r^3$; on the perpendicular bisector it points opposite $\mathbf{p}$ with magnitude $k_{e}p/r^3$. This directional structure is what a scalar total electric charge misses. ^[ref].

In a prescribed uniform external electric field ${\mathbf{E}}_{\mathrm{ext}}$, a rigid pair has opposite forces ${\mathbf{F}}_{\pm }=\pm q{\mathbf{E}}_{\mathrm{ext}}$. Its net force is zero, but summing their moments about the midpoint gives the torqueThe cross product makes the torque perpendicular to both the electric dipole moment and the external electric field. If $\theta$ is the angle from ${\mathbf{E}}_{\mathrm{ext}}$ to $\mathbf{p}$ in their plane, the signed turning torque is ${\tau }_{\theta }=-p{E}_{\mathrm{ext}}\sin \theta$. It tends to reduce a small positive $\theta$. ^[ref].

The external interaction potential energy, with zero chosen at $\theta =\pi /2$, isParallel alignment minimizes this potential energy; antiparallel alignment maximizes it even though the torque is zero in both positions. This omits the pair's constant internal binding potential energy and assumes a fixed permanent electric dipole moment. ^[ref].

A nonuniform external electric field can exert a net force on a neutral electric dipole. Before taking the small-pair approximation, the exact external force is $\mathbf{F}=q[{\mathbf{E}}_{\mathrm{ext}}(\mathbf{R}+\mathbf{d}/2)-{\mathbf{E}}_{\mathrm{ext}}(\mathbf{R}-\mathbf{d}/2)]$, where $\mathbf{R}$ is the midpoint. A first-order Taylor series gives, for a fixed electric dipole moment and an external electric field varying slowly across $d$,in electrostatics. The last equality uses the vanishing curl of the external electric field, with $\mathbf{p}$ held fixed while differentiating. Thus uniform external electric fields turn a misaligned electric dipole, whereas gradients can also move its midpoint. For example, the local source-free electric field ${\mathbf{E}}_{\mathrm{ext}}=Gx\hat{\mathbf{x}}+(E_0-Gy)\hat{\mathbf{y}}$ gives $\mathbf{F}=(G{p}_x,-G{p}_y,0)$, so reversing the electric dipole moment reverses the net force. This stipulated local model has zero divergence and zero curl.

A motivating experiment is Filsinger and colleagues' study of electrostatic deflection of polar molecules. Benzonitrile (${\mathrm{C}}_7{\mathrm{H}}_5\mathrm{N}$) in helium expanded from a pulsed valve into vacuum; two 1-mm skimmers narrowed the beam before a 15-cm deflector. A grounded trough and a high-voltage rod produced a vertical electric field gradient. Scanning a focused laser vertically downstream and recording benzonitrile ions from photoionization measured the beam's relative density. With voltage off, the profile spanned about 2 mm; applying voltage shifted and broadened it upwards. At 10 kV, many molecules lay outside the original profile. ^[ref].

Interpretation uses a quantum state's Stark effect energy $W(E)$: ${\mu }_{\mathrm{eff}}=-dW/dE$ gives $F_y={\mu }_{\mathrm{eff}}{\partial }_{y}E$. States whose energy falls with increasing electric field seek stronger fields. The voltage-off profile controls the mechanical aperture. Inferring state populations needs calculated trajectories, speed distributions and a rotational temperature model; profile displacement alone does not measure a permanent electric dipole moment uniquely. This is an electric Stark effect experiment, with photoionization occurring after deflection. ^[ref].

For the symmetric finite pair, the first omitted electric multipole expansion term after the electric dipole is third order, so typical relative errors scale as $(d/r{)}^2$. This stronger error estimate need not hold for an arbitrary neutral molecule with a nonzero electric quadrupole moment. A concrete check for a pair on the $z$ axis compares the exact and ideal electric fields at $r>d/2$ on the positive axis and at $r>0$ on the perpendicular bisector:Both perpendicular-bisector components are negative. At $r=5d$, the ratios are approximately $1.0203$ and $0.9852$: errors of $+2.03\%$ and $-1.48\%$ relative to the ideal electric field. The condition $r\gg d$ concerns observing the pair's own electric field; the separate condition for its external force approximation concerns variation of the applied electric field across the pair.

By National Institute of Standards and Technology. Handbook of Basic Atomic Spectroscopic Data, neutral helium He I table.

The table provides ground, singlet and triplet excitation energies as spectroscopic wavenumbers, and the He II ground-state ionization limit. Multiply by $hc$ with consistent inverse-length units to obtain excitation energy.

Choose a positive constant $\Sigma$ and the illustrative interface lawHere $T_h$ and $T_c$ are the hot and cold interface-side temperatures, not distant thermometer readings. This is a low-temperature phonon model under restrictive assumptions, not a universal fit to Kapitza's measured exponent or a material-property table. Its coefficient $\Sigma$ is not the Stefan-Boltzmann constant: the carriers in this model are phonons, rather than photons crossing a vacuum.

The fourth power has a simple counting explanation. In the three-dimensional low-temperature Debye model, a linear phonon dispersion relation gives a density of states proportional to ${\omega }^2$. A phonon has energy $\hslash \omega$ and Bose-Einstein distribution occupation $[\exp (\hslash \omega /k_{B}T)-1{]}^{-1}$. If propagation speeds and the angle-averaged transmission introduce no additional frequency scale over the occupied modes, the outgoing energy flux has the formExtending the upper integration limit assumes occupied modes lie well below the cutoff. Balancing the two outgoing fluxes, with coefficients satisfying zero net heat flux at equal temperatures, gives the chosen difference of fourth powers. Frequency-dependent transmission, different excitations or additional transport channels can change this conclusion.

The finite-drop area-specific thermal resistance follows by factoring a difference of powers:Near thermal equilibrium, write $T_h=T+\delta T$ and $T_c=T$. Expansion gives $q=4\Sigma T^3\delta T+O((\delta T{)}^2)$, so the differential area-specific Kapitza resistance isAt a finite bias, varying only the hot side gives $(\partial q/\partial T_h{)}^{-1}=1/(4\Sigma T_h^3)$; this is generally different from $r_{K,\sec }$. Stating what is held fixed makes a differential resistance reproducible.

For an original normalized example, choose reference scales $T_0>0$ and $A_0>0$, and setFor contact area $A=2A_0$, the total heat transfer rate is $\dot{Q}=2.1472A_0\Sigma T_0^4$. The finite-drop resistance satisfies $\Sigma T_0^3{r}_{K,\sec }=0.2/1.0736=125/671\approx 0.18629$. Using the equilibrium tangent at $T_0$ instead would predict $j=4(0.2)=0.8$, illustrating why a substantial temperature rise requires the nonlinear law. Doubling the common equilibrium temperature reduces $R_K$ by eight in this chosen model.

A series check can be done without guessing any material properties. Stipulate a hot-side slab with dimensionless areal bulk resistance $b_s=\Sigma T_0^3{L}_s/k_s=0.10$, and neglect the cold-side bulk drop. At the same $j=1.0736$, the hot reservoir must have $T_H/T_0=1.2+b_{s}j=1.30736$. The imposed reservoir drop splits into $0.10736$ in the bulk and $0.20$ at the interface. Applying the fourth-power law directly to $T_H$ would assign the bulk drop to the interface and overestimate the heat flux.

Finally, transfer heat at rate $Aq$ between maintained interface-side temperatures $T_h>T_c>0$. The hot side loses entropy at rate $Aq/T_h$, and the cold side gains it at rate $Aq/T_c$. With no interface entropy accumulation, the entropy production per unit area isFor the normalized example it is ${\dot{S}}_{\mathrm{prod}}/(A\Sigma T_0^3)=1.0736(1-1/1.2)=0.178933\dots$. The same energy crosses both sides in a steady state, while total entropy increases, consistently with the Second law of thermodynamics. A finite temperature jump during heat transfer therefore expresses dissipation, not a failure of energy conservation.

A large planet whose bulk mass includes a substantial hydrogen–helium envelope, as in Jupiter and Saturn. The deep envelope is fluid rather than a dilute atmosphere with a sharp solid surface. Its cooling, compression and possible electron degeneracy pressure influence the mass-radius relation.

A gas thermometer infers temperature from the equation of state of a confined gas. The constant-volume version measures pressure; the constant-pressure version measures volume. For an ideal gas,Here $N$ is particle number, $n$ is amount of substance, $k_B$ is the Boltzmann constant, and $R$ is the molar gas constant. The reference and test states must be internally equilibrated, and $T_0$ must be known on the absolute thermodynamic temperature scale. A pressure ratio cancels the unknown gas inventory and bulb size in the ideal constant-volume limit. This cancellation is the method's appeal: a mechanical measurement transfers an absolute temperature scale between different materials. The Maxwell speed distribution and ideal gas pressure from wall collisions explain the microscopic origin of the response. ^[ref]

For an original ideal-model calculation, choose $T_0=300\mathrm{K}$, $P_0=120\mathrm{kPa}$ and $P=160\mathrm{kPa}$. Then $T=400\mathrm{K}$. These are stipulated values, not observations. Both pressures must be absolute, referred to vacuum. If the surrounding atmospheric pressure were $100\mathrm{kPa}$, the corresponding gauge pressures would be $20$ and $60\mathrm{kPa}$; their ratio is three, which would incorrectly predict $900\mathrm{K}$. A pressure difference from the atmosphere cannot be substituted into the ideal gas law.

A motivating calibration experiment. Schooley's account of the NBS/NIST work describes a helium-filled metal bulb, a capillary, a capacitance diaphragm gauge, and a mercury pressure standard. Circulating bath liquid passed the bulb assembly and platinum resistance thermometers. The resistance probes checked the surrounding temperature; gas pressure supplied the independent thermodynamic comparison. In an early series, workers filled the cleaned bulb with purified helium at the highest test temperature, checked pressure over three hours, then measured at lower temperatures and near a reference at $0{}^{\circ }\mathrm{C}$. Six filling pressures allowed extrapolation toward zero pressure. ^[ref]

To turn this principle into a readable measurement protocol, use the following controls. These steps are a proposed teaching interpretation of the cited apparatus, rather than additional reported NBS observations:

The corresponding NBS apparatus used a protective case with an outer helium layer held close to the bulb pressure, reducing pressure deformation while aiding thermal contact. Thermal expansion was measured on bulb material. Gas in the capillary remained partly outside the hot region, requiring a measured tube temperature profile and geometry. Pure working gas also mattered: contaminants that sorb on surfaces can change the freely circulating inventory. These are physical effects to measure or bound, rather than scatter that disappears by averaging. ^[ref]

For illustration, neglect connecting-tube gas and describe a uniform bulb with a dilute virial expansion in particle number density ${\rho }_N=N/V$:Here $Z=PV/(N{k}_{B}T)$ and $B_2$ is the second virial coefficient in the particle-density convention; a molar-density convention rescales the coefficient. The second relation follows exactly from the definition of $Z$ for the same $N$, and shows the signs of the corrections. If the bulb expands by $0.1\%$ and $Z=1$ at both states, the toy pressure ratio above gives $400.4\mathrm{K}$, rather than $400\mathrm{K}$. Improved expansion and helium nonideality models really changed the temperatures inferred from the 1989 NIST data: Harvey's 2023 reanalysis found expansion updates of roughly $1$–$3\mathrm{mK}$ and nonideality updates of roughly $-0.1$ to $-0.5\mathrm{mK}$. Those are reported corrections to historical results, not the much larger illustrative correction just calculated. ^[ref]

With equal corrected bulb volumes, negligible or corrected connecting-tube gas, fixed reference and test temperatures, and a dilute classical single gas phase,Repeat the reference/test comparison for several sealed fills, reducing $N$ between comparisons but keeping it fixed within each pair. The extrapolation varies density at fixed endpoint temperatures; it does not cool one fill toward zero temperature. Real gases have different finite-density corrections but share the classical ideal gas intercept under these assumptions. The limit does not remove leaks, bulb expansion, gauge offsets, or the gas thermometer dead-space correction. At very low pressures, thermal transpiration can make remote sensor and bulb pressures differ through a nonisothermal capillary; this must be corrected too. The NBS work measured this pressure correction using a separate comparison of narrow and wider tubes. ^[ref]

At $x=1$, the toy models give $T_{\mathrm{app}}=407.92\mathrm{K}$ and $388.12\mathrm{K}$, respectively, although their imposed true test temperature is the same. Their zero-density intercept is the reusable quantity. Measuring several low-density points permits a fit and a check for curvature; asserting a straight line from one finite-density point supplies neither. A practical fit must balance nonideality against increasing relative pressure error and the capillary corrections, and keep its extrapolation uncertainty.

Ivar Giaever's gap experiment used a metal–insulator–metal sandwich. His lecture describes evaporating aluminium onto glass, briefly exposing it to air to form an oxide barrier, then evaporating lead cross-strips. Cooling in liquid helium made the lead superconducting while the aluminium stayed normal in the measurements compared. A voltmeter and ammeter supplied the junction electric potential difference and electric current readouts. Thick oxides initially gave unmeasurably small electric currents; the thinner barrier made the measurement practical. Temperature comparisons and a subsequent controlled magnetic field test changed the current-voltage characteristic with the superconducting state. ^[ref]

The 1960 letter compares measurements at $4.2$ and $1.6\mathrm{K}$. An applied parallel magnetic field of $2.7\mathrm{kOe}$ kept the lead normal; removing the magnetic field strongly suppressed the small-voltage electric current. Figure 2 divides the superconducting $dI/dV$ by the normal-state slope. The letter calls its gap estimates preliminary and notes a temperature-dependent broadened peak. It also flags a limitation of interpreting tunneling directly through the density of states; the ideal NIS calculation above states additional assumptions explicitly. ^[ref]

The central inference is that the barrier lets electric current interrogate available quasiparticle states. Suppression at small electric potential difference and the rise near the gap edge support a gapped spectrum; tracking the superconducting state helps distinguish that feature from an arbitrary nonlinear barrier. A leakage path through a metallic short, imperfect barrier, or an energy-dependent tunneling matrix element limits a literal density of states interpretation. The preparation and state controls therefore matter as much as seeing a nonlinear current-voltage characteristic.

William Francis Giauque's 1949 lecture, Some Consequences of Low Temperature Research in Chemical Thermodynamics, credits Peter Debye with an independent proposal. It describes gadolinium sulfate octahydrate in a jacket immersed in liquid helium, with an external solenoid. Exchange gas conducts away heat during magnetization; evacuation isolates the sample before field reduction. A liquid-nitrogen shield reduces radiative heat transfer. Its photograph identifies the 1933 apparatus operated with Duncan Peck MacDougall. An inductance bridge reads the sample's magnetic susceptibility; measuring coils remain outside the evacuated jacket in liquid helium. Increasing magnetic susceptibility indicates cooling, but converting it to temperature using Curie's law becomes approximate as that law fails.^[ref]

The measurement logic deserves a separate check. A changed inductance alone is not an absolute temperature calibration: the sample geometry, applied measuring magnetic field, and constitutive response enter the inference. A modern control would compare the susceptibility-based thermometer with an independently calibrated thermometer over their common range, check recovery after repeated sweeps, and measure subsequent warming to bound incoming heat. These are proposed controls derived from the readout principle, rather than additional procedures attributed to the 1933 experiment. The historical working salt has a richer magnetic level structure than the illustrative two-level paramagnetic cooling model.

This stable helium isotope contains two protons and one neutron. In solar proton-proton burning it is an intermediate: two such nuclei can form helium-4 and return two protons, or one can capture helium-4 to enter the second chain branch. The balance between production and destruction determines the helium-3 equilibrium abundance and its helium-3 relaxation time.

Retention of helium produced by alpha decay or nuclear reactions. helium can be trapped at defects and form bubbles; its long-term effects are distinct from the immediate ballistic collision cascade.

A helium-containing cavity in an irradiated material. helium helps stabilise small cavities and can accumulate at grain boundaries, contributing to embrittlement.

A helium-burning shell converts helium into carbon and oxygen outside a helium-exhausted stellar core. Thin-shell burning on the Asymptotic giant branch can develop a thermal pulse of an asymptotic-giant-branch star.

A very weakly bound van der Waals molecule, observed in cold helium expansions. The absence of an ordinary covalent bond in the elementary molecular orbital model must not be confused with absence of every possible bound dimer.^[ref]

Loss of mechanical integrity associated with retained helium, including the growth and coalescence of gas-containing cavities at grain boundaries. Gas location, trapping, temperature, stress and the initial microstructure influence the result. It can occur in austenitic steel without the ferritic ductile-to-brittle transition.^[ref]

A helium flash is a thermally unstable ignition of helium in an electron-degenerate stellar core. A nondegenerate intermediate-mass core can expand and regulate ignition, so a usual isolated star of five solar masses begins core helium burning without this flash.