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Radiative recombination removes the kinetic energy of captured electrons from a plasma. The thermal cooling per event is not the entire emitted photon energy: binding energy had been stored as ionization energy. If recombination is the only cooling and ionization is balanced, the mean removed kinetic energy equals the mean photoelectron heating energy.

For full hydrogen ionization after instantaneous reionization, $n_{e0}=3{\Omega }_b{H}_0^2/(8\pi G{m}_p)$ and $H(z)=H_0(1+z)\sqrt{1+{\Omega }_{m}z}$. Integrating Thomson scattering givesFor ${\Omega }_m=1$, $\kappa ={\Omega }_b{C}_H[(1+z_{\mathrm{re}}{)}^{3/2}-1]$. At fixed baryon density and redshift, lowering the matter density reduces the expansion rate and increases the optical depth.

// Target: cosmology.bigb

Solvation is stabilization through interactions between a solute and its solvent. It involves both the solute's surroundings and reorganization of the solvent: neighbors move, orientations change, and existing interactions compete with new ones. Electric forces, Van der Waals forces, and specific interactions such as hydrogen bonds can contribute. Solvation can also occur at groups on an insoluble material, so it does not require complete dissolution.^[ref]

Solvation by water is called hydration. Hydration of ions is the charged special case, but neutral molecules such as sucrose can also interact with water without becoming ions. A hydrated cation retains its positive net electric charge; orienting neutral water molecules around it does not supply an extra electron. Dissolution, desolvation, and ionization describe different changes, as the examples below show.^[ref][ref]

The useful question is therefore not just whether a solute attracts a solvent, but how much the whole system gains after paying for the rearrangements. Solvation free energy describes a specified transfer into a solvent; solvation thermodynamic balance explains why even cooling during dissolution can accompany favorable mixing. The existing solubility discussion concerns how much remains dissolved at chemical equilibrium.

Solvent-specific stabilization means that the same species need not have the same solvation free energy in different solvents. Chemical polarity, hydrogen bonds, and the cost of reorganizing each solvent all matter. Water can stabilize ions by ion-dipole attractions; a favorable isolated attraction alone still cannot decide the total dissolution balance.^[ref]

With a common gaseous reference and compatible dissolved standard states, subtraction removes that reference:A negative result favors solvent B under those reference conventions. This comparison assumes the same species $X$: if protonation, ionization, or formation of ion pairs changes its identity, include a chemical equilibrium for each corresponding chemical reaction.

The even, spherically symmetric hydrogen atom ground state has no permanent electric dipole, so a uniform electric field gives no first-order energy shift. The electric-dipole selection rules for hydrogen allow odd-parity excited states to mix in at second order, yielding a negative shift quadratic in the field.

Here there is a nondegenerate energy eigenvalue, so second-order nondegenerate perturbation theory applies. With $H^{\prime }=e\mathcal{E}z$, it givesThe notation includes integration over the continuous spectrum, as well as a sum over discrete energy eigenstates. All denominators are negative because $1s$ is the ground state. The positive electric polarizability ${\alpha }_{1s}$ therefore lowers its energy.^[ref]

An independent way to see the coefficient avoids truncating that sum. Define $E_{\mathrm{h}}=e^2/(4\pi {ϵ}_0{a}_0)$ and $f=e{a}_0\mathcal{E}/E_{\mathrm{h}}$. In coordinates $\mathbf{\rho }=\mathbf{r}/a_0$, the dimensionless unperturbed Hamiltonian operator and normalized wavefunction are $h_0=-{\nabla }_{\rho }^2/2-1/\rho$ and ${\psi }_0={\pi }^{-1/2}{e}^{-\rho }$, with energy eigenvalue $-1/2$. Writing the first-order wavefunction correction as $f\chi$, substitution into the time-independent Schrödinger equation givesThe second equality can be checked by differentiation. Its parity makes $\chi$ orthogonal to ${\psi }_0$. The resulting integral issince the normalized radial probability density is $4{\rho }^2{e}^{-2\rho }$, giving $⟨{\rho }^2{⟩}_0=3$ and $⟨{\rho }^3{⟩}_0=15/2$. Converting back to SI units gives the stated electric polarizability. The induced electric dipole moment is $-\partial \Delta E/\partial \mathcal{E}={\alpha }_{1s}\mathcal{E}$ and tends to zero with the electric field, unlike the selected degenerate branches' leading dipoles. Both calculations describe weak-field level shifts; a spatially uniform static electric field also allows eventual quantum tunnelling and ionization, which these real energy corrections do not calculate.

State-selective field ionization distinguishes atomic quantum states through their different electric field ionization thresholds.

A strong electrolyte yields essentially all ions from its dissolved substance in the specified solvent and conditions. Strength does not specify concentration.

For dissolved sodium chloride in dilute water, one dissolved formula unit supplies one sodium cation and one chloride anion. For hydrogen chloride in water, ions instead arise by proton transfer to the solvent, giving hydronium and chloride. Thus separating existing ions by dissociation of an ionic compound and producing ions from neutral molecules by ionization can both give a strong electrolyte.^[ref]

The denominator in “essentially all” is the dissolved inventory. Solubility asks how much enters the chemical solution; electrolyte strength asks what happens to the part that enters. A sparingly soluble ionic compound can therefore have an essentially fully dissociated dissolved portion. Undissolved solid does not count as neutral dissolved chemical species, and low electrical conductivity alone does not establish a weak electrolyte.^[ref]

The diagram's weak electrolyte has the smaller converted fraction but more ions in the same volume: ten charged symbols versus four. Even at equal total ion concentrations, different ion drift velocities can give different electrical conductivities. Strength is useful for predicting dissolved chemical species in chemical equations; concentration and transport properties are additionally needed to predict electric current.

An inspected motivating experiment is Harvard Natural Sciences Lecture Demonstrations' Conductivity of Solutions by Harvard Natural Sciences Lecture Demonstrations, Setting it Up. Separate labeled beakers hold aqueous sodium chloride, sucrose, acetic acid and hydrochloric acid samples specified at $0.10\mathrm{mol}{\mathrm{L}}^{-1}$, with about $150\mathrm{mL}$ in $250\mathrm{mL}$ beakers; distilled water supplies a blank. Metal electrodes enclosed in tubing connect to a lamp. The readout is lamp brightness; the page reports that an ionic chemical solution lights the lamp. A separate deionized-water rinse station limits carryover between samples.^[ref]

For a controlled interpretation, keep electrode spacing and immersed area fixed, compare at the same temperature, and recheck the blank after rinsing. These are additional controls: the page supplies no numerical electrode geometry, temperature or calibration. Brightness supports the presence of mobile ions; it cannot alone measure the converted fraction because concentration, ion mobility and circuit response also matter. Its classification of insoluble compounds as weak should not replace the dissolved-inventory distinction above.^[ref]

The primary model computes cold and warm atomic equilibrium branches and the pressure interval where they coexist. Its heating and cooling depend on the radiation field, ionization and gas abundances. This is a thermal-phase model for the Milky Way disk, not a claim that all interstellar phases share one pressure or a perfectly static configuration.

Space Science Reviews 134, pp. 155–190. The article discusses plasma emission, absorption, ionization and practical X-ray diagnostics. Its institutional NED version includes a cooling-function section with explicit density normalization.

A detector that reconstructs one coordinate from the time required for ionization electrons to drift in an electric field, with the other coordinates measured by the charge or light pattern. A two-phase xenon implementation obtains depth from the delay between prompt particle scintillation and gas-phase charge readout.

A triggered cloud chamber expands in response to an event indicated by other detectors, rather than hoping that a particle arrives during a randomly chosen sensitive interval. The ionization trail is produced before expansion. The apparatus must respond while the ions remain sufficiently localized to make a useful photograph.

Tunnel ionization is ionization through quantum tunnelling across a field-distorted binding barrier.

The upper part of the terrestrial atmosphere includes regions where ionization makes the response to electromagnetic waves depend strongly on mobile electrons. The ionosphere describes that electrical structure; its boundaries need not coincide with the boundaries defined by temperature in the mesosphere and thermosphere.

Diffuse ionized gas near $10^4\mathrm{K}$, maintained by ionization and heating balanced by radiative losses. It is physically distinct from warm gas that remains mostly neutral despite a similar kinetic temperature.

Charles Thomson Rees Wilson's 1927 lecture recalls electron cloud trails produced with X-rays in 1911, then alpha particle trails from a radium-tipped source. The improved 1912 chamber had a wide, shallow space with a suddenly falling floor to limit stirring. A roof–floor clearing electric field removed unwanted ions. Sharp photography required expansion, then the ionizing particle, then illumination.^[ref]

The piston mechanism connects the space beneath the moving floor to an evacuated vessel: opening the valve lets the floor drop. Water in a trough supplies the vapor. The Smithsonian's surviving demonstration object is explicitly a 1961 replica of the 1912 apparatus.^[ref]

After dust removal, X-rays increased droplet formation above a lower expansion threshold; larger expansions also produced dense background cloud.^[ref]

The control logic separates a localized ionization trail from fog. Delays permit ion displacement and drop growth; stirring deforms the trail. A photograph alone supplies neither an absolute event time nor a calibrated particle energy.