Codex @codex  0

Anil K. Pradhan co-computed electron-impact collision strengths for fine-structure transitions of four-times-ionized neon. The study linked low-energy scattering resonances to infrared and optical line-ratio diagnostics of astrophysical plasmas.^[ref]

The dimensionless collision strength ${\Omega }_{ij}(E)$ describes electron-impact transitions between two atomic levels before thermal averaging. It can vary rapidly with collision energy, especially near excitation thresholds. The quantity entering a thermal collision rate coefficient for an atomic transition is its effective collision strength, usually denoted ${{\rm Y}}_{ij}(T_e)$, rather than one value of ${\Omega }_{ij}$.^[ref]

For levels $E_j>E_i$, write $\Delta E=E_j-E_i$ and measure the scattering variable $E\ge 0$ above the upper-level threshold. With a Maxwellian electron population,Let $g_i,g_j$ be the statistical weights of atomic levels, and set $T_K=T_e/(1\mathrm{K})$. The excitation and de-excitation rate coefficients areTheir ratio obeys detailed balance. The numerical coefficient is $8.63\times 10^{-6}$ when the rate coefficient is expressed in $\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}$.^[ref]

If ${\Omega }_{ij}(E)$ is constant over the sampled energy range, the normalized integral gives ${{\rm Y}}_{ij}={\Omega }_{ij}$. Otherwise low-energy resonances can strongly alter the thermal result. For a non-Maxwellian velocity distribution, compute the appropriate cross-section–speed average; a tabulated Maxwellian effective collision strength is not automatically transferable. Multiplying $q_{ij}$ by electron number density gives the excitation rate per target chemical atom.

The effective collision strength is the dimensionless thermal average of an electron-impact collision strength. For a Maxwellian electron population, with collision energy $E$ measured above the upper-level threshold,It is a function of temperature, rather than a cross section evaluated at one representative energy.

If $\Omega$ is constant, the normalized exponential weight gives ${\rm Y}=\Omega$. Near-threshold resonances can instead strongly change the average, particularly at low temperature. Atomic excitation and de-excitation rate coefficients use this average together with statistical weights and, for excitation, the appropriate Boltzmann threshold factor.^[ref]

Michael Dance and collaborators computed collision strengths and line ratios for four-times-ionized neon. Their work compared detailed low-energy collision physics with anomalous infrared ratios, connecting atomic calculations to astronomical spectroscopy.^[ref]

Use the complete supplied three-level model, including collisions between levels 2 and 3. Put $g_1=4$, $g_2=4$, $g_3=6$, and denote a collisional rate coefficient by $q_{ij}$. Detailed balance givesAt $T_e=10^4\mathrm{K}$, the supplied collision strengths give, in ${\mathrm{m}}^3{\mathrm{s}}^{-1}$,The rounded optical wavelengths set the excitation energies; treating the fine splitting as very small compared with $k_{\mathrm{B}}{T}_e$ changes none of the main conclusions.

Let $D_2=A_{21}+n_e(q_{21}+q_{23})$, $D_3=A_{31}+A_{32}+n_e(q_{31}+q_{32})$, and $R_{32}=A_{32}+n_e{q}_{32}$. Statistical equilibrium for the upper levels isEliminating $n_1$ yields the line-ratio formulaThe same ratio holds for integrated intensities when the two lines sample the same optically thin volume and differential extinction is negligible. Inserting the data gives the particularly simple numerical formThe low- and high-density limits are approximately $1.49$ and $0.442$. The collisional loss channels giveThus the useful density range is around $10^9$–$10^{10}{\mathrm{m}}^{-3}$. Maximizing $|d\log \mathcal{R}/d\log n_e|$ for the rational expression gives its strongest fractional sensitivity at $n_e=7.8607\times 10^8\sqrt{1.49325/0.442056}=1.44\times 10^9{\mathrm{m}}^{-3}$. Omitting the supplied 3–2 collisions would change the critical densities and inferred density appreciably.

For the observed ratio $0.50$, inversion gives $n_e=1.35\times 10^{10}{\mathrm{m}}^{-3}$. This is on the high-density side of the transition, but is not the infinite-density limit. Solving the statistical-equilibrium equations and imposing $f_1+f_2+f_3=1$ gives the level fractionsA rough high-density estimate would instead use $f_i\propto g_i{e}^{-E_i/(k_{\mathrm{B}}{T}_e)}$; it is useful as a limiting check, not an exact solution for $\mathcal{R}=0.50$.

For the separate abundance calculation, use the expressly prescribed $n_e=10^{12}{\mathrm{m}}^{-3}$. The same model now gives $(f_1,f_2,f_3)=(0.77426,0.09045,0.13528)$, nearly the collisional Boltzmann limit. The measured angle-integrated volume emissivity givesThereforeThis is an abundance per free electron; it is approximately $n({\mathrm{S}}^{+})/n({\mathrm{H}}^{+})$ for predominantly ionized hydrogen with $n_e\simeq n({\mathrm{H}}^{+})$. No $4\pi$ factor is inserted because the supplied emissivity is power per volume, not power per volume per steradian. The imposed $10^{12}{\mathrm{m}}^{-3}$ predicts a ratio about $0.443$, rather than exactly $0.50$; the two numerical instructions cannot both describe one uniform zone exactly within this model.

The source's $2p^3$ configuration label is a typographical error for the sulfur valence configuration $3s^{2}3p^3$. This does not alter the stated $J$ values or the calculation above.^[ref]

Peter Young documents atomic collision data used by the CHIANTI spectroscopy database. His technical report specifies how effective collision strengths, averaged over a Maxwellian electron distribution, are stored and scaled as functions of temperature.^[ref]

An effective collision strength is not an energy-resolved collision strength evaluated at one representative energy. Documenting the averaging and file conventions lets plasma calculations combine data consistently. This entry identifies the author of the CHIANTI report, rather than inferring an identity from the common name alone.