Input atomic data

As already mentioned in Chapter 2, the rate-equations for all levels ni have to be solved in parallel with the equations of radiative transfer for all required frequencies. To accurately determine the level populations, we basically need a good knowledge of the local temperatures and particle densities, the non-local radiation field, accurate cross-sections, and all transitions between atomic states.

The two first items depend on the model atmosphere used for the NLTE calculations (see Chap- ter 2), whereas the remaining ones are related to the model atoms for the species involved in the calculations. So far, only a small amount of atomic data could be determined experimentally, and the bulk of the atomic data is provided from theoretical calculations, via ab-initio methods. Large databases of transition probabilities and cross-sections for photoionization and excitation via electron impact are available for the community, provided, e.g., by the OPACITY Project (OP, Seaton 1987; Seaton et al. 1994) and the IRON Project (IP; Hummer et al. 1993). Ab-initio data for radiative pro- cesses are usually available for transitions between levels up to n_≤10. For excitation via electron collisions, explicit data are scarce, typically covering transitions up to n≤4, and biased towards se- lected ions of light elements and iron. The remainder of the data has to be calculated by means of different approximations.

In the following, we concentrate on the basic processes that should be considered for the construc- tion of a comprehensive model atom, together with various approximations used for the implementa- tion.

Radiative processes

The non-local character of the radiation field drives the stellar plasma to NLTE conditions, coupling the thermodynamic state of the plasma at different layers in the stellar atmosphere. Only specific radiative transitions are usually allowed. They obey selection rules, which describe the conditions

different spin systems are not allowed.

Radiative bound-bound transitions. The absorption/emission of photons can lead to excitations/ de-excitations of electrons within an ion/atom, giving rise to spectral lines. The strength of the spectral lines is mainly determined by the number of absorbers (or emitters) and the line absorption cross- sections, which is given by

αi j=πe 2

mec

fi jφ(ν), (3.1)

where e is the electron charge, me the electron mass, fi j the oscillator strength, and φ(ν) the line absorption profile. To calculate the contribution of radiative bound-bound transitions to the radiative rates, only the oscillator strengths of the transitions need to be provided as input atomic data.

Radiative bound-free transitions. Photoionization is the process in which an ion absorbs a photon with enough energy to excite a bound electron beyond the ionization threshold, leading to an ion with a higher ionization stage. The inverse process is the recombination.

Photoionization cross-sections are usually taken from ab-initio data from OP and IP. These cross- sections are typically computed by the R-matrix method using the close-coupling approximation and contain complex resonance structures (Seaton 1987). These resonances are either produced by au- toionizing Rydberg series states or by double excited states, which produce the wide PEC (Photo- Excitation of the Core) resonances that are related to the dielectronic recombination processes, see below.

For excited levels with missing detailed data, ‘smooth’ (resonance free) photoionization cross- sections can be used, provided in terms of least square fittings from the non-linear form of the cross- section by Seaton (1958),

αik(ν) =α0[β(ν0/ν)s+ (1−β)(ν0/ν)s+1] (3.2)

withα0the cross-section at the thresholdν0, and fit parametersβ and s.

Collisional processes

Inelastic collisions with particles can lead to excitation and ionization of atoms/ions as well. One of the most important characteristics of the collisional processes is that they drive the stellar plasma to- wards LTE conditions, because the velocity distribution in stellar atmospheres is usually Maxwellian. Typically, only electron collisions are considered since the thermal velocity and hence the collision frequency of heavy particles is much smaller.1 Unlike radiative processes, collisions do not obey any selection rule. Therefore, collisions between all states within an atom/ion are possible and need to be accounted for.

1 _{This is not the case for cool stars, especially for metal-poor objects, where hydrogen collisions may become the dominant}

3.1. CONSTRUCTION OF A COMPREHENSIVE MODEL ATOM 27

Collisional bound-bound transitions. The collisional rates are defined in terms of the effective collision strength via

Ci j=8.631·10− 6

T1/2_gi nee

−u0 _{γ. (3.3)}

The effective collision strength is γi j= Z ∞ 0 Ωi j exp −_kTEj d Ej kT , (3.4)

where Ej is the electron’s kinetic energy after the excitation has occurred, andΩi j is the collision strength, first introduced by Seaton (1953). It is a dimensionless quantity defined by the ratio of two areas (the cross-section Q and the square of the appropriate de Broglie wavelength), weighted by the statistical weight of the transition, gi,

Ωi j =4πgi

λ2

i

Q(i→ j). (3.5)

Thus, the effective collision strength is the thermally averaged collision strengthΩi j.

Only few ab-initio data of collisional excitations are available, usually up to levels with n = 3. . . 4, in terms of tabulations of the effective collision strength. For the bulk of the possible collisional transitions, approximations need to be used which give, at best, order-of-magnitude estimates. In the following, we refer to the two most commonly used ones:

(i) In the radiatively permitted case, the collision rate Ci j might be calculated on the assumption that the collision cross-section is proportional to the oscillator strength of the transition. This scaling is the base of the van Regemorter (1962) approximation,

Ci j=5.465·10−11neT1/2[14.5 fi j(IH/E0)2]u0 exp(−u0)Γ(u0), (3.6)

where E0is the threshold energy of the process, u0=E0/kT , fi jis the oscillator strength, IHis the ionization energy of hydrogen, andΓ(u0)is the maximum of a slowly-varying function of temperature,

Γ(u0) =max[g¯,0.276 exp(u0)E1(u0)]. (3.7)

The parameter ¯g is about 0.7 for transitions nl_→nl’, and about 0.2 for transitions nl_→n’l’, n’

6

=n. E1(x)is the first exponential integral.

(ii) For the radiatively forbidden case, Allen (1973) provides a semi-empirical formula, Ci j = 5.465·10−11neT1/2exp(−u0)Γ,

Γ = Ω

gi IH

kT, (3.8)

Collisional bound-free transitions. Collisional ionizations can2become a dominant factor for the coupling of high-lying levels to the continuum, whilst for the ground-state and low-lying levels they are rather inefficient because only few electrons have energies high enough to overcome the threshold for the reaction.

Cross-sections for ionization by electron impact are even less quantitatively known than the atomic data for the previously discussed processes. Experimentally, only ionization from the ground-state are usually covered, whilst on the theoretical side the situation is not much better. Nevertheless, rea- sonable approximations can be applied in the majority of cases. Collisional ionization cross-sections are usually calculated in terms of the photoionization cross-section at threshold,α0, using the Seaton

(1962) formula. This yields a collisional rate Cik=1.55·1013α0gn¯ e

e−u0 u0T1/2

, (3.9)

where the ¯g-factor is Z (charge) dependent, being 0.1, 0.2, and 0.3 for Z=1, Z=2, and Z_≥3, respec- tively.

Other processes

For the construction of comprehensive model atoms to be used within the analysis of stellar atmo- spheres, the consideration of the above processes is sufficient in most cases.3However, there are other processes that might be relevant in certain cases and might need to be considered as well, e.g., dielec- tronic recombination for the NIIIλλ4634-4640-4642 emission line formation problem, see Chapter 4. Autoionization/Dielectronic recombination. If two electrons are excited within a complex atom/ion with several electrons, they can give rise to states with energies both below and above the ionization potential. States above the ionization limit, under certain selection rules, may preferentially autoionize to the ground state of the ion plus a free electron.

Thus the ionization from an initial bound state A(i) to an ionized final state A+(f) can occur either directly,

hν1+A(i)→A+(f) +e−, (3.10)

or by a (different) transition from the initial bound state A(i) to an intermediate doubly excited state ¯

A above the ionization potential that finally autoionizes to A+(f).

hν2+A(i)→A¯ →A+(f) +e−. (3.11)

Photoionizations of the latter kind are produced by strong resonances, involving Photo-Excitation of the Core (PEC) resonances (Yan & Seaton, 1987). In this case, the photoionization occurs due to intermediate resonant state(s) in the core of the atom leading to autoionization rather than in the direct ionization of an excited electron. In this respect, PEC resonances are particularly strong, because they correspond to a single electron transition in which the outer electron is a spectator and does not change, giving rise to wide resonances in the total photo-ionization cross-section (see Fig. 4.2), which are produced by the strong dipole coupling that exists among the involved states.

2 _{For high enough densities.}