This review presents basic equations for the solution of the NLTE radiative transfer problem for trace elements and methods for its solution are summarized. The importance of frequency coupling in radiative transfer in stellar atmospheres is emphasized.

Tests are presented of the 1D Accelerated Lambda Iteration method, which is widely used for solving the radiative transfer equation for a stellar atmosphere. We use our ARTY code as a reference solution and tables for these tests are provided. We model a static idealized stellar atmosphere, which is illuminated on its inner face and where internal sources are distributed with weak or strong gradients. This is an extension of published tests for a slab without incident radiation and gradients. Typical physical conditions for the continuum radiation and spectral lines are used, as well as typical values for the numerical parameters in order to reach a 1% accuracy. It is shown that the method is able to reach such an accuracy for most cases but the spatial discretization has to be refined for strong gradients and spectral lines, beyond the scope of realistic stellar atmospheres models. Discussion is provided on faster methods.

A brief review of the terms and definitions related to stellar atmosphere modelling and spectral line formation is presented. In particular, two topics are discussed: the chemical elements that can be treated in a given atmosphere as trace elements, and the scattering processes that are taken into account in line formation modelling.

The conditions of thermodynamic equilibrium, local thermodynamic equilibrium, and statistical equilibrium are discussed in detail. The equations of statistical equilibrium and the supplementary equations are shown together with the expressions for radiative and collisional rates with the emphasize on the solution for trace elements.

A brief summary of some aspects of atomic physics theory useful for stellar atmospheres is given.

Non-LTE modelling for a particular atom requires accurate collisional excitation and ionization cross-sections for the entire system of transitions in the atom. This review concerns with inelastic collisions with electrons and neutral hydrogen atoms. For the selected atoms, H i and Ca ii, comparisons are made between electron impact excitation rates from ab initio calculations and various theoretical approximations. The effect of the use of modern data on non-LTE modelling is shown. For most transitions and most atoms, hydrogen collisional rates are calculated using a semi-empirical modification of the classical Thomson formula for ionization by electrons. Approaches used to estimate empirically the efficiency of hydrogenic collisions in the statistical equilibrium of atoms are reviewed.

The oscillator strength is a key parameter in the description of the line absorption coefficient. It can be determined through experiment, abinitio and semi-empirical calculations, and backward analysis of line profiles. Each method has its advantages, and the uncertainty attached to its determination can range from low to indeterminable. For analysis of line profiles or equivalent widths the uncertainty in the oscillator strength can rival or surpass the difference between the derived element abundance from classical LTE and non-LTE analyses. It is therefore important to understand the nature of oscillator strength uncertainties and to assess whether this uncertainty can be a factor in choosing to initiate a non-LTE analysis or in the interpretation of its results. Methods for the determination of the oscillator strength are presented, prioritizing experiments, along with commentary about the sources and impact of the uncertainties. The Se I spectrum is used to illustrate how gf-values can be constructed from published data on atomic lifetimes and line intensities.

Model atoms are an integral part in the solution of non-LTE problems. They comprise the atomic input data that are used to specify the statistical equilibrium equations and the opacities and emissivities of radiative transfer. A realistic implementation of the structure and the processes governing the quantum-mechanical system of an atom is decisive for the successful modelling of observed spectra. We provide guidelines and suggestions for the construction of robust and comprehensive model atoms as required in non-LTE line-formation computations for stellar atmospheres. Emphasis is given on the use of standard stars for testing model atoms under a wide range of plasma conditions.

Resonance broadening is important for the hydrogen lines in the spectra of F-type and later stars. In the corresponding temperature regime, the extended wings of the Balmer lines are used as a stellar effective temperature indicator. We show the effect of the use of two broadening theories, Ali & Griem (1965, 1966) and Barklem et al. (2000a, 2000b), on the effective temperature derived in non-LTE from Hα and Hβ in the Sun and the metal-poor star HD19445. Van der Waals broadening is important for strong spectral lines in the atmospheres of F-type and later stars. For the selected transitions in Ca I and Ca II, line profile comparisons are made between applying the van der Waals damping constants from laboratory measurements, the ABO perturbation theory, and the classic Unsöld approximation.

A very brief introduction to the theory of line broadening is presented.

In NLTE computations of trace elements in stellar atmospheres, background opacities are generally treated in LTE. It is thus important to assess the impact of different methods of including this background opacity on the statistical equilibrium of the trace element and its resulting NLTE abundance. This article illustrates these effects using two examples, nitrogen in Vega and carbon in the Sun.

A-type stars with their shallow convection zones serve as ideal physics laboratories for stellar atmosphere research. In the absence of large scale mixing, processes such as diffusion, mass loss and accretion leave their characteristic imprint on the chemical composition of the photosphere. This characteristic surface pattern can be studied by means of stellar abundance analysis. However, such patterns can be hidden in the large uncertainties of LTE abundances. Thus, detailed NLTE studies that can push stellar abundance analysis beyond the 0.1 dex uncertainty limit are a pre-requisite for using A star atmospheres as physics laboratories.

We describe a self-consistent spectrum analysis technique employing non-LTE line formation, which allows precise atmospheric parameters of massive stars to be derived: 1σ-uncertainties as low as  ~1% in effective temperature and  ~0.05–0.10 dex in surface gravity can be achieved. Special emphasis is given to the minimisation of the main sources of systematic errors in the atmospheric model computation, the observed spectra and the quantitative spectral analysis. Examples of applications are discussed for OB-type stars near the main sequence and their evolved progeny, the BA-type supergiants, covering masses of  ~8 to 25 M⊙ and a range in effective temperature from  ~8000 to 35000 K. Relaxing the assumption of local thermodynamic equilibrium in stellar spectral synthesis has been shown to be decisive for improving the accuracy of quantitative analyses. Despite the present examples, which concentrate on hot, massive stars, the same philosophy can be applied to line-formation calculations for all types of stars, including cooler objects like the Sun, once the underlying stellar atmospheric physics is reproduced consistently.

This paper reviews the various population and depopulation mechanisms that establish the statistical equilibrium of atoms and shows how departures from LTE for level occupation numbers translate in non-LTE effects for spectral lines.

Line-formation calculations in the Rayleigh-Jeans tail of the spectral energy distribution are complicated by an amplification of non-LTE effects. For hot stars this can make quantitative modelling of spectral lines in the near-IR challenging. An introduction to the modelling problems is given and several examples in the context of near-IR line formation for hydrogen and helium are discussed.

Two astrophysical problems are reviewed where non-LTE line formation for the selected atoms is considered in a chemically stratified atmosphere and this leads to better interpretation of observed data. These are an origin of the Mn ii emission lines in the 3He and HgMn stars and a violation of the ionization equilibrium between the second and first ions of rare-earth elements in rapidly oscillating chemically peculiar stars.

Departures from LTE may significantly affect determinations of stellar parameters and chemical abundances of cool stars, in particular, metal-poor stars. We review the application of the non-LTE ionization equilibrium between Ca i and Ca ii to constrain the surface gravity of very metal-poor and hyper metal-poor stars spectroscopically. The few examples from Galactic chemical evolution studies show that our understanding of how nucleosynthesis proceeds throughout Galactic history depends on the accuracy of spectral line formation modelling. Problems of non-LTE modelling of Fe i– Fe ii are also discussed.

We review the literature of high dispersion studies of late B and early A type supergiants to assess the importance of non-LTE abundance calculations for them. Practically we are interested in learning which elements and species have such calculations been performed and then which of these should be implemented in calculations with plane-parallel LTE model atmospheres. The techniques available for the quantitative modeling of these atmospheres are outlined and some recent results are discussed.