A comparison between detailed and configuration-averaged collisional-radiative codes applied to nonlocal thermal equilibrium plasmas

Abstract

A collisional-radiative model describing non-local-thermodynamic-equilibrium plasmas is developed. It is based on the HULLAC suite of codes for the transitions rates, in the zero-temperature radiation field hypothesis. Two variants of the model are presented, the first one is configurationaveraged, while the second one is a detailed level version. Comparisons are made between them in the case of a carbon plasma; they show that the configuration-averaged code gives correct results for an electronic temperature T e = 10 eV (or higher) but fails at lower temperatures such as T e = 1 eV. The validity of the configuration-average approximation is discussed: the intuitive criterion requiring that the average configuration-energy dispersion must be less than the electron thermal energy turns out to be a necessary but far from sufficient condition. Another condition based on the resolution of a modified rate-equation system is proposed. Its efficiency is emphasized in the case of low-temperature plasmas. Finally, it is shown that near-threshold autoionization cascade processes may induce a severe failure of the configuration-average formalism. It is now well-known that in highly-charged hot plasmas, emission and absorption spectra usually display broad structures theoretically described as unresolved transition arrays (UTA) [1,2,3], spin-orbit split arrays (SOSA) [4] or supertransition arrays (STA) [5]. Isolated lines may also be present when transitions occur between two configurations of small degeneracy. The UTA formalism consists in expanding the individual transition energies as a function of the various moments < E n > -where the ponderation is performed using line strengths -with the assumption that levels inside a given configuration are distributed according to thermal equilibrium, the validity of this assumption lying on the condition that the configuration width ∆ c must be smaller than the thermal energy k B T e . Due to its numerous applications, the theory of non-LTE plasmas is nowadays a very active subject [19,20]. For instance, laser-produced and discharged-produced plasmas appear as very promising sources of intense extreme-UV light well suited for 13.5 nm-lithography ([21, 22, 23] and other references in the same volume); for such plasmas of moderate density, the non-LTE condition prevail in most cases. Another domain of application is low-density astrophysics plasmas as well as laboratory coronal plasmas.The aim of this paper is first to present a detailed CR model. One essential feature is that it must be based on a reliable atomic code. To this respect, the HULLAC (Hebrew University Lawrence Livermore Atomic Code) parametric-potential code is known to be efficient in applications dealing with ion spectroscopy and collisional rate calculation. A known limitation of several atomic models used in plasma physics [12,13,14,16,17] is that configuration interaction is ignored. However it has been demonstrated that configuration interaction may play a major role in several radiative effects in pla...

Highly efficient sparse-matrix inversion techniques and average procedures applied to collisional-radiative codes

“…However, the CG algorithm may lack of robustness. For instance, we considered the case where radiative and autoionization rates are canceled: including only the collisional excitation and ionization and the reverse processes, and comparing the solution with Saha-Boltzmann is a way to check the algorithm accuracy [17]. In a krypton plasma at a 500 eV temperature and 10 14 cm −3 electronic density, with 5 662 levels and 14 charge states accounted for, we did obtain 3 populations different from Saha-Boltzmann by 2 × 10 −13 in average (6 × 10 −10 at maximum), but this was obtained after a rather large number of iterations: 1966 with ε = 10 −13 in Eq.…”

“…The matrix condition 4 is checked inside the code and it was found satisfactory in every considered cases. As an accuracy test proposed before [17], we solved with the LU algorithm a kinetic system of 4481 equations including only collisional excitation, collisional ionization plus the reverse processes. One gets then the Saha-Boltzmann solution with a maximum population difference of 1.2 × 10 −12 , and an average difference of 5.4 × 10 −16 , the Gauss algorithm providing the same excellent accuracy.…”

Gershgorin and H theorems applied to collisional-radiative equilibrium model

“…This is what is implicitly hidden in the expression (23). This is not a straightforward task for complex codes such as those based on the superconfiguration approach [7,8] or for approach intending to explicitly check the validity of the configuration average procedure [9,10]. As for the total ion energy, we have other trends for the relaxation towards the steady-state.…”

X-ray spectra of the Fe-L complex

“…As reported in Lucy (2001), the super level method applying to a system with ∼ 1000 levels can reach an accuracy of 0.1 with 6 iterations, and 0.01 with 20 iterations. Meanwhile, Poirier & de Gaufridy de Dortan (2007) showed that the super level method, as adopted in G03, might become less accurate when the rms deviation of transition rates inside one super level increases.…”