1. INTRODUCTION
Radiative properties of hot dense plasmas remain a subject of current interest (Alexiou et al., Reference Alexiou, Calisti, Gauthier, Klein, Leboucher-Dalimier, Lee, Stamm and Talin1997; Csanak & Daughton, Reference Csanak and Daughton2004; Orlov et al., Reference Orlov, Gus'kov, Pikuz, Rozanov, Shelkovenko, Zmitrenko and Hammer2007) since they play an important role in the inertial fusion confinement (ICF) research as well as studies in stellar physics (Bauche et al., Reference Bauche, Bauche-Arnoult and Peyrusse2006). In particular, the understanding of ICF plasmas requires emissivities and opacities both for hydro-simulations and diagnostics (Peyrusse, Reference Peyrusse2004) since, for example, the emission spectra from plasmas under non-local thermodynamic equilibrium (NLTE) may be used for plasma diagnostics, or the spectrally integrated emissivity may be used to determine the evolution of the electronic and radiation temperatures in a hydrodynamic simulation (Bowen et al., Reference Bowen, Decoster, Fontes, Fournier, Peyrusse and Ralchenko2003). Furthermore, modeling of the energy transport in hot dense plasmas relies on radiative opacities (Yuan et al., Reference Yuan, Haynes, Peterson and Moses2003; Mínguez et al., Reference Mínguez, Rodriguez, Gil, Sauvan, Florido, Rubiano, Martel and Mancini2005).
The first step on the calculation of plasma radiative properties is the computation of the required atomic data. However, this is a complex task since the number of the atomic levels involved, and therefore the amount of atomic data to obtain is huge and approximations must be made. For low and intermediate Z-plasmas detailed level accounting (DLA) or detailed term accounting (DTA) models are commonly used. On the other hand, for high-Z elements statistical approaches involving grouping of levels such as the detailed configuration accounting (DCA) approach or super configuration accounting (SCA) approach (Bar-Shalom et al., Reference Bar-Shalom, Oreg and Klapisch1997; Peyrusse, Reference Peyrusse2001) have shown to be very efficient when it is combined with unresolved transition array (UTA/SOSA) (Bauche et al., Reference Bauche, Bauche-Arnoult and Klapisch1987) and/or the super transition array (STA) formalisms (Bar-Shalom et al., Reference Bar-Shalom, Oreg, Goldstein, Shvarts and Zigler1989). Nowadays, new hybrid models that mix detailed levels and configurations are under development (Mazevet & Abdallah, Reference Mazevet and Abdallah2006; Hansen et al., Reference Hansen, Bauche, Bauche-Arnoult and Gu2007).
Once the atomic data are available, we need to compute the atomic level populations in the plasma. The need for accurate computation of radiative properties covers an extensive range of thermodynamic conditions. Depending on the LTE or NLTE character of the plasma under analysis, existing physics models have different uses or implementations (Peyrusse, Reference Peyrusse2004). Under LTE conditions there exists an equation which gives the occupation probabilities of bound states, the Saha-Boltzmann equation, and, moreover, the Kirchoff's gives a simple relationship between the emissivity and the opacity. Therefore, it is only necessary to calculate the opacity from its definition. On the other hand, under NLTE conditions, the problem shows great complexity because there is not an a priori expression for the occupation probabilities of bound states, and therefore one must find the statistical distribution of the ionic states by using the so-called collisional-radiative model (CRM), which implies to solve a set of rate equations, with coupling of configurations, free electrons and photons. Moreover, if we consider that for accurate simulations of the emission and the absorption spectra, it is essential to take into account as many quantum levels as possible, and that the Kirchoff's law is no longer valid in this regime and then the opacity and the emissivity must be calculated on the same footing, sometimes the calculations become unmanageable. For this reason, it results that it is more complex to provide accurate level populations under NLTE than LTE conditions.
Due to the difficulties that arise in the calculation of the radiative properties there is still a lack of complete understanding of these magnitudes and since they are relevant in many areas of the plasma, many groups are currently developing codes and calculations (Serduke et al., Reference Serduke, Mínguez, Davidson and Iglesias2000; Peyrusse, Reference Peyrusse2001; Bar-Shalom et al., Reference Bar-Shalom, Klapisc and Oreg2001; Bowen, Reference Bowen2001; Zeng & Yuan, Reference Zeng and Yuan2002; Iglesias et al., Reference Iglesias, Chen, Sonnad and Wilson2003; Bowen et al., Reference Bowen, Decoster, Fontes, Fournier, Peyrusse and Ralchenko2003; Yuan et al., Reference Yuan, Haynes, Peterson and Moses2003; Rubiano et al., Reference Rubiano, Florido, Rodriguez, Gil, Martel and Mínguez2004; Chung et al., Reference Chung, Chen, Morgan, Ralchenko and Lee2005; Hakel et al., Reference Hakel, Sherrill, Mazevet, Abdallah, Colgan, Kilkrease, Magee, Fontes and Zhang2006; Wu et al., Reference Wu, Pang and Yan2006; Yuan & Moses, Reference Yuan and Moses2006; Mazevet & Abdallah, Reference Mazevet and Abdallah2006; Rodríguez et al., Reference Rodriguez, Gil, Florido, Rubiano, Martel and Mínguez2006; Abdallah et al., Reference Abdallah, Kilcrease, Magee, Mazevet, Hakel and Sherrill2007a, Reference Abadallah, Batani, Desai, Lucchini, Faenov, Pikuz, Magunov and Narayanan2007b; Hansen et al., Reference Hansen, Bauche, Bauche-Arnoult and Gu2007) of plasma radiative properties and new results are always welcomed.
In this paper, a new code is presented, RAPCAL, for calculating radiative properties such as opacities, emissivities, source functions, intensities, transmissions, and radiative power losses for optically thin and thick plasmas, both under LTE and NLTE conditions. The next section is devoted to describing RAPCAL code, in the third section, some results are presented in order to check the validity of the model and finally, some conclusions and general remarks are presented.
2. CODE DESCRIPTION
RAPCAL is an extension of the analytical expressions based kinetics (ABAKO) code (Florido, Reference Florido2007; Florido et al., Reference Florido, Rodriguez, Gil, Rubiano, Martel, Suárez, Mendoza and Mínguez2008) designed for the calculation of plasma radiative properties. It is divided in three modules. The first one was designed to calculate the required atomic data. The second one is devoted to the calculation of the population of optically thin and thick plasmas, both under LTE and NLTE conditions. And, finally, in the third module, radiative properties are calculated for both plasma regimes. In the following, we will describe these modules.
2.1. Atomic Module
RAPCAL contains an internal model to generate atomic magnitudes but it has also been designed to work using external data coming from other computational codes or databases.
At the present time, the external source often used in RAPCAL is the flexible atomic code (FAC) (Gu, Reference Gu2003), which is able to provide atomic data either in DLA or relativistic DCA approaches. The energy of the levels or configurations of an atomic ion with N electrons are obtained by diagonalizing the relativistic Hamiltonian. The basis states which are usually referred to as configuration state functions (CSF), which are built as anti-symmetric sums of products of N one-electron Dirac spinners. In coupling the angular momenta, the standard jj-coupling scheme is used. Finally, the approximate atomic wave functions are evaluated mixing the basis states (configuration interaction, CI) with the same symmetries with the mixing coefficients obtained from diagonalizing the total Hamiltonian.
With respect to the atomic model implemented in RAPCAL, this always works in the DCA approach. In this case, for a given ion, for each relativistic configuration, we solve a Dirac equation for each occupied level which gives us its relativistic mono-electronic wave function, and the corresponding energy level. The total energy is then obtained using a density-functional formalism (Rajagopal, Reference Rajagopal1980; Kohn & Sham, Reference Kohn and Sham1965) assuming the local density approach for the exchange and correlation energy. The effective potential used in the Dirac equation is anyone of the analytical central ones developed by our group. These can model isolated ions (Martel et al., Reference Martel, Doreste, Mínguez and Gil1995)
where N is the number of bound electrons and Z is the nuclear charge and φ(r) is the screening function, given by
where the three parameters a 1, a 2, a 3 were determined by fitting the potential (Martel et al., Reference Martel, Rubiano, Gil, Doreste and Minguez1998) to a self-consistent one used in DAVID code (Liberman et al., Reference Liberman, Cromer and Weber1971). Ions immersed into weakly (Gil et al., Reference Gil, Martel, Minguez, Rubiano, Rodríguez and Ruano2002) and strongly coupled plasmas (Rodriguez et al., Reference Rodriguez, Gil and Florido2007) are also modeled by potentials U I(r) and U II(r), respectively
![\eqalignno{U^{II} \lpar r\rpar &=- {1 \over r} \Big\{ \lpar N - 1\rpar \lpar \phi \lpar r\rpar - \eta \lpar r\rpar \rpar + 1 \Big\} \cr \quad &- {1 \over r} \Big[ Z - N+\lpar N - 1\rpar \eta \lpar 0\rpar \Big] e^{ - ar}\comma \;}](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021083621293-0637:S026303460800044X_eqn3.gif?pub-status=live)
with
where 
is the plasma average ionization, R 0 is the ion-sphere radius, and a is the inverse of the Debye radius. Both for isolated and non-isolated situations, singly and doubly excited configurations can be included (Rodriguez et al., Reference Rodriguez, Rubiano, Gil, Martel, Mínguez and Florido2002). The usefulness and accuracy of this set of analytical potentials in obtaining atomic data for level populations and opacities have been shown previously (Minguez et al., Reference Mínguez, Gil, Martel, Rubiano, Rodriguez and Doreste1998, 2005; Bowen et al., Reference Bowen, Lee and Ralchenko2006; Rubiano et al., Reference Rubiano, Florido, Bowen, Lee and Ralchenko.2007). Furthermore, this atomic model is a very useful tool in order to optimize detailed calculations since it provides us in a very short time with valuable information such as the plasma average ionization and the most abundant ions in the plasma for given conditions. We would like to point out that this atomic module is also ready to perform the atomic calculations with other analytical potentials available in the literature or using a self-consistent potential (Liberman et al., Reference Liberman, Cromer and Weber1971) and it is also implemented a relativistic screened hydrogenic model (Rubiano et al., Reference Rubiano, Florido, Rodriguez, Gil, Martel and Mínguez2004) for the situations wherein very fast atomic calculations are desired, for example in-line hydro-code calculations.
Finally, for dense plasmas, either for the calculations using FAC or the atomic model, the lowering of the ionization potential is taken into account following the model proposed by Stewart and Pyatt (Reference Stewart and Pyatt1966)
![\Delta I_i = {3 \over 2} {\lpar \xi_i + 1\rpar e^2 \over R_0} \left\{\left[1+\left({D \over R_0}\right)^3 \right]^{2/3} - \left({D \over R_0} \right)^2 \right\}\comma \; \eqno\lpar 6\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021083621293-0637:S026303460800044X_eqn6.gif?pub-status=live)
where D is the Debye length, e is the electron charge, and ξi is the charge of the ion, i. When the atomic data are calculated using the analytical potential including plasma effects, the atomic, and the population calculation modules must be solved iteratively until the convergence is achieved.
2.2. Plasma Population Distribution Calculation Module
This module of RAPCAL is basically an ABAKO code (Florido, Reference Florido2007; Florido et al., Reference Florido, Rodriguez, Gil, Rubiano, Martel, Suárez, Mendoza and Mínguez2008). At the moment, the code works under stationary assumption. Therefore, to find the level population it solves the set of steady-state rate equations
where N ξm denotes the population density of the atomic level ξm and R ξ′m′→ξm+ and R −ξm →ξ′m′ take into account all the processes which contribute to populate and depopulate the ξm state, respectively. The collisional-radiative steady-state (CRSS) model in ABAKO is solved level by level (or configuration by configuration, depending on the DLA or DCA approach) and it is applied to low-to-high Z ions under a wide range of laboratory or astrophysical plasma conditions: corona equilibrium, NLTE, or LTE, optically thin and thick plasmas. Special care was taken during the development of our CRSS model to achieve an optimal equilibrium between accuracy and computational cost. Hence, analytical expressions has been employed for the rate coefficients of the atomic processes included in the CRSS model, which yield a substantial saving of computational requirements, but providing satisfactory results in relation to those obtained from more sophisticated codes and experimental data.
The processes included in the CRSS model are the following: collisional ionization (Lotz, Reference Lotz1968) and three-body recombination, spontaneous decay, collisional excitation (Van Regemorter, Reference Van Regemorter1962), and deexcitation, radiative recombination (Kramers, Reference Kramers1923), electron capture and autoionization. We have added between brackets the references wherefrom their approximated analytical rates coefficients have been acquired. The rates of the inverse processes are obtained through the detailed balance principle. It is worth pointing out that the auto-ionizing states are included explicitly. It has been proved that their contribution is critical in the determination of the ionization balance. The cross section of the auto-ionization is evaluated using detailed balance principle from the electron capture cross section. This one is obtained from the collisional excitation cross section using a known approximation (Griem, Reference Griem1997). Only those atomic processes whose rates are independent of the radiation field intensity are explicitly considered in the CRSS model.
RAPCAL also models homogeneous optically thick plasmas. For this situation, the escape factor formalism (Mancini et al., Reference Mancini, Joyce and Hooper1987) for the basic geometries—plane, cylindrical and spherical—is used in order to take into account the bound-bound opacity effects. Nowadays, this module is being improved to model non-uniform plasmas with planar geometry using a new technique for the line transport (Florido et al., Reference Florido, Gil, Rodriguez, Rubiano, Martel and Minguez2006) based on the definition of zone-to-zone radiative coupling coefficients. A full description of this formalism will be exposed in a forthcoming paper.
Finally, since the number of rate equations is huge due to the number of atomic levels involved, we employ the technique of sparse matrices to storage the non-zero elements of the coefficient matrix of the system, which implies substantial savings in memory requirement. For the matrix inversion we use iterative procedures (Florido et al., Reference Florido, Gil, Rodriguez, Rubiano, Martel and Mínguez2005) because they entail much less memory than direct methods and they are also faster. It is worth remarking this fact because, as we said previously, when we include plasma effects through the continuum lowering in our atomic model this module and the atomic one must be solved iteratively.
2.3. Radiative Properties Calculation Module
The total spectrally resolved opacity and emissivity of plasma is the combination of bound-bound, bound-free, free-free, and scattering processes. Besides stimulated emission is taken into account. The bound-bound absorption coefficient and emissivity are evaluated through the following equations
where
![\mu_{\xi_{i\comma \,j}} \lpar \nu\rpar = {\pi \over mc} \left({e^2 \over 4\pi \varepsilon_0} \right)N_{\xi_i\, } f_{\xi_{i\comma \,j}} \phi_{\xi_{i\comma \,j}}^a \lpar \nu\rpar \left[1 - {N_{\xi_{i}} g_{\xi_j} \over N_{\xi_j} g_{\xi_i}} \right]\comma \; \eqno\lpar 10\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021083621293-0637:S026303460800044X_eqn10.gif?pub-status=live)
where m is the electron mass, c is the speed of the light, i and j denote levels of ion ξ, N ξi denotes the population (cm−3) of the i-level of this ion, ν is the frequency of the photon, g denotes the degeneracy of the level, and f ξi,j denotes the oscillator strength of the transition, which is evaluated in the electric dipole approximation as it follows
where P(1) is the electron dipole operator. When it is employed, the atomic model implemented in RAPCAL, the oscillator strength associated to the transition between two relativistic configurations i and j is evaluated in terms of mono-electronic oscillator strengths (Rodriguez et al., Reference Rodriguez, Gil and Florido2007). In Eq. (11), φaξi,j(ν) and φeξj,i(ν) are the absorption and the emission line-shape functions, respectively. In this work, both profiles have been taken as equal, assuming complete redistribution hypothesis (Mihalas, Reference Mihalas1978). In our work, we have considered natural, Doppler, and electron-impact broadenings. The Doppler half width at half maximum (HWHM) is given by Cowan (Reference Cowan1981)
where A is the atomic weight of the ion in grams, and kT (with k the Boltzmann constant and T the electron temperature), hνij (the transition energy), and Γd are in eV. The calculation of the electron-impact HWHM is somewhat complicated and it can be obtained by fully quantum mechanical calculation (Seaton, Reference Seaton1990) or by semi-classical method (Griem, Reference Griem1974), but even the semi-classical method needs elaborate calculation and it is not useful to obtain a large number of such data for the evaluation of the X-ray spectra (Zeng et al., Reference Zeng, Yuan and Lu2001a). Expressed using a semi-empirical formula (Dimitrijevic & Konjevic, Reference Dimitrijevic and Konjevic1980, Reference Dimitrijevic and Konjevic1987) the electron–impact HWHM in eV is given by
where N e is the electron density and n i, n j, l i, l j are the principal and angular quantum numbers of the initial and final orbitals, respectively, related with the transition. In this paper, the width due to collisions between ions has not been considered. Zeng and Yuan (Reference Zeng and Yuan2002) have verified that the semi-empirical formula given in Eq. (14) provides results very similar to those of a quantum mechanical calculation. The line-shape function is applied with the Voigt profile which includes all the broadenings cited before
where 
is the natural HWHM, and 
. The natural and electron-impact HWHM have been added since in the impact approximation, the two contributions have the same Lorentz profile shape.
For the bound-free transitions, the absorption coefficient and the emissivity are evaluated as follows
where
![\mu_{\xi_i\comma \xi+1_j} \lpar \nu\rpar = N_{\xi_i} \sigma \,\lpar \nu\rpar _{\xi_i\comma \xi+1_j} \left[1 - {N_{\xi+1_j} g_{\xi _i} \over N_{\xi_i} g_{\xi+1_j}} \right]N_e n\lpar \varepsilon\comma \, T\rpar \comma \; \eqno\lpar 19\rpar](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021083621293-0637:S026303460800044X_eqn19.gif?pub-status=live)
where n(ɛ, T) is the free electron distribution, ɛ is the free electron energy, f(ɛ, T) = g(ɛ)n(ɛ, T) where g(ɛ) is the free electron degeneracy and σ(ν)ξi,ξ+1j is the photoionization cross section. In Eq. (20), the Milne relation has been considered. When using the FAC code for generating the atomic data, the photoionization cross section is calculated in the distorted wave (DW) approximation without including resonances, improving the efficiency of the calculations by extending the factorization-interpolation procedure of Bar-Shalom et al. (Reference Bar-Shalom, Klapisch and Oreg1988) to the evaluation of the photoionization cross section. Thus, the photoionization cross section is given in terms of the differential strength, in atomic units (Gu, Reference Gu2003)
where α is the fine structure constant, and the differential oscillator strength may be calculated similarly to the bound-bound oscillator strength. When the atomic data are calculated using the atomic model implemented in the code, the cross section is computed using the Kramer expression (Kramers, Reference Kramers1923)
where a 0 is the Bohr radius and E nξi is the threshold ionization energy for the shell n i of the ion ξ in the electronic configuration i.
For the free-free transitions, the absorption coefficient has been evaluated using the semi-classical expressions of Kramer for the cross section (Rose, Reference Rose1992)
where the free-free Gaunt factor g ff is taken as unit. The total free-free absorption coefficient is
where N ion is the ion density and the negative term again corrects for induced processes. Eq. (24) is valid for LTE conditions as long as the velocities are maxwellian. The free-free emissivity is also evaluated through the Kramer approximation (Minguez, Reference Mínguez, Velarde, Ronen and Martínez-Val1993)
Finally, the scattering contribution to the absorption is approximated using the Thomson scattering cross section (Rutten, Reference Rutten1995). Thus, the total opacity and emissivity can be written as follows
where ρ is the density of matter (gcm−3). The source function is then evaluated through the following expression
The Rosseland and Planck mean opacities are given by Serduke et al. (Reference Serduke, Mínguez, Davidson and Iglesias2000)
where B′(ν, T) is the temperature derivative of the normalized Planck function B(ν, T), which is given by
where u = hν/kT is a dimensionless variable. The transmission spectrum, which provides the fraction of radiation transmitted with respect to some incident source of arbitrary intensity, is given by Zeng and Yuan (Reference Zeng and Yuan2002)
where L is the path length transversed by the light source through the plasma.
The radiative power loss, which can be important in understanding energy distributions and spectral characteristics of plasmas, is evaluated as following (eV/s/ion) (Chung et al., Reference Chung, Fournier and Lee2006) for the bound-bound contribution
where A ij is the spontaneous decay rate of the transition. For the bound-free contribution, we have
![\eqalignno{P_{bf} \lpar T_e\comma \,N_e\rpar &= 4\pi \sum_\xi \sum_{ij} N_{\xi_i} \cdot \left[{N_{{\xi - 1}_j} N_\xi \over N_{\xi_{i}} N_{\xi - 1}} \right]^{LTE} \vint_{\nu_0}^\infty \sigma \,\lpar \nu\rpar _{\xi_{i}\comma \xi + 1_j} \cr &\quad \times \left({2\, h\nu^3 \over c^2} \right)\; e^{ - \lpar h\nu / kT_e\rpar } d\nu\comma \;}](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151021083621293-0637:S026303460800044X_eqn34.gif?pub-status=live)
where the LTE population ratio is obtained through the Saha equation and ν0 is the threshold frequency. The free-free contribution is obtained for a pure Coulomb field as (Karzas & Latter, Reference Karzas and Latter1961)
where we have assumed the gaunt factor equal to unity. The total radiative power loss is then obtained as the sum of the three contributions. Finally, the radiative cooling coefficient of an ion can be obtained from the radiative power loss of the ion divided by its number density in the plasma.
3. RESULTS
It is not our aim in this work to carry out an exhaustive study of the radiative properties but rather to present some results that allow us to show the capabilities of the code and also to check the results that it provides with other codes and available experimental data.
We have made calculations in a wide range of plasma conditions, and, therefore, CE, NLTE, and LTE situations are considered. In particular, we have analyzed some interesting radiative magnitudes of plasmas of carbon, aluminum, krypton, and xenon, covering, this way, plasmas of low, intermediate, and high Z. The study of carbon plasmas is a subject of current interest since it is likely to be a major plasma-facing wall component in ITER (Skinner & Federici, Reference Skinner and Federici2006) and it plays a major role in inertial fusion scenarios (Filevich et al., Reference Filevich, Grava, Purvis, Marconi, Rocca, Nilsen, Dunn and Johnson2007). Aluminum has always been considered as a representative of low Z element and much experimental and theoretical effort has been made to investigate the opacities through these plasmas during the past two decades (Zeng & Yuan, Reference Zeng and Yuan2002; Keskinen & Schmitt, Reference Keskinen and Schmitt2007; Lomonosov, Reference Lomonosov2007) and, therefore, it is a benchmark element. Furthermore, both aluminum and carbon plasmas have interest in astrophysics. Finally, radiation in the outermost region of magnetically confined fusion plasmas from impurities seems to be a possible mechanism for controlling the heat load deposited onto the plasma facing components of a fusion reactor (Fournier et al., Reference Fournier, May, Pacella, Finkenthel, Gregory and Goldstein2000; Post, Reference Post1995). Noble gases such as krypton and xenon are potentially useful in this regard (Mandrekas et al., Reference Mandrekas, Stacey and Kelly1996) showing the advantage that they do not perturb the plasma core.
For carbon and aluminum plasmas, we have performed calculations both under DLA approach (RAPCAL-DLA), using atomic data from FAC code including CI among levels belonging to the same configuration, and under relativistic DCA approach with atomic data obtained from the atomic module implemented in RAPCAL based on the analytical potential given by Eqs. (1) and (2) (RAPCAL-AP). The comparison of the results will permit us to show the accuracy of the simple atomic model implemented in RAPCAL. Finally, for the krypton and xenon cases, the calculations were carried out in the relativistic DCA approach, using for the former the data provided by the analytical potential and for the latter atomic data obtained from FAC code.
Both the level populations and radiative properties depend strongly on the atomic configurations included in the calculations. It is still an open question (Peyrusse et al., Reference Peyrusse, Bauche-Arnoult and Bauche2006; Hansen et al., Reference Hansen, Bauche, Bauche-Arnoult and Gu2007; Chung et al., Reference Chung, Chen and Lee2007) which is the most suitable election of them. However, our experience with RAPCAL has led us to consider a “complete” set of configurations, in the sense that the addition of new configurations will not change the results, which allows us to obtain accurate results.
3.1. Carbon Plasmas
In Table 1, we listed the set of configurations and the number of relativistic configurations and resulting levels considered for the carbon ions. In the table, (n)w denotes all the possible relativistic configurations that arise from the shell n with w bound electrons.
Table 1. Set of configurations and number of relativistic configurations and levels considered in the calculations performed by RAPCAL code for carbon ions (n ≤ 10, n' ≤ 6)
Since radiative properties depend on the average ionization, we started by checking this magnitude. Thus, in Table 2, we compare the average ionization for low temperatures and moderate densities, which correspond to NLTE situations, calculated using RAPCAL with those provided by Colgan et al. (Reference Colgan, Fontes and Abdallah2006) using ATOMIC code, both under DCA (ATOMIC-DCA), and DLA (ATOMIC-DLA) approaches. This code is a detailed kinetics model developed at the Los Alamos National Laboratory (LANL), which includes configuration interaction effects for atomic data as well as quantum mechanical calculation for cross sections of the atomic processes. As we can see there is a general agreement between the RAPCAL and ATOMIC results both in DCA and DLA situations, obtaining relative errors that are in general lower than 5%. This is a remarkable result, since our kinetic model is based on analytical expressions for the cross sections and therefore this reduces considerably the complexity and the computing time. Moreover, it is also observed that RAPCAL-AP also gives acceptable results. This fact has been proved in the Fourth NLTE Kinetics Codes Comparison Workshop (Rubiano et al., Reference Rubiano, Florido, Bowen, Lee and Ralchenko.2007). Since the analytical potentials of the atomic module of RAPCAL avoid the iterative procedures of the self-consistent calculations, RAPCAL-AP allow us to make fast computations of plasma average ionization and ion populations.
Table 2. Average ionization for different conditions of temperature and electron density. Results from ATOMIC (Colgan et al., Reference Colgan, Fontes and Abdallah2006) and RAPCAL codes for configuration average (ATOMIC-DCA, RAPCAL-AP) and detailed level (ATOMIC-DLA, RAPCAL-DLA) accounting approximations
Taking into account the simplicity of the RAPCAL-PA calculations and its acceptable accuracy, this was used in a previous work (Gil et al., Reference Gil, Rodriguez, Florido, Rubiano, Martel and Minguez2008) to obtain maps of the average ionization and the plasma regimes (LTE, NLTE, and CE) for optically thin carbon plasmas, as a function of electron density and temperature. These maps are illustrated in Figure 1. The map of plasma regimes was elaborated by applying a criterion on the ion populations and it was proved that its predictions agree with those obtained using Griems criterion (Griem, Reference Griem1963), though our method can state the regime of the whole plasma. This map gives useful a priori information for many topics of plasma. Moreover, from the point of view of computational time, it also implies a considerable saving, since the resolution of CE and Saha equations for carbon plasmas only requires to solve 6 × 6 matrices, whereas the rate equations entail to handle with matrices of very high order because the number of levels (and, therefore, transitions) needed to get accurate results under NLTE conditions is usually very large (in our case, for example, 31928 × 31928).
Fig. 1. (Color online) Carbon plasma regimes (a) and average ionization (b) as a function of electron density and temperature.
In order to check the results of radiative properties we plotted in Figure 2, the spectrally resolved emissivity for given plasma conditions versus the calculations given in Colgan et al. (Reference Colgan, Fontes and Abdallah2006). The line profile considered is a Doppler one. As we can see, the emissivities obtained using both codes under DLA approach are very similar, showing the same structures at comparable energy positions. Furthermore, we have also plotted the emissivity calculated using RAPCAL-AP. We can observe that there is a diminution of lines with respect to RAPCAL-DLA calculations and also that some lines are shifted. These results are expected since the DCA approach entails a lower number of transitions and, moreover, the atomic data provided by the atomic module implemented in RAPCAL are not so good than those given by FAC code. This fact produces a shift on the position of the peaks. Therefore, the simple atomic model is able to provide accurate results of atomic magnitudes such as the average ionization and ion populations and due to its simplicity and low computing cost, it is useful to optimize more complex calculations (by means of the maps given in Figs. 1a and 1b, for instance). On the contrary, when very detailed calculations are required, as the spectrally resolved emissivities, the RAPCAL-AP model can only provide qualitatively results and RAPCAL-DLA calculations are required.
Fig. 2. (Color online) (a) Emissivity of carbon at an electron temperature of 10 eV and density of 1015 cm−3. (b) Comparisons with results from Colgan et al. (Reference Colgan, Fontes and Abdallah2006).
Finally, in Table 3, Planck and Rosseland mean opacities for two densities and temperatures are compared with the LTE opacities calculation code LEDCOP (Magee et al., Reference Magee, Abdallah and Clark1995) developed at LANL. This code uses a basis set of detailed LS terms including interactions with the plasma which are treated as perturbations. With the help of Figure 1a we have checked that the LTE assumption is appropriate for all the conditions shown in the table. It is observed to be a good agreement between both results.
Table 3. Rosseland and Planck mean opacities for optically thin carbon plasmas assuming LTE approach. The results are compared with those provided by LEDCOP code (Magee et al., Reference Magee, Abdallah and Clark1995)
3.2. Aluminum Plasmas
In Table 4, are listed the number of configurations and levels considered for the seven first ions of aluminum and the set of configurations selected. For the rest of the ions, they were presented in Table 1. Once more and using RAPCAL-AP calculations, we carried out the same maps for carbon cases which are displayed in Figure 3.
Fig. 3. (Color online) Aluminum plasma regimes (a) and average ionization (b) as a function of electron density and temperature.
Table 4. Set of configurations and numbers of relativistic configurations and levels considered in the calculations performed by RAPCAL code for the seven first aluminum ions (n ≤ 10, n' ≤ 6)
As we did for carbon plasmas, we start analyzing the results obtained with RAPCAL for the average ionization. In this case, we also compared the integer charge state distribution (see Table 5). The plasma conditions are T = 40 eV and ρ = 0.0135 gcm−3, in which we can assume LTE conditions according to Figure 3a. We have performed calculations both with RAPCAL-AP and RAPCAL-DLA. The results are compared with those provided by Zeng and Yuan (Reference Zeng and Yuan2002) that solve the Saha-Boltzmann equations in the DTA approach, including a large number of levels, and Faussurier et al. (Reference Faussurier, Blancard and Decoster1997) who proposed a method based on the average atom (AA) model. The first feature that we can observe is the good agreement among the different calculations for the average ionization, which is excellent between the two detailed calculations. For these calculations it is also found a good accord for the integer charge distributions. The other two models, which are cruder, show shifts toward lower (RAPCAL-DCA) and higher (Faussurier et al., Reference Faussurier, Blancard and Decoster1997) charge state distributions, although their results are also competitive. This fact is very relevant because these kinds of calculations usually require enormous amounts of computing time and this is considerably reduced when these simple models are used.
Table 5. Integer charge state distribution (in percentage) and average ionization (
) of an aluminum plasma (T = 40 eV, ρ = 0.0135 gcm−3) assuming LTE conditions. For comparison our results in DLA approach (RAPCAL-DLA), using the atomic module of RAPCAL (RAPCAL-AP), and those of Zeng and Yuan (Reference Zeng and Yuan2002) and Faussurier et al. (Reference Faussurier, Blancard and Decoster1997) are given
As it is known, the plasma transmission is a useful tool for spectroscopic diagnostics of plasma temperature, and since the transmission is related straight to the opacity, its analysis permits to check the opacity models. In particular, we have studied the transmission spectrum of an aluminum plasma experimentally obtained with the iodine laser ASTERIX at the Max-Planck-Institut für Quantenoptik reported by Winhart et al. (Reference Winhart, Eidmann, Iglesias, Bar-Shalom, Mínguez, Rickert and Rose1995). The plasma conditions were T ~ 20 eV and ρ ~ 0.01 gcm−3, the spectral range 70 ≤ hν ≤ 280 eV and the sample layer was 1075 Å. According to that work, LTE is a reasonable assumption for these conditions. Taking into account the average ionization provided, about 4.3, it gives an electron density around 1021 cm−3 and Figure 3a also corroborates the validity of the assumption. In Figure 4, we present our result calculated under DLA approach compared with the experimental one and the obtained by the widely known opacity code OPAL (Rogers et al., Reference Rogers, Iglesias and Wilson1992).
Fig. 4. Comparison of the experimental aluminum transmission spectrum (Winhart et al., Reference Winhart, Eidmann, Iglesias, Bar-Shalom, Mínguez, Rickert and Rose1995) (black solid) with RAPCAL (gray solid) and OPAL (Rogers et al., Reference Rogers, Iglesias and Wilson1992) (dotted). Plasma conditions are T = 20 eV, ρ = 0.01 gcm−3.
From the figure, we can observe that in general our agreement with the experimental data and OPAL results is quite good. This agreement is better both at low and high wavelengths. However, it can be observed that our model predicts a greater transmission, and therefore a lower opacity, in the range 95–105 Å. On the contrary, OPAL seems to reproduce better this region of the spectrum although it overestimates the transmission in the higher range of wavelengths. Some of the differences between both theoretical results can be explained taking into account that the average ionization predicted by OPAL is 4.30 whereas we obtain 4.18. Although they are very similar, discrepancies in the average ionization imply differences in the ion and level populations which have a dramatically influence in the radiative properties. Zeng et al. (Reference Zeng, Yuan and Lu2001b) also carried out a study of this experiment. They solved the Saha-Boltzmann equations under DTA approach and performed an elaborated calculation of the opacity, obtaining a good agreement with the experiment. In Table 6, we display the population fraction of the most abundant ions obtained by them and using RAPCAL, as well as the average ionization, and the mean opacities, which are also compared with LEDCOP. We can see that our average ionization is very similar to the obtained by the other two calculations and that the population fractions show a good agreement with those reported in Zeng et al. (Reference Zeng, Yuan and Lu2001b). However, our model presents a small shift of the ion population to lower charge states. This confirms that small changes in the average ionization could entail large ones in the level populations and therefore in the radiative properties. From Table 6, it is worth pointing out the excellent agreement among the mean opacities obtained by the three computations.
Table 6. Population of the most abundant ions, average ionization and mean opacities for an aluminum plasma (T = 20 eV, ρ = 0.01 gcm−3). Comparisons between RAPCAL and results from Zeng et al. (Reference Zeng, Yuan and Lu2001) and LEDCOP code (Magee et al., Reference Magee, Abdallah and Clark1995)
We also present some calculations of radiative properties under NLTE conditions carried out using RAPCAL-DLA. Thus, in Figures 5 and 6, are displayed the absorption coefficient, and the emissivity for the plasma conditions 400 eV and 0.1778 gcm−3 and 160 eV and 0.0056 gcm−3, respectively. The calculations are compared with THERMOS (Nikiforov et al., Reference Nikiforov, Novikov, Uvarov, Dragalov and Solomyannaya1995), which is a widely known opacity code that uses a self-consistent field model with Dirac equations for generating the atomic data and it calculates the NLTE opacities by numerical solution of the rate equations following an improved AA approximation, and DESNA code (Nikiforov et al., Reference Nikiforov, Novikov and Uvarov2000) which is a detailed CRSS level-by-level kinetics model that also calculates NLTE emissivities and opacities.
Fig. 5. Absorption coefficient for aluminum plasma under NLTE situation at 400 eV and 0.1778 gcm−3. Comparison of RAPCAL code with THERMOS (Nikiforov et al., Reference Nikiforov, Novikov, Uvarov, Dragalov and Solomyannaya1995) and DESNA (Nikiforov et al., Reference Nikiforov, Novikov and Uvarov2000) models.
Fig. 6. Emissivity for aluminum plasma under NLTE situation at 160 eV and 0.0056 gcm−3. Comparison of RAPCAL code with THERMOS (Nikiforov et al., Reference Nikiforov, Novikov, Uvarov, Dragalov and Solomyannaya1995) and DESNA (Nikiforov et al., Reference Nikiforov, Novikov and Uvarov2000) models.
The first thing that we can observe from both figures is that THERMOS results show a lesser line structure than the other two. This fact is expected since, as it was said above, the rate equations in this code are solved in the AA approximation. On the other hand, our results and those of DESNA models show a general agreement with respect to the structure of lines and their positions in the photon energy scale. However, some differences in the number of lines and their intensity are observed, which could obey the fact that in DESNA the detailed structure of ion level configurations is taken into account in LS-link approximation subject to splitting into terms whereas in our case FAC works in the DLA approximation, obtaining the atomic energies by diagonalizing the relativistic Hamiltonian.
3.3. Krypton Plasmas
For the case of krypton, we have analyzed the radiative power loss in the optically thin limit. As it is known, this magnitude is of importance for high-Z plasmas since it allows us to know the radiation emitted and krypton plasmas are employed in multiple applications, where the plasmas range electron temperatures of a few eV to a few keV and electron densities from 1014 to 1023 cm−3. In the first place, we studied the radiative power loss for low-electron-density (≤2 × 1014 cm−3) krypton plasmas for a wide range of temperatures. Our calculations were performed using the atomic module of RAPCAL under relativistic DCA approach (i.e., RAPCAL-AP) and the results are listed in Table 7. They are compared with those obtained using the analytical expression, valid for low density regime only, proposed by Fournier et al. (Reference Fournier, May, Pacella, Finkenthel, Gregory and Goldstein2000). This expression was fitted to total radiative cooling coefficients computed by them and validated with krypton cooling rates derived from tokamak experiments. From the table, we can observe that both results are in general quite similar. Obviously, there are discrepancies but they are small and derive from the differences in the atomic models employed. In these low density situations, the corona equilibrium can be assumed. Under this regime, the plasma average ionization and level populations are almost independent of the density and, therefore, the radiative cooling coefficient does not change. In Table 7, the results shown are obtained as the total radiative cooling coefficient multiplied by the electron and ion densities, i.e., by n e2/
. For this reason, the dependence of the radiative power loss in Table 7 with the density is only on n e2, increasing two orders of magnitude when the density increases one. However, we can observe for the highest density and the two lowest temperatures shown, that our results exhibit a different behavior. This fact implies that for these temperatures, the CE assumption is not accurate enough. Obviously, the calculation performed using the analytical fit does not present this discrepancy since it does not depend on the density.
Table 7. Radiative power loss (ergs/s/cm3) for low densities and several temperatures. Comparison between RAPCAL results and those obtained using the fit provided by Fournier et al. (Reference Fournier, May, Pacella, Finkenthel, Gregory and Goldstein2000)
We have also carried out an analysis for higher density and temperatures. In Table 8, we present the cases considered and the average ionization obtained for those cases. The calculations were made using RAPCAL-AP. The results are compared with those reported by Chung et al. (Reference Chung, Fournier and Lee2006). In that work, the calculations performed were very sophisticated. The atomic data were generated by the HULLAC suite of codes (Bar-Shalom et al., Reference Bar-Shalom, Klapisc and Oreg2001). For atomic structure data HULLAC calculates the multi-configuration, intermediate coupling energy eigenvalues of the fine structure levels, and configuration interaction is taken into account for energy level calculations and oscillator strengths (Klapisch, Reference Klapisch1971). Autoionization rates (Oreg et al., Reference Oreg, Goldstein and Klapisch1991) and photoionization cross sections were computed using the multi-configuration wave functions and the collisional excitation and ionization were obtained in the distorted wave approximation (Bar-Shalom et al., Reference Bar-Shalom, Klapisch and Oreg1988). A good agreement between our results and those provided by Chung et al. (Reference Chung, Fournier and Lee2006) is found, which is a remarkable fact since, as it was stated before, their calculations are very complex. We can observe that for the two lowest densities and for all the temperatures considered, the plasma average ionization is density independent. This fact implies that in these cases the krypton plasma can be assumed under CE conditions. Hence, the total radiative cooling coefficients scaled by the electron densities for these plasma conditions are quite similar, as it is displayed in Figure 7. On the other hand, for the highest temperature considered, the CE approach could be assumed for temperatures higher than 7 keV, as we can see both form Table 8 and Figure 7. Finally, we would like to point out that results displayed in Figure 7 are very similar, both qualitatively and quantitatively, to those reported by Chung et al. (Reference Chung, Fournier and Lee2006). As a conclusion, we have obtained that the simple atomic model implemented in RAPCAL can be used to compute radiative power losses and cooling coefficients for high Z elements in a wide range of plasma conditions with reasonable accuracy. This is remarkable result since, as it was said previously, these magnitudes are essential in many topics of the plasma research.
Fig. 7. Total radiative cooling rates scaled by electron density at electron densities of 1014 cm−3, 1018 cm−3 and 1022 cm−3.
Table 8. Average charge state of krypton ions. Our results are compared with those given by Chung et al. (Reference Chung, Fournier and Lee2006)
3.4. Xenon Plasmas
In this last section, we present a simulation of the intensity spectrum of an optically thick xenon plasma obtained by the experiment carried out by Chenais-Popovics et al. (Reference Chenais-Popovics, Malka, Gauthier, Gary, Peyrusse, Rabec-Le Gloahec, Matsushima, Bauche-Arnoult, Bachelier and Bauche2002) at the Laboratoire pour l'Utilisasion des Laseres Intenses (LULI). From the experiment they determined that the plasma was under stationary situation during the laser pulse, where the ion density is 4.75 × 1018 cm−3. The electron temperature, average ionization and electron densities were obtained by fitting simultaneously the electron and ion Thomson scattering spectra and they are, respectively, 415±40 eV, 27.4±1.5, and 1.30±0.05×1020 cm−3.
Our calculation was performed assuming an homogeneous plasma with planar symmetry. The calculations were done under relativistic DCA-UTA approach (Bauche-Arnoult et al., Reference Bauche-Arnoult, Bauche and Klapisch1985) using FAC code and including corrections to the oscillator strengths due to CI. The configurations included in our simulation for xenon ions involved are listed in Table 9. Furthermore, we also include in the table the number of relativistic configurations and line transitions. According to the experimental description, a spectral resolution of 50 mÅ was fixed and the reabsorption effect was introduced assuming a 160 µm path length by means of the escape factor formalism. The simulation was carried out at 450 eV, the upper limit of the estimated error bar in the Thomson scattering measurements, since the agreement obtained for the average ionization at 450 eV is better than for 415 eV. This also happens with the simulation of AVERROES/TRANSPEC, a sophisticated collisional-radiative super-configuration code (Peyrusse, Reference Peyrusse2000). Thus, the average ionization obtained for this temperature by RAPCAL is 27.1 whereas AVERROES gives 26.8 (Chenais-Popovics et al., Reference Chenais-Popovics, Malka, Gauthier, Gary, Peyrusse, Rabec-Le Gloahec, Matsushima, Bauche-Arnoult, Bachelier and Bauche2002). Both of them are in agreement with the Thomson scattering results. In Figure 8, we present the intensity spectrum obtained using RAPCAL compared with the experimental one (Chenais-Popovics et al., Reference Chenais-Popovics, Malka, Gauthier, Gary, Peyrusse, Rabec-Le Gloahec, Matsushima, Bauche-Arnoult, Bachelier and Bauche2002) and the one provided by ATOMIC (Fontes et al., Reference Fontes, Colgan, Zhang and Abdallah2006). The latter calculation was carried out with the relativistic UTA formalism, and therefore, it is comparable to RAPCAL results. The main features of the experimental spectrum appear in the region of wavelengths larger than 12.5 Å associated to the 3d–4f transition array. The higher peaks are related to Co-like and Fe-like ions. In the spectrum calculated by RAPCAL they are also the main ones although the relation between the intensities of the peaks disagrees with the experimental situation. This fact is due to our calculations under-populate the Fe-like ion. On the other hand, in the spectrum provided by ATOMIC the transitions associated to Ni-like and Cu-like ions are larger than the Fe-like and the peak linked to the transition array of the Mn-like ion is not present. This one appears in the RAPCAL calculation but our code underestimates the populations of Ni and Cu-like ions. It seems like the ATOMIC code provides a lower ionization than the RAPCAL code. In Fontes et al. (Reference Fontes, Colgan, Zhang and Abdallah2006), the authors considered the possibility that some differences between the theoretical and experimental spectra could come from the fact that the temperature might be higher than the one experimentally measured. However, their calculations at temperatures of 500 and 600 eV improve the range of lower wavelengths but increase the discrepancies in the opposite range of wavelengths. We would like to point out that the experimental spectrum is a time-integrated one and, therefore, it could occur that we were not able to fit the whole spectrum under the assumption of uniform plasma.
Fig. 8. (Color online) Experimental, ATOMIC and RAPCAL xenon intensity data. The numerical calculations were performed assuming T = 450 eV and n i = 4.75 × 1018 cm−3.
Table 9. Set of configurations selected and number of relativistic configurations and line transitions included in the calculations performed by RAPCAL code for the xenon ions involved (n ≤ 8, n' ≤ 5, n'' ≤ 6)
Another source of discrepancy is the spectral detail of the calculation. In order to obtain better results, a DLA description including full CI is required. However, a detailed level calculation for an ion as complex as xenon would include a huge amount of levels and transitions and the problem would become intractable. Therefore, hybrid models that combine DLA description including CI for the most relevant levels and transitions and other approaches for the rest seem to be the solution. In particular, in Fontes et al. (Reference Colgan, Fontes and Abdallah2006), a calculation is performed of the xenon spectra making use of an hybrid model based on the “dual method” described in Abadallah et al. (Reference Abdallah, Zhang, Fontes, Kilcrease and Archer2001) and the spectrum obtained presents a better agreement with the experimental data than in the DCA-UTA calculation. Therefore, taking into account that the RAPCAL DCA-UTA calculation provides an accurate value for the average ionization and an estimation of the spectra similar to the ATOMIC DCA-UTA, it is also expected that an hybrid model can improve our results dramatically. T = 450 eV and n i = 4.75 × 1018 cm−3.
4. CONCLUSIONS
A new flexible code to obtain accurate optical properties for optically thin and thick plasmas in a wide range of densities and temperatures has been presented. In this code, the level populations are obtained solving a CRSS model which employs analytical expressions for the rate coefficients of the atomic processes, which yield a substantial saving of computational requirements but providing satisfactory results as it has been shown. It used the matrix sparse technique for the storage, which allow us to include a large number of atomic levels, and iterative procedures in order to accelerate the resolution of the set of equations. The self-absorption effects in homogeneous plasmas are modeled using the escape factor formalism for planar, cylindrical, and spherical symmetries. The atomic data required can be provided to the code both as an input file or they can be generated using the atomic module implemented in RAPCAL, which is based on analytical potentials under the relativistic DCA approach. This last option is very useful when we are interested in providing radiative properties to simulations that require opacities and emissivities to hydro-codes and, therefore, celerity is desired although some accuracy is lost.
We have presented results for ion populations, plasma average ionization, spectrally resolved opacities and emissivities, mean opacities, transmission and intensity spectrum, and radiative power loss. We have considered low-, medium- and high-Z elements (C, Al, Kr, and Xe), both in LTE and NLTE situations, in a wide range of plasma conditions. Furthermore, we have analyzed the opacity effects in the ion populations and the intensity spectrum in an optically thick homogeneous plasma. We have performed calculations using both very accurate atomic data provided by FAC code and using the simple model implemented in RAPCAL. For all the calculations shown, we have made comparisons with experimental data or with theoretical ones provided by well known numerical codes. We have obtained that the simple atomic model based in analytical potentials implemented in RAPCAL is useful and accurate in order to compute ion populations, plasma average ionization or radiative power losses and also to optimize more detailed and complex calculations by means of the maps of the average ionization and plasma regimes presented. However, for magnitudes such as spectrally resolved opacities and emissivities, this simple model is not accurate enough and it only gives qualitatively conclusions. On the other hand, the calculations made using RAPCAL with atomic data from FAC show good agreements for all the magnitudes analyzed.
In this work, we have presented calculations using the atomic module of RAPCAL using as effective potential the isolated analytical one. However, as it was said previously, we have also developed analytical potentials that model strongly and weakly coupled plasmas. It is our purpose in a future work to carry out calculations of radiative properties using these potentials.
Finally, there are two tasks that need to be considered. The first one is the development of a hybrid model that allows us to handle high Z elements with more precision, mixing DLA approach for the ions which has more influence on the spectrum and DCA approach for the others. The second one is to extend the opacity effects to non-homogeneous plasmas.
ACKNOWLEDGMENTS
This work has been supported by a Research Project of the Spanish Science and Education Ministry and also by the “Keep in touch” Project of the European Union.
