Opacity calculations of plasmas by using parametric potentials

Laser and Particle Beams (1996), vol. 14, no. 4, pp. 631-635 Printed in the United States of Arnerica By P. MARTEL,*f J.M. GIL,*? R. RODRIGUEZ,* E. MÍNGUEZ,? AND L. DORESTE* *Departamento de Física, Universidad de Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria tInstituto de Fusión Nuclear, Universidad Politécnica de Madrid, 28006 Madrid, Spain (Received 20 October 1995; Revised 30 May 1996; Accepted 1 July 1996) A numerical model for opacity calculations by using a family of analytical potentials for each configuration in the plasma is presented. The obtained numerical opacity results with this model are compared with those obtained by using a self-consistent potential model. In the opacity calculations of hot dense matter, it is usually necessary to consider many configurations and also thousands of line transitions. This requires the knowledge of the atomic structure of the ions in the plasma. Several computer codes are currently available t^ determine these ~temic dit2, which r!!ews imprwement ef ?he rrdiitien npicity endes. Models using both self-consistent methods or analytical potentials have been recently tested for severa1 elements in a short range of densities and temperatures (Rickert et al. 1995), and have shown important differences in the frequency-dependent opacity. Also, some of these models tried to simulate experimental results (Winhart et al. 1995), showing deviations from the experimental RosseIand opacity by not more than a factor of two. For high-Z elements it is necessary to treat a large number of configurations, so that analytical potentials seem to be useful to generate the atomic physics data for opacity calculation. In a previous work we described a detailed opacity model (Mínguez & Falquina 1992) in which an average ion was first solved. Then, using the procedure reported by Goldberg et al. (1986), denoting and promoting electrons in turn to the average, the probability of each configuration in the plasma is determined according to Argo and Huebner (1976). Finally, for each configuration with higher probability than lop5, the radial Dirac equation is solved again using a self-consistent potential and obtaining the atomic data for opacity ci!cii!atinn. This prncediire was tested with ether me de!^ d~ring the Third Qpacity Workshop (Rickert et al. 1995). However, this model is large and time-consuming, essentially for high-Z elements. Later on, we proposed a new family of parametric potentials (Martel et al. 1995) with three parameters in the general form, which can be reduced to two-parameter and oneparameter potentiais, accoráing to ciifferent situations. lhe parameters of these potentiais were determined by fitting to a self-consistent potential. Also, an important feature is that the parameters of the potential were fitted by a simple function of the nuclear charge, and it is available for intermediate and highly ionized atoms from helium to uranium sequences. Transitions energies and oscillator strengths obtained with this proposed model were in good agreement with other models. 5 1996 Carnbridge University Press 0263-0346/96 $1 1 .00 + .10 632 P. Martel er al. In this work the main goal is to use this family of parametric potentials to create a numerical model usetu1 for opaCity cakulafions. With this model it is possible to treat a large number of configurations with a small calculation time, and to obtain results close to those obtained with the self-consistent model explained above. Numerical results of opacities for iron at several densities and temperatures are reported. 2. Opacity model using parametric potential The first part of the numerical model to be proposed is based on the average ion model, being that the detailed configurations determined from it follow a well-known procedure. The average ion model for the given density and temperature is solved using the JIMENA computer code (Mínguez & Falquina 1992). This gives for each n subshell the occupation number NnU. Then, using the procedure of Goldberg et al. (1986), a ser of configurations p in turn to the average are determined. The probability Ptp) o1 each configuration in the plasma is created by means of the binomial formula (Argo & Huebner 1976). These configurations in which P(p) is larger than are now taken into account for the next phase of the calculation, where we introduce the family of parametric potentials U(r) given by Martel et al. (1995) where the screening function, +(r), has different values depending on N (number of bound electrons), and Z (atomic number). We use: These parameters a,, a2, and a, are determined by fitting the equation (2) to the selfconsistent potential by means of a nonlinear simplex method, giving as a result a fourthdegree polynom: The coefficients, of this expression were obtained for the ground state of He-like to Fe-like ions, and they can be found in Martel et al. (1995). Also, there are unpublished coefficients, from Co-like to U-like ions, available that follow the same procedure. These coefficients will be included in a future paper. Using this parametric potential in the Dirac equation for each detailed configuration, the energy levels and oscillator strengths are obtained. Because the number of electrons and the parametric potential were assumed fixed, the calculation time is rather low in comparison with the self-consistent calculation, by a factor that depends on the material and the computer. Finally, with al1 the atomic data generated for al1 the configurations, the opacity calculation is fast and simple. In this multifrequency opacity calculation, bound-bound, boundfree, free-free, and scattering processes are included, assuming that also included in the line broadening is the Moszkowski formulism (1979). Opacity calculations o f plasmas 633 € I I 1iiI Calculated by using analytical potential Calculated by using self-consislent potentials . --/ 1 FIGURE 1. Bound-bound opacity for Fe at kT 20 eV and density lop3 g cmp3. 3. Analysis of numerical results for iron plasmas Severa1 cases of iron were simulated with this model, mainly those cases already included in previous references, such as: at 20-eV temperature and densities g cm-3 and lo-' g cmp3; and higher temperatures and densities as 200 eV and 7.86 g cmp3. Results were compared with those obtained with JIMENA code by using a self-consistent detailed model. The reason was that this model has already been checked with other models and with experiments, and the parametric potential was fitted with this self-consistent one. Figures 1 through 3 show only the bound-bound opacities for iron at several temperatures and densities, obtained with the JIMENA code using a self-consistent potential, and AL- ---l-.*.--l --*--*.-l t... L ---- /*\ LIK a~~a~~~i~al pu~cll~ial gi~c~~ uy cqua~iu~~ (1). A similar piofiie is o'oiaineci wiih iess caiculation time For the above cases, mean opacities, Planck and Rosseland, are obtained with both models, and the final results are shown in table 1, KRs, and Kp,, being the Rosseland and TABLE 1. Rosseland and Planck mean opacities at several temperatures and densities with a self-consistent potential (K,,, K,,,) and the analytical potential (KRa, Kpa) Incident photon energy (eV) FIGURE 2. Bound-bound opacity for Fe at kT 20 eV and density loT2 g cm-' - Calculated by using analytical otential ---- Calculated by using self-consisfent potentiais 10 -' 1 1 o 10 10 ' 10 Incident photon energy (eV) FIGURE 3. Bound-bound opacity for Fe at kT 200 eV and density 7.86 g cm-3 Opacity calculations of plasmas 63 5 Planck opacities obtained with the self-consistent potential, and KR, and K,, with the analytical potential. The agreement is better at high densities, because the parameters in equation (3) fit the self-consistent potential better, as was reported by Martel et al. (1995). In this work we show how the use of parametric potential allows detailed calculations of opacities to be made with considerable savings of computation time. We think that this fact could be useful in calculations of high-Z plasmas in which the large number of configurations will make a self-consistent detailed configuration calculation unapproachable. Also, this model permits one to get a set of atomic data for the analytical rate equations used in non-LTE numerical models' calculations. REFERENCES ARCO, M.F. & HUEBNER, W.F. 1976 JQSRT 16, 1091. GOLDBERG, A. et al. 1986 Phys. Rev. A 34, 421. MARTEL, P. et al. 1995 J. Quant. Spectrosc. Radiat. Transfer 54, 621. M~NGUEZ, E. & FALQUINA, R. 1992 Laser Part. Beams 10, 651. Mosz~ows~~, S.A. 1962 Prog. Theor. Phys. 28, 1. RICKERT, A. et al. (eds). 1995 Max Planck Institut für Quantenoptik Report, MPQ-204. WINHART, G. et al. 1995 J. Quant. Spectrosc. Radiat. Transfer 54, 437.