2 The line intensity

The number of photons emitted in a spectral line ($i \rightarrow j$) for an optically thin coronal plasma is given by:

$$I_{ij} = \int_V N_j{\left({X^m}\right)} A_{ji} {\rm d}V$$ (1)

$$= \int_V G_{ij}(T, N_{\rm e}) N_{\rm e}^2 {\rm d}V \ \rm ph~cm^{-3}~s^{-1}$$

where

• $G_{ij} (T,N_{\rm e})$ is the contribution function of the line and depends on the electron temperature (T) mainly through the ion abundance, and on the electron density ($N_{\rm e}$), mainly through the level population.

The function $G_{ij} (T,N_{\rm e})$ may be expressed as

$${G{\left({T,\lambda_{i,j}}\right)}} = {N_j{\left({X^m}\right... ...}}~{N{\left({H}\right)}\over N_{\rm e}}~{A_{ji}\over N_{\rm e}}$$ (2)

and

• ${N_j{\left({X^{+m}}\right)}\over{N{\left({X^{+m}}\right)}}}$ is the relative upper level population;
• ${{N{\left({X^{+m}}\right)}}\over{N{\left({X}\right)}}}$ is the relative abundance of the ion X^+m;
• ${{N{\left({X}\right)}}\over{N{\left({H}\right)}}}$ is the abundance of the element X relative to Hydrogen;
• ${N{\left({H}\right)}\over N_{\rm e}}$ is the hydrogen abundance relative to the electron density ($\approx 0.8$).

The element abundance and the ionisation balance population of each ion are known from the literature (Allen 1973; Feldman et al. 1992; Meyer 1985; Grevesse & Anders 1992; Waljeski et al. 1994 for the element abundances, Shull & Van Steenberg 1982; Arnaud & Rothenflug 1985; Arnaud & Raymond 1992; Landini & Monsignori Fossi 1991 for the ionisation equilibrium), while the level population must be calculated solving the statistical equilibrium equation including all the important processes involved in level excitation and de-excitation. In low density plasmas the most important populating and de-populating processes are spontaneous radiative decay and excitation and de-excitation from electron-ion collisions, since they are generally faster than ionisations and recombinations. In hot plasmas proton collision rates can also be important in determining the level population.

The statistical equilibrium equations take the form

 

$$N_j{\left({N_{\rm e}\Sigma_iC^{\rm e}_{j,i}+N_{\rm p}\Sigma_iC^{\rm p}_{j,i}+\Sigma_{i<j}A_{j,i}}\right)} =$$ (3)

$$\Sigma_iN_i{\left({N_{\rm e}C^{\rm e}_{i,j}+N_{\rm p}C^{\rm p}_{i,j}}\right)}+\Sigma_{i\gt j}N_iA_{i,j}$$

with $C^{\rm e}_{j,i}$ and $C^{\rm p}_{j,i}$ the electron and proton collisional excitation rates ($\rm cm^{3}s^{-1}$), $C^{\rm e}_{i,j}$ and $C^{\rm p}_{i,j}$ the electron and proton collisional de-excitation rates and A_ji $(\rm s^{-1})$ are radiative decay probabilities from level j to level i.

The collisional excitation rate for a Maxwellian electron velocity distribution can be expressed as

$$C^{\rm e}_{i,j}={{8.63\cdot10^{-6}}\over{T_{\rm e}^{1/2}}} {... ...a_i}} \exp{\left({-{{\Delta E_{i,j}}\over{kT_{\rm e}}}}\right)}$$ (4)

where $\omega_i$ is the statistical weight of level i, k is the Boltzmann constant and $\Upsilon_{i,j}$ is the thermally-averaged collision strength (effective collision strength):

$$\Upsilon_{i,j} = \int_0^\infty \Omega_{i,j} \exp{\left({-{E\over{kT_{\rm e}}}}\right)}{\rm d}{E\over{kT_{\rm e}}}$$ (5)

where $\Omega_{i,j}$ is the collision strength, related to the electron excitation cross section and E is the energy of the scattered electron relative to the final energy state of the ion. In the evaluation of the effective collision strength the scaling laws of Burgess & Tully (1992) have been adopted.

For solving the statistical equilibrium in Eq. (3) it is necessary to have a large dataset which includes an atomic model with experimental energy levels and radiative and collisional transition probabilities for all the possible transitions within the levels of the adopted atomic model. For this reason in the recent past several extensive databases of theoretical data have been developed, such as ADAS (Summers et al. 1996) and CHIANTI (Dere et al. 1997). Both these databases allow the complete solution of the system of equation described above for the most important ions, though both of them neglect, at the moment, the proton collision rates.

The aim of the present update of the Arcetri spectral code is to renew the entire set of theoretical data which composed the old version of the Code, as described in Landini & Monsignori Fossi (1990) and Monsignori Fossi & Landini (1996), which included the most important emission lines in the range 1 - 2000 Å. For the evaluation of their emission the assumption was made that the population of the upper level of the transition j occurs mainly by collisional excitation from the ground level g and the radiative spontaneous decay dominates any other depopulation process. The old version also included a full set of atomic data which allowed the solution of the statistical equilibrium equation for the ions of the Beryllium, Carbon and Nitrogen isoelectronic sequences and for the Iron Ions from Fe IX to Fe XXIV.

In the present update we have included the whole CHIANTI dataset in the Arcetri spectral code, and we have also added some original data:

• Atomic data for the so-called minor elements (Na, Al, Cl, P, K, Ti,Cr, Mn, Co and Zn). The ions of these elements may provide several observed lines some of which are weak in most spectra but can prove useful in spectral analysis and DEM studies;
• Fe III atomic model, collisional and radiative dataset.
• The atomic data for the lines in the range 1 - 2000 Å which are included neither in CHIANTI nor in the minor elements dataset; they have been renewed updating the transition probabilities from the literature.

The program evaluates level population and contribution functions for temperature ranging between 10⁴ and 10⁸ K and any electron density, usually assumed between 10⁶ and 10¹⁵ $\rm cm^{-3}$.Also continuum emission from free-free, free-bound and two-photon processes is evaluated.

An example of the continuum emission is shown in Fig. 1.

  

Figure 1: The continuum emission per unit emission measure for a thin plasma at temperature 3  104 K and electron density $1~ 10^8~ \rm cm^{-3}$between 1 Å  and 2000 Å. Free-free (dotted line), free-bound (dashed line) and two-photon continuum (long-dashed line) are indicated

The selection of the data for the minor elements has been carried on using the excellent reviews given in the CHIANTI paper (Dere et al. 1997), Pradhan & Gallagher (1992), Itikawa et al. (1984) and Itikawa (1991) and (1996). The paucity of theoretical calculations in the literature (especially for collisional data) has often severely restricted the range of possible choices. The minor elements dataset have been developed mainly adopting theoretical results calculated using the University College of London distorted wave program (Eissner & Seaton 1972) and the close coupling (CC) R-Matrix package (Burke et al. 1971; Berrington et al. 1978) developed at Queens University of Belfast. Like the CHIANTI data, the minor elements dataset allows to evaluate the statistical equilibrium level population. Nevertheless no collision data are available for the Magnesium-like, Oxygen-like, and nitrogen-like minor elements, so it has been necessary to interpolate the effective collision strengths and the radiative data from the isoelectronic ions which were available in the CHIANTI database. The quality of the interpolation depends heavily on the accuracy of the existing data, and in some cases there are severe limitations to the reliability of the interpolated results.