3 The super-level concept

The numerical solution of the statistical equations is limited to model atoms with a few hundred energy levels. However, the complex electron configurations of the iron group elements would require thousands of energy levels for a detailed description. For this reason, a simplified treatment is necessary. In the present work we adopt the concept of super-levels, which has originally been proposed by Anderson (1989).

In this approximation a large number of atomic energy levels (termed "sub-levels'' in the following) is represented by a small number of "super-levels''. Each transition between two super-levels comprises a large number of atomic line transitions ("sub-lines''). In the radiation transport, all sub-lines must be treated at their proper frequency because of the frequency coupling in expanding atmospheres. So in contrast to static atmospheres, no sampling or re-ordering techniques can be applied. Instead, each super-line has a complicated "profile function'' which is a superposition of all sub-lines involved.

The preparation of the model atom and the transition cross sections in the present work is based on the work of Dreizler & Werner (1993). The complete iron group data of Kurucz (1991), with some 10⁷ line transitions between several thousands of energy levels, are represented by model atoms with 10-20 super-levels per ionization stage.

In the following, we describe the construction of the model atom (Sect. 3.1), the representation of opacities and emissivities in the radiation transport (Sect. 3.2), and the corresponding transition probabilities in the statistical equations (Sect. 3.3).

   
3.1 The model atom

 

 

Table 2: Solar abundances (number ratios) of iron group elements with respect to $n_{\rm Fe}$. In the generic model atom for the iron group a relative weighting according to these values is adopted.

Element X	Z	$n_X/n_{\rm Fe}$
Sc	21	$3.09\times 10^{-5}$
Ti	22	$2.40\times 10^{-3}$
V	23	$2.95\times 10^{-4}$
Cr	24	$1.29\times 10^{-2}$
Mn	25	$9.33\times 10^{-3}$
Fe	26	1.0
Co	27	$2.24\times 10^{-3}$
Ni	28	$4.79\times 10^{-2}$

Figure 1: Ionization stage V of the generic model atom. The sum of the statistical weights of theconsidered atomic energy levels is plotted in bins of $100~ {\rm cm}^{-1}$. The contribution of levels with even parity is illustrated in grey. The boundaries of the energy bands are indicated by horizontal lines. For this model ion 19 804 atomic levels are represented by 19 super-levels.

For the construction of the model atom, the atomic energy levels from the Kurucz data (Kurucz 1991) are divided into energy bands B_L, each of which is represented by a super-level L. The relative occupation of the sub-levels whithin each energy band is assumed to be in LTE, corresponding to a temperature T. In the present section we do not specify this temperature and work out the expressions for the super-level energies and statistical weights for the general case (in Sect. 3.2 we will give a detailed discussion of this point).

Owing to the similarities in the electron configurations, it is even possible to represent the whole iron group by one generic model atom (Dreizler & Werner 1993; Haas et al. 1996). In this case, the energy levels of different elements i are weighted corresponding to their relative abundances a_i(Table 2). The advantage of this approach is the computationally cheap consideration of all important iron group opacities. The disadvantage is the loss of accuracy concerning the modeling of the detailed atomic processes.

Under the assumption of LTE, the relative occupation of the sub-levels within an energy band is

$$% g_{il}(T) = a_i\ g_l\ {\rm e}^{(E_L-E_l) / kT},$$ (15)

where the statistical weight g_l and energy E_l refer to sub-level l, and E_L is the mean energy of super-level L (see below). The statistical weight of super-level L is defined as the sum over the relative occupation probabilities g_il(T) of the atomic levels contained in B_L

$$% G_L(T)=\sum_{i,l~ \in B_L}g_{il}(T).$$ (16)

For the energy E_L of super-level L, we take the averaged energy over the relative occupations g_il(T)

$$\frac{\sum_{i,l~ \in B_L} g_{il}(T) E_l}{G_L(T)}$$                    E_L(T) =

$$\frac{\sum_{i,l~ \in B_L} E_l\ a_i\ g_l\ {\rm e}^{- E_l / k T}} {\sum_{i,l~ \in B_L} a_i\ g_l\ {\rm e}^{-E_l / k T}}\cdot$$ = (17)

With these definitions the occupation probabilities n_il of the sub-levels in energy band B_L can be calculated from the given occupation probability n_L of the corresponding super-level:

$$% n_{il} = n_L~\frac{g_{il}(T)}{G_L(T)}\cdot$$ (18)

An example for a typical model ion is given in Fig. 1. The boundaries between the energy bands are chosen to group atomic levels with similar electron configurations, and to lie in regions with a relatively low level density. Furthermore it is important to detach the low lying energy levels with even parity from regions where levels of both parities are mixed. In the example shown in Fig. 1, the lowest 4 super-levels (up to $160~000~{\rm cm}^{-1}$) only contain sub-levels of even parity. Therefore, super-levels 2-4 cannot decay radiatively, and behave metastable. Because of their relatively long lifetime, recombinations to these levels lead very likely to a re-ionization, which strongly reduces the effective recombination rate of the ion.

   
3.2 Opacities and emissivities

For a sub-line between sub-levels l and u the opacity $\kappa$ and emissivity $\eta$ are given by

$$% \kappa_{lu}(\nu) = n_l\sigma_{lu}(\nu)~\left( 1 - \frac{n_u}{n_l}\frac{g_l}{g_u}\right)$$ (19)

and

$$% \eta_{lu}(\nu) = n_u\frac{g_l}{g_u}\frac{2h\nu_{lu}^3}{c^2}\sigma_{lu}(\nu) .$$ (20)

The corresponding cross section $\sigma$ is given by

$$% \sigma_{lu}(\nu) = \frac{h\nu_{lu}}{4\pi} B_{lu} \phi(\nu-\nu_{lu}) = \frac{\pi~e^2}{m_{\rm e} c} f_{lu} \phi(\nu-\nu_{lu})$$ (21)

with the Einstein coefficient B_lu or the oscillator strength f_lu. For the profile function $\phi$ a Doppler profile with Gaussian shape and a broadening velocity $v_{\rm D}$ is assumed.

The opacities and emissivities for transitions between super-levels ("super-lines'') are obtained by adding up the opacities and emissivities of the involved sub-lines. For given super-level populations n_L and n_U the sub-level populations n_l and n_u are calculated from Eq. (18), and we obtain

 

$${\kappa_{LU}(\nu, T) = \sum_{i,l,u~\in B_L, B_U} \kappa_{lu}(\nu) =}$$

$${ \sum_{i,l,u~\in B_L, B_U} n_L \frac{g_{il}(T)}{G_L(T)} \sigma_{... ...{ \frac{g_{iu}(T)}{g_{il}(T)}}\frac{G_L(T)}{G_U(T)} {\frac{g_l}{g_u}} \right) }$$

$${= n_L {\sigma_{LU}(\nu, T)} \left(1-\frac{n_U}{n_L}\frac{G_L(T)}{G_U(T)} {{\rm e}^{ {h(\nu_{LU}-\nu)}/{k T}}}\right) }$$ (22)

 

$$\eta_{LU}(\nu, T) \sum_{i,l,u~\in B_L, B_U} \eta_{lu}(\nu)$$ =

$$\sum_{i,l,u~\in B_L, B_U} n_U \frac{g_{iu}(T)}{G_U(T)} \sigma_{lu}(\nu) \frac{2 h \nu^3}{c^2}\frac{g_l}{g_u}$$ =

$$n_U {\sigma_{LU}(\nu,T)}\frac{2 h \nu^3}{c^2}\frac{G_L(T)}{G_U(T)} {{\rm e}^{ {h(\nu_{LU}-\nu)}/{k T}}},$$ = (23)

with the composite cross section

$$% \sigma_{LU}(\nu,T) = \sum_{i,l,u~\in B_L, B_U} \frac{g_{il}(T)}{G_L(T)}~ \sigma_{lu}(\nu),$$ (24)

and the mean frequency

$$% \nu_{LU} = (E_U - E_L) / h.$$ (25)

These cross sections serve as a kind of profile functions for the super-lines. An example is displayed in Fig. 2.

Note that the energies, statistical weights and cross sections for super-levels and super-lines depend on the temperature T, which has not yet been specified. Due to the large number of spectral lines accounted for ($\approx$10⁷), a relatively large amount of computing time is needed for the calculation of the  $\sigma_{LU}$. Therefore, we calculate the  $\sigma_{LU}$, E_L and G_L in advance to the atmosphere calculations for fixed excitation temperatures. By this, we also avoid to handle the super-level energies, statistical weights and composite cross sections as being dependent on temperature and hence on radius. In our subsequent notation we will omit the explicit temperature dependences in these expressions.

The excitation temperatures are chosen as a typical ionization temperature for each ion. Starting from values obtained from the Saha equation for typical densities, they are iterated in the present work to match the local electron temperature $T_{\rm e}$ for the main ionization stages of the model presented in Sect. 4. The exact choice of the excitation temperatures turned out not to be critical.

With T set to a fixed excitation temperature, the direct application of Eqs. (22) and (23) is not possible anymore, because in the LTE-limit the line source function $S_{LU} = \eta_{LU}/\kappa_{LU}$ does not match the Planck function $B_\nu(T_{\rm e})$, which would lead to severe problems at large depth. We circumvent these problems by inserting the local electron temperature $T_{\rm e}$ into the exponential terms in these equations, i.e. the E_L, G_L and $\sigma_{LU}$ are evaluated for the excitation temperature T, but $\kappa_{LU}$ and $\eta_{LU}$ remain compatible to $T_{\rm e}$

$$% \kappa_{LU}(\nu) = n_L \sigma_{LU}(\nu) \left(1-\frac{n_U}... ...G_L}{G_U} {{\rm e}^{ {h(\nu_{LU}-\nu)}/{k T_{\rm e}}}}\right)$$ (26)

$$% \eta_{LU}(\nu) = n_U \sigma_{LU}(\nu)\frac{2 h \nu^3}{c^2}\frac{G_L}{G_U} {{\rm e}^{ {h(\nu_{LU}-\nu)}/{k T_{\rm e}}}}.$$ (27)

It is easily shown that for relative LTE populations of n_U and n_L the line source function S_LU now equals the Planck function $B_\nu(T_{\rm e})$, i.e. the LTE-limit is completely recovered.

Hence our super-level treatment implies the following approximations: (1) relative population numbers within each super-level according to Boltzmann's formula (LTE), (2) evaluation of these relative population numbers with an approximate temperature, (3) neglect of lines between sub-levels within the same super-level. Clearly, the higher the number of super-levels, i.e. the smaller their energy bandwidth, the less is the error. Note that the approximations affect the way how the atomic cross sections are combined, but do not introduce any inconsistencies.

Figure 2: Super-line cross section $\sigma _{LU}(\nu )$ between super-levels L=10 and U=15 of the generic ion presented in Fig. 1.

In analogy to the lines, the continuum cross sections of the iron group elements are also added up to a composed cross section for each super level. If available, data from the Opacity Project (Seaton et al. 1992; Cunto & Mendoza 1992) are used. Otherwise the continua are treated in hydrogenic approximation under the assumption of an effective principal quantum number

$$% n_{\rm eff} = Z~ \sqrt{{\frac{R_{\rm f}}{\nu_{\rm th}}}},$$ (28)

with the ion charge Z. In this approximation the cross section has $\nu^{-3}$ dependence and a threshold value of

$$% \sigma_{\rm th} = \frac{64~\pi^4 Z^4~m~e^{10}}{3\sqrt{3}~... ...1\times 10^{-10}}{Z \sqrt{\nu_{\rm th}/{\rm Hz}}}~{\rm cm}^2 .$$ (29)

   
3.3 Transition rates

The radiative rate coefficients are calculated consistently to the opacities and emissivities in Eqs. (26) and (27) by dividing the terms under the integral by $h\nu$ in oder to convert energies into photon numbers. For spontaneous emission processes we get from Eq. (27)

$$% R_{UL}^{\rm spon} = \frac{G_L}{G_U} \int \frac{8\pi\nu^2}... ...(\nu_{LU}-\nu)}/{k T_{\rm e}}}}~ \sigma_{LU}(\nu)~{\rm d}\nu.$$ (30)

From Eq. (26) we obtain the rate coefficients for absorption processes

$$% R_{LU}^{\rm rad} = \int \frac{4\pi}{h\nu} J_\nu~ {\sigma_{LU}(\nu)}~{\rm d}\nu,$$ (31)

and for the induced emission

$$% R_{UL}^{\rm rad} = \frac{G_L}{G_U} \int \frac{4\pi}{h\nu}... ...LU}-\nu)}/{k T_{\rm e}}}~J_\nu~{\sigma_{LU}(\nu)}~{\rm d}\nu .$$ (32)

In our implementation of the ALI method, the exact rate integrals (Eqs. (30)-(32)) are calculated only once per iteration cycle with the radiation field obtained from the formal solution (see Sect. 2.2). For the correction term (by the local feedback of the new population numbers, see Sect. 2.3) which enters the internal iteration cycle when solving the non-linear rate equations, we use a simpler formulation which is in full analogy to normal line transitions. This formulation saves computing time with respect to an exact treatment, but recovers the same solution at convergence.

For that purpose, we define effective Einstein coefficients A_UL, mean intensities ${{J}}_{LU}(\vec{n})$, and their derivatives ${\partial {{J}}_{LU}}/{\partial \vec{n}}$. The rate integrals (Eqs. (30), (31), and (32)) are simplified by substituting $\nu$ by the mean frequency $\nu_{LU}$ (Eq. (25)) in all slowly varying terms (i.e. except in $J_\nu$ and $\sigma _{LU}(\nu )$). Because the cross sections have a limited bandwidth, in most cases with a pronounced maximum around $\nu_{LU}$, this substitution does not affect the rate coefficients considerably. The approximate rate coefficients are

$$% R_{LU}=B_{LU}~{{J}}_{LU}(\vec{n})$$ (33)

and

$$% R_{UL}=A_{UL}+B_{UL}~{{J}}_{LU}(\vec{n}),$$ (34)

with the effective Einstein coefficients

$$% A_{UL} = \frac{G_L}{G_U} \frac{8\pi\nu^2_{LU}}{c^2} \int \sigma_{LU}(\nu)~{\rm d}\nu,$$ (35)

$$% B_{UL} = \frac{c^2}{2h\nu_{LU}^3} A_{UL},\; B_{LU} = B_{UL} \frac{G_U}{G_L},$$ (36)

the profile functions

$$% \phi_{LU}(\nu) = \frac{\sigma_{LU}(\nu)} {\int \sigma_{LU}(\nu)~{\rm d}\nu},$$ (37)

and the mean intensities

$$% {{J}}_{LU}(\vec{n}) = \int J_\nu(\vec{n})~\phi_{LU}(\nu)~{\rm d}\nu.$$ (38)

The mean intensities ${{J}}_{LU}(\vec{n})$ are again calculated on basis of the ${J}_{LU}^{\rm fs}$, which are derived in the previous radiation transport from the old populations $\vec{n}^{\rm old}$. In contrast to our treatment of ordinary line transitions, a linear extrapolation with respect to the difference vector $\vec{n} - \vec{n}^{\rm old}$ is performed. Utilizing the diagonal element $D_\nu$ from Eq. (10) we get the approximate radiation field

$$% J_\nu (\vec{n}) = J_\nu^{\rm fs} + D_\nu \frac{\partial S_\nu}{\partial \vec{n}} \cdot (\vec{n} - \vec{n}^{\rm old}),$$ (39)

and the mean intensities ${{J}}_{LU}(\vec{n})$ (Eq. (38))

 

$${{J}}_{LU}(\vec{n}) = {{J}}_{LU}^{\rm fs} + (\vec{n} - \vec{n}^{\... ...t \int D_\nu \frac{\partial S_\nu}{\partial \vec{n}} \phi_{LU}(\nu)~{\rm d}\nu.$$ (40)

The integral on the right hand side of Eq. (40), which corresponds directly to the derivative ${\partial {{J}}_{LU}}/{\partial \vec{n}}$, is prepared in the radiation transport together with the ${J}_{LU}^{\rm fs}$. In the present implementation only derivatives with respect to n_L and n_U are taken into account.

Collisional cross sections are calculated by application of the generalized formula of van Regemorter (1962) to the effective Einstein coefficients A_UL (Eq. (35)).