2 Description of the method

The broadening of spectral lines by a plasma is due to the interactions between the radiating element and the free ions and electrons of the plasma. These two contributions can be described in terms of interaction potentials with the corresponding electronic and ionic plasma microfields. These microfields are stochastic processes, whose static properties are described by distribution functions P(E). In the Model Microfield Method Frisch & Brissaud (1971), the dynamics of the microfield fluctuations are treated by a statistical process model where the microfield (electronic or ionic) is assumed to be constant during a given time interval. The microfield then jumps instantaneously to another constant value for the next time interval. The jumping times follow Poisson statistics, with a field dependent frequency $\nu(E)$. The calculation requires the knowledge of the electronic and ionic field distribution functions, for which Hooper's distribution functions (Hooper 1966; Hooper 1968) were used. The other input parameter required is the jumping frequency $\nu(E)$, which is chosen so as to reproduce the field autocorrelation function correctly Brissaud & Frisch (1971); Stehlé (1994b).

This model is the only universal model, currently available, which can reproduce correctly both the line widths and the intensity profile in the line wings. However, in the line centre, it might be interesting to compare the MMM linewidths with the results of computer simulations Gigosos & Cardenoso (1996); Cardenoso & Gigosos (1997), which do not include the Doppler broadening. Such simulations can reach accurate results for the line width. On the other hand, they suffer from two limitations, neither of which affect MMM modelling. First, perturbing charges (electrons and ions) are assumed to move along straight lines, which is incorrect at high densities. Secondly, the numerical procedure converges with difficulty at low densities and in the line wings, where the contributions of rare close interactions become important.

The present calculation, summarized below, follows the procedure already described by Stehlé (1994a, 1994b). Let the initial and final states of the optical transition be denoted by i,i'... and f, f'..., (with energies E_i, E_i', E_f, E_f'). The line profile, normalized to unit area, $I(\omega)$, is defined in the Liouville space spanned by the states |if>> by

$$I(\omega) ={1 \over \pi} \sum_{if,i'f'} \vec d_{if} \vec d_{i'f'} < U(\omega,\vec F_i,\vec F_{\rm e})>_{ei;\,if,i'f'}$$ (1)

where $<U(\omega)>_{ei}$ is the Fourier transform of the hydrogen evolution operator, averaged over the realizations of the stochastic dynamic electronic and ionic microfields $\vec F_i, \vec F_{\rm e}$, and $\vec d$ is the dipole operator of the hydrogen bound electron (normalized to $\sum_{if} d_{if}^{2}$).

We shall use the "no quenching approximation", which means that the interactions between the plasma and the radiating element mix together only those eigenstates belonging to the same principal quantum number. Neither do we include the fine structure effects. Thus $E_f -E_i=E_{f'}-E_{i'} = \hbar \omega_0$.

The typical fluctuation times of the ionic and electronic microfields, respectively given by the inverse of the electronic and ionic plasma frequencies $\omega_{{\rm pe}},\omega_{{\rm pi}}$, are very different. Thus, during an interval of constancy of the ionic microfield $\vec F_i$, the electronic microfield presents large time fluctuations. This justifies averaging the evolution operator over these electronic fluctuations for a given ionic microfield value. The procedure has been applied to the Model Microfield Method by Brissaud & Frisch (1971) and discussed by Frerichs (1989). It requires first calculating, for the static ionic microfield $\vec F_i$, the evolution operator $<U(\omega,\vec F_i)>_{\rm e}$, averaged over electrons, and then performing the dynamical MMM average over the ions.

In practice, we express $<U(\omega,\vec F_i)>_{\rm e}$ in terms of the frequency dependent relaxation operator $\gamma_{\rm e}(\omega, \vec F_i)$ (Greene 1982) defined by

$$<U(\omega,{\vec F_i})>_{\rm e} = i \, [\Delta \omega I - \vec d \vec F_i + i \gamma_{\rm e}(\omega, \vec F_i)] ^{-1}$$ (2)

where $\Delta \omega$ is the detuning from the line center $\omega_0$.

It is thus equivalent whether we calculate $<U(\omega,\vec F_i)>_{\rm e}$ or $\gamma_{\rm e}(\omega, \vec F_i)$. We shall suppose that $\gamma_{\rm e}(\omega, \vec F_i)$ is independent of the ionic microfield value, or $\gamma_{\rm e}(\omega, \vec F_i)= \gamma_{\rm e}(\omega)$. This approximation is justified because the linear ionic splitting is almost negligible compared to the electronic plasma frequency.

To obtain $\gamma_{\rm e}(\omega)$, it is thus sufficient to calculate $<U(\omega,\vec F_i)>_{\rm e}$ for F_i=0. This electronic average is performed by the Model Microfield Method, using the appropriate electronic frequency jump $\nu_{\rm e}(F_{\rm e})$ and the MMM equation for $<U>_{\rm e}$,

$$<U(\omega)>_{{\rm MMM,e}} \lbrace U(\omega + i \nu_{\rm e}\rbrace + \lbrace \nu_{\rm e} U( \omega + i \nu_{\rm e})) \rbrace$$ =

$$\times \lbrace \nu_{\rm e} I - \nu_{\rm e}^2 {U}(\omega + i \nu_{\rm e})\rbrace^{-1}$$

$$\times \lbrace\nu_{\rm e} U(\omega + i \nu_{\rm e})\rbrace$$ (3)

where $U(\omega + i \nu_{\rm e}(F_{\rm e}))$ is defined by

$$U(\omega+ i \nu_{\rm e}(F_e))=i \, (\Delta \omega I - \vec d. \vec F_{\rm e} +i \nu_{\rm e}(F_{\rm e}) I)^{-1}.$$ (4)

The notation $\lbrace..\rbrace$ denotes the static average over the electronic plasma microfield $P_{\rm e}(F_{\rm e})$ (i.e. $\int_0^\infty P_{\rm e}(F_{\rm e})...~{\rm d}F_{\rm e}$) and $\nu_{\rm e}(F_{\rm e})$ is defined by

$$\nu_{\rm e}(F_{{\rm e}})= {\omega_{{\rm pe}} \over x+1} \left... ...x)^{1/5} + x (x+1) +{3 \over 2} ({\pi \over 2})^{1/2}\right ]$$ (5)

with

$$x=x(F_{\rm e})= \int^{F_{\rm e}}_0 E^2 P_{\rm e}(E) {\rm d}E .$$ (6)

From $<U(\omega)>_{{\rm MMM,e}}$ the electronic damping rate $\gamma_{\rm e}(\omega)$ is extracted, using the relation 2, with F_i =0. The dynamic average over all the realizations of the ionic microfields is then performed in the frame of the Model Microfield Method. The method and equations are similar to the previous ones, if one uses the appropriate ionic frequency jump $\nu_i(F_i)$, (Eq. (17) of Stehlé (1994b), with correction of the erroneous factor (160 x)^1/5 which should be replaced by (40 x)^1/5), the ionic field distribution function P_i and the following expression of $U(\omega +{i} \nu_i(F_i))$ for the fixed value of the ionic field $\vec F_i$,

$$U(\omega+ i \nu_i(F_i)\,)=i \, [\Delta \omega I - \vec d. \vec F_i +i (\gamma_{\rm e}(\omega)+ \nu_i(F_i) I)]^{-1}.$$ (7)

One then obtains the plasma averaged evolution operator $<U(\omega)>_{{\rm MMM,ei}}$which includes the contributions of both the ions and the electrons. The final Stark profile $I(\omega)$ is obtained after weighting of the averaged plasma evolution operator $<U(\omega)>$ using Eq. (1).

In general $\gamma_{\rm e}(\omega)$ is not diagonal in if and i'f'. This leads in practice to the numerical inversion of a non diagonal matrix, which is difficult to perform when the final state has a large principal quantum number. In this case, in order to make the calculation more tractable, we perform the "isotropic'' calculation , in which $\gamma_{\rm e}(\omega)$ is approximated by its scalar mean defined as

$$\overline\gamma_{\rm e} (\omega) = \sum_{if,i'f'} \vec d_{if}... ...e};\,if,i'f'} (\omega) / \sum_{if} \vec d_{if} \vec d_{if} \ .$$ (8)

This approximation gives exactly the same line wings intensity as the "standard'' calculation. At low densities, it has been proved (Stehlé 1996a; Cassini 1997), that the centres of hydrogen lines are lorentzian, with an halfwidth equal to $\overline\gamma (\omega)$. Thus this isotropic approximation has a (small) effect only in the line centres and at large densities.

Even in this approximation, the calculation may be too time consuming and one uses the "isotropic, with no lower state interaction'' approximation, but in a different way from that used by Stehlé (1994a). In both the present calculation and in Stehlé (1994a), this approximation means essentially that the lower state broadening is negligible compared with that of the upper state. In Stehlé (1994a) expression (1) was evaluated for all the dipole matrix elements d_ifd_i'f' of the optical transitions. Our method, in the present paper, differs in that we impose a more severe approximation, by assuming that for large n', the line shape (without Doppler broadening) does not depend at all on the lower level n of the transition. This approximation also means that for the case where the Doppler broadening is negligible (i.e. at large densities), the Lyman n', Balmer n' and Paschen n' transitions have the same profile. Doppler broadening must be fully taken into account, as this obviously depends on the line. The error introduced by this approximation has been discussed by Vidal et al. (1971) and Lemke (1997) in the static ion approximation. We illustrated the effects of the approximation, in the context of the MMM, for the Balmer 6, 7 and 8 transitions in Fig. 1.

In practice, we have carried out the calculation of the Lyman 2, 3, 4, 5 and Balmer 3, 4, 5 with the "anisotropic'' approximation, and that for the Lyman 6 and higher, for Balmer 6 to 7 and for Paschen 4 to 8 with the "isotropic'' approximation. The higher Balmer and Paschen lines are approximated by "Lyman'' Stark profiles convolved (if necessary) by the appropriate Doppler broadening.

Figure 1: The line shapes of Ly6, Ly7, Ly8 (full lines) and H6, H7, H8 (dashed lines), without Doppler broadening, in the "isotropic'' approximation for $N_{\rm e}$ = 1014 cm-3 and T = 10000 K. Detunings are given in frequency units

For any given density, there is an effective series limit beyong which individual lines are no longer resolved. This limit falls with density and can be estimated qualitatively by the Inglis-Teller formula (Inglis & Teller 1939; Vidal 1966), as

$$\log_{10}(n_{{\rm IT}}) = 3.10 - 0.13 \log_{10}( 2 \, N_{\rm e} / {\rm cm}^{-3}).$$ (9)

The present calculations are carried out for the Lyman and Balmer lines up to the Inglis-Teller limit for 20 electronic densities $N_{\rm e}$ (given by $\log_{10} (N_{\rm e}({\rm cm}^{-3})) =$ 10, 10.5, 11,... 18, 18.5, 19, 19.5), and 10 temperatures T in K (given by $\log_{10} (T({\rm K}))$ = 3.4, 3.7, 4.0 ... 5.5, 5.8, 6.1).

Although the MMM can perform line shape calculation for correlated plasmas, the present tabulation is limited to density-temperature range where the mean interelectronic distance r₀ is smaller that the electronic Debye length $\lambda_{\rm d}$. This allows the use of Hooper's high frequency and low frequency field distribution functions Hooper (1966); Hooper (1968), which are necessary to calculate respectively the electronic and ionic contributions to the line. Thus tabulations are limited to the density-temperature range, where

$$a= {r_0 \over \lambda_{\rm d}}= 0.0898 \,\, { (N_{\rm e} / {\rm cm}^{-3} )^{1/6}\over (T /{\rm K} )^{1/2}} <1.$$ (10)