3 The collision rate methods

Quantum calculations if they apply, provide good results, but they do not constitute a rapid way to obtain the linewidths, especially for ion broadening. On the other hand, it is known (Sobel'man et al. 1981) that for highly excited states ($n,n^{\prime }\gg 1$ and $\mid n-n^{\prime }\mid \ll n)$, the electron impact widths can be expressed approximately in terms of the total transition rates of the level n and $n^{\prime }.$

3.1 Electron broadening

For highly excited states, the electron-produced width can be approximated by:

$$\gamma _{\rm e}\sim N_{\rm e}\left( \langle v \sigma \left( ... ...le +\langle v \sigma \left( n^{\prime }\right) \rangle \right)$$ (21)

where $\langle$v$\sigma \left( n\right) \rangle$ is the total transition rate for level n, and $\sigma \left( n\right)$ the total transition cross section for the same level, given by:

$$\sigma \left( n\right) =\sum_{k=1-n}^\infty {\sigma \left( n,n+k\right) }$$ (22)

n+k varies from 1 to $\infty$ so that k varies from 1-n to $\infty .$

In (21), the interference term is not taken into account. Its contribution, which is more important for $n_{\alpha}$ transitions is considered to be cancelled by the adiabatic contribution, since the major contribution to electron broadening comes from inelastic collisions. Thus, the summation in Eq. (22) must be taken over $k\neq 0$.

Gee et al. (1976) have proposed a semi-empirical formula for the calculation of e^-- atom excitation cross section $\left( \sigma \left( n,n+k\right) ,k\gt 1\right)$ - de-excitation cross section being obtained via the detailed balance equation - for electrons energies obeying:

$$\frac 4{n^2}\leq \frac \varepsilon {R_y}\ll \left( 137\right) ^2.$$ (23)

They have also proposed a formula for the calculation of transition rates $\alpha \left( n,n+k\right)$ for temperatures and energy levels satisfying:

$$\left( \frac{10^6}{n^2}\leq \frac TK\ll 3~10^9\right).$$ (24)

This last condition does not permit us to determine the total transition rates for all the the lines and temperatures we are interested in $\left( n\sim 5-20,~T\sim 10^3-10^5\right) .$ Thus, using the cross sections, we have defined a numerical method for the calculation of the total transition rates, based on a Gauss-Laguerre integration. The merit of this approach is that it can provide transition rates out of the validity domain of Gee et al.

In fact, putting $x=\frac \varepsilon {kT}$ where $\varepsilon =\frac 12mv ^2$ is the perturber's kinetic energy (here the electron), one easily obtains:

$$\begin{array} {c} \langle v \sigma \left( n\right) \rangle =... ... \\ =\left( \frac{8kT}{\pi m} \right) ^{1/2}G(T) \end{array}$$ (25)

where

$$G(T)=\int_{0}^\infty {x\sigma \left( n,\frac{kTx}{R_y}\right) \exp \left( -x\right) {\rm d}x.}$$ (26)

can be calculated by Gauss-Laguerre with good precision.

There are three collision rate methods. The first (Method A) consists of determining total transition rates by summation of transition rates provided by the semi-em-pirical formula of Gee et al. (1976). The second (Method B) is based on the semi-empirical formula of Sobel'man et al. The third (Method C) is the numerical method developed in this work, which uses accurate transition cross sections available in the literature.

Using these methods, we have calculated the total transition rates for a few hydrogen lines. The results obtained were compared in Motapon & Tran Minh (1995). Good agreement was found between methods A and C using the conditions of Gee et al.'s semi-empirical formula. A good agreement was also found between methods B and C, except for high n (about 100) at high energies ($T\gt\linebreak 25000$ K).

3.2 Ion broadening

In the atomic energy level range we are concerned with, elastic collisions dominate ion broadening. Thus, the collision rate method for ions requires cross sections for elastic scaterring. On the other hand, the approximation of complete degeneracy yields divergent cross sections for the process $nl\longrightarrow nl\pm 1$ (Toshima 1977a,b); Pengelly & Seaton (1964), so that this degeneracy must be removed by the fine structure and the Lamb shift, the hyperfine structure being neglected here. The total cross section $\sigma \left( n\right)$ is given by:

$$\sigma \left( n\right) =\frac 1{n^2}\sum_{l,l^{\prime }}\sigma \left( nl\longrightarrow nl^{\prime }\right).$$ (27)

As

$$\begin{eqnarray} \sigma \left( nl\longrightarrow nl^{\prime }\right) =\frac 1{2(... ...es \sigma \left( nlj\longrightarrow nl^{\prime }j^{\prime }\right)\end{eqnarray}$$ (28)

one has:

$$\begin{eqnarray} \sigma \left( n\right) =\frac 1{2n^2}\sum_{l=0}^{n-1} \sum_{l^{... ...es \sigma \left( nlj\longrightarrow nl^{\prime} j^{\prime }\right)\end{eqnarray}$$ (29)

where $\sigma \left( nlj\longrightarrow nl^{\prime }j^{\prime }\right)$ is the proton-hydrogen atom excitation cross section from state $\mid i\rangle \equiv \mid nlj\rangle$ to state $\mid f\rangle \equiv \mid n^{\prime }l^{\prime }j^{\prime }\rangle ,$ given, for large impact parameters by Seaton (1962), Pengelly & Seaton (1963)

$$\sigma \left( nlj\longrightarrow nl^{\prime }j^{\prime }\rig... ...ft\{ \ln \left( \frac{R_{\rm c}}{R_1}\right) +\frac 12\right\}$$ (30)

where $R_{\rm c}$ is the smallest value of $0.72\tau v$, $1.12\hbar v/\Delta E$and $R_{\rm D}$ the Debye radius. R₁ is given by

$$R_1^2=\frac{8Z^2e^2S}{3\hbar ^2v^2}$$ (31)

where $S=e^2\mid \langle nlj\mid {\bf r}\mid nl^{\prime }j^{\prime }\rangle \mid ^2,\langle nlj\mid {\bf r}\mid nl^{\prime }j^{\prime }\rangle$ being a dipole matrix element in the j-representation (Condon & Shortley 1959).

$\Delta E=\Delta E\left( nlj\longrightarrow nl^{\prime }j^{\prime }\right)$is the energy splitting between $\mid nlj\rangle$ and $\mid nl^{\prime }j^{\prime }\rangle .$

Transition rates can be carried out in a similar way as in the electronic case.

From a general point of view, estimations give $1.12 \frac{\hbar v}{\Delta E}$ to be less than $0.72\tau v$ for the atomic levels involved in the transitions we are interested in. On the other hand, in the ordinary conditions ($T\sim 5000$ to 40000 K) the Debye radius $R_{\rm D}$ ($=6.90 (T/N_{\rm e})^{1/2}$ in centimetres) is greater than $1.12 \frac{\hbar v}{\Delta E}$ unless the density exceeds 10⁹ cm^-3. Hence, the density range for application of the collision rate method to ions is reduced since the impact approximation for ions impose $N_{\rm e}<10^{11}$ cm^-3. Furthermore, there is a formal analogy between this description and the traditional semi-classical one. It finally comes out that one can merely apply the formulae (5) to (8) of Hoang Binh et al. (1987) to the calculation of $\Delta \omega _i$, especially because in this density range, the semi-classical description (given above) does not yet break down.