4 Comparison of both approaches and discussion

Before comparing the widths provided by the semi-clas-sical method and the collision rate method (respectively hereafter referred to as SCM and CRM), we must first determine the limits in density of the impact approximation, that is the densities above which the impact approximation does not apply. It is well known (Sobel'man et al. 1981) that the impact approximation is valid for the level n if the dimensionless quantity $h_{\rm e}=N_{\rm e}\left( n(n-1)\frac \hbar {m_{\rm e}}\right) ^3$ is very small compared to unity. This gives the "limiting'' values mentioned in Table 1 below.

  

Table 1: Limiting densities of the impact approximation for electron broadening

Further, the density must be taken below the Inglis-Teller limit given in Griem (1974) by:

$$\begin{eqnarray} N_{{\rm IT}}\approx \left( Z^{9/2}/120a_0^3Z_{\rm p}^{3/2}\righ... ...^2\right) ^{3/2}\nonumber \\ \times \left( m^2-n^2\right) ^{-3/2} \end{eqnarray}$$ (32)

where $Z_{\rm p}e$ is the perturber's charge, Ze the radiator's charge: m is the upper atomic level for the transition and n the lower level.

The cut-offs on the impact parameters are:

$$\rho _{\max }\approx \min \left( \rho _{\rm D},\lambda /2\pi \right)$$ (33)

and, for $\triangle \omega$ greater than the ion plasma frequency $\omega _{{\rm p_i}}$ the Lewis cut-off at $\rho _{\max }$ $\left( \approx v/\mid \Delta \omega \mid \right)$ is more appropriate.

$$\rho _{\min }\approx \left( \frac 23\right) ^{1/2}\frac \hbar {mv_{\rm e}}\left[ G\left( n,\Delta n\right) \right] ^{1/2}$$ (34)

where $G\left( n,\Delta n\right)$ is given by Eq. (9).

The other characteristic radii are:

i)

$\rho _{{\rm eff}},$ the radius above which the collision is complete, i.e. $\rho _{{\rm eff}}^2=\sigma \left( n\right) /\pi .$ $\rho _{{\rm eff}}$ could be a possible lower cut-off on the impact parameters; but n scaling from 5 to 20 is of the order of $\rho _{\min }$ - as chosen in (34) - which is about two to five times the atomic mean radius $\left( n^2a_0\right)$;

ii)

the mean radius r₀ of the sphere occupied by one perturber, which is given by $\frac 43\pi r_0^3N_{\rm e}=1.$ r₀ is large, as compared to the lower cut-off $\rho _{\min }$ at low densities, but gets closer to $\rho _{\min }$when the density increases. In some critical cases, the perturber is too close to the Rydberg atom or even inside it. In such cases, the dipole approximation as well as the impact approximation break down, and the SCM and CMR are problematical. At higher densities (above the impact approximation limit and the Inglis-Teller limit), there may be many perturbers in the Rydberg atom; line dissolution is important; the problem becomes more complicated and deserves particular attention.

Following the two approaches described in the previous sections, we have calculated the electron impact linewidths of a few $n_{\gamma}$ lines at densities varying from 10⁹ to 10¹⁷ cm^-3 and T=10⁴ K. The results are given in Table 2 below.

  

Table 2: Comparison of the linewidths of $n_{\gamma}$ transitions. CRM is referred to the collision method and SCM to the semi-classical

Table 2 shows discrepancies between CRM and SCM at nearly all densities, especially at low densities which correspond to large values of the Debye radius. Analogous results are found for $n_{\alpha}$ and $n_{\beta}$transitions. Nevertheless, for $n_{\gamma}$ transitions, the interference term in the electron collision operator $\Phi _{mn}$ is known to be less significant, so that the CRM could provide satisfactory results for the linewidths. Hence, the SCM with the chosen lower cut-off overestimates the linewidths for $n\sim 5-20,$ especially at low densities. Another cut-off must be determined such as to obtain agreement with the CRM. This occurs when $\rho _{\min }$ is designated to be larger. This would be in agreement with the validity of the SCM. Finally, the CRM described in this work remains an alternative tool for determining linewidths at low electron densities.

Electron and ion impact widths are compared quantitatively in Table 3 for $N_{\rm e}=10^{10}$ and $N_{\rm e}=10^{12}$ cm^-3.

  

Table 3: Electron and ion impact linewidths at $N_{\rm e}=10^{10}$ and $N_{{\rm e}}=10^{11 }$cm^-3 for T=10000 K

$$\begin{eqnarray} \Delta \omega _i &=&\frac 43\pi \left( \frac \hbar {m_{\rm e}}\... ...\rho _{\max }/\rho _{\min }+0.5\right] F\left( n,\Delta n\right). \end{eqnarray}$$ (35)

For protons,

$$\rho _{\min }=\left( \frac 23\right) ^{1/2}\frac \hbar {m_{\... ...gle \frac 1{v}\right\rangle [F\left( n,\Delta n\right) ]^{1/2}$$ (36)

$$\begin{eqnarray} v=\left[ 2kT\pi /\mu \right] ^{1/2} & = & \left[ 2kT\pi /\left(... ...) \right] ^{1/2}\nonumber \\ & = & 2\left[ kT\pi /M\right] ^{1/2} \end{eqnarray}$$ (37)

$$\left\langle \frac 1{v}\right\rangle =\left( 2\mu /\pi kT\right) ^{1/2}=(M/\pi kT)^{1/2}$$ (38)

$$\begin{eqnarray} \Delta \omega _i = \frac 43\pi \left( \frac \hbar {m_{\rm e}}\r... ...ft[ \ln C+0.5\right]\nonumber \\ \times F\left( n,\Delta n\right)\end{eqnarray}$$ (39)

$$\begin{eqnarray} =\frac 43 \sqrt{\pi }\left( \frac \hbar {m_{\rm e}}\right) ^2\left( \frac Mk\right) ^{1/2}10^6 \equiv 3.487~10^{-4}\end{eqnarray}$$ (40)

$$\begin{eqnarray} \Delta \omega _i=3.487~10^{-4}T^{-1/2}N_i\left[ \ln C+0.5\right]\nonumber \\ \times F\left( n,\Delta n\right) \end{eqnarray}$$ (41)

in s^-1, N_i in cm^-3 and T in K.

The Table 3 shows a domination of ion contribution to impact broadening of $n_\alpha ,$ $n_{\beta}$ and $n_{\gamma}$ lines for all the atomic levels concerned with this work. It is observed that electron broadening is expected to take over for n>20, as claimed by Griem (1967).

For electron densities above the impact approximation limit - for ions - and below the Inglis-Teller limit, the quasistatic approximation can be applied to ions, but the impact approximation still applies to electrons for n varying between 5 and 20.

In this case, theoreticians of lines broadening commonly define the profile of the line emitted during the transition from the level m to the level n, and broadened by electrons and ions, by:

$$\begin{eqnarray} I\left( \Delta \omega \right) =\frac 1\pi Re\int_{0}^{\infty } ... ...\alpha ^{\prime }\right\rangle \mid \beta ^{\prime }\right\rangle \end{eqnarray}$$ (42)

where:

$$\Theta =i\left( \omega -\omega _0-\frac 1\hbar \left( H^m\left( F\right) -H^n\left( F\right) \right) \right) +\Phi _{mn}$$ (43)

W(F) is the ion microfield distribution and the electron $\Phi _{mn}$ the electron impact operator; $\mid \alpha \rangle ,\mid \beta \rangle ,\mid \alpha ^{\prime }\rangle$ and $\mid \beta ^{\prime }\rangle$ are parabolic states. In the description given by formula (42), one considers that the ions create a "static'' microfield which induces a lifting of the degeneracy of the atomic energy levels - linear Stark effect - and the interaction with electrons, mainly by collisions provokes a broadening of so-formed Stark components.

Since $\Phi _{mn}$ is not diagonal in the parabolic representation, the difficulty in this formulation lies in the inversion of large matrices. For memory, the size of matrices to be inverted is $\left( m\times n\right) ^2\times \left( m\times n\right) ^2$. Nevertheless, we can simplify the problem by building from $\Phi _{mn}$ and operator $\stackrel{}{\stackrel{\sim }{\Phi }_{mn}}$ which would be diagonal in the parabolic representation (see Seaton 1990). A diagonal operator can also be obtained by simply neglecting the off-diagonal elements of $\Phi _{mn}.$According to Griem (1960) and Sobel'man (1963, 1972), an approximate profile can be obtained by replacing $\Phi _{mn}$ by half-width $\gamma /2$ of the line i.e.:

$$\begin{eqnarray} I\left( \Delta \omega \right) =\int_0^\infty {W\left( F\right) ... ...elta \omega -C_{\alpha \beta }F\right) ^2+\frac{\gamma ^2}4}\cdot \end{eqnarray}$$ (44)

An approximate method for the evaluation of the lineshape of formula (44) consists of using the Griem's table (Griem 1960). The agreement between this method and formula (44)-based calculations has been found to be particularly good for transitions $n+\Delta n\longrightarrow n$ with $\Delta n\gt 1,$ but not for $n_{\alpha}$ transitions. A regularly spaced extended table from the original Griem's table, permits, via a double interpolation program, a rapid determination of approximate lineshapes of $n_\beta ,n_\gamma ,$ ... lines.