2 The semi-classical method

The traditional semi-classical treatment of line broadening depends on two main hypothesis:

(i) the classical path approximation for the perturber, i.e. the perturber is supposed to follow a well known path, which is a straight line for the broadening of neutral hydrogen;

(ii) the impact approximation, the validity conditions of which have been largely treated in the literature (Griem 1974; Sobel'man 1981).

Furthermore, we make the dipole approximation for the interaction potential.

During the transition from level m to level n, the electron collision operator is given (Sobel'man 1981; Lisitsa 1977; Griem 1974) by:

 

$$\Phi _{mn}=N_{\rm e}\int_0^\infty v f\left( v \right) \int_0^\infty 2\pi \rho {\rm d}\rho \left\{ 1-S_m^{\dagger }S_n\right\}$$ (1)

where $\left\{ ...\right\}$ represents the mean over the angles of the perturber, and f(v) is Maxwell's distribution. S_m (resp. S_n) is the collision matrix for level m (resp. n) and $S_m^{\dagger }$ is the hermitian conjugate of S_m. S_m reads in terms of $\tilde V_m\left( t\right) =\exp \left( -iH_0t/\hbar \right) V\exp \left( iH_0t/\hbar \right)$, the potential in the interaction representation:

$$S_m=\exp \left[ -\frac i\hbar \int_{-\infty }^\infty \tilde V_m\left( t\right) {\rm d}t\right]$$ (2)

V being the perturber-radiator interaction potential.

After the average over the angles has been carried out as well as the two integrations in (1), the collision operator is approximated by (Lisitsa 1977):

$$\begin{eqnarray} \Phi _{mn} & = & \frac{16}{3}N_{\rm e}\frac{e_{}^4}{\hbar ^2}\l... ...mes \left[ 0.215+\ln \frac{\rho _{\max }} {\rho _{\min }}\right] \end{eqnarray}$$ (3)

where ${\bf r}_m$ and ${\bf r}_n$ are the position operators in levels m and n, $\rho _{\max }$ and $\rho _{\min }$ being the maximum and minimum impact parameters. The linewidths are given by the real and the imaginary parts of the matrix elements of $\Phi _{mn}$. In the dipole approximation, the shifts vanish since the collision operator is real. In Eq. (3), 0.215 is the contribution of strong collision while $\ln \frac{\rho _{\max }}{\rho _{\min }}$, which is that of weak collisions, is supposed - by hypothesis - to be the major contribution to line broadening. ${\bf r}_m.{\bf r}_m$ and ${\bf r}_n.{\bf r}_n$ are the broadening terms for both upper and lower levels, and $-2{\bf r}_m.{\bf r}_n$ is the interference term.

2.1 Ion impact widths

For ions, the quantity v$_i/\rho _{\min }^i$ is of the order of the angular frequency separating two close levels (Griem 1967). Hence, only collisions with impact parameters $\rho$ less than $\rho _{\min }^i$ can lead to states of different principal quantum numbers, i.e. non-adiabatic transitions, and those with $\rho$ greater than $\rho _{\min }^i$ are adiabatic. This explains why, in ion broadening, within the impact approximation, the inelastic contribution which is not significant in the dominant term - that of weak collisions - is neglected.

Let us consider the transition $m=n+\Delta n\rightarrow n$. According to Griem the contribution of weak collisions to the ion impact width is:

$$\Delta \omega _i\approx \frac 43\pi \left\langle \frac 1{v_i... ...\frac{\rho _{\max }}{\rho _{\min }} G\left( n,\Delta n\right)$$ (4)

where $m_{\rm e}$ is the electron mass. Owing to the major contribution of elastic collisions to broadening, $G(n,\Delta n)$ corresponds to the F-function of Hoang Binh et al.:

$$\begin{eqnarray} F(n,\Delta n) & = & \sum_{l,ml^{\prime },m^{\prime }}{\mid \lan... ...le \mid ^2}\nonumber \\ & & \hspace{2cm}\times f(n,l,\Delta n) \end{eqnarray}$$ (5)

where

$$\sum_{l,ml^{\prime },m^{\prime }} {\mid \langle n+\Delta n, l+1, m\mid Z\mid n,l,m\rangle \mid ^2}$$ (6)

is normalized to unity, with

$$\begin{eqnarray} \lefteqn{f(n,l,\Delta n) = \frac 94n^2(n^2-l^2-l-1)} \nonumber ... ...l+2}R_{n+\Delta n,l+2}^{nl+1}\right\} / R_{n+\Delta n,l+1}^{nl} } \end{eqnarray}$$ (7)

where the $R_{nl}^{n^{\prime }l^{\prime }\prime }$s are the the dipole radial integrals.

2.2 Electron impact widths

In the electron case, the velocities are greater. The inelastic contribution to the linewidth becomes important, and dominates the elastic one in most cases. The contribution of weak collisions to the electronic width is:

$$\Delta \omega _i\approx \frac 43\pi \frac 1{v_{\rm e}}\left(... ...n \frac{\rho _{\max }}{\rho _{\min }}G\left( n,\Delta n\right)$$ (8)

where $G\left( n,\Delta n\right)$ is a function containing the elastic and inelastic contributions to broadening. To take these two contributions into account, let us introduce the intermediate states in the summation over the matrix elements of the operators ${\bf r}$.${\bf r}$. $G\left( n,\Delta n\right)$ is:

$$\begin{eqnarray} G\left( n,\Delta n\right) & = & \sum_{l,ml^{\prime },m^{\prime ... ...gle \mid ^2} \nonumber \\ & & \hspace{2cm}\times g(n,l,\Delta n) \end{eqnarray}$$ (9)

where

$$g(n,l,\Delta n)=g_{{\rm elast}}(n,l,\Delta n)+g_{{\rm inel}}(n,l,\Delta n)$$ (10)

and

$$g_{{\rm elast}}(n,l,\Delta n)=f(n,l,\Delta n).$$ (11)

One then has:

$$g(n,\Delta n)=F(n,\Delta n)+H(n,\Delta n)$$ (12)

where $F(n,\Delta n)$ accounts for the elastic part, given by Eq. (5), and $H(n,\Delta n)$ for the inelastic part:

$$\begin{eqnarray} H\left( n,\Delta n\right) & = & \sum_{l,m}{\mid \langle n+\Delt... ...gle \mid ^2} \nonumber \\ & & \hspace{2cm}\times h(n,l,\Delta n) \end{eqnarray}$$ (13)

with

$$h\left( n,l,\Delta n\right) =\sum_{l^{\prime },m^{\prime }} {g_{{\rm inel}}(n,l,\Delta n)}$$ (14)

$$\begin{eqnarray} =\sum_{l^{\prime },m^{\prime }}\sum_{n^{\prime }\neq n} \sum_{n... ...\prime \prime },l^{\prime }+1,m^{\prime }\rangle ^2\right\}\cdot} \end{eqnarray}$$ (15)

On the other hand,

$$\begin{eqnarray} & & \sum_{l^{\prime },m^{\prime }} {\langle nlm\mid r\mid n^{\p... ...ight) ^2+\frac{l+1}{2l+1}\left( R_{nl}^{n^{\prime }l+1}\right) ^2 \end{eqnarray}$$ (16)

and

$$\begin{eqnarray} & & \sum_{l^{\prime },m^{\prime }}{\langle n+\Delta n,l+1,m\mid... ...ac{l+2}{2l+3}\left( R_{n+\Delta n,l+1}^{n^{\prime }l+2}\right) ^2 \end{eqnarray}$$ (17)

so

$$\begin{eqnarray} h\left( n,l,\Delta n\right) =\sum_{n^{\prime }\neq n} {\left\{ ... ...3}\left( R_{n+\Delta n,l+1}^{n^{\prime }l+2}\right) ^2\right\}. } \end{eqnarray}$$ (18)

Using sum rules for dipole radial integrals (Bethe & Salpeter 1957) one finds:

$$\begin{eqnarray} & & h\left( n,l,\Delta n\right) =\frac{n^2}4\left\{ n^2+3l^2+3l... ...ac{(n+\Delta n)^2}4\left\{ (n+\Delta n)^2+3l^2+9l+17\right\}\cdot \end{eqnarray}$$ (19)

The details for the calculation of $F(n,\Delta n)$ and $H(n,\Delta n)$ are given in the Appendix. This calculation, for a few transitions, has led to three main observations:

i)

For n greater than 5, the inelastic contribution is dominant for the first transitions ($n_\alpha ,n_\beta ,$ and $n_\gamma).$

ii)

For large values of n, $H(n,\Delta n)$ is practically independent of n.

iii)

As $\Delta n$ increases at low values of n, $F(n,\Delta n)$ becomes greater than $H(n,\Delta n),$ and elastic collisions become a significant contribution.

Finally, it turns out that inelastic electron-atom collisions constitute the dominant factor of electron broadening for the lines we are interested in. Therefore, the inelastic width may be a satisfactory approximation to the total linewidth in most cases.

2.3 Limitation of the semi-classical method - Need for other approaches for linewidths determination

In the impact approximation, it is supposed that the mean effect of the collision is weak. Thus, the strong collision contribution (Eq. 3) is small compared to that of weak collisions $\left( \ln \frac{\rho _{\max }}{\rho _{\min }}\right)$. For Rydberg lines, the linewidth from Eq. (7) may be "uncertain'' at some values of the electron densities. In fact $\rho _{\max }$ is generally taken to be the Debye radius $\left( \sqrt{\frac{kT}{4\pi N_{\rm e}e^2}}\right)$or the Lewis cut-off while $\rho _{\min }$ can be chosen similarly to the lower cut-off of Hoang-Binh et al., i.e.:

$$\rho _{\min }=\left( \frac 23\right) ^{1/2}\frac \hbar {mv_{\rm e}}\left[ G\left( n,\Delta n\right) \right] _{}^{1/2}\cdot$$ (20)

However, at typical densities, for the lines we are interested in, $\rho _{\min }$ and $\rho _{\max }$ are sometimes too close, providing a weak contribution comparable to that of strong collisions. Such a situation is non sensical, owing to the basic hypothesis of the semi-classical method. A higher order in Born series is needed or, at least, close coupling calculations would be required. In this case, particular attention must be paid to the description of the strong collision's contribution; a well adapted cut-off must be chosen. Another or other methods are needed for comparison.