2 Theory for isolated lines

Using the Sahal-Bréchot code based on the Sahal-Bréchot theory (Sahal-Bréchot 1969, cf. also Baranger 1958a-c for the basic theory) and constanly updated (cf. for example Ben Nessib et al. 1996), which assumes: the impact approximation, the semi-classical approximation (the radiator is treated quantically and the perturbers follow classical paths, which are straight lines in the case of neutral radiating atoms) and the perturbation theory, we have calculated the full half width 2W and the shift d of the line emitted between the initial level i and the final level f which are given by the following formulae (Sahal-Bréchot 1969):

$$2W=N\int^{\infty}_{0}vf(v){\rm d}v\bigg (\sum_{i'\neq i}\sigma_{ii'}(v)+\sum_{f'\neq f}\sigma_{ff'}(v)+\sigma_{\rm el}\bigg )$$ (1)

$${d}=\int^{\infty}_{0}vf(v){\rm d}v\int^{R_{\rm d}}_{R_3}2\pi \rho {\rm d}\rho \sin 2\phi_{\rm p}$$ (2)

i' and f' are the perturbing levels, N and v are respectively the density and the velocity of perturbers, f(v) is the Maxwellian distribution of velocities.

The inelastic cross sections $\sigma_{ii'}(v)$ (respectively $\sigma_{ff'}(v)$)can be expressed by an integration over the impact parameter $\rho$ of the transition probability P_ii' as

$$\sum_{i'\neq i}\sigma_{ii'}(v)={1\over 2}\pi R_{1}^{2}+\int^... ... d}}_{R_1}2\pi \rho {\rm d}\rho \sum_{i'\neq i}P_{ii'}(\rho,v).$$ (3)

The elastic collision contribution to the width is given by:

$$\sigma_{\rm el}= 2\pi R_{2}^{2}+\int^{R_{\rm d}}_{R_2}8\pi \rho {\rm d}\rho \sin^{2} \delta$$ (4)

$$\delta = (\phi _{\rm p}^{2}+\phi _{\rm q}^{2})^{1\over 2}\cdot$$ (5)

The phase shifts $\rm \phi _{p}$ and $\rm \phi _{q}$ are due respectively to the polarisation and the quadrupole potential (Sahal-Bréchot 1969).

The cut-offs R₁, R₂, R₃, the Debye cut-off $R_{\rm d}$ and the symmetrization procedures are described by Sahal-Bréchot (1969) and further papers and will not be rediscussed therein.

As already discussed by Baranger (1958a-c, 1962) and Sahal-Bréchot (1969), the impact approximation is valid when the average effect of collisions is weak, in other words when strong collisions are separated in time, or equivalently, when the duration of a collision is very small compared to the separation time between strong collisions. The validity criterion for the impact approximation is given by:

$$C_{1}=\tau W_{\rm strong} \ll 1$$ (6)

where $\tau$ is a typical collision duration and $W_{\rm strong}$ is the strong collision contribution to the collisional half-width. It can also be written as (Ben Nessib et al. 1996):

$$C_{1}=N\pi \rho_{\rm typ} ^{3}\ll 1$$ (7)

where $\rho_{\rm typ}$ is a typical impact parameter for strong collisions.

The resulting profiles are Lorentzian. This condition is well verified by electronic collisions for a large range of densities.

For ionic collisions the impact approximation fails, especially for high densities. Then we can apply the quasistatic approximation. If the quadratic interaction potential is dominant, the quasistatic total width and shift are given by (Griem 1974):

$$W_{\rm iq}=1.75 10^{-4}N_{\rm e}^{1\over 4}A\big [1-0.068N_{\rm e}^{1\over 6}T^{-{1\over 2}}\big ]W_{\rm e}$$ (8)

$${d}_{\rm iq}=10^{-4}N_{\rm e}^{1\over 4}A\big [1-0.068N_{\rm e}^{1\over 6}T^{-{1\over 2}}\big ]W_{\rm e}$$ (9)

$W_{\rm e}$ is the impact contribution of electrons to the total width.

A is the quasistatic quadratic parameter, defined by Griem (1964), Traving (1968) and expressed by Ben Nessib et al. (1996) as follows:

$$A=\left({eF_{0}^{2}\over \hbar W_{\rm e}}\mid \alpha_{i}-\alpha_{f}\mid \right)^{3\over 4}$$ (10)

$F_{0}=2\pi (4/15)^{(2/3)}eN_{\rm e}^{2/3}$ is the normal field strength (Griem 1974). T is in Kelvin and $N_{\rm e}$ in cm^-3.

$\alpha_{i}$ is the polarizability of the initial level (resp. $\alpha_{\rm f}$ is the polarisability of the final level). It is expressed as:

$$\alpha_{i}=4a_{0}^{3}\sum_{i'\neq i}f_{ii'}\left({I_{\rm H}\over \Delta E_{ii'}}\right)^{2}$$ (11)

a₀ is the Bohr radius and $I_{\rm H}$ is the ionisation energy of hydrogen.