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Astron. Astrophys. 363, 1091-1105 (2000)

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4. Self-broadening theory

In the case of hydrogen, self-broadening refers to the broadening of hydrogen lines by collisions with other neutral hydrogen atoms. It has been known for some time, that similar atoms undergo a resonance interaction when the states of the two atoms are capable of optical combination (Eisenschitz & London 1930). It is also well known that two neutral atoms have dispersive and inductive interactions, often called the van der Waals interaction. The dispersive interaction corresponds to the simultaneous fluctuation of the atoms brought about by the repulsive electrostatic interaction of the electrons in each atom which promotes and demotes the electrons to virtual states. At long range the dispersive interaction dominates and a multipole expansion of the electrostatic interaction is valid whose first term leads to an interaction of the form [FORMULA]. There are also components of the interaction which correspond to the induction of virtual transitions in one atom due to the static field of the other, such interactions are usually less important.

We now outline a new theory of self-broadening in the impact approximation which includes both resonance and dispersive and inductive interactions, in a single theory.

4.1. Overlapping lines in the impact approximation

As already explained, in stellar spectra the hydrogen lines are split into components by the quasistatic ion field and these components are then impact broadened by electrons, hydrogen atoms, and to a negligible extent, helium atoms. The combined impact broadening of the Stark components by electrons and hydrogen can be handled in a consistent way by the use of overlapping line theory.

Baranger (1958) was first to examine the problem of pressure broadening of overlapping lines in the impact approximation. The review by Peach (1981) covers many aspects of line broadening theory including the case of overlapping lines. Adopting the notation of Peach the line shape for a transition between states of principle quantum number [FORMULA] and [FORMULA] is given by


in the reduced line/doubled-atom space, with [FORMULA] the operator corresponding to the electric dipole operator, and [FORMULA] the Hamiltonian. The operator [FORMULA] is the operator corresponding to [FORMULA] in line space, where N is the perturber density and [FORMULA] indicates averaging over all possible orientations of the collision. [FORMULA] is the collision or scattering matrix, and here the subscript refers to the two different upper and lower state subspaces.

The dipole operator determines the strength of each component's contribution to the complete line, including possible interference between components. In the reduced line space the matrix elements of [FORMULA] are related to the reduced state space matrix elements by


where via the Wigner-Eckhart theorem (e.g. Edmonds 1960) one can find


where [FORMULA] is now simply the radial component which can be computed by standard methods such as those discussed by Condon & Shortley (1935). We use the readily available computer code from Vidal et al. (1971).

4.2. Semi-classical treatment

The second matrix in Eq. (1) determines the line profile shape characteristics for each component. In the semi-classical theory, assuming straight line trajectories we can show that


where b is the impact parameter of the collision, and [FORMULA] the distribution of velocities v. This matrix is a complex square matrix of order [FORMULA], and once computed can be inverted easily by standard numerical techniques to give the matrix required in Eq. (1).

Determination of the [FORMULA] matrices can be simplified if it is assumed that there are no [FORMULA]-changing collision-induced transitions so that the [FORMULA] matrices are block diagonal. We compute these matrices via the method proposed by Roueff (1974) which accounts for changes in the orientation of the atoms during the collision relative to the single orientation in which the potentials are computed, via the [FORMULA] matrix. The relevant expressions for the evolution have been presented in Anstee & O'Mara (1991) and Barklem & O'Mara (1997).

In our treatment of the broadening of metallic lines it is assumed that there are no collision-induced transitions, an assumption which is justified by the collisions being too slow. In hydrogen, due to the accidental near [FORMULA] degeneracy, this assumption may break down. This is discussed in Sect. 4.5.

4.3. The interaction potential

We model the interaction between two hydrogen atoms as shown in Fig. 1. With reference to Fig. 1 the electrostatic interaction is


[FIGURE] Fig. 1. Model for electrostatic interaction between the two hydrogen atoms.

The resonance interaction occurs between like atoms due to a possible exchange of excitation. When one considers the interaction of a ground state hydrogen atom with an excited hydrogen atom the states [FORMULA] and [FORMULA] are degenerate. If the excited state is a p state it has an allowed dipole transition to the ground state and the off-diagonal elements are quite large compared to the diagonal matrix elements of V which at large separations can be neglected. It can be shown (see for example Margenau & Kestner 1969; Fontana 1961) that the strength of the resonance interaction [FORMULA] to first order is given by the matrix elements of V between these two states. Analytic expressions for these matrix elements, to within a final numerical integration over the radial co-ordinate have been calculated from V in Eq. (5) with the assistance of the Mathematica package without resort to a multipole expansion of V. These matrix elements must, at long range, reduce to the form


where [FORMULA] is a constant that must be computed. Table 1 shows values computed for [FORMULA] for lower lying states of hydrogen interacting with a ground state perturber. These were computed by direct evaluation of the dipole-dipole matrix element integrals using Mathematica. The [FORMULA] and [FORMULA] elements were computed entirely analytically. The higher states required some numerical evaluation. These values are in excellent agreement with Stephens & Dalgarno (1974) and Kolos (1967) for the 1s2p and 1s3p interactions and with our calculations based on an unexpanded V when the interatomic separation is large. Note that the interaction can be of either sign and declines dramatically with increasing principal quantum number which results in a multipole expansion of V being invalid for all but 2p states when employed in line broadening calculations.


Table 1. The computed values of [FORMULA] for resonance interactions due to ground state hydrogen perturbers.

For those perturbed atom states which are not connected to the ground state by an allowed dipole transition, the off diagonal matrix elements of V (corresponding to forbidden transitions) are zero or negligible. Thus the interaction potential to second order for these states is


The summation excludes the states [FORMULA] and [FORMULA]. At large separation this expression is dominated by the second term and behaves as


where [FORMULA] is always negative implying an attractive force unlike the situation for the resonance interaction where the force can be attractive or repulsive.

For p states, only the second order term above applies. Thus, as suggested by Margenau & Kestner (1969), the interaction energy to second order is the sum of the resonance interaction and the second order dispersive-inductive term such that


the summation excluding the degenerate states as above. For these states however, the first order term dominates due to the resonance interaction.

In order to simplify the infinite sum over all product states of the system in the above second order expressions we employ the first of two approximations suggested by Unsöld (1927) where, at a fixed separation R between the atoms, the energy denominator is replaced by a constant value [FORMULA]. The infinite sum in the above second order expression can then be completed using the closure relation reducing the expression to the simpler form


Furthermore we make the approximation that we may use the value of [FORMULA] at infinite separation, [FORMULA], at all separations R.

For the first few states of hydrogen we have inferred the value of [FORMULA] from previous calculations of the van der Waals coefficient [FORMULA] by Stephens & Dalgarno (1974), and these are tabulated in Table 2. For higher states the value of [FORMULA] is well approximated by [FORMULA] atomic units, a value obtained by neglecting the contribution to the energy denominator made by virtual states of the excited atom, the second approximation suggested by Unsöld (1955). In the case of the long range H-H interaction being considered here it is expected that the value of [FORMULA] will converge towards this value for higher lying states, and this is seen to be the case in Table 2, particularly for states with [FORMULA]-symmetry which make the largest contribution to the interaction and hence the line broadening.


Table 2. Implied [FORMULA] values for long range H-H interactions computed from the [FORMULA] calculations of Stephens & Dalgarno (1974).

In Paper I an [FORMULA] value of [FORMULA] was used for all states. When [FORMULA] values from Table 2 are used we do not find any significant change to the broadening and consequently the estimate [FORMULA] is used for all higher states.

4.4. Potential curves

The first and second order dispersive-inductive terms, in the context of the Unsöld approximation, were computed using methods described by Anstee & O'Mara (1991 , 1995) and Barklem & O'Mara (1997). For p states the total interaction can be obtained by simply adding the second order interactions to the resonance interaction.

We have made comparisons of a number of our spin-averaged type potential curves with appropriate molecular-type spin dependent curves from the literature. Our potential curves fulfilled our expectations from previous experience. That is, they show excellent agreement at long range. They then have reasonable agreement with molecular curves at "intermediate" separations (we define these separations as where the potential starts to deviate from the long range asymptotic expansion behaviour), far better agreement than the multipole expansion results. At shorter range we see generally poor agreement as the use of perturbation theory breaks down at these separations. For 1s[FORMULA] interactions good agreement was observed for R greater than 10-15 [FORMULA]. It will be shown however that our curves are of acceptable accuracy in the region that is predicted by the model to be important in broadening.

Following Anstee & O'Mara (1991) the interatomic separations important in the determination of the line broadening cross-sections have been identified by multiplying potentials by a Gaussian "lump" of unit peak amplitude and width of 2 Bohr radii and adding them to original potentials thus amplifying them by up to a factor of two. The line broadening cross-section can then be calculated as a function of the lump position and plotted against the lump position. A corresponding lump appears in this plot which clearly identifies the interatomic separations important in the line broadening.

Using this procedure it was found that the broadening of the 3d state is most sensitive to potentials at intermediate separations, ie. those where the potential curve starts to deviate from the long range behaviour, around 10-30 [FORMULA] in this case. This is the same as has been observed in the broadening of metallic lines by hydrogen collisions (Anstee & O'Mara 1991; Barklem & O'Mara 1998). Due to the strong resonance interaction with a [FORMULA] dependence at long range for 2p states it was found that the broadening is much more sensitive to the long range interaction. Even at a lump position of 60 [FORMULA] the cross-section still showed some sensitivity, not yet having converged to the value of 1180 atomic units. It was also observed that the model appears to be more sensitive to the 1s2p[FORMULA] curve than the 1s2p[FORMULA] at intermediate separations, however, we see that at larger separations the broadening is more sensitive to the 1s2p[FORMULA] curve. This behaviour was somewhat unexpected and may be result of the dispersive-inductive interaction being attractive for each of these states while the resonance interaction is attractive for [FORMULA] states and repulsive for [FORMULA] states. In conclusion, it has been shown that the model used here is insensitive to the accuracy of the potentials at small separations where the curves used in this work are known to be inaccurate.

The dependence of the cross-section for the p-d component of H[FORMULA] with [FORMULA] for the 2p and 4d states was also investigated. This transition was chosen as the [FORMULA] values for the upper state are the most uncertain. It was seen that the broadening is practically independent of the 2p state [FORMULA] value, since this potential is dominated by the first order resonance component which is not dependent on [FORMULA]. The dependence on the [FORMULA] of the upper state is actually reasonably strong when considering say [FORMULA] varying over the range -0.8 to -0.4, however, it can be safely assumed that the [FORMULA] values for the 1s4d interactions lie between -0.457 and -0.444. Within this range the cross-section was found to deviate by only around 1 per cent.

Our calculations do not include exchange effects. Our investigations of metallic lines suggest that exchange effects start to become important when [FORMULA] where [FORMULA] is the effective principal quantum number which equals n in hydrogen. As shown by Lortet & Roueff (1969), the p-d transition dominates the Balmer lines and exchange effects thus should not be important in the broadening for H[FORMULA], H[FORMULA] and H[FORMULA].

4.5. Validity of approximations

The validity of approximations used in the calculations, is now considered. All of the assumptions or approximations used in previous work for metallic lines are retained (Anstee & O'Mara 1991). We have mentioned the various approximations in the previous discussion of the theory. The approximations made are the impact approximation (including the binary collision assumption), use of Rayleigh-Schrödinger perturbation theory, the classical straight path approximation and the neglect of collision-induced transitions. In hydrogen lines the impact approximation and neglect of collision-induced transitions may breakdown. The impact approximation may be suspect in the far line wings due to the fact that the lines are often so broad. The neglect of collision-induced transitions becomes doubtful as the levels are [FORMULA] degenerate. Hence we will discuss these two approximations.

The validity of the impact approximation for self-broadening of hydrogen lines was discussed by Lortet & Roueff (1969). In view of our new calculations we can revisit this analysis, now without the need to split the conditions into resonance and van der Waals parts. The impact approximation is strictly valid when the detuning (in angular frequency units) is far less than the inverse collision duration. If one considers the detuning for which these quantities are equal, the absolute maximum detuning for which the impact approximation is valid can be estimated by


where as usual [FORMULA] is the broadening cross-section for collision velocity v. Using the cross-section data which we discuss in the next section (see Table 3) we have computed [FORMULA] for the lower Balmer lines and these are shown in Table 3 for a collision speed of 14000 m s-1. Such a collision speed corresponds approximately to 5000 K temperature. For the sun, hydrogen line wings are formed in regions of the atmosphere which are typically hotter than this, and thus [FORMULA] is greater as [FORMULA] and [FORMULA] with [FORMULA], where [FORMULA] is defined in the caption to Table 3. The impact approximation is only strictly valid when the detuning is far less, say around five times less, than the inverse collision duration. Using this as the criterion the impact approximation is only secure at detunings of less than about 7.0 Å for H[FORMULA] and about 2.6 Å for H[FORMULA] in these conditions.


Table 3. The broadening characteristics of the p-d component of lower Balmer lines. The cross-section [FORMULA] is given in atomic units for a collision speed of [FORMULA] m s-1. The velocity parameter [FORMULA] gives the velocity dependence assuming [FORMULA]. The approximate maximum detuning for validity of the impact approximation for the self-broadening, computed for [FORMULA] m s-1, is given.

Examining the extent of the solar profiles one sees that the approximation is valid for most of the H[FORMULA] profile but is not valid in the outer wings of the H[FORMULA] and H[FORMULA] profiles. Outside of the limits set in Table 3 one could use methods which are reviewed by Allard & Kielkopf (1982). However, outside the impact regime collisions at very short range become important where the method we use to calculate the interatomic interaction is no longer valid.

In our calculations it is assumed that the collisions do not cause transitions between nearly degenerate states of the same n but different [FORMULA]. Such transitions are only likely when the duration of the collision is comparable with the Bohr period for transitions between these nearly degenerate states whose splitting is brought about by the quasistatic ion field. This condition is often termed the Massey criterion. If the collision duration is either much greater (the adiabatic approximation) or much smaller (the sudden approximation) than the Bohr period the probability of a transition occurring is very low. In the present context the collision duration is given by [FORMULA], where [FORMULA] can be estimated from the cross-section data in Table 3 and at unit optical depth in the sun a typical collision speed is about 14000 m s-1. The appropriate Bohr periods for H[FORMULA], H[FORMULA] and H[FORMULA] can be estimated from linear Stark shift parameters for hydrogen (for example see Condon & Shortley 1935) and an estimate of the quasistatic ion field at unit optical depth in the sun. A comparison of the collision durations and Bohr periods show that the Bohr period is about 400 times greater for H[FORMULA], 100 times greater for H[FORMULA], and 50 times greater for H[FORMULA], than the collision duration at unit optical depth in the solar photosphere. These results indicate that above unit optical depth in the sun, the sudden approximation is valid and that for these lines [FORMULA]-changing collisions can be neglected. Due to the increasing Stark effect [FORMULA]-changing collisions may become important for the higher Balmer lines. Due to the lower quasistatic ion field in metal deficient stars [FORMULA]-changing collisions will be even more unlikely.

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Online publication: December 5, 2000