## 4. Self-broadening theoryIn 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 . 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 approximationAs 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 and is given by in the reduced line/doubled-atom space, with
the operator corresponding to the
electric dipole operator, and the
Hamiltonian. The operator is the
operator corresponding to in line
space, where 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 are related to the reduced state space matrix elements by where via the Wigner-Eckhart theorem (e.g. Edmonds 1960) one can find where 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 treatmentThe 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 Determination of the matrices can be simplified if it is assumed that there are no -changing collision-induced transitions so that the 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 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 degeneracy, this assumption may break down. This is discussed in Sect. 4.5. ## 4.3. The interaction potentialWe model the interaction between two hydrogen atoms as shown in Fig. 1. With reference to Fig. 1 the electrostatic interaction is
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
and
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 where is a constant that must be
computed. Table 1 shows values computed for
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 and
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
For those perturbed atom states which are not connected to the
ground state by an allowed dipole transition, the off diagonal matrix
elements of The summation excludes the states and . At large separation this expression is dominated by the second term and behaves as where 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 Furthermore we make the approximation that we may use the value of
at infinite separation,
, at all separations For the first few states of hydrogen we have inferred the value of from previous calculations of the van der Waals coefficient by Stephens & Dalgarno (1974), and these are tabulated in Table 2. For higher states the value of is well approximated by 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 will converge towards this value for higher lying states, and this is seen to be the case in Table 2, particularly for states with -symmetry which make the largest contribution to the interaction and hence the line broadening.
In Paper I an value of was used for all states. When values from Table 2 are used we do not find any significant change to the broadening and consequently the estimate is used for all higher states. ## 4.4. Potential curvesThe 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 interactions good agreement was
observed for 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 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 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 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 curve than the 1s2p at intermediate separations, however, we see that at larger separations the broadening is more sensitive to the 1s2p 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 states and repulsive for 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 with for the 2p and 4d states was also investigated. This transition was chosen as the values for the upper state are the most uncertain. It was seen that the broadening is practically independent of the 2p state value, since this potential is dominated by the first order resonance component which is not dependent on . The dependence on the of the upper state is actually reasonably strong when considering say varying over the range -0.8 to -0.4, however, it can be safely assumed that the 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 where
is the effective principal quantum
number which equals ## 4.5. Validity of approximationsThe 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 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 is the broadening
cross-section for collision velocity
Examining the extent of the solar profiles one sees that the approximation is valid for most of the H profile but is not valid in the outer wings of the H and H 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 © European Southern Observatory (ESO) 2000 Online publication: December 5, 2000 |