P. S. Barklem, S. D. Anstee, B. J. O'Mara, PASA, 15 (3), 336
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Theory Outline
The line broadening theory used has been developed from the initial work on the sodium D lines by Anstee and O'Mara (1991), which itself is a development of the work of Brueckner (1971) and O'Mara (1976). The theory has been further extended to general transitions by Anstee and O'Mara (1995), Barklem and O'Mara (1997) and Barklem, O'Mara and Ross (1998). The theory is well described in these papers, and so here only a brief review is presented.
Collisions with hydrogen atoms are sufficiently fast for the impact approximation of collision broadening to be applicable. This approximation results in a Lorentz profile for the line. For a hydrogen atom number density N, and temperature T, the line has a half width at half maximum given by 
where 
is the line broadening cross-section, and f(v) is the Maxwellian distribution of velocities for the given temperature. In this work the perturber trajectory is treated classically, and hence the semi-classical cross-section is defined by 
where b is the impact parameter defining the distance of closest approach of the atoms, and the line broadening efficiency factor is the real part of a complex efficiency factor 
averaged over all orientations of the perturbed atom. The efficiency factor can be expressed in terms of the S-matrix for the collision, where the S-matrix is dependent on the interaction potentials between the atoms. Most broadening theories follow this treatment with the only difference being the method of finding the interaction potential.
In the theory developed by Anstee and O'Mara (1991) spin is neglected and Rayleigh-Schrödinger perturbation theory taken to second order is used to calculate the interaction energy. The perturbed atom is simply modelled as an atomic core with overall unit charge, with a single valence electron. The Unsöld approximation (1955) is used to simplify the second order expression. Anstee and O'Mara (1991) have shown that line broadening is dominated by interactions at intermediate separations making perturbation theory particularly appropriate. This is a very important result with implications for all line broadening theories.
The main advantage of the method is that calculations can be made for general states of the perturbed atoms, without requiring any prior knowledge of the species of the atom or energies of states. The potentials may be calculated using Coulomb wavefunctions for the perturbed atom, simply dependent on the effective principal quantum numbers and azimuthal quantum numbers of the two levels of the transition. Anstee and O'Mara (1995) have tabulated line broadening cross-sections for s-p and p-s transitions, Barklem and O'Mara (1997) for p-d and d-p transitions and Barklem, O'Mara and Ross (1998) for d-f and f-d transitions. In all of these cases, line broadening cross-sections were tabulated against effective principal quantum numbers of the upper and lower states for a perturber velocity of 
ms
. Cross-sections were calculated for a range of velocities and were found to obey the velocity law 
The parameter 
was determined by regression and similarly tabulated.
For cross-sections obeying this relationship, Anstee and O'Mara (1995) found that the line width per unit hydrogen atom density could be expressed 
where 
, and 
is the reduced mass of the two atoms.
In order to make this data more accessible, fortran code has been written to interpolate these tables. The code simply requires the effective principal quantum numbers and the azimuthal quantum numbers of the upper and lower levels of the transition. Code is also included to compute the line width per unit hydrogen atom density from the data, given the temperature.
The code is suitably broken into subroutines which should allow easy integration into other programs if necessary. The program has been successfully tested under DEC fortran. The code uses a bicubic spline interpolation routine from the Numerical Recipes package (Press et al 1992) for the interpolation of tables. The Gamma function routine is from the Netlib archive (http://netlib.bell-labs.com/netlib/master/readme.html).
The code is available on the World Wide Web at
http://www.physics.uq.edu.au/people/barklem/barklem.html.The code can also be obtained by contacting the authors.
It should be noted that this program was not used for the examples given in the papers cited above. These were interpolated using a Matlab program. There may be small differences between the two methods of interpolation.
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