3. Collision broadening formalisms
Under the Van der Waals interaction, the interaction constant for an ion making the transition between states lo and hi and its H perturber is computed as
where is the mean square radius of excitation state i in cm, is the polarizability of the H atom, e is the electronic charge, and h is Planck's constant (e.g. Woolley & Stibbs 1953, Eq. (IX-45)). Expressing the mean square radii as in Bohr radii2 () rather than in cm2 gives
Once the interaction constant is obtained, the damping per perturber can be computed as
where v = average relative velocity of the atom and perturber (Unsöld 1955, Eq. (82,48); Gray 1992, Eq. (11.31,34)).
Most of the variety amongst computations of collision broadening arises in the approximation used for computing the mean square radii of the levels. Several of these are described below.
3.1. Scaled hydrogenic approximation
The scaled hydrogenic approximation for the mean square radius of excitation state i is
where Z is the effective nuclear charge seen by the electron (1 for Fe I, 2 for Fe II, etc.), l is the orbital angular momentum quantum number of state i, and is the effective principal quantum number given by
I being the ionisation energy of the atom (in eV), and the excitation energy of the state (in eV) (e.g. Woolley & Stibbs 1953, Eq. (IX-48)).
3.2. Unsöld's approximation
A widely adopted approximation is that due to Unsöld (1955), which differs from the hydrogenic approximation by not retaining the term . This has the computational advantage that one no longer needs to know the orbital angular momentum quantum number l, and can compute the mean square radius of a state for essentially any line, needing to know only its excitation energy, ionisation energy, and ionisation state (which gives the effective nuclear charge).
3.3. Simmons & Blackwell enhancement
Based on fits to the wings of strong lines with accurately known gf values, Simmons & Blackwell (1982) derived the damping coefficients of a number of neutral iron lines empirically, and determined what enhancements were required to Unsöld's formalism. Their enhancements to ranged from 1.0 to 1.5, being higher in general for higher excitation levels. In the present work, is used as an approximation to the Simmons & Blackwell result.
3.4. WIDTH6 approximation for the iron peak elements
Kurucz's WIDTH6 program is widely used for computing the abundances of stellar absorption lines. The approximation used there for lines in the iron peak elements (scandium to nickel inclusive), based on Kurucz's own computations of wave-functions, is
where S is the number of electrons in the ion under consideration. Furthermore, WIDTH6 neglects the lower level entirely, adopting . This approach is retained in WIDTH9 (Kurucz 1993).
This approximation differs considerably from those presented earlier. It needs to be borne in mind that Kurucz was motivated to compute synthetic spectra of strong lines of the iron group whose upper states were 4p (Kurucz & Furenlid 1979). Consequently his approximation was tailored to these strong lines for which broadening clearly mattered, rather than to much weaker lines.
3.5. Anstee & O'Mara formulation
An alternative formulation by Brueckner (1971), which retains the full interaction potential rather than treating it as a series expansion in , was developed further by O'Mara (1976) and Anstee & O'Mara (1991). Anstee & O'Mara (1995) have now computed the broadening cross sections for s-p and p-s transitions, tabulating these for a large range of quantum numbers. 1 2 An application of those results led Anstee, O'Mara & Ross (1997) to derive a solar iron abundance matching the meteoritic value.
Anstee, O'Mara & Ross argue the value of using strong (damped) lines in preference to weak lines for abundance analyses once the broadening is known, since strong line wings are not sensitive to microturbulence or NLTE effects in line cores, and because of the relative ease of measuring strong lines. The last of these depends on the degree of line blending and on the reliability of the continuum placement. Despite these advantages, the absence of strong lines for many elements, especially in metal poor stars, means that weak-line analyses will retain their importance.
© European Southern Observatory (ESO) 1998
Online publication: March 3, 1998