Widths and Shifts of Isolated Lines of Neutral and Ionized Atoms Perturbed by Collisions With Electrons and Ions: An Outline of the Semiclassical Perturbation (SCP) Method and of the Approximations Used for the Calculations
Abstract
:1. Introduction
2. Collisional Line Broadening in the Impact Approximation
3. The Semiclassical Perturbation Approximation (SCP) for Stark-Broadening Studies
3.1. The Semiclassical Approximation and the Parametric Representation of the Orbits
Attractive Hyperbola | Repulsive Hyperbola | Straight Path | |
---|---|---|---|
x | ρ | ||
y | |||
t | |||
distance of closest approach | ρ |
3.2. The Time-Dependent Perturbation Approximation for the Calculation of the S (or T) Matrix
3.3. The Atom-Perturber Interaction Potential
3.4. Determination of the T-Matrix Elements Using Equations (22) and (24) and the Coordinates of the Perturber
3.5. Expressions of , Symmetrization and Some Asymptotic Limits
3.5.1. Case of Neutral Atoms (Straight Path)
Contribution of the dipolar interaction: expressions of the transition probabilities, of the inelastic cross-sections and of
Symmetrization
Remark concerning the shift
Asymptotic limits and series expansions of , ,
3.5.2. Case of Ionized Atoms (Hyperbolic Path)
Contribution of the dipolar interaction: expressions of the transition probabilities, of the inelastic cross-sections and of
Symmetrization of the transition probabilities and cross-sections
Asymptotic and series expansions for A, a and B, used in the computer code
- (1)
- First, we recall that:
- (2)
- At high energies, E, or small , zero, one obtains:At very high energies, the Coulomb attraction (or repulsion) becomes weak; the contribution of high impact parameters predominates, , and the straight path case is recovered.
- (3)
- (4)
- When (or ), we have used the expansion ([26] p. 202):
- (5)
- Beyond the validity of these above expansions, the and functions have been calculated with an unpublished Fortran subroutine developed in [24]. The and follow.
- (6)
- Asymptotic expansions for for the attractive case (collisions with electrons). We have used two expansions. The first one is valid for high values of ξ, and the first two terms were calculated by [6]. We have added the third term of the expansion in the numerical SCP code. It reads:
- (7)
- Beyond the validity of these expansions, has been calculated by means of a numerical integration of the Hilbert transform (Equation (50)) or by a numerical integration of the second imaginary term of the Dyson series (Equation (52)).
- (8)
- Asymptotic expansions for for the repulsive case (collisions with positive ions). Due to the mass effect, ξ is always high in typical conditions for Stark broadening studies. Therefore, an expansion valid for high values of ξ is sufficient and has been introduced in the SCP computer code. It reads [6]:
Contribution of the quadrupolar interaction: expression of for radiating ions
4. Results and Discussion
4.1. Validity of the Impact Approximation
T | |||
Full width at half maximum | |||
Strong collisions contribution | |||
Inelastic collision contribution from the upper level | |||
Inelastic collision contribution from the lower level | |||
Feshbach resonances contribution from the upper level | |||
Feshbach resonances contribution from the lower level | |||
Elastic collisions contribution (polarization + quadrupole) | |||
Elastic collisions contributions (without quadrupole) |
T | |||
Full width at half maximum | |||
Strong collisions contribution | |||
Inelastic collision contribution from the upper level | |||
Inelastic collision contribution from the lower level | |||
Elastic collisions contribution (polarization + quadrupole) | |||
Elastic collisions contribution (without quadrupole) |
T | |||
Full width at half maximum | |||
Strong collisions contribution | |||
Inelastic collision contribution from the upper level | |||
Inelastic collision contribution from the lower level | |||
Elastic collisions contribution (polarization + quadrupole) | |||
Elastic collisions contribution (without quadrupole) |
T | |||
Full width at half maximum | |||
Strong collisions contribution | |||
Inelastic collision contribution from the upper level | |||
Inelastic collision contribution from the lower level | |||
Elastic collisions contribution (polarization + quadrupole) | |||
Elastic collisions contribution (without quadrupole) |
4.2. Validity of the Isolated Line Approximation
4.3. Comparisons of the Different Contributions
4.3.1. Electron Collisions
4.3.2. Ion Collisions
4.4. Accuracy of the SCP Method
4.5. Ab Initio and Automatic Codes for Obtaining a Great Number of Data in a Same Run and the Influence of the Chosen Atomic Structure
4.6. Modifications in Progress and Prospects
4.7. Regularities and Systematic Trends
4.7.1. Behavior with temperature
- (1)
- High temperatures:At high temperatures (or very small ), the Coulomb attraction or repulsion for ion emitters is small; the behavior is the same for neutrals and ions: behaves as , and thus, the cross-sections as . With a rough reasoning, it can be deduced that the widths decrease as ; cf. in particular [51,52]. In addition, due to the mass effect, the contribution of the ion colliders can be greater than that of electrons, e.g., [30].
- (2)
- Low temperatures:At low temperatures, the behavior is different for neutral and ion emitters. For ions colliding with electrons, the collision strengths tend towards a finite limit; the cross-sections decrease as near the threshold, and the width decreases as . For neutrals colliding with electrons, the width begins to increase with the temperature.
4.7.2. Behavior with the charge of the perturber
4.7.3. Behavior with the charge of the radiating ion
4.7.4. Behavior of the width of a spectral series of transitions of a given neutral or ionized atom with increasing principal quantum number n
4.7.5. Importance of the fine-structure splitting
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
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Sahal-Bréchot, S.; Dimitrijević, M.S.; Nessib, N.B. Widths and Shifts of Isolated Lines of Neutral and Ionized Atoms Perturbed by Collisions With Electrons and Ions: An Outline of the Semiclassical Perturbation (SCP) Method and of the Approximations Used for the Calculations. Atoms 2014, 2, 225-252. https://doi.org/10.3390/atoms2020225
Sahal-Bréchot S, Dimitrijević MS, Nessib NB. Widths and Shifts of Isolated Lines of Neutral and Ionized Atoms Perturbed by Collisions With Electrons and Ions: An Outline of the Semiclassical Perturbation (SCP) Method and of the Approximations Used for the Calculations. Atoms. 2014; 2(2):225-252. https://doi.org/10.3390/atoms2020225
Chicago/Turabian StyleSahal-Bréchot, Sylvie, Milan S. Dimitrijević, and Nabil Ben Nessib. 2014. "Widths and Shifts of Isolated Lines of Neutral and Ionized Atoms Perturbed by Collisions With Electrons and Ions: An Outline of the Semiclassical Perturbation (SCP) Method and of the Approximations Used for the Calculations" Atoms 2, no. 2: 225-252. https://doi.org/10.3390/atoms2020225