1. Univ. Madeira Dept. Física - 1 - Universidade da Madeira Departamento de Física COST 529 Meeting Eindhoven, March 31st, 2006 project Modes of current transfer to cathodes of high-pressure arc discharges and their stability of FCT and FEDER, action 529 of the programme COST of the EC. Acknowledgements Pedro G. C. Almeida Departamento de Física, Universidade da Madeira, Portugal Calculation of mobilities of ions in high electric fields under conditions of HID lamps

2. Univ. Madeira Dept. Física - 2 - Motivation There is a well known tool that calculates transport properties of electrons in weakly ionized plasma: BOLSIG – built by Boeuf, Pitchford, and Morgan. However, so far there exists no similar tool for the transport properties of ions. We developed a tool that fills this gap for a number of ion-atom systems, in particular for those typical of HID lamps. This tool is available for download in our website: http://fisica.uma.pt The aim of this talk is to present such a tool and its physical foundations.

3. Univ. Madeira Dept. Física - 3 - The ions in a uniform plasma under an electric field acquire an oriented movement along the field direction with velocity: Generally, the ion mobility depends on the electric field strength. Knowledge of the ion mobility is necessary to calculate the ambipolar diffusion coefficient and to model the plasma- cathode interaction.  i is the ion mobility. In HID lamps this occurs in the near-cathode layer. Motivation

4. Univ. Madeira Dept. Física - 4 - For most hot gases, no experimental information is available and one must use a kinetic theory in order to determine the ion mobility. For ionized gases close to room temperature, there is vast experimental information on the mobility of ions available in the literature. Motivation The rigorous calculation of the ion mobility is a classical problem of the plasma kinetic theory.

5. Univ. Madeira Dept. Física - 5 - This assumption holds provided that the ion mass is much smaller than the mass of the neutral particals. The simplest tool in the kinetic theory: first approximation in the two-temperature theory. It is based on the assumption that the ion distribution function is close to a Maxwellian, with a temperature that can be different from the neutral temperature. Motivation

6. Univ. Madeira Dept. Física - 6 - These difficulties can be overcome by assuming that the ions may acquire a mean velocity that is, in general, different from the plasma mean mass velocity: the ion distribution becomes non-isotropic and the accuracy of the two-temperature theory decreases. When This approach is called two-temperature displaced-distribution theory. Motivation

7. Univ. Madeira Dept. Física - 7 - This last approach is usually termed three-temperature theory. The next move would be to introduce different temperatures in the (shifted) ion distribution function: one along the field direction and the other along the transversal directions. The three-temperature theory is more complex than the two- temperature displaced-distribution theory. On the other hand, it is unclear whether the introduction of a temperature anisotropy significantly improves the accuracy of the theory. There are indications that a velocity shift plays a more important role, especially for heavy ions. Motivation

8. Univ. Madeira Dept. Física - 8 - The ion distribution function is approximated by a shifted Maxwellian function: The neutral particle distribution function is approximated by a Maxwellian function: With this information it is possible to calculate rates of momentum and energy exchange in collisions of ions and atoms, which contain information on the ion mobility. The two-temperature displaced-distribution theory

9. Univ. Madeira Dept. Física - 9 - Equations of conservation of ion momentum and energy: r ia (m) and r ia (e) are the rates of exchange of momentum and energy in collisions between ions and atoms. The two-temperature displaced-distribution theory

10. Univ. Madeira Dept. Física - 10 - The integrals can be evaluated to give: The mobility is: The two-temperature displaced-distribution theory

11. Univ. Madeira Dept. Física - 11 - The two-temperature displaced-distribution theory

12. Univ. Madeira Dept. Física - 12 - It is possible to combine the equations of conservation of ion momentum and energy to obtain: To find the ion mobility it is necessary to solve the above equations for v i and T i for given E/n and T. The equations for the ion mobility and temperature are identical, with accuracy of the coefficients  and , to the respective equations of the first-approximation of the two-temperature theory. If these coefficients take the unitary value, then the formulae of the present model will coincide with those of the first approximation of the two-temperature theory. The two-temperature displaced-distribution theory

13. Univ. Madeira Dept. Física - 13 - Both coefficients are finite for a general potential of interaction between ions and neutrals For some interaction models they can be found analytically (e.g. rigid spheres) For Maxwell molecules they are equal to unity The two-temperature displaced-distribution theory

14. Univ. Madeira Dept. Física - 14 - Percentage deviation of high- field rigid-spheres ion mobility from the exact results by Skullerud 1976. Triangles: first-approximation in the two- temperature theory. Crosses: two-temperature displaced- distribution theory. Comparison with other theoretical results

15. Univ. Madeira Dept. Física - 15 - Reduced velocity of ions as a function of the reduced field strength. : first and fifth approximation in the two-temperature theory. - - - -: two-temperature displaced-distribution theory. Points: Monte Carlo simulation. Comparison with other theoretical results

16. Univ. Madeira Dept. Física - 16 - Percentage deviation of high- field rigid-sphere ion energy normalized by m a v i 2 from exact results by Skullerud 1976. Triangles: first- approximation in the two- temperature theory. Crosses: two-temperature displaced- distribution theory. Comparison with other theoretical results

17. Univ. Madeira Dept. Física - 17 - Comparison with experimental data

18. Univ. Madeira Dept. Física - 18 - Comparison with experimental data The ion mobility calculation tool

19. Univ. Madeira Dept. Física - 19 - The presented method: Allows a rapid calculation of ion mobilities in hot gases and high electric fields; is not much more complex than the first-approximation in the two- temperature theory; it is better justified in the case m i /m a ≥ O(1) ; yields results in excellent agreement with experimental data; gives results in good agreement with those given by more elaborate approaches. Conclusions