Univ. Madeira Dept. Física - 1 - Universidade da Madeira Departamento de Física COST 529 Meeting Eindhoven, March 31st, 2006 project Modes of current transfer

- 1 - Univ. MadeiraDept. Física

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COST 529 MeetingEindhoven, 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

Calculation of mobilities of ions in high electric fields under conditions of HID lamps


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.


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


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.


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


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


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


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


Equations of conservation of ion momentum and energy:

ria(m) and ria

(e) are the rates of exchange of momentum and energy in collisions between ions and atoms.



The integrals can be evaluated to give:

The mobility is:





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 vi and Ti 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.



• 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



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


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.



Percentage deviation of high-field rigid-sphere ion energy normalized by mavi

2 from exact

results by Skullerud 1976. Triangles: first-approximation in the two-temperature theory. Crosses: two-temperature displaced-distribution theory.



Comparison with experimental data


Comparison with experimental data

The ion mobility calculation tool


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 mi/ma ≥ O(1);

• yields results in excellent agreement with experimental data;• gives results in good agreement with those given by more

elaborate approaches.

Conclusions

Documents

Univ. Madeira Dept. Física - 1 - Universidade da Madeira Departamento de Física COST 529 Meeting Eindhoven, March 31st, 2006 project Modes of current transfer