We have calculated the thermalisation time for an electron swarm in gaseous xenon using a multi-term time-dependent Boltzmann equation (BE), for a range of instantaneously applied reduced electric fields 1 Td < E/N < 1000 Td. Starting from a Maxwellian electron energy distribution function (EEDF) at room temperature for a given E/N, the time-evolution of the EEDF and associated electron swarm parameters (drift velocity We, mean energy 〈ε〉, ionisation coefficient ki, excitation coefficient kex) are followed as they converge to steady-state values. For all values of E/N considered, the individual swarm parameters are found to converge at different rates. For E/N > 5 Td, they converge in order We (fastest), 〈ε〉, kex, and ki (slowest). The time taken for the slowest swarm parameter to converge to an acceptable level (e.g. to within 10% of its steady-state value) is used universally as the benchmark for evaluating the thermalisation time τth. This time is found to be strongly dependent on the value of the reduced electric field E/N, dropping by almost 5 orders of magnitude for increasing E/N fields 1 Td < E/N < 1000 Td. As a key outcome from this work, the calculated thermalisation times τth·p are expressed as a general formula, as a function of both the reduced electric field E/N and a user defined convergence level between 1% and 20%. We show that ballpark estimates of thermalisation times, based on the inverse of the collision frequency for energy dissipation 1/νe(ε) at typical average electron energies, are likely to be unreliable if applied to the heating phase. We also undertake a brief analysis of the cooling phase when the electric field is instantaneously removed from the plasma (i.e. field-free) after it evolves to steady-state conditions during the previous heating phase. Finally, we compare calculated thermalisation times with the typical risetimes of the voltage pulse waveforms for several experimental ‘nanosecond’ pulse excited plasma discharge devices.
- Multi-term Boltzmann equation
- Nanosecond discharge