After years of experimenting, modelling and theorizing, ARCNL researchers demonstrate in a paper in Physical Review Letters a rather simple, universal power law to govern the complex behavior of an expanding plasma of highly charged particles.
‘The notion that something as complex as an expanding, hot and dense plasma can be described with such simple mathematics, keeps bewildering me.’ John Sheil, ARCNL group leader and Assistant Professor at Vrije Universiteit Amsterdam, is still amazed by the universality and simplicity of the power law he and his team discovered to be governing the average charge state of expanding tin plasmas.
Plasmas – matter so hot that electrons are stripped from the atoms – are of interest for a myriad of applications. For ARCNL, the fact that the extreme ultraviolet radiation sources in ASML’s lithography machines consist of laser-driven tin plasmas, is the general motivation behind extensive research into plasma physics.
In ASML’s EUV source a droplet of tin is hit by a powerful laser, resulting in a plasma that emits the desired EUV radiation. ‘The plasma generated in an EUV source reaches temperatures of some 500.000 Kelvin, resulting in highly ionized tin,’ explains Sheil. ‘These fast moving, highly charged particles should be suppressed to avoid damaging optical components in the machine. In order to design schemes to slow down these particles, you need to understand how many of them are produced, what their speed is, and where they are heading.’
Establish universality
In a series of experiments, ARCNL and VU Amsterdam researchers studied the properties of the particles flying away from the plasma. They varied all possible parameters in their set-up, ranging from the properties of the laser used to create the plasma to the geometry of the tin source and the angle under which they detected the particles. ‘To our major surprise, the average charge state of the plasma turned out to always scale with their kinetic energy to the power of 0.4.’ And this not only held true for their own experiments, he emphasizes. ‘The ASML team in San Diego also sent us their data, and even with an entirely different set up this power law still holds!’
Analytical and numerical proof
It was a serendipitous experimental finding, Sheil says. ‘And it took us years and a vast amount of discussions on the blackboard to be able to explain it.’ The researchers started reading old books from the sixties and seventies about plasma physics. How does a plasma expand in the first place? ‘But it wasn’t until we organized a code comparison workshop that we forced a breakthrough.’ The participants of the workshop studied the question of how ionized a plasma gets at a certain temperature. ‘They found that, in general, the average charge state scales with the temperature to the power of 0.6. Coupling this with an analytic model of plasma expansion, we ended up with the exact same 0.4 power law that we had observed from our experiments.’ To further back up their finding, the researchers also conducted an extensive numerical simulation. This resulted in the same power law as well.
Break the law
As far as the research goes, many follow up questions remain, he says. ‘For example, what happens when you do not use tin, but look at plasmas made out of other elements? In a preliminary calculation we predicted that this behavior will be general for any other very hot plasma consisting of heavy ions. But it will be interesting to find out which properties and conditions will be instrumental in breaking the power law.’
Do you want to know more about this research? Then contact John Sheil (j.sheil@vu.nl, j.sheil@arcnl.nl).
Reference
J. Sheil, L. Poirier, A.C. Lassise, S. Schouwenaars, N. Braaksma, A. Frenzel, R. Hoekstra, and O. O. Versolato, Power law scaling relating the average charge state and kinetic energy in expanding laser-driven plasmas, Phys. Rev. Lett. 133, 125101 (2024)
Doi:10.1103/PhysRevLett.133.125101