Mode – The improved operating regime of a M. K. Lilley1, N. Niasse1 1 Physics Department, Imperial College, London, SW7 2AZ, UK, [email protected]

1 Summary

It was found in 1997, by G. H. Miley [1], that [2] can operate in a regime where the Particle orbits – Quantitative: effective transparency of the accelerating grid is greatly enhanced over the value one would traditionally expect from considering the fraction of area taken up by the grid wires. This • After 1ms particle collisions with wires “Star Mode” reduces the heating of the grid wires and so should in principle allow smaller essentially stop devices to be constructed. At present there is no complete explanation for this Star Mode. In this presentation we revisit some of the basic ideas and offer some new insights into the • Look what fraction are still remaining problem by considering the stability of the individual particle orbits • Particle orbits are surprisingly stable (don’t hit the wires) for a large range of initial radii 2 Experimental signatures • Sharp transition between stable and • Quasi-spherical DC electrostatic potential unstable orbits for wire number >4 create by charged wire grid (a) (b) • Gas filled chamber

• Glow discharge subsequently forms Dynamical systems perspective: • Characteristics depend on gas presure and voltage • 1 particle simulation - Top left • (a) Low voltage – High pressure (c) (d) to bottom right increasing • (b) High voltage – High pressure initial x position for fixed y • (c) High voltage – Low pressure • Stable bands in which particle oscillates about the • (d & e) STAR MODE – High voltage – Low axis. This band changes pressure (~10kV and ~20mTorr) Figures reproduced from [3] character and eventually STAR MODE - Claimed ends in chaos

(e) • Longer particle lifetime and less heating of • Poincaré plots – Find the the grid outer turning point for the particles 푟 = 0 and plot the • Better radial particle focussing and higher (휃, 푝 ) values (100 particles) fusion reaction rate θ

• Existence of mode due to shape of vacuum potential STAR MODE - Questions • What causes the transition to star mode • Phase space islands show existence of when varying voltage and gas pressure? stable orbits for large initial radii • Why is the plasma shielding effect • Destabilising nature of the potential

apparently so small? ripples can only be seen for more Figure reproduced from [4] inwardly initialised particles where it is more pronounced. Localised instability of the orbits manifests as a figure of 8 3 Single particle modelling in 2D surrounded by islands of stability Potential: • Formally one should solve 훻2푉 =0 with 푉 = 0 at outer boundary and 푉 = 푉푤 on circular wire surface • If wire radius much smaller than distance • Expansion of equation of motion about between them we can instead solve 푥 = 0 to understand stability of orbits (all 2 훻 푉 = 1/4휋푛 푞훿 풓 − 풓푤 with 푉 = 0 variables normalised to outer radius) boundary condition only (can use method of images or greens function 푞 푉 = ln 퐴 − ln 퐵 method) 4휋푛

2푛 푛 2푛 • Potential is symmetric to a large degree 퐴 = 푟푤 − 2 푟푟푤 cos 푛휃 + 푟 and only very close to the wires are the 푛 2푛 perturbations noticeable 퐵 = 1 − 2 푟푟푤 cos 푛휃 + (푟푟푤) • Along radial lines passing through the 2 푛 푛 2 푞 푛 푦 푟푤 1 1 푛 1 1 푛 푛 1 1 3 centre of the gaps in the wires the 푚푥 = 2 − 푥 − 2 − − 6 푦 푟푤 2 − 2 푥 2휋 푦 퐴0 퐵0 푦 퐴0 퐵0 퐴0 퐵0 potential “ripples” appear to act to push particles towards the wires, not focus. • Destabilising linear term, but nonlinearity is stabilising as long as the particles start far enough away from the wires – suggests why sharp transition exists Particle orbits – Visual: • For wires positioned too close to the boundary nonlinearity always destabilising so • Symplectic leap frog scheme solves no star mode should form Newton's equations – good for long timescale orbital motion 4 Interpretation and Next steps • Particles initialised with zero velocity

uniformly in angle at a given initial • The existence of a sharp transition in initial radius (푅푡푟푎푛푠) from stable to unstable radius orbits produces a natural length scale 푅푡푟푎푛푠 − 푅푤푖푟푒 • If particle hits the wire or outer • If the ionisation mean free path is larger than this then an initially unstable particle orbit boundary kill it will quickly find its way onto a stable path after its initial impact with the wire, forming • Time integrated density of particle part of the star. trajectories - Top left to bottom right • This would explain the appearance of star mode at low gas pressures increasing initial particle radius . • Need to understand the role that voltage plays – it doesn’t appear to affect potential • Transition into apparent star like shape pattern – seems to contradict repelling feature of the potential at gap • Addressing the plasma self fields – simple Debye length arguments suggest complete shielding of the vacuum field, yet we see interesting features and produce !

REFERENCES ACKNOWLEDGEMENTS

[1] G. H. Miley, et al., IEEE Trans. Plas. Sci 25 733 (1997) [3] R. Jimenez, Amateur , p23 (2008) The authors would like to acknowledge the stimulating and useful conversations with Prof. Jeroen Lamb and Prof. Dimitry Turaev from the Imperial College Department of Mathematics [2] R. L. Hirsch, J. Applied Physics, 38, 4522 (1967) [4] R. Meyer, PhD Thesis, University of Missouri, p10 (2007)