Relativistic Electron Vortices

Vortices are ubiquitous whenever we have a fluid in motion or a field. For the electromagnetic field these take the form of a line in space around which the phase accumulates an integer multiple, l, of 2 pi, analogous to the manner in which the height changes as we climb a spiral staircase. It is well-established that this phenomenon is associated with an orbital angular momentum for the light in which each photon carries l times h-bar units of orbital angular momentum.

Recent developments have demonstrated, beyond reasonable doubt, that propagating electrons can also be prepared with an on-axis vortex corresponding to a phase-singularity in the wave function. As with light, we can associate this with an orbital angular momentum.

While the phenomenon of electron vortices and the associated orbital angular momentum may be said to be well understood in the non-relativistic, Schroedinger, domain the same cannot be said to be true for relativistic electrons governed by the Dirac equation. In the Dirac theory, for example, the local velocity of an electron is not proportional to the gradient of the phase of the wave function and for this reason the appealing link between the existence of vortices and electron angular momentum is brought into question.

There are three reasons why this problem is important: one practical and two fundamental. The first derives from the requirement to be able to describe, as simply as possible, experiments with shaped electron beams as they move towards higher energies. The second is the question of whether electron vortices are real and what happens to these topological features as we move into the relativistic regime. Finally, we know that the spin and orbital parts of the electron angular momentum are not separately conserved so we need to know how to interpret the mechanical consequences of relativistic electron vortices. We note that analogous difficulties arose in the study of optical angular momentum but have been resolved. This encourages us to apply similar methods to electrons and we shall do so by writing Maxwell's equations for light and the Dirac equation for the electron in similar forms. This should allow us to apply insights and practical techniques devised and tested in optics to electrons.

The initial parts of the project will be devoted to obtaining a clear understanding of the fundamentals of relativistic electron optics. With this we shall move on to develop methods to model, accurately, the effects of external magnetic and electric fields. This will necessitate the writing of some numerical codes and we intend, in the longer term, to apply these to the experimental arrangements at Glasgow, York and Oregon.

Once fully developed, we will have a product that can be passed on to other users. There is precedent for adapting the current setup of electron microscopes with commercially marketed add-ons that have been developed within academia, for example detector chips developed by the particle physics community at CERN (such as Timepix and Medipix). Similarly, this project has the potential to result in a commercially-viable and licensable numerical code that can be integrated into existing (and future) electron microscope technology. To this end, close collaboration with academic experimental groups will be nurtured throughout the project via regular visits and the results with be disseminated to wider research and commercial partners by attendance at international conferences including (but not limited to) the annual Microscopy & Microanalysis Meeting and Microscience Microscopy Conference.

It is too early to tell whether the study of electron vortices will be anything like as fruitful and influential as optical angular momentum has been, but this short programme represents a first step in this direction. A successful investigation will lay the ground work for further, more expansive funding applications involving both theoretical and experimental partners. These will constitute both academic grants and knowledge exchange and transfer partnerships to enable us to continue the fundamental research while also delivering a tangible upgrade to existing technology.

All members of the team have an extensive track-record of public outreach, which will allow us to integrate this project into our existing programme of school and adult education level interaction. The obvious applications to electron microscope imaging and high-profile experiments such as Diamond Light Source will provide an instantly engaging platform from which we can widen the impact of this research to the public.