Speaker
Description
Under suitable conditions, particles with the same sign of charge confined in a potential can form ordered structures stabilized by Coulomb repulsion, known as Coulomb crystals. This phase of matter has been extensively studied and observed in systems ranging from cooled ions in storage rings [1] to electrons in one- and two-dimensional materials [2,3]. The formation of a Coulomb crystal is primarily governed by strong Coulomb coupling, namely a large ratio between the average interparticle Coulomb energy and the average kinetic energy, associated with temperature in classical systems. Recently, strong electron-electron energy correlations have been observed in photoemitted few-electron pulses, hinting at the possibility of realizing such conditions in free electron beams in an electron microscope [4].
In this work, we challenge the paradigm that associates Coulomb crystals exclusively with trapped particles by showing that free electron beams in an electron microscope can form transient ordered chains during propagation. Using Monte Carlo simulations, we track the evolution of the beam parameters within the phase diagram for the formation of spatial order and disorder. We find that the beam is created in the allowed crystallization region through the combined action of acceleration and time-dependent photoemission and subsequently exits this region as it expands and diverges. Analysis of the longitudinal phase-space distribution shows that photoemitted electron pulses evolve from an initially stochastic state to a transient Coulomb crystal that is time-ordered but energy-disordered, before expanding deterministically under Coulomb repulsion and developing well-defined peaks in both energy and time. Based on these simulations and phase-space arguments, we interpret the well-defined energy-energy correlations measured in few-electron pulses photoemitted from a thermal-field emission gun in a transmission electron microscope equipped with a Timepix3 detector, containing 4 to 8 particles, as evidence of transient ordered Coulomb crystals formed during propagation through the column.
We also extend the concept of Coulomb crystallization to pulsed and continuous electron beams transversely confined either by a sequence of alternating static quadrupoles or by a laser-induced ponderomotive potential. For pulsed beams, we predict strongly correlated electron pulses with up to 10 electrons per bunch at trapping frequencies of 10 μeV and for a trap length of 50 mm placed after the tip extractor. For continuous beams with current I, we implement molecular-dynamics simulations to compute the emittance and brightness across gas, liquid, and solid phases, revealing a significant enhancement of beam brightness in the crystalline regime compared with its gas-phase counterpart. Furthermore, we study the ability of the self-bunched beam to coherently enhance cathodoluminescence emission, showing how the structured beam density-density correlation function yields a 70% superradiant radiation build-up at the first bunching harmonic, with photon energy 2πℏI/e. We also connect the production of such beams to electron-source properties, thereby providing practical constraints for their realization. These results represent a fundamental step toward the exploration of correlated phases in free-electron systems and the development of self-organized high-brightness beams for electron microscopy.
[1] G. Birkl, K. Kassner, and H. Walther, Nature 357, 310-313 (1992).
[2] I. Shapir1, A. Hamo1, S. Pecker, C. P. Moca, Ö. Legeza, G. Zarand, S. Ilani, Science 364, 870-875 (2019).
[3] H. Li1, Z. Xiang, A. P. Reddy, T. Devakul, R. Sailus, R. Banerjee, T. Taniguchi, K. Watanabe, S. Tongay, A. Zettl,
L. Fu, M. F. Crommie, F. Wang, Science 285, 86-91 (2024).
[4] Rudolf Haindl, Armin Feist, Till Domröse, Marcel Möller, John H. Gaida, Sergey V. Yalunin, Claus Ropers, Nat. Phys. 19, 1410-1417 (2023).