Relativisitc Effect in Photon-Induced Near Field Electron Microscopy
Electrons and photons, when interacting via a nanostructure, produce a new way of imaging in space and time, termed photon-induced near field electron microscopy or PINEM [Barwick et al. Nature2009, 462, 902]. The phenomenon was described by considering the evanescent field produced by the nanostructure, but quantification of the experimental results was achieved by solving the Schrödinger equation for the interaction of the three bodies. The question remained, is the nonrelativistic formulation sufficient for this description? Here, relativistic and nonrelativistic quantum mechanical formulations are compared for electron–photon interaction mediated by nanostructures, and it is shown that there is an exact equivalence for the two formulations. The nonrelativistic formulation was found to be valid in the relativistic regime when using in the former formulation the relativistically corrected velocity (and the corresponding values of momentum and energy). In the PINEM experiment, 200 keV electrons were utilized, giving the experimental (relativistically corrected) velocity to be 0.7c(v without relativistic correction is 0.885c). When this value (0.7c), together with those of the corresponding momentum (pc = mv) and energy (Ec = (1/2)mv2), is used in the first order solution of the Schrödinger formulation, an exact equivalence is obtained.
Additional Information© 2012 American Chemical Society. Received: May 9, 2012; Revised: June 26, 2012; Published: June 28, 2012. This article is part of the Jörn Manz Festschrift special issue. This work was supported by the National Science Foundation and the Air Force Office of Scientific Research in the Center for Physical Biology funded by the Gordon and Betty Moore Foundation. The authors declare no competing financial interest.
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