Hamiltonian Approach to Electron Diffraction

Peter Hawkes, Erwin K. Kasper

Armen G. Hayrapetyan, Karen K. Grigoryan, Jörg B. Götte +1 more

We report on the possibility of diffracting electrons from light waves traveling inside a dielectric medium. We show that, in the frame of reference which moves with the group velocity of light, the traveling wave acts as a stationary diffraction grating from which electrons can diffract, similar to the conventional Kapitza–Dirac effect. To characterize the Kapitza–Dirac effect with traveling light waves , we make use of the Hamiltonian Analogy between electron optics and quantum mechanics and apply the Helmholtz–Kirchhoff theory of diffraction. Introduction and background. The Kapitza-Dirac effect, i.e. the diffraction of electrons from a periodic light wave, has been predicted in the 1930's. Only now, with the advent of sufficiently intense lasers, it became possible to observe the effect experimentally. Commonly a standing wave is used to create the diffraction grating, but also other configurations, involving counter propagating waves of different frequencies have been considered in the past. In this paper, we address the question whether it is possible to observe the effect with a single light beam, travelling with a reduced group velocity within a dielectric medium. Main results. We confirm that the Kapitza-Dirac effect can be observed with a single beam of light, provided the longitudinal velocity of the electrons is commensurate with the reduced group velocity. Using Lorentz transformations between a laboratory and reference frame associated with the travelling light field, we show that in this frame of reference the light acts as a stationary diffraction grating for electrons. In contrast to a standing wave, a single, travelling light wave can give rise to inelastic diffraction when electrons exchanges energy with the light field. Wider implications. Diffraction phenomena occur for all kinds of waves and represent a signature of fundamental interference processes. Our extension of the standard Kapitza-Dirac effect to the case of a single light beam enlarges the scope to observe such effects, and might lead to simplified schemes in interferometry and microscopy. Figure. Geometry of the electron diffraction in the laboratory (a) and wave (b) frames of reference. Electrons overtake the slowed light and experience diffraction for specific angles (c). The "butterfly wings" represent the regions where the Kapitza-Dirac effect with traveling waves occurs (c).

Francisco Carlos Lovey

Xiang-Yao Wu, Zhang Baijun, Haibo Li +7 more

Phenomena of electron, neutron, atomic and molecular diffraction have been studied in many experiments, and these experiments have been explained by some theoretical works. We study electronic double-slit diffraction with a quantum mechanical approach and obtain the following results: (1) When the ratio of ( d + a )/ a = n ( n = 1,2,3,…), orders n ,2 n ,3 n ,… are missing in diffraction pattern. (2) When the ratio of ( d + a )/ a ≠ n ( n = 1,2,3,…), there is not missing order in diffraction pattern. (3) The slit thickness c has a large affect on the electronic diffraction pattern, which is a new quantum effect. We believe that all the predictions in our work can be tested by the electronic double slit diffraction experiment.

M. F. Olekseeva, K. D. Tovstyuk

Lai Wang

In this paper, single slit diffraction and dowble slit diffraction of electron are taken for the examples of analysis of electron diffraction. Feynman's path integral theory is used to analyze quantitatively the electron diffractive state. This method is brief and clear.