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Journal of Experimental and Theoretical Physics

Classical and quantum versions of the Kapitza-Dirac effect

Maxim A. Efremov, Mikhail Vladimirovich Fedorov

Published Sep 1, 1999• 15 citations
Publisher

Abstract

The scattering of electrons on a standing electromagnetic wave (the Kapitza-Dirac effect) is considered within the quantum-mechanical and classical descriptions of the motion of electrons. The mean scattering angle and the distribution function of the electrons after scattering are found. It is shown that assignment of the initial electronic wave function in the form of a plane wave in the quantum-mechanical picture gives rise to the characteristic parameter β = mc 2 /ħ ω 2 τ (where ω and τ are the frequency of the field and the duration of the interaction), which separates the regions of Bragg ( β ≪1) and nearly classical ( β ≫1) scattering. When β ≫1, the mean scattering angle does not depend on the choice of the description method, but the distribution functions of the electrons after scattering in the quantum and classical approaches differ significantly. When β ≪1, both the distribution function and the mean scattering angle differ. Modification of the quantum theory and assignment of the initial electronic wave function in the form of a more or less localized wave packet are apparently needed to eliminate the differences discovered. The results obtained can be used to determine the dimensions of the wave packet which characterizes the state of electrons in a beam from experiments on scattering from a standing light wave.

DOI
10.1134/1.559004

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The scattering of electrons on a standing electromagnetic wave (the Kapitza-Dirac effect) is considered within the quantum-mechanical and classical descriptions of the motion of electrons. The mean scattering angle and the distribution function of the electrons after scattering are found. It is shown that assignment of the initial electronic wave function in the form of a plane wave in the quantum-mechanical picture gives rise to the characteristic parameter β = mc 2 /ħ ω 2 τ (where ω and τ are the frequency of the field and the duration of the interaction), which separates the regions of Bragg ( β ≪1) and nearly classical ( β ≫1) scattering. When β ≫1, the mean scattering angle does not depend on the choice of the description method, but the distribution functions of the electrons after scattering in the quantum and classical approaches differ significantly. When β ≪1, both the distribution function and the mean scattering angle differ. Modification of the quantum theory and assignment of the initial electronic wave function in the form of a more or less localized wave packet are apparently needed to eliminate the differences discovered. The results obtained can be used to determine the dimensions of the wave packet which characterizes the state of electrons in a beam from experiments on scattering from a standing light wave.
The Kapitza-Dirac effect (KDE), or the scattering of electrons by a standing light wave, is considered quantum mechanically with electron states before scattering taken in the form of wavepackets with homogeneously distributed `centres of mass'. The angular distribution functions of electrons after scattering and the average angle of scattering are found. This approach is used to establish the applicability conditions of purely classical, purely quantum-mechanical and intermediate results in the theory of KDE, as well as to describe some new regimes of scattering, characteristics of which depend explicitly on the size of the wavepackets before scattering.
The Bragg diffraction of electrons off of a standing electromagnetic wave, first predicted by Kapitza and Dirac, is discussed. The original derivation of Kapitza and Dirac is reviewed. This is followed by a more general derivation, which uses quantum mechanical, Born approximation, scattering theory. A review of the recent experiment results is given. Because the Kapitza-Dirac effect reverses the usual roles of “particles” and “waves” in Bragg diffraction, the discussion emphasizes the physical insight that this phenomenon gives into light and the particle-wave duality of quantum mechanics.
The diffraction of electrons by a standing wave of light was first proposed by Kapitza and Dirac in 1933. We observed this effect for the first time in September 2001 by showing separate electron diffraction peaks. It was first observed for a focused laser beam, where the standing wave of light serves as a thin light crystal. Later, the effect was also observed with an unfocused laser beam, where the standing wave serves as a thick light crystal, where scattering can only occur at the Bragg angle. The theory behind the effect has been formulated within both atomic optics and electron optics communities. It involves the ponderomotive potential that an electron experiences within the electromagnetic field of two counter-propagating laser beams. The derivations of the differential equations given by both communities have identical results with their own benefits when making numerical calculations. These theoretical calculations fit well to our data, which falls between the limits of the diffraction and Bragg regimes. To analyze the data collected in these experiments, the statistical behavior of the arrival of electrons, governed by the Poisson distribution, was taken into account. The Gaussian nature of the laser beam and its consequences on focusing needed to be addressed, along with how far from diffraction limited the laser beam actually was. There may be a possibility of new effects arising from the interaction of electrons and light. The effect is like KDE, but modified to include two counter-propagating lasers of different frequency, the consequence of the longitudinal velocity of the electrons, and the role that the polarization of the laser light plays.
The Kapitza-Dirac effect is the scattering of electrons by a standing wave of very intense light, which can nowadays be realized by using laser beam. A theory is presented, in which no assumptions are necessary as to the strength of the field and to the length of the interaction time. Basic is the assumption that the laser beam should have a gentle cross-sectional profile. It allows one to separate the electron motion into two components: The one parallel to the beam is the motion in the periodic field of light, which generates “adiabatic potential” for the motion transverse to the beam. This picture can be compared to that of hydrogen molecule in which electron motion generates the adiabatic potential for nuclear vibration. In this way, it is shown that the problem can be reduced to that of one-dimensional scattering. The present theory is shown to embrace the predecessor theories of Bartell and of Fedorov.
The transition probability for electron scattering in the Kapitza-Dirac experiment is calculated using nonrelativistic theory as a function of the time of flight ttt of the electron through the light beam, and as a function of the intensity. It is found that for intensities up to approximately 107{10}^{7}107 W/cm2{\mathrm{cm}}^{2}cm2 the probability equals sin2a0t{sin}^{2}{a}_{0}tsin2a0​t, where a0{a}_{0}a0​ is proportional to the intensity. At intensities of the order of 109{10}^{9}109 W/cm2{\mathrm{cm}}^{2}cm2, the dependence of the probability on the time is more complicated but it also oscillates between zero and nearly unity as the time increases. Both first- and second-order Bragg reflections are considered as well as conditions off the Bragg maximum.