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1 Classical Interpretations of Kapitza-Dirac Effect 司徒树平中山大学理工学院物理系 July 2006, Lanzhou

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2 Abstract A classical theory of scattering of an electron beam by a laser standing wave (the Kapitza-Dirac effect) is presented. The prediction of the theory is compared to the experimental results. The results of this theoretical approach is in good agreement with the experimental measurements.

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3 Introduction In 1923, de.Broglie proposed the concept of wave-particle duality In 1923, de.Broglie proposed the concept of wave-particle duality In 1927, Davisson and Germer Observed the diffraction of electron by a periodic material structure. In 1927, Davisson and Germer Observed the diffraction of electron by a periodic material structure. In 1933, Kapitza and Dirac suggested that a beam of electron should also be diffracted by a standing light wave, known as “Kapitza-Dirac effect”. In 1933, Kapitza and Dirac suggested that a beam of electron should also be diffracted by a standing light wave, known as “Kapitza-Dirac effect”.

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4 They predicted that the deflection is required to obey the first- order Bragg condition with a lattice-spacing,and the probability of an electron being deflected is proportional to the square of the light beam ， They predicted that the deflection is required to obey the first- order Bragg condition with a lattice-spacing,and the probability of an electron being deflected is proportional to the square of the light beam ， They also pointed out that the intensity of ordinary light source is too low for any possible experimental observations. After the advent of laser, the experimental investigation of this effect became possible. Several experimental observations were reported.

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6 The experiment in 1988 demonstrated the Kapitza-Dirac effect with high intensity laser power. The experiment in 1988 demonstrated the Kapitza-Dirac effect with high intensity laser power. ‘The electron motion is most easily analyzed by classical electrodynamics.’ ‘The electron motion is most easily analyzed by classical electrodynamics.’ P.H.Bucksbaum, D.W.Schumacher, and M. Bashkansky, Phys.Rev. Lett,61,1182,(1988) P.H.Bucksbaum, D.W.Schumacher, and M. Bashkansky, Phys.Rev. Lett,61,1182,(1988)

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7 Recent Experiment Recently, An experimental observation of scattering of electron by the standing wave was reported. A series of electron peaks at different scattering angles were clearly shown and is in beautiful agreement with the Kapitza–Dirac theory. Recently, An experimental observation of scattering of electron by the standing wave was reported. A series of electron peaks at different scattering angles were clearly shown and is in beautiful agreement with the Kapitza–Dirac theory. Freimund, D. L., Aflatooni, K. & Batelaan, H. “Observation of the Kapitza–Dirac effect”,Nature 413, 142 - 143 (2001). Freimund, D. L., Aflatooni, K. & Batelaan, H. “Observation of the Kapitza–Dirac effect”,Nature 413, 142 - 143 (2001). Freimund, D. L & Batelaan, H. “Bragg Scattering of Free Electrons Using the Kapitza–Dirac Effect”,Phys. Rev.Lett. 89, 28 (283602). Freimund, D. L & Batelaan, H. “Bragg Scattering of Free Electrons Using the Kapitza–Dirac Effect”,Phys. Rev.Lett. 89, 28 (283602).

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8 Schematic of the Batelaan’s experimental apparatus

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9 Experi- ment results:

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10 Experimental conditions: electron beam: energy of 380-eV energy of 380-eV laser beam: Nd:YAG laser with 10-ns pulses and an energy of 0.2 J per pulse focused to a beam waist 125 μm in diameter, Nd:YAG laser with 10-ns pulses and an energy of 0.2 J per pulse focused to a beam waist 125 μm in diameter, wavelength of the laser light, 532 nm; wavelength of the laser light, 532 nm;

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11 Classical Interpretations Obviously, the experimental results gave the support to the suggestion that the interaction of electron with coherent radiation is basically a classical process. Obviously, the experimental results gave the support to the suggestion that the interaction of electron with coherent radiation is basically a classical process. An electron of initial velocity,small incident angle is passing across a laser standing wave of diameter D, An electron of initial velocity,small incident angle is passing across a laser standing wave of diameter D, electric field, electric field, magnetic field, magnetic field,

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12 Deflection of an electron by a laser standing wave

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13 The non-relativistic motion of election is governed by the Lorentz-force equation: It is then reduced to a simple pendulum equation: or the invariant relation form: Further integration gives the interaction time t for an electron to pass through the laser beam of diameter D: an incomplete elliptic integral of the first kind. Finally we obtained the corresponding value and the angle of deflection is calculated by:

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14 Phase-space diagram of simple pendulum equation

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15 Assuming that the incident phases of the electrons are uniformly distributed from to, Assuming that the incident phases of the electrons are uniformly distributed from to, The phases of electron’s motion will evolve to a new distribution after a certain time of interaction. The phases of electron’s motion will evolve to a new distribution after a certain time of interaction.

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16 Time evolution of phase distribution of the electron

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17 Classical prediction of the electron deflection for Bartell’s experiment Angular distribution of electron probability for Bartell’s experiment Angular distribution of electron probability for Bartell’s experiment

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18 Prelimiary calculation result of the electron beam profile for the Batelaan’s experiment in 2001 Prelimiary calculation result of the electron beam profile for the Batelaan’s experiment in 2001

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19 Conclusions Classical theory give a good explanation to the experiments, comparing to the quantum treatment also semi-quantum treatment, this classical approach is still competent. Classical theory give a good explanation to the experiments, comparing to the quantum treatment also semi-quantum treatment, this classical approach is still competent. Yet it is still needed to be confirmed by the new beautiful Batelaan’s experiment. Maybe this is a challenge. Yet it is still needed to be confirmed by the new beautiful Batelaan’s experiment. Maybe this is a challenge. The Batelaan’s experiment opens a new field that worth pursuing, such as using it as a spectroscopic tool, qutanum computing. The Batelaan’s experiment opens a new field that worth pursuing, such as using it as a spectroscopic tool, qutanum computing. More work should be done in the understanding about the scattering of electron by coherent wave, not only the standing wave, but also the motion wave More work should be done in the understanding about the scattering of electron by coherent wave, not only the standing wave, but also the motion wave

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