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Measurements of Long-range Electronic Correlations During Femtosecond Diffraction Experiments Performed on Nanocrystals of Buckminsterfullerene
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A surprise in the first Born approximation for electron scattering.

M M J Treacy1, D Van Dyck

  • 1Department of Physics, Arizona State University, Tempe, Arizona 85287, USA. treacy@asu.edu

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|December 31, 2011
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Summary

The first Born approximation incorrectly predicts electron scattering phase. Correcting this approximation to conserve electrons yields the expected π/2 phase shift, resolving issues with the Optical Theorem and scattering cross-section calculations.

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Area of Science:

  • Quantum mechanics
  • Electron scattering
  • Wave optics

Background:

  • Standard textbook derivations of electron scattering using the first Born approximation predict in-phase scattered waves.
  • This contradicts the known π/2 phase shift for waves scattered by weak phase objects.
  • The first Born approximation leads to a zero total scattering cross-section, violating the Optical Theorem.

Purpose of the Study:

  • To resolve the discrepancy in phase prediction for electron scattering under the first Born approximation.
  • To demonstrate the failure of the first Born approximation in conserving electrons.
  • To present a modified derivation that incorporates electron conservation and yields the correct phase shift.

Main Methods:

  • Revisiting the standard textbook derivation of the first Born approximation for electron scattering.
  • Modifying the derivation to enforce electron conservation.
  • Analyzing far-field and near-field expansions for scattered waves.

Main Results:

  • The first Born approximation fails to conserve electrons, even to first order.
  • A modified derivation that conserves electrons introduces the correct π/2 phase shift without altering the scattering amplitude.
  • Far-field expansions are inappropriate for calculating exit waves; near-field expansions yield the correct phase-shifted result.

Conclusions:

  • The standard first Born approximation is fundamentally flawed due to its failure to conserve electrons.
  • Electron conservation is crucial for accurately describing wave scattering, particularly phase shifts.
  • Near-field expansions are more appropriate than far-field expansions for calculating scattered waves from samples.