### Abstract

A standard textbook derivation for the scattering of electrons by a weak potential under the first Born approximation suggests that the far-field scattered wave should be in phase with the incident wave. However, it is well known that waves scattered from a weak phase object should be phase-shifted by π/2 relative to the incident wave. A disturbing consequence of this missing phase is that, according to the Optical Theorem, the total scattering cross section would be zero in the first Born approximation. We resolve this mystery pedagogically by showing that the first Born approximation fails to conserve electrons even to first order. Modifying the derivation to conserve electrons introduces the correct phase without changing the scattering amplitude. We also show that the far-field expansion for the scattered waves used in many texts is inappropriate for computing an exit wave from a sample, and that the near-field expansion also give the appropriately phase-shifted result.

Original language | English (US) |
---|---|

Pages (from-to) | 57-62 |

Number of pages | 6 |

Journal | Ultramicroscopy |

Volume | 119 |

DOIs | |

State | Published - Aug 2012 |

### Fingerprint

### Keywords

- First Born approximation
- Weak phase object

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics
- Instrumentation
- Electronic, Optical and Magnetic Materials

### Cite this

*Ultramicroscopy*,

*119*, 57-62. https://doi.org/10.1016/j.ultramic.2011.11.012

**A surprise in the first Born approximation for electron scattering.** / Treacy, Michael; Van Dyck, D.

Research output: Contribution to journal › Article

*Ultramicroscopy*, vol. 119, pp. 57-62. https://doi.org/10.1016/j.ultramic.2011.11.012

}

TY - JOUR

T1 - A surprise in the first Born approximation for electron scattering

AU - Treacy, Michael

AU - Van Dyck, D.

PY - 2012/8

Y1 - 2012/8

N2 - A standard textbook derivation for the scattering of electrons by a weak potential under the first Born approximation suggests that the far-field scattered wave should be in phase with the incident wave. However, it is well known that waves scattered from a weak phase object should be phase-shifted by π/2 relative to the incident wave. A disturbing consequence of this missing phase is that, according to the Optical Theorem, the total scattering cross section would be zero in the first Born approximation. We resolve this mystery pedagogically by showing that the first Born approximation fails to conserve electrons even to first order. Modifying the derivation to conserve electrons introduces the correct phase without changing the scattering amplitude. We also show that the far-field expansion for the scattered waves used in many texts is inappropriate for computing an exit wave from a sample, and that the near-field expansion also give the appropriately phase-shifted result.

AB - A standard textbook derivation for the scattering of electrons by a weak potential under the first Born approximation suggests that the far-field scattered wave should be in phase with the incident wave. However, it is well known that waves scattered from a weak phase object should be phase-shifted by π/2 relative to the incident wave. A disturbing consequence of this missing phase is that, according to the Optical Theorem, the total scattering cross section would be zero in the first Born approximation. We resolve this mystery pedagogically by showing that the first Born approximation fails to conserve electrons even to first order. Modifying the derivation to conserve electrons introduces the correct phase without changing the scattering amplitude. We also show that the far-field expansion for the scattered waves used in many texts is inappropriate for computing an exit wave from a sample, and that the near-field expansion also give the appropriately phase-shifted result.

KW - First Born approximation

KW - Weak phase object

UR - http://www.scopus.com/inward/record.url?scp=84865380024&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84865380024&partnerID=8YFLogxK

U2 - 10.1016/j.ultramic.2011.11.012

DO - 10.1016/j.ultramic.2011.11.012

M3 - Article

C2 - 22206601

AN - SCOPUS:84865380024

VL - 119

SP - 57

EP - 62

JO - Ultramicroscopy

JF - Ultramicroscopy

SN - 0304-3991

ER -