^{1}, E. Ya. Sherman

^{1,2}and J. E. Sipe

^{1}

### Abstract

We present a general formalism for scattering of electron bunches used in ultrafast electron diffraction experiments. To perform the scattering calculation, we associate the classical distribution function that describes the electron bunch just before scattering with the asymptotic-in Wigner distribution. Using single scattering and far-field approximations, we derive an expression for the diffracted signal and discuss the effects of the different bunch parameters on the measured diffracted flux. We identify the transverse and longitudinal coherence lengths and discuss the importance of these length scales in diffraction pattern formation. We present sample numerical calculations for scattering by nanosize particles based on our model and discuss the results in terms of bunch and scattering target parameters.

We would like to thank R. J. Dwayne Miller and members of his research group for their insightful discussions on UED experiments. This work was supported by the Natural Sciences and Engineering Research Council of Canada. E.Ya. Sherman acknowledges support of the Ikerbasque foundation and the University of the Basque Country, grant GIU07/40.

I. INTRODUCTION

II. THE WIGNER FUNCTION AND ITS VARIANCES

III. THE SCATTERED ELECTRON CURRENT

IV. DIFFRACTED FLUX IN THE FAR FIELD

A. A standard analysis

B. A more general analysis involving the role of bunch parameters

V. DIFFRACTION PATTERNS FOR GAUSSIAN BUNCHES

A. Gaussian model

B. Numerical calculations and analysis of diffraction patterns

VI. CONCLUSION

### Key Topics

- Electron scattering
- 77.0
- Atom scattering
- 30.0
- Molecule scattering
- 25.0
- Coherence
- 15.0
- Electron diffraction
- 9.0

## Figures

A simple sketch of the UED experiment.

A simple sketch of the UED experiment.

A phase-space picture of a distribution function showing some of the characteristic parameters. The index is a coordinate label.

A phase-space picture of a distribution function showing some of the characteristic parameters. The index is a coordinate label.

A schematic view of the asymptotic-in and asymptotic-out scattering states. The free electron Hamiltonian is defined as .

A schematic view of the asymptotic-in and asymptotic-out scattering states. The free electron Hamiltonian is defined as .

Coordinates in Eq. (28) used in the far-field asymptotic approximation.

Coordinates in Eq. (28) used in the far-field asymptotic approximation.

Thin film geometry. Sketched is the electron density associated with . The transverse size of the electron bunch is on the order , while the thin film can extend up to centimeters. The magnitude of , the coordinate of a typical atom, can take on values as large or greater than the size of the electron bunch.

Thin film geometry. Sketched is the electron density associated with . The transverse size of the electron bunch is on the order , while the thin film can extend up to centimeters. The magnitude of , the coordinate of a typical atom, can take on values as large or greater than the size of the electron bunch.

Coordinates of integration of Eq. (41). The vector ranges from 0 to and remains strictly in the plane perpendicular to the direction.

Coordinates of integration of Eq. (41). The vector ranges from 0 to and remains strictly in the plane perpendicular to the direction.

An electron at a position within the electron bunch has an outgoing scattered momentum . The angle between and the direction of observation is small if the distance to the detector is large [Eq. (42a)].

An electron at a position within the electron bunch has an outgoing scattered momentum . The angle between and the direction of observation is small if the distance to the detector is large [Eq. (42a)].

The gray vector is the maximum possible variation of the average local momentum vector . This variation is limited by the local widths of the Wigner function . The actual variation of the component of the vector for a fixed energy , denoted by the dark gray strip, is much less than . The vector makes an angle with , while makes an angle with the axis.

The gray vector is the maximum possible variation of the average local momentum vector . This variation is limited by the local widths of the Wigner function . The actual variation of the component of the vector for a fixed energy , denoted by the dark gray strip, is much less than . The vector makes an angle with , while makes an angle with the axis.

Detail of angles defined in Eq. (49) and a simple sketch of the Huygens–Fresnel principle. The solid lines represent the incident electron waves and the dashed lines represent the scattered electron waves.

Detail of angles defined in Eq. (49) and a simple sketch of the Huygens–Fresnel principle. The solid lines represent the incident electron waves and the dashed lines represent the scattered electron waves.

New coordinate system (52) defined with respect to , used in Eq. (53).

New coordinate system (52) defined with respect to , used in Eq. (53).

The variables and geometry for Eq. (53).

The variables and geometry for Eq. (53).

Definition of the domain position and the lattice cites used in calculations of in Eq. (56): (a) square lattices and (b) linear molecules.

Definition of the domain position and the lattice cites used in calculations of in Eq. (56): (a) square lattices and (b) linear molecules.

Definition of .

Definition of .

A sample numerical calculation of the diffracted signal using the final expression for (53) and a Gaussian model for the Wigner distribution with an electron energy of 50 keV . Main plots: (a) and . The targets are linear clusters, 32 unit cells in length, unit cell . (b) and . The targets are square clusters, cells, unit cell . Insets: , narrow peak , wider peak . All other parameters are the same except the target radius is in size. The local bunch spread and the diffraction-limited Bragg spot width are of the same order of magnitude and both significantly to the width of the peaks in the insets.

A sample numerical calculation of the diffracted signal using the final expression for (53) and a Gaussian model for the Wigner distribution with an electron energy of 50 keV . Main plots: (a) and . The targets are linear clusters, 32 unit cells in length, unit cell . (b) and . The targets are square clusters, cells, unit cell . Insets: , narrow peak , wider peak . All other parameters are the same except the target radius is in size. The local bunch spread and the diffraction-limited Bragg spot width are of the same order of magnitude and both significantly to the width of the peaks in the insets.

The broadening of the diffraction peak. Only part of the diffraction picture is shown. The darkness of the elliptic spot corresponds to the signal intensity at the screen. Every cluster is hit with a local mean electron momentum as shown in the figure. The Gaussian on the left-hand side shows the electron density distribution in the bunch.

The broadening of the diffraction peak. Only part of the diffraction picture is shown. The darkness of the elliptic spot corresponds to the signal intensity at the screen. Every cluster is hit with a local mean electron momentum as shown in the figure. The Gaussian on the left-hand side shows the electron density distribution in the bunch.

Thin film geometry used in the expansion of the phase factors.

Thin film geometry used in the expansion of the phase factors.

## Tables

(A) Typical values of UED experimental parameters. (B) Typical parameters for 30 keV electron bunches. The variances are calculated for an electron bunch time of flight of 0.5 ns, with an initial bunch size of and , and a momentum spread corresponding to (full width at half maximum). The label *T* indicates any direction in the plane, and the average velocity is in the direction.

(A) Typical values of UED experimental parameters. (B) Typical parameters for 30 keV electron bunches. The variances are calculated for an electron bunch time of flight of 0.5 ns, with an initial bunch size of and , and a momentum spread corresponding to (full width at half maximum). The label *T* indicates any direction in the plane, and the average velocity is in the direction.

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