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Theory of ultrafast electron diffraction: The role of the electron bunch properties
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Image of FIG. 1.
FIG. 1.

A simple sketch of the UED experiment.

Image of FIG. 2.
FIG. 2.

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

Image of FIG. 3.
FIG. 3.

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

Image of FIG. 4.
FIG. 4.

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

Image of FIG. 5.
FIG. 5.

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.

Image of FIG. 6.
FIG. 6.

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

Image of FIG. 7.
FIG. 7.

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)].

Image of FIG. 8.
FIG. 8.

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.

Image of FIG. 9.
FIG. 9.

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.

Image of FIG. 10.
FIG. 10.

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

Image of FIG. 11.
FIG. 11.

The variables and geometry for Eq. (53).

Image of FIG. 12.
FIG. 12.

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

Image of FIG. 13.
FIG. 13.

Definition of .

Image of FIG. 14.
FIG. 14.

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.

Image of FIG. 15.
FIG. 15.

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.

Image of FIG. 16.
FIG. 16.

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


Generic image for table
Table I.

(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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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Theory of ultrafast electron diffraction: The role of the electron bunch properties