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A classroom demonstration of reciprocal space
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Image of Fig. 1.
Fig. 1.

Optical light scattering on a periodic array of nanowires. (a) Classroom setup of the experiment to illustrate reciprocal space. (b) Optical image of a substrate with different densities of nanowires. (c and d) SEM images of InAs nanowires positioned using e-beam lithography. The image is taken from a angle, and the height is given by the vertical scalebar. (e) Sketch of Bragg's law. When equals an integer number of wavelengths , constructive interference leads to a diffraction spot.

Image of Fig. 2.
Fig. 2.

Ewald construction in the 2D case. The incident wave vector defines the radius of the Ewald circle, and the elastic outgoing wave vectors are given for all spots were Bragg's condition is fulfilled.

Image of Fig. 3.
Fig. 3.

Schematic diagram of the scattering experiment. Monochromatic light comes in from the left, scatters on nanowires positioned in an array, and produces diffraction spots to the right.

Image of Fig. 4.
Fig. 4.

Experimental setup. (a) Image showing the full setup and the alignment of the laser on the sample and the diffraction pattern. (b) Sample mounted on a two-axis goniometer on top of a turntable. (c) A schematic diagram of the setup. The first lens is used to obtain a collimated beam and the two others are set in a telescope configuration to decrease the beam size by a factor of 0.4. The two mirrors next to the sample are for elevating the beam and pointing it down at the sample with an angle , close to Brewster's angle for InAs.

Image of Fig. 5.
Fig. 5.

Measurement on a rectangular grid of nanowires. (a) Raw data. (b)Data with background subtracted. (c) Diffraction spots on the screen. (d)Experimental [red (light gray) dots with errorbars] and theoretically calculated [blue (dark gray) dots without errorbars] diffraction spots.

Image of Fig. 6.
Fig. 6.

Deviations of the diffraction points. (a) The movement of the specular spot when the sample is rotated. It can be seen that the sample is very well aligned in both the horizontal and vertical directions. (b) Vertical and (c) horizontal deviations of the experimentally observed diffraction spots from the theoretically derived values.

Image of Fig. 7.
Fig. 7.

Images of the diffraction patterns for different rotation angles of the sample. The alignment of the sample is optimized such that the position of the specular spot (0, 0) is static under rotation. See the online article for a video made from a collection of images showing a full rotation at intervals. The video is also available as supplementary material (see Ref. 12 ) (enhanced online). [URL: http://dx.doi.org/10.1119/1.4773979.1]doi: 10.1119/1.4773979.1.


Generic image for table
Table I.

Experimental parameters. These values are used throughout the experiment unless otherwise stated.



The following multimedia file is available, if you log in: 1.4773979.original.v1.mp4

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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: A classroom demonstration of reciprocal space