Argand plot: a sensitive fingerprint for electron channelling

Abstract

In this abstract it will be shown that exit waves are very useful in order to obtain quantitative information on the projected atom column positions and on the composition of these columns. According to the channelling theory [1], an atom column acts as a channel for the incoming electrons in which the electrons scatter dynamically without leaving the column. This theory gives a simple mathematical expression for the complex exit wave including only the lowest bound state of the column potential, which is called the 1s state. Its amplitude is peaked at the atom column positions whereas the phase, which is a constant over the column, is proportional to the average mass density of the column [2]. A convenient way to visualize these effects is by plotting the complex value of the exit wave for each pixel located at a projected atom column position in a so-called Argand plot with the x- and y-coordinate corresponding to the real and imaginary value, respectively [2, 3]. It turns out to be more convenient to shift the exit wave over a vector corresponding with the entrance wave so as to yield a fingerprint of the electron interactions only. The column position is chosen since its value is most sensitive to absorption, the Debye-Waller factor and the scattering factors. For a wedge-shape structure containing different thicknesses, the Argand plot is expected to be a circular locus which starts at the origin (for zero thickness) and which has its center on the x-axis. The radius of the circle is then the amplitude of the scattering wave, which should be identical for different atom column types if the columns are well-separated. This can also be proved using an empirical scaling behaviour which is valid for all the 1s states [4].