Fluctuation electron microscopy analysis


Pixelated STEM can be used as a basis for fluctuation electron microscopy (FEM) measurements. By analyzing the variance amongst a large set of electron diffraction patterns, the medium-range order present in highly disordered materials can be statistically assessed. Pixstem’s FEM functionality is based on the methods outlined in:

T L Daulton, et al., Ultramicroscopy 110 (2010) 1279-1289.


Briefly, the input data set consists of a series of 2-D nanobeam electron diffraction patterns which is used to determine several measures of structural variance. These measures are:

  • V-Omegak: normalized variance of the annular mean
  • V-rk: mean of normalized variances of rings
  • Vrek: normalized variance of ring ensemble
  • Omega-Vi: normalized variance image
  • Omega-Vk: Annular mean of the variance image

Performing the analysis

This example will use a test dataset that approximates a FEM-style signal. The dataset is a PixelatedSTEM signals that consists of a 10x10 array of electron diffraction patterns (100x100 pixels), each of which contain a central bright disk, two rings of varying, intensity, and Poissonian noise. The test data is loaded using pixstem.dummy_data.get_fem_signal().

>>> import pixstem.api as ps
>>> s = ps.dummy_data.get_fem_signal()
>>> fem_results = s.fem_analysis(centre_x=50, centre_y=50, show_progressbar=False)

Visualizing the results

Each calculated statistical measure is stored as an independent signal in the results dictionary and can easily be plotted for visual inspection. For example:

>>> fem_results['V-Omegak'].plot()

Alternatively, a plotting function: pixstem.fem_tools.plot_fem() is available to plot all calculated measures in a single Matplotlib figure.

>>> import pixstem.fem_tools as ft
>>> fig = ft.plot_fem(s, fem_results)
>>> fig.savefig('FEM_Results.png')

Storing and recalling the results

Since each of the calculated statistical measures is a distinct PixelatedSTEM signal, they can each be saved manually the same as any other signal.

>>> fem_results['V-Omegak'].save('V-Omegak.hdf5')

For convenience, a dedicated function is also available: pixstem.fem_tools.save_fem(). This will create a separate folder and save all of the calculated measures in separate HDF5 files with a common root name.

>>> ft.save_fem(fem_results, 'fem_results')

The saved files can be read back into memory using pixstem.fem_tools.load_fem(). This will return a dictionary consisting of the previously saved signals.

>>> fem_results_loaded = ft.load_fem('fem_results')