Relrod - Sample Geometry and Excitation Error

Relrod - Sample Geometry and Excitation Error#

part of

MSE672: Introduction to Transmission Electron Microscopy

Spring 2025
by Gerd Duscher

Microscopy Facilities
Institute of Advanced Materials & Manufacturing
Materials Science & Engineering
The University of Tennessee, Knoxville

Background and methods to analysis and quantification of data acquired with transmission electron microscopes.

import sys
if 'google.colab' in sys.modules:
    !{sys.executable} -m pip install ipympl
    
%matplotlib  widget
import matplotlib.pyplot as plt
import numpy as np


from mpl_toolkits.mplot3d import Axes3D  # 3D plotting

__notebook_version__ = '2021.02.09'
print('notebook version: ', __notebook_version__)

notebook version:  2021.02.09

Goals#

Here we want to invesitate the influence of sample shape onto the diffraction pattern.

Generally, we have to consider the sample to be a thin disk-like shape, which in reciprocal space that is a rod-like shape.

It is this rod, together with dynamic scattering, that makes diffraction spots with excitation error possible.

In the following, we perform Fourier Transforms on disks with different ratios of height to radius.

Make a disk shaped sample#

Please, change width and height to see the effect in the Fourier Transform

radius = 50

height= 8

data = np.zeros([100,100,100])

x, y = np.mgrid[-data.shape[0]/2:data.shape[0]/2,-data.shape[1]/2:data.shape[1]/2]

data[x**2+y**2<radius**2, int(data.shape[2]/2-height/2): int(data.shape[2]/2+1+height/2)  ] = 1
X, Y, Z = np.where(data>0)

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')

# Plot the surface
ax.scatter(X, Y, Z, color='r', label='real space')
#ax.scatter(Xf, Yf, Zf, color='b', label='reciprocal space')

ax.set_xlim(0,data.shape[0])
ax.set_ylim(0,data.shape[1])
ax.set_zlim(0,data.shape[2]);

Perform Fourier transform#

r_data = np.fft.fftshift(np.fft.fftn(np.fft.fftshift(data)))

r_data = np.log(1+np.abs(r_data))
r_data[r_data<r_data.max()*.8] = 0
r_data[r_data>0]=1

Xf, Yf, Zf = np.where(r_data>0)

Plot Data Fast#

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')

# Plot the surface
#ax.scatter(X, Y, Z, color='r', label='real space')
ax.scatter(Xf, Yf, Zf, color='b', label='reciprocal space')

ax.set_xlim(0,data.shape[0])
ax.set_ylim(0,data.shape[1])
ax.set_zlim(0,data.shape[2]);

Slow Plotting Routine#

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')

# Plot the surface
ax.voxels(r_data, color='b')
ax.voxels(data, color='r')

ax.set_xlim(0,data.shape[0])
ax.set_ylim(0,data.shape[1])
ax.set_zlim(0,data.shape[2]);

Conclusion#

The shape of the Fourier Transform of the sample is called relrod.

This relrod replaces any point in recirpocal lattice.

We can conclude a few points from this notebook:

The relrod is perpendicular to the sample
The relrod is the major contribution to the excitation error.

The surface of the sample may not be exactly perpendicular to the zone axis and has to be considered in our calulations