Unic Cell Determination and Stereographic Projection

2.10. Unic Cell Determination and Stereographic Projection#

part of

MSE672: Introduction to Transmission Electron Microscopy

Spring 2024

Gerd Duscher	Khalid Hattar
Microscopy Facilities	Tennessee Ion Beam Materials Laboratory
Materials Science & Engineering	Nuclear Engineering
Institute of Advanced Materials & Manufacturing
The University of Tennessee, Knoxville

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

2.10.1. Load relevant python packages#

2.10.1.1. Check Installed Packages#

import sys
import importlib.metadata
def test_package(package_name):
    """Test if package exists and returns version or -1"""
    try:
        version = importlib.metadata.version(package_name)
    except importlib.metadata.PackageNotFoundError:
        version = '-1'
    return version

if test_package('pyTEMlib') < '0.2024.1.0':
    print('installing pyTEMlib')
    !{sys.executable} -m pip install git+https://github.com/pycroscopy/pyTEMlib.git@main -q --upgrade

if 'google.colab' in sys.modules:
    !{sys.executable} -m pip install numpy==1.24.4
print('done')

done

2.10.2. Import numerical and plotting python packages#

Import the python packages that we will use:

Beside the basic numerical (numpy) and plotting (pylab of matplotlib) libraries,

and some libraries from pyTEMlib

kinematic scattering library.
animation

# import matplotlib and numpy
#                       use "inline" instead of "notebook" for non-interactive plots
%matplotlib  widget
import matplotlib.pyplot as plt
import numpy as np
import sys
if 'google.colab' in sys.modules:
    from google.colab import output
    output.enable_custom_widget_manager()
    
# additional package 
import  itertools 
from matplotlib import patches

# Import libraries from pyTEMlib
import pyTEMlib
import pyTEMlib.kinematic_scattering as ks         # Kinematic scattering Library
                             # Atomic form factors from Kirklands book
from pyTEMlib import animation        # Provide some functions to plot schematic plots
__notebook_version__ = '2024.01.23'
print('pyTEM version: ', pyTEMlib.__version__)
print('notebook version: ', __notebook_version__)

You don't have igor2 installed.     If you wish to open igor files, you will need to install it     (pip install igor2) before attempting.
You don't have gwyfile installed.     If you wish to open .gwy files, you will need to      install it (pip install gwyfile) before attempting.
Symmetry functions of spglib enabled
Using kinematic_scattering library version {_version_ }  by G.Duscher
pyTEM version:  0.2023.9.1
notebook version:  2024.01.23

2.10.3. Unit Cell Determination#

The HOLZ rings will give the lattice repeat vector (reciprocal vector parallel to the zone axis).
So tilting in [001] zone axis, the ZOLZ pattern will give you the [100] and [010] distance
and the HOLZ ring radius the [001] distance.

This is the determination of the lattice parameter of a unit cell.

Thus, we see that one can determine 3D information from a single two dimensional pattern.

It might be necessary to use other low order zone axes.

2.10.3.1. Measurements#

Record HOLZ and ZOLZ patterns, if possible in one picture (use double illumination with different exposure times to enhance dynamic range), but with different convergence angles.
If the angle of the ring is too large then your measurements may suffer from lens distortions.
If the HOLZ ring is split measure the inner one.

2.10.3.2. Z-Component of Unit Cell#

If $H$ is the distance between the reciprocal-lattice planes parallel to the beam and $G_n$ is the projected radius of the HOLZ ring, then $$ G_1 = \left( \frac{2H}{\lambda}\right)^{1/2} = \sqrt{\frac{2H}{\lambda}}\\ G_2 = 2\left( \frac{H}{\lambda}\right)^{1/2} = 2 \sqrt{\frac{H}{\lambda}} $$ for FOLZ and SOLZ.

With $\lambda$ wavelength of incident electrons

Similar expressions can be developed for higher order HOLZ rings.

In real space you get for example for FOLZ: \begin{equation} \frac{1}{H}=\frac{2}{\lambda G_1^2} = \frac{2}{\lambda} \left(\frac{\lambda L}{r}\right)^{2} \end{equation}

If you did use a zone axis which is not ${100}$, then you have to compare your result to calculated values.

Assuming you are looking down $[UVW]$ then we know: \begin{equation} \frac{1}{H}= |[UVW]| \end{equation}

Now we have to calculate this $|[UVW]|$ for different structures:\

2.10.3.2.1. for fcc:#

\begin{equation} \frac{1}{H}= \frac{a_0}{p(U^2+V^2+W^2)} \end{equation} with $a_0$ is the lattice parameter and $p=1$ for $U+V+W$ is odd; $p=2$ for $U+V+W$ is even.

2.10.3.2.2. for bcc:#

the same relationship as for fcc is true for bcc but $p$ is different: $p=2$ for $U$, $V$, and $W$ all odd; $p=1$ otherwise.

Look up other crystal systems.

If a ring is forbidden: you have to multiply your measurement $1/H_m$ with an integer $n$ to obtain the distance of the crystal.

2.10.4. Lattice Centering#

We are going to look at cubic structures to

fcc
bcc
a-face
b-face
primitive or simple cubic

The maximal excitation error is chosen so that ZOLZ and FOLZ overlap and we can see the different centering

def plot_spots(tags, ax):
    """Simple plotting for spot pattern"""
    
    points = tags['allowed']['g']
    ix = np.argsort((points**2).sum(axis=1))
    p = points[ix]
    Laue_zones = np.unique(p[:,2])     
    ZOLZ = np.where(p[:,2] == Laue_zones[0])
    FOLZ = np.where(p[:,2] == Laue_zones[1])
    SOLZ = np.where(p[:,2] == Laue_zones[2])

    ax.scatter(p[ZOLZ,0], p[ZOLZ,1], color='red')
    ax.scatter(p[FOLZ,0], p[FOLZ,1], color='blue', alpha = 0.3)
    ax.scatter(p[SOLZ,0], p[SOLZ,1], color='green', alpha = 0.3)

    ax.set_aspect('equal')
    ax.set_title(tags['crystal_name'])
    ax.set_xlim(-2, 8)
    ax.set_ylim(-2, 8)
    
# load structure
atoms = ks.structure_by_name('FCC Fe')

# add necessary parameters for kinematic scattering calculation
tags = {}
tags['acceleration_voltage_V'] = 200000
tags['convergence_angle_mrad'] = 0
tags['zone_hkl'] = [0, 0, 1] # incident neares zone axis: defines Laue Zones!!!!
tags['mistilt']  = np.array([0,0,0])  # mistilt in degrees

tags['Sg_max'] = 0.25 # 1/nm  maximum allowed excitation error ; This parameter is related to the thickness
tags['hkl_max'] = 45   # Highest evaluated Miller indices
# calulcuate kinematic scattering data
atoms.info['experimental'] = tags.copy()
ks.kinematic_scattering(atoms, False)

fig, ax = plt.subplots(nrows=2, ncols=3, figsize=(10,6))

# plot diffraction pattern
atoms.info['diffraction']['crystal_name'] = 'FCC or I'
plot_spots(atoms.info['diffraction'], ax[0, 0])
plot_spots(atoms.info['diffraction'], ax[1, 2])


atoms = ks.structure_by_name('BCC Fe')
atoms.info['experimental'] = tags.copy()
ks.kinematic_scattering(atoms, False)
atoms.info['diffraction']['crystal_name'] = 'BCC'
# plot diffraction pattern
plot_spots(atoms.info['diffraction'], ax[1, 0])

atoms.set_scaled_positions([[0, 0, 0], [0, 1/2, 1/2]])
ks.kinematic_scattering(atoms, False)
atoms.info['diffraction']['crystal_name'] = 'a-face'
plot_spots(atoms.info['diffraction'], ax[0, 1])

atoms.set_scaled_positions([[0. , 0. , 0. ], [1/2, 0, 1/2]])
ks.kinematic_scattering(atoms, False)
atoms.info['diffraction']['crystal_name'] = 'b-face'
plot_spots(atoms.info['diffraction'], ax[1, 1])

del atoms[1]
ks.kinematic_scattering(atoms, False)
atoms.info['diffraction']['crystal_name'] = 'simple cubic'
plot_spots(atoms.info['diffraction'], ax[0, 2])

To analyse an experimental pattern:

Extend the pattern for the ZOLZ into the HOLZ ring and look for discrepancies.

2.10.5. Laue Circle#

The mistilt (angles in degrees) leads to a circular pattern in the ZOLZ.

Any mistilt will cause the Ewald sphere to cut through the projection plane in a circle: the Laue circle. The maximal excitation error $S_{g_{max}}$ has to be rather small for this effect to appear.

The nearest zone axis will always be in the middle of the Laue circle.

If you encounter such a Laue circle try to minimize the circle by tilting towards the center. You can try this out below in rhe

# -----Input ----------
mistilt_alpha  = -0.0  # in degree
mistilt_beta = 0.0     # in degree
S_g = .02 # 1/Ang  maximum allowed excitation error ; This parameter is related to the thickness
# ---------------------


atoms = ks.structure_by_name('FCC Fe')

# add necessary parameters for kinematic scattering calculation
tags = {'acceleration_voltage_V': 100000,
        'convergence_angle_mrad': 0,
        'zone_hkl': [0, 0, 1], # incident neares zone axis: defines Laue Zones!!!!
        'mistilt_alpha degree': mistilt_alpha,  # mistilt in degrees
        'mistilt_beta degree': mistilt_beta,
        'Sg_max': S_g, # 1/Ang  maximum allowed excitation error ; This parameter is related to the thickness
        'hkl_max': 13   # Highest evaluated Miller indices
       }
# calulcuate kinematic scattering data
atoms.info['experimental'] = tags
 
#tags['crystal_name'] = f"FCC with mistilt {tags['mistilt alpha']}"
ks.kinematic_scattering(atoms, verbose=False)
atoms.info['output'] = pyTEMlib.diffraction_plot.plotSAED_parameter()
atoms.info['output']['plot_labels'] = False
atoms.info['output']['plot_Kikuchi'] = False

pyTEMlib.diffraction_plot.plot_diffraction_pattern(atoms)

The next graph shows a cross section through the reciprocal space (with Ewald sphere).

The tilt out of zone axis (blue) leaves some spots in the middle with an high excitation error $s_g$ larger than the maximum allowed one. These spots (in the figure from 2 1/nm to 8 1/nm) are invisible in such a case. Because the Ewald sphere is a 3D object the cut of a sphere with a plane will give a circle.

plt.figure()
animation.deficient_holz_line(exact_bragg=False, laue_zone=0, color='black')
animation.deficient_holz_line(exact_bragg=True, laue_zone=0, color='blue')

2.10.6. Stereographic Projection#

There are a lot of problems in materials science you can solve with diffraction patterns in the TEM.

For instance the orientation relationship two crystals have to each other. What is the grain boundary plane and so on.

The method to visualize such orientation relationships is the stereographic projection.

2.10.6.1. Construction#

The Schematic below shows the construction of Stereographic Projection for cubic systems.

Steps of Construction:

Draw the crystal in the middle of a sphere.
Draw a line from the center of the sphere through the middle of each plane (must be normal to the plane).
Mark where this line intersects the sphere (it is named P in figure above).
From this point draw a line to the south or north pole so that you intersect the equatorial plane. If you have to go to the south pole, mark the intersection of this line with equatorial plane with a dot; if you go to the north pole mark this intersection with a circle.We construct the point P’.

Prperties of Stereograpic Projection

This point represents uniquely one plane.
The relevant area of the equatorial plane is a disk.
Now we can also project circumference of a circle. Note that all the planes perpendicular to a low order zone axis lay on such a circle.
These circles show up as lines or as ovals in the stereographic projection .

Change the Miller indices (reflection) to see the change

# ------Input ----------
reflection = np.array([3,0, 1])
# -----------------------
if reflection[1] != 0:
    print('we only use a cross section so y is set to 0')
    reflection[1] = 0

R = 90 # 90 degrees projection sphere
x,y,z =reflection/np.linalg.norm(reflection)*R # Coordinates on sphere surface
x_projeted = (x*R/(R+z)) # x coordinate on stereographic projection plane
print(f'projected x-coordinate is {x_projeted:.2f} degree')

plt.figure()
plt.title(f'Cross Section of Stereographic Projection of {reflection} ')
sphere = plt.Circle(( 0. , 0. ), R , fill=False, linewidth=2) 
plt.gca().set_aspect( 'equal') 
plt.gca().add_artist(sphere) 

plt.plot([-R*1.1,R*1.1], [0,0])
plt.text(0.04, 0, 'O', horizontalalignment='center', verticalalignment='bottom')
plt.text(-50, 0, 'projection plane', horizontalalignment='center', verticalalignment='bottom')
plt.ylim(-R*1.1,R*1.1)
plt.scatter(0,-R)
plt.text(0,-R*1.02, 'S', horizontalalignment='center', verticalalignment='top')
plt.scatter(0,0)

plt.plot([0, x], [0, z], label='diffracted wave vector') 
plt.scatter(x, z)
plt.text(x*1.04, z, 'P', horizontalalignment='center', verticalalignment='bottom')

plt.plot([x, 0], [z, -R], label='connection to south pole') 
plt.scatter(x_projeted, 0)
plt.text(x_projeted, -0.4, 'P\'', horizontalalignment='left', verticalalignment='top', )
plt.legend(loc='upper left');

projected x-coordinate is 64.87 degree

Some features of the stereographic projection:

We can represent plane normals and directions in the same projection.
The can read off the angles between the directions, because the angles are preserved in this projection. Possibly the most important feature of this projection.
The zone axis is always 90$^{\rm o} $ away from any plane normal to its zone.
All the planes normal to a particular zone will lay on a great circle (oval). The zone of the centrale pole is on the circumference of the whole projection.
The angle between two planes is the angle between their normals measured with the Wulff net.
We can add the symmetry elements of any particular crystal system.

2.10.6.2. Wulff Plot#

The result of the stereographic Projection of the holeprojeciton sphere is shown below. It is convenient to show the ** circles of the sphere** as a grid the Wulff Plot.

But first we define some helper functions.

## ## Some helper functions first
def circumcenter(a,b,c):
    ax, ay = a
    bx, by = b
    cx, cy = c
    d = 2 * (ax * (by - cy) + bx * (cy - ay) + cx * (ay - by))
    ux = ((ax * ax + ay * ay) * (by - cy) + (bx * bx + by * by) * (cy - ay) + (cx * cx + cy * cy) * (ay - by)) / d
    uy = ((ax * ax + ay * ay) * (cx - bx) + (bx * bx + by * by) * (ax - cx) + (cx * cx + cy * cy) * (bx - ax)) / d
    return (ux, uy)

def wulff_net(ax, density=10):
    
    outer_ring = plt.Circle(( 0. , 0. ), 90 , fill=False, linewidth=2) 

    ax.set_aspect( 'equal') 
    ax.add_artist( outer_ring ) 
    ax.spines['left'].set_position(('data', 0))
    ax.spines['bottom'].set_position(('data', 0))
    ax.spines['top'].set_visible(False)
    ax.spines['right'].set_visible(False)
    
    ax.set_xlim(-94,94)
    ax.set_ylim(-94,94)
    
    for phi in range(density,90,density):
        phi_r = np.radians(phi)
        x,y = 90*np.sin(phi_r), 90*np.cos(phi_r)
        u,v = circumcenter([x,y],[-x,y],[0,90-phi])
        theta = np.degrees(np.arctan2( y-v, x-u))
        ax.add_patch(patches.Arc((0. , v), (v-90+phi)*2, (v-90+phi)*2, fill=False, edgecolor = 'gray', linewidth=.5, theta1=180-theta , theta2=theta))
        ax.add_patch(patches.Arc((0. , -v), (v-90+phi)*2, (v-90+phi)*2, fill=False, edgecolor = 'gray', linewidth=.5, theta1=-theta , theta2=180+theta))
        u,v = circumcenter([0,90],[0,-90],[phi,0])
        theta = np.degrees(np.arctan2(u, phi))
        radius = np.sqrt(u**2+ 90**2)
        
        theta = np.degrees(np.arctan2(  90, u))
        radius = np.abs(u-phi)
        ax.add_patch(patches.Arc((u, 0), radius*2, radius*2, fill=False, edgecolor = 'gray', linewidth=.5, theta1=180+theta , theta2=-180-theta))
        ax.add_patch(patches.Arc((-u, 0), radius*2, radius*2, fill=False, edgecolor = 'gray', linewidth=.5, theta1=theta , theta2=-theta))

    
def add_main_circles(ax):
    phi = 45
    phi_r = np.radians(phi)
    x,y = 90*np.sin(phi_r), 90*np.cos(phi_r)

    ax.plot([x,-x], [y,-y], color='blue')
    ax.plot([x,-x], [-y,y], color='blue')

    phi_r = np.radians(phi)
    x,y = 90*np.sin(phi_r), 90*np.cos(phi_r)
    u,v = circumcenter([x,y],[-x,y],[0,90-phi])
    theta = np.degrees(np.arctan2( y-v, x-u))
    ax.add_patch(patches.Arc((0., 90), np.sqrt(2)*180, np.sqrt(2)*180, fill=False, edgecolor = 'blue', linewidth=1, theta1=180+45 , theta2=-45))
    ax.add_patch(patches.Arc((0., -90), np.sqrt(2)*180, np.sqrt(2)*180, fill=False, edgecolor = 'blue', linewidth=1, theta1=45, theta2=180-45))
    ax.add_patch(patches.Arc((90., 0), np.sqrt(2)*180, np.sqrt(2)*180, fill=False, edgecolor = 'blue', linewidth=1, theta1=180-45, theta2=180+45))
    ax.add_patch(patches.Arc((-90., 0), np.sqrt(2)*180, np.sqrt(2)*180, fill=False, edgecolor = 'blue', linewidth=1, theta1=-45 , theta2=45))

Again change the Miller indices around to see where the reflections lands.

# ------Input ----------
reflection1 = np.array([1, 0, 0])
reflection2 = np.array([2, 1, 1])

# -----------------------
R = 90 # 90 degrees Ewald sphere
projected1 =(reflection1/np.linalg.norm(reflection1)*R) # Coordinates on ewals sphere surface
projected1 = projected1*R/(R+projected1[2]) # x coordinate on stereographic projection plane
projected2 =(reflection2/np.linalg.norm(reflection2)*R) # Coordinates on ewals sphere surface
projected2 = projected2*R/(R+projected2[2]) # x coordinate on stereographic projection plane

if projected1[2]>=0:
    print(f'Projected coordinates: {projected1[:2]}')
else:
    print('negative l Miller index is not supported')
plt.figure()
plt.title(f'Stereographic Projection of {reflection1} and {reflection2}')
wulff_net(plt.gca(), density=10)
add_main_circles(plt.gca())
if projected1[2]>=0:
    plt.scatter(projected1[0], projected1[1], color='red')
if projected2[2]>=0:
    plt.scatter(projected2[0], projected2[1], color='orange')

Projected coordinates: [90.  0.]

2.10.6.3. Cubic Crystal Reflections#

change the maximum Miller index around to see what happens

# ------Input ----------
hkl_max = 7
# -----------------------
h  = np.linspace(-hkl_max,hkl_max,2*hkl_max+1)  # all evaluated single Miller Indices
hkl  = np.array(list(itertools.product(h,h,h)), dtype=int) # all evaluated Miller indices
zero = np.where(np.linalg.norm(hkl)==0)
R = 90 # 90 degrees projection sphere

projected = []
reflections = []
for reflection in hkl:
    if reflection[2]>=0:
        if np.linalg.norm(reflection) >0:
            p = reflection/np.linalg.norm(reflection)*R# Coordinates on sphere surface
            projected.append(p*R/(R+p[2])) # x coordinate on stereographic projection plane
            reflections.append(reflection)
projected = np.array(projected)
reflections = np.array(reflections, dtype=int)

plt.figure()
plt.title(f'Stereographic Projection of hkl up to [{hkl_max}{hkl_max}{hkl_max}]' )
wulff_net(plt.gca(), density=10)
add_main_circles(plt.gca())
color=['orange', 'green', 'red'] + ['blue']*200
for index, spot in enumerate(projected):
    color_index = int(np.abs(reflections[index]).sum()-1)
    plt.scatter(spot[0], spot[1], color=color[color_index])
    if color_index<3:
        plt.text(spot[0], spot[1], f'{reflections[index]}',  horizontalalignment='left', verticalalignment='top')

2.10.6.4. Stereographic Projections for Any Symmertry and Any Orientation#

We did already all the work in the earlier notebooks and now we can just plot those the projections of the allowed $\vec{g}$ vectors.

The stereographic projection is after all only a projection of allowed reflections.

Use a high hkl_max parameter (about 15) and you start seeing the Kikuchi bands (next notebook)

Also see whether you can detect the 3-fold symmetry in [111] zone axis.

# ---Input ---------
hkl_max = 8
zone_axis = [0,0,1]
# ------------------

#Initialize the dictionary of the input
tags = {}
### Define Crystal
atoms = ks.structure_by_name('silicon')

# add necessary parameters for kinematic scattering calculation
tags = {}
### Define experimental parameters:
tags['acceleration_voltage_V'] = 200.0 *1000.0 #V
tags['new_figure'] = False
tags['plot FOV'] = 30
tags['convergence_angle_mrad'] = 0
tags['zone_hkl'] = np.array(zone_axis)  # incident neares zone axis: defines Laue Zones!!!!
tags['mistilt']  = np.array([0,0,0])  # mistilt in degrees
tags['Sg_max'] = 20 # 1/nm  maximum allowed excitation error ; This parameter is related to the thickness
tags['hkl_max'] = hkl_max   # Highest evaluated Miller indices

atoms.info['experimental'] = tags
######################################
# Diffraction Simulation of Crystal #
######################################

 
ks.kinematic_scattering(atoms, verbose=False)

hkl = atoms.info['diffraction']['allowed']['g'][atoms.info['diffraction']['allowed']['g'][:,2]>=0]

projected = []
reflections = []
for reflection in hkl:
    p = reflection/np.linalg.norm(reflection)*R# Coordinates on sphere surface
    projected.append(p*R/(R+p[2])) # x coordinate on stereographic projection plane
    reflections.append(reflection)
projected = np.array(projected)
reflections = np.array(reflections, dtype=int)

plt.figure()
plt.title(f'Stereographic Projection of hkl up to [{hkl_max}{hkl_max}{hkl_max}]' )
wulff_net(plt.gca(), density=10)
# add_main_circles(plt.gca())
color=['orange', 'green', 'red'] + ['blue']*100
alpha = [1, 1, 1] + [0.2]*100
for index, spot in enumerate(projected):
    color_index = int(np.abs(reflections[index]).sum()-1)
    plt.scatter(spot[0], spot[1], color=color[color_index], alpha = alpha[color_index])
    if color_index<3:
        #plt.text(spot[0], spot[1], f'{reflections[index]}',  horizontalalignment='left', verticalalignment='top')
        pass

2.10.6.5. Just a Pretty Plot#

def add_main_planes(ax):
    ax.scatter(0, 0, color='blue', s=50)
    ax.text(0, 0, '[001]', horizontalalignment='left',verticalalignment='top')
    ax.scatter(-90, 0, color='blue', s=50)
    ax.text(-90, 0, r'[0$\bar{1}$0]', horizontalalignment='left',verticalalignment='top')
    ax.scatter(90, 0, color='blue', s=50)
    ax.text(90, 0, '[010]', horizontalalignment='left',verticalalignment='top')
    ax.scatter(0, 90, color='blue', s=50)
    ax.text(0, 90, r'[$\bar{1}$00]', horizontalalignment='left', verticalalignment='top')
    ax.scatter(0, -90, color='blue', s=50)
    ax.text(0, -90, '[100]', horizontalalignment='left', verticalalignment='top')
    
    phi_r = np.radians(45)
    r = 46.6# 1/np.tan(phi_r/2)*20
    x,y = r*np.sin(phi_r), r*np.cos(phi_r)
    ax.scatter(x,-y, color='red', s=50)
    ax.text(x,-y, '[111]', horizontalalignment='left',verticalalignment='top')
    ax.scatter(x,y, color='red', s=50)
    ax.text(x,y, r'[$\bar{1}$11]', horizontalalignment='left',verticalalignment='top')
    ax.scatter(-x,y, color='red', s=50)
    ax.text(-x,y, r'[$\bar{1}\bar{1}$1]', horizontalalignment='left',verticalalignment='top')
    ax.scatter(-x,-y, color='red', s=50)
    ax.text(-x,-y, r'[1,$\bar{1}$,1]', horizontalalignment='left',verticalalignment='top')

    ax.scatter(37.2, 0, color='green', s=50)
    ax.text(37.2, 0, '[011]', horizontalalignment='left',verticalalignment='top')
    ax.scatter(-37.2, 0, color='green', s=50)
    ax.text(-37.2, 0, r'[0$\bar{1}$1]', horizontalalignment='left',verticalalignment='top')
    ax.scatter(0, -37.2, color='green', s=50)
    ax.text(0, -37.2, r'[101]', horizontalalignment='left',verticalalignment='top')
    ax.scatter(0, 37.2, color='green', s=50)
    ax.text(0,37.2 , r'[$\bar{1}$01]', horizontalalignment='left',verticalalignment='top')
    
    phi = 45
    phi_r = np.radians(phi)
    x,y = 90*np.sin(phi_r), 90*np.cos(phi_r)

    ax.scatter(-x, -y, color='green', s=50)
    ax.text(-x, -y, '[110]', horizontalalignment='left',verticalalignment='top')
    ax.scatter(x,-y, color='green', s=50)
    ax.text(x,-y, r'[1$\bar{1}$0]', horizontalalignment='left',verticalalignment='top')
    ax.scatter(x,y, color='green', s=50)
    ax.text(x,y, r'[$\bar{1}$10]', horizontalalignment='left',verticalalignment='top')
    ax.scatter(-x,y, color='green', s=50)
    ax.text(-x,y, r'[$\bar{1}\bar{1}$0]', horizontalalignment='left',verticalalignment='top')

plt.figure(figsize=(5, 5))

wulff_net(plt.gca(), density=10)
add_main_planes(plt.gca())
add_main_circles(plt.gca())

2.10.7. Summary#

Lot’s of information can be gained with basic crystallogrpahy tools and trigonometry.