Generating RHEED Patterns from Arbitrary Directions¶

RHEED patterns depend strongly on the crystallographic direction of the incident electron beam. This guide covers how to simulate RHEED from any azimuthal direction or surface orientation using rheedium.

Ewald sphere top view — Top-down view of the Ewald sphere showing how different azimuthal angles \(\phi\) change which reciprocal lattice rods are intersected. Rotating the beam direction probes different in-plane periodicities.¶

Beam Direction Parameters¶

The kinematic_simulator function accepts two angle parameters that control the beam direction:

Parameter	Description	Typical Range
`theta_deg`	Grazing incidence angle (from surface plane)	1-5 degrees
`phi_deg`	Azimuthal angle (in-plane rotation)	0-360 degrees

Geometry Definition¶

The coordinate system is defined as:

x-axis: Default beam direction when phi_deg=0
y-axis: Perpendicular to beam, in the surface plane
z-axis: Surface normal (pointing up)

from rheedium.plots import plot_grazing_incidence_geometry
import matplotlib.pyplot as plt

fig, ax = plt.subplots(figsize=(14, 8))
plot_grazing_incidence_geometry(
    theta_deg=2.0,
    ax=ax,
)
plt.savefig("grazing_geometry.png", dpi=150, bbox_inches="tight")
plt.show()

Grazing Incidence Geometry

The incident wavevector components are:

\[\begin{split} \mathbf{k}_{in} = k \begin{pmatrix} \cos\theta \cos\phi \\ \cos\theta \sin\phi \\ -\sin\theta \end{pmatrix} \end{split}\]

where \(k = 2\pi/\lambda\) is the wavevector magnitude.

Azimuthal Rotation (phi_deg)¶

Rotating the azimuthal angle changes which crystallographic direction the beam is aligned with.

Basic Usage¶

import rheedium as rh

crystal = rh.inout.parse_cif("structure.cif")

# Beam along [100] direction (phi = 0)
pattern_100 = rh.simul.kinematic_simulator(
    crystal=crystal,
    voltage_kv=20.0,
    theta_deg=2.0,
    phi_deg=0.0,
)

# Beam along [110] direction (phi = 45 for cubic)
pattern_110 = rh.simul.kinematic_simulator(
    crystal=crystal,
    voltage_kv=20.0,
    theta_deg=2.0,
    phi_deg=45.0,
)

# Beam along [010] direction (phi = 90)
pattern_010 = rh.simul.kinematic_simulator(
    crystal=crystal,
    voltage_kv=20.0,
    theta_deg=2.0,
    phi_deg=90.0,
)

Azimuthal Dependence¶

Different azimuths probe different in-plane periodicities:

Azimuth	Cubic Crystal	Pattern Characteristics
0 degrees	[100]	Vertical streaks from (h00) rods
45 degrees	[110]	Denser pattern from (hh0) rods
90 degrees	[010]	Same as 0 degrees (4-fold symmetry)

Azimuthal Scan¶

To study the full azimuthal dependence:

import jax.numpy as jnp

# Scan azimuthal angle
phi_values = jnp.linspace(0, 90, 19)  # 0 to 90 in 5-degree steps

patterns = []
for phi in phi_values:
    pattern = rh.simul.kinematic_simulator(
        crystal=crystal,
        voltage_kv=20.0,
        theta_deg=2.0,
        phi_deg=float(phi),
    )
    patterns.append(pattern)

Miller index planes in a cubic unit cell: (100), (110), and (111). Each surface orientation corresponds to a specific set of Miller indices defining the crystal plane parallel to the surface.¶

Non-(001) Surface Orientations¶

For surfaces other than (001), you need to reorient the crystal so the desired surface normal is along z.

Using bulk_to_slice¶

The bulk_to_slice function creates a surface slab with the specified Miller plane as the surface:

from rheedium.types import bulk_to_slice
import jax.numpy as jnp

# Load bulk crystal
bulk = rh.inout.parse_cif("structure.cif")

# Create (111)-oriented surface
slab_111 = bulk_to_slice(
    bulk_crystal=bulk,
    orientation=jnp.array([1, 1, 1]),
    depth=30.0,  # Slab thickness in Angstroms
)

# Simulate RHEED from (111) surface
pattern_111 = rh.simul.kinematic_simulator(
    crystal=slab_111,
    voltage_kv=20.0,
    theta_deg=2.0,
    phi_deg=0.0,
)

Common Surface Orientations¶

Surface	Miller Indices	Typical Applications
(001)	[0, 0, 1]	Default, most common
(111)	[1, 1, 1]	Close-packed planes in FCC
(110)	[1, 1, 0]	Open surfaces
(011)	[0, 1, 1]	Equivalent to (110) in cubic
(112)	[1, 1, 2]	Vicinal/stepped surfaces

Example: (110) Surface¶

# Create (110)-oriented surface slab
slab_110 = bulk_to_slice(
    bulk_crystal=bulk,
    orientation=jnp.array([1, 1, 0]),
    depth=25.0,
)

# RHEED along different azimuths on (110)
# phi=0 is along [001] in-plane direction
pattern_110_001 = rh.simul.kinematic_simulator(
    crystal=slab_110,
    voltage_kv=20.0,
    theta_deg=2.0,
    phi_deg=0.0,
)

# phi=90 is along [1-10] in-plane direction
pattern_110_1m10 = rh.simul.kinematic_simulator(
    crystal=slab_110,
    voltage_kv=20.0,
    theta_deg=2.0,
    phi_deg=90.0,
)

Combining Surface Orientation and Azimuth¶

For complete control, combine surface reorientation with azimuthal rotation:

import jax.numpy as jnp
from rheedium.types import bulk_to_slice

# 1. Create surface slab with desired orientation
bulk = rh.inout.parse_cif("SrTiO3.cif")
slab = bulk_to_slice(
    bulk_crystal=bulk,
    orientation=jnp.array([1, 1, 1]),  # (111) surface
    depth=20.0,
)

# 2. Simulate from multiple azimuths
azimuths = [0, 30, 60, 90]
patterns = {}

for phi in azimuths:
    patterns[phi] = rh.simul.kinematic_simulator(
        crystal=slab,
        voltage_kv=20.0,
        theta_deg=2.0,
        phi_deg=float(phi),
    )

Grazing Angle Effects¶

The grazing angle theta_deg affects:

Penetration depth: Lower angles = more surface sensitive
Ewald sphere intersection: Changes which reflections are excited
Pattern geometry: Affects streak spacing and curvature

The Ewald sphere construction for RHEED geometry:

from rheedium.plots import plot_ewald_sphere_2d
import matplotlib.pyplot as plt

fig, ax = plt.subplots(figsize=(12, 8))
plot_ewald_sphere_2d(
    voltage_kv=20.0,
    theta_deg=2.0,
    lattice_spacing=3.905,
    n_rods=9,
    ax=ax,
)
plt.savefig("ewald_sphere.png", dpi=150, bbox_inches="tight")
plt.show()

Ewald Sphere 2D

Typical Values¶

Material Type	Recommended theta_deg
Metals	1-2 degrees
Oxides	2-3 degrees
Semiconductors	2-4 degrees

Angle Scan¶

# Study grazing angle dependence
theta_values = [1.0, 1.5, 2.0, 2.5, 3.0]

for theta in theta_values:
    pattern = rh.simul.kinematic_simulator(
        crystal=crystal,
        voltage_kv=20.0,
        theta_deg=theta,
        phi_deg=0.0,
    )
    rh.plots.plot_rheed(pattern)

Visualization¶

The plot_rheed function renders the simulated pattern:

# Default visualization with Gaussian spot broadening
rh.plots.plot_rheed(pattern)

# Customize appearance
rh.plots.plot_rheed(
    pattern,
    grid_size=400,           # Higher resolution
    spot_width=0.05,         # Sharper spots
    cmap_name="phosphor",    # Phosphor screen colormap
)

Comparing Multiple Azimuths¶

import matplotlib.pyplot as plt

fig, axes = plt.subplots(1, 3, figsize=(15, 6))

for ax, phi in zip(axes, [0, 45, 90]):
    pattern = rh.simul.kinematic_simulator(
        crystal=crystal,
        voltage_kv=20.0,
        theta_deg=2.0,
        phi_deg=phi,
    )
    # Use lower-level plotting for subplots
    coords = np.asarray(pattern.detector_points)
    intensities = np.asarray(pattern.intensities)
    ax.scatter(coords[:, 0], coords[:, 1], c=intensities, s=10)
    ax.set_title(f"phi = {phi} degrees")
    ax.set_xlabel("x_d (mm)")
    ax.set_ylabel("y_d (mm)")

plt.tight_layout()
plt.show()

3D Ewald sphere perspective — 3D perspective view of the Ewald sphere with reciprocal lattice rods. The combination of surface orientation and azimuthal angle determines which rods the sphere intersects and thus which reflections appear in the RHEED pattern.¶

Physical Considerations¶

Symmetry¶

The pattern symmetry depends on both surface and azimuth:

(001) surface, phi=0: Pattern has vertical mirror plane
(001) surface, phi=45: Different symmetry for [110] azimuth
(111) surface: 3-fold or 6-fold symmetry depending on termination

Zone Axis Alignment¶

For strong diffraction features, align the beam along a low-index zone axis:

Surface	Strong Azimuths
(001) cubic	0 degrees ([100]), 45 degrees ([110])
(111) cubic	0, 60, 120 degrees (equivalent)
(110) cubic	0 degrees ([001]), 90 degrees ([1-10])

Reconstruction Effects¶

Surface reconstructions change the in-plane periodicity:

# For a 2x1 reconstruction, you might see:
# - Extra streaks at half-integer positions
# - Different intensity along different azimuths
#
# The simulation uses bulk-terminated positions;
# for reconstructions, modify the surface layer positions
# before simulation.

Complete Workflow Example¶

import rheedium as rh
import jax.numpy as jnp
from rheedium.types import bulk_to_slice, SurfaceConfig

# 1. Load structure
bulk = rh.inout.parse_cif("GaAs.cif")

# 2. Create (110) surface slab
slab = bulk_to_slice(
    bulk_crystal=bulk,
    orientation=jnp.array([1, 1, 0]),
    depth=25.0,
)

# 3. Configure surface atom identification
surface_config = SurfaceConfig(method="layers", n_layers=2)

# 4. Simulate RHEED along [001] azimuth
pattern = rh.simul.kinematic_simulator(
    crystal=slab,
    voltage_kv=15.0,
    theta_deg=2.5,
    phi_deg=0.0,  # [001] in-plane direction
    surface_config=surface_config,
    temperature=300.0,
    surface_roughness=0.5,
)

# 5. Visualize
rh.plots.plot_rheed(pattern, grid_size=300)

Quick Reference¶

Goal	Parameters
Change beam azimuth	`phi_deg=45.0`
Change grazing angle	`theta_deg=2.5`
(111) surface	`bulk_to_slice(..., orientation=[1,1,1])`
(110) surface	`bulk_to_slice(..., orientation=[1,1,0])`
Azimuthal scan	Loop over `phi_deg` values

Key Source Files¶

simul/simulator.py - kinematic_simulator with angle parameters
simul/simul_utils.py - incident_wavevector calculation
types/crystal_types.py - bulk_to_slice for surface reorientation
plots/figuring.py - plot_rheed visualization