Laue Simulator & Indexer — Help & Theory

Core Functionality

Simulation: Define a lattice and detector to predict a Laue pattern (virtual goniometer + polychromatic beam).
Indexing: Load a real image, pick peaks, and solve the unknown crystal orientation; then simulate and overlay spots for verification.
Orientation: Calculate the goniometer angles required to geometrically align a specific (h,k,l) plane normal with the detector normal.

Two Orientations: how rotations are applied

Base Orientation (unknown): The sample’s as-mounted orientation found by the search. Selecting a solution sets this as the internal base matrix.

Goniometer Rotation (knobs): The X,Y,Z sliders in the Set tab. They rotate on top of the Base Orientation and also respond to canvas drag.

Total rotation used in simulation = Base then Goniometer (XYZ Euler). After you choose a solution, the sliders are reset to 0 so the display shows the solved orientation. Moving a slider then offsets from that base.

Typical workflow (indexing)

Analysis → Load Image. If spots are white on black, check Invert Image. Toggle Show Spots to view overlays.
1. Set Center. Click the beam center. This enables Set Scale and shows a blue reference crosshair.
Set Scale. Drag the blue crosshair to a known reference. Enter its U (mm), V (mm) from the center. (The app uses separate px/mm for U and V; V is inverted screen-to-mm.)
Experimental Peaks. Use Find Peaks or Add Peak. Prefer 3–5 peaks of high symmetry far apart and across quadrants. You can refine (U, V) values directly in the list.
Set. Choose Bravais type, enter lattice parameters, wavelength band. Set the Detector Angle (0°=transmission, 180°=back-scatter) and Distance (mm) exactly as in the experiment.
Search Orientation. Wait for results; then open Solutions, click the best row (lowest Score, best Matches/RMS). The view updates with a yellow overlay.
(Optional) Orient tab. Enter target (h,k,l) and click Calculate Goniometer Angles. The tool calculates the X/Y/Z sliders needed to geometrically align the (h,k,l) plane normal to be parallel with the detector normal (based on the current Detector Angle).
Generate Report. Saves a PDF with parameters, peak table, best solution matches (assigned hkl), and a summary of all solutions.

UI reference (tabs)

Set

Lattice & parameters. Bravais choice enables the right fields (e.g., only a for cubic; a,c for tetragonal; monoclinic fixes α=γ=90°, etc.).
X-ray Source (Laue). λ_min–λ_max (Å) define the polychromatic band.
Detector Geometry. Angle (°) defines the detector plane's orientation relative to the beam (0° is transmission, 180° is back-scattering). Dist. (mm) is the distance from the sample to the detector plane's center.
Goniometer Rotation (°). X/Y/Z act relative to the chosen Base Orientation; dragging the canvas adjusts Y/Z.

Analysis

Set Center → Set Scale. After center, enter a second point’s real coordinates (U, V) to establish px→mm.
Experimental Peaks. Automatic finder prefers bright, well-separated spots; manual picks allow precise control. Edit (U, V) values inline; click a row label to select and drag on the image.

Solutions

Rows show Score (lower is better), Matches, RMS (mm), and Rot (X,Y,Z). Multiple nearly identical rows are symmetry-equivalent orientations—choose the best group’s top row.

Orient

Computes goniometer angles to geometrically align the chosen (h,k,l) plane normal (its reciprocal vector g) to be parallel with the detector's normal vector.
• This calculation is dynamic: it reads the Detector Angle from the 'Set' tab and uses its normal as the alignment target.
• Note: This is a purely geometric alignment. A diffraction spot will generally not appear at the center, as this function no longer solves for the Bragg condition.
• This calculation is valid for all angles from 0° (aligns g with the beam) to 180° (aligns g anti-parallel to the beam).

Mathematical model

1) Unit cell → reciprocal basis

Given cell a,b,c and angles α,β,γ, direct vectors are

// direct basis (Å)
a = (a, 0, 0)
b = (b cosγ, b sinγ, 0)
c = (c cosβ, c (cosα − cosβ cosγ)/sinγ, V/(a b sinγ))

V = a b c √(1 − cos²α − cos²β − cos²γ + 2 cosα cosβ cosγ)

Reciprocal basis vectors (Å⁻¹) are

a* = (b × c)/V ,  b* = (c × a)/V ,  c* = (a × b)/V

Any reflection (h,k,l) maps to g₀ = h a* + k b* + l c*.

2) Systematic absences (Bravais rules)

Type	Condition
P	all (h,k,l) allowed
I	`h + k + l` even
F	h, k, l all even or all odd
C	`h + k` even
R	`−h + k + l ≡ 0 (mod 3)`

3) Orientation stacking

Reflections are rotated first by the Base Orientation U (from the solver), then by the goniometer Euler rotation R(X,Y,Z):

g = R(X,Y,Z) · U · g₀

4) Laue condition, wavelength band, and detector hit

Beam points along +X. With unit wavevectors k̂_in = (1,0,0) and k̂_out, define the (normalized) scattering vector ĝ = k̂_out − k̂_in. For a given lattice vector g (Å⁻¹), elastic diffraction requires

2 k_in · g + |g|² = 0 ,  with k_in = (1/λ, 0, 0)

Solving for the required wavenumber gives

k = − |g|² / (2 gₓ)

The polychromatic band is k ∈ [1/λ_max, 1/λ_min]. If k falls outside, the spot is culled.

The diffracted ray is k_out = g + (k,0,0). The detector plane is defined by its angle φ (0° to 180°) and distance D.

// Detector plane normal and u-axis (horizontal)
n̂ = (cosφ, sinφ, 0)
û = (-sinφ, cosφ, 0)

The intersection P_hit is found by solving t for the ray P(t) = t · k_out and the plane P · n̂ = D.

This gives t = D / (k_out · n̂).
A hit is valid only if t > 0 (intersection is in front of the sample, pointing toward the detector).
The (U,V) coordinates on the detector are the projection of P_hit:

v = P_hit_z   // Vertical coordinate is the Z component
u = P_hit · û   // Horizontal coordinate
(x₂d, y₂d) = (u, v) // These are the plotted (mm) coordinates

5) Experimental geometry → scattering vectors

From a measured spot at detector coordinates (u_mm, v_mm) relative to the center, we reconstruct the 3D scattering vector.

// φ is the detector angle, D is distance
n̂ = (cosφ, sinφ, 0)
û = (-sinφ, cosφ, 0)
v̂ = (0, 0, 1)

// Reconstruct 3D spot position from (u,v)
P_center = D · n̂
P_hit = P_center + (u_mm · û) + (v_mm · v̂)

k̂_out = normalize( P_hit )
ĝ_exp = normalize( k̂_out − (1,0,0) )  // used for orientation solving

6) Pixel ↔ millimeter conversion

After you click the beam center (pixels) and place a scale point with known (mm), the app sets independent scale factors for U and V:

pixels_per_mm_x = (x_px − x0_px) / U_mm
pixels_per_mm_y = −(y_px − y0_px) / V_mm   // minus because image Y is down

Orientation search (what “Search Orientation” does)

Gather experimental peaks ...
Build HKL library ...
Angle pairing: ...
Candidate orientation: ...
Score it: ...
Keep top-N ...

Note on high-index reflections: The search uses two different (h,k,l) ranges.

Seeding (Step 2-3): A small library (e.g., |h,k,l| ≤ 2) is used to quickly find pairs of angles that match your experimental data.
Scoring (Step 5): A large library (e.g., |h,k,l| ≤ 12) is used to generate a full simulation pattern to score the candidate orientation.

This means a high-index experimental spot (like a (3,1,0)) will not help seed the search, but it will be correctly identified and used to confirm and score a good solution. The search only requires at least one pair of your experimental spots to fall within the small "seeding" range.

This strategy is robust to small peak sets and naturally yields symmetry-related orientations (e.g., cubic equivalences).

Peak finder details

Grayscale image; optional inversion. Threshold at ~90% of the frame’s max intensity.
Local-max test in a small neighborhood; exclude a central disk (~5 mm radius).
Balanced selection: take strongest candidates per quadrant to seed indexing.
You can always refine by manual adding/dragging spots.

Report contents

Detector snapshot with overlays.
Simulation + analysis parameters (lattice, λ band, D, Angle, center/scale).
Experimental peak list in (U, V) mm (+ raw pixels).
Best solution details: each peak’s assigned (h,k,l), simulated (U, V) and error (mm).
All solutions summary.

Troubleshooting

No or poor solutions

Verify Bravais type and lattice parameters.
Check detector Distance and Angle (e.g., 0° for transmission, 180° for back-scatter).
Re-do center/scale carefully; a wrong scale ruins geometry.
Use clear, well-separated peaks (3+ across different quadrants). Try Invert Image if the finder misses peaks.

A spot doesn't appear after “Orient to (h,k,l)”

This is the expected behavior. The 'Orient' function now performs a purely geometric alignment, making the (h,k,l) plane normal parallel to the detector normal. It no longer calculates the angles required for a diffraction spot (the Bragg condition).
A spot will only appear at the center if this geometric alignment also happens to satisfy the Bragg condition for a wavelength in your source's range (which is only guaranteed at 180°).

Conventions & hints

Coordinate System: The incident beam is along the +X axis. The lab's vertical direction is +Z (mapping to the detector's +V coordinate). The detector plane's position is defined by its Angle (0° = transmission, plane at x = +D; 90° = side, plane at y = +D; 180° = back-scatter, plane at x = -D).
In the detector's (U, V) coordinate system in mm (handled by the scale).
The (U, V) coordinates can be manually edited (so it is the scale).
Dragging the canvas rotates Y/Z goniometer axes; use sliders for precise values.
Use a realistic λ band (e.g., 0.2–3.0 Å) and a correct D to reduce false solutions.