Enter lattice data
The calculator keeps a single-column page flow, while the form itself uses three columns on large screens, two on medium screens, and one on mobile.
Example data table
| Material style | Lattice family | Cell parameters | Typical path | Use case |
|---|---|---|---|---|
| Graphitic layered solid | Hexagonal | a = 2.46 Å, c = 6.71 Å, γ = 120° | Γ-M-K-Γ-A-L-H-A-L-M-K-H | Band mapping across basal and c-axis directions |
| Rock-salt oxide | Face-centered cubic | a = 4.21 Å | Γ-X-W-K-Γ-L-U-W-L-K-U-X | Electronic screening for ionic crystals |
| Framework molecular crystal | Orthorhombic | a = 6.20 Å, b = 7.10 Å, c = 8.30 Å | Γ-X-S-Y-Γ-Z-U-R-T-Z-Y-T-U-X-S-R | Anisotropic transport comparison |
Formula used
The calculator builds the direct lattice vectors from the supplied cell lengths and angles, then converts them to reciprocal space. For a general cell, the direct basis is a1, a2, and a3, where a1 = (a, 0, 0) and the other two vectors are assembled from the angular relationships.
- Direct cell volume:
V = a1 · (a2 × a3) - Reciprocal vectors:
b1 = s(a2 × a3) / V,b2 = s(a3 × a1) / V,b3 = s(a1 × a2) / V - Scale factor
sis either1or2π, depending on your convention. - Cartesian k-point:
k = u·b1 + v·b2 + w·b3from fractional coordinates(u, v, w). - Segment length: Euclidean distance between consecutive cartesian k-points.
The path library uses common high-symmetry routes for cubic, tetragonal, orthorhombic, and hexagonal cells. In advanced workflows, exact paths can differ by convention, so this tool is best treated as a fast, transparent planning calculator.
How to use this calculator
- Choose the lattice family that best matches your crystal or teaching example.
- Enter cell lengths and angles. Preset families auto-align common constraints such as
a = borγ = 120°. - Set the number of points per segment to control interpolation density for plotting or export.
- Decide whether your reciprocal basis should include the
2πfactor. - Press Submit. The result block appears below the header and above the form.
- Review the summary, special k-points, segment lengths, and interpolated path rows.
- Use the CSV export for spreadsheet work and the PDF export for reports or lab notes.
Reciprocal Framework
Brillouin-zone path selection starts with a stable reciprocal basis. This calculator derives b1, b2, and b3 from the entered cell lengths and angles, then converts fractional symmetry points into cartesian k-space coordinates. For chemistry workflows, that matters because orbital overlap, anisotropy, and dispersion trends are all interpreted along these reciprocal directions.
Symmetry Point Coverage
The built-in point library supports simple cubic, face-centered cubic, body-centered cubic, tetragonal, orthorhombic, hexagonal, and custom primitive cells. Each family is paired with a practical high-symmetry route such as Γ-M-K-Γ-A-L-H for layered hexagonal systems. This lets researchers move from lattice geometry to a reviewable path without rebuilding coordinates manually.
Distance Interpretation
Segment length is computed from the Euclidean separation between consecutive cartesian k-points. The cumulative distance column then tracks progress along the entire route. That is useful when preparing band-structure plots, because x-axis spacing should reflect actual reciprocal travel rather than equally naming each label without geometric weighting.
Sampling Density Control
Points per segment determine interpolation density. Low values are useful for classroom demonstrations and rapid checks, while larger values provide smoother sampling for plotting, comparison studies, or export into simulation notes. When screening many compounds, consistent sampling density also improves side-by-side interpretation across related crystal families.
Convention Management
The option to include or exclude the 2π factor helps match differing conventions used in textbooks, electronic-structure codes, and laboratory reports. Keeping that choice visible reduces transcription errors. It also makes the calculator suitable for mixed teams where computational chemists, materials scientists, and students may follow different reciprocal-space notations.
Export-Ready Review
The summary table, special-point table, segment table, interpolated path table, and plotted trajectory give a compact audit trail for each calculation. In practice, this supports method appendices, teaching handouts, and pre-simulation checks. Example datasets further show how hexagonal, cubic, and orthorhombic cells map into usable reciprocal paths.
Because the calculator exposes both fractional and cartesian coordinates, users can validate whether published labels align with their chosen convention before exporting data. That verification step is valuable for chemistry projects involving doped phases, polymorph comparisons, adsorption studies, or phonon and electron analyses, where an incorrect path can misplace extrema, flatten trends, or complicate interpretation during collaborative review.
This improves reproducibility across iterative structure refinement and reporting.
FAQs
1. What does this calculator actually compute?
It converts direct-lattice inputs into reciprocal vectors, assigns built-in symmetry points for the selected cell family, interpolates a recommended k-path, and reports distances for plotting or export.
2. Why is the 2π option important?
Some texts and software define reciprocal vectors with 2π, while others omit it. Matching the correct convention prevents scaling errors when comparing tables, plots, and simulation outputs.
3. Can I use the custom cell for nonstandard chemistry problems?
Yes. The custom primitive option accepts general lengths and angles, making it useful for exploratory studies, distorted cells, and teaching cases where standard presets are too restrictive.
4. How many points per segment should I choose?
Use lower values for quick inspection and higher values for smoother plots. For many reporting tasks, 10 to 30 points per segment gives a practical balance.
5. Are these paths universal for every crystal study?
No. They are practical standard routes for common lattice families. Published studies or code-specific conventions may use alternative labels or extra segments for special cases.
6. What can I do with the CSV and PDF exports?
CSV is useful for spreadsheet review, custom plotting, and shared datasets. PDF works well for lab notes, method appendices, classroom material, and quick reporting snapshots.