Map reciprocal space from lattice inputs for clear first-zone boundaries and vectors. Export tables and summaries for plotting, validation, teaching, and band calculations today.
Choose general lattice for full a, b, c and angles, or cubic mode for quick band-structure setups.
Sample inputs and typical outputs for a cubic crystal with a = 5.43 A.
| Case | a (A) | Mode | |b1| (1/A) | BZ boundary |b1|/2 (1/A) | V* (1/A^3) |
|---|---|---|---|---|---|
| Silicon-like cubic | 5.43 | Cubic | ~ 1.157 | ~ 0.579 | ~ 1.548 |
Real cell volume for a general lattice is:
V = a b c * sqrt(1 + 2cos(alpha)cos(beta)cos(gamma) - cos^2(alpha) - cos^2(beta) - cos^2(gamma))
Reciprocal lattice vectors (2*pi convention) are:
b1 = 2*pi (a2 x a3)/V, b2 = 2*pi (a3 x a1)/V, b3 = 2*pi (a1 x a2)/V
Reciprocal lattice parameters are:
a* = 2*pi b c sin(alpha) / V, b* = 2*pi a c sin(beta) / V, c* = 2*pi a b sin(gamma) / V
Reciprocal volume is:
V* = (2*pi)^3 / V
First-zone boundary estimate along each direction uses:
k_boundary,i ~ |bi| / 2
A focused overview of Brillouin-zone metrics and how to use them in analysis workflows.
Brillouin zones are Wigner–Seitz cells of the reciprocal lattice. They organize electron and phonon wavevectors, making symmetry and periodicity explicit. The first zone encloses all distinct k states before repetition. Use zone geometry to compare materials, interpret diffraction, and choose efficient sampling meshes in practice.
This calculator builds reciprocal vectors with the 2*pi convention. If real-space lengths are entered in angstroms, outputs are in 1/angstrom. Vector lengths |b1|, |b2|, and |b3| set spacings in k-space and compare to scattering vectors in X-ray experiments.
Accurate reciprocal results require a reliable real-space volume. For triclinic cells the volume depends on all three angles through the standard determinant expression. Small rounding errors can inflate reciprocal parameters when the volume is tiny. The tool clamps trigonometric expressions and guards square roots to keep outputs stable for near-degenerate geometries.
A quick estimate of the nearest zone boundary along each reciprocal direction is |bi|/2. This is not the full polyhedral shape, but it is informative for scale setting. It helps decide plotting limits, phonon cutoff ranges, and whether a k-grid is fine enough to resolve narrow dispersion features.
In cubic quick mode, a single lattice constant defines geometry and enforces 90-degree angles. Simple cubic, body-centered cubic, and face-centered cubic are common reference cases in condensed matter. Although their real-space cells differ, the reciprocal lattices map between these types in well-known ways, which is why standard symmetry points are widely reused.
Band-structure plots usually follow paths connecting high-symmetry points such as G, X, M, R, L, W, K, H, N, and P. The included table provides these coordinates in 2*pi/a units for cubic systems. Use them to construct consistent k-paths, compare published results, and ensure that symmetry-related degeneracies appear where expected.
Density-functional calculations, tight-binding models, and lattice-dynamics codes need reciprocal parameters. Use the exported CSV to document inputs and reproduce runs, or attach the PDF summary to reports. Diffraction users can compare |bi| scales with experimental Q ranges and check detector feasibility.
Always confirm that the length unit label matches your model, especially when mixing angstrom and nanometer conventions. For general cells, verify angles are measured between the correct lattice vectors. As a sanity check, orthogonal cells should yield b1, b2, b3 aligned with axes and V* equal to (2*pi)^3/V.
It reports reciprocal vectors, their magnitudes, reciprocal parameters, and an approximate boundary distance of |bi|/2 along each direction. These values help set k-space scales, but they do not list the full polyhedral faces for arbitrary lattices.
The 2*pi convention is used, so b1, b2, and b3 satisfy ai·bj = 2*pi*delta_ij. If your software uses a different convention, rescale outputs accordingly before building k-grids or comparing literature tables.
Yes. Enter any positive lengths and set the unit label to nm or bohr. Outputs automatically show reciprocal units as 1/unit. Keep the same unit consistently across inputs to avoid hidden scaling errors.
Standard symmetry points depend on lattice type and conventional cell definitions. The calculator supplies widely used cubic coordinates in 2*pi/a units for sc, bcc, and fcc, which are common starting points for band-structure paths.
Recheck angles and lengths for typos and confirm the cell is physically valid. Near-degenerate cells amplify numerical noise in reciprocal parameters. If you truly need such geometry, increase precision and validate results against an independent tool.
Try an orthogonal cell with alpha=beta=gamma=90 degrees. The reciprocal vectors should align with axes, and magnitudes should match 2*pi/a, 2*pi/b, and 2*pi/c. Also confirm V* equals (2*pi)^3/V.
No. It provides rigorous reciprocal vectors and useful boundary scales, plus standard cubic k-points. For full zone polyhedra in low symmetry, use dedicated geometry libraries or crystallography packages and compare their face distances to |bi|/2.
Accurate reciprocal-space results help you plan simulations confidently now.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.