Reciprocal Lattice Vector Calculator
Compute reciprocal lattice vectors a*, b*, c* and their magnitudes from any set of direct lattice vectors a, b, c.
🧭 What is a Reciprocal Lattice Vector Calculator?
The reciprocal lattice is a lattice constructed in Fourier (spatial-frequency) space from a crystal's direct lattice vectors a, b, c, using the relations a* = 2π(b×c)/V, b* = 2π(c×a)/V, and c* = 2π(a×b)/V, where V = a·(b×c) is the direct unit cell volume. This calculator computes the three reciprocal vectors, their magnitudes, and the direct cell volume from any set of three direct lattice vectors, cubic, orthorhombic, or fully non-orthogonal.
Crystallographers and condensed matter physicists work in reciprocal space constantly, because it is the natural coordinate system for diffraction. A diffraction peak at Miller indices (h,k,l) corresponds directly to the reciprocal lattice vector G = h·a* + k·b* + l·c*, and the Bragg diffraction condition is exactly the geometric statement that this vector's tip touches the Ewald sphere. A solid-state physics student computing a Brillouin zone starts from these same reciprocal vectors. A materials scientist analyzing a non-cubic crystal's diffraction pattern needs the full cross-product formula, not just the simple cubic shortcut.
A common point of confusion is the 2π factor: this calculator uses the physics (solid-state) convention, a* = 2π(b×c)/V, while many crystallography textbooks use the convention a* = (b×c)/V without the 2π. Both are correct within their own field, the difference is purely a scaling convention, not a physical disagreement. Another important property is that a* is always perpendicular to both b and c, a direct consequence of the cross product, regardless of how skewed the original direct lattice is.
This calculator is useful for solid-state physics and crystallography coursework, for computing reciprocal vectors of non-cubic crystals where the simple formulas don't apply, and as a companion to the Ewald Sphere Geometry Calculator and Bragg Law Calculator for full diffraction geometry analysis.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Simple Cubic Lattice
a = (5,0,0), b = (0,5,0), c = (0,0,5) Å, an orthogonal cubic lattice
Example 2 — Orthorhombic Lattice
a = (4,0,0), b = (0,5,0), c = (0,0,6) Å, three unequal orthogonal edges
Example 3 — Non-Orthogonal (Hexagonal-Like) Lattice
a = (3,0,0), b = (-1.5,2.598076211,0), c = (0,0,5) Å, a 120° in-plane angle
❓ Frequently Asked Questions
🔗 Related Calculators
What is the reciprocal lattice?
The reciprocal lattice is a lattice in Fourier (spatial-frequency) space, constructed from a crystal's direct lattice vectors a, b, c using a* = 2π(b×c)/V, b* = 2π(c×a)/V, c* = 2π(a×b)/V, where V = a·(b×c) is the direct cell volume. It is the natural coordinate system for describing X-ray and neutron diffraction.
How do you calculate the reciprocal lattice vector a*?
a* = 2π(b×c) / [a·(b×c)], where b×c is the cross product of the other two direct lattice vectors and a·(b×c) is the direct unit cell volume V. The vectors b* and c* follow the same pattern with the vectors cycled.
Why is a* perpendicular to b and c?
Because a* is built directly from the cross product b×c, and any cross product u×v is perpendicular to both u and v by definition. This orthogonality is a defining geometric property of reciprocal lattice vectors, regardless of how skewed or non-orthogonal the direct lattice itself is.
What units does the reciprocal lattice vector use?
Inverse length, angstroms to the power -1 (Å⁻¹) when the direct lattice vectors are given in angstroms. This reflects that reciprocal space describes spatial frequency, how rapidly a periodic pattern repeats per unit distance.
What does 'coplanar lattice vectors' mean and why is it an error?
If the three direct lattice vectors a, b, c all lie in the same plane, they cannot define a real three-dimensional unit cell, the volume V = a·(b×c) works out to exactly zero. Since every reciprocal vector formula divides by V, no reciprocal lattice can be constructed, so this calculator blocks that input with an explanatory error.
Is there a factor of 2π in the reciprocal lattice formula or not?
It depends on convention. The physics (solid-state) convention, used by this calculator, includes 2π: a* = 2π(b×c)/V. The crystallography convention commonly used in diffraction textbooks omits it: a* = (b×c)/V. Both are correct within their own field, just be consistent about which one you're using.
How does the reciprocal lattice relate to diffraction?
Every diffraction peak (h,k,l) corresponds to a specific reciprocal lattice vector G(hkl) = h·a* + k·b* + l·c*. The Bragg condition for diffraction is exactly the geometric statement that this reciprocal lattice point touches the Ewald sphere, connecting reciprocal lattice geometry directly to measurable diffraction angles.
What is the reciprocal lattice of a cubic lattice?
For a cubic direct lattice with edge length a, the reciprocal lattice is also cubic, with edge length 2π/a (in the 2π convention), and the reciprocal vectors point along the same x, y, z axes as the direct vectors. This is the simplest possible reciprocal lattice relationship.
How do I find the direct cell volume from this calculator?
The direct unit cell volume V = a·(b×c) is shown alongside the reciprocal vectors in the step-by-step working, no extra input needed, it is computed as part of finding the reciprocal lattice.
Can this calculator handle non-orthogonal (triclinic) lattices?
Yes. The cross-product formula works for any set of three linearly independent lattice vectors, cubic, orthorhombic, monoclinic, or fully triclinic, without any special-case formulas needed, unlike d-spacing calculations which require a different formula per crystal system.