Reciprocal Cell Calculator

The reciprocal cell of a direct cell: a*, b*, c*, α*, β*, γ* and the reciprocal volume, together with both metric tensors. The reciprocal lattice is where diffraction happens — every reflection hkl is one of its points.

You supply: The six constants of the direct cell. No crystal system and no space group: the reciprocal cell is a function of those six numbers whatever the cell is called.
Reading it: This page uses the crystallographic convention, a · a* = 1, so a* is 1/d(100) in Å⁻¹; the physics convention puts a 2π in and makes every reciprocal length larger by that factor. The starred angles are angles between plane normals, so α* is not in general 180° − α — that holds for the unique angle of a monoclinic cell and for nothing else.

Worked examples: NaCl – halite (rock salt) · Cu – copper, face-centred cubic · α-Fe – ferrite, body-centred cubic · CsCl – caesium chloride · quartz – α-quartz, SiO₂ · rutile – rutile, TiO₂ · aragonite – aragonite, CaCO₃ · ZrO₂ – baddeleyite, monoclinic zirconia · albite – low albite, NaAlSi₃O₈

Input

This page uses the crystallographic convention, a · a* = 1, in which a* is measured in Å⁻¹ and is directly the reciprocal of a spacing in Å. Solid-state physics more often writes a · a* = 2π, which makes every reciprocal length here larger by a factor of 2π and every reciprocal volume larger by (2π)³. The starred angles and the shape of the reciprocal lattice are the same in both.

Reciprocal cell

a*	0.23501 Å⁻¹
b*	0.23501 Å⁻¹
c*	0.185007 Å⁻¹
α*	90.000°
β*	90.000°
γ*	60.000°
Cell volume V	113.007 Å³
V*	0.00884898 Å⁻³

Each reciprocal axis is one over a d-spacing: a* is 1/d(100), b* is 1/d(010) and c* is 1/d(001) — the perpendicular spacings of the three families of planes the cell faces lie in. The starred angles are the angles between those plane normals, which is why α* is not in general 180° − α: that holds for the unique angle of a monoclinic cell and for nothing else.

Metric tensors

The metric tensor turns Miller indices into a d-spacing without any trigonometry: 1/d² is the row (h k l) times G* times the same column, which is exactly the sum the HKL calculator evaluates. The two are inverses of each other, G* = G⁻¹, and each is symmetric because a dot product does not care which vector comes first.

$G = (\begin{matrix} 24.1415 & -12.0707 & 0 \\ -12.0707 & 24.1415 & 0 \\ 0 & 0 & 29.2162 \end{matrix}) Å^{2}$

$G^{*} = (\begin{matrix} 0.0552299 & 0.027615 & 0 \\ 0.027615 & 0.0552299 & 0 \\ 0 & 0 & 0.0342276 \end{matrix}) Å^{- 2}$

How this is calculated

$V = a b c \sqrt{1 - \cos^{2} (α) - \cos^{2} (β) - \cos^{2} (γ) + 2 \cos (α) \cos (β) \cos (γ)}$

$V = 4.9134 Å \times 4.9134 Å \times 5.4052 Å \times \sqrt{0.75} = 113.007 Å^{3}$

Each reciprocal axis is one over the perpendicular spacing of the planes the other two axes lie in — so a* comes from b and c, and the other two follow by cycling a → b → c and α → β → γ together. $a^{*} = \frac{b c \sin (α)}{V}$

$a^{*} = \frac{4.9134 Å \times 5.4052 Å \times \sin (90 °)}{113.007 Å^{3}} = 0.23501 Å^{- 1}$

$b^{*} = \frac{5.4052 Å \times 4.9134 Å \times \sin (90 °)}{113.007 Å^{3}} = 0.23501 Å^{- 1}$

$c^{*} = \frac{4.9134 Å \times 4.9134 Å \times \sin (120 °)}{113.007 Å^{3}} = 0.185007 Å^{- 1}$

The starred angles are the angles between the reciprocal axes, which are the normals to the cell faces. They cycle the same way. $\cos (α^{*}) = \frac{\cos (β) \cos (γ) - \cos (α)}{\sin (β) \sin (γ)}$

$α^{*} = \arccos (\frac{\cos (90 °) \times \cos (120 °) - \cos (90 °)}{\sin (90 °) \times \sin (120 °)}) = 90.000 °$

$β^{*} = \arccos (\frac{\cos (120 °) \times \cos (90 °) - \cos (90 °)}{\sin (120 °) \times \sin (90 °)}) = 90.000 °$

$γ^{*} = \arccos (\frac{\cos (90 °) \times \cos (90 °) - \cos (120 °)}{\sin (90 °) \times \sin (90 °)}) = 60.000 °$

$V^{*} = \frac{1}{V} = \frac{1}{113.007 Å^{3}} = 0.00884898 Å^{- 3}$