HKL Calculator

https://xraytools.com/pxrdcalc

For a unit cell and a Miller index: the cell volume, the d-spacing of that reflection, the angle it appears at, the reflections symmetry makes equivalent to it, and the powder pattern the whole cell would give.

Before this: A Miller index is the label for one family of parallel lattice planes, and every answer here is about that family rather than about an atom. Bragg’s law is where the spacing turns into an angle.
You supply: A crystal system, the cell parameters that system leaves free, a wavelength — or an anode preset — and h k l.
Reading it: Results update as you type. The equivalent reflections assume the holohedral Laue class, and the panel beneath them says what that costs a structure of lower symmetry. Stick heights are multiplicity for a cell with no atoms in it, and real relative intensities once there are some — from a named structure, from an atom list you paste in, or from a CIF the parser has read. The chart says which.

Worked examples: cubic – NaCl, Fm3m (200) · cubic – Cu, Fm3m (111) · cubic – α-Fe, Im3m (110) · cubic – CsCl, Pm3m (110) · cubic – ZnS, F43m (111) · hexagonal – quartz, P3₂21 (101) · tetragonal – cristobalite, P4₁2₁2 (101) · hexagonal – berlinite, P3₁21 (101) · tetragonal – rutile, P4₂/mnm (110) · orthorhombic – aragonite, Pmcn (111) · monoclinic – ZrO₂, P2₁/c (111) · triclinic – albite, C1 (111) · tetragonal – urea, P42₁m (110)

Earlier on the path: Bragg Calculator Next on the path: Reciprocal Cell Calculator On From planes to a powder pattern, step 3 of 7

Notation here: d · θ, 2θ · (hkl) · {hkl}, ⟨uvw⟩ · I — what each one means here

Terms here: asymmetric unit · centring · point group · holohedry · Laue class · multiplicity · setting

What each input changes

Reflection: Which family of planes. Larger indices mean closer planes, so a smaller d and a larger angle — and past a point, no angle at all, because sin θ cannot exceed 1.
Crystal system: How many cell constants are independent, and therefore which reflections symmetry makes equivalent to the one you asked about. It changes the multiplicity, which changes peak heights.
Lattice centring: Which reflections are systematically absent by the LATTICE alone. It removes lines from the pattern; it never moves one.
Space group: Optional, and it supersedes the radios above: name a group and the screw-axis and glide absences are applied as well as the centring, with the lattice taken from the symbol. A group the Tables give at two origins takes a suffix, and which one you want is decided by the coordinates you are pasting.

Teaching with this page

Objective: After this page a learner can read a Miller index, find the reflections symmetry makes equivalent to it, and predict where the line appears.
Start from: this worked example
Ask first: A powder pattern shows one maximum at 43.3°. How many reflections produced it?
Watch for: “One — one maximum, one reflection”
Then: Reciprocal Cell Calculator

Check yourself: A powder pattern shows one maximum at 43.3°. How many reflections produced it?

Cannot be told from the position alone One — one maximum, one reflection

Any reflections with the same d land at the same angle and add together. In a cubic cell (333) and (511) have identical spacings, so they are one peak and always will be — no instrument separates them. A reflection is a contribution indexed by hkl; a peak is what the pattern shows. The equivalents this page lists for your index are exactly the ones that will arrive together.

Result

Results

a =	5.640 Å
b =	5.640 Å
c =	5.640 Å
α =	90°
β =	90°
γ =	90°
V =	179.4 Å³
d =	2.820 Å
2θ =	31.73°

The cell, and the plane

a1b1c1 axesdrag to rotate · scroll to zoomLook along

200 cuts the axes at a at 0.5, parallel to b, parallel to c.

The spheres are lattice points, not atoms. A lattice point is a position the whole structure repeats from; how many atoms sit on each one, and where, is the basis and is not shown here. NaCl and copper have the same face‑centred lattice and quite different structures.

Powder pattern

23 reflections between 5° and 90° 2θ, lattice P (no reflection condition). Hover a stick to identify it. Click one to select it, or drag across the chart to zoom into that range.

200 is the highlighted stick.

The first 12 of 23 reflections, lowest angle first. Click a row to select its reflection.
2θ (°)	d (Å)	hkl	multiplicity
15.71	5.6400	100	6
22.29	3.9881	110	12
27.39	3.2563	111	8
31.73	2.8200	200	6
35.59	2.5223	210	24
39.12	2.3025	211	24
45.49	1.9940	220	12
48.42	1.8800	300 (6) + 221 (24)	30
51.22	1.7835	310	24
53.92	1.7005	311	24
56.52	1.6281	222	8
59.05	1.5643	320	24

The two numeric columns are fixed at two and four decimal places so the points line up and the rows can be compared down the column — that is a format, not a claim about how well the spacings are known. The d and 2θ in the results above are rounded to the figures your cell constants carry, which is the site's answer for the selected reflection.

Stick heights are multiplicity only — how many symmetry-equivalent reflections coincide at that angle. They are not intensities: there is no structure factor, no Lorentz–polarisation factor and no temperature factor here, so a measured pattern will have the peaks in these positions with quite different heights. Positions and absences are the part to read. Load one of the named structures above, or give an atom list, to get real intensities.

Whole classes of reflection can be systematically absent, and there are three kinds. Integral absences come from the lattice centring and affect every hkl. Zonal absences come from glide planes and empty one plane of the pattern — 0kl, h0l, hk0 or hhl. Serial absences come from screw axes and empty one row — h00, 0k0 or 00l. You have named a lattice rather than a space group, so only the first kind is applied here: for a structure with a screw axis or a glide plane this chart draws some reflections the crystal does not produce, and they are the ones in those rows and planes. Name a space group above and all three are applied.

Why

How this is calculated

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

$V = 5.640 Å \times 5.640 Å \times 5.640 Å \times \sqrt{1} = 179.4 Å^{3}$

For a cubic cell (a = b = c, α = β = γ = 90°) the general form reduces to $\frac{1}{d^{2}} = \frac{h^{2} + k^{2} + l^{2}}{a^{2}}$

$\frac{1}{d^{2}} = \frac{2^{2} + 0^{2} + 0^{2}}{{(5.640 Å)}^{2}} = 0.125748 Å^{- 2}$

$d = \frac{1}{\sqrt{0.125748 Å^{- 2}}} = 2.820 Å$

λ = 1.541838 Å is Cu Kα:

$2 θ = 2 \times \arcsin (\frac{λ}{2 \times d}) = 2 \times \arcsin (\frac{1.541838 Å}{2 \times 2.820 Å}) = 31.73 °$

Volume and d are quoted to 4 significant figures — the fewest that a, b and c carry — and 2θ to 4, which counts the wavelength as well.

The general form, and what this cell strikes out

$σ = S_{11} h^{2} + S_{22} k^{2} + S_{33} l^{2} + 2 S_{12} h k + 2 S_{23} k l + 2 S_{13} h l$

$\frac{1}{d^{2}} = \frac{σ}{V^{2}}$

S₁₁ = b²c² sin²α, S₂₂ = a²c² sin²β, S₃₃ = a²b² sin²γ, S₁₂ = abc²(cos α cos β − cos γ), S₂₃ = a²bc(cos β cos γ − cos α), S₁₃ = ab²c(cos γ cos α − cos β). This cell strikes out 3 of the six. A struck term is zero for every reflection in this crystal system, which is exactly what having fewer free cell parameters buys you.

equivalent reflections: 6

200020002002020200

A diffraction pattern is centrosymmetric whether or not the crystal is — I(hkl) = I(−h−k−l), which is Friedel’s law — so a reflection and its opposite always appear together in this list. It follows that reflection positions and intensities can tell apart only 11 of the 32 crystal classes, the Laue classes. This form asks for a crystal system, which does not fix the point group, so the list assumes the holohedral — highest-symmetry — Laue class of the system you chose; a structure in a lower class has fewer equivalents than shown here, which does not mean fewer reflections at that angle. They still all arrive together — quartz’s (101) and (011) are not equivalent in 3̄m1 and coincide in d regardless — they simply no longer share one |F|, so a powder peak is the sum over them rather than one of them times a count. Friedel’s law is itself an approximation: resonant scattering breaks it slightly, which is what makes absolute structure determinable at all, and how much of that contrast a composition offers is what the Friedif calculator estimates.

Limits

What this calculation assumes

Which emission line the angles are computed at
Which classes of reflection the space group removed
The overall displacement parameter in force
Whether atom positions were read, and what happens without them
What the simulated intensities include, and what they leave out
What counts as an equivalent reflection here

Every one of these is stated in full beside the panel it applies to. This list is here so you can see how many there are.

Where this comes from

Rietveld refinement of Debye-Scherrer synchrotron X-ray data from Al2O3
P. Thompson, D. E. Cox and J. B. Hastings, J. Appl. Cryst. 1987, 20, 79–83 · doi:10.1107/S0021889887087090
The pseudo-Voigt drawn when you ask for a profile, and the mixing parameter η that runs it from Gaussian to Lorentzian. The title is about a refinement; the peak shape is the part used here.
The Scherrer formula for X-ray particle size determination
A. L. Patterson, Phys. Rev. 1939, 56, 978–982 · doi:10.1103/PhysRev.56.978
Where the crystallite-size broadening comes from, derived rather than quoted — including what the shape constant K actually depends on, which is the part usually left out.