xraytools.

Bragg Calculator

https://xraytools.com/bragg

Powder or single crystal

Converts between the angle a reflection is observed at and the spacing of the lattice planes that produced it, through λ = 2d sin θ.

You supply
Either an angle (θ or 2θ) or a d-spacing, and the anodes you want it answered for.
Reading it
One row per anode, quoted to the figures you typed — never fewer than four significant figures and never more than the wavelength constant carries. n.a. means that radiation cannot reach the reflection at all: nothing with d < λ/2 diffracts.

Worked examples: NaCl (200) · quartz (101)

Earlier on the path: Lattice explorer Next on the path: HKL Calculator On From planes to a powder pattern, step 2 of 7

Notation here: d · θ, 2θ — what each one means here

See also: Lattice explorer · HKL Calculator · Peak Finding · Line Broadening · Powder Indexing · The Ewald Construction

What each input changes
Convert
Which direction the conversion runs. Everything else on the page is the same one relation; this only says which end you are holding.
Emission line
Which wavelength the angles are computed at. Kα1 moves every angle slightly against the Kα mean; Kβ moves them a long way, which is what an unfiltered pattern shows.
Teaching with this page
Objective
After this page a learner can convert between a measured angle and a lattice spacing in either direction, and say which quantity the instrument supplied.
Start from
this worked example
Ask first
A powder diffractometer reports a line at 31.7°. Is that the angle to put into Bragg’s law?
Watch for
“Yes — it is the angle the instrument measured”
Then
HKL Calculator
Check yourself: A powder diffractometer reports a line at 31.7°. Is that the angle to put into Bragg’s law?

Not as it stands — the instrument reports 2θ Yes — it is the angle the instrument measured

What a diffractometer reports is the angle between the incident and the diffracted beam, 2θ. Bragg’s law is written with θ, the angle each of those beams makes with the planes, so the reported figure is halved before it goes in. Halving twice, or not at all, moves every d by about a factor of two, and it is the commonest arithmetic mistake in the subject. This page offers both as separate modes rather than guessing which you have.

Input

X-ray sources
Emission line
Convert
°
°
Å
Å

Results

Mo: 0.8409 Å
Cu: 1.824 Å

Wavelengths are Kα.

How the reflection works

the plane spacing d = 0.8409 Ådthe upper ray, arriving at θ = 25° to the planesthe upper ray, leaving at θ = 25° to the planesthe lower ray, arriving at θ = 25° to the planesthe lower ray, leaving at θ = 25° to the planesA̅D̅ = d sin θ = 0.3554 Å of extra pathD̅C̅ = d sin θ = 0.3554 Å of extra patha wavefront: every point on it is the same distance along the beama wavefront: every point on it is the same distance along the beamB̅D̅ = d = 0.8409 Å, the side both triangles stand onBDa right angle: B̅A̅ meets A̅D̅ square, which is why the side opposite θ is d sin θa right angle: B̅C̅ meets C̅D̅ square, which is why the side opposite θ is d sin θwhere the upper ray reflectswhere the wavefront meets the lower raywhere the lower ray reflectswhere the wavefront meets it againABCDwhere the beam would have gone if the crystal were not thereθ = 25°, measured from the PLANE and not from its normalθ2θ = 50° — the angle a diffractometer reads2θAD + DC = 2d sin θ = 0.7107 Å

BA and BC are wavefronts: every point on one is the same distance along the beam, so the lower ray’s extra path is exactly what lies between them — AD on the way in and DC on the way out.

BD = d joins the two reflection points, and the wavefronts meet the lower ray square — the right angles are marked at A and at C. So ABD and CBD are right-angled triangles with BD as the hypotenuse and the angle at B equal to θ, which puts AD and DC opposite it: AD = DC = BD sin θ = d sin θ = 0.3554 Å. AD + DC = 2d sin θ = 0.7107 Å, and that is λ = 0.7107 Å. Setting the two equal is Bragg’s law with nothing in between: the waves come away in step here, and away from this angle they fall out of step. How fast they do so is what sets the width of the line — the drop is abrupt only for a perfect crystal of unlimited extent, which is what this figure draws.

The path difference above is one wavelength, and any whole number of them would do as well: the general law is nλ = 2d sin θ, with n the order of the reflection. Crystallography folds n away rather than carrying it: the nth order from (hkl) is written as the first order from (nh nk nl), whose spacing is d(hkl)/n. The second order from (100) is the reflection called 200. That is why every other page here writes λ = 2d sin θ with no n in it, and why a d-spacing always belongs to a named set of indices.

The angle, the plane spacing and the wavelength along each ray are to scale; nothing else in the figure is — the height of the wave stands for no length here. The atoms are drawn evenly along each plane at a spacing this page does not know and does not need — Bragg’s law says nothing about the arrangement within a plane, which is why it holds whatever that arrangement is. θ is measured from the plane and not from its normal, so the angle between the incoming and outgoing beams — the one a diffractometer reads — is 2θ.

Drawn for Mo Kα. Every other row is the same figure at that anode’s own angle.

How this is calculated

d=λ2×sin(θ)

d(Mo Kα)=0.710730Å2×sin(25°)=0.8409Å

Every other row is the same equation with that anode’s wavelength in place of this one. Results are quoted to the figures you typed — never fewer than four significant figures and never more than the wavelength constant carries.

Characteristic wavelengths
Every anode this page offers, with the line in use marked. CSV
Anode Energy Kα / keV Kα / Å in use Kα1 / Å Kα2 / Å Kβ / Å
Ag22.110.5608680.55940750.5637890.497069
Mo17.440.7107300.7093000.7135900.632288
Cu8.041.5418381.5405621.5443901.392218
Co6.931.7902601.7889651.7928501.62079
Fe6.401.9373551.9360421.9399801.75661
Cr5.412.291002.289702.2936062.08487

Kα here is the 2:1 weighted mean of Kα1 and Kα2. An unmonochromated tube delivers the two lines separately; the mean is the single effective wavelength to compute with when the doublet is not resolved, which is ordinary laboratory data, and it is what this site uses unless you say otherwise. Choose Kα1 if a monochromator or a Johansson mirror removes the Kα2 component. Kα2 is not offered on its own, because no experiment runs on it alone — it is listed only because the Kα1/Kα2 splitting is why high-angle peaks look doubled. Kβ is what the filter or monochromator removes; compute with it to find out where the ghost peaks of an unfiltered pattern fall.

Kα is not an independent constant: it is (2 × Kα1 + Kα2) / 3, which reproduces every Kα value in this table to its own last digit.

Values taken from B. B. He, Two-Dimensional X-Ray Diffraction, John Wiley & Sons, Hoboken, NJ, Second edition, 2018.

Where this comes from