The Ewald Construction

https://xraytools.com/ewald

Bragg’s law as a picture. A sphere of radius 1/λ in reciprocal space, the lattice drawn under it, and one reflection sitting on the surface — which is what “in diffracting position” means. The construction is drawn to scale from your own cell.

Before this: The picture is drawn in reciprocal space, so a point on it is a reflection and the distance from the origin is 1/d. Without that the sphere is just a circle.
You supply: A unit cell, a wavelength and a Miller index — the same inputs the HKL calculator takes, resolved by the same code. The section drawn is hk0, so l = 0 is what shows the point landing on the sphere.
Reading it: This page works entirely in reciprocal space, and the construction panel says what that makes of the sphere and the dots. The limiting sphere of radius 2/λ is the practical reading — nothing with d < λ/2 can be measured at any orientation, and the reachable volume goes as (2/λ)³.

Worked examples: NaCl (220), further out on the sphere · copper (111) with Mo radiation · quartz (600), too fine for Cu Kα · a detector to 140°, where the blind cone is a third of the data · a Laue camera at 60 kV

Notation here: [uvw] · a*, b*, c* — what each one means here

Terms here: centring

What each input changes

Radiation: The radius of the sphere, 1/λ. A shorter wavelength gives a larger sphere and so brings more of the reciprocal lattice within reach — which is the whole reason for using Mo.
Detector 2θ_max: Where the detector stops, which is what really sets the resolution: d_min = λ/(2 sin θ_max), always coarser than the λ/2 the table above counts to. It also sets the width of the cone a single-axis rotation never reaches, because that cone’s half-angle is the reflection’s own θ and is widest at the edge of the data.
Laue tube voltage: The accelerating voltage, which fixes the SHORT end of a white beam at hc/eU and nothing else about it. Raising it grows the outer sphere of the Laue crescent, so a stationary crystal reaches finer spacings without the wavelength anybody names changing.
Laue λ_max: The long end of the band, which this page does not compute: it is set by absorption in the tube window, the air path and the sample. 2 Å is the usual working figure for a laboratory camera, and every count in the Laue arm scales with it.

Teaching with this page

Objective: After this page a learner can say what a rotation, a powder and a white beam each do to the same sphere, and what each one costs.
Start from: this worked example
Ask first: A full 360° rotation about one spindle leaves the data 96 % complete. Where is the rest?
Watch for: “Lost at the detector edges and to shadowed frames”

Check yourself: A full 360° rotation about one spindle leaves the data 96 % complete. Where is the rest?

In a cone about the spindle, which no rotation about it reaches Lost at the detector edges and to shadowed frames

Turning about one axis keeps a reciprocal lattice point’s component along that axis fixed, so the point travels on a circle rather than over a sphere. It meets the Ewald sphere only if the angle ψ between d* and the spindle is at least that reflection’s own θ — so everything within θ of the axis is blind, and the cone widens as the resolution goes up. Symmetry does not always rescue it: a rotation about the spindle preserves that angle, so a crystal mounted with a symmetry axis along the spindle keeps every equivalent inside the same cone. The fix is a second goniometer circle or a remount, which is what a κ or four-circle geometry is for.

Input

Unit cell

Crystal system

Bravais centring

Radiation

This wavelength is Cu Kα.

Reflection

The reciprocal lattice section drawn is hk0, so a reflection with l = 0 is the one you will see land on the sphere.

Experiment

Detector 2θ_max

Laue tube voltage

Laue λ_max

Only the three experiments at the foot of the page read these. The customary detector limit is about 55° for Mo Kα and about 140° for Cu Kα — the same angular reach buys very different resolution.

The construction

2 0 0: d = 2.82 Å, 2θ = 31.7305°. On the sphere: this reflection is diffracting right now.

Everything here is drawn in reciprocal space: the sphere is not the crystal and the dots are not atoms, they are reciprocal lattice points, one per family of lattice planes. The sphere has radius 1/λ and the incident beam runs through its centre to the origin. A point sitting on the sphere is a reflection in diffracting position; the diffracted beam leaves the centre through it, at 2θ to the incident beam.

Ewald sphere radius 1/λ	0.648577	Å⁻¹
\|d*\| = 1/d	0.35461	Å⁻¹
d	2.82	Å
θ	15.8652	°
2θ, the angle between the beams	31.7305	°

How this is calculated

A reciprocal lattice point is d* from the origin, and the origin sits on the sphere. Both lengths are in reciprocal Ångström: $d^{*} = \frac{1}{d} = \frac{1}{2.82 Å} = 0.35461 Å^{- 1}$

The sphere’s radius is one over the wavelength: $r = \frac{1}{λ} = \frac{1}{1.54184 Å} = 0.648577 Å^{- 1}$

The chord from the origin to a point on the sphere is 2r sin θ — plane geometry, with no diffraction in it. So: $d^{*} = 2 r \sin (θ)$

Substituting both lengths turns that into Bragg’s law, and nothing has been assumed on the way: $\frac{1}{d} = \frac{2}{λ} \sin (θ) \Leftrightarrow λ = 2 d \sin (θ)$

$\sin (θ) = \frac{λ}{2 d} = \frac{1.54184 Å}{2 \times 2.82 Å} = 0.273376$

The diffracted beam leaves the centre through the point, so the angle between the two beams is twice the angle in Bragg’s law: $2 θ = 2 \times \arcsin (0.273376) = 31.7305 °$

What this wavelength can reach

Rotating the crystal moves a reciprocal lattice point anywhere on a sphere of its own radius about the origin, so a reflection can be brought onto the Ewald sphere exactly when |d*| ≤ 2/λ — that is, when d ≥ λ/2. For this wavelength that is 0.770919 Å, and nothing finer can be measured with it at any orientation, on any instrument. It is the same limit the Bragg calculator prints as n.a., reached from the other side.

What each anode’s wavelength can reach for this cell. CSV
Anode	λ / Å	d_min = λ/2 / Å	reflections in range	allowed by the lattice	against Cu
Ag	0.560868	0.2804	34,048	34,048	20.69×
Mo	0.710730	0.3554	16,830	16,830	10.22×
Cu	1.541838	0.7709	1,646	1,646	1.00×
Co	1.790260	0.8951	1,020	1,020	0.62×
Fe	1.937355	0.9687	798	798	0.48×
Cr	2.29100	1.1455	484	484	0.29×

The reachable volume of reciprocal space goes as (2/λ)³, so halving the wavelength gives eight times the data. That is the whole argument for a shorter wavelength on a structure that needs high-resolution data — and it is independent of the absorption argument on the Moseley page, which points the same way for heavy atoms and for a different reason.

Counted one point at a time. The closed form (4/3)π(2/λ)³V — the limiting sphere’s volume divided by the reciprocal cell’s — gives 1,640 for this wavelength, against the 1,646 actually counted. They are different derivations and neither is used to compute the other.

One construction, three experiments

The construction above puts one reciprocal lattice point on the sphere, for one wavelength and one orientation. Every diffraction experiment is a way of getting more points onto it, and there are only three moves: turn the crystal, use every orientation at once, or open the wavelength. The sphere itself never changes.

Rotation. A section holding the spindle; spin it about the vertical. The shaded cones never come round.

Powder. The dashed sphere is every orientation of this reflection; where it cuts the Ewald sphere is the ring, edge-on.

Laue. Every wavelength in the band has its own sphere. The crescent is what they sweep between them.

Rotation. Turning the crystal moves a reciprocal lattice point on a sphere of radius |d*| about the origin, so it can be brought onto the Ewald sphere whenever |d*| ≤ 2/λ. One axis is weaker than that. A point at an angle ψ to the spindle keeps its component along the spindle, so it reaches the sphere only if sin ψ ≥ sin θ — a reflection lying within its own θ of the axis never comes round, whatever you do with φ. At 55° that blind double cone has a half-angle of 27.5° and holds 6.7 % of the reciprocal space this instrument reaches. The 170 reflections above d = 1.66956 Å are collected one at a time, each as it crosses. A rotation about the spindle preserves the angle to the spindle, so mounting a symmetry axis along it keeps every equivalent of a blind reflection inside the same cone: filling the cusp means moving the crystal off that axis, with a second goniometer circle or a remount.

Powder. A powder holds crystallites in every orientation at once, so d* is not a point but the whole sphere of radius |d*| about the origin. Where that sphere cuts the Ewald sphere is a circle of radius |d*| cos θ = 0.341102 Å⁻¹ centred on the incident beam, and the beams leaving it form a cone of half-angle 2θ = 31.7305° — the Debye–Scherrer cone. Nothing has to be turned: all 170 reflections above d = 1.66956 Å are on their own cones at the same instant. What is lost is direction. The ring records |d*| and nothing else, so every reflection with the same d lands in the same place: in a cubic pattern (333) and (511) are one line, and no amount of counting statistics separates them.

Laue. Open the wavelength instead of moving the crystal. Each λ has its own Ewald sphere and they all touch the reciprocal origin, so a white beam between 0.30996 and 2 Å sweeps out the crescent between the sphere of radius 1/λ_max and the one of radius 1/λ_min. Every reciprocal lattice point in it lies on exactly one of those spheres and diffracts, from a crystal that never moves — about 25,100 of them for this cell. The short end is the tube’s own short-wavelength limit, hc/eU at 40 kV, which is the Duane–Hunt limit drawn on the tube page; the long end is set by absorption in the window, the air path and the sample, and is a working figure rather than anything this page computes. The band reaches d ≥ 0.15498 Å, far finer than any single wavelength in it. The price is that (hkl) at λ and (nh nk nl) at λ/n leave in exactly the same direction, so one spot is a superposition of up to 6 orders — which is why Laue photographs orient crystals and follow reactions, and are not how intensities are measured.

Where this comes from

Elucidations on the reciprocal lattice and the Ewald sphere
J. Foadi and G. Evans, Eur. J. Phys. 2008, 29, 1059–1068 · doi:10.1088/0143-0807/29/5/017
The construction drawn above, taken slowly and with the algebra shown. It is written for a reader meeting it for the first time, which is unusual for a paper about it.