The Ewald Construction

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.

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 is reciprocal space: the sphere is not the crystal and the dots are not atoms. 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

Input

Unit cell

Crystal system

Å

Bravais centring

Radiation

Å

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.

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.

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.