The **Ewald sphere** is a mathematical construct that relates the geometry of the incident and scattered wave vectors to the reciprocal lattice.

Consider a crystal at *A* being irradiated by an incident wave vector **s**_{0}, scattering a wave vector ** s** that makes an angle 2θ with

**s**_{0}, thereby satisfying Bragg’s law (see diagram below).

Since X-ray scattering is elastic, I** s**I = I

**s**_{0}I and the two wave vectors become radii of a sphere called the

**. The vector**

*Ewald sphere**OP*is the reciprocal lattice vector (

**–**

*s*

**s**_{0}) that is denoted by

**with the origin at O.**

*h*Since ** h** = (

**–**

*s*

**s**_{0}), eq24 becomes

From eq11, . Substitute and eq24f in eq24e, we have:

This means that *IA* = *AO* = *AP* = *1/λ*. Therefore, to satisfy Bragg’s law that results in constructive interference of scattered X-rays, the head of the reciprocal lattice vector must lie on any point on the surface of the Ewald sphere. From eq24f and eq11,

Since *-1* ≤ *sinθ* ≤ *1*,

As the maximum magnitude of the reciprocal lattice vector is *2/λ*, all reciprocal lattice vectors that potentially satisfy the Laue equations or Bragg’s law are enclosed in a sphere of radius known as the ** limiting sphere**. The Ewald sphere is used to visualize different X-ray diffraction techniques including single crystal X-ray diffraction.